Solutions of the optical Bloch equations

The optical Bloch equations provide a time-dependent quantum description of a spontaneously emitting atom driven by a classical electromagnetic field. Considerable insight into the physical processes involved can be gained by studying these equations in the transient excitation limit, as well as the steady-state limit, as we see in this section. We begin by considering the coherent part of the evolution, then extend this to re-visit the Bloch sphere picture of the optical Bloch equations, which provides useful visualizations of transient responses and steady state solutions.

▼ Category:8.422

► Category:Quantum Light

► Category:Optical Bloch Equations

► Category:Bose-Einstein condensates

► Category:Ion traps and quantum information

► Category:Light forces

► Category:Photon-atom interactions

  Introduction to Atomic Physics

  Quantum light: states and dynamics

  Optical Bloch equations

  Photon-atom interactions

  Frontiers in Atomic Physics

  Ion traps and quantum information

  Quantum gases

Eigenstates of the Jaynes-Cummings Hamiltonian

The optical Bloch equations are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\rho} = -\frac{i}{\hbar}[H,\rho] -\frac{\Gamma}{2} \left[ { \sigma_+ \sigma_- \rho - 2 \sigma_- \rho \sigma_+ + \rho \sigma_+ \sigma_- } \right] \,, }

where the Hamiltonian 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 H = \frac{\hbar\omega_0}{2}(|e \rangle \langle e|-|g \rangle \langle g|) + \frac{\hbar\Omega}{2} (|g \rangle \langle e|+|e \rangle \langle g|) \,. }

Useful limits to this equation of motion can be obtained, for example, by solving 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 \rho} in the steady state, when ${\dot {\rho }}=0$. Both the Hamiltonian part of this equation, and the damping part, are important to consider, and we begin here by reviewing the coherent evolution under $H$. $H$ arises from the Jaynes-Cummings interaction we have previously considered in the context of cavity QED, describing a single two-level atom interacting with a single mode of the electromagnetic field:

$H_{JC}={\frac {\hbar \omega _{0}}{2}}\left[{|e\rangle \langle e|-|g\rangle \langle g|}\right]+\hbar \omega a^{\dagger }a+{\frac {\hbar \Omega }{2}}\left[{|g\rangle \langle e|+|e\rangle \langle g|}\right]\left[{a+a^{\dagger }}\right]\,.$

The last term in this expression 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 H_I} , the dipole interaction between atom and field. By defining Failed to parse (MathML with SVG or PNG fallback (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_+ = |e \rangle \langle g|} 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 \sigma_- =|g \rangle \langle e|} , we may write this interaction as

$H_{I}={\frac {\hbar \Omega }{2}}\left[{\sigma _{+}+\sigma _{-}}\right]\left[{a+a^{\dagger }}\right]\,.$

In the frame of reference of the atom and field, recall 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 \begin{array}{rcl} \sigma_+ a^\dagger &\rightarrow& e^{i(\omega+\omega_0)t} \sigma_+ a^\dagger \\ \sigma_- a &\rightarrow& e^{-i(\omega+\omega_0)t} \sigma_- a \\ \sigma_+ a &\rightarrow& e^{i(\omega_0-\omega)t} \sigma_+ a \\ \sigma_- a^\dagger &\rightarrow& e^{-i(\omega_0-\omega_0)t} \sigma_- a^\dagger \,. \end{array}}

When near resonance, Failed to parse (MathML with SVG or PNG fallback (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_0} , and because 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 \sigma_+ a^\dagger } 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 \sigma_- a} terms oscillate at nearly twice the frequency 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 \omega} , those terms can be dropped. Doing so is known as the rotating wave approximation, and it gives us a simplified interaaction Hamiltonian

Failed to parse (MathML with SVG or PNG fallback (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_I = \frac{\hbar\Omega}{2} \left[ { a^\dagger \sigma_+ + a \sigma_- } \right] \,. }

