Difference between revisions of "Interaction of an atom with an electromagnetic field"

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imported>Ichuang
(fixed figure, equation, and question cross-references)
imported>Ichuang
(fixed figure, equation, and question cross-references)
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Einstein considered a system of <math>N</math> atoms in thermal equilibrium with a radiation field. The system has two levels (an energy level consists of all of the states that have a given energy; the number of quantum states in a given level is its multiplicity.) with energies <math>E_ b</math> and <math>E_ a</math>, with <math>E_ b > E_ a</math>, and <math>E_ b - E_ a =\hbar \omega </math>. The numbers of atoms in the two levels are related by <math>N_ b + N_ a = N</math>. Einstein assumed the Planck radiation law for the spectral energy density temperature. For radiation in thermal equilibrium at temperature <math>T</math>, the energy per unit volume in wavelength range <math>d\omega </math> is:  
 
Einstein considered a system of <math>N</math> atoms in thermal equilibrium with a radiation field. The system has two levels (an energy level consists of all of the states that have a given energy; the number of quantum states in a given level is its multiplicity.) with energies <math>E_ b</math> and <math>E_ a</math>, with <math>E_ b > E_ a</math>, and <math>E_ b - E_ a =\hbar \omega </math>. The numbers of atoms in the two levels are related by <math>N_ b + N_ a = N</math>. Einstein assumed the Planck radiation law for the spectral energy density temperature. For radiation in thermal equilibrium at temperature <math>T</math>, the energy per unit volume in wavelength range <math>d\omega </math> is:  
  
−
{{EqL
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:<equation id="erad1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  \rho _ E (\omega )d\omega = \frac{\hbar \omega ^3}{\pi ^2 c^3} \frac{1}{{\rm exp} (\hbar \omega /kT) -1 }d\omega . \end{align}</math>
+
<math>\begin{align} \  \rho _ E (\omega )d\omega = \frac{\hbar \omega ^3}{\pi ^2 c^3} \frac{1}{{\rm exp} (\hbar \omega /kT) -1 }d\omega . \end{align}</math>
−
|num=erad1
+
</equation>
−
}}
 
  
 
The mean occupation number of a harmonic oscillator at temperature <math>T</math>, which can be interpreted as the mean number of photons in one mode of the radiation field, is  
 
The mean occupation number of a harmonic oscillator at temperature <math>T</math>, which can be interpreted as the mean number of photons in one mode of the radiation field, is  
  
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{{EqL
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:<equation id="erad2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  \bar{n} = \frac{1}{{\rm exp} (\hbar \omega /kT) -1}. \end{align}</math>
+
<math>\begin{align} \  \bar{n} = \frac{1}{{\rm exp} (\hbar \omega /kT) -1}. \end{align}</math>
−
|num=erad2
+
</equation>
−
}}
 
  
 
According to the Boltzmann Law of statistical mechanics, in thermal equilibrium the populations of the two levels are related by  
 
According to the Boltzmann Law of statistical mechanics, in thermal equilibrium the populations of the two levels are related by  
  
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{{EqL
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:<equation id="erad3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  \frac{N_ b}{N_ a} = \frac{g_ b}{g_ a} e^{-(E_ b -E_ a)/kT} = \frac{g_ b}{g_ a} e^{-\hbar \omega /kT} . \end{align}</math>
+
<math>\begin{align} \  \frac{N_ b}{N_ a} = \frac{g_ b}{g_ a} e^{-(E_ b -E_ a)/kT} = \frac{g_ b}{g_ a} e^{-\hbar \omega /kT} . \end{align}</math>
−
|num=erad3
+
</equation>
−
}}
 
  
 
Here <math>g_ b</math> and <math>g_ a</math> are the multiplicities of the two levels. The last step assumes the Bohr frequency condition, <math>\omega = (E_ b -E_ a)\  \hbar </math>. However, Einstein's paper actually derives this relation independently.  
 
Here <math>g_ b</math> and <math>g_ a</math> are the multiplicities of the two levels. The last step assumes the Bohr frequency condition, <math>\omega = (E_ b -E_ a)\  \hbar </math>. However, Einstein's paper actually derives this relation independently.  
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According to classical theory, an oscillator can exchange energy with the radiation field at a rate that is proportional to the spectral density of radiation. The rates for absorption and emission are equal. The population transfer rate equation is thus predicted to be  
 
According to classical theory, an oscillator can exchange energy with the radiation field at a rate that is proportional to the spectral density of radiation. The rates for absorption and emission are equal. The population transfer rate equation is thus predicted to be  
  
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{{EqL
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:<equation id="erad4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  \dot{N}_ b = - { \rho _ E (\omega ) B_{ba}} N_ b + \rho _ E (\omega ) B_{ab} N_ a = -\dot{N}_ a . \end{align}</math>
+
<math>\begin{align} \  \dot{N}_ b = - { \rho _ E (\omega ) B_{ba}} N_ b + \rho _ E (\omega ) B_{ab} N_ a = -\dot{N}_ a . \end{align}</math>
−
|num=erad4
+
</equation>
−
}}
 
  
 
This equation is incompatible with Eq. [[{{SUBPAGENAME}}#erad3|erad3]]. (This can be seen by setting <math>A_{ba}=0 </math> in Eq. [[{{SUBPAGENAME}}#erad5|erad5]] which then leads to <math>B_{ba}=B_{ab}=0</math>.) To overcome this problem, Einstein postulated that atoms in state b must spontaneously radiate to state a, with a constant radiation rate <math>A_{ba}</math>. Today such a process seems quite natural: the language of quantum mechanics is the language of probabilities and there is nothing jarring about asserting that the probability of radiating in a short time interval is proportional to the length of the interval. At that time such a random fundamental process could not be justified on physical principles. Einstein, in his characteristic Olympian style, brushed aside such concerns and merely asserted that the process is analagous to radioactive decay. With this addition, Eq. [[{{SUBPAGENAME}}#erad4|erad4]] becomes  
 
This equation is incompatible with Eq. [[{{SUBPAGENAME}}#erad3|erad3]]. (This can be seen by setting <math>A_{ba}=0 </math> in Eq. [[{{SUBPAGENAME}}#erad5|erad5]] which then leads to <math>B_{ba}=B_{ab}=0</math>.) To overcome this problem, Einstein postulated that atoms in state b must spontaneously radiate to state a, with a constant radiation rate <math>A_{ba}</math>. Today such a process seems quite natural: the language of quantum mechanics is the language of probabilities and there is nothing jarring about asserting that the probability of radiating in a short time interval is proportional to the length of the interval. At that time such a random fundamental process could not be justified on physical principles. Einstein, in his characteristic Olympian style, brushed aside such concerns and merely asserted that the process is analagous to radioactive decay. With this addition, Eq. [[{{SUBPAGENAME}}#erad4|erad4]] becomes  
  
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{{EqL
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:<equation id="erad5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align}  \  \dot{N}_ b = - {\left[ \rho _ E (\omega ) B_{ba} + A_{ba} \right]} N_ b + \rho _ E (\omega ) B_{ab} N_ a = -\dot{N}_ a . \end{align}</math>
+
<math>\begin{align}  \  \dot{N}_ b = - {\left[ \rho _ E (\omega ) B_{ba} + A_{ba} \right]} N_ b + \rho _ E (\omega ) B_{ab} N_ a = -\dot{N}_ a . \end{align}</math>
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|num=erad5
+
</equation>
−
}}
 
  
 
By combining Eqs. [[{{SUBPAGENAME}}#eq:plancklaw|eq:plancklaw]], [[{{SUBPAGENAME}}#eq:frac|eq:frac]], [[{{SUBPAGENAME}}#eq:rad2|eq:rad2]]  it follows that  
 
By combining Eqs. [[{{SUBPAGENAME}}#eq:plancklaw|eq:plancklaw]], [[{{SUBPAGENAME}}#eq:frac|eq:frac]], [[{{SUBPAGENAME}}#eq:rad2|eq:rad2]]  it follows that  
  
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{{EqL
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:<equation id=" erl5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align}  \  g_ b B_{ba} & =&  g_ a B_{ab} \\ \frac{\hbar \omega ^3}{\pi ^2 c^3} B_{ba} & =&  A_{ba}  \\  \rho _ E (\omega ) B_{ba} & =&  \bar{n} A_{ba}  \\  \end{align}</math>
+
<math>\begin{align}  \  g_ b B_{ba} & =&  g_ a B_{ab} \\ \frac{\hbar \omega ^3}{\pi ^2 c^3} B_{ba} & =&  A_{ba}  \\  \rho _ E (\omega ) B_{ba} & =&  \bar{n} A_{ba}  \\  \end{align}</math>
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|num=EQ_ erl5
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</equation>
−
}}
 
  
 
Consequently, the rate of transition <math>b\rightarrow a</math> is  
 
Consequently, the rate of transition <math>b\rightarrow a</math> is  
  
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{{EqL
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:<equation id=" erl6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  B_{ba} \rho _ E (\omega ) + A_{ba} = (\bar{n} +1 )A_{ba}, \end{align}</math>
+
<math>\begin{align} \  B_{ba} \rho _ E (\omega ) + A_{ba} = (\bar{n} +1 )A_{ba}, \end{align}</math>
−
|num=EQ_ erl6
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</equation>
−
}}
 
  
 
while the rate of absorption is  
 
while the rate of absorption is  
  
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{{EqL
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:<equation id=" erl7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  B_{ab} \rho _ E (\omega ) = \frac{g_ b}{g_ a} \bar{n} A_{ba} \end{align}</math>
+
<math>\begin{align} \  B_{ab} \rho _ E (\omega ) = \frac{g_ b}{g_ a} \bar{n} A_{ba} \end{align}</math>
−
|num=EQ_ erl7
+
</equation>
−
}}
 
  
 
If we consider emission and absorption between single states by taking <math>g_ b = g_ a = 1</math>, then the ratio of rate of emission to rate of absorption is <math>(\bar{n} + 1) /\bar{n}</math>.  
 
If we consider emission and absorption between single states by taking <math>g_ b = g_ a = 1</math>, then the ratio of rate of emission to rate of absorption is <math>(\bar{n} + 1) /\bar{n}</math>.  
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Our starting point is Maxwell's equations (S.I. units):  
 
Our starting point is Maxwell's equations (S.I. units):  
  
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{{EqL
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:<equation id="Maxwell" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align}  \  \nabla \cdot {\bf E} &  = &  \rho /\epsilon _0 \\  \nabla \cdot {\bf B} &  = &  0 \\  \nabla \times {\bf E} &  = &  - \frac{\partial {\bf B}}{\partial t} \\  \nabla \times {\bf B} &  = &  \frac{1}{c^2} \frac{\partial \bf { E}}{\partial t} + \mu _0 \bf {J} \end{align}</math>
+
<math>\begin{align}  \  \nabla \cdot {\bf E} &  = &  \rho /\epsilon _0 \\  \nabla \cdot {\bf B} &  = &  0 \\  \nabla \times {\bf E} &  = &  - \frac{\partial {\bf B}}{\partial t} \\  \nabla \times {\bf B} &  = &  \frac{1}{c^2} \frac{\partial \bf { E}}{\partial t} + \mu _0 \bf {J} \end{align}</math>
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|num=eq:Maxwell
+
</equation>
−
}}
 
  
 
The charge density <math>\rho </math> and current density '''J''' obey the continuity equation  
 
The charge density <math>\rho </math> and current density '''J''' obey the continuity equation  
  
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{{EqL
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:<equation id=" wd2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  \nabla \cdot {\bf J} + \frac{\partial \rho }{\partial t} = 0 \end{align}</math>
+
<math>\begin{align} \  \nabla \cdot {\bf J} + \frac{\partial \rho }{\partial t} = 0 \end{align}</math>
−
|num=EQ_ wd2
+
</equation>
−
}}
 
  
 
Introducing the vector potential '''A''' and the scalar potential <math>\psi </math>, we have  
 
Introducing the vector potential '''A''' and the scalar potential <math>\psi </math>, we have  
  
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{{EqL
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:<equation id=" wd3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  {\bf E} &  = &  - \nabla \psi - \frac{\partial {\bf A}}{\partial t} \\ {\bf B} &  = &  \nabla \times {\bf A}  \end{align}</math>
+
<math>\begin{align} \  {\bf E} &  = &  - \nabla \psi - \frac{\partial {\bf A}}{\partial t} \\ {\bf B} &  = &  \nabla \times {\bf A}  \end{align}</math>
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|num=EQ_ wd3
+
</equation>
−
}}
 
  
 
We are free to change the potentials by a gauge transformation:  
 
We are free to change the potentials by a gauge transformation:  
  
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{{EqL
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:<equation id=" wd4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  {\bf A}^\prime = {\bf A} + \nabla \Lambda , ~ ~ ~ ~ ~ \psi ^\prime = \psi - \frac{\partial \Lambda }{\partial t} \end{align}</math>
+
<math>\begin{align} \  {\bf A}^\prime = {\bf A} + \nabla \Lambda , ~ ~ ~ ~ ~ \psi ^\prime = \psi - \frac{\partial \Lambda }{\partial t} \end{align}</math>
−
|num=EQ_ wd4
+
</equation>
−
}}
 
  
 
where <math>\Lambda </math> is a scalar function. This transformation leaves the fields invariant, but changes the form of the dynamical equation. We shall work in the <i>
 
where <math>\Lambda </math> is a scalar function. This transformation leaves the fields invariant, but changes the form of the dynamical equation. We shall work in the <i>
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</i> (often called the radiation gauge), defined by  
 
</i> (often called the radiation gauge), defined by  
  
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{{EqL
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:<equation id=" wd5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  \nabla \cdot {\bf A} = 0 \end{align}</math>
+
<math>\begin{align} \  \nabla \cdot {\bf A} = 0 \end{align}</math>
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|num=EQ_ wd5
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</equation>
−
}}
 
  
 
In free space, '''A''' obeys the wave equation  
 
In free space, '''A''' obeys the wave equation  
  
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{{EqL
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:<equation id=" wd6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  \nabla ^2 {\bf A} = \frac{1}{c^2} \frac{\partial ^2 {\bf A}}{\partial t^2} \end{align}</math>
+
<math>\begin{align} \  \nabla ^2 {\bf A} = \frac{1}{c^2} \frac{\partial ^2 {\bf A}}{\partial t^2} \end{align}</math>
−
|num=EQ_ wd6
+
</equation>
−
}}
 
  
 
Because <math>\nabla \cdot {\bf A}= 0</math>, '''A''' is transverse. We take a propagating plane wave solution of the form  
 
Because <math>\nabla \cdot {\bf A}= 0</math>, '''A''' is transverse. We take a propagating plane wave solution of the form  
  
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{{EqL
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:<equation id="A-field" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  {\bf A}(r, t) = A{\bf \hat{e}} \cos ({\bf k}\cdot {\bf r} -\omega t) = A{\bf \hat{e}} \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} + e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right], \end{align}</math>
+
<math>\begin{align} \  {\bf A}(r, t) = A{\bf \hat{e}} \cos ({\bf k}\cdot {\bf r} -\omega t) = A{\bf \hat{e}} \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} + e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right], \end{align}</math>
−
|num=eq:A-field
+
</equation>
−
}}
 
  
 
where <math>k^2 =\omega ^2 / c^2</math> and <math>{\bf \hat{e}}\cdot {\bf k}= 0</math>. For a linearly polarized field, the polarization vector <math>{\bf \hat{e}}</math> is real. For an elliptically polarized field it is complex, and for a circularly polarized field it is given by <math>{\bf \hat{e}} = ({\bf \hat{ x}} \pm i {\bf \hat{ y}} ) /\sqrt {2}</math> , where the + and <math>-</math> signs correspond to positive and negative helicity, respectively. (Alternatively, they correspond to left and right hand circular polarization, respectively, the sign convention being a tradition from optics.) The electric and magnetic fields are then given by  
 
where <math>k^2 =\omega ^2 / c^2</math> and <math>{\bf \hat{e}}\cdot {\bf k}= 0</math>. For a linearly polarized field, the polarization vector <math>{\bf \hat{e}}</math> is real. For an elliptically polarized field it is complex, and for a circularly polarized field it is given by <math>{\bf \hat{e}} = ({\bf \hat{ x}} \pm i {\bf \hat{ y}} ) /\sqrt {2}</math> , where the + and <math>-</math> signs correspond to positive and negative helicity, respectively. (Alternatively, they correspond to left and right hand circular polarization, respectively, the sign convention being a tradition from optics.) The electric and magnetic fields are then given by  
  
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{{EqL
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:<equation id="E-field" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
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|math=<math>\begin{align} \  {\bf E}(r, t) = \omega A{\bf \hat{e}} \sin ({\bf k}\cdot {\bf r} -\omega t) = - i \omega A {\bf \hat{e}} \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} - e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right]. \end{align}</math>
+
<math>\begin{align} \  {\bf E}(r, t) = \omega A{\bf \hat{e}} \sin ({\bf k}\cdot {\bf r} -\omega t) = - i \omega A {\bf \hat{e}} \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} - e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right]. \end{align}</math>
−
|num=eq:E-field
+
</equation>
−
}}
 
  
−
{{EqL
+
:<equation id="B-field" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  {\bf B}(r, t) = k ({\bf \hat{k}} \times {\bf \hat{ e}}) \sin ({\bf k}\cdot {\bf r} -\omega t) = - i k A ({\bf \hat{k}} \times {\bf \hat{ e}}) \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} - e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right]. \end{align}</math>
+
<math>\begin{align} \  {\bf B}(r, t) = k ({\bf \hat{k}} \times {\bf \hat{ e}}) \sin ({\bf k}\cdot {\bf r} -\omega t) = - i k A ({\bf \hat{k}} \times {\bf \hat{ e}}) \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} - e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right]. \end{align}</math>
−
|num=eq:B-field
+
</equation>
−
}}
 
  
 
The time average Poynting vector is  
 
The time average Poynting vector is  
  
−
{{EqL
+
:<equation id=" wd9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  {\bf S} = \frac{ \epsilon _0 c^2}{2} ( {{\bf E} \times {\bf B}^* )} = \frac{\epsilon _0 c}{2} \omega ^2 A^2 {\bf \hat{k}} . \end{align}</math>
+
<math>\begin{align} \  {\bf S} = \frac{ \epsilon _0 c^2}{2} ( {{\bf E} \times {\bf B}^* )} = \frac{\epsilon _0 c}{2} \omega ^2 A^2 {\bf \hat{k}} . \end{align}</math>
−
|num=EQ_ wd9
+
</equation>
−
}}
 
  
 
The average energy density in the wave is given by  
 
The average energy density in the wave is given by  
  
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{{EqL
+
:<equation id="energy-density" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  u = \omega ^2 \frac{\epsilon _0 }{2} A^2 {\bf \hat{k}} . \end{align}</math>
+
<math>\begin{align} \  u = \omega ^2 \frac{\epsilon _0 }{2} A^2 {\bf \hat{k}} . \end{align}</math>
−
|num=eq:energy-density
+
</equation>
−
}}
 
  
 
<br style="clear: both" />
 
<br style="clear: both" />
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The behavior of charged particles in an electromagnetic field is correctly described by Hamilton's equations provided that the canonical momentum is redefined:  
 
The behavior of charged particles in an electromagnetic field is correctly described by Hamilton's equations provided that the canonical momentum is redefined:  
  
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{{EqL
+
:<equation id=" int1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  {\bf p}_{\rm can} = {\bf p}_{\rm kin} + q {\bf A} \end{align}</math>
+
<math>\begin{align} \  {\bf p}_{\rm can} = {\bf p}_{\rm kin} + q {\bf A} \end{align}</math>
−
|num=EQ_ int1
+
</equation>
−
}}
 
  
 
The kinetic energy is <math>{\bf p}_{\rm kin}^2 /2 m</math>. Taking <math>q = - e</math>, the Hamiltonian for an atom in an electromagnetic field in free space is  
 
The kinetic energy is <math>{\bf p}_{\rm kin}^2 /2 m</math>. Taking <math>q = - e</math>, the Hamiltonian for an atom in an electromagnetic field in free space is  
  
−
{{EqL
+
:<equation id=" int2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H = \frac{1}{2m} \sum _{j=1}^{N} {\left( {\bf p}_ j + e {\bf A} (r_ j )\right)^2} + \sum _{j=1}^{N} V ({\bf r}_ j ), \end{align}</math>
+
<math>\begin{align} \  H = \frac{1}{2m} \sum _{j=1}^{N} {\left( {\bf p}_ j + e {\bf A} (r_ j )\right)^2} + \sum _{j=1}^{N} V ({\bf r}_ j ), \end{align}</math>
−
|num=EQ_ int2
+
</equation>
−
}}
 
  
 
where <math>V ( {\bf r}_ j )</math> describes the potential energy due to internal interactions. We are neglecting spin interactions.  
 
where <math>V ( {\bf r}_ j )</math> describes the potential energy due to internal interactions. We are neglecting spin interactions.  
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Expanding and rearranging, we have  
 
Expanding and rearranging, we have  
  
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{{EqL
+
:<equation id=" int3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H & =&  \sum _{j=1}^{N} \frac{{\bf p}_ j^2}{2m} + V ({\bf r}_ j ) + \frac{e}{2m} \sum _{j=1}^{N} {\left({\bf p}_ j \cdot {\bf A} ( {\bf r}_ j) + {\bf A} ({\bf r}_ j ) \cdot {\bf p}_ j \right)} + \frac{e^2}{2m} \sum _{j=1}^{N} A_ j^2 ({\bf r} ) \\ &  = &  H_0 + H_{\rm int} + H^{(2)} .  \end{align}</math>
+
<math>\begin{align} \  H & =&  \sum _{j=1}^{N} \frac{{\bf p}_ j^2}{2m} + V ({\bf r}_ j ) + \frac{e}{2m} \sum _{j=1}^{N} {\left({\bf p}_ j \cdot {\bf A} ( {\bf r}_ j) + {\bf A} ({\bf r}_ j ) \cdot {\bf p}_ j \right)} + \frac{e^2}{2m} \sum _{j=1}^{N} A_ j^2 ({\bf r} ) \\ &  = &  H_0 + H_{\rm int} + H^{(2)} .  \end{align}</math>
−
|num=EQ_ int3
+
</equation>
−
}}
 
  
 
Here, <math>{\bf p}_ j = - i\hbar \nabla _ j </math>. Consequently, <math>H_0</math> describes the unperturbed atom. <math>H_{\rm int}</math> describes the atom's interaction with the field. <math>H^{(2)}</math>, which is second order in '''A''', plays a role only at very high intensities. (In a static magnetic field, however, <math>H^{(2)}</math> gives rise to diamagnetism.)  
 
