Measurement and Damping in Quantum Optics

Quantum optics is the natural laboratory for open quantum systems. Photons leave cavities, atoms emit into radiation modes, detectors produce clicks and currents, and measured output fields condition the state of the source. The same experiment can be described as a master equation, a Langevin equation, an input–output relation, or a quantum trajectory depending on which degrees of freedom and records are retained.

The examples below connect damping models to the light that reaches a detector. For the broader theory of optical states and light–matter interaction, see Quantum Optics.

Optical Sources, Loss, and Detectors

A typical quantum-optical setup has:

• a localized system: atom, ion, molecule, cavity mode, mechanical mode, or nonlinear optical mode;
• one or more electromagnetic reservoirs;
• coherent drives supplied by lasers;
• detectors that monitor selected output modes;
• unmonitored loss channels that produce decoherence and damping.

The unconditional state often obeys a quantum optical master equation,

$$\dot\rho = - \frac{i}{\hbar}[H,\rho] + \sum_j \mathcal D[L_j]\rho,$$

where

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12\{L^\dagger L,\rho\}.$$

The collapse operators $L_j$ are not just mathematical terms. In quantum optics they often correspond directly to physical output channels: spontaneous emission, cavity leakage, absorption, scattering, or detector-monitored fluorescence.

Cavity Damping

For a single cavity mode with annihilation operator $a$ and photon-number decay rate $\kappa$ (field-amplitude decay rate $\kappa/2$), zero-temperature damping is described by

$$\dot\rho = - \frac{i}{\hbar}[H,\rho] + \kappa\mathcal D[a]\rho.$$

If the reservoir has thermal occupation $\bar n$, the damping terms become

$$\kappa(\bar n+1)\mathcal D[a]\rho + \kappa\bar n\,\mathcal D[a^\dagger]\rho.$$

At optical frequencies and ordinary laboratory temperatures, $\bar n$ is usually negligible. At microwave frequencies it may not be. This distinction matters when comparing optical cavity QED with circuit QED.

The same cavity coupling appears in input–output theory:

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

up to phase convention. The output field is what detectors actually see. See Input–Output Theory.

Spontaneous Emission

For a two-level atom with lowering operator $\sigma_-=\lvert g\rangle\langle e\rvert$, spontaneous emission into unobserved modes is modeled by

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

With no drive or re-excitation, the excited-state population decays as

$$\dot\rho_{ee} = -\Gamma\rho_{ee},$$

while the optical coherence decays at rate $\Gamma/2$ before adding any extra pure dephasing. This is the optical prototype behind the Amplitude Damping Master Equation.

Adding a coherent near-resonant drive gives the standard Optical Bloch Equations for saturation, resonance fluorescence, and damped Rabi dynamics.

If emitted photons are monitored, the same channel can be unraveled into quantum jumps. If the detector sees a photon, the conditioned state updates by

$$\rho_c \longmapsto \frac{ L\rho_c L^\dagger }{ \operatorname{Tr}(L^\dagger L\rho_c) }.$$

If every dissipative channel is counted perfectly, no-click evolution is generated by a non-Hermitian Hamiltonian and normalization. Undetected channels must still contribute their recycling terms; inefficient no-click conditioning generally gives a mixed state. Averaging over all records returns the unconditional master equation.

Photon Counting

For an ideal detector with vacuum incident field in the counted channel and collapse operator $L$, the probability of a click in a short interval $dt$ is

$$p_{\mathrm{click}} = \operatorname{Tr}(L^\dagger L\rho_c)\,dt.$$

Direct detection is therefore naturally modeled by a Poisson increment $dN_t$ satisfying

$$dN_t^2=dN_t, \qquad \mathbb E[dN_t\mid\rho_c] = \operatorname{Tr}(L^\dagger L\rho_c)\,dt.$$

Photon counting is the operational basis for antibunching, resonance fluorescence records, photon-correlation measurements, and many single-emitter experiments. See Photon Counting and Quantum Jump Trajectories.

Homodyne and Heterodyne Detection

Homodyne detection mixes the output field with a strong local oscillator and measures one field quadrature. For a monitored channel $c$ and detection efficiency $\eta$, a common normalized stochastic master equation is

$$d\rho_c = \mathcal L\rho_c\,dt + \sqrt{\eta}\, \mathcal H[e^{-i\phi}c]\rho_c\,dW_t,$$

where $\phi$ is the local-oscillator phase and

$$\mathcal H[c]\rho = c\rho+\rho c^\dagger - \operatorname{Tr}[(c+c^\dagger)\rho]\rho.$$

The measured current has the schematic form

$$dY_t = \sqrt{\eta}\, \langle e^{-i\phi}c+e^{i\phi}c^\dagger\rangle_c\,dt + dW_t.$$

Heterodyne detection measures both quadratures at the cost of added vacuum noise. Both descriptions are diffusive rather than jump-like; the measurement record is a noisy continuous current. See Homodyne Detection, Heterodyne Detection, and Diffusive Trajectories.

