Homodyne Detection

Homodyne detection measures one quadrature of an optical or microwave field by interfering the signal with a strong phase reference called the local oscillator. In balanced homodyne detection, the two outputs of a beam splitter are sent to photodetectors and the photocurrents are subtracted. The large local-oscillator amplitude converts a field quadrature into an intensity difference.

Beam Splitters explains the lossless mixer and its phase conventions. For mode selection, vacuum calibration, and receiver imperfections, see Homodyne and Heterodyne Detection. Here we connect that detector to a continuous record and a conditional state.

In continuous-measurement language, homodyne detection produces a diffusive record:

$$dY_t = \text{conditional quadrature signal}\,dt + dW_t.$$

The Wiener increment $dW_t$ is not a technical afterthought. It is the shot-noise or vacuum-noise part of the record, and it is what makes homodyne-conditioned states follow stochastic trajectories rather than deterministic expectation-value curves.

The Phase-Selected Quadrature

For a single mode with annihilation operator $a$, define a quadrature convention

$$X_\phi = \frac{ e^{-i\phi}a+e^{i\phi}a^\dagger }{\sqrt2}.$$

The phase $\phi$ is the local-oscillator phase. Changing $\phi$ rotates the measured quadrature:

$$X_{\phi+\pi/2} = \frac{ -i e^{-i\phi}a+i e^{i\phi}a^\dagger }{\sqrt2},$$

which obeys $[X_\phi,X_{\phi+\pi/2}]=i$. Each quadrature has vacuum variance $1/2$ in this normalization.

Continuous-output records often absorb factors of $\sqrt2$ into the definition of $dY_t$ and the coupling operator. Always check the convention before comparing formulas. The operational statement is convention independent: a homodyne detector measures one phase-selected quadrature of the output field.

Balanced Detection Derivation

Treat the local oscillator as a large coherent amplitude

$$\alpha_{\mathrm{LO}} = |\alpha_{\mathrm{LO}}|e^{i\phi}.$$

Let $a_s$ be the signal mode entering the other input port of a balanced beam splitter. With one common phase convention, the two output modes are

$$a_+ = \frac{a_s+\alpha_{\mathrm{LO}}}{\sqrt2}, \qquad a_- = \frac{a_s-\alpha_{\mathrm{LO}}}{\sqrt2}.$$

The photon-number difference is

$$\begin{aligned} n_+ - n_- &= a_+^\dagger a_+ - a_-^\dagger a_- \\ &= \alpha_{\mathrm{LO}}^*a_s + \alpha_{\mathrm{LO}}a_s^\dagger . \end{aligned}$$

Dividing by the local-oscillator amplitude gives

$$\frac{n_+ - n_-}{|\alpha_{\mathrm{LO}}|} = e^{-i\phi}a_s + e^{i\phi}a_s^\dagger .$$

Thus the difference current measures the signal quadrature selected by $\phi$. The sum current is dominated by the local-oscillator intensity and is usually used for normalization and diagnostics rather than as the quantum signal.

The replacement of the oscillator by a coherent amplitude is the strong-local-oscillator approximation. It does not turn both beam-splitter outputs into independent canonical modes after that replacement; it is used to obtain the leading measurement signal. Vacuum fluctuations still set the difference-current noise.

This derivation also explains why balance matters. Subtracting the two detector outputs cancels the common large local-oscillator intensity to leading order, while preserving the interference term linear in the signal field.

Continuous Output Field

For a Markovian output port, Input–Output Theory writes

$$b_{\mathrm{out}}(t) = b_{\mathrm{in}}(t)+L(t),$$

where $L$ is the system coupling operator for that port. With vacuum input and known coherent amplitudes subtracted, a common homodyne record convention is

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

Here $\eta$ is the total detection efficiency, including collection loss, propagation loss, mode mismatch, detector efficiency, and any added-noise penalty that is represented as loss in the model. The conditional mean is the output quadrature signal; the innovation is

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

The record is a property of the traveling output field. It is usually not correct to replace $b_{\mathrm{out}}$ by an intracavity operator without the input–output relation and the relevant phase convention.

Conditional State Update

The same monitored output channel gives a diffusive stochastic master equation. In a common normalized convention,

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

where

$$\mathcal H[C]\rho = C\rho+\rho C^\dagger - \operatorname{Tr} \!\left[ (C+C^\dagger)\rho \right]\rho.$$

The unconditional generator $\mathcal L$ contains the Hamiltonian and all monitored and unmonitored Lindblad channels. Averaging over homodyne records recovers the unconditional master equation.

Stochastic Master Equations explains how normalized conditional equations follow from instruments, and the broader family of noisy-current trajectories is treated in Diffusive Trajectories.

Homodyne Versus Photon Counting

Photon counting measures output intensity by registering discrete detection events. Homodyne detection measures a field quadrature through interference with a phase reference. The difference is not cosmetic:

• photon counting gives a point process $dN_t$;
• homodyne detection gives a diffusive current $dY_t$;
• photon counting is naturally sensitive to $L^\dagger L$;
• homodyne detection is sensitive to $e^{-i\phi}L+e^{i\phi}L^\dagger$;
• the same unmonitored dissipator $\mathcal D[L]\rho$ can underlie either record.

Thus a Lindblad operator $L$ is not automatically a detector click. How the output field is measured determines the unraveling.