Under this approximation, it is useful to note that this interaction merely exchanges one quantum of excitation from atom to field, and back, so that the total number of excitations Failed to parse (MathML with SVG or PNG fallback (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 + |e \rangle \langle e|} is a constant of the motion. We may thus write the total Hamiltonian, in the rotating wave approximation, 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 H = \hbar\omega N + \delta \sigma_z + \frac{\hbar\Omega}{2}( a^\dagger \sigma_- +a \sigma_+ ) \,, }

where we have defined Failed to parse (MathML with SVG or PNG fallback (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_z = |e \rangle \langle e|-|g \rangle \langle g|} , 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 \delta = \omega_0-\omega} . Below, we may use Failed to parse (MathML with SVG or PNG fallback (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 = \hbar\Omega/2} to simplify writing. What are the eigenstates of this Hamiltonian? It describes a two-level system coupled to a simple harmonic oscillator; when uncoupled, if $\delta =0$, then the eigenstates are simply those 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 |e{\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 |g{\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 |n{\rangle}} , as shown here:

When coupled, degenerate energy levels split, with harmonic oscillator levels Failed to parse (MathML with SVG or PNG fallback (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{\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 |n+1{\rangle}} splitting into two energy levels separated by $2g{\sqrt {n+1}}$. Since the coupling only pairs levels separated by one quantum of excitation, it is straightforward to show that the eigenstates of the Jaynes-Cummings Hamiltonian fall into well defined pairs of states, which we may label 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 |\pm,n{\rangle}} ; these are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\pm,n \rangle = \frac{1}{\sqrt{2}} \left[ { |e,n \rangle \pm |g,n+1 \rangle } \right] \,, }

and they have 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_{\pm,n} = \hbar\omega(n+1) \pm g\sqrt{n+1} \,. }

When $\delta \neq 0$, similar physics result, but with slightly more complicated expressions describing the eigenstates, as we shall see when we later return to the "dressed states" picture.

Strongly driven atom: Mollow triplet

An atom strongly coupled to a single mode electromagnetic field, or an atom driven strongly by a single mode field, will thus have an emission spectrum described by the coupled energy level diagram:

where, to good approximation, the energy level differences are Failed to parse (MathML with SVG or PNG fallback (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} 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\pm\Omega} . These three lines which appear in the spectrum are known as the Mollow triplet:

The Mollow triplet is experimentally observed in a wide variety of systems. However, while our energy eigenstate analysis has predicted the number and frequencies of the emission lines, it fails to explain a key characteristic: the widths are not the same. If the central peak 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_0} has 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 \Gamma/2} , the two sidebands each have a width 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 3\Gamma/4} . To explain this, we need the optical Bloch equations.

Bloch vector evolution

We have previously seen that an arbitrary qubit 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 |\psi \rangle = \cos\theta|0 \rangle + e^{i\phi}\sin\phi|1{\rangle}} can be represented as being a point on a unit sphere, located 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 (\theta,\phi)} in polar coordinates. Similarly, a density matrix Failed to parse (MathML with SVG or PNG fallback (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} may be depicted as being a point inside or on the unit sphere, using

Failed to parse (MathML with SVG or PNG fallback (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 = \frac{I+\vec{r}\cdot{\sigma}}{2} \,, }

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 \vec{r}} is the Bloch vector representation 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 \rho} .

Explicitly, if we 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 \rho = \left[ \begin{array}{cc}{\rho_{ee}}&{\rho_{eg}}\\{\rho_{ge}}&{\rho_{gg}}\end{array}\right] \,, }

then

${\begin{array}{rcl}r_{x}&=&\rho _{ge}+\rho _{eg}\\r_{y}&=&(\rho _{ge}-\rho _{eg})/i\\r_{z}&=&\rho _{gg}-\rho _{ee}\,.\end{array}}$