Here, <math>{\bf p}_ j = - i\hbar \nabla _ j </math>. Consequently, <math>H_0</math> describes the unperturbed atom. <math>H_{\rm int}</math> describes the atom's interaction with the field. <math>H^{(2)}</math>, which is second order in '''A''', plays a role only at very high intensities. (In a static magnetic field, however, <math>H^{(2)}</math> gives rise to diamagnetism.)  
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Because we are working in the Coulomb gauge, <math>\nabla \cdot {\bf A} =0</math> so that '''A''' and '''p''' commute. We have  
 
Because we are working in the Coulomb gauge, <math>\nabla \cdot {\bf A} =0</math> so that '''A''' and '''p''' commute. We have  
  
−
{{EqL
+
:<equation id=" int4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{\rm int} = \frac{eA}{m} \hat{\bf {e}} \cdot {\bf p} \cos ({\bf k}\cdot {\bf r} -\omega t) . \end{align}</math>
+
<math>\begin{align} \  H_{\rm int} = \frac{eA}{m} \hat{\bf {e}} \cdot {\bf p} \cos ({\bf k}\cdot {\bf r} -\omega t) . \end{align}</math>
−
|num=EQ_ int4
+
</equation>
−
}}
 
  
 
It is convenient to write the matrix element between states <math> | a \rangle </math> and <math> | b \rangle </math> in the form  
 
It is convenient to write the matrix element between states <math> | a \rangle </math> and <math> | b \rangle </math> in the form  
  
−
{{EqL
+
:<equation id=" int5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \langle b | H_{\rm int} | a \rangle = \frac{1}{2} H_{ba} e^{-i\omega t} + \frac{1}{2} H_{ba} e^{+i\omega t}, \end{align}</math>
+
<math>\begin{align} \  \langle b | H_{\rm int} | a \rangle = \frac{1}{2} H_{ba} e^{-i\omega t} + \frac{1}{2} H_{ba} e^{+i\omega t}, \end{align}</math>
−
|num=EQ_ int5
+
</equation>
−
}}
 
  
 
where  
 
where  
  
−
{{EqL
+
:<equation id=" int6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{ba} = \frac{eA}{m} {\bf \hat{e}} \,  \langle b |{\bf p} \,  e^{i {\bf k} \cdot {\bf r}} | a \rangle . \end{align}</math>
+
<math>\begin{align} \  H_{ba} = \frac{eA}{m} {\bf \hat{e}} \,  \langle b |{\bf p} \,  e^{i {\bf k} \cdot {\bf r}} | a \rangle . \end{align}</math>
−
|num=EQ_ int6
+
</equation>
−
}}
 
  
 
Atomic dimensions are small compared to the wavelength of radiation involved in optical transitions. The scale of the ratio is set by <math>\alpha \approx 1/137</math>. Consequently, when the matrix element in Eq. [[{{SUBPAGENAME}}#EQ_int6|EQ_int6]] is evaluated, the wave function vanishes except in the region where <math>{\bf k}\cdot {\bf r} = 2 \pi r /\lambda \ll 1</math>. It is therefore appropriate to expand the exponential:  
 
Atomic dimensions are small compared to the wavelength of radiation involved in optical transitions. The scale of the ratio is set by <math>\alpha \approx 1/137</math>. Consequently, when the matrix element in Eq. [[{{SUBPAGENAME}}#EQ_int6|EQ_int6]] is evaluated, the wave function vanishes except in the region where <math>{\bf k}\cdot {\bf r} = 2 \pi r /\lambda \ll 1</math>. It is therefore appropriate to expand the exponential:  
  
−
{{EqL
+
:<equation id=" int7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{ba} = \frac{eA}{m} {\bf \hat{e}} \cdot \langle b | {\bf p} (1 + i{\bf k} \cdot {\bf r} - 1/2 ({\bf k}\cdot {\bf r} )^2 + \cdots ) | a \rangle \end{align}</math>
+
<math>\begin{align} \  H_{ba} = \frac{eA}{m} {\bf \hat{e}} \cdot \langle b | {\bf p} (1 + i{\bf k} \cdot {\bf r} - 1/2 ({\bf k}\cdot {\bf r} )^2 + \cdots ) | a \rangle \end{align}</math>
−
|num=EQ_ int7
+
</equation>
−
}}
 
  
 
Unless <math>\langle b | {\bf p} | a \rangle </math> vanishes, for instance due to parity considerations, the leading term dominates and we can neglect the others. For reasons that will become clear, this is called the dipole approximation. This is by far the most important situation, and we shall defer consideration of the higher order terms. In the dipole approximation we have  
 
Unless <math>\langle b | {\bf p} | a \rangle </math> vanishes, for instance due to parity considerations, the leading term dominates and we can neglect the others. For reasons that will become clear, this is called the dipole approximation. This is by far the most important situation, and we shall defer consideration of the higher order terms. In the dipole approximation we have  
  
−
{{EqL
+
:<equation id=" int8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{ba} = \frac{eA}{m} {\bf \hat{e}} \cdot \langle b | {\bf p} | a \rangle = \frac{-ieE}{m\omega } {\bf \hat{e}} \cdot \langle b | {\bf p} | a \rangle \end{align}</math>
+
<math>\begin{align} \  H_{ba} = \frac{eA}{m} {\bf \hat{e}} \cdot \langle b | {\bf p} | a \rangle = \frac{-ieE}{m\omega } {\bf \hat{e}} \cdot \langle b | {\bf p} | a \rangle \end{align}</math>
−
|num=EQ_ int8
+
</equation>
−
}}
 
  
 
where we have used, from Eq. [[{{SUBPAGENAME}}#eq:E-field|eq:E-field]], <math>A = -iE/\omega </math>. It can be shown (i.e. left as exercise) that the matrix element of '''p''' can be transfomred into a matrix element for <math>{\bf r}</math>:  
 
where we have used, from Eq. [[{{SUBPAGENAME}}#eq:E-field|eq:E-field]], <math>A = -iE/\omega </math>. It can be shown (i.e. left as exercise) that the matrix element of '''p''' can be transfomred into a matrix element for <math>{\bf r}</math>:  
  
−
{{EqL
+
:<equation id=" int9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \langle b | {\bf p} | a \rangle = - i m \omega _{ab} \langle b | {\bf r} | a \rangle = + i m \omega _{ba} \langle b | {\bf r} | a \rangle \end{align}</math>
+
<math>\begin{align} \  \langle b | {\bf p} | a \rangle = - i m \omega _{ab} \langle b | {\bf r} | a \rangle = + i m \omega _{ba} \langle b | {\bf r} | a \rangle \end{align}</math>
−
|num=EQ_ int9
+
</equation>
−
}}
 
  
 
This results in  
 
This results in  
  
−
{{EqL
+
:<equation id=" int10" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{ba} = \frac{e E \omega _{ba}}{\omega } {\bf \hat{e}} \cdot \langle b | {\bf r} | a \rangle \end{align}</math>
+
<math>\begin{align} \  H_{ba} = \frac{e E \omega _{ba}}{\omega } {\bf \hat{e}} \cdot \langle b | {\bf r} | a \rangle \end{align}</math>
−
|num=EQ_ int10
+
</equation>
−
}}
 
  
 
We will be interested in resonance phenomena in which <math>\omega \approx \omega _{ba}</math>. Consequently,  
 
We will be interested in resonance phenomena in which <math>\omega \approx \omega _{ba}</math>. Consequently,  
  
−
{{EqL
+
:<equation id=" int11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{ba} = + e {\bf E}_0 \cdot \langle b | {\bf r} | a \rangle = - {\bf d}_{ba} \cdot {\bf E} \end{align}</math>
+
<math>\begin{align} \  H_{ba} = + e {\bf E}_0 \cdot \langle b | {\bf r} | a \rangle = - {\bf d}_{ba} \cdot {\bf E} \end{align}</math>
−
|num=EQ_ int11
+
</equation>
−
}}
 
  
 
where '''d ''' is the dipole operator, <math>{\bf d} = - e {\bf r}</math>. Displaying the time dependence explictlty, we have  
 
where '''d ''' is the dipole operator, <math>{\bf d} = - e {\bf r}</math>. Displaying the time dependence explictlty, we have  
  
−
{{EqL
+
:<equation id=" int12" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{ba}^\prime = - {\bf d}_{ba}\cdot {\bf E}_0 e^{-i\omega t}. \end{align}</math>
+
<math>\begin{align} \  H_{ba}^\prime = - {\bf d}_{ba}\cdot {\bf E}_0 e^{-i\omega t}. \end{align}</math>
−
|num=EQ_ int12
+
</equation>
−
}}
 
  
 
However, it is important to bear in mind that this is only the first term in a series, and that if it vanishes the higher order terms will contribute a perturbation at the driving frequency.  
 
However, it is important to bear in mind that this is only the first term in a series, and that if it vanishes the higher order terms will contribute a perturbation at the driving frequency.  
Line 276: Line 245:
 
We shall consider a single mode of the radiation field. This means a single value of the wave vector '''k''', and one of the two orthogonal transverse polarization vectors <math>{\bf \hat{e}}</math>. The radiation field is described by a plane wave vector potential of the form Eq. [[{{SUBPAGENAME}}#eq:A-field|eq:A-field]]. We assume that '''k''' obeys a periodic boundary or condition, <math>k_ x L_ x = 2\pi n_ x</math>, etc. (For any '''k''', we can choose boundaries <math>L_ x , L_ y , L_ z</math> to satisfy this.) The time averaged energy density is given by Eq. [[{{SUBPAGENAME}}#eq:energy-density|eq:energy-density]], and the total energy in the volume V defined by these boundaries is  
 
We shall consider a single mode of the radiation field. This means a single value of the wave vector '''k''', and one of the two orthogonal transverse polarization vectors <math>{\bf \hat{e}}</math>. The radiation field is described by a plane wave vector potential of the form Eq. [[{{SUBPAGENAME}}#eq:A-field|eq:A-field]]. We assume that '''k''' obeys a periodic boundary or condition, <math>k_ x L_ x = 2\pi n_ x</math>, etc. (For any '''k''', we can choose boundaries <math>L_ x , L_ y , L_ z</math> to satisfy this.) The time averaged energy density is given by Eq. [[{{SUBPAGENAME}}#eq:energy-density|eq:energy-density]], and the total energy in the volume V defined by these boundaries is  
  
−
{{EqL
+
:<equation id="energy-total" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  U = \frac{\epsilon _0 }{2}\omega ^2 A^2 V, \end{align}</math>
+
<math>\begin{align} \  U = \frac{\epsilon _0 }{2}\omega ^2 A^2 V, \end{align}</math>
−
|num=eq:energy-total
+
</equation>
−
}}
 
  
 
where <math>A^2</math> is the mean squared value of <math>A</math> averaged over the spatial mode. We now make a formal connection between the radiation field and a harmonic oscillator. We define variables Q and P by  
 
where <math>A^2</math> is the mean squared value of <math>A</math> averaged over the spatial mode. We now make a formal connection between the radiation field and a harmonic oscillator. We define variables Q and P by  
  
−
{{EqL
+
:<equation id=" qrd5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  A = \frac{1}{\omega } \sqrt {\frac{1}{\epsilon _ o V}} (\omega Q + iP ), ~ ~ A^* =\frac{1}{\omega }\sqrt {\frac{1}{\epsilon _ o V}} (\omega Q - iP ). \end{align}</math>
+
<math>\begin{align} \  A = \frac{1}{\omega } \sqrt {\frac{1}{\epsilon _ o V}} (\omega Q + iP ), ~ ~ A^* =\frac{1}{\omega }\sqrt {\frac{1}{\epsilon _ o V}} (\omega Q - iP ). \end{align}</math>
−
|num=EQ_ qrd5
+
</equation>
−
}}
 
  
 
Then, from Eq. [[{{SUBPAGENAME}}#eq:energy-total|eq:energy-total]], we find  
 
Then, from Eq. [[{{SUBPAGENAME}}#eq:energy-total|eq:energy-total]], we find  
  
−
{{EqL
+
:<equation id=" qrd6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  U = \frac{1}{2} (\omega ^2 Q^2 + P^2 ). \end{align}</math>
+
<math>\begin{align} \  U = \frac{1}{2} (\omega ^2 Q^2 + P^2 ). \end{align}</math>
−
|num=EQ_ qrd6
+
</equation>
−
}}
 
  
 
This describes the energy of a harmonic oscillator having unit mass. We quantize the oscillator in the usual fashion by treating Q and P as operators, with  
 
This describes the energy of a harmonic oscillator having unit mass. We quantize the oscillator in the usual fashion by treating Q and P as operators, with  
  
−
{{EqL
+
:<equation id=" qrd7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  P = - i\hbar \frac{\partial }{\partial Q}, ~ ~ ~ [Q,P] = i\hbar . \end{align}</math>
+
<math>\begin{align} \  P = - i\hbar \frac{\partial }{\partial Q}, ~ ~ ~ [Q,P] = i\hbar . \end{align}</math>
−
|num=EQ_ qrd7
+
</equation>
−
}}
 
  
 
We introduce the operators <math>a</math> and <math>a^\dagger </math> defined by  
 
We introduce the operators <math>a</math> and <math>a^\dagger </math> defined by  
  
−
{{EqL
+
:<equation id=" qrd8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  a = \frac{1}{\sqrt {2\hbar \omega }} (\omega Q + iP ) \end{align}</math>
+
<math>\begin{align} \  a = \frac{1}{\sqrt {2\hbar \omega }} (\omega Q + iP ) \end{align}</math>
−
|num=EQ_ qrd8
+
</equation>
−
}}
 
  
−
{{EqL
+
:<equation id=" qrd9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  a^\dagger = \frac{1}{\sqrt {2\hbar \omega }} (\omega Q - iP ) \end{align}</math>
+
<math>\begin{align} \  a^\dagger = \frac{1}{\sqrt {2\hbar \omega }} (\omega Q - iP ) \end{align}</math>
−
|num=EQ_ qrd9
+
</equation>
−
}}
 
  
 
The fundamental commutation rule is  
 
The fundamental commutation rule is  
  
−
{{EqL
+
:<equation id=" qrd10" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  [a, a^\dagger ] = 1 \end{align}</math>
+
<math>\begin{align} \  [a, a^\dagger ] = 1 \end{align}</math>
−
|num=EQ_ qrd10
+
</equation>
−
}}
 
  
 
from which the following can be deduced:  
 
from which the following can be deduced:  
  
−
{{EqL
+
:<equation id=" qrd11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H = \frac{1}{2} \hbar \omega [a^\dagger a + a a^\dagger ] = \hbar \omega \left[a^\dagger a + \frac{1}{2} \right] = \hbar \omega \left[N+ \frac{1}{2} \right] \end{align}</math>
+
<math>\begin{align} \  H = \frac{1}{2} \hbar \omega [a^\dagger a + a a^\dagger ] = \hbar \omega \left[a^\dagger a + \frac{1}{2} \right] = \hbar \omega \left[N+ \frac{1}{2} \right] \end{align}</math>
−
|num=EQ_ qrd11
+
</equation>
−
}}
 
  
 
where the number operator <math>N = a^\dagger a </math> obeys  
 
where the number operator <math>N = a^\dagger a </math> obeys  
  
−
{{EqL
+
:<equation id=" qrd12" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  N| n \rangle = n| n \rangle \end{align}</math>
+
<math>\begin{align} \  N| n \rangle = n| n \rangle \end{align}</math>
−
|num=EQ_ qrd12
+
</equation>
−
}}
 
  
 
We also have  
 
We also have  
  
−
{{EqL
+
:<equation id=" qrd13" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \langle n-1| a | n \rangle & =&  \sqrt {n}  \\ \langle n+1| a^\dagger | n \rangle & =&  \sqrt {n +1}  \\ \langle n| a^\dagger a | n \rangle & =&  n  \\ \langle n |a a^\dagger | n \rangle & =&  n+1  \\ \langle n| H | n \rangle & =&  \hbar \omega \left(n+ \frac{1}{2} \right)  \\ \  \langle n| a | n \rangle & =&  \langle n | a^\dagger | n \rangle = 0 \end{align}</math>
+
<math>\begin{align}  \langle n-1| a | n \rangle & =&  \sqrt {n}  \\ \langle n+1| a^\dagger | n \rangle & =&  \sqrt {n +1}  \\ \langle n| a^\dagger a | n \rangle & =&  n  \\ \langle n |a a^\dagger | n \rangle & =&  n+1  \\ \langle n| H | n \rangle & =&  \hbar \omega \left(n+ \frac{1}{2} \right)  \\ \  \langle n| a | n \rangle & =&  \langle n | a^\dagger | n \rangle = 0 \end{align}</math>
−
|num=EQ_ qrd13
+
</equation>
−
}}
 
  
 
The operators <math>a </math> and <math>a^\dagger </math> are called the annihilation and creation operators, respectively. We can express the vector potential and electric field in terms of <math>a</math> and <math>a^\dagger </math> as follows  
 
The operators <math>a </math> and <math>a^\dagger </math> are called the annihilation and creation operators, respectively. We can express the vector potential and electric field in terms of <math>a</math> and <math>a^\dagger </math> as follows  
  
−
{{EqL
+
:<equation id=" part1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  A = \frac{1}{ \omega \sqrt {\epsilon _ o V}} (\omega Q + iP) = \sqrt {\frac{2 \hbar }{ \omega \epsilon _ o V}} a \end{align}</math>
+
<math>\begin{align} \  A = \frac{1}{ \omega \sqrt {\epsilon _ o V}} (\omega Q + iP) = \sqrt {\frac{2 \hbar }{ \omega \epsilon _ o V}} a \end{align}</math>
−
|num=EQ_ part1
+
</equation>
−
}}
 
  
−
{{EqL
+
:<equation id=" part2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  A^\dagger = \frac{1}{ \omega \sqrt {\epsilon _ o V}} (\omega Q - iP) = \sqrt {\frac{2 \hbar }{ \omega \epsilon _ o V}} a^\dagger \end{align}</math>
+
<math>\begin{align} \  A^\dagger = \frac{1}{ \omega \sqrt {\epsilon _ o V}} (\omega Q - iP) = \sqrt {\frac{2 \hbar }{ \omega \epsilon _ o V}} a^\dagger \end{align}</math>
−
|num=EQ_ part2
+
</equation>
−
}}
 
  
−
{{EqL
+
:<equation id=" part3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  {\bf E} = - i \sqrt {\frac{ \hbar \omega }{2 \epsilon _ o V} } {\left[ a{\bf \hat{e}} e^{i({\bf k}\cdot {\bf r} - \omega t)} - a^\dagger {\bf \hat{e}}^* e^{-i({\bf k}\cdot {\bf r} -\omega t)}\right]} \end{align}</math>
+
<math>\begin{align} \  {\bf E} = - i \sqrt {\frac{ \hbar \omega }{2 \epsilon _ o V} } {\left[ a{\bf \hat{e}} e^{i({\bf k}\cdot {\bf r} - \omega t)} - a^\dagger {\bf \hat{e}}^* e^{-i({\bf k}\cdot {\bf r} -\omega t)}\right]} \end{align}</math>
−
|num=EQ_ part3
+
</equation>
−
}}
 
  
 
In the dipole limit we can take <math>e^{i {\bf k}\cdot {\bf r}} = 1</math>. Then  
 
In the dipole limit we can take <math>e^{i {\bf k}\cdot {\bf r}} = 1</math>. Then  
  
−
{{EqL
+
:<equation id=" part3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  {\bf E} = - i \sqrt {\frac{ \hbar \omega }{2 \epsilon _ o V} } \left[ a {\bf \hat e} e^{-i \omega t}- a^\dagger {\bf {\hat e}}^* e^{i \omega t}\right] \end{align}</math>
+
<math>\begin{align} \  {\bf E} = - i \sqrt {\frac{ \hbar \omega }{2 \epsilon _ o V} } \left[ a {\bf \hat e} e^{-i \omega t}- a^\dagger {\bf {\hat e}}^* e^{i \omega t}\right] \end{align}</math>
−
|num=EQ_ part3
+
</equation>
−
}}
 
  
 
The interaction Hamiltonian is,  
 
The interaction Hamiltonian is,  
  
−
{{EqL
+
:<equation id=" qrd16" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{\rm int}= -ie \sqrt {\frac{\hbar \omega }{2\epsilon _ o V}}{\bf r}\cdot {\left[ a{\bf \hat{e}} e^{-i\omega t} - a^\dagger {\bf \hat{e}}^* e^{+i\omega t}\right]}, \end{align}</math>
+
<math>\begin{align} \  H_{\rm int}= -ie \sqrt {\frac{\hbar \omega }{2\epsilon _ o V}}{\bf r}\cdot {\left[ a{\bf \hat{e}} e^{-i\omega t} - a^\dagger {\bf \hat{e}}^* e^{+i\omega t}\right]}, \end{align}</math>
−
|num=EQ_ qrd16
+
</equation>
−
}}
 
  
 
where we have written the dipole operator as <math>{\bf d} = - e {\bf r}</math>.  
 
where we have written the dipole operator as <math>{\bf d} = - e {\bf r}</math>.  
Line 383: Line 337:
 
We consider a two-state atomic system <math> | a \rangle </math>,  <math>| b \rangle </math> and a radiation field described by <math>| n \rangle ,\  n = 0,1,2 \dots </math> The states of the total system can be taken to be  
 
We consider a two-state atomic system <math> | a \rangle </math>,  <math>| b \rangle </math> and a radiation field described by <math>| n \rangle ,\  n = 0,1,2 \dots </math> The states of the total system can be taken to be  
  
−
{{EqL
+
:<equation id=" vac1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  | I \rangle = | a,\  n \rangle = | a \rangle \  | n \rangle , ~ ~ ~  | F \rangle = | b,\  n^\prime \rangle = |b \rangle \  |n^\prime \rangle . \end{align}</math>
+
<math>\begin{align} \  | I \rangle = | a,\  n \rangle = | a \rangle \  | n \rangle , ~ ~ ~  | F \rangle = | b,\  n^\prime \rangle = |b \rangle \  |n^\prime \rangle . \end{align}</math>
−
|num=EQ_ vac1
+
</equation>
−
}}
 
  
 
We shall take <math>{\bf \hat{e}} = {\bf \hat{ z}} </math>. Then  
 
We shall take <math>{\bf \hat{e}} = {\bf \hat{ z}} </math>. Then  
  
−
{{EqL
+
:<equation id=" vac2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \langle F |H_{\rm int} | I \rangle = -i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} \langle n^\prime | a e^{-i\omega t} - a^\dagger e^{i\omega t} | n \rangle e^{-i\omega _{ab} t} \end{align}</math>
+
<math>\begin{align} \  \langle F |H_{\rm int} | I \rangle = -i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} \langle n^\prime | a e^{-i\omega t} - a^\dagger e^{i\omega t} | n \rangle e^{-i\omega _{ab} t} \end{align}</math>
−
|num=EQ_ vac2
+
</equation>
−
}}
 
  
 
The first term in the bracket obeys the selection rule <math>n^\prime = n - 1</math>. This corresponds to loss of one photon from the field and absorption of one photon by the atom. The second term obeys <math>n^\prime = n + 1</math>. This corresponds to emission of a photon by the atom. Using Eq. [[{{SUBPAGENAME}}#EQ_qrd13|EQ_qrd13]], we have  
 
The first term in the bracket obeys the selection rule <math>n^\prime = n - 1</math>. This corresponds to loss of one photon from the field and absorption of one photon by the atom. The second term obeys <math>n^\prime = n + 1</math>. This corresponds to emission of a photon by the atom. Using Eq. [[{{SUBPAGENAME}}#EQ_qrd13|EQ_qrd13]], we have  
  
−
{{EqL
+
:<equation id=" vac3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \langle F | H_{\rm int} | I \rangle = -i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} {\left( \sqrt {n}\, \delta _{n\prime ,n-1} \  e^{-i \omega t} - \sqrt {n+1}\, \delta _{n\prime ,n+1} e^{+i\omega t} \right)} \  e^{-i\omega _{ab} t} \end{align}</math>
+
<math>\begin{align} \  \langle F | H_{\rm int} | I \rangle = -i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} {\left( \sqrt {n}\, \delta _{n\prime ,n-1} \  e^{-i \omega t} - \sqrt {n+1}\, \delta _{n\prime ,n+1} e^{+i\omega t} \right)} \  e^{-i\omega _{ab} t} \end{align}</math>
−
|num=EQ_ vac3
+
</equation>
−
}}
 
  
 
Transitions occur when the total time dependence is zero, or near zero. Thus absorption occurs when <math>\omega =- \omega _{ab}</math>, or <math>E_ a + \hbar \omega = E_ b</math>. As we expect, energy is conserved. Similarly, emission occurs when <math>\omega = + \omega _{ab}</math>, or <math>E_ a - \hbar \omega = E_ b</math>.  
 
