Maxwell-Bloch Equations

The Maxwell-Bloch equations describe the dynamics of a two-state quantum system interacting with the electromagnetic mode of an optical resonator. The equations can be derived either semiclassically, or with the field fully quantized under certain approximations.

Derivation from Cavity Quantum Electrodynamics

Beginning with the Jaynes-Cummings Hamiltonian under coherent drive

$H=\omega_c a^\dagger a + \omega_a \sigma^\dagger\sigma+ig(a^\dagger\sigma-a\sigma^\dagger)+iJ(a^\dagger e^{-i\omega_l t}-a e^{i\omega_l t})$

where a is the lowering operator for the cavity field, and $\sigma=\frac{1}{2}\left(\sigma_x - i\sigma_y\right)$ is the atomic lowering operator written as a combination of Pauli matrices. The time dependence can be removed by transforming the wavefunction according to $|\psi\rangle\rightarrow \operatorname{e}^{-i\omega_l t\left(a^\dagger a + \sigma^\dagger\sigma\right)}|\psi\rangle$, leading to a transformed Hamiltonian

$H=\Delta_c a^\dagger a + \Delta_a \sigma^\dagger\sigma+ig(a^\dagger\sigma-a\sigma^\dagger)+iJ( a^\dagger-a)$

where Δ_i = ω_i − ω_l. As it stands now, the Hamiltonian has four terms. The first two are the self energy of the atom (or other two level system) and field. The third term is an energy conserving interaction term allowing the cavity and atom to exchange population and coherence. These three terms alone give rise to the Jaynes-Cummings ladder of dressed states, and the associated anharmonicity in the energy spectrum. The last term models coupling between the cavity mode and a classical field, ie a laser. The drive strength J is given in terms of the power transmitted through the empty two-sided cavity as $J=\sqrt{2P(\Delta_c^2 + \kappa^2)/(\hbar\omega_c \kappa)}$, where 2κ is the cavity linewidth. This brings to light a crucial point concerning the role of dissipation in the operation of a laser or other cqed device; dissipation is the means by which the system (coupled atom/cavity) interacts with its environment. To this end, dissipation is included by framing the problem in terms of the master equation, where the last two terms are in the Lindblad form

$\dot{\rho}=-i[H,\rho] + 2\kappa\left(a\rho a^\dagger -\frac{1}{2}\left(a^\dagger a \rho + \rho a^\dagger a\right)\right) + 2\gamma\left(\sigma\rho \sigma^\dagger -\frac{1}{2}\left(\sigma^\dagger \sigma\rho + \rho \sigma^\dagger \sigma\right)\right)$

The equations of motion for the expectation values of the operators can be derived from the master equation by the formula $\langle\dot{O}\rangle = \operatorname{tr}\left(O\rho\right)$. The equations of motion for $\langle a\rangle$, $\langle\sigma\rangle$, and $\langle\sigma_z\rangle$, the cavity field, atomic ground state population, and atomic inversion respectively, are

$\frac{d}{dt}\langle a \rangle = i\left(-\Delta_c \langle a \rangle - ig\langle \sigma\rangle - iJ\right) -\kappa \langle a \rangle$

$\frac{d}{dt}\langle \sigma \rangle = i\left(-\Delta_a \langle \sigma \rangle - ig\langle a \sigma_z \rangle\right) -\gamma \langle \sigma \rangle$

$\frac{d}{dt}\langle \sigma_z \rangle = -2g\left(\langle a^\dagger \sigma \rangle+\langle a \sigma^\dagger \rangle\right) -2\gamma \langle \sigma_z\rangle-2\gamma$

At this point, we have produced three of an infinite ladder of coupled equations. As can be seen from the third equation, higher order correlations are necessary. The differential equation for the time evolution of $\langle a^\dagger \sigma \rangle$ will contain expectation values of higher order products of operators, thus leading to an infinite set of coupled equations. We heuristically make the approximation that the expectation value of a product of operators is equal to the product of expectation values of the individual operators. This is akin to assuming that the operators are uncorrelated, and is a good approximation in the classical limit. It turns out that the resulting equations give the correct qualitative behavior even in the single excitation regime. Additionally, to simplify the equations we make the following replacements

$\langle a \rangle = (\gamma/\sqrt{2} g)x$

$\langle \sigma \rangle = -p/\sqrt{2}$

$\langle \sigma_z\rangle = -D$

Θ = Δ_c / κ

C = g² / 2κγ

$y = \sqrt{2} g J/\kappa\gamma$

Δ = Δ_a / γ

And the Maxwell-Bloch equations can be written in their final form

$\dot{x}=\kappa\left(-2Cp+y-(i\Theta+1)x\right)$

$\dot{p} = \gamma\left( -(1+i\Delta)p + xD\right)$

$\dot{D}=\gamma\left(2(1-D)-(x^*p+xp^*)\right)$