Cavity QED as an Open Quantum System

Cavity quantum electrodynamics studies atoms, ions, molecules, quantum dots, or other emitters coupled to a small number of quantized electromagnetic modes. For open quantum systems it is a near-ideal testbed: the cavity enhances selected light-matter interactions, the dominant loss channels often have measurable rates, and emitted fields can be monitored as photon clicks or continuous quadrature records.

A cavity provides both a coherent interaction and an output channel. The following examples show how to distinguish excitation exchange, decay into a selected port, and the conditional state inferred from that port. See Jaynes–Cummings Model for the closed-system spectrum and Cavity QED for mode geometry and coupling regimes.

Coupled Emitter and Cavity

The simplest single-emitter cavity-QED Hamiltonian is the Jaynes–Cummings Hamiltonian,

$$\frac{H_{\mathrm{JC}}}{\hbar} = \omega_c a^\dagger a + \frac{\omega_a}{2}\sigma_z + g \left( a^\dagger\sigma_- + a\sigma_+ \right).$$

Here $a$ annihilates a cavity photon, $\omega_c$ is the cavity frequency, $\omega_a$ is the atomic transition frequency, and $g$ is the single-photon coupling rate. The detuning is

$$\Delta = \omega_a-\omega_c .$$

The model assumes a near-resonant transition, a rotating-wave approximation, and a single relevant cavity mode. It is the right starting point when the coupling is weak compared with optical or microwave carrier frequencies but strong enough to matter relative to decay and detuning.

In a driven experiment one adds terms such as

$$\frac{H_{\mathrm{drive}}}{\hbar} = i \left( \mathcal E a^\dagger e^{-i\omega_d t} - \mathcal E^* a e^{i\omega_d t} \right),$$

or an equivalent rotating-frame drive. The drive is a work source, not a thermal reservoir.

Open-System Master Equation

For near-resonant coupling, low-temperature broad reservoirs, and weak reservoir coupling, a common Markovian cavity-QED model is

$$\dot\rho = - \frac{i}{\hbar}[H,\rho] + \kappa\,\mathcal D[a]\rho + \gamma\,\mathcal D[\sigma_-]\rho + \frac{\gamma_\phi}{2}\mathcal D[\sigma_z]\rho,$$

where

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac{1}{2} \{L^\dagger L,\rho\}.$$

The three displayed channels have direct physical meanings:

• $\kappa\,\mathcal D[a]$ describes cavity-field energy decay through mirrors, ports, or internal loss;
• $\gamma\,\mathcal D[\sigma_-]$ describes spontaneous emission into noncavity modes;
• $\gamma_\phi\,\mathcal D[\sigma_z]/2$ describes pure dephasing of the emitter, when relevant.

This local damping model is an approximation; dressed-state bath transitions may be needed when hybridization and reservoir frequency dependence are resolved. The emitter coherence linewidth in the uncoupled model is $\gamma/2+\gamma_\phi$. The collapse operators are part of the experimental model. A cavity photon emitted into a monitored output mode is not equivalent to an unobserved absorption channel, even if both reduce intracavity energy.

Cavity Decay and Input–Output Fields

For a single-sided cavity with coupling rate $\kappa$ to a transmission line or optical continuum, choose the input–output convention

$$a_{\mathrm{out}}(t) = a_{\mathrm{in}}(t) + \sqrt{\kappa}\,a(t),$$

with $a_{\mathrm{in/out}}$ in units of inverse square-root time and $a$ dimensionless. The matching Langevin input term is $-\sqrt{\kappa}\,a_{\mathrm{in}}$. This relation says that the measured outgoing field contains the incident field plus the field radiated by the cavity mode.

If the input is vacuum and the output is monitored ideally, the photon flux in that output channel is

$$\Phi_{\mathrm{out}} = \kappa \langle a^\dagger a\rangle .$$

Only the monitored part of $\kappa$ contributes to the detector record. If

$$\kappa = \kappa_{\mathrm{mon}} + \kappa_{\mathrm{loss}},$$

then the useful output flux is controlled by $\kappa_{\mathrm{mon}}$, while $\kappa_{\mathrm{loss}}$ is unobserved damping. This distinction is crucial in quantum trajectories and measurement efficiency.

See Input–Output Theory for the canonical open-system language.

Strong Coupling and Cooperativity

Cavity QED is often organized by comparing coherent coupling to loss. Strong coupling means that the atom and cavity exchange excitations faster than those excitations decay. A rough condition is

$$g \gt \kappa,\gamma,$$

with convention-dependent factors of $2$ in linewidth definitions.

A common dimensionless figure of merit is the cooperativity

$$C = \frac{4g^2} {\kappa\gamma}.$$

Some authors omit the factor of $4$ and define $C=g^2/(\kappa\gamma)$. In the resonant bad-cavity limit with negligible pure dephasing, $C=\Gamma_{\mathrm P}/\gamma$ compares cavity-mediated and free-space emission. The useful monitored branching fraction is approximately $[C/(1+C)]\kappa_{\mathrm{mon}}/\kappa$. Thus high cooperativity alone does not guarantee high collection efficiency or resolved coherent exchange.