Transitions occur when the total time dependence is zero, or near zero. Thus absorption occurs when <math>\omega =- \omega _{ab}</math>, or <math>E_ a + \hbar \omega = E_ b</math>. As we expect, energy is conserved. Similarly, emission occurs when <math>\omega = + \omega _{ab}</math>, or <math>E_ a - \hbar \omega = E_ b</math>.  
Line 406: Line 357:
 
A particularly interesting case occurs when <math>n = 0</math>, i.e.  the field is initially in the vacuum state, and <math>\omega = \omega _{ab}</math>. Then  
 
A particularly interesting case occurs when <math>n = 0</math>, i.e.  the field is initially in the vacuum state, and <math>\omega = \omega _{ab}</math>. Then  
  
−
{{EqL
+
:<equation id=" vac4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \langle F | H_{\rm int} | I \rangle = i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} \equiv H_{FI}^0 \end{align}</math>
+
<math>\begin{align} \  \langle F | H_{\rm int} | I \rangle = i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} \equiv H_{FI}^0 \end{align}</math>
−
|num=EQ_ vac4
+
</equation>
−
}}
 
  
 
The situation describes a constant perturbation <math>H_{FI}^0</math> coupling the two states <math>I = | a , n = 0 \rangle </math> and <math>F = | b, n^\prime = 1 \rangle </math>. The states are degenerate because <math>E_ a = E_ b + \hbar \omega </math>. Consequently, <math>E_ a</math> is the upper of the two atomic energy levels.  
 
The situation describes a constant perturbation <math>H_{FI}^0</math> coupling the two states <math>I = | a , n = 0 \rangle </math> and <math>F = | b, n^\prime = 1 \rangle </math>. The states are degenerate because <math>E_ a = E_ b + \hbar \omega </math>. Consequently, <math>E_ a</math> is the upper of the two atomic energy levels.  
Line 415: Line 365:
 
The system is composed of two degenerate eigenstates, but due to the coupling of the field, the degeneracy is split. The eigenstates are symmetric and antisymmetric combinations of the initial states, and we can label them as  
 
The system is composed of two degenerate eigenstates, but due to the coupling of the field, the degeneracy is split. The eigenstates are symmetric and antisymmetric combinations of the initial states, and we can label them as  
  
−
{{EqL
+
:<equation id=" vac5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  | \pm \rangle = \frac{1}{\sqrt {2}} (|I \rangle \pm | F \rangle ) = \frac{1}{\sqrt {2}} ( | a , 0 \rangle \pm | b, 1 \rangle ). \end{align}</math>
+
<math>\begin{align} \  | \pm \rangle = \frac{1}{\sqrt {2}} (|I \rangle \pm | F \rangle ) = \frac{1}{\sqrt {2}} ( | a , 0 \rangle \pm | b, 1 \rangle ). \end{align}</math>
−
|num=EQ_ vac5
+
</equation>
−
}}
 
  
 
The energies of these states are  
 
The energies of these states are  
  
−
{{EqL
+
:<equation id=" vac6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  E_{\pm } = \pm | H_{FI}^0 | \end{align}</math>
+
<math>\begin{align} \  E_{\pm } = \pm | H_{FI}^0 | \end{align}</math>
−
|num=EQ_ vac6
+
</equation>
−
}}
 
  
 
If at <math>t = 0</math>, the atom is in state <math>| a \rangle </math> which means that the radiation field is in state <math>| 0 \rangle </math> then the system is in a superposition state:  
 
If at <math>t = 0</math>, the atom is in state <math>| a \rangle </math> which means that the radiation field is in state <math>| 0 \rangle </math> then the system is in a superposition state:  
  
−
{{EqL
+
:<equation id=" vac7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \psi (0) = \frac{1}{\sqrt {2}} ( | + \rangle + | - \rangle ) . \end{align}</math>
+
<math>\begin{align} \  \psi (0) = \frac{1}{\sqrt {2}} ( | + \rangle + | - \rangle ) . \end{align}</math>
−
|num=EQ_ vac7
+
</equation>
−
}}
 
  
 
The time evolution of this superposition is given by  
 
The time evolution of this superposition is given by  
  
−
{{EqL
+
:<equation id=" vac8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \psi (t) = \frac{1}{\sqrt {2}} \left(| + \rangle e^{i\Omega /2t} + | - \rangle e^{-i\Omega /2t} \right) \end{align}</math>
+
<math>\begin{align} \  \psi (t) = \frac{1}{\sqrt {2}} \left(| + \rangle e^{i\Omega /2t} + | - \rangle e^{-i\Omega /2t} \right) \end{align}</math>
−
|num=EQ_ vac8
+
</equation>
−
}}
 
  
 
where <math>\Omega / 2 = | H_{FI}^0 | / \hbar = e z_{ab}\sqrt {\omega / (e \epsilon _ o V \hbar )}</math>. The probability that the atom is in state <math> | b \rangle </math> at a later time is  
 
where <math>\Omega / 2 = | H_{FI}^0 | / \hbar = e z_{ab}\sqrt {\omega / (e \epsilon _ o V \hbar )}</math>. The probability that the atom is in state <math> | b \rangle </math> at a later time is  
  
−
{{EqL
+
:<equation id=" vac9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  P_ b = \frac{1}{2} (1 + \cos \Omega t ). \end{align}</math>
+
<math>\begin{align} \  P_ b = \frac{1}{2} (1 + \cos \Omega t ). \end{align}</math>
−
|num=EQ_ vac9
+
</equation>
−
}}
 
  
 
The frequency <math>\Omega </math> is called the vacuum Rabi frequency.  
 
The frequency <math>\Omega </math> is called the vacuum Rabi frequency.  
Line 456: Line 401:
 
The atom-vacuum interaction <math>H_{FI}^0</math>, Eq. [[{{SUBPAGENAME}}#EQ_vac4|EQ_vac4]], has a simple physical interpretation. The electric field amplitude associated with the zero point energy in the cavity is given by  
 
The atom-vacuum interaction <math>H_{FI}^0</math>, Eq. [[{{SUBPAGENAME}}#EQ_vac4|EQ_vac4]], has a simple physical interpretation. The electric field amplitude associated with the zero point energy in the cavity is given by  
  
−
{{EqL
+
:<equation id=" vac10" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \epsilon _ o E^2 V = \frac{1}{2} \hbar \omega \end{align}</math>
+
<math>\begin{align} \  \epsilon _ o E^2 V = \frac{1}{2} \hbar \omega \end{align}</math>
−
|num=EQ_ vac10
+
</equation>
−
}}
 
  
 
Consequently, <math>| H_{FI}^0 | = E d_{ab}= ez_{ab} E</math>. The interaction frequency <math>| H_{FI}^0 | / \hbar </math> is sometimes referred to as the vacuum Rabi frequency, although, as we have seen, the actual oscillation frequency is <math>2 \times H_{FI}^0 /\hbar </math>.  
 
Consequently, <math>| H_{FI}^0 | = E d_{ab}= ez_{ab} E</math>. The interaction frequency <math>| H_{FI}^0 | / \hbar </math> is sometimes referred to as the vacuum Rabi frequency, although, as we have seen, the actual oscillation frequency is <math>2 \times H_{FI}^0 /\hbar </math>.  
Line 465: Line 409:
 
Absorption and emission are closely related. Because the rates are proportional to <math>| \langle F | H_{\rm int} | I \rangle |^2</math>, it is evident from Eq. [[{{SUBPAGENAME}}#EQ_vac3|EQ_vac3]] that  
 
Absorption and emission are closely related. Because the rates are proportional to <math>| \langle F | H_{\rm int} | I \rangle |^2</math>, it is evident from Eq. [[{{SUBPAGENAME}}#EQ_vac3|EQ_vac3]] that  
  
−
{{EqL
+
:<equation id=" vac11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \frac{\rm Rate~ of~ emission}{\rm Rate~ of~ absorption} = \frac{n+1}{n} \end{align}</math>
+
<math>\begin{align} \  \frac{\rm Rate~ of~ emission}{\rm Rate~ of~ absorption} = \frac{n+1}{n} \end{align}</math>
−
|num=EQ_ vac11
+
</equation>
−
}}
 
  
 
This result, which applies to radiative transitions between any two states of a system, is general. In the absence of spontaneous emission, the absorption and emission rates are identical.  
 
This result, which applies to radiative transitions between any two states of a system, is general. In the absence of spontaneous emission, the absorption and emission rates are identical.  
Line 484: Line 427:
 
In Chapter 6, first-order perturbation theory was applied to find the response of a system initially in state <math>|a\rangle </math> to a perturbation of the form <math>( H_{ba}/2 ) e^{-i\omega t}</math>. The result is that the amplitude for state <math>|b \rangle </math> is given by  
 
In Chapter 6, first-order perturbation theory was applied to find the response of a system initially in state <math>|a\rangle </math> to a perturbation of the form <math>( H_{ba}/2 ) e^{-i\omega t}</math>. The result is that the amplitude for state <math>|b \rangle </math> is given by  
  
−
{{EqL
+
:<equation id=" abem1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  a_ b (t) = \frac{1}{2 i\hbar } \int _0^ t H_{ba} e^{-i(\omega - \omega _{ba} )t^\prime } dt^\prime = \frac{H_{ba}}{2\hbar } {\left[ \frac{e^{-i(\omega - \omega _{ba} )t} -1}{\omega - \omega _{ba}} \right]} \end{align}</math>
+
<math>\begin{align} \  a_ b (t) = \frac{1}{2 i\hbar } \int _0^ t H_{ba} e^{-i(\omega - \omega _{ba} )t^\prime } dt^\prime = \frac{H_{ba}}{2\hbar } {\left[ \frac{e^{-i(\omega - \omega _{ba} )t} -1}{\omega - \omega _{ba}} \right]} \end{align}</math>
−
|num=EQ_ abem1
+
</equation>
−
}}
 
  
 
There will be a similar expression involving the time-dependence <math>e^{+ i \omega t}</math>. The <math>- i \omega </math> term gives rise to resonance at <math>\omega = \omega _{ba}</math>; the <math>+ i \omega </math> term gives rise to resonance at <math>\omega = \omega _{ab}</math>. One term is responsible for absorption, the other is responsible for emission.  
 
There will be a similar expression involving the time-dependence <math>e^{+ i \omega t}</math>. The <math>- i \omega </math> term gives rise to resonance at <math>\omega = \omega _{ba}</math>; the <math>+ i \omega </math> term gives rise to resonance at <math>\omega = \omega _{ab}</math>. One term is responsible for absorption, the other is responsible for emission.  
Line 493: Line 435:
 
The probability that the system has made a transition to state <math>| b \rangle </math> at time <math>t</math> is  
 
The probability that the system has made a transition to state <math>| b \rangle </math> at time <math>t</math> is  
  
−
{{EqL
+
:<equation id=" abem2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{a\rightarrow b} = | a_ b (t)|^2 = \frac{| H_{ba}|^2}{4 \hbar ^2} \frac{\sin ^2 [(\omega - \omega _{ba} )t/2]}{((\omega - \omega _{ba} )t/2)^2}t^2 \end{align}</math>
+
<math>\begin{align} \  W_{a\rightarrow b} = | a_ b (t)|^2 = \frac{| H_{ba}|^2}{4 \hbar ^2} \frac{\sin ^2 [(\omega - \omega _{ba} )t/2]}{((\omega - \omega _{ba} )t/2)^2}t^2 \end{align}</math>
−
|num=EQ_ abem2
+
</equation>
−
}}
 
  
 
In the limit <math>\omega \rightarrow \omega _{ba}</math>, we have  
 
In the limit <math>\omega \rightarrow \omega _{ba}</math>, we have  
  
−
{{EqL
+
:<equation id=" abem3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{a\rightarrow b} \approx \frac{| H_{ba}|^2}{4 \hbar ^2} t^2 . \end{align}</math>
+
<math>\begin{align} \  W_{a\rightarrow b} \approx \frac{| H_{ba}|^2}{4 \hbar ^2} t^2 . \end{align}</math>
−
|num=EQ_ abem3
+
</equation>
−
}}
 
  
 
So, for short time, <math>W_{a\rightarrow b}</math> increases quadratically. This is reminiscent of a Rabi resonance in a 2-level system in the limit of short time.  
 
So, for short time, <math>W_{a\rightarrow b}</math> increases quadratically. This is reminiscent of a Rabi resonance in a 2-level system in the limit of short time.  
Line 509: Line 449:
 
However, Eq. [[{{SUBPAGENAME}}#EQ_abem2|EQ_abem2]] is only valid provided <math>W_{a\rightarrow b} \ll 1</math>, or for time <math>T \ll \hbar /H_{ba}</math>. For such a short time, the incident radiation will have a spectral width <math>\Delta \omega \sim 1/T</math>. In this case, we must integrate Eq. [[{{SUBPAGENAME}}#EQ_abem2|EQ_abem2]] over the spectrum. In doing this, we shall make use of the relation  
 
However, Eq. [[{{SUBPAGENAME}}#EQ_abem2|EQ_abem2]] is only valid provided <math>W_{a\rightarrow b} \ll 1</math>, or for time <math>T \ll \hbar /H_{ba}</math>. For such a short time, the incident radiation will have a spectral width <math>\Delta \omega \sim 1/T</math>. In this case, we must integrate Eq. [[{{SUBPAGENAME}}#EQ_abem2|EQ_abem2]] over the spectrum. In doing this, we shall make use of the relation  
  
−
{{EqL
+
:<equation id=" abem4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \int _{-\infty }^{+\infty } \frac{\sin ^2 (\omega - \omega _{ba})t/2}{[(\omega - \omega _{ba})/2]^2} d \omega = 2t \int _{-\infty }^{+\infty } \frac{\sin ^2 (u - u_ o)}{(u - u_ o)^2} d u \rightarrow 2 \pi t \int _{-\infty }^{+\infty } \delta (\omega - \omega _{ba} ) d \omega . \end{align}</math>
+
<math>\begin{align} \  \int _{-\infty }^{+\infty } \frac{\sin ^2 (\omega - \omega _{ba})t/2}{[(\omega - \omega _{ba})/2]^2} d \omega = 2t \int _{-\infty }^{+\infty } \frac{\sin ^2 (u - u_ o)}{(u - u_ o)^2} d u \rightarrow 2 \pi t \int _{-\infty }^{+\infty } \delta (\omega - \omega _{ba} ) d \omega . \end{align}</math>
−
|num=EQ_ abem4
+
</equation>
−
}}
 
  
 
Eq. [[{{SUBPAGENAME}}#EQ_abem2|EQ_abem2]] becomes  
 
Eq. [[{{SUBPAGENAME}}#EQ_abem2|EQ_abem2]] becomes  
  
−
{{EqL
+
:<equation id=" abem5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{a\rightarrow b} = \frac{| H_{ba}|^2}{\hbar ^2} 2\pi t \delta (\omega - \omega _{ba} ) \end{align}</math>
+
<math>\begin{align} \  W_{a\rightarrow b} = \frac{| H_{ba}|^2}{\hbar ^2} 2\pi t \delta (\omega - \omega _{ba} ) \end{align}</math>
−
|num=EQ_ abem5
+
</equation>
−
}}
 
  
 
The <math>\delta </math>-function requires that eventually <math>W_{a\rightarrow b}</math> be integrated over a spectral distribution function. Absorbing an <math> \hbar </math> into the delta function, <math>W_{a\rightarrow b}</math> can be written  
 
The <math>\delta </math>-function requires that eventually <math>W_{a\rightarrow b}</math> be integrated over a spectral distribution function. Absorbing an <math> \hbar </math> into the delta function, <math>W_{a\rightarrow b}</math> can be written  
  
−
{{EqL
+
:<equation id=" abem6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{a\rightarrow b} = \frac{| H_{ba}|^2}{\hbar} 2\pi t \delta (E_ b - E_ a - \hbar \omega ). \end{align}</math>
+
<math>\begin{align} \  W_{a\rightarrow b} = \frac{| H_{ba}|^2}{\hbar} 2\pi t \delta (E_ b - E_ a - \hbar \omega ). \end{align}</math>
−
|num=EQ_ abem6
+
</equation>
−
}}
 
  
 
Because the transition probability is proportional to the time, we can define the transition rate  
 
Because the transition probability is proportional to the time, we can define the transition rate  
  
−
{{EqL
+
:<equation id=" abem7a" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = \frac{d}{dt} W_{a\rightarrow b} = 2\pi \frac{| H_{ba}|^2}{\hbar} \delta (\omega - \omega _{ba}) \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = \frac{d}{dt} W_{a\rightarrow b} = 2\pi \frac{| H_{ba}|^2}{\hbar} \delta (\omega - \omega _{ba}) \end{align}</math>
−
|num=EQ_ abem7a
+
</equation>
−
}}
 
  
−
{{EqL
+
:<equation id=" abem7b" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  = 2\pi \frac{| H_{ba}|^2}{\hbar } \delta (E_ b - E_ a - \hbar \omega ) \end{align}</math>
+
<math>\begin{align} \  = 2\pi \frac{| H_{ba}|^2}{\hbar } \delta (E_ b - E_ a - \hbar \omega ) \end{align}</math>
−
|num=EQ_ abem7b
+
</equation>
−
}}
 
  
 
The <math>\delta </math>-function arises because of the assumption in first order perturbation theory that the amplitude of the initial state is not affected significantly. This will not be the case, for instance, if a monochromatic radiation field couples the two states, in which case the amplitudes oscillate between 0 and 1. However, the assumption of perfectly monochromatic radiation is in itself unrealistic.  
 
The <math>\delta </math>-function arises because of the assumption in first order perturbation theory that the amplitude of the initial state is not affected significantly. This will not be the case, for instance, if a monochromatic radiation field couples the two states, in which case the amplitudes oscillate between 0 and 1. However, the assumption of perfectly monochromatic radiation is in itself unrealistic.  
Line 548: Line 483:
 
where <math>S_0</math> is the incident Poynting vector, and f(<math>\omega ^\prime </math>) is a normalized line shape function centered at the frequency <math>\omega ^\prime </math> which obeys <math>\int f (\omega ^\prime ) d\omega ^\prime = 1</math>. We can define a characteristic spectral width of <math>f(\omega ^\prime )</math> by  
 
where <math>S_0</math> is the incident Poynting vector, and f(<math>\omega ^\prime </math>) is a normalized line shape function centered at the frequency <math>\omega ^\prime </math> which obeys <math>\int f (\omega ^\prime ) d\omega ^\prime = 1</math>. We can define a characteristic spectral width of <math>f(\omega ^\prime )</math> by  
  
−
{{EqL
+
:<equation id=" abem8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Delta \omega = \frac{1}{f(\omega _{ab} )} \end{align}</math>
+
<math>\begin{align} \  \Delta \omega = \frac{1}{f(\omega _{ab} )} \end{align}</math>
−
|num=EQ_ abem8
+
</equation>
−
}}
 
  
 
Integrating Eq. [[{{SUBPAGENAME}}#EQ_abem7b|EQ_abem7b]] over the spectrum of the radiation gives  
 
Integrating Eq. [[{{SUBPAGENAME}}#EQ_abem7b|EQ_abem7b]] over the spectrum of the radiation gives  
  
−
{{EqL
+
:<equation id=" abem9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = \frac{2\pi | H_{ba}|^2}{\hbar ^2} f(\omega _{ab} ) \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = \frac{2\pi | H_{ba}|^2}{\hbar ^2} f(\omega _{ab} ) \end{align}</math>
−
|num=EQ_ abem9
+
</equation>
−
}}
 
  
 
If we define the effective Rabi frequency by  
 
If we define the effective Rabi frequency by  
  
−
{{EqL
+
:<equation id=" abem10" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Omega _ R = \frac{| H_{ba}| }{\hbar } \end{align}</math>
+
<math>\begin{align} \  \Omega _ R = \frac{| H_{ba}| }{\hbar } \end{align}</math>
−
|num=EQ_ abem10
+
</equation>
−
}}
 
  
 
then  
 
then  
  
−
{{EqL
+
:<equation id=" abem11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = {2 \pi } \frac{\Omega _ R^2}{\Delta \omega } \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = {2 \pi } \frac{\Omega _ R^2}{\Delta \omega } \end{align}</math>
−
|num=EQ_ abem11
+
</equation>
−
}}
 
  
 
Another situation that often occurs is when the radiation is monochromatic, but the final state is actually composed of many states spaced close to each other in energy so as to form a continuum. If such is the case, the density of final states can be described by  
 
Another situation that often occurs is when the radiation is monochromatic, but the final state is actually composed of many states spaced close to each other in energy so as to form a continuum. If such is the case, the density of final states can be described by  
  
−
{{EqL
+
:<equation id=" abem12" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  dN= \rho (E) dE \end{align}</math>
+
<math>\begin{align} \  dN= \rho (E) dE \end{align}</math>
−
|num=EQ_ abem12
+
</equation>
−
}}
 
  
 
where <math>dN</math> is the number of states in range <math>dE</math>. Taking <math>\hbar \omega = E_ b - E_ a</math> in Eq. [[{{SUBPAGENAME}}#EQ_abem7b|EQ_abem7b]], and integrating gives  
 
where <math>dN</math> is the number of states in range <math>dE</math>. Taking <math>\hbar \omega = E_ b - E_ a</math> in Eq. [[{{SUBPAGENAME}}#EQ_abem7b|EQ_abem7b]], and integrating gives  
  
−
{{EqL
+
:<equation id=" abem13" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = 2\pi \frac{| H_{ba}|^2}{\hbar^2 } \rho (E_ b ) \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = 2\pi \frac{| H_{ba}|^2}{\hbar^2 } \rho (E_ b ) \end{align}</math>
−
|num=EQ_ abem13
+
</equation>
−
}}
 
  
 
This result remains valid in the limit <math>E_ b\rightarrow E_ a</math>, where <math>\omega \rightarrow 0</math>. In this static situation, the result is known as <i>
 
This result remains valid in the limit <math>E_ b\rightarrow E_ a</math>, where <math>\omega \rightarrow 0</math>. In this static situation, the result is known as <i>
Line 594: Line 523:
 
Note that Eq. [[{{SUBPAGENAME}}#EQ_abem9|EQ_abem9]] and Eq. [[{{SUBPAGENAME}}#EQ_abem13|EQ_abem13]] both describe a uniform rate process in which the population of the initial state decreases exponentially in time. If the population of the initial state is <math>P(0)</math>, then  
 
Note that Eq. [[{{SUBPAGENAME}}#EQ_abem9|EQ_abem9]] and Eq. [[{{SUBPAGENAME}}#EQ_abem13|EQ_abem13]] both describe a uniform rate process in which the population of the initial state decreases exponentially in time. If the population of the initial state is <math>P(0)</math>, then  
  
−
{{EqL
+
:<equation id=" abem14" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  P(t) = P(0) e^{-\Gamma _{ba} t} \end{align}</math>
+
<math>\begin{align} \  P(t) = P(0) e^{-\Gamma _{ba} t} \end{align}</math>
−
|num=EQ_ abem14
+
</equation>
−
}}
 
  
 
Applying this to the dipole transition described in Eq. [[{{SUBPAGENAME}}#EQ_int11|EQ_int11]], we have  
 
Applying this to the dipole transition described in Eq. [[{{SUBPAGENAME}}#EQ_int11|EQ_int11]], we have  
  
−
{{EqL
+
:<equation id=" abem15" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = 2\pi \frac{E^2 d_{ba}^2}{\hbar ^2} f(\omega ) \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = 2\pi \frac{E^2 d_{ba}^2}{\hbar ^2} f(\omega ) \end{align}</math>
−
|num=EQ_ abem15
+
</equation>
−
}}
 
  
 
The arguments here do not distinguish whether <math>E_ a < E_ b</math> or <math>E_ a > E_ b</math> (though the sign of <math>\omega = ( E_ b - E_ a )/\hbar </math> obviously does). In the former case the process is absorption, in the latter case it is emission.  
 
