Homodyne and Heterodyne Detection

Direct photodetection reports one number — the mean photon flux $\langle\hat a^\dagger\hat a\rangle$ — and is deaf to optical phase and blind to any field fluctuation weaker than the mean intensity. Homodyne and heterodyne detection are the cure, assembled entirely from three results this chapter already owns: the field quantized into modes carrying $\hat a,\hat a^\dagger$ ; the conjugate quadratures $\hat X_0,\hat X_{\pi/2}$ obeying $[\hat X_0,\hat X_{\pi/2}]=i/2$ ; and the coherent state as the bright, minimum-uncertainty light a laser emits. Overlap a weak signal on a beam splitter with a strong coherent local oscillator split from the same laser, so the two stay phase-locked. Because the local oscillator is macroscopically populated, the photocurrent gains a cross term linear in the signal field and proportional to one quadrature $\hat X_\theta$ chosen by the phase $\theta$ . That linear-in-the-field response — not the quadratic intensity of direct detection — makes the measurement phase-sensitive and able to resolve a single mode's fluctuations down to the vacuum level. Balanced homodyne detects both output ports and subtracts, cancelling the huge local-oscillator term and its classical noise, leaving a shot-noise-limited readout of $\hat X_\theta$ . Scanning $\theta$ rotates which axis of phase space you project onto, so a squeezed state betrays itself as a dip below the flat vacuum trace. Detuning the local oscillator turns homodyne into heterodyne — both quadratures at once, at a mandatory 3 dB cost. Collect the histograms $\mathrm{Pr}(x_\theta)$ at every $\theta$ and you hold the marginals of the Wigner function: homodyne detection is quantum-state tomography.

Key takeaways

Mixing a weak signal with a bright coherent local oscillator on a beam splitter makes the photocurrent carry the signal field linearly, not as an intensity — the route to phase-sensitive, quantum-limited measurement.
Balanced homodyne subtracts the two output currents: the intensity $|\alpha_{\mathrm{LO}}|^2$ and its classical noise cancel, leaving $\hat i_-=2|\alpha_{\mathrm{LO}}|\,\hat X_\theta$ — one quadrature scaled by the local-oscillator amplitude.
The local-oscillator phase $\theta$ selects the quadrature ( $\theta=0$ amplitude, $\theta=\pi/2$ phase); sweeping $\theta$ traces the state's noise ellipse.
Heterodyne (detuned local oscillator) reads both conjugate quadratures at once but pays a fixed 3.01 dB penalty from image-band vacuum, and the histograms $\mathrm{Pr}(x_\theta)$ are marginals of the Wigner function that reconstruct the full state.
Key idea: a strong local oscillator converts one linear quadrature $\hat X_\theta$ of a weak signal into a macroscopic photocurrent; balanced subtraction removes local-oscillator noise, leaving a shot-noise-limited, phase-selective measurement whose statistics are marginals of the Wigner function.

The core insight

A strong local oscillator converts one linear quadrature $\hat X_\theta$ of a weak signal into a macroscopic photocurrent; balanced subtraction cancels the local-oscillator noise, leaving a shot-noise-limited, phase-selective measurement whose outcome statistics are the marginals of the signal's Wigner function.

Where this lives today

The most visible home of homodyne readout is gravitational-wave astronomy: Advanced LIGO and Advanced Virgo inject squeezed vacuum into the interferometer's dark port and read it out with balanced-homodyne-type detection referenced to the carrier, so the strain-carrying phase quadrature is measured below the standard quantum limit, and filter-cavity squeezing now buys several dB of broadband sensitivity. The same instrument is the universal state analyzer of continuous-variable quantum information — cluster-state photonics, Gaussian boson sampling, and quantum key distribution all read quadratures with balanced homodyne, and tomography images Fock and cat states with negative Wigner functions. Heterodyne detection dominates wherever the signal frequency is unknown or must be band-selected: coherent optical receivers recover both amplitude and phase for QPSK and QAM far beyond direct detection, FMCW LIDAR beats a returned echo against a local oscillator for range and velocity at once, and frequency-comb metrology locks a laser to a comb tooth through the beat note — the technique behind optical atomic clocks. In every case the strong local oscillator does the same job as in a squeezing experiment, lifting a handful of photons above thermal and electronic noise and turning phase into current.

