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

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

where the Hamiltonian is

$H={\frac {\hbar \omega _{0}}{2}}(|e\rangle \langle e|-|g\rangle \langle g|)+{\frac {\hbar \Omega }{2}}(|g\rangle \langle e|+|e\rangle \langle g|)\,.$

Useful limits to this equation of motion can be obtained, for example, by solving for $\rho$ 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 $H_{I}$, 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 $\sigma _{-}=|g\rangle \langle e|$, we may write this interaction 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_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

${\begin{array}{rcl}\sigma _{+}a^{\dagger }&\rightarrow &e^{i(\omega +\omega _{0})t}\sigma _{+}a^{\dagger }\\\sigma _{-}a&\rightarrow &e^{-i(\omega +\omega _{0})t}\sigma _{-}a\\\sigma _{+}a&\rightarrow &e^{i(\omega _{0}-\omega )t}\sigma _{+}a\\\sigma _{-}a^{\dagger }&\rightarrow &e^{-i(\omega _{0}-\omega _{0})t}\sigma _{-}a^{\dagger }\,.\end{array}}$

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 Failed to parse (MathML with SVG or PNG fallback (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} , 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:

\noindent 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $|\pm ,n{\rangle }$; 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 Failed to parse (MathML with SVG or PNG fallback (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\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:

\noindent 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.

Rotating frame of reference for atom + field

A simplification worth using in the study of the optical Bloch equations is a transformation into the rotating frame of the light field. The Hamiltonian for the atom + classical field may be written in general 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 = \frac{\omega_0}{2} Z + g (X\cos\omega t + Y \sin\omega t) \,, }

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 g} parameterizes the strength 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 \omega_0} is the atomic transition frequency, and $X,Y,Z$ are the Pauli matrices as usual. Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi(t) \rangle = e^{i\omega t Z/2} |\chi(t){\rangle}} , such that the Schr\"odinger 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 i \partial_t |\chi(t) \rangle = H |\chi(t){\rangle} }

can be re-expressed as

$i\partial _{t}|\phi (t){\rangle }=\left[{e^{i\omega Zt/2}He^{-i\omega Zt/2}-{\frac {\omega }{2}}Z}\right]|\phi (t){\rangle }\,.$

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 e^{i\omega Z t/2} X e^{-i\omega Z t/2} = (X\cos\omega t - Y \sin\omega t) \,, }

Eq.(\ref{eq:nmr:schrB}) simplifies to become

Failed to parse (MathML with SVG or PNG fallback (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 \partial_t |\phi(t){\rangle} = \left[ { \frac{\omega_0 - \omega}{2} Z + g X } \right] |\phi(t){\rangle} \,, }

where the terms on the right multiplying the state can be identified as the effective `rotating frame' Hamiltonian. The solution to this equation 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 |\phi(t) \rangle = e^{i \left[ { \frac{\omega_0 - \omega}{2} Z + g X } \right] t} |\phi(0){\rangle} \,. }

The concept of resonance arises from the behavior of this time evolution, which can be understood as being a single qubit rotation about the axis

Failed to parse (MathML with SVG or PNG fallback (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 n = \frac{\hat z + \frac{2g}{\omega_0 - \omega} \,\hat x} {\sqrt{1+ \left( {\frac{2g}{\omega_0-\omega}} \right) ^2}} }

by an angle

$|{\vec {n}}|=t{\sqrt {\left({\frac {\omega _{0}-\omega }{2}}\right)^{2}+g^{2}}}\,.$

When $\omega$ is far from $\omega _{0}$, the qubit is negligibly affected by the laser field; the axis of its rotation is nearly parallel with ${\hat {z}}$, and its time evolution is nearly exactly that of the free atom Hamiltonian. On the other hand, 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_0\approx \omega} , the free atom contribution becomes negligible, and a small laser field can cause large changes in the state, corresponding to rotations about 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} axis. The enormous effect a small field can have on the atom, when tuned to the appropriate frequency, is responsible for the concept of atomic `resonance,' as well as nuclear magnetic resonance.

Bloch vector evolution

We have previously seen that an arbitrary qubit state $|\psi \rangle =\cos \theta |0\rangle +e^{i\phi }\sin \phi |1{\rangle }$ 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

$\rho ={\frac {I+{\vec {r}}\cdot {\sigma }}{2}}\,,$

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 $\rho$. 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

Failed to parse (MathML with SVG or PNG fallback (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 &=& \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 ${\vec {r}}$. 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} )

Failed to parse (MathML with SVG or PNG fallback (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{\delta}{2} Z + g X \,. }

This gives us the equations of motion

${\begin{array}{rcl}{\dot {r}}_{x}&=&\delta \,r_{y}-{\frac {\Gamma }{2}}r_{x}\\{\dot {r}}_{y}&=&-\delta \,r_{x}-g\,r_{z}-{\frac {\Gamma }{2}}r_{y}\\{\dot {r}}_{z}&=&g\,r_{y}-\Gamma (r_{z}-1)\,.\end{array}}$

Note how these equations of motion provide a simple set of flows on the Bloch sphere: the Failed to parse (MathML with SVG or PNG fallback (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} 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, $g$ corresponds to a rotation in the ${\hat {y}}-{\hat {z}}$ 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 ${\hat {y}}$ 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 $r_{z}=1$. 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} , $r_{y}$, 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

${\begin{array}{rcl}\langle d\rangle &=&d_{ge}{\rm {Tr}}(\rho (|e\rangle \langle g|+|g\rangle \langle e|))\\&=&d_{ge}(\rho _{ge}e^{i\omega t}+\rho _{eg}e^{-i\omega t})\\&=&d_{ge}(r_{x}\cos \omega t+r_{y}\sin \omega t)\,.\end{array}}$

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,

${\begin{array}{rcl}{\dot {r}}_{x}&=&\delta \,r_{y}-{\frac {\Gamma }{2}}r_{x}\\{\dot {r}}_{y}&=&-\delta \,r_{x}-g\,r_{z}-{\frac {\Gamma }{2}}r_{y}\\{\dot {r}}_{z}&=&g\,r_{y}-\Gamma (r_{z}-1)\,,\end{array}}$

we may note that when Failed to parse (MathML with SVG or PNG fallback (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\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 $r_{z}=1$, 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 Failed to parse (MathML with SVG or PNG fallback (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} , 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 ${\hat {y}}$ 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 Failed to parse (MathML with SVG or PNG fallback (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}} 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 $r_{x}$ 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 Failed to parse (MathML with SVG or PNG fallback (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} . Computation of the eigenvalues of the equations of motion verify this qualitative picture, and show that for $g\gg \Gamma$, 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 $3\Gamma /4$, 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):

Failed to parse (MathML with SVG or PNG fallback (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 &=& (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 $r_{x}$ solution (the component in-phase with the dipole) corresponds to a dispersion curve. And under a strong driving 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 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

$\rho _{ee}={\frac {1-r_{z}}{2}}={\frac {g^{2}/4}{\delta ^{2}+g^{2}/2+\Gamma ^{2}/4}}\,,$

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 {\em saturated}, and the linewidth of the transition broadens from its natural value Failed to parse (MathML with SVG or PNG fallback (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} , becoming Failed to parse (MathML with SVG or PNG fallback (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\sqrt{1+s}} on resonance, at $\delta =0$.