Homodyne Versus Heterodyne

Homodyne detection measures one quadrature selected by a phase reference. Heterodyne detection records both quadratures by introducing an additional mode or frequency offset, which brings extra vacuum noise. Homodyne is therefore preferable when one quadrature contains the signal of interest and the orthogonal quadrature is not needed.

In phase-space language, homodyne samples a marginal quadrature distribution, while Heterodyne Detection gives a noisy complex-amplitude record. The two methods answer different experimental questions even when they monitor the same output port.

Examples

In resonance fluorescence, homodyne detection of emitted light can monitor a dipole quadrature of a driven atom. The resulting trajectories are diffusive rather than jump-like: the state is steered by a noisy photocurrent instead of by isolated photon detections.

In cavity QED, homodyne detection of a transmitted or reflected field measures a quadrature of the cavity output. If the cavity field depends on the state of an atom or qubit, the homodyne record becomes an indirect continuous measurement of that internal degree of freedom.

In circuit QED, microwave homodyne or near-homodyne detection after amplification is the basis of dispersive qubit readout. After eliminating the resonator in an appropriate regime, the record is often written schematically as

$$dY_t = 2\sqrt{\eta\Gamma_{\mathrm m}}\, \langle\sigma_z\rangle_c\,dt + dW_t.$$

Here the effective collapse operator is $\sqrt{\Gamma_{\mathrm m}}\sigma_z$, so its unread coherence-decay rate is $2\Gamma_{\mathrm m}$. The reduction assumes that the resonator response, amplifier noise, demodulation phase, integration window, and leakage are adequately modeled.

Calibration and Imperfections

A realistic homodyne record depends on more than the ideal stochastic equation. Important experimental ingredients include:

• local-oscillator phase and phase noise;
• mode matching or visibility between signal and local oscillator;
• detector quantum efficiency and propagation loss;
• electronic noise and amplifier added noise;
• finite detector bandwidth;
• saturation or nonlinearity in the photodetectors or amplifiers;
• imperfect subtraction of the two photocurrents;
• offsets and slow drifts in the recorded current.

Balanced subtraction removes common classical intensity noise but does not remove quantum shot noise. Many of these effects can be folded into an effective efficiency $\eta$ only if they behave like loss or independent Gaussian noise over the bandwidth of interest. Phase drift, detector saturation, and filtering generally require more explicit modeling.

Exercises

1. Starting from

$$a_\pm = \frac{a_s\pm\alpha_{\mathrm{LO}}}{\sqrt2},$$

derive the difference signal $n_+ - n_-$ used in balanced homodyne detection.

Solution

Compute

$$n_+ = \frac12 \left( a_s^\dagger+\alpha_{\mathrm{LO}}^* \right) \left( a_s+\alpha_{\mathrm{LO}} \right),$$

and

$$n_- = \frac12 \left( a_s^\dagger-\alpha_{\mathrm{LO}}^* \right) \left( a_s-\alpha_{\mathrm{LO}} \right).$$

Subtracting cancels $a_s^\dagger a_s$ and $|\alpha_{\mathrm{LO}}|^2$, leaving

$$n_+ - n_- = \alpha_{\mathrm{LO}}^*a_s + \alpha_{\mathrm{LO}}a_s^\dagger.$$

With $\alpha_{\mathrm{LO}}=|\alpha_{\mathrm{LO}}|e^{i\phi}$, the normalized difference is

$$\frac{n_+ - n_-}{|\alpha_{\mathrm{LO}}|} = e^{-i\phi}a_s+e^{i\phi}a_s^\dagger.$$

2. Show that changing the local-oscillator phase by $\pi$ reverses the sign of the measured quadrature.

Solution

Using

$$X_\phi = \frac{ e^{-i\phi}a+e^{i\phi}a^\dagger }{\sqrt2},$$

one has

$$\begin{aligned} X_{\phi+\pi} &= \frac{ e^{-i(\phi+\pi)}a + e^{i(\phi+\pi)}a^\dagger }{\sqrt2} \\ &= \frac{ -e^{-i\phi}a - e^{i\phi}a^\dagger }{\sqrt2} = -X_\phi. \end{aligned}$$

3. For the homodyne record

$$dY_t = \sqrt{\eta} \langle C+C^\dagger\rangle_c dt + dW_t,$$

what is the conditional mean and variance of $dY_t$?

Solution

Since $\mathbb E[dW_t|\rho_c(t)]=0$ and $dW_t^2=dt$,

$$\mathbb E[dY_t|\rho_c(t)] = \sqrt{\eta} \langle C+C^\dagger\rangle_c dt,$$

and

$$\operatorname{Var}(dY_t|\rho_c(t)) = dt.$$

The mean is order $dt$, while the noise scale is order $\sqrt{dt}$.

4. Why does inefficient homodyne detection generally produce less pure conditional states than ideal homodyne detection, even when the same unconditional dissipator is present?

Solution

Efficiency less than one means that only part of the output information reaches the observer. A useful model splits the channel into a monitored part and an unmonitored loss part. The monitored part contributes to the stochastic innovation, while the unmonitored part still contributes to decoherence in the unconditional generator. The observer therefore lacks some information about the backaction, so the conditional state is generally more mixed than in an ideal measurement.

References

• H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed., Wiley-VCH, 2004.
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• U. Leonhardt, Measuring the Quantum State of Light, 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.