The physical picture

Picture the bright local oscillator as a ruler laid alongside a tiny signal wave. Alone the signal is too weak to register above dark current and amplifier noise, but its interference with the ruler produces a beat of amplitude $2|\alpha_{\mathrm{LO}}|$ times the signal — the local oscillator amplifies one quadrature into a macroscopic current for free. That is why it is essential: a noiseless single-quadrature amplifier lifting a single-photon-scale signal into the milliamp range where ordinary electronics can read it. The subtraction of the two outputs keeps the amplification quantum-limited. Each detector sees the enormous local-oscillator power and all of the laser's technical intensity noise, but the interference term has opposite sign at the two ports, so subtracting cancels the common intensity and its classical fluctuations exactly while the interference adds. What survives is only the beat of the local oscillator with vacuum entering the signal port — irreducible shot noise, the floor against which everything is calibrated.

Figure

Balanced homodyne detection. A weak signal â_s and a bright local oscillator of amplitude α_LO at phase θ meet on a 50:50 beam splitter. The two output photocurrents are subtracted, so the local-oscillator intensity term cancels and only the signal times local-oscillator cross term survives as î_- ≈ 2 α_LO X̂_θ, a single field quadrature selected by θ

The phase $\theta$ rotates which phase-space axis is projected onto the current. A coherent or vacuum state is a symmetric disk, so its projection has the same width at every $\theta$ — a flat trace. A squeezed state is an ellipse, and sweeping $\theta$ traces its width, dipping below the shot-noise line where the state is squeezed and rising above it a quarter-turn away — the working diagnostic of every squeezing experiment.

Figure

The homodyne diagnostic: noise power versus local-oscillator phase θ for a squeezed state, relative to the vacuum shot-noise line at 0 dB. A source producing 10 dB of squeezing viewed through 90 percent efficiency dips to −7.2 dB below shot noise at θ = 0 and π and rises to +9.6 dB of anti-squeezing at θ = π/2. A coherent or vacuum state gives a flat line at 0 dB

Heterodyne detection differs only in that the local-oscillator ruler now rotates at the beat frequency $\Delta=\omega_{\mathrm{LO}}-\omega_s$ . Because the projection axis spins through every angle, one record yields both $\hat X_0$ and $\hat X_{\pi/2}$ at once. The price is a second empty mode — the image band, mirror-symmetric about the local oscillator — that beats down to the same frequency in vacuum, adding half a quantum per quadrature: the unavoidable 3 dB toll for measuring two non-commuting observables together.

The core equations

The observable the scheme delivers is a rotated quadrature of the signal mode, a Hermitian combination of $\hat a$ and $\hat a^\dagger$ weighted by an externally chosen phase.

\hat X_\theta = \tfrac{1}{2}\big(\hat a\, e^{-i\theta} + \hat a^\dagger e^{i\theta}\big) \qquad (1)

where: $\hat X_\theta$ — quadrature selected by the local-oscillator phase $\theta$ ; $\hat a,\hat a^\dagger$ — annihilation and creation operators of the signal mode; $\theta$ — local-oscillator phase. At $\theta=0$ this is the amplitude quadrature $\hat X_0$ , at $\theta=\pi/2$ the phase quadrature $\hat X_{\pi/2}$ . The two do not commute, $[\hat X_0,\hat X_{\pi/2}]=i/2$ , so $\Delta X_0\,\Delta X_{\pi/2}\ge1/4$ with vacuum (shot-noise) variance $1/4$ per quadrature — the fact behind the noise floor, "sub-shot-noise", and the heterodyne penalty. A lossless 50:50 beam splitter maps the two inputs onto two output modes; the relative sign between ports is the convention that makes the intensity terms cancel on subtraction.

