Heterodyne Detection

Heterodyne detection records both quadratures of a field in one measurement stream. Operationally, this is done by mixing the signal with a local oscillator at a different frequency, or by splitting the signal and performing two homodyne measurements with orthogonal phases. The price is extra vacuum noise: heterodyne detection does not evade the noncommutativity of conjugate quadratures.

Homodyne and Heterodyne Detection owns the practical optical receiver, image-band and dual-homodyne pictures, coherent-state POVM, calibration, and tomography connection. This page owns the continuous complex record and the conditional stochastic evolution it drives.

In trajectory language, heterodyne detection produces a complex diffusive record. Schematically,

$$dZ_t = \sqrt{\eta}\,\langle L\rangle_c\,dt + d\zeta_t,$$

where $L$ is the monitored output coupling operator and $d\zeta_t$ is complex Wiener noise in a convention specified below. The exact factors depend on quadrature normalization, but the physical content is stable: one noisy stream carries the in-phase component and another noisy stream carries the quadrature component.

Why It Is Not Two Perfect Measurements

A single-mode field has conjugate quadratures. In one common normalization,

$$X = \frac{a+a^\dagger}{\sqrt2}, \qquad P = \frac{-i(a-a^\dagger)}{\sqrt2},$$

with

$$[X,P]=i.$$

No measurement can return noiseless preexisting values of both $X$ and $P$. Heterodyne detection instead returns a noisy complex amplitude whose distribution is broader than an ideal quadrature marginal. The additional noise is not a defect of a poor detector; it is the vacuum noise introduced by the measurement arrangement.

This is why heterodyne detection is often described as measuring coherent-state phase space rather than performing two sharp quadrature measurements.

Two-Homodyne Picture

One way to model heterodyne detection is to split the signal on a balanced beam splitter and homodyne the two outputs with local-oscillator phases separated by $\pi/2$. The unused input port of the splitter contributes vacuum noise. Each homodyne detector receives only part of the signal amplitude, and the vacuum port supplies the added noise required for simultaneous quadrature readout.

For a monitored output operator $L$, a common real-record convention is

$$\begin{aligned} dY_x &= \sqrt{\frac{\eta}{2}}\, \langle L+L^\dagger\rangle_c\,dt + dW_x, \\ dY_y &= \sqrt{\frac{\eta}{2}}\, \langle -i(L-L^\dagger)\rangle_c\,dt + dW_y, \end{aligned}$$

where

$$\mathbb E[dW_x]=\mathbb E[dW_y]=0, \qquad dW_x^2=dW_y^2=dt, \qquad dW_xdW_y=0.$$

The factor of $1/\sqrt2$ in the signal terms is a common way to represent the signal split between two quadrature measurements. Other conventions absorb this factor into the definition of the complex record or the efficiency. The convention must be stated before comparing measurement rates.

Complex Record

Define

$$dZ_t = \frac{dY_x+i\,dY_y}{\sqrt2}, \qquad d\zeta_t = \frac{dW_x+i\,dW_y}{\sqrt2}.$$

Then the real-record convention above becomes

$$dZ_t = \sqrt{\eta}\,\langle L\rangle_c\,dt + d\zeta_t.$$

The complex noise satisfies

$$\mathbb E[d\zeta_t]=0, \qquad d\zeta_t\,d\zeta_t^*=dt, \qquad d\zeta_t^2=0.$$

These rules are the complex form of two independent Wiener records. They are also a useful diagnostic in simulations: the real and imaginary innovations should have the correct variance and no spurious cross-correlation after calibration.

Conditional State Update

In the two-homodyne convention, the normalized stochastic master equation can be written schematically as

$$\begin{aligned} d\rho_c =& \mathcal L\rho_c\,dt + \sqrt{\frac{\eta}{2}}\, \mathcal H[L]\rho_c\,dW_x \\ &+ \sqrt{\frac{\eta}{2}}\, \mathcal H[-iL]\rho_c\,dW_y . \end{aligned}$$

Here

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

Averaging over the two noise streams gives back the same unconditional master equation generated by $\mathcal L$. The individual conditioned trajectory, however, differs from a homodyne trajectory because the observer has a different record.

The broader stochastic calculus is covered in Diffusive Trajectories and Stochastic Master Equations.

Coherent-State POVM

For a single mode, ideal heterodyne detection is closely related to the coherent-state POVM

$$E(\alpha)\,d^2\alpha = \frac{d^2\alpha}{\pi} |\alpha\rangle\langle\alpha|.$$

The probability density for outcome $\alpha$ is

$$p(\alpha) = \operatorname{Tr}[\rho E(\alpha)] = \frac{1}{\pi} \langle\alpha|\rho|\alpha\rangle.$$

This is the Husimi $Q$ distribution in a common convention. The distribution is positive and experimentally accessible, but it is smoothed relative to sharper quasiprobability representations. The smoothing is another way to see the added vacuum noise in heterodyne readout.

The POVM language is most useful for finite-mode or tomography discussions. Continuous heterodyne monitoring of an output field is usually written as a stochastic record and a stochastic master equation.

Optical and Microwave Implementations

In optical heterodyne detection, the local oscillator is detuned from the signal carrier. The measured photocurrent oscillates at an intermediate frequency. Demodulating the beat note produces two baseband records, often called in-phase and quadrature components.

In microwave circuit QED, phase-preserving amplification and IQ demodulation naturally produce a complex record. The useful record may be written as

$$dZ_t = \text{complex conditional signal}\,dt + d\zeta_t,$$

after offsets, gains, mixer imbalance, and amplifier noise are calibrated. In many microwave settings, the amplifier adds noise beyond the quantum-limited heterodyne noise. That extra noise reduces the effective measurement efficiency.

