Quantum Optical Master Equation

The quantum optical master equation is the standard Markovian master equation for atoms, cavities, and light-matter systems coupled to broadband electromagnetic reservoirs. It is the workhorse behind spontaneous emission, cavity loss, driven two-level atoms, cavity QED damping, resonance fluorescence, and many photon-counting models.

The dipole matrix elements, $N$ versus $N+1$ occupation factors, photon mode sum, and free-space golden-rule rate are derived at Transition Rates in Light–Matter Interaction. This page owns the reduced Markovian dynamics that uses those rates.

Spontaneous Emission owns the complementary atom–field state, Wigner–Weisskopf memory equation, dipole pattern, lifetime and linewidth conventions, emitted photon mode, and Purcell-effect preview.

In its simplest zero-temperature two-level form,

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \Gamma\mathcal D[\sigma_-]\rho,$$

where

$$\sigma_-=\lvert g\rangle\langle e\rvert, \qquad \mathcal D[L]\rho = L\rho L^\dagger - \frac12\{L^\dagger L,\rho\}.$$

The same framework also describes cavity damping with $L=a$, thermal photons with both $a$ and $a^\dagger$, and composite atom-cavity systems with several loss channels.

Scope

The phrase “quantum optical master equation” usually refers to a Lindblad–GKSL equation obtained after:

• quantizing the relevant atomic, molecular, or cavity degrees of freedom;
• coupling them to electromagnetic reservoir modes;
• using a dipole or rotating-wave interaction where appropriate;
• assuming a broadband reservoir with short correlation time;
• applying Born, Markov, and secular approximations;
• tracing over the unobserved radiation modes.

The resulting equation is an unconditional equation for the system density operator. If emitted or transmitted photons are actually monitored, the appropriate conditioned description is a quantum trajectory or filtering equation. The unconditional master equation is recovered by averaging over records.

Spontaneous Emission

For a two-level atom with transition frequency $\omega_0$, a common system Hamiltonian is

$$H_0 = \hbar\omega_0\sigma_+\sigma_-,$$

where

$$\sigma_+=\lvert e\rangle\langle g\rvert, \qquad \sigma_-=\lvert g\rangle\langle e\rvert.$$

In free space at zero temperature, the quantum optical master equation is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H_0+H_{\mathrm{LS}},\rho] + \Gamma\mathcal D[\sigma_-]\rho.$$

The Lamb-shift Hamiltonian $H_{\mathrm{LS}}$ shifts the transition frequency. The dissipator describes irreversible emission into modes that are not retained.

For an electric dipole transition in free space, a standard spontaneous-emission rate is

$$\Gamma = \frac{ \omega_0^3 \lvert \mathbf d_{eg}\rvert^2 }{ 3\pi\epsilon_0\hbar c^3 },$$

where $\mathbf d_{eg}$ is the transition dipole matrix element. Structured reservoirs, cavities, photonic crystals, waveguides, and boundaries modify this rate and may invalidate the broadband Markov approximation.

Population and Coherence Decay

For

$$\dot\rho = \Gamma\mathcal D[\sigma_-]\rho,$$

the excited-state population obeys

$$\dot\rho_{ee} = -\Gamma\rho_{ee}, \qquad \rho_{ee}(t) = e^{-\Gamma t}\rho_{ee}(0).$$

The optical coherence obeys

$$\dot\rho_{eg} = -\frac{\Gamma}{2}\rho_{eg}$$

in the interaction picture, before adding extra pure dephasing. Population decay therefore contributes $\Gamma/2$ to the transverse coherence decay rate.

This is the continuous-time generator behind the zero-temperature Amplitude-Damping Channel. See Amplitude Damping Master Equation for the general two-level generator and $T_1$ conventions.

Driven Two-Level Atom

In a frame rotating at a classical drive frequency $\omega_L$, the driven two-level Hamiltonian is often written

$$H_{\mathrm{rot}} = - \hbar\Delta\,\sigma_+\sigma_- + \frac{\hbar\Omega}{2} (\sigma_+ + \sigma_-),$$

where

$$\Delta=\omega_L-\omega_0$$

is the detuning and $\Omega$ is the Rabi frequency.