\hat c = \tfrac{1}{\sqrt2}(\hat a_s + \hat a_{\mathrm{LO}}), \quad \hat d = \tfrac{1}{\sqrt2}(\hat a_s - \hat a_{\mathrm{LO}}) \qquad (2)

where: $\hat c,\hat d$ — output-port modes on the detectors; $\hat a_s$ — signal input; $\hat a_{\mathrm{LO}}$ — local-oscillator input. Each detector reads a current proportional to its port's photon number. Subtracting the two photocurrents is the defining move of balanced detection: the single-beam intensities cancel and only the interference survives.

\hat i_- = \hat c^\dagger \hat c - \hat d^\dagger \hat d = \hat a_s^\dagger \hat a_{\mathrm{LO}} + \hat a_{\mathrm{LO}}^\dagger \hat a_s \qquad (3)

where: $\hat i_-$ — difference of the two current operators. The large $\hat a_{\mathrm{LO}}^\dagger\hat a_{\mathrm{LO}}$ term — carrier of the laser's classical intensity noise — has vanished identically, with the negligible $\hat a_s^\dagger\hat a_s$ . Treat the bright local oscillator as a classical amplitude $\alpha_{\mathrm{LO}}=|\alpha_{\mathrm{LO}}|e^{i\theta}$ , its own fluctuations negligible beside its size; the current then collapses onto a single quadrature.

\hat i_- \;\longrightarrow\; |\alpha_{\mathrm{LO}}|\big(\hat a_s e^{-i\theta} + \hat a_s^\dagger e^{i\theta}\big) = 2|\alpha_{\mathrm{LO}}|\,\hat X_\theta \qquad (4)

where: $\alpha_{\mathrm{LO}}=|\alpha_{\mathrm{LO}}|e^{i\theta}$ — classical amplitude; $\hat X_\theta$ — the quadrature read out, scaled by $2|\alpha_{\mathrm{LO}}|$ . Tuning $\theta$ chooses the quadrature; more power gives more gain. The measured noise is the current variance — the signal's quantum fluctuations magnified into a recordable spectrum.

\langle \Delta \hat i_-^2\rangle = 4|\alpha_{\mathrm{LO}}|^2\,\langle \Delta \hat X_\theta^2\rangle \qquad (5)

where: $\langle \Delta \hat X_\theta^2\rangle$ — signal-quadrature variance, $1/4$ for vacuum or coherent light, so $\langle\Delta\hat i_-^2\rangle=|\alpha_{\mathrm{LO}}|^2$ — a shot-noise floor linear in local-oscillator power, beneath which squeezing dips. Sweep $\theta$ and each outcome histogram is a geometric projection of the state's Wigner distribution onto the rotated axis — the Radon transform, the geometry of a CT scan. Inverting it (filtered backprojection, or a maximum-likelihood estimate) recovers the full state.

\mathrm{Pr}(x_\theta) = \int_{-\infty}^{\infty} W\big(x_\theta\cos\theta - p\sin\theta,\; x_\theta\sin\theta + p\cos\theta\big)\,\mathrm{d}p \qquad (6)

where: $\mathrm{Pr}(x_\theta)$ — measured probability density of homodyne outcomes at phase $\theta$ ; $W$ — the signal state's Wigner function. The full set over $\theta\in[0,\pi)$ determines $W$ , hence the density matrix.

Derivation ABalanced homodyne cancels the local-oscillator noise and reads a quadrature

The goal is to start from the unitary 50:50 beam-splitter relation and the bright-local-oscillator approximation and arrive at $\hat i_-=2|\alpha_{\mathrm{LO}}|\hat X_\theta$ , showing exactly where the enormous local-oscillator intensity and its technical noise disappear and why the surviving current is linear in the signal field — the calculation that makes a beam splitter plus two photodiodes a quantum-limited quadrature meter.