Visulization of the evolution of a density matrix under the optical Bloch equations is thus helped by rewriting them in terms of a differential equation 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 \vec{r}} . A convenient starting point for this is the optical Bloch equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\rho} = -\frac{i}{\hbar}[H,\rho] -\frac{\Gamma}{2} \left[ { \sigma_+ \sigma_- \rho - 2 \sigma_- \rho \sigma_+ + \rho \sigma_+ \sigma_- } \right] \,, }

using the rotating frame Hamiltonian (suppressing Failed to parse (MathML with SVG or PNG fallback (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} )

$H={\frac {\delta }{2}}Z+gX\,.$

This gives us the equations of motion

Failed to parse (MathML with SVG or PNG fallback (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{array}{rcl} \dot{r}_x &=& \delta\, r_y - \frac{\Gamma}{2} r_x \\ \dot{r}_y &=& -\delta\, r_x - g\, r_z - \frac{\Gamma}{2} r_y \\ \dot{r}_z &=& g\, r_y - \Gamma (r_z -1) \,. \end{array}}

Note how these equations of motion provide a simple set of flows on the Bloch sphere: the $\delta$ terms correspond to a rotation in 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 \hat{x}-\hat{y}} plane, Failed to parse (MathML with SVG or PNG fallback (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} corresponds to a rotation in 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 \hat{y}-\hat{z}} plane, 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 \Gamma} drives a relaxation process which shrinks Failed to parse (MathML with SVG or PNG fallback (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{x}} 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 \hat{y}} components of the Bloch vector, while moving 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 \hat{z}} component toward Failed to parse (MathML with SVG or PNG fallback (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_z=1} .

Physically, what is the meaning 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_x} , Failed to parse (MathML with SVG or PNG fallback (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_y} , 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_z} ? Failed to parse (MathML with SVG or PNG fallback (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_z} is manifestly the population difference between the excited and ground states. The other two components may be interpreted by recognizing that the average dipole moment of the atom 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 \begin{array}{rcl} \langle d \rangle &=& d_{ge} {\rm Tr}(\rho(|e \rangle \langle g|+|g \rangle \langle e|)) \\ &=& d_{ge} (\rho_{ge} e^{i\omega t} + \rho_{eg} e^{-i\omega t}) \\ &=& d_{ge} (r_x \cos\omega t + r_y \sin\omega t) \,. \end{array}}

Thus, Failed to parse (MathML with SVG or PNG fallback (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_x} 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_y} correspond to the phase components of the atomic dipole moment which are in-phase and in quadrature with the incident electromagnetic field.

Transient repsonse of the optical Bloch equations

The optical Bloch equations allow us to study the internal state of the atom as it changes due to the external driving field, and due to spontaneous emission. Starting from the time-independent form of the equations,

Failed to parse (MathML with SVG or PNG fallback (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{array}{rcl} \dot{r}_x &=& \delta\, r_y - \frac{\Gamma}{2} r_x \\ \dot{r}_y &=& -\delta\, r_x - g\, r_z - \frac{\Gamma}{2} r_y \\ \dot{r}_z &=& g\, r_y - \Gamma (r_z -1) \,, \end{array}}

we may note that when $g\rightarrow 0$ and at resonance, Failed to parse (MathML with SVG or PNG fallback (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=0} , the Bloch vector exhibits pure damping behavior, towards Failed to parse (MathML with SVG or PNG fallback (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_z=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_x=r_y=0} .

When $g\gg \Gamma$, Rabi oscillations occur, represnted by rapid rotations of the Bloch vector about Failed to parse (MathML with SVG or PNG fallback (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{x}} . Since the relaxation along Failed to parse (MathML with SVG or PNG fallback (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{y}} occurs at 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 \Gamma/2} , and the relaxation about ${\hat {z}}$ occurs at 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 \Gamma} , we might expect that the average relaxation rate of the rotating components under such a strong driving field would be Failed to parse (MathML with SVG or PNG fallback (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+\Gamma/2)/2 = 3\Gamma/4} . The remaining component Failed to parse (MathML with SVG or PNG fallback (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_x} does not rotate, because it sits along Failed to parse (MathML with SVG or PNG fallback (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{x}} , the axis of rotation. Thus, it relaxes with rate $\Gamma /2$. Computation of the eigenvalues of the equations of motion verify this qualitative picture, and show that 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 g\gg \Gamma} , 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 \delta=0} , the eigenvalues of motion are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm ig + 3\Gamma/4} 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 \Gamma/2} .

