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.

Helpful background. Lindblad operators describe loss channels, and thermal master equations explain occupation factors. Transition rates in light–matter interaction derives the free-space dipole rate; Spontaneous emission develops the atom–field state, emitted mode, and lifetime–linewidth relation.

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.

Electromagnetic reservoirs and the reduced equation

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, in SI units for an isolated electric-dipole transition in three-dimensional free space. For multiple allowed final states, sum their partial rates. This formula assumes the dipole and weak-coupling approximations; it is not a universal rate for every electromagnetic environment. 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 a real Rabi frequency after choosing the drive phase. The rotating-wave approximation requires the drive and detuning scales to be small compared with the carrier frequency; the Markov approximation additionally requires a broad reservoir over the frequencies sampled by the driven system.

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,

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

For a state with finite occupation moment, 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 5 GHz the energy scale is $hf/k_B\simeq0.240$ K: a 300 K bath has about $1.25\times10^3$ thermal photons per mode, whereas a 20 mK equilibrium bath has $\bar n\simeq6.2\times10^{-6}$. The temperature of the relevant input field, including poorly thermalized lines, matters more than the nominal refrigerator temperature.

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

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

This commonly used local-loss model assumes approximately flat independent reservoir couplings over the relevant dressed transitions. Strong internal coupling does not itself invalidate a Markov description if the cavity is retained, but dressed-state and thermal consistency may require the global construction. In the ultrastrong regime the Jaynes–Cummings interaction and bare loss operators require further revision.

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 full matrix $\Gamma$ must be Hermitian positive semidefinite for this dissipator to be GKSL. Reservoir-induced coherent exchange also belongs in the Hamiltonian. 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

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

For a positive-probability, ideally resolved 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.

Cavity amplitude, energy, and linewidth: worked example

The adjoint master equation gives

$$\frac{d\langle a\rangle}{dt}=-(i\omega_c+\kappa/2)\langle a\rangle, \qquad \frac{d\langle n\rangle}{dt}=-\kappa\langle n\rangle$$

at zero temperature. Thus field amplitude and energy do not have the same ringdown time. With $\kappa/(2\pi)=1$ MHz, the energy lifetime is $1/\kappa\simeq159$ ns and the amplitude lifetime is $2/\kappa\simeq318$ ns.

For a weak coherent input at detuning $\Delta=\omega_L-\omega_c$, the steady cavity intensity is proportional to

$$\frac{1}{(\kappa/2)^2+\Delta^2}.$$

The half-maximum detunings are $\pm\kappa/2$. Its full width is therefore $\kappa$ in angular frequency, or 1 MHz in ordinary frequency for the example. These are the conventions to translate when a source writes a dissipator $2\kappa\mathcal D[a]$.

A near-resonant narrow mode should be retained as part of the system when its dynamics is resolved. Tracing it out as a broad reservoir as well would double count it. Band edges, long delays, or structured modes left outside the system can require a memory equation even when this local-loss description works for a larger retained system.

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 $\beta$, compare a reverse pair of adjacent-level transitions and show that the upward/downward rate ratio 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 $J=\sqrt{\Gamma}\sigma_-$ and ideal unit-efficiency detection of this channel, 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.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002). Publisher.
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley (1992).
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004). Publisher.
• S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons, Oxford University Press (2006).
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010). DOI.