The Maxwell–Bloch equations, also called the optical Bloch equations^[1] describe the dynamics of a two-state quantum system interacting with the electromagnetic mode of an optical resonator. They are analogous to (but not at all equivalent to) the Bloch equations which describe the motion of the nuclear magnetic moment in an electromagnetic field. The equations can be derived either semiclassically or with the field fully quantized when certain approximations are made.

Semi-classical formulation

The derivation of the semi-classical optical Bloch equations is nearly identical to solving the two-state quantum system (see the discussion there). However, usually one casts these equations into a density matrix form. The system we are dealing with can be described by the wave function:

${\displaystyle \psi =c_{g}\psi _{g}+c_{e}\psi _{e}}$

${\displaystyle \left|c_{g}\right|^{2}+\left|c_{e}\right|^{2}=1}$

The density matrix is

${\displaystyle \rho ={\begin{bmatrix}\rho _{ee}&\rho _{eg}\\\rho _{ge}&\rho _{gg}\end{bmatrix}}={\begin{bmatrix}c_{e}c_{e}^{*}&c_{e}c_{g}^{*}\\c_{g}c_{e}^{*}&c_{g}c_{g}^{*}\end{bmatrix}}}$

(other conventions are possible; this follows the derivation in Metcalf (1999)).^[2] One can now solve the Heisenberg equation of motion, or translate the results from solving the Schrödinger equation into density matrix form. One arrives at the following equations, including spontaneous emission:

${\displaystyle {\frac {d\rho _{gg}}{dt}}=\gamma \rho _{ee}+{\frac {i}{2}}(\Omega ^{*}{\bar {\rho }}_{eg}-\Omega {\bar {\rho }}_{ge})}$

${\displaystyle {\frac {d\rho _{ee}}{dt}}=-\gamma \rho _{ee}+{\frac {i}{2}}(\Omega {\bar {\rho }}_{ge}-\Omega ^{*}{\bar {\rho }}_{eg})}$

${\displaystyle {\frac {d{\bar {\rho }}_{ge}}{dt}}=-\left({\frac {\gamma }{2}}+i\delta \right){\bar {\rho }}_{ge}+{\frac {i}{2}}\Omega ^{*}(\rho _{ee}-\rho _{gg})}$

${\displaystyle {\frac {d{\bar {\rho }}_{eg}}{dt}}=-\left({\frac {\gamma }{2}}-i\delta \right){\bar {\rho }}_{eg}+{\frac {i}{2}}\Omega (\rho _{gg}-\rho _{ee})}$

In the derivation of these formulae, we define ${\displaystyle {\bar {\rho }}_{ge}\equiv \rho _{ge}e^{-i\delta t}}$ and ${\displaystyle {\bar {\rho }}_{eg}\equiv \rho _{eg}e^{i\delta t}}$. It was also explicitly assumed that spontaneous emission is described by an exponential decay of the coefficient ${\displaystyle \rho _{eg}(t)}$ with decay constant ${\displaystyle {\frac {\gamma }{2}}}$. ${\displaystyle \Omega }$ is the Rabi frequency, which is

${\displaystyle \Omega ={\vec {d_{g,e}}}\cdot {\vec {E_{0}}}/\hbar }$,

and ${\displaystyle \delta =\omega -\omega _{0}}$ is the detuning and measures how far the light frequency, ${\displaystyle \omega }$, is from the transition, ${\displaystyle \omega _{0}}$. Here, ${\displaystyle {\vec {d}}_{g,e}}$ is the transition dipole moment for the ${\displaystyle g\rightarrow e}$Zdroj:https://en.wikipedia.org?pojem=Maxwell–Bloch_equations
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