Step 1. Send $\hat a_s$ and $\hat a_{\mathrm{LO}}$ into a 50:50 beam splitter, giving $\hat c=\tfrac{1}{\sqrt2}(\hat a_s+\hat a_{\mathrm{LO}})$ and $\hat d=\tfrac{1}{\sqrt2}(\hat a_s-\hat a_{\mathrm{LO}})$ . Why: energy conservation forces unitary mixing, and the relative minus sign between ports is what later produces an exact cancellation rather than a doubling.

Step 2. Each detector reads a photon number; expanding gives $\hat n_c=\tfrac12(\hat a_s^\dagger\hat a_s+\hat a_{\mathrm{LO}}^\dagger\hat a_{\mathrm{LO}}+\hat a_s^\dagger\hat a_{\mathrm{LO}}+\hat a_{\mathrm{LO}}^\dagger\hat a_s)$ and $\hat n_d$ the same with the cross terms sign-flipped. Why: separating each port into single-beam intensities plus a cross term exposes what is common-mode and what carries the signal.

Step 3. Subtract: $\hat i_-=\hat n_c-\hat n_d=\hat a_s^\dagger\hat a_{\mathrm{LO}}+\hat a_{\mathrm{LO}}^\dagger\hat a_s$ ; the intensities cancel identically. Why: the local-oscillator intensity carries the laser's amplitude noise, so removing it in common-mode rejection is the entire reason balanced detection beats a single photodiode.

Step 4. Replace the bright local oscillator by its eigenvalue $\hat a_{\mathrm{LO}}\to\alpha_{\mathrm{LO}}=|\alpha_{\mathrm{LO}}|e^{i\theta}$ , valid because $|\alpha_{\mathrm{LO}}|^2\sim10^{16}$ gives fractional fluctuations of order $10^{-8}$ , yielding $\hat i_-=2|\alpha_{\mathrm{LO}}|\hat X_\theta$ . Why: linearity is the payoff — a quadratic intensity is blind to phase and to sub-mean fluctuations, whereas this beat carries both, amplified into a macroscopic current. *Verified: reproduces Loudon, Quantum Theory of Light, 3rd ed., Ch. 6 and Bachor and Ralph, Ch. 8; conventions consistent with $[\hat X_0,\hat X_{\pi/2}]=i/2$ and vacuum variance $1/4$ .*

Go deeper · Derivation B · 1 minThe 3 dB heterodyne penalty from the image band

Derivation BThe 3 dB heterodyne penalty from the image band

The goal is to show why heterodyne detection cannot match homodyne on a single quadrature — that detuning the local oscillator to read both quadratures at once necessarily admits a second, empty mode whose vacuum fluctuations double the noise floor, giving a fixed, source-independent 3.01 dB penalty that is the concrete face of the statement that non-commuting observables resist simultaneous sharp measurement.

Step 1. Detune the local oscillator to $\omega_{\mathrm{LO}}=\omega_s+\Delta$ ; the beat now oscillates, $\hat i_{\mathrm{het}}(t)\propto \hat a_s^\dagger\alpha_{\mathrm{LO}}e^{i\Delta t}+\text{h.c.}$ , carrying the signal on a radio-frequency carrier. Why: a carrier is what lets one record be demodulated into two orthogonal channels.

Step 2. Mixing the beat with $\cos\Delta t$ and $\sin\Delta t$ extracts $\hat X_0$ and $\hat X_{\pi/2}$ from the same record. Why: this is the operational meaning of reading both quadratures — the rotating axis visits every angle within one beat period.

Step 3. A second mode $\hat a_{\mathrm{im}}$ at the image frequency $\omega_{\mathrm{LO}}+\Delta$ , mirror-symmetric about the local oscillator, beats down to the same $\Delta$ and injects an extra $1/4$ of variance per quadrature, so the vacuum floor becomes $|\alpha_{\mathrm{LO}}|^2(\tfrac14+\tfrac14)$ — twice the homodyne value, i.e. $10\log_{10}2=3.01$ dB. Why: any measurement at $\Delta$ collects both local-oscillator sidebands, and in a signal-only experiment the image sideband holds nothing but vacuum, whose uncorrelated variance adds.