A common quantum optical master equation is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H_{\mathrm{rot}},\rho] + \Gamma\mathcal D[\sigma_-]\rho + \frac{\gamma_\phi}{2}\mathcal D[\sigma_z]\rho.$$

The $\Gamma$ term describes radiative decay. The $\gamma_\phi$ term describes additional pure dephasing, if present. This equation is the density-matrix starting point for Optical Bloch Equations.

Cavity Damping

For one cavity mode with Hamiltonian

$$H_c=\hbar\omega_c a^\dagger a,$$

coupling to external modes through imperfect mirrors gives photon loss. At zero temperature,

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H_c,\rho] + \kappa\mathcal D[a]\rho.$$

The mean photon number decays as

$$\frac{d}{dt}\langle a^\dagger a\rangle = -\kappa\langle a^\dagger a\rangle.$$

At finite temperature, the thermal version is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H_c,\rho] + \kappa(\bar n+1)\mathcal D[a]\rho + \kappa\bar n\,\mathcal D[a^\dagger]\rho,$$

where

$$\bar n = \frac{1} {e^{\beta\hbar\omega_c}-1}.$$

At optical frequencies and room temperature, $\bar n$ is usually negligible. At microwave frequencies, it may not be.

When the leaking field is retained as a traveling mode, the same damping channel is described by Quantum Langevin Equations and Input–Output Theory through relations such as $b_{\mathrm{out}}=b_{\mathrm{in}}+\sqrt{\kappa}a$. The finite-time channel distinction between flagged erasure and unflagged attenuation is summarized in Erasure and Loss Channels, while the covariance-matrix form of attenuation is collected in Gaussian Channels.

Atom-Cavity Systems

For a single two-level emitter coupled to a cavity mode, the Jaynes–Cummings Hamiltonian is

$$H_{\mathrm{JC}} = \hbar\omega_c a^\dagger a + \hbar\omega_a\sigma_+\sigma_- + \hbar g(a^\dagger\sigma_-+a\sigma_+).$$

See Jaynes–Cummings Model for the Hamiltonian reference entry.

A common dissipative cavity-QED master equation is

$$\begin{aligned} \frac{d\rho}{dt} =& - \frac{i}{\hbar}[H_{\mathrm{JC}},\rho] \\ &+ \kappa\mathcal D[a]\rho + \Gamma\mathcal D[\sigma_-]\rho + \frac{\gamma_\phi}{2}\mathcal D[\sigma_z]\rho. \end{aligned}$$

Here $\kappa$ is the cavity energy-decay rate, $\Gamma$ is the emitter’s radiative decay rate into modes outside the retained cavity mode, and $\gamma_\phi$ is additional pure dephasing. Which terms should be present depends on the device, temperature, and system-environment boundary.

Collective Emission

For several emitters coupled to a common electromagnetic reservoir, the dissipator can include cross decay:

$$\sum_{i,j} \Gamma_{ij} \left( \sigma_-^{(i)}\rho\sigma_+^{(j)} - \frac12 \{\sigma_+^{(j)}\sigma_-^{(i)},\rho\} \right).$$

The diagonal rates $\Gamma_{ii}$ describe individual spontaneous emission. The off-diagonal rates $\Gamma_{ij}$ encode reservoir-mediated collective decay and can produce superradiant and subradiant modes.

Replacing this collective dissipator by independent local terms

$$\sum_i\Gamma_i\mathcal D[\sigma_-^{(i)}]\rho$$

is an additional approximation. It is not always valid when emitters are close compared with the relevant wavelength or share a structured reservoir.

Photon Counting Interpretation

The operator

$$J=\sqrt{\Gamma}\,\sigma_-$$

is a jump operator for an ideal photon-counting unraveling of spontaneous emission. In a short interval $dt$, the emission probability is

$$p_{\mathrm{jump}} = dt\, \operatorname{Tr} \left( J^\dagger J\rho \right) = \Gamma dt\,\rho_{ee}.$$

After a detected jump, the conditioned state updates as

$$\rho \longmapsto \frac{ \sigma_-\rho\sigma_+ }{ \operatorname{Tr}(\sigma_+\sigma_-\rho) }.$$

The same unconditional master equation can have other unravelings, such as homodyne or heterodyne detection of the emitted field. A Lindblad operator is not automatically an observed detector click without specifying the monitoring scheme.