When the atom and cavity are nearly resonant, the dressed one-excitation eigenfrequencies are split by approximately

$$2g$$

in the ideal lossless model. With loss included, observing a resolved vacuum Rabi splitting requires the splitting to be large compared with the relevant linewidths.

Purcell Effect Preview

The Purcell effect is the modification of spontaneous emission by the electromagnetic environment. In cavity QED, an atom can decay into a selected cavity mode at an enhanced rate.

In the weak-excitation, bad-cavity regime, the cavity mode can often be eliminated. On resonance, a common estimate for the cavity-enhanced decay rate is

$$\Gamma_{\mathrm P} \approx \frac{4g^2}{\kappa}.$$

With detuning,

$$\Gamma_{\mathrm P}(\Delta) \approx \frac{g^2\kappa} {\Delta^2+(\kappa/2)^2}.$$

These formulas assume that the cavity responds faster than the emitter: near resonance one needs $g\ll\kappa$ and emitter linewidths small compared with the cavity response scale. The output continuum is Markovian. Here $\Gamma_{\mathrm P}$ counts all cavity decay ports; multiply by $\kappa_{\mathrm{mon}}/\kappa$ for the collected part. They are estimates, not universal definitions. In structured reservoirs, multimode cavities, or strong-coupling regimes, the full dynamics must be modeled rather than reduced to a single rate.

The Purcell effect links cavity QED to Amplitude Damping: the decay channel is still amplitude damping, but the environment has been engineered so that the rate and emitted mode are controlled.

Quantum Jumps

When the cavity output is photon-counted, the conditional state evolves by no-click evolution interrupted by jumps. For a monitored cavity channel with operator

$$L_{\mathrm{mon}} = \sqrt{\kappa_{\mathrm{mon}}}\,a,$$

a detected photon updates the state as

$$\rho \mapsto \frac{ L_{\mathrm{mon}}\rho L_{\mathrm{mon}}^\dagger } { \operatorname{Tr} \left( L_{\mathrm{mon}}^\dagger L_{\mathrm{mon}}\rho \right) }.$$

Between detected clicks, unobserved emission must still be averaged over. If $\mathcal M$ denotes counted channels and $\mathcal U$ unobserved channels, the unnormalized no-click state obeys

$$\dot{\widetilde\rho} =-\frac{i}{\hbar}(H_{\mathrm{eff}}\widetilde\rho-\widetilde\rho H_{\mathrm{eff}}^\dagger) +\sum_{j\in\mathcal U}L_j\widetilde\rho L_j^\dagger, \qquad H_{\mathrm{eff}}=H-\frac{i\hbar}{2}\sum_{j\in\mathcal M\cup\mathcal U}L_j^\dagger L_j.$$

Only when all channels are monitored perfectly can the recycling sum be omitted. Normalize by $\operatorname{Tr}\widetilde\rho$ to condition on the observed silence. See Quantum-Jump Trajectories for detection inefficiency and waiting times.

Cavity-QED jump records can reveal antibunching, photon blockade, atomic state changes, or single-photon generation. They also make the link between measurement backaction and emitted radiation very concrete.

Homodyne and Heterodyne Measurement

Instead of counting photons, one can interfere the output field with a strong local oscillator and measure a quadrature. In homodyne detection, the measured current has the schematic form

$$dY_t = \sqrt{\eta\kappa_{\mathrm{mon}}} \, \langle e^{-i\phi}a+e^{i\phi}a^\dagger \rangle_t dt + dW_t,$$

after subtracting the known coherent input baseline, where $\eta$ is the detector efficiency for the monitored port, $\phi$ is the local-oscillator phase, and $dW_t$ is a Wiener noise increment.

The corresponding conditional state obeys a stochastic master equation. This is the diffusive counterpart of photon-counting trajectories. It is central in continuous cavity-field measurement, feedback, and weak measurement of atoms through transmitted or reflected light.

Heterodyne detection measures both quadratures at the price of added vacuum noise. It is often the natural description when the output field is analyzed as a complex amplitude rather than a single quadrature.

See Homodyne Detection, Diffusive Trajectories, and Stochastic Master Equations.

Dispersive Readout

When the atom and cavity are far detuned,

$$|\Delta| \gg g\sqrt{n+1},$$

direct excitation exchange is suppressed. To second order in $g/\Delta$, the interaction becomes approximately dispersive:

$$\frac{H_{\mathrm{disp}}}{\hbar} \approx \left( \omega_c+\chi\sigma_z \right) a^\dagger a + \frac{\omega_a+\chi}{2}\sigma_z, \qquad \chi \approx \frac{g^2}{\Delta}.$$

The atomic state shifts the cavity resonance, and the cavity photon number shifts the atomic transition. Measuring the phase or amplitude of the transmitted field can therefore measure the atomic state, while photons in the cavity can dephase the atom.