Step 4. This penalty is the Arthurs–Kelly bound in action: because $[\hat X_0,\hat X_{\pi/2}]\ne0$ , no joint measurement of both quadratures reaches the single-quadrature vacuum limit for each. Why: the commutator that sets the uncertainty relation also sets the minimum cost of measuring around it, tying the 3 dB to quantum mechanics, not hardware. *Verified: the factor-of-two (3.01 dB) penalty matches Yuen and Chan and Yuen and Shapiro; recomputed as $10\log_{10}2=3.010$ dB.*

Go deeper · Derivation C · 1 minDetection efficiency admits vacuum and caps observable squeezing

Derivation CDetection efficiency admits vacuum and caps observable squeezing

The aim is to model every imperfection between source and readout — finite quantum efficiency, imperfect signal-to-local-oscillator overlap, and passive loss — as a single beam splitter of transmission $\eta$ that mixes vacuum into the signal, and to show that this admixture pulls any measured variance toward the shot-noise level, capping how much squeezing can be seen — the reason laboratory squeezing records lag ideal source values.

Step 1. Represent loss by $\hat X_{\mathrm{out}}=\sqrt\eta\,\hat X_{\mathrm{sig}}+\sqrt{1-\eta}\,\hat X_{\mathrm{vac}}$ . Why: any linear loss is unitarily equivalent to a beam splitter coupling to an unoccupied mode, so vacuum leaks in wherever signal leaks out.

Step 2. Signal and injected vacuum are uncorrelated, so variances add: $V_{\mathrm{obs}}=\eta V_{\mathrm{sig}}+(1-\eta)\tfrac14$ . Why: the vacuum port always carries exactly the shot-noise variance, so the observed noise is a weighted average of source and vacuum, never below either.

Step 3. Let the source squeeze perfectly, $V_{\mathrm{sig}}\to0$ ; then $V_{\mathrm{obs}}\to(1-\eta)\tfrac14$ , a floor set by loss alone. Why: even an infinitely squeezed source cannot beat $(1-\eta)/4$ , so at $\eta=0.9$ the best observable squeezing is $10\log_{10}(1-\eta)=-10$ dB — efficiency, not the source, is the bottleneck. *Verified: matches the standard loss model in Leonhardt, Ch. 5; the $\eta=0.9$ floor recomputed as $-10.0$ dB.*

Limiting cases

Case	Result	Meaning
Strong local oscillator	$\hat i_-\to 2\lvert\alpha_{\mathrm{LO}}\rvert\,\hat X_\theta$	For $\lvert\alpha_{\mathrm{LO}}\rvert\gg\langle\hat a_s^\dagger\hat a_s\rangle^{1/2}$ the current is linear in the quadrature; the local oscillator is a noiseless amplifier.
Vacuum signal input	$\langle\Delta\hat i_-^2\rangle=\lvert\alpha_{\mathrm{LO}}\rvert^2$	The variance equals the shot-noise level set by the local oscillator — the calibration reference.
Heterodyne (detuned)	$\langle\Delta\hat i_{\mathrm{het}}^2\rangle=\lvert\alpha_{\mathrm{LO}}\rvert^2(1+\tfrac12)$	Reading both quadratures adds a half-quantum from the image band: a 3.01 dB penalty.
Finite efficiency	$V_{\mathrm{obs}}=\eta V_{\mathrm{sig}}+(1-\eta)\tfrac14$	Loss admits vacuum, pulling any variance toward $1/4$ ; squeezing is capped at $(1-\eta)/4$ .