Heterodyne Versus Homodyne

Homodyne detection is better when one quadrature is known to carry the relevant signal and the local-oscillator phase can be stabilized. It concentrates the measurement strength into that quadrature.

Heterodyne detection is better when the phase is unknown, drifting, intentionally rotating, or when the complex field amplitude itself is the desired record. It gives both quadratures at once but with extra noise in each.

A useful comparison is:

Feature	Homodyne	Heterodyne
Record	one real noisy current	two real currents or one complex current
Phase reference	fixed measured quadrature	rotating or two-quadrature readout
Noise	one quadrature shot-noise stream	added vacuum noise for simultaneous quadratures
Trajectory	one Wiener increment	two independent Wiener increments
Typical use	phase-sensitive measurement	complex-amplitude measurement

Common Mistakes

• Describing heterodyne detection as noiseless simultaneous measurement of $X$ and $P$.
• Forgetting the factor associated with splitting the signal between two quadrature measurements.
• Comparing a homodyne measurement rate with a heterodyne measurement rate without matching conventions.
• Treating the complex record $dZ_t$ as the field amplitude itself rather than a noisy measurement increment.
• Ignoring image-band vacuum noise or amplifier added noise.
• Assuming the Husimi $Q$ distribution is the Wigner function.
• Dropping correlations introduced by real electronics, such as IQ imbalance, filtering, or phase drift, while using an ideal two-Wiener model.

Cross-Links

• Optical Phase-Space Distributions for the Husimi $Q$ outcome density, ordering rules, state examples, and receiver-noise caveats.
• Homodyne Detection for one-quadrature local-oscillator measurement.
• Measurement Records for calibrated real and complex stochastic records.
• Diffusive Trajectories for two-Wiener stochastic equations and ensemble recovery.
• Unravelings for comparison with photon-counting and homodyne monitoring of the same master equation.
• Input–Output Theory for the output field coupled to a system operator.
• Gaussian Channels for attenuation, amplification, and added-noise constraints in continuous-variable language.
• Quantum Optics and Circuit QED for common experimental settings.

References

• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• U. Leonhardt, Measuring the Quantum State of Light, Cambridge University Press, 1997.
• H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed., Wiley-VCH, 2004.
• A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Reviews of Modern Physics 93, 025005, 2021.

Exercises

1. Starting from two independent Wiener increments $dW_x$ and $dW_y$, define $d\zeta=(dW_x+i\,dW_y)/\sqrt2$. Show that $d\zeta\,d\zeta^*=dt$ and $d\zeta^2=0$.

Solution

Using $dW_x^2=dW_y^2=dt$ and $dW_xdW_y=0$,

$$\begin{aligned} d\zeta\,d\zeta^* &= \frac12 (dW_x+i\,dW_y)(dW_x-i\,dW_y) \\ &= \frac12(dW_x^2+dW_y^2) = dt. \end{aligned}$$

Similarly,

$$\begin{aligned} d\zeta^2 &= \frac12 (dW_x+i\,dW_y)^2 \\ &= \frac12(dW_x^2-dW_y^2+2i\,dW_xdW_y) = 0. \end{aligned}$$

2. In the real-record convention

$$\begin{aligned} dY_x &= \sqrt{\frac{\eta}{2}}\, \langle L+L^\dagger\rangle_c\,dt+dW_x, \\ dY_y &= \sqrt{\frac{\eta}{2}}\, \langle -i(L-L^\dagger)\rangle_c\,dt+dW_y, \end{aligned}$$

show that $dZ=(dY_x+i\,dY_y)/\sqrt2$ has drift $\sqrt{\eta}\langle L\rangle_cdt$.

Solution

The drift of $dZ$ is

$$\frac{1}{\sqrt2} \sqrt{\frac{\eta}{2}} \left( \langle L+L^\dagger\rangle_c + i\langle -i(L-L^\dagger)\rangle_c \right)dt.$$

Since

$$i[-i(L-L^\dagger)]=L-L^\dagger,$$

the expression in parentheses is

$$\langle L+L^\dagger+L-L^\dagger\rangle_c = 2\langle L\rangle_c.$$

Therefore the drift is

$$\sqrt{\eta}\,\langle L\rangle_cdt.$$

3. Why does splitting a signal and homodyning two outputs not give two perfect quadrature measurements of the original field?

Solution

The beam splitter has another input port. If no intentional field is injected there, that port contributes vacuum fluctuations. Those fluctuations enter the two output fields and add noise to the two homodyne records. The signal is also split, so each homodyne detector receives only part of the signal amplitude. Together these effects enforce the quantum limit on simultaneous quadrature readout.

4. The heterodyne POVM has effects $E(\alpha)d^2\alpha=d^2\alpha\,|\alpha\rangle\langle\alpha|/\pi$. What probability density is obtained for an input coherent state $|\beta\rangle$?

Solution

For $\rho=|\beta\rangle\langle\beta|$,

$$p(\alpha) = \frac1\pi |\langle\alpha|\beta\rangle|^2.$$

The coherent-state overlap satisfies

$$|\langle\alpha|\beta\rangle|^2 = \exp(-|\alpha-\beta|^2),$$

so

$$p(\alpha) = \frac1\pi \exp(-|\alpha-\beta|^2).$$

The outcome is centered at $\beta$, but it has finite width because heterodyne detection is a noisy coherent-state measurement.