The arguments here do not distinguish whether <math>E_ a < E_ b</math> or <math>E_ a > E_ b</math> (though the sign of <math>\omega = ( E_ b - E_ a )/\hbar </math> obviously does). In the former case the process is absorption, in the latter case it is emission.  
Line 616: Line 543:
 
The rate of absorption, in CGS units, for the transition <math>a \rightarrow b</math>, where <math>E_ b > E_ a</math>, is, from Eq. [[{{SUBPAGENAME}}#EQ_qrd16|EQ_qrd16]] and Eq. [[{{SUBPAGENAME}}#EQ_abem7b|EQ_abem7b]],  
 
The rate of absorption, in CGS units, for the transition <math>a \rightarrow b</math>, where <math>E_ b > E_ a</math>, is, from Eq. [[{{SUBPAGENAME}}#EQ_qrd16|EQ_qrd16]] and Eq. [[{{SUBPAGENAME}}#EQ_abem7b|EQ_abem7b]],  
  
−
{{EqL
+
:<equation id=" sem1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = \frac{4\pi ^2}{\hbar V} | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 n\omega \delta (\omega _0 -\omega ) . \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = \frac{4\pi ^2}{\hbar V} | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 n\omega \delta (\omega _0 -\omega ) . \end{align}</math>
−
|num=EQ_ sem1
+
</equation>
−
}}
 
  
 
where <math>\omega _0 = ( E_ b - E_ a ) /\hbar </math>. To evaluate this we need to let <math>n \rightarrow n (\omega )</math>, where <math>n (\omega ) d\omega </math> is the number of photons in the frequency interval <math>d\omega </math>, and integrate over the spectrum. The result is  
 
where <math>\omega _0 = ( E_ b - E_ a ) /\hbar </math>. To evaluate this we need to let <math>n \rightarrow n (\omega )</math>, where <math>n (\omega ) d\omega </math> is the number of photons in the frequency interval <math>d\omega </math>, and integrate over the spectrum. The result is  
  
−
{{EqL
+
:<equation id=" sem2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = \frac{4\pi ^2}{\hbar V} | {\bf \hat{e}}\cdot {\bf d}_{ba} |^2 \omega _0 n(\omega _0 ) \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = \frac{4\pi ^2}{\hbar V} | {\bf \hat{e}}\cdot {\bf d}_{ba} |^2 \omega _0 n(\omega _0 ) \end{align}</math>
−
|num=EQ_ sem2
+
</equation>
−
}}
 
  
 
To calculate <math>n (\omega )</math>, we first calculate the mode density in space by applying the usual periodic boundary condition  
 
To calculate <math>n (\omega )</math>, we first calculate the mode density in space by applying the usual periodic boundary condition  
  
−
{{EqL
+
:<equation id=" sem3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  k_ j L = 2\pi n_ j , ~ ~ ~ j = x,y,z. \end{align}</math>
+
<math>\begin{align} \  k_ j L = 2\pi n_ j , ~ ~ ~ j = x,y,z. \end{align}</math>
−
|num=EQ_ sem3
+
</equation>
−
}}
 
  
 
The number of modes in the range <math>d^3 k = dk_ x dk_ y dk_ z</math> is  
 
The number of modes in the range <math>d^3 k = dk_ x dk_ y dk_ z</math> is  
  
−
{{EqL
+
:<equation id=" sem4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  dN = dn_ x dn_ y dn_ z = \frac{V}{{\left(2 \pi \right)^3} } d^3 k=\frac{V}{{\left(2 \pi \right)^3} }k^2 dk \  d\Omega = \frac{V}{{\left(2 \pi \right)^3} } \frac{\omega ^2\,  d\omega \  d\Omega }{c^3} \end{align}</math>
+
<math>\begin{align}  \  dN = dn_ x dn_ y dn_ z = \frac{V}{{\left(2 \pi \right)^3} } d^3 k=\frac{V}{{\left(2 \pi \right)^3} }k^2 dk \  d\Omega = \frac{V}{{\left(2 \pi \right)^3} } \frac{\omega ^2\,  d\omega \  d\Omega }{c^3} \end{align}</math>
−
|num=EQ_ sem4
+
</equation>
−
}}
 
  
 
Letting <math>\bar{n} = \bar{n (\omega ) }</math> be the average number of photons per mode, then  
 
Letting <math>\bar{n} = \bar{n (\omega ) }</math> be the average number of photons per mode, then  
  
−
{{EqL
+
:<equation id=" sem5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  n (\omega ) = \bar{n} \frac{dN}{d\omega } = \frac{\bar{n} V\omega ^2 d\Omega }{(2\pi )^3 c^3} \end{align}</math>
+
<math>\begin{align} \  n (\omega ) = \bar{n} \frac{dN}{d\omega } = \frac{\bar{n} V\omega ^2 d\Omega }{(2\pi )^3 c^3} \end{align}</math>
−
|num=EQ_ sem5
+
</equation>
−
}}
 
  
 
Introducing this into Eq. [[{{SUBPAGENAME}}#EQ_sem2|EQ_sem2]] gives  
 
Introducing this into Eq. [[{{SUBPAGENAME}}#EQ_sem2|EQ_sem2]] gives  
  
−
{{EqL
+
:<equation id=" sem6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = \frac{\bar{n}\omega ^3}{2\pi \hbar c^3} | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 d\Omega \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = \frac{\bar{n}\omega ^3}{2\pi \hbar c^3} | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 d\Omega \end{align}</math>
−
|num=EQ_ sem6
+
</equation>
−
}}
 
  
 
We wish to apply this to the case of isotropic radiation in free space, as, for instance, in a thermal radiation field. We can take <math>{\bf d}_{ba}</math> to lie along the <math>z</math> axis and describe '''k''' in spherical coordinates about this axis. Since the wave is transverse, <math>{\bf \hat{e}} \cdot {\bf \hat{D}} = \sin \theta </math> for one polarization, and zero for the other one.  Consequently,  
 
We wish to apply this to the case of isotropic radiation in free space, as, for instance, in a thermal radiation field. We can take <math>{\bf d}_{ba}</math> to lie along the <math>z</math> axis and describe '''k''' in spherical coordinates about this axis. Since the wave is transverse, <math>{\bf \hat{e}} \cdot {\bf \hat{D}} = \sin \theta </math> for one polarization, and zero for the other one.  Consequently,  
  
−
{{EqL
+
:<equation id=" sem7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \int | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 d\Omega = | {\bf d}_{ba} |^2 \int \sin ^2 \theta d\Omega = \frac{8\pi }{3} | {\bf d}_{ba}|^2 \end{align}</math>
+
<math>\begin{align} \  \int | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 d\Omega = | {\bf d}_{ba} |^2 \int \sin ^2 \theta d\Omega = \frac{8\pi }{3} | {\bf d}_{ba}|^2 \end{align}</math>
−
|num=EQ_ sem7
+
</equation>
−
}}
 
  
 
Introducing this into Eq. [[{{SUBPAGENAME}}#EQ_sem6|EQ_sem6]] yields the absorption rates  
 
Introducing this into Eq. [[{{SUBPAGENAME}}#EQ_sem6|EQ_sem6]] yields the absorption rates  
  
−
{{EqL
+
:<equation id=" sem8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ab} = \frac{4}{3} \frac{\omega ^3}{\hbar c^3} | {\bf d}_{ba} |^2 \bar{n} \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ab} = \frac{4}{3} \frac{\omega ^3}{\hbar c^3} | {\bf d}_{ba} |^2 \bar{n} \end{align}</math>
−
|num=EQ_ sem8
+
</equation>
−
}}
 
  
 
It follows that the emission rate for the transition <math>b\rightarrow a</math> is  
 
It follows that the emission rate for the transition <math>b\rightarrow a</math> is  
  
−
{{EqL
+
:<equation id=" sem9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ba} = \frac{4}{3} \frac{\omega ^3}{\hbar c^3} | {\bf d}_{ba} |^2 (\bar{n} + 1) \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ba} = \frac{4}{3} \frac{\omega ^3}{\hbar c^3} | {\bf d}_{ba} |^2 (\bar{n} + 1) \end{align}</math>
−
|num=EQ_ sem9
+
</equation>
−
}}
 
  
 
If there are no photons present, the emission rate—called the rate of spontaneous emission—is  
 
If there are no photons present, the emission rate—called the rate of spontaneous emission—is  
  
−
{{EqL
+
:<equation id=" sem10" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ba}^0 = \frac{4}{3} \frac{ \omega ^3}{\hbar c^3} | {\bf d}_{ba}|^2 = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} | \langle b| {\bf r} | a \rangle |^2 \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ba}^0 = \frac{4}{3} \frac{ \omega ^3}{\hbar c^3} | {\bf d}_{ba}|^2 = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} | \langle b| {\bf r} | a \rangle |^2 \end{align}</math>
−
|num=EQ_ sem10
+
</equation>
−
}}
 
  
 
In atomic units, in which <math>c = 1 / \alpha </math>, we have  
 
In atomic units, in which <math>c = 1 / \alpha </math>, we have  
  
−
{{EqL
+
:<equation id=" sem11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Gamma _{ba}^0 = \frac{4}{3} \alpha ^3 \omega ^3 | {\bf r}_{ba} |^2 . \end{align}</math>
+
<math>\begin{align} \  \Gamma _{ba}^0 = \frac{4}{3} \alpha ^3 \omega ^3 | {\bf r}_{ba} |^2 . \end{align}</math>
−
|num=EQ_ sem11
+
</equation>
−
}}
 
  
 
Taking, typically, <math>\omega = 1</math>, and <math>r_{ba}= 1</math>, we have <math>\Gamma ^0 \approx \alpha ^3</math>. The “<math>Q</math>'' of a radiative transition is <math>Q =\omega /\Gamma \approx \alpha ^{-3}\approx </math> <math>3 \times 10^6</math>. The <math>\alpha ^3</math> dependence of <math>\Gamma </math> indicates that radiation is fundamentally a weak process: hence the high <math>Q</math> and the relatively long radiative lifetime of a state, <math>\tau = 1 /\Gamma </math>. For example, for the <math>2P\rightarrow 1S</math> transition in hydrogen (the <math>L_{\alpha }</math> transition), we have <math>\omega = 3/8</math>, and taking <math>r_{2p,1s} \approx 1</math>, we find <math>\tau = 3.6\times 10^7</math> atomic units, or 0.8 ns. The actual lifetime is 1.6 ns.  
 
Taking, typically, <math>\omega = 1</math>, and <math>r_{ba}= 1</math>, we have <math>\Gamma ^0 \approx \alpha ^3</math>. The “<math>Q</math>'' of a radiative transition is <math>Q =\omega /\Gamma \approx \alpha ^{-3}\approx </math> <math>3 \times 10^6</math>. The <math>\alpha ^3</math> dependence of <math>\Gamma </math> indicates that radiation is fundamentally a weak process: hence the high <math>Q</math> and the relatively long radiative lifetime of a state, <math>\tau = 1 /\Gamma </math>. For example, for the <math>2P\rightarrow 1S</math> transition in hydrogen (the <math>L_{\alpha }</math> transition), we have <math>\omega = 3/8</math>, and taking <math>r_{2p,1s} \approx 1</math>, we find <math>\tau = 3.6\times 10^7</math> atomic units, or 0.8 ns. The actual lifetime is 1.6 ns.  
Line 703: Line 619:
 
Because the absorption and stimulated emission rates are proportional to the spontaneous emission rate, we shall focus our attention on the Einstein A coefficient:  
 
Because the absorption and stimulated emission rates are proportional to the spontaneous emission rate, we shall focus our attention on the Einstein A coefficient:  
  
−
{{EqL
+
:<equation id=" lines1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} | \langle b | {\bf r} | a \rangle |^2 \end{align}</math>
+
<math>\begin{align} \  A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} | \langle b | {\bf r} | a \rangle |^2 \end{align}</math>
−
|num=EQ_ lines1
+
</equation>
−
}}
 
  
 
where  
 
where  
  
−
{{EqL
+
:<equation id=" lines2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  | \langle b | {\bf r} | a \rangle |^2 = | \langle b | x | a \rangle |^2 + | \langle b | y | a \rangle |^2 + | \langle b | z | a \rangle |^2 \end{align}</math>
+
<math>\begin{align} \  | \langle b | {\bf r} | a \rangle |^2 = | \langle b | x | a \rangle |^2 + | \langle b | y | a \rangle |^2 + | \langle b | z | a \rangle |^2 \end{align}</math>
−
|num=EQ_ lines2
+
</equation>
−
}}
 
  
 
For an isolated atom, the initial and final states will be eigenstates of total angular momentum. (If there is an accidental degeneracy, as in hydrogen, it is still possible to select angular momentum eigenstates.) If the final angular momentum is <math>J_ a</math>, then the atom can decay into each of the <math>2 J_ a + 1</math> final states, characterized by the azimuthal quantum number <math>m_ a = -J_ a , -J_ a + 1,\dots , +J_ a</math>. Consequently,  
 
For an isolated atom, the initial and final states will be eigenstates of total angular momentum. (If there is an accidental degeneracy, as in hydrogen, it is still possible to select angular momentum eigenstates.) If the final angular momentum is <math>J_ a</math>, then the atom can decay into each of the <math>2 J_ a + 1</math> final states, characterized by the azimuthal quantum number <math>m_ a = -J_ a , -J_ a + 1,\dots , +J_ a</math>. Consequently,  
  
−
{{EqL
+
:<equation id=" lines3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3}\sum _{m_ a} | \langle b, J_ b | {\bf r} |a, J_ a, m_ a \rangle |^2 \end{align}</math>
+
<math>\begin{align} \  A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3}\sum _{m_ a} | \langle b, J_ b | {\bf r} |a, J_ a, m_ a \rangle |^2 \end{align}</math>
−
|num=EQ_ lines3
+
</equation>
−
}}
 
  
 
The upper level, however, is also degenerate, with a (<math>2 J_ b + 1</math>)–fold degeneracy. The lifetime cannot depend on which state the atom happens to be in. This follows from the isotropy of space: <math>m_ b</math> depends on the orientation of <math>{\bf J}_ b</math> with respect to some direction in space, but the decay rate for an isolated atom can't depend on how the atom happens to be oriented. Consequently, it is convenient to define the <i>
 
The upper level, however, is also degenerate, with a (<math>2 J_ b + 1</math>)–fold degeneracy. The lifetime cannot depend on which state the atom happens to be in. This follows from the isotropy of space: <math>m_ b</math> depends on the orientation of <math>{\bf J}_ b</math> with respect to some direction in space, but the decay rate for an isolated atom can't depend on how the atom happens to be oriented. Consequently, it is convenient to define the <i>
Line 726: Line 639:
 
</i> <math>S_{ba}</math>, given by  
 
</i> <math>S_{ba}</math>, given by  
  
−
{{EqL
+
:<equation id=" lines4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  S_{ba} = S_{ab} = \sum _{m_ b} \sum _{m_ a} | \langle b, J_ b, m_ b | {\bf r} | a, J_ a, m_ a \rangle |^2 \end{align}</math>
+
<math>\begin{align} \  S_{ba} = S_{ab} = \sum _{m_ b} \sum _{m_ a} | \langle b, J_ b, m_ b | {\bf r} | a, J_ a, m_ a \rangle |^2 \end{align}</math>
−
|num=EQ_ lines4
+
</equation>
−
}}
 
  
 
Then,  
 
Then,  
  
−
{{EqL
+
:<equation id=" lines5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} \frac{S_{ba}}{g_ b} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} \frac{S_{ba}}{2J_ b +1} \end{align}</math>
+
<math>\begin{align} \  A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} \frac{S_{ba}}{g_ b} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} \frac{S_{ba}}{2J_ b +1} \end{align}</math>
−
|num=EQ_ lines5
+
</equation>
−
}}
 
  
 
The line strength is closely related to the average oscillator strength <math>\bar{f}_{ab}</math>. <math>\bar{f}_{ab}</math> is obtained by averaging <math>f_{ab}</math> over the initial state <math>|b\rangle </math>, and summing over the values of <math>m</math> in the final state, <math>|a\rangle </math>. For absorption, <math>\omega _{ab} > 0</math>, and  
 
The line strength is closely related to the average oscillator strength <math>\bar{f}_{ab}</math>. <math>\bar{f}_{ab}</math> is obtained by averaging <math>f_{ab}</math> over the initial state <math>|b\rangle </math>, and summing over the values of <math>m</math> in the final state, <math>|a\rangle </math>. For absorption, <math>\omega _{ab} > 0</math>, and  
  
−
{{EqL
+
:<equation id=" line11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \bar{f}_{ab} = \frac{2m}{3\hbar } \omega _{ab} \frac{1}{2J_ b + 1} \sum _{m_ b} \sum _{m_ a} |\langle b, J_ b, m_ b |{\bf r} | a, J_ a, m_ a \rangle |^2 \end{align}</math>
+
<math>\begin{align} \  \bar{f}_{ab} = \frac{2m}{3\hbar } \omega _{ab} \frac{1}{2J_ b + 1} \sum _{m_ b} \sum _{m_ a} |\langle b, J_ b, m_ b |{\bf r} | a, J_ a, m_ a \rangle |^2 \end{align}</math>
−
|num=EQ_ line11
+
</equation>
−
}}
 
  
 
It follows that  
 
It follows that  
  
−
{{EqL
+
:<equation id=" line12" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \bar{f}_{ba} = - \frac{2J_ b + 1}{2J_ a +1} \bar{f}_{ab} . \end{align}</math>
+
<math>\begin{align} \  \bar{f}_{ba} = - \frac{2J_ b + 1}{2J_ a +1} \bar{f}_{ab} . \end{align}</math>
−
|num=EQ_ line12
+
</equation>
−
}}
 
  
 
In terms of the oscillator strength, we have  
 
In terms of the oscillator strength, we have  
  
−
{{EqL
+
:<equation id=" line13" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \bar{f}_{ab} = \frac{2m}{3\hbar }\omega _{ab} \frac{1}{2J_ b + 1} {S}_{ab} . \end{align}</math>
+
<math>\begin{align} \  \bar{f}_{ab} = \frac{2m}{3\hbar }\omega _{ab} \frac{1}{2J_ b + 1} {S}_{ab} . \end{align}</math>
−
|num=EQ_ line13
+
</equation>
−
}}
 
  
−
{{EqL
+
:<equation id=" line14" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \bar{f}_{ba} = - \frac{2m}{3\hbar } | \omega _{ab} | \frac{1}{2J_ a + 1} {S}_{ab} . \end{align}</math>
+
<math>\begin{align} \  \bar{f}_{ba} = - \frac{2m}{3\hbar } | \omega _{ab} | \frac{1}{2J_ a + 1} {S}_{ab} . \end{align}</math>
−
|num=EQ_ line14
+
</equation>
−
}}
 
  
 
<br style="clear: both" />
 
<br style="clear: both" />
Line 774: Line 681:
 
For an electric dipole transition, the radiation interaction is  
 
For an electric dipole transition, the radiation interaction is  
  
−
{{EqL
+
:<equation id=" broad1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  | H_{ba} | = e | {\bf r}_{ba} |\cdot {\bf \hat{e}} E/2, \end{align}</math>
+
<math>\begin{align} \  | H_{ba} | = e | {\bf r}_{ba} |\cdot {\bf \hat{e}} E/2, \end{align}</math>
−
|num=EQ_ broad1
+
</equation>
−
}}
 
  
 
where <math>E </math> is the amplitude of the field. The transition rate, from Eq. [[{{SUBPAGENAME}}#EQ_sem7|EQ_sem7]], is  
 
where <math>E </math> is the amplitude of the field. The transition rate, from Eq. [[{{SUBPAGENAME}}#EQ_sem7|EQ_sem7]], is  
  
−
{{EqL
+
:<equation id=" broad2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{ab} = \frac{\pi }{2} \frac{e^2 | {\bf \hat{e}} \cdot {\bf r}_{ba} |^2 E^2}{\hbar ^2} f (\omega _0 ) = \frac{\pi }{2} \frac{e^2 | {\bf \hat{e}} \cdot {\bf r}_{ba} |^2 E^2}{\hbar } f(E_ b - E_ a ) \end{align}</math>
+
<math>\begin{align} \  W_{ab} = \frac{\pi }{2} \frac{e^2 | {\bf \hat{e}} \cdot {\bf r}_{ba} |^2 E^2}{\hbar ^2} f (\omega _0 ) = \frac{\pi }{2} \frac{e^2 | {\bf \hat{e}} \cdot {\bf r}_{ba} |^2 E^2}{\hbar } f(E_ b - E_ a ) \end{align}</math>
−
|num=EQ_ broad2
+
</equation>
−
}}
 
  
 
where <math>\omega _0 = ( E_ b - E_ a )/\hbar </math> and <math>f (\omega )</math> is the normalized line shape function, or alternatively, the normalized density of states, expressed in frequency units. The transition rate is proportional to the intensity <math>I_0</math> of a monochromatic radiation source. <math>I_0</math> is given by the Poynting vector, and can be expressed by the electric field as <math>E^2 = 8 \pi I_0 / c</math>. Consequently,  
 
where <math>\omega _0 = ( E_ b - E_ a )/\hbar </math> and <math>f (\omega )</math> is the normalized line shape function, or alternatively, the normalized density of states, expressed in frequency units. The transition rate is proportional to the intensity <math>I_0</math> of a monochromatic radiation source. <math>I_0</math> is given by the Poynting vector, and can be expressed by the electric field as <math>E^2 = 8 \pi I_0 / c</math>. Consequently,  
  
−
{{EqL
+
:<equation id=" broad3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{ab} = \frac{4\pi ^2}{c} \frac{e^2 | {\bf \hat{e}} \cdot {\bf r}_{ba} |^2}{\hbar ^2} I_0 f (\omega _0 ) \end{align}</math>
+
<math>\begin{align} \  W_{ab} = \frac{4\pi ^2}{c} \frac{e^2 | {\bf \hat{e}} \cdot {\bf r}_{ba} |^2}{\hbar ^2} I_0 f (\omega _0 ) \end{align}</math>
−
|num=EQ_ broad3
+
</equation>
−
}}
 
  
 
In the case of a Lorentzian line having a FWHM of <math>\Gamma _0</math> centered on frequency <math>\omega _0</math>,  
 
In the case of a Lorentzian line having a FWHM of <math>\Gamma _0</math> centered on frequency <math>\omega _0</math>,  
  
−
{{EqL
+
:<equation id=" broad4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  f(\omega ) = \frac{1}{\pi } \frac{(\Gamma _0 /2)}{(\omega - \omega _0 )^2 + (\Gamma _0 /2)^2} \end{align}</math>
+
<math>\begin{align} \  f(\omega ) = \frac{1}{\pi } \frac{(\Gamma _0 /2)}{(\omega - \omega _0 )^2 + (\Gamma _0 /2)^2} \end{align}</math>
−
|num=EQ_ broad4
+
</equation>
−
}}
 
  
 
In this case,  
 
In this case,  
  
−
{{EqL
+
:<equation id=" broad5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{ab} = \frac{8\pi e^2}{c\hbar ^2 \Gamma _0} | \langle b | {\bf \hat{e}} \cdot {\bf r} | a \rangle |^2 I_0 \end{align}</math>
+
<math>\begin{align} \  W_{ab} = \frac{8\pi e^2}{c\hbar ^2 \Gamma _0} | \langle b | {\bf \hat{e}} \cdot {\bf r} | a \rangle |^2 I_0 \end{align}</math>
−
|num=EQ_ broad5
+
</equation>
−
}}
 