Common trapSub-shot-noise means the quadrature variance drops below 1/4, not that any variance goes below zero

The homodyne current measures a field quadrature $\hat X_\theta$ , not a photon number, and its noise is referenced to the vacuum level $1/4$ . "Sub-shot-noise" or "squeezed" means one quadrature variance falls below $1/4$ while its conjugate rises to keep the product at or above $1/16$ — never that any variance becomes negative, which is impossible. Two further traps: mode-matching is everything, since only the signal component in the exact spatial, temporal, and polarization mode of the local oscillator is detected, so imperfect overlap acts as loss, rescaling $\hat X\to\sqrt\eta\,\hat X+\sqrt{1-\eta}\,\hat X_{\mathrm{vac}}$ ; and a squeezing figure in dB is meaningless without stating the efficiency, since $(1-\eta)/4$ silently caps what any source can show.

Worked examples

1 · The local oscillator sets the shot-noise floor (confidence builder)

A balanced homodyne receiver runs a local oscillator of $P_{\mathrm{LO}}=4\ \mathrm{mW}$ at $\lambda=1064\ \mathrm{nm}$ (the Nd:YAG line used in gravitational-wave interferometers). Find the photon flux, the DC photocurrent per diode at unit efficiency, and the shot-noise current spectral density.

Step 1. The energy per photon is $E=hc/\lambda=(6.626\times10^{-34})(2.998\times10^8)/(1.064\times10^{-6})=\mathbf{1.87\times10^{-19}\ J}$ , i.e. $1.17\ \mathrm{eV}$ .

Step 2. The flux is $\Phi=P_{\mathrm{LO}}/E=(4\times10^{-3})/(1.87\times10^{-19})=2.14\times10^{16}\ \mathrm{s^{-1}}$ , so the DC photocurrent at unit efficiency is $I=e\Phi=(1.602\times10^{-19})(2.14\times10^{16})=\mathbf{3.4\times10^{-3}\ A}=3.4\ \mathrm{mA}$ .

Step 3. The shot-noise spectral density is $i_N=\sqrt{2eI}=\sqrt{2(1.602\times10^{-19})(3.4\times10^{-3})}=\mathbf{3.3\times10^{-11}\ A/\sqrt{Hz}}\approx33\ \mathrm{pA}/\sqrt{Hz}$ . Since $i_N\propto\sqrt{P_{\mathrm{LO}}}$ , the shot-noise power grows linearly with local-oscillator power, exactly as $\langle\Delta\hat i_-^2\rangle=|\alpha_{\mathrm{LO}}|^2$ predicts — so raising $P_{\mathrm{LO}}$ lifts the quantum floor above any fixed electronic noise. The receiver runs at 3.4 mA with a 33 pA/√Hz shot-noise floor, scaling linearly with LO power.

2 · How loss caps observable squeezing

A source produces $10\ \mathrm{dB}$ of squeezing ( $V_{\mathrm{sig}}^-=0.025$ , anti-squeezed $V_{\mathrm{sig}}^+=2.5$ , vacuum $=0.25$ ), detected at efficiency $\eta=0.90$ . What is observed, and what efficiency is needed to ever see $6\ \mathrm{dB}$ ?

Step 1. Apply $V_{\mathrm{obs}}=\eta V_{\mathrm{sig}}+(1-\eta)\tfrac14$ to the squeezed quadrature: $V_{\mathrm{obs}}^-=0.90(0.025)+0.10(0.25)=0.0475$ , i.e. $10\log_{10}(0.0475/0.25)=\mathbf{-7.2\ dB}$ — the $10\ \mathrm{dB}$ source shows only $7.2\ \mathrm{dB}$ .

Step 2. The anti-squeezed quadrature dilutes less: $V_{\mathrm{obs}}^+=0.90(2.5)+0.10(0.25)=2.275$ , i.e. $\mathbf{+9.6\ dB}$ . Loss makes the ellipse asymmetric — the dip shrinks toward shot noise faster than the peak grows.

Step 3. For $-6\ \mathrm{dB}$ the observed variance must reach $0.25\times10^{-0.6}=0.0628$ ; even an infinitely squeezed source gives $V_{\mathrm{obs}}^-\to(1-\eta)/4$ , so $(1-\eta)/4\le0.0628$ requires $\eta\ge0.75$ . The 10 dB source reads −7.2 dB at 90% efficiency, and no source can reach −6 dB below about 75% efficiency.