Coherent States and Drives

A coherent optical drive is often treated as a classical amplitude plus quantum vacuum fluctuations:

$$b_{\mathrm{in}}(t) = \beta_{\mathrm{in}}(t) + \xi_{\mathrm{in}}(t).$$

For a driven cavity, the mean field obeys a damped oscillator equation. In a frame rotating at the drive frequency,

$$\dot\alpha = - \left( i\Delta_c+\frac{\kappa}{2} \right)\alpha + \mathcal E,$$

Here $\Delta_c=\omega_c-\omega_d$ is cavity minus drive frequency. With the plus-sign input–output convention above, $\mathcal E=-\sqrt{\kappa_{\mathrm{ext}}}\beta_{\mathrm{in}}$, in inverse-time units. Direct counting of a coherently illuminated output includes interference with the incident field; it is not described by the cavity-emission rate alone.

For linear loss into vacuum, a coherent state remains coherent with decaying amplitude. Thermal noise instead adds mixed-state fluctuations. In covariance language this is a Gaussian attenuation channel; see Gaussian Channels for the open-system convention. This is one reason classical optical fields emerge naturally from lossy quantum optical modes. Nonlinearities, single-photon sources, photon blockade, and conditional measurements are what make the field strongly nonclassical.

Squeezed Light

For a single mode, define the quadrature

$$X_\theta = \frac12 \left( ae^{-i\theta} + a^\dagger e^{i\theta} \right).$$

In the vacuum state,

$$(\Delta X_\theta)^2 = \frac14$$

for every $\theta$. A squeezed state has reduced noise in one quadrature,

$$(\Delta X_{\theta_0})^2 \lt \frac14,$$

with increased noise in the conjugate quadrature. Squeezed light is therefore both a nonclassical state and a practical measurement resource: it can reduce imprecision in one quadrature while increasing backaction or noise elsewhere.

Open-system descriptions of squeezed reservoirs modify the usual damping master equation by adding phase-sensitive noise correlations. The physical message is that “the bath” need not be thermal or vacuum; engineered optical reservoirs can carry squeezing, correlations, and directionality.

Correlation Functions

Quantum optics often characterizes output fields through normally ordered correlation functions. The second-order coherence is

$$g^{(2)}(\tau) = \frac{ \langle a^\dagger(0)a^\dagger(\tau)a(\tau)a(0) \rangle }{ \langle a^\dagger a\rangle^2 }$$

for a stationary single mode with nonzero mean occupation. For a vacuum-input fluorescence channel, replace $a$ by the appropriate source operator, such as $\sigma_-$. Standard reference values are:

Field	$g^{(2)}(0)$	Interpretation
coherent state	$1$	Poissonian photon statistics
thermal state	$2$	bunching
ideal two-level fluorescence, without background	$0$	zero-delay antibunching

The quantum regression theorem connects such multitime correlations to the same Liouvillian used in the master equation, under Markovian assumptions. When those assumptions fail, correlation functions can reveal memory, filtering, detector bandwidth, and mode mismatch.

What Is Actually Measured

Quantum optics forces a useful distinction:

• the intracavity or atomic state is a model variable;
• the output field is the propagating quantum system that reaches the detector;
• the detector record is a classical stochastic process derived from the measured field.

Confusing these levels leads to mistakes. A click record is not the same object as the intracavity photon number. A homodyne current is not a projective measurement of a field quadrature at a single instant. Unmonitored output modes can retain information about the source, producing reduced-state decoherence without a recorded detector outcome.

Exercises

Photon-Counting Update

For a monitored spontaneous-emission channel $L=\sqrt{\Gamma}\sigma_-$, write the conditioned state immediately after a detected photon.

Solution

The jump update is

$$\rho_c \longmapsto \frac{ L\rho_c L^\dagger }{ \operatorname{Tr}(L^\dagger L\rho_c) }.$$

Substituting $L=\sqrt{\Gamma}\sigma_-$ gives

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

For an ideal two-level atom, this leaves the atom in the ground state whenever the denominator is nonzero.

Coherent-State Damping

A cavity mode obeys $\dot\rho=\kappa\mathcal D[a]\rho$ with no drive. What happens to an initial coherent amplitude $\alpha(0)$?

Solution

The mean amplitude obeys

$$\frac{d}{dt}\langle a\rangle = - \frac{\kappa}{2} \langle a\rangle.$$

Therefore

$$\alpha(t) = \alpha(0)e^{-\kappa t/2}.$$

For linear damping into vacuum, an initial coherent state remains coherent while its amplitude decays. The photon number decays as $e^{-\kappa t}$.

Antibunching

Why does an ideal single-photon source have $g^{(2)}(0)=0$?

Solution

For a single mode with support only on $|0\rangle$ and $|1\rangle$, $a^2\rho=0$, hence the normally ordered numerator $\langle a^{\dagger2}a^2\rangle$ vanishes. For ideal two-level fluorescence the corresponding identity is $\sigma_-^2=0$. Provided the mean intensity is nonzero,

$$g^{(2)}(0)=0.$$

This is antibunching and has no classical analogue for an ordinary positive-intensity random field.

Homodyne Phase

In a homodyne measurement with channel $c$, what quadrature appears in the measurement signal when the local-oscillator phase is $\phi$?

Solution

The signal term is proportional to

$$\langle e^{-i\phi}c + e^{i\phi}c^\dagger \rangle_c.$$

Changing $\phi$ rotates the measured quadrature. The stochastic noise term remains a Wiener increment in the ideal Markovian model.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Physical Review Letters 68, 580-583 (1992).
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).
• R. J. Glauber, “The quantum theory of optical coherence,” Physical Review 130, 2529-2539 (1963).
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer (2008).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).