  
 
Note that <math>W_{ab}</math> is the rate of transition between two particular quantum states, not the total rate between energy levels. Naturally, we also have <math> W_{ab} = W_{ba}</math>.\
 
Note that <math>W_{ab}</math> is the rate of transition between two particular quantum states, not the total rate between energy levels. Naturally, we also have <math> W_{ab} = W_{ba}</math>.\
Line 811: Line 713:
 
An alternative way to express Eq. [[{{SUBPAGENAME}}#EQ_broad2|EQ_broad2]] is to introduce the Rabi frequency,  
 
An alternative way to express Eq. [[{{SUBPAGENAME}}#EQ_broad2|EQ_broad2]] is to introduce the Rabi frequency,  
  
−
{{EqL
+
:<equation id=" broad6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \Omega _ R = \frac{2 H_{ba}}{\hbar } = \frac{e |{\bf \hat{e}}\cdot {\bf r}_{ba} | E}{\hbar } \end{align}</math>
+
<math>\begin{align} \  \Omega _ R = \frac{2 H_{ba}}{\hbar } = \frac{e |{\bf \hat{e}}\cdot {\bf r}_{ba} | E}{\hbar } \end{align}</math>
−
|num=EQ_ broad6
+
</equation>
−
}}
 
  
 
In which case  
 
In which case  
  
−
{{EqL
+
:<equation id=" broad7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{ab} = \frac{\pi }{2} \Omega _ R^2 f (\omega _0 ) = \Omega _ R^2 \frac{1}{\Gamma _0} \end{align}</math>
+
<math>\begin{align} \  W_{ab} = \frac{\pi }{2} \Omega _ R^2 f (\omega _0 ) = \Omega _ R^2 \frac{1}{\Gamma _0} \end{align}</math>
−
|num=EQ_ broad7
+
</equation>
−
}}
 
  
 
If the width of the final state is due soley to spontaneous emission, <math>\Gamma _0 = A = ( 4 e^2 \omega ^3 / 3 \hbar c^3 ) | r_{ba} |^2</math>. Since <math>W_{ab}</math> is proportional to <math> | r_{ba} |^2 /A_0</math>, it is independent of <math> | r_{ba} |^2</math>. It is left as a problem to find the exact relationship, but it can readily be seen that it is of the form  
 
If the width of the final state is due soley to spontaneous emission, <math>\Gamma _0 = A = ( 4 e^2 \omega ^3 / 3 \hbar c^3 ) | r_{ba} |^2</math>. Since <math>W_{ab}</math> is proportional to <math> | r_{ba} |^2 /A_0</math>, it is independent of <math> | r_{ba} |^2</math>. It is left as a problem to find the exact relationship, but it can readily be seen that it is of the form  
  
−
{{EqL
+
:<equation id=" broad8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_{ab} = X\lambda ^2 I_0 /\hbar \omega \end{align}</math>
+
<math>\begin{align} \  W_{ab} = X\lambda ^2 I_0 /\hbar \omega \end{align}</math>
−
|num=EQ_ broad8
+
</equation>
−
}}
 
  
 
where X is a numerical factor. <math>I/ \hbar \omega </math> is the photon flux—i.e. the number of photons per second per unit area in the beam. Since <math>W_{ab}</math> is an excitation rate, we interpret <math>X\lambda ^2</math> as the resonance absorption cross section for the atom, <math>\sigma _0</math>.  
 
where X is a numerical factor. <math>I/ \hbar \omega </math> is the photon flux—i.e. the number of photons per second per unit area in the beam. Since <math>W_{ab}</math> is an excitation rate, we interpret <math>X\lambda ^2</math> as the resonance absorption cross section for the atom, <math>\sigma _0</math>.  
Line 840: Line 739:
 
We now discuss broad band excitation. Using the result of the last section, finding the excitation rate or the absorption cross section for broad band excitation is trivial. From Eq. [[{{SUBPAGENAME}}#EQ_broad2|EQ_broad2]], the absorption rate is proportional to <math>f(\omega _0 )</math>. For monochromatic excitation, <math>f (\omega _0 ) = (2/ \pi ) A^{-1} </math> and <math>W_{\rm mono}= X\lambda ^2 I_0/\hbar \omega </math>. For a spectral source having linewidth <math>\Delta \omega _ s</math>, defined so that the normalized line shape function is <math>f (\omega _0 ) = (2/ \pi ) {\Delta \omega _ s}^{-1} </math>, then the broad band excitation rate is obtained by replacing <math>\Gamma _0</math> with <math>\Delta \omega _ s</math> in Eq. [[{{SUBPAGENAME}}#EQ_broad8|EQ_broad8]]. Thus  
 
We now discuss broad band excitation. Using the result of the last section, finding the excitation rate or the absorption cross section for broad band excitation is trivial. From Eq. [[{{SUBPAGENAME}}#EQ_broad2|EQ_broad2]], the absorption rate is proportional to <math>f(\omega _0 )</math>. For monochromatic excitation, <math>f (\omega _0 ) = (2/ \pi ) A^{-1} </math> and <math>W_{\rm mono}= X\lambda ^2 I_0/\hbar \omega </math>. For a spectral source having linewidth <math>\Delta \omega _ s</math>, defined so that the normalized line shape function is <math>f (\omega _0 ) = (2/ \pi ) {\Delta \omega _ s}^{-1} </math>, then the broad band excitation rate is obtained by replacing <math>\Gamma _0</math> with <math>\Delta \omega _ s</math> in Eq. [[{{SUBPAGENAME}}#EQ_broad8|EQ_broad8]]. Thus  
  
−
{{EqL
+
:<equation id=" band1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  W_ B = {\left( X\lambda ^2 \frac{\Gamma _0}{\Delta \omega _ s}\right)} \frac{I_0}{\hbar \omega } \end{align}</math>
+
<math>\begin{align} \  W_ B = {\left( X\lambda ^2 \frac{\Gamma _0}{\Delta \omega _ s}\right)} \frac{I_0}{\hbar \omega } \end{align}</math>
−
|num=EQ_ band1
+
</equation>
−
}}
 
  
 
Similarly, the effective absorption cross section is  
 
Similarly, the effective absorption cross section is  
  
−
{{EqL
+
:<equation id=" band2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \sigma _{\rm eff} = \sigma _0 \frac{\Gamma _0}{\Delta \omega _ s} \end{align}</math>
+
<math>\begin{align} \  \sigma _{\rm eff} = \sigma _0 \frac{\Gamma _0}{\Delta \omega _ s} \end{align}</math>
−
|num=EQ_ band2
+
</equation>
−
}}
 
  
 
This relation is valid provided <math>\Delta \omega _ s \gg \Gamma _0</math>. If the two widths are comparable, the problem needs to be worked out in detail, though the general behavior would be for <math>\Delta \omega _ s \rightarrow ( \Delta \omega _ s^2 + \Gamma _0^2 )^{1/2}</math>. Note that <math>\Delta \omega _ s</math> represents the actual resonance width. Thus, if Doppler broadening is the major broadening mechanism then  
 
This relation is valid provided <math>\Delta \omega _ s \gg \Gamma _0</math>. If the two widths are comparable, the problem needs to be worked out in detail, though the general behavior would be for <math>\Delta \omega _ s \rightarrow ( \Delta \omega _ s^2 + \Gamma _0^2 )^{1/2}</math>. Note that <math>\Delta \omega _ s</math> represents the actual resonance width. Thus, if Doppler broadening is the major broadening mechanism then  
  
−
{{EqL
+
:<equation id=" band3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \sigma _{\rm eff} = \sigma _0 \Gamma _0 /\Delta \omega _{\rm Doppler} . \end{align}</math>
+
<math>\begin{align} \  \sigma _{\rm eff} = \sigma _0 \Gamma _0 /\Delta \omega _{\rm Doppler} . \end{align}</math>
−
|num=EQ_ band3
+
</equation>
−
}}
 
  
 
Except in the case of high resolution laser spectroscopy, it is generally true that <math>\Delta \omega _ s \gg \Gamma _0</math>, so that <math>\sigma _{\rm eff}\ll \sigma _0</math>.  
 
Except in the case of high resolution laser spectroscopy, it is generally true that <math>\Delta \omega _ s \gg \Gamma _0</math>, so that <math>\sigma _{\rm eff}\ll \sigma _0</math>.  
Line 865: Line 761:
 
When the external light intensity is strong, the population in the excited state is no longer negligible, and the transition is saturated. We define the saturated absorption rate <math>R^s </math> as the net transfer from initial state <math>a </math> to final state <math>b </math>, and <math>R^u </math> is the unsaturated rate for the stimulated absorption and emission,  
 
When the external light intensity is strong, the population in the excited state is no longer negligible, and the transition is saturated. We define the saturated absorption rate <math>R^s </math> as the net transfer from initial state <math>a </math> to final state <math>b </math>, and <math>R^u </math> is the unsaturated rate for the stimulated absorption and emission,  
  
−
{{EqL
+
:<equation id=" sat1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  R^s (n_a+n_b) = R^u (n_a-n_b). \end{align}</math>
+
<math>\begin{align} \  R^s (n_a+n_b) = R^u (n_a-n_b). \end{align}</math>
−
|num=EQ_ sat1
+
</equation>
−
}}
 
  
 
When the system reaches steady state,  
 
When the system reaches steady state,  
  
−
{{EqL
+
:<equation id=" sat2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  \dot{n_b}&=&-n_b(R^u+\Gamma)+n_aR^u =0\\ \dot{n_a}&=&n_b(R^u+\Gamma)-n_aR^u =0  \\  \end{align}</math>
+
<math>\begin{align}  \  \dot{n_b}&=&-n_b(R^u+\Gamma)+n_aR^u =0\\ \dot{n_a}&=&n_b(R^u+\Gamma)-n_aR^u =0  \\  \end{align}</math>
−
|num=EQ_ sat2
+
</equation>
−
}}
 
  
 
which gives
 
which gives
  
−
{{EqL
+
:<equation id=" sat3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  \frac{n_b}{n_a}=\frac{R^u}{R^u+\Gamma}  \end{align}</math>
+
<math>\begin{align}  \  \frac{n_b}{n_a}=\frac{R^u}{R^u+\Gamma}  \end{align}</math>
−
|num=EQ_ sat3
+
</equation>
−
}}
 
  
 
From Eq. [[{{SUBPAGENAME}}#EQ_sat1|EQ_sat1]], we have
 
From Eq. [[{{SUBPAGENAME}}#EQ_sat1|EQ_sat1]], we have
  
−
{{EqL
+
:<equation id=" sat4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  R^s=\frac{\Gamma}{2}\frac{S}{1+S}=\frac{R^u}{1+S}  \end{align}</math>
+
<math>\begin{align}  \  R^s=\frac{\Gamma}{2}\frac{S}{1+S}=\frac{R^u}{1+S}  \end{align}</math>
−
|num=EQ_ sat4
+
</equation>
−
}}
 
  
 
where <math>S </math> is the saturation parameter and is defined as <math>S=2R^u/\Gamma </math>. The transition rate is reduced by a factor of <math>1+S</math> due to saturation.
 
where <math>S </math> is the saturation parameter and is defined as <math>S=2R^u/\Gamma </math>. The transition rate is reduced by a factor of <math>1+S</math> due to saturation.
Line 897: Line 789:
 
For the case of monochromatic radiation, as discussed above, the unsaturated transition rate
 
For the case of monochromatic radiation, as discussed above, the unsaturated transition rate
  
−
{{EqL
+
:<equation id=" sat5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  R^u=W_{ab}=\frac{\pi }{2}\omega_R^2 f(\omega )= \frac{\omega_R^2 }{\Gamma} \frac{1}{1+(2\delta/\Gamma)^2} \end{align}</math>
+
<math>\begin{align}  \  R^u=W_{ab}=\frac{\pi }{2}\omega_R^2 f(\omega )= \frac{\omega_R^2 }{\Gamma} \frac{1}{1+(2\delta/\Gamma)^2} \end{align}</math>
−
|num=EQ_ sat5
+
</equation>
−
}}
 
 
where <math>\delta </math> the detuning with respect to the center frequency <math>\omega_0 </math>.
 
where <math>\delta </math> the detuning with respect to the center frequency <math>\omega_0 </math>.
  
 
Thus in general the saturated transition rate  
 
Thus in general the saturated transition rate  
  
−
{{EqL
+
:<equation id=" sat6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  R^s= \frac{\omega_R^2 }{\Gamma} \frac{1}{1+(2\delta/\Gamma)^2+2\omega_R^2/\Gamma^2} \end{align}</math>
+
<math>\begin{align}  \  R^s= \frac{\omega_R^2 }{\Gamma} \frac{1}{1+(2\delta/\Gamma)^2+2\omega_R^2/\Gamma^2} \end{align}</math>
−
|num=EQ_ sat6
+
</equation>
−
}}
 
 
and the saturation parameter
 
and the saturation parameter
−
{{EqL
+
:<equation id=" sat7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  S= \frac{S_{res}}{1+(2\delta/\Gamma)^2} \end{align}</math>
+
<math>\begin{align}  \  S= \frac{S_{res}}{1+(2\delta/\Gamma)^2} \end{align}</math>
−
|num=EQ_ sat7
+
</equation>
−
}}
 
  
 
with the resonant saturation parameter <math>S_{res}=2\omega_R^2/\Gamma^2</math>.  
 
with the resonant saturation parameter <math>S_{res}=2\omega_R^2/\Gamma^2</math>.  
Line 919: Line 808:
 
The saturated rate <math>R^s </math> has a Lorentzian line with FWHM  
 
The saturated rate <math>R^s </math> has a Lorentzian line with FWHM  
  
−
{{EqL
+
:<equation id=" sat8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  \delta_{FWHM }=\frac{\Gamma}{2}\sqrt{1+S_{res}}  \end{align}</math>
+
<math>\begin{align}  \  \delta_{FWHM }=\frac{\Gamma}{2}\sqrt{1+S_{res}}  \end{align}</math>
−
|num=EQ_ sat8
+
</equation>
−
}}
 
  
 
=== Power Broadening ===
 
=== Power Broadening ===
Line 930: Line 818:
 
The saturation intensity <math>I_{sat} </math> is the light field intensity corresponding to the saturation parameter <math>S_{res}=1 </math> for a resonant light, and that is when <math>R^u=\omega_R^2/\Gamma=\Gamma/2</math>. Since the Rabi frequency <math>\omega_R^2\propto I</math>, we have the linear relation
 
The saturation intensity <math>I_{sat} </math> is the light field intensity corresponding to the saturation parameter <math>S_{res}=1 </math> for a resonant light, and that is when <math>R^u=\omega_R^2/\Gamma=\Gamma/2</math>. Since the Rabi frequency <math>\omega_R^2\propto I</math>, we have the linear relation
  
−
{{EqL
+
:<equation id=" sat9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  \omega_R^2=\frac{\Gamma^2}{2}\frac{I}{I_{sat}}  \end{align}</math>
+
<math>\begin{align}  \  \omega_R^2=\frac{\Gamma^2}{2}\frac{I}{I_{sat}}  \end{align}</math>
−
|num=EQ_ sat9
+
</equation>
−
}}
 
  
 
and that gives
 
and that gives
  
−
{{EqL
+
:<equation id=" sat10" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  I_{sat}=\frac{\Gamma^2}{2}\frac{I}{\omega_R^2}=\frac{\hbar \omega^3}{12\pi c^2}\Gamma  \end{align}</math>
+
<math>\begin{align}  \  I_{sat}=\frac{\Gamma^2}{2}\frac{I}{\omega_R^2}=\frac{\hbar \omega^3}{12\pi c^2}\Gamma  \end{align}</math>
−
|num=EQ_ sat10
+
</equation>
−
}}
 
 
for example, <math>I_{sat}=6\; mW/cm^2</math> for Na D line.
 
for example, <math>I_{sat}=6\; mW/cm^2</math> for Na D line.
  
Line 947: Line 833:
 
A quick derivation for the saturation intensity is to express the light intensity <math>I </math> and the Rabi frequency <math>\omega_R </math> in terms of the number of photons <math>n </math>,
 
A quick derivation for the saturation intensity is to express the light intensity <math>I </math> and the Rabi frequency <math>\omega_R </math> in terms of the number of photons <math>n </math>,
  
−
{{EqL
+
:<equation id=" sat11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \  I=\frac{Energy}{Area\times Time}=\frac{\hbar\omega n}{V/c}=\frac{\hbar\omega nc}{V}  \end{align}</math>
+
<math>\begin{align}  \  I=\frac{Energy}{Area\times Time}=\frac{\hbar\omega n}{V/c}=\frac{\hbar\omega nc}{V}  \end{align}</math>
−
|num=EQ_ sat11
+
</equation>
−
}}
+
:<equation id=" sat12" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
{{EqL
+
<math>\begin{align}  \  \omega_R^2=(n+1)\omega_1^2\simeq n (\vec{d}\cdot \hat{e})^2 \left(\frac{2}{\hbar}\right)^2 \left(\frac{\hbar\omega}{2\epsilon_0 V}\right)=n\Gamma\frac{6\pi c^3}{\omega^2 V}  \end{align}</math>
−
|math=<math>\begin{align}  \  \omega_R^2=(n+1)\omega_1^2\simeq n (\vec{d}\cdot \hat{e})^2 \left(\frac{2}{\hbar}\right)^2 \left(\frac{\hbar\omega}{2\epsilon_0 V}\right)=n\Gamma\frac{6\pi c^3}{\omega^2 V}  \end{align}</math>
+
</equation>
−
|num=EQ_ sat12
 
−
}}
 
  
 
thus
 
thus
−
{{EqL
+
:<equation id=" sat13" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ \frac{I}{\omega^2_R}=\frac{\hbar \omega^3}{6\pi c^2\Gamma }  \end{align}</math>
+
<math>\begin{align}  \ \frac{I}{\omega^2_R}=\frac{\hbar \omega^3}{6\pi c^2\Gamma }  \end{align}</math>
−
|num=EQ_ sat13
+
</equation>
−
}}
 
 
and pluging this into Eq. [[{{SUBPAGENAME}}#EQ_ sat9|EQ_ sat9]] gives the saturation intensity.
 
and pluging this into Eq. [[{{SUBPAGENAME}}#EQ_ sat9|EQ_ sat9]] gives the saturation intensity.
  
 
For the case of broadband radiation, we define the average intensity per frequency interval as <math>\bar{I} </math>, and when the saturation parameter <math>S=1 </math>, <math>\bar{I}=\bar{I}_{sat} </math>
 
For the case of broadband radiation, we define the average intensity per frequency interval as <math>\bar{I} </math>, and when the saturation parameter <math>S=1 </math>, <math>\bar{I}=\bar{I}_{sat} </math>
  
−
{{EqL
+
:<equation id=" sat14" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ W_{ge}=B_{ge}\frac{\bar{I}}{c}=\frac{\Gamma}{2}  \end{align}</math>
+
<math>\begin{align}  \ W_{ge}=B_{ge}\frac{\bar{I}}{c}=\frac{\Gamma}{2}  \end{align}</math>
−
|num=EQ_ sat14
+
</equation>
−
}}
 
 
thus
 
thus
  
−
{{EqL
+
:<equation id=" sat15" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ \bar{I}_{sat}=\frac{c}{2}\frac{A}{B_{ge}}=\frac{\hbar\omega_{eg}^3}{6\pi^2 c^2}  \end{align}</math>
+
<math>\begin{align}  \ \bar{I}_{sat}=\frac{c}{2}\frac{A}{B_{ge}}=\frac{\hbar\omega_{eg}^3}{6\pi^2 c^2}  \end{align}</math>
−
|num=EQ_ sat15
+
</equation>
−
}}
 
 
which is independent of matrix element! For visible light, <math>\bar{I}_{sat}\approx \frac{12 \;W}{cm^2}\frac{1}{cm^{-1}}</math>, where <math>1 \;cm^{-1}\simeq 30 \;GHz </math>.
 
which is independent of matrix element! For visible light, <math>\bar{I}_{sat}\approx \frac{12 \;W}{cm^2}\frac{1}{cm^{-1}}</math>, where <math>1 \;cm^{-1}\simeq 30 \;GHz </math>.
  
Line 983: Line 864:
 
For monochromatic radiation,  
 
For monochromatic radiation,  
  
−
{{EqL
+
:<equation id=" sat16" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ W_{ge}=n_{phot}\sigma c=\frac{I\sigma}{\hbar\omega}  \end{align}</math>
+
<math>\begin{align}  \ W_{ge}=n_{phot}\sigma c=\frac{I\sigma}{\hbar\omega}  \end{align}</math>
−
|num=EQ_ sat16
+
</equation>
−
}}
 
 
in the low intensity limit <math>W_{ge}=R^u </math>. If we extrapolate it to saturation parameter <math>S=1 </math>, then <math>I=I_{sat} </math>, and <math>W_{ge}=R^u=\Gamma/2 </math>
 
in the low intensity limit <math>W_{ge}=R^u </math>. If we extrapolate it to saturation parameter <math>S=1 </math>, then <math>I=I_{sat} </math>, and <math>W_{ge}=R^u=\Gamma/2 </math>
  
−
{{EqL
+
:<equation id=" sat17" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ \frac{\Gamma}{2}=\frac{I_{sat}\sigma}{\hbar \omega}  \end{align}</math>
+
<math>\begin{align}  \ \frac{\Gamma}{2}=\frac{I_{sat}\sigma}{\hbar \omega}  \end{align}</math>
−
|num=EQ_ sat17
+
</equation>
−
}}
 
  
 
and from Eq. [[{{SUBPAGENAME}}#EQ_ sat10|EQ_ sat10]], we have  
 
and from Eq. [[{{SUBPAGENAME}}#EQ_ sat10|EQ_ sat10]], we have  
  
−
{{EqL
+
:<equation id=" sat18" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ \sigma=6\pi\frac{c^2}{\omega^2}=6\pi (\lambda/2\pi)^2  \end{align}</math>
+
<math>\begin{align}  \ \sigma=6\pi\frac{c^2}{\omega^2}=6\pi (\lambda/2\pi)^2  \end{align}</math>
−
|num=EQ_ sat18
+
</equation>
−
}}
 
 
This is the resonant cross section for weak radiation, and it is usually much larger than the size of the atom, and independent of matrix element. If we plot the cross section as a function of detuning, it is a Lorentzian line. Strong transitions have a larger widths, but the cross section on resonance is always the same.
 
This is the resonant cross section for weak radiation, and it is usually much larger than the size of the atom, and independent of matrix element. If we plot the cross section as a function of detuning, it is a Lorentzian line. Strong transitions have a larger widths, but the cross section on resonance is always the same.
  