Physics of the portrait

Balanced homodyne detection is a rare instrument that is also a theorem: a beam splitter, two photodiodes, and a subtraction turn the abstract operator $\hat X_\theta$ into a voltage, and turn the Wigner function from a calculational device into something you can photograph.

Origins · 1 minbalanced homodyne detection and how this was discovered

balanced homodyne detection

the local-oscillator readout of a single field quadrature X̂_θ · quantum optics and quantum measurement

The idea that a strong reference beam could make a weak field's phase measurable is old — optical beat notes were seen with thermal light in 1955 — but its quantum form belongs to the 1970s and 1980s. Horace Yuen's 1976 theory of two-photon coherent (squeezed) states reframed detection around the quadratures $\hat X_0,\hat X_{\pi/2}$ , and Yuen and Vincent Chan's 1983 proposal of balanced homodyne detection supplied the decisive trick: split the output, detect both ports, and subtract, so the local oscillator's classical noise cancels while its amplitude amplifies one quadrature. Yuen and Jeffrey Shapiro showed that heterodyne detection must pay a 3 dB penalty — the optical face of the Arthurs–Kelly limit. The deepest turn came in 1993, when Michael Raymer's group (Smithey, Beck, Raymer, and Faridani) realized the phase-scanned histograms are Radon projections of the Wigner function and inverted them to reconstruct a squeezed state's phase-space portrait, building on the insight of Vogel and Risken. What began as hearing a beat note became the standard method for measuring the complete quantum state of light — and today the same readout, fed with squeezed vacuum, sharpens gravitational-wave interferometers below the standard quantum limit.

History

1955

— Forrester, Gudmundsen and Johnson demonstrate photoelectric mixing of light

1962

— laser sources make optical heterodyne detection practical

1976

— Yuen introduces two-photon coherent (squeezed) states and quadrature measurement

1983

— Yuen and Chan propose balanced homodyne detection with a strong local oscillator

1993

— Smithey, Beck, Raymer and Faridani reconstruct a Wigner function by homodyne tomography

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 6 · Bachor and Ralph, A Guide to Experiments in Quantum Optics, 2nd ed., Ch. 8–9 · Leonhardt, Measuring the Quantum State of Light, Ch. 3–5 · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 24 · Gerry and Knight, Introductory Quantum Optics, Ch. 5–7.

Provenance. A verifier should recompute: (1) the difference current $\hat i_-=\hat a_s^\dagger\hat a_{\mathrm{LO}}+\hat a_{\mathrm{LO}}^\dagger\hat a_s$ and the exact cancellation of intensity terms — Loudon Ch. 6; (2) the strong-LO reduction to $2|\alpha_{\mathrm{LO}}|\hat X_\theta$ and vacuum noise $|\alpha_{\mathrm{LO}}|^2$ — Bachor and Ralph Ch. 8; (3) the 3.01 dB penalty, $10\log_{10}2$ — Yuen and Chan, Opt. Lett. 8, 177 (1983); (4) the loss model $V_{\mathrm{obs}}=\eta V_{\mathrm{sig}}+(1-\eta)/4$ and the $-10$ dB cap at $\eta=0.9$ — Leonhardt Ch. 5; (5) the histogram–Wigner-marginal relation and inverse Radon reconstruction — Smithey et al., Phys. Rev. Lett. 70, 1244 (1993). Convention: $[\hat X_0,\hat X_{\pi/2}]=i/2$ , vacuum variance $1/4$ ; dB values are $10\log_{10}$ of variance ratios. Worked-example anchors ( $E=1.87\times10^{-19}$ J at 1064 nm; $3.4$ mA and $33$ pA/√Hz at 4 mW; $-7.2$ dB from a 10 dB source at $\eta=0.9$ ) are recomputable.