This dispersive logic is especially prominent in superconducting Circuit QED, but it also appears in optical and microwave cavity QED. The open-system lesson is general: information gain and measurement-induced dephasing are two sides of the same coupling to an output field.

Measuring Cavity Fields

Cavity QED also allows the field itself to be measured. Examples include:

• photon counting of cavity leakage;
• homodyne tomography of output quadratures;
• dispersive probing of photon number by atoms crossing the cavity;
• quantum nondemolition monitoring of selected observables;
• correlation measurements such as $g^{(2)}(\tau)$.

A field measurement is never just “looking inside the cavity.” It is a coupling to another system or continuum. The measured observable depends on the detection scheme, bandwidth, efficiency, and mode matching.

For example, an idealized photon-number quantum nondemolition measurement should commute with the measured number operator while still entangling the field with a meter. Real implementations must also control loss, finite interaction time, detector noise, and backaction on conjugate field variables.

Regime Map

Useful cavity-QED regimes include:

• bad-cavity or Purcell regime, where $\kappa$ is large and the cavity acts as an engineered decay channel;
• strong-coupling regime, where coherent atom-cavity exchange is visible;
• dispersive regime, where state-dependent frequency shifts enable measurement;
• photon-blockade regime, where nonlinearity suppresses multiple occupancy;
• many-emitter regime, where collective coupling scales like $\sqrt N g$ under suitable symmetry assumptions.

These regimes are not sharply separated by slogans. They are approximations controlled by inequalities among $g$, $\kappa$, $\gamma$, detuning, drive strength, temperature, and mode structure.

Exercises

Output Flux

For a one-sided cavity with monitored decay operator $L=\sqrt{\kappa}a$ and vacuum input, show that the detected photon flux is

$$\Phi_{\mathrm{out}} = \kappa\langle a^\dagger a\rangle .$$

Solution

In a photon-counting unraveling, the probability for a jump in a short interval $dt$ is

$$dp = \operatorname{Tr}(L^\dagger L\rho)\,dt.$$

With $L=\sqrt{\kappa}a$,

$$L^\dagger L = \kappa a^\dagger a.$$

Therefore

$$\frac{dp}{dt} = \kappa \operatorname{Tr}(a^\dagger a\rho) = \kappa\langle a^\dagger a\rangle .$$

This rate is the ideal detected photon flux for that monitored output channel.

Purcell Estimate

Use the detuned Purcell estimate

$$\Gamma_{\mathrm P}(\Delta) \approx \frac{g^2\kappa} {\Delta^2+(\kappa/2)^2}$$

to recover the resonant bad-cavity expression.

Solution

Set $\Delta=0$:

$$\Gamma_{\mathrm P}(0) \approx \frac{g^2\kappa} {(\kappa/2)^2} = \frac{g^2\kappa} {\kappa^2/4} = \frac{4g^2}{\kappa}.$$

The expression assumes that the cavity mode can be eliminated as a quickly relaxing intermediate degree of freedom.

Dispersive Condition

Why does the dispersive approximation require a condition like

$$|\Delta| \gg g\sqrt{n+1}$$

rather than merely $|\Delta|\gg g$?

Solution

In the Jaynes–Cummings model, the coupling between $|e,n\rangle$ and $|g,n+1\rangle$ is enhanced by the matrix element

$$g\sqrt{n+1}.$$

The perturbative expansion compares this coupling to the detuning. If the cavity contains many photons, the relevant coupling is not $g$ but $g\sqrt{n+1}$. The dispersive approximation can fail at high photon number even when $|\Delta|$ is larger than $g$.

Conditional Versus Unconditional Evolution

Explain why a no-click evolution generated by $H_{\mathrm{eff}}$ is not the same as the unconditional master equation.

Solution

The no-click evolution is conditioned on a specific measurement record: no monitored photon was detected during the interval. The state update includes information gained from the absence of a click. Its trace is the no-click probability, so the lost trace is the probability of at least one click. Unobserved channels still require their recycling terms.

The unconditional master equation averages over all records, including click and no-click alternatives. The jump terms restore trace and represent the ensemble of possible emissions. Therefore the no-click trajectory is one conditioned branch, not the ensemble state.

References

• H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Physica Scripta T76, 127-137 (1998).
• H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: Coherence in context,” Science 298, 1372-1377 (2002).
• J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Reviews of Modern Physics 73, 565-582 (2001).
• A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.-M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Physical Review Letters 83, 5166-5169 (1999).
• A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Reviews of Modern Physics 87, 1379-1418 (2015).
• H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Reports on Progress in Physics 69, 1325-1382 (2006).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).