Line 1,006: Line 884:
 
For broadband radiation,  
 
For broadband radiation,  
  
−
{{EqL
+
:<equation id=" sat19" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ W_{ge}&=&\int \sigma(\omega)\frac{\bar{I}(\omega)}{\hbar\omega}d\omega  \\ &=& \frac{6\pi (\lambda/2\pi)^2}{\hbar\omega_{eg}}\bar{I}(\omega_{eg})\frac{\pi\Gamma}{2} \end{align}</math>
+
<math>\begin{align}  \ W_{ge}&=&\int \sigma(\omega)\frac{\bar{I}(\omega)}{\hbar\omega}d\omega  \\ &=& \frac{6\pi (\lambda/2\pi)^2}{\hbar\omega_{eg}}\bar{I}(\omega_{eg})\frac{\pi\Gamma}{2} \end{align}</math>
−
|num=EQ_ sat19
+
</equation>
−
}}
 
  
 
at saturation <math>S=1 </math>,
 
at saturation <math>S=1 </math>,
  
−
{{EqL
+
:<equation id=" sat20" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ \frac{\Gamma}{2}=\frac{6\pi (\lambda/2\pi)^2}{\hbar\omega_{eg}}\bar{I}_{sat}\frac{\pi\Gamma}{2}  \end{align}</math>
+
<math>\begin{align}  \ \frac{\Gamma}{2}=\frac{6\pi (\lambda/2\pi)^2}{\hbar\omega_{eg}}\bar{I}_{sat}\frac{\pi\Gamma}{2}  \end{align}</math>
−
|num=EQ_ sat20
+
</equation>
−
}}
 
 
thus
 
thus
−
{{EqL
+
:<equation id=" sat21" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align}  \ \bar{I}_{sat}=\frac{\hbar\omega_{eg}}{6\pi^2 (\lambda/2\pi)^2}=\frac{\hbar \omega_{eg}^3}{6\pi^2 c^2}  \end{align}</math>
+
<math>\begin{align}  \ \bar{I}_{sat}=\frac{\hbar\omega_{eg}}{6\pi^2 (\lambda/2\pi)^2}=\frac{\hbar \omega_{eg}^3}{6\pi^2 c^2}  \end{align}</math>
−
|num=EQ_ sat21
+
</equation>
−
}}
 
 
which is the same as we have derived in Eq. [[{{SUBPAGENAME}}#EQ_ sat15|EQ_ sat15]].
 
which is the same as we have derived in Eq. [[{{SUBPAGENAME}}#EQ_ sat15|EQ_ sat15]].
  
Line 1,051: Line 926:
 
k r \approx \frac{\hbar\omega}{\hbar c}a_0\approx\frac{e^2/a_0}{\hbar c}a_0\approx\frac{e^2}{\hbar c}=\alpha,
 
k r \approx \frac{\hbar\omega}{\hbar c}a_0\approx\frac{e^2/a_0}{\hbar c}a_0\approx\frac{e^2}{\hbar c}=\alpha,
 
</math>
 
</math>
−
the expansion in <xr id="eq:hor3"/> is effectively an expansion in <math>\alpha</math>.
+
the expansion in eq:hor3/> is effectively an expansion in <math>\alpha</math>.
 
We can rewrite the second term as follows:
 
We can rewrite the second term as follows:
 
:<math>
 
:<math>
 
   p_z x = (p_z x - zp_x )/2 + (p_z x + zp_x )/2 .
 
   p_z x = (p_z x - zp_x )/2 + (p_z x + zp_x )/2 .
 
</math>
 
</math>
−
The first term of Eq. <xr id="eq:hor4"/> is <math>- \hbar L_y/2</math>, and the matrix element becomes
+
The first term of Eq. eq:hor4/> is <math>- \hbar L_y/2</math>, and the matrix element becomes
 
:<math>
 
:<math>
 
   -\frac{ieAk}{2 m} \langle b | \hbar L_y |
 
   -\frac{ieAk}{2 m} \langle b | \hbar L_y |
Line 1,064: Line 939:
 
where <math>\mu_B = e\hbar /2 m</math> is the Bohr magneton.
 
where <math>\mu_B = e\hbar /2 m</math> is the Bohr magneton.
 
The magnetic field is <math>B = - i k A \hat{y}</math>.
 
The magnetic field is <math>B = - i k A \hat{y}</math>.
−
Consequently, Eq.\ <xr id="eq:hor5"/> can be written in the more
+
Consequently, Eq.\ eq:hor5/> can be written in the more
 
familiar form <math>-\vec{\mu} \cdot B</math>. (The orbital magnetic moment is <math>\vec{\mu}
 
familiar form <math>-\vec{\mu} \cdot B</math>. (The orbital magnetic moment is <math>\vec{\mu}
 
   = -\mu_B L</math>: the minus sign arises from our convention that <math>e</math> is
 
   = -\mu_B L</math>: the minus sign arises from our convention that <math>e</math> is
Line 1,072: Line 947:
 
   H_{\rm int}(M1) = B \cdot \mu_B\langle b |L + g_sS| a\rangle,
 
   H_{\rm int}(M1) = B \cdot \mu_B\langle b |L + g_sS| a\rangle,
 
</math>
 
</math>
−
where we have added the spin dependent term from Eq. <xr id="eq:hor_Hint"/>.  <math>M1</math> indicates that the matrix element is for a magnetic dipole transition.  The strength of the <math>M1</math> transition is set by
+
where we have added the spin dependent term from Eq. eq:hor_Hint/>.  <math>M1</math> indicates that the matrix element is for a magnetic dipole transition.  The strength of the <math>M1</math> transition is set by
 
:<math>
 
:<math>
 
\mu_B/c = \frac{1}{2}\frac{e\hbar}{mc}=\frac{1}{2}\frac{e^2}{\hbar c}\frac{\hbar^2}{e m} = \frac{1}{2}\alpha e a_0,
 
\mu_B/c = \frac{1}{2}\frac{e\hbar}{mc}=\frac{1}{2}\frac{e^2}{\hbar c}\frac{\hbar^2}{e m} = \frac{1}{2}\alpha e a_0,
Line 1,078: Line 953:
 
so it is indeed a factor of <math>\alpha</math> weaker than a dipole transition, as we argued above.
 
so it is indeed a factor of <math>\alpha</math> weaker than a dipole transition, as we argued above.
  
−
The second term in Eq.\ <xr id="eq:hor4"/> involves <math>( p_z x + z p_x
+
The second term in Eq.\ eq:hor4/> involves <math>( p_z x + z p_x
 
   )/2</math>.
 
   )/2</math>.
 
Making use of the commutator relation <math>[ r, H_0 ] = i\hbar
 
Making use of the commutator relation <math>[ r, H_0 ] = i\hbar
Line 1,104: Line 979:
 
more general expression involving the matrix element <math>\langle b |r:r|a\rangle</math> of a tensor product.  It is straightforward to verify that the electric quadrupole interaction is also of order <math>\alpha</math>.
 
more general expression involving the matrix element <math>\langle b |r:r|a\rangle</math> of a tensor product.  It is straightforward to verify that the electric quadrupole interaction is also of order <math>\alpha</math>.
  
−
The total matrix element of the second term in the expansion of Eq.\ <xr id="eq:hor3"/> can be written
+
The total matrix element of the second term in the expansion of Eq.\ eq:hor3/> can be written
 
:<math>
 
:<math>
 
   H_{\rm int}^{(2)} = H_{\rm int} (M1) + H_{\rm int} (E2).
 
   H_{\rm int}^{(2)} = H_{\rm int} (M1) + H_{\rm int} (E2).
Line 1,178: Line 1,053:
 
The dipole matrix element for a particular polarization of the field, <math>\hat{\bf {e}}</math>, is  
 
The dipole matrix element for a particular polarization of the field, <math>\hat{\bf {e}}</math>, is  
  
−
{{EqL
+
:<equation id=" select1" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  {\bf \hat{e}} \cdot {\bf r}_{ba} = {\bf \hat{e}} \cdot \langle b, J_ b, m_ b | {\bf r} | a, J_ a , m_ a \rangle . \end{align}</math>
+
<math>\begin{align} \  {\bf \hat{e}} \cdot {\bf r}_{ba} = {\bf \hat{e}} \cdot \langle b, J_ b, m_ b | {\bf r} | a, J_ a , m_ a \rangle . \end{align}</math>
−
|num=EQ_ select1
+
</equation>
−
}}
 
  
 
It is straightforward to calculate <math>x_{ba}, y_{ba}, z_{ba},</math> but a more general approach is to write '''r''' in terms of a spherical tensor. This yields the selection rules directly, and allows the matrix element to be calculated for various geometries using the Wigner-Eckart theorem as discussed above.
 
It is straightforward to calculate <math>x_{ba}, y_{ba}, z_{ba},</math> but a more general approach is to write '''r''' in terms of a spherical tensor. This yields the selection rules directly, and allows the matrix element to be calculated for various geometries using the Wigner-Eckart theorem as discussed above.
Line 1,189: Line 1,063:
 
The spherical harmonics of rank 1 are  
 
The spherical harmonics of rank 1 are  
  
−
{{EqL
+
:<equation id=" select2" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  Y_{1,0} = \sqrt {\frac{3}{4\pi }} \cos \theta ; \qquad Y_{1, +1} = - \sqrt {\frac{3}{8\pi }} \sin \theta e^{+i\phi }\qquad Y_{1,-1} = \sqrt {\frac{3}{8\pi }} \sin \theta e^{-i\phi } \end{align}</math>
+
<math>\begin{align} \  Y_{1,0} = \sqrt {\frac{3}{4\pi }} \cos \theta ; \qquad Y_{1, +1} = - \sqrt {\frac{3}{8\pi }} \sin \theta e^{+i\phi }\qquad Y_{1,-1} = \sqrt {\frac{3}{8\pi }} \sin \theta e^{-i\phi } \end{align}</math>
−
|num=EQ_ select2
+
</equation>
−
}}
 
  
 
These are normalized so that  
 
These are normalized so that  
  
−
{{EqL
+
:<equation id=" select3" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \int Y_{1,m^\prime }^* Y_{1,m} \sin \theta d\theta d\phi = \delta _{m^\prime , m} \end{align}</math>
+
<math>\begin{align} \  \int Y_{1,m^\prime }^* Y_{1,m} \sin \theta d\theta d\phi = \delta _{m^\prime , m} \end{align}</math>
−
|num=EQ_ select3
+
</equation>
−
}}
 
  
 
We can write the vector '''r''' in terms of components <math>r_ m ,\  m = +1, 0, -1</math>,  
 
We can write the vector '''r''' in terms of components <math>r_ m ,\  m = +1, 0, -1</math>,  
  
−
{{EqL
+
:<equation id=" select4" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  r_0 = r\sqrt {\frac{4\pi }{3}} Y_{1,0} ,\qquad r_{\pm } = r\sqrt {\frac{4\pi }{3}} Y_{1,\pm 1} , \end{align}</math>
+
<math>\begin{align} \  r_0 = r\sqrt {\frac{4\pi }{3}} Y_{1,0} ,\qquad r_{\pm } = r\sqrt {\frac{4\pi }{3}} Y_{1,\pm 1} , \end{align}</math>
−
|num=EQ_ select4
+
</equation>
−
}}
 
  
 
or, more generally  
 
or, more generally  
  
−
{{EqL
+
:<equation id=" select5" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  r_ M = rT_{1,M} (\theta , \phi ) \end{align}</math>
+
<math>\begin{align} \  r_ M = rT_{1,M} (\theta , \phi ) \end{align}</math>
−
|num=EQ_ select5
+
</equation>
−
}}
 
  
 
Consequently,  
 
Consequently,  
  
−
{{EqL
+
:<equation id=" select6" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  \langle b, J_ b, m_ b | r_ M | a, J_ a, m_ a \rangle = \langle b, J_ b, m_ b | rT_{1,M} | a, J_ a, m_ a \rangle \end{align}</math>
+
<math>\begin{align} \  \langle b, J_ b, m_ b | r_ M | a, J_ a, m_ a \rangle = \langle b, J_ b, m_ b | rT_{1,M} | a, J_ a, m_ a \rangle \end{align}</math>
−
|num=EQ_ select6
+
</equation>
−
}}
 
  
−
{{EqL
+
:<equation id=" select7" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  = \langle b, J_ b | r | a, J_ a \rangle \langle J_ b, m_ b |  T_{1,M} | J_ a, m_ a \rangle \end{align}</math>
+
<math>\begin{align} \  = \langle b, J_ b | r | a, J_ a \rangle \langle J_ b, m_ b |  T_{1,M} | J_ a, m_ a \rangle \end{align}</math>
−
|num=EQ_ select7
+
</equation>
−
}}
 
  
 
The first factor is independent of <math>m</math>. It is  
 
The first factor is independent of <math>m</math>. It is  
  
−
{{EqL
+
:<equation id=" select8" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  r_{ba} = \int _0^{\infty } R_{b,J_ b}^* (r) r R_{a,J_ a} (r) r^2 dr \end{align}</math>
+
<math>\begin{align} \  r_{ba} = \int _0^{\infty } R_{b,J_ b}^* (r) r R_{a,J_ a} (r) r^2 dr \end{align}</math>
−
|num=EQ_ select8
+
</equation>
−
}}
 
  
 
where <math>r_{ba}</math> contains the radial part of the matrix element. It vanishes unless <math>| b \rangle </math> and <math>| a \rangle </math> have opposite parity. The second factor in Eq. [[{{SUBPAGENAME}}#EQ_select7|EQ_select7]] yields the selection rule  
 
where <math>r_{ba}</math> contains the radial part of the matrix element. It vanishes unless <math>| b \rangle </math> and <math>| a \rangle </math> have opposite parity. The second factor in Eq. [[{{SUBPAGENAME}}#EQ_select7|EQ_select7]] yields the selection rule  
  
−
{{EqL
+
:<equation id=" select9" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  | J_ b - J_ a | = 0, 1; ~ ~ ~ m_ b = m_ a \pm M = m_ a, m_ a \pm 1 \end{align}</math>
+
<math>\begin{align} \  | J_ b - J_ a | = 0, 1; ~ ~ ~ m_ b = m_ a \pm M = m_ a, m_ a \pm 1 \end{align}</math>
−
|num=EQ_ select9
+
</equation>
−
}}
 
  
 
Similarly, for magnetic dipole transition, Eq. [[{{SUBPAGENAME}}#EQ_hor6|EQ_hor6]], we have  
 
Similarly, for magnetic dipole transition, Eq. [[{{SUBPAGENAME}}#EQ_hor6|EQ_hor6]], we have  
  
−
{{EqL
+
:<equation id=" select10" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  H_{ba} (M1) = \mu _ B B \langle b, J_ b, m_ b , | T_{LM} (L) | a, J_ a , m_ a \rangle \end{align}</math>
+
<math>\begin{align} \  H_{ba} (M1) = \mu _ B B \langle b, J_ b, m_ b , | T_{LM} (L) | a, J_ a , m_ a \rangle \end{align}</math>
−
|num=EQ_ select10
+
</equation>
−
}}
 
  
 
It immediately follows that parity is unchanged, and that  
 
It immediately follows that parity is unchanged, and that  
  
−
{{EqL
+
:<equation id=" select11" noautocaption><span style="float:right; display:block;"><caption>(%i)</caption></span>
−
|math=<math>\begin{align} \  | \Delta J | = 0,1 ~ ~ ~ (J=0\rightarrow J= 0~ \mbox{forbidden}); ~ ~ | \Delta m | = 0,1 \end{align}</math>
+
<math>\begin{align} \  | \Delta J | = 0,1 ~ ~ ~ (J=0\rightarrow J= 0~ \mbox{forbidden}); ~ ~ | \Delta m | = 0,1 \end{align}</math>
−
|num=EQ_ select11
+
</equation>
−
}}
 
  
 
This discussion of matrix elements, selection rules, and radiative processes barely skims the subject. For an authoritative treatment, the books by Shore and Manzel, and Sobelman are recommended.
 
This discussion of matrix elements, selection rules, and radiative processes barely skims the subject. For an authoritative treatment, the books by Shore and Manzel, and Sobelman are recommended.

Revision as of 00:18, 26 October 2015

This section introduces the interaction of atoms with radiative modes of the electromagnetic field.

▼ Category:8.421
► Category:Atoms in electric fields
  Atoms
  Atoms in electric fields
  Atoms in Magnetic Fields
  Coherence
  Effects of the Nucleus on Atomic Structure
  Fine Structure and Lamb Shift
  Higher-Order Radiation Processes
  Interaction of an atom with an electromagnetic field
  Old Coherence
  Resonances
  Two-Photon Excitation

Introduction: Spontaneous and Stimulated Emission

Einstein's 1917 paper on the theory of radiation [EIN17a] provided seminal concepts for the quantum theory of radiation. It also anticipated devices such as the laser, and pointed the way to the field of laser-cooling of atoms. In it, he set out to answer two questions:

1) How do the internal states of an atom that radiates and absorbs energy come into equilibrium with a thermal radiation field? (In answering this question Einstein invented the concept of spontaneous emission)

2) How do the translational states of an atom in thermal equilibrium (i.e. states obeying the Maxwell-Boltzmann Law for the distribution of velocities) come into thermal equilibrium with a radiation field? (In answering this question, Einstein introduced the concept of photon recoil. He also demonstrated that the field itself must obey the Planck radiation law.)

The first part of Einstein's paper, which addresses question 1), is well known, but the second part, which addresses question 2), is every bit as germane to contemporary atom/optical physics. Because the paper preceded the creation of quantum mechanics there was no way for him to calculate transition rates. However, his arguments are based on general statistical principles and provide the foundation for interpreting the quantum mechanical results.

Einstein considered a system of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} atoms in thermal equilibrium with a radiation field. The system has two levels (an energy level consists of all of the states that have a given energy; the number of quantum states in a given level is its multiplicity.) with energies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_ b} and , with , and . The numbers of atoms in the two levels are related by . Einstein assumed the Planck radiation law for the spectral energy density temperature. For radiation in thermal equilibrium at temperature , the energy per unit volume in wavelength range is:

<equation id="erad1" noautocaption>(%i)

</equation>

The mean occupation number of a harmonic oscillator at temperature , which can be interpreted as the mean number of photons in one mode of the radiation field, is

<equation id="erad2" noautocaption>(%i)

</equation>

According to the Boltzmann Law of statistical mechanics, in thermal equilibrium the populations of the two levels are related by

<equation id="erad3" noautocaption>(%i)

</equation>

Here and are the multiplicities of the two levels. The last step assumes the Bohr frequency condition, . However, Einstein's paper actually derives this relation independently.

According to classical theory, an oscillator can exchange energy with the radiation field at a rate that is proportional to the spectral density of radiation. The rates for absorption and emission are equal. The population transfer rate equation is thus predicted to be

<equation id="erad4" noautocaption>(%i)

</equation>

This equation is incompatible with Eq. erad3. (This can be seen by setting in Eq. erad5 which then leads to .) To overcome this problem, Einstein postulated that atoms in state b must spontaneously radiate to state a, with a constant radiation rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ba}} . Today such a process seems quite natural: the language of quantum mechanics is the language of probabilities and there is nothing jarring about asserting that the probability of radiating in a short time interval is proportional to the length of the interval. At that time such a random fundamental process could not be justified on physical principles. Einstein, in his characteristic Olympian style, brushed aside such concerns and merely asserted that the process is analagous to radioactive decay. With this addition, Eq. erad4 becomes

<equation id="erad5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \dot{N}_ b = - {\left[ \rho _ E (\omega ) B_{ba} + A_{ba} \right]} N_ b + \rho _ E (\omega ) B_{ab} N_ a = -\dot{N}_ a . \end{align}} </equation>

By combining Eqs. eq:plancklaw, eq:frac, eq:rad2  it follows that

<equation id=" erl5" noautocaption>(%i)

</equation>

Consequently, the rate of transition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\rightarrow a} is

<equation id=" erl6" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ B_{ba} \rho _ E (\omega ) + A_{ba} = (\bar{n} +1 )A_{ba}, \end{align}} </equation>

while the rate of absorption is

<equation id=" erl7" noautocaption>(%i)

</equation>

If we consider emission and absorption between single states by taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_ b = g_ a = 1} , then the ratio of rate of emission to rate of absorption is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\bar{n} + 1) /\bar{n}} .

This argument reveals the fundamental role of spontaneous emission. Without it, atomic systems could not achieve thermal equilibrium with a radiation field. Thermal equilibrium requires some form of dissipation, and dissipation is equivalent to having an irreversible process. Spontaneous emission is the fundamental irreversible process in nature. The reason that it is irreversible is that once a photon is radiated into the vacuum, the probability that it will ever be reabsorbed is zero: there are an infinity of vacuum modes available for emission but only one mode for absorption. If the vacuum modes are limited, for instance by cavity effects, the number of modes becomes finite and equilibrium is never truly achieved. In the limit of only a single mode, the motion becomes reversible.

The identification of the Einstein coefficient with the rate of spontaneous emission is so well established that we shall henceforth use the symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ba}} to denote the spontaneous decay rate from state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} to . The radiative lifetime for such a transition is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau _{ba} = A_{ba}^{-1}} .

Here, Einstein came to a halt. Lacking quantum theory, there was no way to calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ba}} .


Quantum Theory of Absorption and Emission

We shall start by describing the behavior of an atom in a classical electromagnetic field. Although treating the field classically while treating the atom quantum mechanically is fundamentally inconsistent, it provides a natural and intuitive approach to the problem. Furthermore, it is completely justified in cases where the radiation fields are large, in the sense that there are many photons in each mode, as for instance, in the case of microwave or laser spectroscopy. There is, however, one important process that this approach cannot deal with satisfactorily. This is spontaneous emission, which we shall treat later using a quantized field. Nevertheless, phenomenological properties such as selection rules, radiation rates and cross sections, can be developed naturally with this approach.


The classical E-M field

Our starting point is Maxwell's equations (S.I. units):

<equation id="Maxwell" noautocaption>(%i)

</equation>

The charge density Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho } and current density J obey the continuity equation

<equation id=" wd2" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \nabla \cdot {\bf J} + \frac{\partial \rho }{\partial t} = 0 \end{align}} </equation>

Introducing the vector potential A and the scalar potential , we have

<equation id=" wd3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ {\bf E} & = & - \nabla \psi - \frac{\partial {\bf A}}{\partial t} \\ {\bf B} & = & \nabla \times {\bf A} \end{align}} </equation>

We are free to change the potentials by a gauge transformation:

<equation id=" wd4" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ {\bf A}^\prime = {\bf A} + \nabla \Lambda , ~ ~ ~ ~ ~ \psi ^\prime = \psi - \frac{\partial \Lambda }{\partial t} \end{align}} </equation>

where is a scalar function. This transformation leaves the fields invariant, but changes the form of the dynamical equation. We shall work in the Coulomb gauge (often called the radiation gauge), defined by

<equation id=" wd5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \nabla \cdot {\bf A} = 0 \end{align}} </equation>

In free space, A obeys the wave equation

<equation id=" wd6" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \nabla ^2 {\bf A} = \frac{1}{c^2} \frac{\partial ^2 {\bf A}}{\partial t^2} \end{align}} </equation>

Because , A is transverse. We take a propagating plane wave solution of the form

<equation id="A-field" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ {\bf A}(r, t) = A{\bf \hat{e}} \cos ({\bf k}\cdot {\bf r} -\omega t) = A{\bf \hat{e}} \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} + e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right], \end{align}} </equation>

where and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf \hat{e}}\cdot {\bf k}= 0} . For a linearly polarized field, the polarization vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf \hat{e}}} is real. For an elliptically polarized field it is complex, and for a circularly polarized field it is given by , where the + and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -} signs correspond to positive and negative helicity, respectively. (Alternatively, they correspond to left and right hand circular polarization, respectively, the sign convention being a tradition from optics.) The electric and magnetic fields are then given by

<equation id="E-field" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ {\bf E}(r, t) = \omega A{\bf \hat{e}} \sin ({\bf k}\cdot {\bf r} -\omega t) = - i \omega A {\bf \hat{e}} \frac{1}{2} \left[ e^{i({\bf k}\cdot {\bf r} -\omega t)} - e^{-i({\bf k}\cdot {\bf r} -\omega t)} \right]. \end{align}} </equation>

<equation id="B-field" noautocaption>(%i)

</equation>

The time average Poynting vector is

<equation id=" wd9" noautocaption>(%i)

</equation>

The average energy density in the wave is given by

<equation id="energy-density" noautocaption>(%i)

</equation>


Interaction of an electromagnetic wave and an atom

The behavior of charged particles in an electromagnetic field is correctly described by Hamilton's equations provided that the canonical momentum is redefined:

<equation id=" int1" noautocaption>(%i)

</equation>

The kinetic energy is . Taking , the Hamiltonian for an atom in an electromagnetic field in free space is

<equation id=" int2" noautocaption>(%i)

</equation>

where describes the potential energy due to internal interactions. We are neglecting spin interactions.