Prerequisites. Quantization of the Electromagnetic Field · Quadrature Operators and the Uncertainty Relation · Coherent States · Vacuum State and Zero-Point Energy. Leads to. Squeezed States of Light · Phase-Space Quasiprobability Distributions · Photon Statistics and the Mandel Q.

Ask the tutor: ask me to walk through why the balanced difference current cancels the local-oscillator noise but not the signal beat, or to reconstruct a Wigner function from a scanned set of quadrature histograms step by step.

Why does the local oscillator have to be strong, and where does its power go?

It does two jobs: its phase selects a quadrature, its amplitude amplifies it. The beat scales as $2|\alpha_{\mathrm{LO}}|$ , lifting a single-photon-scale field into a macroscopic current, and the shot-noise floor grows with local-oscillator power so it can be pushed above any fixed electronic noise. The power is not "used up" on the signal — it flows to the detectors as the large common-mode current that balanced subtraction discards, leaving only the tiny modulation, the beat that carries your quadrature.

Doesn't the local oscillator add its own quantum noise?

Its amplitude noise is cancelled and its phase is what you deliberately read, so ideally neither corrupts the result. Balanced subtraction removes the local oscillator's classical and quantum intensity fluctuations in common mode, because they appear with the same sign at both ports. What survives is the beat of the local oscillator against whatever enters the signal port — vacuum, if nothing else — exactly the shot-noise limit. In practice it must be clean at the sideband frequency and well mode-matched, since imbalance or phase jitter reintroduces classical noise.

What exactly does "sub-shot-noise" mean, and is anything going below zero?

Nothing goes below zero. The shot-noise level is the vacuum quadrature variance, $1/4$ in the convention with $[\hat X_0,\hat X_{\pi/2}]=i/2$ . A squeezed state has one quadrature below $1/4$ and, necessarily, its conjugate above $1/4$ , so the product stays at or above $1/16$ . "Sub-shot-noise" names that dip of one quadrature below the vacuum line, seen directly when you scan the phase. Variances stay positive; squeezing merely redistributes the unavoidable fluctuation out of the quadrature you care about and into the one you do not.

When should I use homodyne and when heterodyne?

Use homodyne when you know the signal frequency and want the lowest noise on one quadrature — measuring squeezing, reading a gravitational-wave interferometer, anywhere 3 dB matters and you can phase-lock the local oscillator. Use heterodyne when the frequency is unknown, drifting, or must be band-selected, or when you need both quadratures from one record — spectroscopy, comb beat notes, coherent communications, LIDAR. The trade is fixed: heterodyne buys the full complex amplitude and frequency agility at a mandatory 3.01 dB cost, while homodyne gives one quadrature at the true shot-noise limit but needs a phase-locked local oscillator.

Why can't heterodyne beat the vacuum limit on both quadratures at once?

Because the two quadratures do not commute, $[\hat X_0,\hat X_{\pi/2}]=i/2$ , they cannot both be measured with vacuum-limited precision in one shot — the Arthurs–Kelly theorem. Physically, reading both quadratures at a beat frequency forces the detector to accept two local-oscillator sidebands, and the image sideband contributes an independent vacuum mode whose fluctuations add the half-quantum per quadrature the uncertainty relation demands. You can measure one quadrature at the vacuum limit (homodyne) or both at once with a 3 dB penalty (heterodyne), but not both each at the single-quadrature limit.

How does scanning one phase knob give the whole quantum state?

Each phase $\theta$ projects the state's phase-space distribution onto one axis, and the histogram of outcomes at that $\theta$ is the marginal $\mathrm{Pr}(x_\theta)$ of the Wigner function along it. Collect these marginals for every $\theta$ from $0$ to $\pi$ and you have the complete Radon transform of $W$ — the same projections a CT scanner collects around a patient. Inverting it (filtered backprojection, or a maximum-likelihood estimate that enforces a physical density matrix) reconstructs $W(x,p)$ and hence the full state — one knob suffices because a half-turn of the axis exposes every phase-space direction.