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 describes the optical receiver, image-band vacuum noise, calibration, and tomography. Here the emphasis is the continuous complex record and its stochastic state update.

In trajectory language, heterodyne detection produces a complex diffusive record. In the convention used here,

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

where $L$ is the monitored output coupling operator and $d\zeta_t$ is complex Wiener noise in a convention specified below. With the normalization specified below, one noisy stream carries the in-phase component and another noisy stream carries the quadrature component.

Two Conjugate Field Quadratures

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. Here $\eta$ is the total detection efficiency before accounting for the balanced split. For $\eta=1$, both outputs are observed: the factor $1/\sqrt2$ is not an additional unobserved loss channel. This distinction prevents double counting a factor of two in the inferred measurement efficiency.

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 is

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

Here

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

The generator $\mathcal L$ contains the full $\mathcal D[L]$, along with the Hamiltonian and any other channels. Averaging over the two innovations gives back that generator. Ideal monitoring of both quadratures can preserve a pure initial state even though each quadrature separately has added imprecision. 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

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 $\beta$. Its real and imaginary parts each have variance $1/2$. Therefore the calibrated quadrature estimates $X_{\mathrm{est}}=\sqrt2\operatorname{Re}\alpha$ and $P_{\mathrm{est}}=\sqrt2\operatorname{Im}\alpha$ each have variance $1$, compared with the coherent state’s ideal homodyne variance $1/2$. The extra $1/2$ in each quadrature is the added vacuum noise.

References

• 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.
• 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.