Expanding and rearranging, we have

<equation id=" int3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ H & =& \sum _{j=1}^{N} \frac{{\bf p}_ j^2}{2m} + V ({\bf r}_ j ) + \frac{e}{2m} \sum _{j=1}^{N} {\left({\bf p}_ j \cdot {\bf A} ( {\bf r}_ j) + {\bf A} ({\bf r}_ j ) \cdot {\bf p}_ j \right)} + \frac{e^2}{2m} \sum _{j=1}^{N} A_ j^2 ({\bf r} ) \\ & = & H_0 + H_{\rm int} + H^{(2)} . \end{align}} </equation>

Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf p}_ j = - i\hbar \nabla _ j } . Consequently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} describes the unperturbed atom. describes the atom's interaction with the field. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^{(2)}} , which is second order in A, plays a role only at very high intensities. (In a static magnetic field, however, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^{(2)}} gives rise to diamagnetism.)

Because we are working in the Coulomb gauge, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot {\bf A} =0} so that A and p commute. We have

<equation id=" int4" noautocaption>(%i)

</equation>

It is convenient to write the matrix element between states Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | a \rangle } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | b \rangle } in the form

<equation id=" int5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \langle b | H_{\rm int} | a \rangle = \frac{1}{2} H_{ba} e^{-i\omega t} + \frac{1}{2} H_{ba} e^{+i\omega t}, \end{align}} </equation>

where

<equation id=" int6" noautocaption>(%i)

</equation>

Atomic dimensions are small compared to the wavelength of radiation involved in optical transitions. The scale of the ratio is set by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \approx 1/137} . Consequently, when the matrix element in Eq. EQ_int6 is evaluated, the wave function vanishes except in the region where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf k}\cdot {\bf r} = 2 \pi r /\lambda \ll 1} . It is therefore appropriate to expand the exponential:

<equation id=" int7" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ H_{ba} = \frac{eA}{m} {\bf \hat{e}} \cdot \langle b | {\bf p} (1 + i{\bf k} \cdot {\bf r} - 1/2 ({\bf k}\cdot {\bf r} )^2 + \cdots ) | a \rangle \end{align}} </equation>

Unless vanishes, for instance due to parity considerations, the leading term dominates and we can neglect the others. For reasons that will become clear, this is called the dipole approximation. This is by far the most important situation, and we shall defer consideration of the higher order terms. In the dipole approximation we have

<equation id=" int8" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ H_{ba} = \frac{eA}{m} {\bf \hat{e}} \cdot \langle b | {\bf p} | a \rangle = \frac{-ieE}{m\omega } {\bf \hat{e}} \cdot \langle b | {\bf p} | a \rangle \end{align}} </equation>

where we have used, from Eq. eq:E-field, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = -iE/\omega } . It can be shown (i.e. left as exercise) that the matrix element of p can be transfomred into a matrix element for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf r}} :

<equation id=" int9" noautocaption>(%i)

</equation>

This results in

<equation id=" int10" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ H_{ba} = \frac{e E \omega _{ba}}{\omega } {\bf \hat{e}} \cdot \langle b | {\bf r} | a \rangle \end{align}} </equation>

We will be interested in resonance phenomena in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \approx \omega _{ba}} . Consequently,

<equation id=" int11" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ H_{ba} = + e {\bf E}_0 \cdot \langle b | {\bf r} | a \rangle = - {\bf d}_{ba} \cdot {\bf E} \end{align}} </equation>

where d is the dipole operator, . Displaying the time dependence explictlty, we have

<equation id=" int12" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ H_{ba}^\prime = - {\bf d}_{ba}\cdot {\bf E}_0 e^{-i\omega t}. \end{align}} </equation>

However, it is important to bear in mind that this is only the first term in a series, and that if it vanishes the higher order terms will contribute a perturbation at the driving frequency.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{ba}} appears as a matrix element of the momentum operator p in Eq. EQ_int8, and of the dipole operator r in Eq. EQ_int11. These matrix elements look different and depend on different parts of the wave function. The momentum operator emphasizes the curvature of the wave function, which is largest at small distances, whereas the dipole operator evaluates the moment of the charge distribution, i.e. the long range behavior. In practice, the accuracy of a calculation can depend significantly on which operator is used.


Quantization of the radiation field

We shall consider a single mode of the radiation field. This means a single value of the wave vector k, and one of the two orthogonal transverse polarization vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf \hat{e}}} . The radiation field is described by a plane wave vector potential of the form Eq. eq:A-field. We assume that k obeys a periodic boundary or condition, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_ x L_ x = 2\pi n_ x} , etc. (For any k, we can choose boundaries to satisfy this.) The time averaged energy density is given by Eq. eq:energy-density, and the total energy in the volume V defined by these boundaries is

<equation id="energy-total" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ U = \frac{\epsilon _0 }{2}\omega ^2 A^2 V, \end{align}} </equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2} is the mean squared value of averaged over the spatial mode. We now make a formal connection between the radiation field and a harmonic oscillator. We define variables Q and P by

<equation id=" qrd5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ A = \frac{1}{\omega } \sqrt {\frac{1}{\epsilon _ o V}} (\omega Q + iP ), ~ ~ A^* =\frac{1}{\omega }\sqrt {\frac{1}{\epsilon _ o V}} (\omega Q - iP ). \end{align}} </equation>

Then, from Eq. eq:energy-total, we find

<equation id=" qrd6" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ U = \frac{1}{2} (\omega ^2 Q^2 + P^2 ). \end{align}} </equation>

This describes the energy of a harmonic oscillator having unit mass. We quantize the oscillator in the usual fashion by treating Q and P as operators, with

<equation id=" qrd7" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ P = - i\hbar \frac{\partial }{\partial Q}, ~ ~ ~ [Q,P] = i\hbar . \end{align}} </equation>

We introduce the operators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and defined by

<equation id=" qrd8" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ a = \frac{1}{\sqrt {2\hbar \omega }} (\omega Q + iP ) \end{align}} </equation>

<equation id=" qrd9" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ a^\dagger = \frac{1}{\sqrt {2\hbar \omega }} (\omega Q - iP ) \end{align}} </equation>

The fundamental commutation rule is

<equation id=" qrd10" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ [a, a^\dagger ] = 1 \end{align}} </equation>

from which the following can be deduced:

<equation id=" qrd11" noautocaption>(%i)

</equation>

where the number operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = a^\dagger a } obeys

<equation id=" qrd12" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ N| n \rangle = n| n \rangle \end{align}} </equation>

We also have

<equation id=" qrd13" noautocaption>(%i)

</equation>

The operators and are called the annihilation and creation operators, respectively. We can express the vector potential and electric field in terms of and as follows

<equation id=" part1" noautocaption>(%i)

</equation>

<equation id=" part2" noautocaption>(%i)

</equation>

<equation id=" part3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ {\bf E} = - i \sqrt {\frac{ \hbar \omega }{2 \epsilon _ o V} } {\left[ a{\bf \hat{e}} e^{i({\bf k}\cdot {\bf r} - \omega t)} - a^\dagger {\bf \hat{e}}^* e^{-i({\bf k}\cdot {\bf r} -\omega t)}\right]} \end{align}} </equation>

In the dipole limit we can take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i {\bf k}\cdot {\bf r}} = 1} . Then

<equation id=" part3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ {\bf E} = - i \sqrt {\frac{ \hbar \omega }{2 \epsilon _ o V} } \left[ a {\bf \hat e} e^{-i \omega t}- a^\dagger {\bf {\hat e}}^* e^{i \omega t}\right] \end{align}} </equation>

The interaction Hamiltonian is,

<equation id=" qrd16" noautocaption>(%i)

</equation>

where we have written the dipole operator as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf d} = - e {\bf r}} .


Interaction of a two-level system and a single mode of the radiation field

We consider a two-state atomic system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | a \rangle } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | b \rangle } and a radiation field described by The states of the total system can be taken to be

<equation id=" vac1" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ | I \rangle = | a,\ n \rangle = | a \rangle \ | n \rangle , ~ ~ ~ | F \rangle = | b,\ n^\prime \rangle = |b \rangle \ |n^\prime \rangle . \end{align}} </equation>

We shall take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf \hat{e}} = {\bf \hat{ z}} } . Then

<equation id=" vac2" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \langle F |H_{\rm int} | I \rangle = -i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} \langle n^\prime | a e^{-i\omega t} - a^\dagger e^{i\omega t} | n \rangle e^{-i\omega _{ab} t} \end{align}} </equation>

The first term in the bracket obeys the selection rule Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^\prime = n - 1} . This corresponds to loss of one photon from the field and absorption of one photon by the atom. The second term obeys . This corresponds to emission of a photon by the atom. Using Eq. EQ_qrd13, we have

<equation id=" vac3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \langle F | H_{\rm int} | I \rangle = -i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} {\left( \sqrt {n}\, \delta _{n\prime ,n-1} \ e^{-i \omega t} - \sqrt {n+1}\, \delta _{n\prime ,n+1} e^{+i\omega t} \right)} \ e^{-i\omega _{ab} t} \end{align}} </equation>

Transitions occur when the total time dependence is zero, or near zero. Thus absorption occurs when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega =- \omega _{ab}} , or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_ a + \hbar \omega = E_ b} . As we expect, energy is conserved. Similarly, emission occurs when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = + \omega _{ab}} , or .

A particularly interesting case occurs when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 0} , i.e. the field is initially in the vacuum state, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \omega _{ab}} . Then

<equation id=" vac4" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \langle F | H_{\rm int} | I \rangle = i e z_{ab} \sqrt {\frac{2\pi \hbar \omega }{V}} \equiv H_{FI}^0 \end{align}} </equation>

The situation describes a constant perturbation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{FI}^0} coupling the two states and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = | b, n^\prime = 1 \rangle } . The states are degenerate because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_ a = E_ b + \hbar \omega } . Consequently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_ a} is the upper of the two atomic energy levels.

The system is composed of two degenerate eigenstates, but due to the coupling of the field, the degeneracy is split. The eigenstates are symmetric and antisymmetric combinations of the initial states, and we can label them as

<equation id=" vac5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ | \pm \rangle = \frac{1}{\sqrt {2}} (|I \rangle \pm | F \rangle ) = \frac{1}{\sqrt {2}} ( | a , 0 \rangle \pm | b, 1 \rangle ). \end{align}} </equation>

The energies of these states are

<equation id=" vac6" noautocaption>(%i)

</equation>

If at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = 0} , the atom is in state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | a \rangle } which means that the radiation field is in state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | 0 \rangle } then the system is in a superposition state:

<equation id=" vac7" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \psi (0) = \frac{1}{\sqrt {2}} ( | + \rangle + | - \rangle ) . \end{align}} </equation>

The time evolution of this superposition is given by

<equation id=" vac8" noautocaption>(%i)

</equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega / 2 = | H_{FI}^0 | / \hbar = e z_{ab}\sqrt {\omega / (e \epsilon _ o V \hbar )}} . The probability that the atom is in state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | b \rangle } at a later time is

<equation id=" vac9" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ P_ b = \frac{1}{2} (1 + \cos \Omega t ). \end{align}} </equation>

The frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega } is called the vacuum Rabi frequency.

The dynamics of a 2-level atom interacting with a single mode of the vacuum were first analyzed in [JAC63] and the oscillations are sometimes called Jaynes-Cummings oscillations.

The atom-vacuum interaction , Eq. EQ_vac4, has a simple physical interpretation. The electric field amplitude associated with the zero point energy in the cavity is given by

<equation id=" vac10" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \epsilon _ o E^2 V = \frac{1}{2} \hbar \omega \end{align}} </equation>

Consequently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | H_{FI}^0 | = E d_{ab}= ez_{ab} E} . The interaction frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | H_{FI}^0 | / \hbar } is sometimes referred to as the vacuum Rabi frequency, although, as we have seen, the actual oscillation frequency is .

Absorption and emission are closely related. Because the rates are proportional to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \langle F | H_{\rm int} | I \rangle |^2} , it is evident from Eq. EQ_vac3 that

<equation id=" vac11" noautocaption>(%i)

</equation>

This result, which applies to radiative transitions between any two states of a system, is general. In the absence of spontaneous emission, the absorption and emission rates are identical.

The oscillatory behavior described by Eq. EQ_vac8 is exactly the opposite of free space behavior in which an excited atom irreversibly decays to the lowest available state by spontaneous emission. The distinction is that in free space there are an infinite number of final states available to the photon, since it can go off in any direction, but in the cavity there is only one state. The natural way to regard the atom-cavity system is not in terms of the atom and cavity separately, as in Eq. EQ_vac1, but in terms of the coupled states and (Eq. EQ_vac5). Such states, called dressed atom states, are the true eigenstates of the atom-cavity system.


Absorption and emission

In Chapter 6, first-order perturbation theory was applied to find the response of a system initially in state to a perturbation of the form . The result is that the amplitude for state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |b \rangle } is given by

<equation id=" abem1" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ a_ b (t) = \frac{1}{2 i\hbar } \int _0^ t H_{ba} e^{-i(\omega - \omega _{ba} )t^\prime } dt^\prime = \frac{H_{ba}}{2\hbar } {\left[ \frac{e^{-i(\omega - \omega _{ba} )t} -1}{\omega - \omega _{ba}} \right]} \end{align}} </equation>

There will be a similar expression involving the time-dependence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{+ i \omega t}} . The term gives rise to resonance at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \omega _{ba}} ; the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + i \omega } term gives rise to resonance at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \omega _{ab}} . One term is responsible for absorption, the other is responsible for emission.

The probability that the system has made a transition to state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | b \rangle } at time is

<equation id=" abem2" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{a\rightarrow b} = | a_ b (t)|^2 = \frac{| H_{ba}|^2}{4 \hbar ^2} \frac{\sin ^2 [(\omega - \omega _{ba} )t/2]}{((\omega - \omega _{ba} )t/2)^2}t^2 \end{align}} </equation>

In the limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \rightarrow \omega _{ba}} , we have

<equation id=" abem3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{a\rightarrow b} \approx \frac{| H_{ba}|^2}{4 \hbar ^2} t^2 . \end{align}} </equation>

So, for short time, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{a\rightarrow b}} increases quadratically. This is reminiscent of a Rabi resonance in a 2-level system in the limit of short time.

However, Eq. EQ_abem2 is only valid provided , or for time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T \ll \hbar /H_{ba}} . For such a short time, the incident radiation will have a spectral width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \omega \sim 1/T} . In this case, we must integrate Eq. EQ_abem2 over the spectrum. In doing this, we shall make use of the relation

<equation id=" abem4" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \int _{-\infty }^{+\infty } \frac{\sin ^2 (\omega - \omega _{ba})t/2}{[(\omega - \omega _{ba})/2]^2} d \omega = 2t \int _{-\infty }^{+\infty } \frac{\sin ^2 (u - u_ o)}{(u - u_ o)^2} d u \rightarrow 2 \pi t \int _{-\infty }^{+\infty } \delta (\omega - \omega _{ba} ) d \omega . \end{align}} </equation>

Eq. EQ_abem2 becomes

<equation id=" abem5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{a\rightarrow b} = \frac{| H_{ba}|^2}{\hbar ^2} 2\pi t \delta (\omega - \omega _{ba} ) \end{align}} </equation>

The -function requires that eventually Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{a\rightarrow b}} be integrated over a spectral distribution function. Absorbing an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar } into the delta function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{a\rightarrow b}} can be written

<equation id=" abem6" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{a\rightarrow b} = \frac{| H_{ba}|^2}{\hbar} 2\pi t \delta (E_ b - E_ a - \hbar \omega ). \end{align}} </equation>

Because the transition probability is proportional to the time, we can define the transition rate

<equation id=" abem7a" noautocaption>(%i)

</equation>

<equation id=" abem7b" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ = 2\pi \frac{| H_{ba}|^2}{\hbar } \delta (E_ b - E_ a - \hbar \omega ) \end{align}} </equation>

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta } -function arises because of the assumption in first order perturbation theory that the amplitude of the initial state is not affected significantly. This will not be the case, for instance, if a monochromatic radiation field couples the two states, in which case the amplitudes oscillate between 0 and 1. However, the assumption of perfectly monochromatic radiation is in itself unrealistic.

Radiation always has some spectral width. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | H_{ba}|^2} is proportional to the intensity of the radiation field at resonance. The intensity can be written in terms of a spectral density function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} S(\omega ^\prime ) = S_0 f(\omega ^\prime ) \end{align}}

where is the incident Poynting vector, and f(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega ^\prime } ) is a normalized line shape function centered at the frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega ^\prime } which obeys Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int f (\omega ^\prime ) d\omega ^\prime = 1} . We can define a characteristic spectral width of by

<equation id=" abem8" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Delta \omega = \frac{1}{f(\omega _{ab} )} \end{align}} </equation>

Integrating Eq. EQ_abem7b over the spectrum of the radiation gives

<equation id=" abem9" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Gamma _{ab} = \frac{2\pi | H_{ba}|^2}{\hbar ^2} f(\omega _{ab} ) \end{align}} </equation>

If we define the effective Rabi frequency by

<equation id=" abem10" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Omega _ R = \frac{| H_{ba}| }{\hbar } \end{align}} </equation>

then

<equation id=" abem11" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Gamma _{ab} = {2 \pi } \frac{\Omega _ R^2}{\Delta \omega } \end{align}} </equation>

Another situation that often occurs is when the radiation is monochromatic, but the final state is actually composed of many states spaced close to each other in energy so as to form a continuum. If such is the case, the density of final states can be described by

<equation id=" abem12" noautocaption>(%i)

</equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dN} is the number of states in range Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dE} . Taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar \omega = E_ b - E_ a} in Eq. EQ_abem7b, and integrating gives

<equation id=" abem13" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Gamma _{ab} = 2\pi \frac{| H_{ba}|^2}{\hbar^2 } \rho (E_ b ) \end{align}} </equation>

This result remains valid in the limit , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \rightarrow 0} . In this static situation, the result is known as Fermi's Golden Rule .

Note that Eq. EQ_abem9 and Eq. EQ_abem13 both describe a uniform rate process in which the population of the initial state decreases exponentially in time. If the population of the initial state is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(0)} , then

<equation id=" abem14" noautocaption>(%i)

</equation>

Applying this to the dipole transition described in Eq. EQ_int11, we have

<equation id=" abem15" noautocaption>(%i)

</equation>

The arguments here do not distinguish whether or (though the sign of obviously does). In the former case the process is absorption, in the latter case it is emission.


Spontaneous emission rate

The rate of absorption, in CGS units, for the transition , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_ b > E_ a} , is, from Eq. EQ_qrd16 and Eq. EQ_abem7b,

<equation id=" sem1" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Gamma _{ab} = \frac{4\pi ^2}{\hbar V} | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 n\omega \delta (\omega _0 -\omega ) . \end{align}} </equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega _0 = ( E_ b - E_ a ) /\hbar } . To evaluate this we need to let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \rightarrow n (\omega )} , where is the number of photons in the frequency interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\omega } , and integrate over the spectrum. The result is

<equation id=" sem2" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Gamma _{ab} = \frac{4\pi ^2}{\hbar V} | {\bf \hat{e}}\cdot {\bf d}_{ba} |^2 \omega _0 n(\omega _0 ) \end{align}} </equation>

To calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n (\omega )} , we first calculate the mode density in space by applying the usual periodic boundary condition

<equation id=" sem3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ k_ j L = 2\pi n_ j , ~ ~ ~ j = x,y,z. \end{align}} </equation>

The number of modes in the range is

<equation id=" sem4" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ dN = dn_ x dn_ y dn_ z = \frac{V}{{\left(2 \pi \right)^3} } d^3 k=\frac{V}{{\left(2 \pi \right)^3} }k^2 dk \ d\Omega = \frac{V}{{\left(2 \pi \right)^3} } \frac{\omega ^2\, d\omega \ d\Omega }{c^3} \end{align}} </equation>

Letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{n} = \bar{n (\omega ) }} be the average number of photons per mode, then

<equation id=" sem5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ n (\omega ) = \bar{n} \frac{dN}{d\omega } = \frac{\bar{n} V\omega ^2 d\Omega }{(2\pi )^3 c^3} \end{align}} </equation>

Introducing this into Eq. EQ_sem2 gives

<equation id=" sem6" noautocaption>(%i)

</equation>

We wish to apply this to the case of isotropic radiation in free space, as, for instance, in a thermal radiation field. We can take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf d}_{ba}} to lie along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} axis and describe k in spherical coordinates about this axis. Since the wave is transverse, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf \hat{e}} \cdot {\bf \hat{D}} = \sin \theta } for one polarization, and zero for the other one. Consequently,

<equation id=" sem7" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \int | {\bf \hat{e}} \cdot {\bf d}_{ba} |^2 d\Omega = | {\bf d}_{ba} |^2 \int \sin ^2 \theta d\Omega = \frac{8\pi }{3} | {\bf d}_{ba}|^2 \end{align}} </equation>

Introducing this into Eq. EQ_sem6 yields the absorption rates

<equation id=" sem8" noautocaption>(%i)

</equation>

It follows that the emission rate for the transition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\rightarrow a} is

<equation id=" sem9" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Gamma _{ba} = \frac{4}{3} \frac{\omega ^3}{\hbar c^3} | {\bf d}_{ba} |^2 (\bar{n} + 1) \end{align}} </equation>

If there are no photons present, the emission rate—called the rate of spontaneous emission—is

<equation id=" sem10" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Gamma _{ba}^0 = \frac{4}{3} \frac{ \omega ^3}{\hbar c^3} | {\bf d}_{ba}|^2 = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} | \langle b| {\bf r} | a \rangle |^2 \end{align}} </equation>

In atomic units, in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = 1 / \alpha } , we have

<equation id=" sem11" noautocaption>(%i)

</equation>

Taking, typically, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = 1} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{ba}= 1} , we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma ^0 \approx \alpha ^3} . The “ of a radiative transition is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q =\omega /\Gamma \approx \alpha ^{-3}\approx } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \times 10^6} . The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha ^3} dependence of indicates that radiation is fundamentally a weak process: hence the high Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} and the relatively long radiative lifetime of a state, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = 1 /\Gamma } . For example, for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2P\rightarrow 1S} transition in hydrogen (the transition), we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = 3/8} , and taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{2p,1s} \approx 1} , we find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = 3.6\times 10^7} atomic units, or 0.8 ns. The actual lifetime is 1.6 ns.

The lifetime for a strong transition in the optical region is typically 10–100 ns. Because of the dependence of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma ^0} , the radiative lifetime for a transition in the microwave region—for instance an electric dipole rotational transition in a molecule—is longer by the factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( \lambda _{\rm microwave} /\lambda _{\rm optical} )^3 \approx 10^{15}} , yielding lifetimes on the order of months. Furthermore, if the transition moment is magnetic dipole rather than electric dipole, the lifetime is further increased by a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha ^{-2}} , giving a time of thousands of years.