These correspond to a main peak at $\omega _{0}$ with 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 \Gamma/2} , and two sidebands 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_0\pm \Omega} , with widths Failed to parse (MathML with SVG or PNG fallback (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\Gamma/4} , thus explaining the widths of the observed Mollow triplet lines.

Steady-state solution of the optical Bloch equations

The steady state solution of the optical Bloch equations are found by setting all the time derivatives to zero, giving a set of three simultaneous equations,

Failed to parse (MathML with SVG or PNG fallback (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{array}{rcl} 0 &=& \delta\, r_y - \frac{\Gamma}{2} r_x \\ 0 &=& -\delta\, r_x - g\, r_z - \frac{\Gamma}{2} r_y \\ 0 &=& g\, r_y - \Gamma (r_z -1) \,. \end{array}}

The solutions are (up to overall minus signs which can be absorbed into definitions):

${\begin{array}{rcl}r_{x}&=&(g\delta ){\frac {1}{\delta ^{2}+g^{2}/2+(\Gamma /2)^{2}}}\\r_{y}&=&\left({\frac {g\Gamma }{2}}\right){\frac {1}{\delta ^{2}+g^{2}/2+(\Gamma /2)^{2}}}\\r_{z}&=&\left(\delta ^{2}+{\frac {\Gamma ^{2}}{4}}\right){\frac {1}{\delta ^{2}+g^{2}/2+(\Gamma /2)^{2}}}\end{array}}$

Physically, these are Lorentzians; 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 r_y} solution (the component in quadrature with the dipole) corresponds to an absorption curve with half-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 \sqrt{\frac{\Gamma^2}{4}+\frac{g^2}{2}} \,, }

and 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 r_x} solution (the component in-phase with the dipole) corresponds to a dispersion curve. And under a strong driving field, as $g\rightarrow \infty$, Failed to parse (MathML with SVG or PNG fallback (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_z\rightarrow 0} , indicating that the populations in the excited and ground states are equalizing. The steady-state population in the excited 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 \rho_{ee} = \frac{1-r_z}{2} = \frac{g^2/4}{\delta^2+g^2/2+\Gamma^2/4} \,, }

an important result that will later be used in studying light forces.

These solutions can be re-expressed in a simplified manner by defining 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 = \frac{g^2/2}{\delta^2+\Gamma^2/4} \,, }

in terms of which 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 \begin{array}{rcl} r_x &=& \frac{2\delta}{g} \frac{s}{1+s} \\ r_y &=& \frac{\Gamma}{g} \frac{s}{1+s} \\ r_z &=& \frac{1}{1+s} \,. \end{array}}

As $s\rightarrow \infty$, the atomic transitions become saturated, and the linewidth of the transition broadens from its natural value $\Gamma$, becoming $\Gamma '=\Gamma {\sqrt {1+s}}$ on resonance, at $\delta =0$.

Damped vacuum Rabi oscillations

The optical Bloch equations can be generalized to describe not just an atom interacting with the vacuum, but also an atom and a single cavity mode, each interacting with its own reservoir. This is the master equation for cavity QED, and using such a master equation we can revisit the phenomenon of vacuum Rabi oscillations and see what happens in the presence of damping.

Generalization of the optical Bloch equations

The starting point for generalizing the optical Bloch equations is the Lindblad form we previously saw in the full-derivation walkthrough,

${\dot {\rho }}_{A}=-{\frac {i}{\hbar }}[H,\rho _{A}]-{\frac {\Gamma }{2}}\left[{\sigma _{+}\sigma _{-}\rho _{A}-2\sigma _{-}\rho _{A}\sigma _{+}+\rho _{A}\sigma _{+}\sigma _{-}}\right]\,.$

In this expression, $\sigma _{+}$ 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 \sigma_-} are "jump operators," and represent changes that occur to the atom when distinct dissipative events happen ("distinct" meaning that the environment changes between orthogonal states).