Assumptions and Limits

The quantum optical master equation is powerful because electromagnetic reservoirs are often broad and weakly coupled. But it is not automatic.

Check the following:

• Is the reservoir correlation time short compared with system evolution?
• Is the rotating-wave approximation valid near the relevant transition?
• Is the spectral density smooth near the transition frequency?
• Are thermal photons negligible or included?
• Are cavity modes retained in the system rather than traced out?
• Are collective decay terms important?
• Is a structured reservoir, band edge, waveguide delay, or strong coupling producing memory?
• Are the reported rates population-decay rates, field-amplitude linewidths, or full-width linewidths?

The same symbol $\kappa$ is used with different linewidth conventions in different communities. Always check whether $\langle a\rangle$ decays as $e^{-\kappa t/2}$ or $e^{-\kappa t}$ in the source’s convention.

Common Mistakes

Treating every photon loss as zero temperature

Optical reservoirs are often effectively at zero temperature, but microwave and mechanical reservoirs may have appreciable thermal occupation. Then the $a^\dagger$ or $\sigma_+$ terms cannot be ignored.

Double-counting a cavity mode

A near-resonant cavity mode should be included in the system Hamiltonian if it is dynamically resolved. Do not also trace it out as a featureless reservoir.

Confusing unconditional decay with detected photons

The master equation describes the state after ignoring the emitted field. Photon-counting trajectories require a specified detection model.

Dropping collective terms without checking geometry

Independent-emitter dissipators can miss superradiance, subradiance, and dipole-dipole reservoir effects.

Using Markovian optics near a structured reservoir

Photonic band edges, high-Q modes outside the retained system, long delay lines, and waveguide feedback can create non-Markovian dynamics.

Exercises

Spontaneous-emission population

For

$$\dot\rho = \Gamma\mathcal D[\sigma_-]\rho,$$

derive the equation for $\rho_{ee}$.

Solution

Use $\sigma_+\sigma_-=\lvert e\rangle\langle e\rvert$ and $\sigma_-\rho\sigma_+=\rho_{ee}\lvert g\rangle\langle g\rvert$. Then

$$\dot\rho_{ee} = \Gamma \langle e|\mathcal D[\sigma_-]\rho|e\rangle = -\Gamma\rho_{ee}.$$

Thus $\rho_{ee}(t)=e^{-\Gamma t}\rho_{ee}(0)$.

Cavity photon loss

For

$$\dot\rho = \kappa\mathcal D[a]\rho,$$

show that $\frac{d}{dt}\langle a^\dagger a\rangle=-\kappa\langle a^\dagger a\rangle$.

Solution

Use the adjoint dissipator:

$$\mathcal D^\dagger[a]N = a^\dagger N a - \frac12\{a^\dagger a,N\}, \qquad N=a^\dagger a.$$

Using $[N,a]=-a$, one finds

$$\mathcal D^\dagger[a]N=-N.$$

Therefore

$$\frac{d}{dt}\langle N\rangle = \kappa\langle \mathcal D^\dagger[a]N\rangle = -\kappa\langle N\rangle.$$

Thermal photon occupation

For a cavity mode at inverse temperature $\beta$, show that the ratio of upward to downward thermal damping rates is $e^{-\beta\hbar\omega_c}$.

Solution

The downward coefficient is $\kappa(\bar n+1)$ and the upward coefficient is $\kappa\bar n$, with

$$\bar n = \frac{1} {e^{\beta\hbar\omega_c}-1}.$$

Thus

$$\frac{\kappa\bar n} {\kappa(\bar n+1)} = \frac{\bar n}{\bar n+1} = e^{-\beta\hbar\omega_c}.$$

Jump probability

For $J=\sqrt{\Gamma}\sigma_-$, compute the probability of a detected spontaneous-emission jump in a short interval $dt$.

Solution

The probability is

$$p_{\mathrm{jump}} = dt\,\operatorname{Tr}(J^\dagger J\rho) = \Gamma dt\,\operatorname{Tr}(\sigma_+\sigma_-\rho).$$

Since $\sigma_+\sigma_-=\lvert e\rangle\langle e\rvert$,

$$p_{\mathrm{jump}} = \Gamma dt\,\rho_{ee}.$$

References

• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley (1992).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons, Oxford University Press (2006).