Line Strength

Because the absorption and stimulated emission rates are proportional to the spontaneous emission rate, we shall focus our attention on the Einstein A coefficient:

<equation id=" lines1" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} | \langle b | {\bf r} | a \rangle |^2 \end{align}} </equation>

where

<equation id=" lines2" noautocaption>(%i)

</equation>

For an isolated atom, the initial and final states will be eigenstates of total angular momentum. (If there is an accidental degeneracy, as in hydrogen, it is still possible to select angular momentum eigenstates.) If the final angular momentum is , then the atom can decay into each of the final states, characterized by the azimuthal quantum number . Consequently,

<equation id=" lines3" noautocaption>(%i)

</equation>

The upper level, however, is also degenerate, with a (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 J_ b + 1} )–fold degeneracy. The lifetime cannot depend on which state the atom happens to be in. This follows from the isotropy of space: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_ b} depends on the orientation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf J}_ b} with respect to some direction in space, but the decay rate for an isolated atom can't depend on how the atom happens to be oriented. Consequently, it is convenient to define the line strength , given by

<equation id=" lines4" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ S_{ba} = S_{ab} = \sum _{m_ b} \sum _{m_ a} | \langle b, J_ b, m_ b | {\bf r} | a, J_ a, m_ a \rangle |^2 \end{align}} </equation>

Then,

<equation id=" lines5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ A_{ba} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} \frac{S_{ba}}{g_ b} = \frac{4}{3} \frac{e^2\omega ^3}{\hbar c^3} \frac{S_{ba}}{2J_ b +1} \end{align}} </equation>

The line strength is closely related to the average oscillator strength Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{f}_{ab}} . is obtained by averaging Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ab}} over the initial state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |b\rangle } , and summing over the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} in the final state, . For absorption, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega _{ab} > 0} , and

<equation id=" line11" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \bar{f}_{ab} = \frac{2m}{3\hbar } \omega _{ab} \frac{1}{2J_ b + 1} \sum _{m_ b} \sum _{m_ a} |\langle b, J_ b, m_ b |{\bf r} | a, J_ a, m_ a \rangle |^2 \end{align}} </equation>

It follows that

<equation id=" line12" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \bar{f}_{ba} = - \frac{2J_ b + 1}{2J_ a +1} \bar{f}_{ab} . \end{align}} </equation>

In terms of the oscillator strength, we have

<equation id=" line13" noautocaption>(%i)

</equation>

<equation id=" line14" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \bar{f}_{ba} = - \frac{2m}{3\hbar } | \omega _{ab} | \frac{1}{2J_ a + 1} {S}_{ab} . \end{align}} </equation>


Excitation by narrow and broad band light sources

We have calculated the rate of absorption and emission of an atom in a thermal field, but a more common situation involves interaction with a light beam, either monochromatic or broad band. Here broad band means having a spectral width that is broad compared to the natural line width of the system—the spontaneous decay rate.

For an electric dipole transition, the radiation interaction is

<equation id=" broad1" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ | H_{ba} | = e | {\bf r}_{ba} |\cdot {\bf \hat{e}} E/2, \end{align}} </equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E } is the amplitude of the field. The transition rate, from Eq. EQ_sem7, is

<equation id=" broad2" noautocaption>(%i)

</equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega _0 = ( E_ b - E_ a )/\hbar } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f (\omega )} is the normalized line shape function, or alternatively, the normalized density of states, expressed in frequency units. The transition rate is proportional to the intensity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_0} of a monochromatic radiation source. is given by the Poynting vector, and can be expressed by the electric field as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^2 = 8 \pi I_0 / c} . Consequently,

<equation id=" broad3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{ab} = \frac{4\pi ^2}{c} \frac{e^2 | {\bf \hat{e}} \cdot {\bf r}_{ba} |^2}{\hbar ^2} I_0 f (\omega _0 ) \end{align}} </equation>

In the case of a Lorentzian line having a FWHM of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma _0} centered on frequency ,

<equation id=" broad4" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ f(\omega ) = \frac{1}{\pi } \frac{(\Gamma _0 /2)}{(\omega - \omega _0 )^2 + (\Gamma _0 /2)^2} \end{align}} </equation>

In this case,

<equation id=" broad5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{ab} = \frac{8\pi e^2}{c\hbar ^2 \Gamma _0} | \langle b | {\bf \hat{e}} \cdot {\bf r} | a \rangle |^2 I_0 \end{align}} </equation>

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{ab}} is the rate of transition between two particular quantum states, not the total rate between energy levels. Naturally, we also have .\

An alternative way to express Eq. EQ_broad2 is to introduce the Rabi frequency,

<equation id=" broad6" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \Omega _ R = \frac{2 H_{ba}}{\hbar } = \frac{e |{\bf \hat{e}}\cdot {\bf r}_{ba} | E}{\hbar } \end{align}} </equation>

In which case

<equation id=" broad7" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{ab} = \frac{\pi }{2} \Omega _ R^2 f (\omega _0 ) = \Omega _ R^2 \frac{1}{\Gamma _0} \end{align}} </equation>

If the width of the final state is due soley to spontaneous emission, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma _0 = A = ( 4 e^2 \omega ^3 / 3 \hbar c^3 ) | r_{ba} |^2} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{ab}} is proportional to , it is independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | r_{ba} |^2} . It is left as a problem to find the exact relationship, but it can readily be seen that it is of the form

<equation id=" broad8" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{ab} = X\lambda ^2 I_0 /\hbar \omega \end{align}} </equation>

where X is a numerical factor. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I/ \hbar \omega } is the photon flux—i.e. the number of photons per second per unit area in the beam. Since is an excitation rate, we interpret Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\lambda ^2} as the resonance absorption cross section for the atom, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma _0} .

At first glance it is puzzling that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma _0} does not depend on the structure of the atom; one might expect that a transition with a large oscillator strength—i.e. a large value of —should have a large absorption cross section. However, the absorption rate is inversely proportional to the linewidth, and since that also increases with , the two factors cancel out. This behavior is not limited to electric dipole transitions, but is quite general.

There is, however, an important feature of absorption that does depend on the oscillator strength. is the cross section assuming that the radiation is monochromatic compared to the natural line width. As the spontaneous decay rate becomes smaller and smaller, eventually the natural linewidth becomes narrower than the spectral width of the laser, or whatever source is used. In that case, the excitation becomes broad band.

Broad Band Excitation

We now discuss broad band excitation. Using the result of the last section, finding the excitation rate or the absorption cross section for broad band excitation is trivial. From Eq. EQ_broad2, the absorption rate is proportional to . For monochromatic excitation, and . For a spectral source having linewidth , defined so that the normalized line shape function is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f (\omega _0 ) = (2/ \pi ) {\Delta \omega _ s}^{-1} } , then the broad band excitation rate is obtained by replacing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma _0} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \omega _ s} in Eq. EQ_broad8. Thus

<equation id=" band1" noautocaption>(%i)

</equation>

Similarly, the effective absorption cross section is

<equation id=" band2" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \sigma _{\rm eff} = \sigma _0 \frac{\Gamma _0}{\Delta \omega _ s} \end{align}} </equation>

This relation is valid provided Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \omega _ s \gg \Gamma _0} . If the two widths are comparable, the problem needs to be worked out in detail, though the general behavior would be for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \omega _ s \rightarrow ( \Delta \omega _ s^2 + \Gamma _0^2 )^{1/2}} . Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \omega _ s} represents the actual resonance width. Thus, if Doppler broadening is the major broadening mechanism then

<equation id=" band3" noautocaption>(%i)

</equation>

Except in the case of high resolution laser spectroscopy, it is generally true that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \omega _ s \gg \Gamma _0} , so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma _{\rm eff}\ll \sigma _0} .

Saturation and Saturated Absorption Rates

When the external light intensity is strong, the population in the excited state is no longer negligible, and the transition is saturated. We define the saturated absorption rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^s } as the net transfer from initial state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } to final state , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^u } is the unsaturated rate for the stimulated absorption and emission,

<equation id=" sat1" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ R^s (n_a+n_b) = R^u (n_a-n_b). \end{align}} </equation>

When the system reaches steady state,

<equation id=" sat2" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \dot{n_b}&=&-n_b(R^u+\Gamma)+n_aR^u =0\\ \dot{n_a}&=&n_b(R^u+\Gamma)-n_aR^u =0 \\ \end{align}} </equation>

which gives

<equation id=" sat3" noautocaption>(%i)

</equation>

From Eq. EQ_sat1, we have

<equation id=" sat4" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ R^s=\frac{\Gamma}{2}\frac{S}{1+S}=\frac{R^u}{1+S} \end{align}} </equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } is the saturation parameter and is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=2R^u/\Gamma } . The transition rate is reduced by a factor of due to saturation.

For low intensity light, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\ll 1} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^s=R^u} ; for very high intensity light, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\gg 1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^s=\Gamma/2} .

For the case of monochromatic radiation, as discussed above, the unsaturated transition rate

<equation id=" sat5" noautocaption>(%i)

</equation> where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta } the detuning with respect to the center frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 } .

Thus in general the saturated transition rate

<equation id=" sat6" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ R^s= \frac{\omega_R^2 }{\Gamma} \frac{1}{1+(2\delta/\Gamma)^2+2\omega_R^2/\Gamma^2} \end{align}} </equation> and the saturation parameter

<equation id=" sat7" noautocaption>(%i)

</equation>

with the resonant saturation parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{res}=2\omega_R^2/\Gamma^2} .

The saturated rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^s } has a Lorentzian line with FWHM

<equation id=" sat8" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \delta_{FWHM }=\frac{\Gamma}{2}\sqrt{1+S_{res}} \end{align}} </equation>

Power Broadening

This resultant increase in the spectrum width is called saturation (or power) broadening.

The saturation intensity is the light field intensity corresponding to the saturation parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{res}=1 } for a resonant light, and that is when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^u=\omega_R^2/\Gamma=\Gamma/2} . Since the Rabi frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R^2\propto I} , we have the linear relation

<equation id=" sat9" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \omega_R^2=\frac{\Gamma^2}{2}\frac{I}{I_{sat}} \end{align}} </equation>

and that gives

<equation id=" sat10" noautocaption>(%i)

</equation> for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{sat}=6\; mW/cm^2} for Na D line.

Saturation Intensity

A quick derivation for the saturation intensity is to express the light intensity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I } and the Rabi frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R } in terms of the number of photons ,

<equation id=" sat11" noautocaption>(%i)

</equation>

<equation id=" sat12" noautocaption>(%i)

</equation>

thus

<equation id=" sat13" noautocaption>(%i)

</equation> and pluging this into Eq. EQ_ sat9 gives the saturation intensity.

For the case of broadband radiation, we define the average intensity per frequency interval as , and when the saturation parameter , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{I}=\bar{I}_{sat} }

<equation id=" sat14" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{ge}=B_{ge}\frac{\bar{I}}{c}=\frac{\Gamma}{2} \end{align}} </equation> thus

<equation id=" sat15" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \bar{I}_{sat}=\frac{c}{2}\frac{A}{B_{ge}}=\frac{\hbar\omega_{eg}^3}{6\pi^2 c^2} \end{align}} </equation> which is independent of matrix element! For visible light, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{I}_{sat}\approx \frac{12 \;W}{cm^2}\frac{1}{cm^{-1}}} , where .

Absorption Cross Section

Cross section is the effective area that represents the probability of some scattering or absorption event. In the case of atom-photon interaction, the absorption rate is the collision rate of an atom with the incoming photons, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=n_{phot}\sigma c } .

For monochromatic radiation,

<equation id=" sat16" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{ge}=n_{phot}\sigma c=\frac{I\sigma}{\hbar\omega} \end{align}} </equation> in the low intensity limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{ge}=R^u } . If we extrapolate it to saturation parameter , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I=I_{sat} } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{ge}=R^u=\Gamma/2 }

<equation id=" sat17" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \frac{\Gamma}{2}=\frac{I_{sat}\sigma}{\hbar \omega} \end{align}} </equation>

and from Eq. EQ_ sat10, we have

<equation id=" sat18" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \sigma=6\pi\frac{c^2}{\omega^2}=6\pi (\lambda/2\pi)^2 \end{align}} </equation> This is the resonant cross section for weak radiation, and it is usually much larger than the size of the atom, and independent of matrix element. If we plot the cross section as a function of detuning, it is a Lorentzian line. Strong transitions have a larger widths, but the cross section on resonance is always the same.

When the transition is saturated at high intensity, the resonant cross section goes as . The transition bleaches out Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma\rightarrow 0 } when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\gg 1 } .

For broadband radiation,

<equation id=" sat19" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ W_{ge}&=&\int \sigma(\omega)\frac{\bar{I}(\omega)}{\hbar\omega}d\omega \\ &=& \frac{6\pi (\lambda/2\pi)^2}{\hbar\omega_{eg}}\bar{I}(\omega_{eg})\frac{\pi\Gamma}{2} \end{align}} </equation>

at saturation ,

<equation id=" sat20" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \frac{\Gamma}{2}=\frac{6\pi (\lambda/2\pi)^2}{\hbar\omega_{eg}}\bar{I}_{sat}\frac{\pi\Gamma}{2} \end{align}} </equation> thus

<equation id=" sat21" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \bar{I}_{sat}=\frac{\hbar\omega_{eg}}{6\pi^2 (\lambda/2\pi)^2}=\frac{\hbar \omega_{eg}^3}{6\pi^2 c^2} \end{align}} </equation> which is the same as we have derived in Eq. EQ_ sat15.


Higher-order radiation processes

Beyond the dipole approximation: Recall that the interaction Hamiltonian for an atom in an electromagnetic field is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_\mathrm{int} = -\frac{e}{mc} p\cdot A+\frac{e^2}{2mc^2}|A|^2+g_s\mu_B S\cdot({\bf\nabla}\times A), }

where the last term we have so far considered only for static magnetic fields. Neglecting, as before, the term, which is appreciable only for very intense fields, we now consider more fully the dominant term in the atom-field interaction,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{ba} = \frac{e}{\rm mc} \langle b | p \cdot A (r) | a\rangle. }

For concreteness, we shall take A(r) to be a plane wave of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A (r) = A\hat{z} e^{ikx}. }

Expanding the exponential, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{ba} = \frac{eA}{\rm mc} \langle b | p_z (1+ikx + (ikx)^2/2 + \dots ) | a\rangle. }

Thus far in the course, we have considered only the first term, the dipole term. If dipole radiation is forbidden, for instance if and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | b \rangle} have the same parity, then the second term in the parentheses becomes important. Usually, it is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} times smaller. In particular, since

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k r \approx \frac{\hbar\omega}{\hbar c}a_0\approx\frac{e^2/a_0}{\hbar c}a_0\approx\frac{e^2}{\hbar c}=\alpha, }

the expansion in eq:hor3/> is effectively an expansion in . We can rewrite the second term as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_z x = (p_z x - zp_x )/2 + (p_z x + zp_x )/2 . }

The first term of Eq. eq:hor4/> is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \hbar L_y/2} , and the matrix element becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{ieAk}{2 m} \langle b | \hbar L_y | a \rangle = - iAk \langle b | \mu_B L_y | a \rangle, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B = e\hbar /2 m} is the Bohr magneton. The magnetic field is . Consequently, Eq.\ eq:hor5/> can be written in the more familiar form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\vec{\mu} \cdot B} . (The orbital magnetic moment is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\mu} = -\mu_B L} : the minus sign arises from our convention that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} is positive.) We can readily generalize the matrix element to

where we have added the spin dependent term from Eq. eq:hor_Hint/>. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1} indicates that the matrix element is for a magnetic dipole transition. The strength of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1} transition is set by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B/c = \frac{1}{2}\frac{e\hbar}{mc}=\frac{1}{2}\frac{e^2}{\hbar c}\frac{\hbar^2}{e m} = \frac{1}{2}\alpha e a_0, }

so it is indeed a factor of weaker than a dipole transition, as we argued above.

The second term in Eq.\ eq:hor4/> involves Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( p_z x + z p_x )/2} . Making use of the commutator relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [ r, H_0 ] = i\hbar p / m } , we have

So, the contribution of this term to is

where we have taken . This is an electric quadrupole interaction, and we shall denote the matrix element by

The prime indicates that we are considering only one component of a more general expression involving the matrix element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle b |r:r|a\rangle} of a tensor product. It is straightforward to verify that the electric quadrupole interaction is also of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} .

The total matrix element of the second term in the expansion of Eq.\ eq:hor3/> can be written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm int}^{(2)} = H_{\rm int} (M1) + H_{\rm int} (E2). }

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm int} (M1)} is real, whereas is imaginary. Consequently,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | H_{\rm int}^{(2)} |^2 = | H_{\rm int} (M1)|^2 + | H_{\rm int}(E2) |^2. }

The magnetic dipole and electric quadrupole terms do not interfere.

Because transition rates depend on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |H_{ba} |^2} , the magnetic dipole and electric quadrupole rates are both smaller than the dipole rate by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha^2 \sim 5 \times 10^{-5}} . For this reason they are generally referred to as {\it forbidden} processes. However, the term is used somewhat loosely, for there are transitions which are much more strongly suppressed due to other selection rules, as for instance triplet to singlet transitions in helium. \begin{table}

Transition Operator Parity
Electric Dipole Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E1} -
Magnetic Dipole Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\mu_B(L+g_sS)} +
Electric Quadrupole Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E2} +

\caption{Summary of dipole and higher-order radiation processes.} \end{table}

Selection rules

A forbidden transition, then, is one that is weaker than an electric dipole-allowed transition by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha^n} and only appears in some higher-order approximation. Examples of such higher-order effects are the magnetic dipole and electric quadrupole terms described above, multiphoton processes, the relativistic effects which allow singlet to triplet transitions in helium, and hyperfine interactions within the nucleus. To derive selection rules for the transitions we have discussed above, it is useful to express the matrix elements in terms of spherical tensor operators:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int}(T_{l,m}) = \langle n J M | T_{l,m} | n' J' M'\rangle, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{l,m}} is a spherical tensor operator of rank . The operators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{l,m}} transform under rotations like the spherical harmonics Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{l,m}} , and any operator can be written as a linear combination of these spherical tensors. By the Wigner-Eckart Theorem, we can express the matrix element

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle n J M | T_{l,m} | n' J' M'\rangle = \frac{\langle n J \| T_l \| n' J' \rangle}{\sqrt{2J+1}}\langle J' l, M', m| J M\rangle }

in terms of a reduced matrix element and a Clebsch-Gordan coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle J' l, M', m| J M\rangle} . In order for the latter to be nonzero, the triangle rule requires that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |J'-J| \leq l \leq |J'+J|} , while conservation of angular momentum requires Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = M' + m} . Since the operators and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B B} responsible for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1} transitions are both vectors, i.e. tensors of rank , these transitions are both governed by the dipole selection rules

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} |\Delta J| &= 0, 1;\\ |\Delta m| &= 0, 1. \end{align}}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bf r} is a polar vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bf L} is an axial vector, transitions are allowed only between states of opposite parity and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} transitions are allowed only between states of the same parity. The operator responsible for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E2} transitions is a spherical tensor of rank 2. For example,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xz = (T_{2,-1}-T_{2,1})/4. }

In general, then, we expect that the quadrupole moment can be expressed in terms of . Thus, electric quadrupole transitions are allowed only between states connected by tensors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2,m}(r)} , requiring:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} |\Delta J| & = 0, 1, 2; \\ |\Delta m| &= 0, 1, 2. \end{align}}

and parity unchanged.

In addition, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J=0\rightarrow J'=0} transitions are forbidden in all of the cases considered above, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J=J'=0} requires (for any interaction that does not couple to spin) whereas absorption or emission of a photon implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Delta L|=1} .

We now illustrate the use of the spherical tensor for the case of a vector. The dipole matrix element for a particular polarization of the field, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\bf {e}}} , is

<equation id=" select1" noautocaption>(%i)

</equation>

It is straightforward to calculate but a more general approach is to write r in terms of a spherical tensor. This yields the selection rules directly, and allows the matrix element to be calculated for various geometries using the Wigner-Eckart theorem as discussed above.

The orbital angular momentum operator of a system with total angular momentum can be written in terms of a spherical harmonic . Consequently, the spherical harmonics constitute spherical tensor operators. A vector can be written in terms of spherical harmonics of rank 1. This permits the vector operator r to be expressed in terms of the spherical tensor

The spherical harmonics of rank 1 are

<equation id=" select2" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ Y_{1,0} = \sqrt {\frac{3}{4\pi }} \cos \theta ; \qquad Y_{1, +1} = - \sqrt {\frac{3}{8\pi }} \sin \theta e^{+i\phi }\qquad Y_{1,-1} = \sqrt {\frac{3}{8\pi }} \sin \theta e^{-i\phi } \end{align}} </equation>

These are normalized so that

<equation id=" select3" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \int Y_{1,m^\prime }^* Y_{1,m} \sin \theta d\theta d\phi = \delta _{m^\prime , m} \end{align}} </equation>

We can write the vector r in terms of components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_ m ,\ m = +1, 0, -1} ,

<equation id=" select4" noautocaption>(%i)

</equation>

or, more generally

<equation id=" select5" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ r_ M = rT_{1,M} (\theta , \phi ) \end{align}} </equation>

Consequently,

<equation id=" select6" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ \langle b, J_ b, m_ b | r_ M | a, J_ a, m_ a \rangle = \langle b, J_ b, m_ b | rT_{1,M} | a, J_ a, m_ a \rangle \end{align}} </equation>

<equation id=" select7" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ = \langle b, J_ b | r | a, J_ a \rangle \langle J_ b, m_ b | T_{1,M} | J_ a, m_ a \rangle \end{align}} </equation>

The first factor is independent of . It is

<equation id=" select8" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ r_{ba} = \int _0^{\infty } R_{b,J_ b}^* (r) r R_{a,J_ a} (r) r^2 dr \end{align}} </equation>

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{ba}} contains the radial part of the matrix element. It vanishes unless Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | b \rangle } and have opposite parity. The second factor in Eq. EQ_select7 yields the selection rule

<equation id=" select9" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ | J_ b - J_ a | = 0, 1; ~ ~ ~ m_ b = m_ a \pm M = m_ a, m_ a \pm 1 \end{align}} </equation>

Similarly, for magnetic dipole transition, Eq. EQ_hor6, we have

<equation id=" select10" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ H_{ba} (M1) = \mu _ B B \langle b, J_ b, m_ b , | T_{LM} (L) | a, J_ a , m_ a \rangle \end{align}} </equation>

It immediately follows that parity is unchanged, and that

<equation id=" select11" noautocaption>(%i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ | \Delta J | = 0,1 ~ ~ ~ (J=0\rightarrow J= 0~ \mbox{forbidden}); ~ ~ | \Delta m | = 0,1 \end{align}} </equation>

This discussion of matrix elements, selection rules, and radiative processes barely skims the subject. For an authoritative treatment, the books by Shore and Manzel, and Sobelman are recommended.

References

<thebibliography> <attributes> <widelabel>99</widelabel> </attributes> <bibitem> <attributes> <key>JAC63</key> <label>None</label> </attributes> E.T. Jaynes and F.W. Cummings, Proc. IEEE, 51, 89 (1963).


</bibitem><bibitem> <attributes> <key>EIN17</key> <label>None</label> </attributes> A. Einstein, Z. Phys. 18, 121 (1917), reprinted in English by D. ter Haar, <it> The Old Quantum Theory </it>, Pergammon, Oxford.


</bibitem> </thebibliography>

<bibitem>

</bibitem>[EIN17a] A. Einstein, Z. Phys. 18, 121 (1917), translated in Sources of Quantum Mechanics, B. L. Van der Waerden, Cover Publication, Inc., New York, 1967. This book is a gold mine for anyone interested in the development of quantum mechanics.



\begin{thebibliography}{99}

\bibitem{JAC63} E.T. Jaynes and F.W. Cummings, Proc. IEEE, 51, 89 (1963).

\bibitem{EIN17} A. Einstein, Z. Phys. 18, 121 (1917), reprinted in English by D.\ ter Haar, {\it The Old Quantum Theory}, Pergammon, Oxford.

\end{thebibliography}

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