We may write the master equation for our more general scenario by replacing the atomic density matrix Failed to parse (MathML with SVG or PNG fallback (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_A} by a general density matrix Failed to parse (MathML with SVG or PNG fallback (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} representing the atom and cavity field, and by replacing the atomic jump operators with general jump 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 L_k} ,

${\dot {\rho }}=-{\frac {i}{\hbar }}[H,\rho ]+\sum _{k}\left[{L_{k}\rho L_{k}^{\dagger }-{\frac {1}{2}}(L_{k}^{\dagger }L_{k}\rho +\rho L_{k}^{\dagger }L_{k})}\right]\,.$

Note that 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 L_k} include normalization factors which reflect their probabilities of occurrence. In other words, for the atom + vacuum model, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \sqrt{\Gamma} \sigma_-} .

Master equation for cavity QED system

For the cavity QED model, the atom and cavity field each have possible jump operators. In general, the atom and cavity may both couple to a thermal field 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 n_{th}} average photons. In such a case, the jump operators are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_1 = \sqrt{\Gamma(n_{th}+1)}\sigma_-} , $L_{3}={\sqrt {\Gamma n_{th}}}\sigma _{+}$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2 = \sqrt{\kappa(n_{th}+1)} a} , 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 L_4 = \sqrt{\kappa n_{th}} a^\dagger } , 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 \Gamma} parameterizes the spontaneous emission rate of the atom in free space, 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 \kappa} is parameterizes the cavity $Q$ factor.

Experimentally, typically the environment, the vacuum, is essentially at zero temperature, so Failed to parse (MathML with SVG or PNG fallback (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_{th}=0} , in which case the only two relevant jump operators are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_1 = \sqrt{\Gamma}\sigma_-} 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 L_2 = \sqrt{\kappa}a} .

Damped vacuum Rabi oscillations

Vacuum Rabi oscillations, in the absence of damping, involve only two states of the atom and cavity: Failed to parse (MathML with SVG or PNG fallback (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,0{\rangle}} and $|g,1{\rangle }$. When damping is added, 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 |g,0{\rangle}} state must be included, since both the atom and cavity states can decay and loose their quanta of energy. Moreover, because only one quantum of excitation is involved in this system, we can observe the essential physics by considering the case 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 \Gamma} is zero (no spontaneous emission), but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} is nonzero (the cavity is leaky). Let $\Omega _{0}$ be the vacuum Rabi frequency, and denote this three state space 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 |1{\rangle}=|e,0{\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 |2{\rangle}=|g,1{\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 |3{\rangle}=|g,0{\rangle}} . Written out explicitly in terms 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 3\times3} density matrix elements $\rho _{jk}$, the master equation is (for the case Failed to parse (MathML with SVG or PNG fallback (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=0} with the system starting in the 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 |1\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 \begin{array}{rcl} \dot{\rho}_{11} &=& i\frac{\Omega_0}{2}(\rho_{12} - \rho_{21}) \\ \dot{\rho}_{22} &=& -i\frac{\Omega_0}{2}(\rho_{12} - \rho_{21}) -\kappa \rho_{22} \\ \dot{\rho}_{12} &=& i\frac{\Omega_0}{2}(\rho_{11} - \rho_{22}) -\frac{\kappa}{2} \rho_{12} \\ \dot{\rho}_{21} &=& -i\frac{\Omega_0}{2}(\rho_{11} - \rho_{22}) -\frac{\kappa}{2} \rho_{21} \\ \dot{\rho}_{33} &=& \kappa \rho_{22} \,. \end{array}}

When the cavity damping rate is small, $\kappa \ll \Omega _{0}$, then the vacuum Rabi oscillations are damped, with average damping 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 \kappa/2} since the atoms spend roughly half their time in the excited state.

When the cavity damping rate is large, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa\gg\Omega_0} , then the atomic excitation is irreversibly damped, and no oscillations occur. 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 P_e} be the probability of being in 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 |1 \rangle = |e,0{\rangle}} state. We can combine two of the above equations to write

${\begin{array}{rcl}{\dot {\rho }}_{12}-{\dot {\rho }}_{21}&=&i\Omega _{0}(\rho _{11}-\rho _{22})-{\frac {\kappa }{2}}(\rho _{12}-\rho _{21})\,.\end{array}}$

Since in this case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\rho}_{12} + \dot{\rho}_{21}\approx 0} , 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 \rho_{22}\approx 0} , it follows 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 (\rho_{12}-\rho_{21}) \approx \frac{2i\Omega_0}{\kappa}(\rho_{11}-\rho_{22}) \,, }

Since in this regime, we slowly try to populate state $|2{\rangle }$ while it quickly decays away, we can approximate Failed to parse (MathML with SVG or PNG fallback (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_{22}\ll\rho_{11}} giving

Failed to parse (MathML with SVG or PNG fallback (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_{12}-\rho_{21}) \approx \frac{2i\Omega_0}{\kappa}\rho_{11} \,, }

Finally plugging this back into the optical bloch equation for $\rho _{11}$ yields

${\dot {\rho }}_{11}=-{\frac {\Omega _{0}^{2}}{\kappa }}\rho _{11}\,,$

so $P_{e}$ decays exponentially, with rate $\Gamma _{c}=\Omega _{0}^{2}/\kappa$. Note that a large cavity damping rate slows down the decay of the atom's excited state. This is due to the fact that the stimulated transition (Rabi oscillation) to the lower state is slowed down by the cavity damping (which is a manifestation of the quantum Zeno effect).

Purcell factor: cavity enhanced spontaneous emission

How does this compare with the free space spontaneous emission rate $\Gamma$? Recall that the vacuum Rabi frequency is (for the case that the dipole matrix element vector and cavity mode polarization are aligned)

Failed to parse (MathML with SVG or PNG fallback (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 = \frac{d E_0}{\hbar} \,, }

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 d} is the atomic dipole moment 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 E_0} is the electric field amplitude of a single photon at the atomic transition 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} (Note that this is a standing wave in a cavity so it is a factor of two larger than the field for a running wave),

$E_{0}={\sqrt {\frac {2\hbar \omega _{0}}{\epsilon _{0}V}}}\,,$

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 V} is the cavity volume. This gives

Failed to parse (MathML with SVG or PNG fallback (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 = d \sqrt{\frac{2\omega_0}{\epsilon_0 \hbar V}} \,. }

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 Q=\omega_0/\kappa} , we thus find 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 \Gamma_c = \frac{2d^2}{\epsilon_0\hbar} \frac{Q}{V} \, }

as the decay rate of the atom in the cavity.

Recall that the spontaneous emission rate of an atom in free space, as determined by Fermi's golden rule, is

$\Gamma ={\frac {d^{2}\omega _{0}^{3}}{3\pi \epsilon _{0}\hbar c^{3}}}$

The ratio of this rate to the decay rate in the cavity 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 \eta = \frac{\Gamma_c}{\Gamma} = \frac{3}{4\pi^2} \frac{Q \lambda^3}{V} \,, }

where we 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 \lambda = 2\pi c/\omega_0} as being the wavelength of the cavity field, which is assumed to be resonant with the atomic transition 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} . Note that $\eta$ is independent of the atomic dipole strength, and determined solely by cavity parameters. Moreover, note that for small, 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} cavities, the decay rate of the atom in the cavity can be much larger than the free space spontaneous emission rate. This "cavity enhanced" spontaneous emission rate was predicted by Purcell (1946), an observation credited as being the starting point of cavity QED.