Interferometers

An optical interferometer converts a relative phase into a probability distribution over detector outcomes. Its essential operations are coherent splitting, phase accumulation, recombination, and readout. Quantum optics adds a precise state and measurement model to this familiar wave-optics sequence: the input can be a coherent, number, squeezed, or entangled state, and the output record has state-dependent fluctuations even when every optical component is ideal.

The Mach–Zehnder interferometer is the canonical two-path example. Two balanced beam splitters surround a differential phase shift. With one photon at the input, output clicks sample complementary sinusoidal probabilities. With coherent light, the output counts are Poissonian and the local shot-noise-limited phase uncertainty scales as the inverse square root of the detected photon number. Squeezed vacuum in the unused port can reduce the relevant quadrature noise, but loss, phase jitter, anti-squeezing, and measurement backaction determine the usable improvement.

The phase response, input state, measurement, and photon budget together determine sensitivity; fringe visibility alone does not.

Required background. Beam Splitters supplies the two-mode unitary, Coherent Light explains Poisson output counts, and Fisher Information defines statistical sensitivity. Mach–Zehnder Interferometry develops global phase estimation and comparisons at fixed probe resources.

Mach–Zehnder Interferometer

Mode convention

Let $\hat a$ and $\hat b$ be input modes, $\hat u$ and $\hat \ell$ the upper and lower arm modes, and $\hat c$ and $\hat d$ the output modes. Use the real balanced beam-splitter matrix

$$B = \frac1{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}, \qquad B^2=I.$$

This Hadamard convention is related by port phases to the symmetric matrix with reflection amplitude $i/\sqrt2$. All predictions below are unchanged when a different convention is propagated consistently through the complete interferometer.

Assign the differential arm phase symmetrically:

$$P(\phi) = \begin{pmatrix} e^{-i\phi/2}&0\\ 0&e^{i\phi/2} \end{pmatrix}.$$

A phase applied to only one arm differs from this by an overall common phase, which does not affect output counting probabilities. Common and differential phases should nevertheless be kept separate when an external phase reference is present.

The full annihilation-operator map is

$$\begin{pmatrix} \hat c\\ \hat d \end{pmatrix} = S_{\mathrm{MZI}}(\phi) \begin{pmatrix} \hat a\\ \hat b \end{pmatrix},$$

where

$$\begin{aligned} S_{\mathrm{MZI}}(\phi) &= B P(\phi) B \\ &= \begin{pmatrix} \cos(\phi/2)&-i\sin(\phi/2)\\ -i\sin(\phi/2)&\cos(\phi/2) \end{pmatrix}. \end{aligned}$$

The transformation is unitary for every real $\phi$. At $\phi=0$, the two balanced mixers undo one another. At $\phi=\pi$, the input ports are swapped up to phase.

Differential phase is the signal

For a monochromatic field of vacuum wavelength $\lambda$ and wavenumber $k_0=2\pi/\lambda$, a nondispersive single-pass optical path difference $\Delta(nL)=n_\ell L_\ell-n_uL_u$ gives

$$\phi = k_0\Delta(nL).$$

This sign follows from propagation amplitudes $e^{ik_0nL}$ with temporal convention $e^{-i\omega t}$ and the displayed matrix $P$. Reversing the arm subtraction reverses the sign of $\phi$, while leaving count fringes unchanged. The interferometer measures this relative phase, not the absolute phase of one isolated field. If the source, propagation, or detector supplies an external phase reference, a common phase can become relevant through that larger comparison and must be included in the model.

For broadband light,

$$\phi \longrightarrow \phi(\omega),$$

and the detector may average over frequency-dependent fringes. A scalar phase is justified only when dispersion and group-delay mismatch are negligible over the detected mode or are included in the estimator.

SU(2) form

Define

$$\hat J_z = \frac12 \left( \hat n_u-\hat n_\ell \right).$$

The arm phase is generated by

$$\hat U_\phi = e^{-i\phi\hat J_z}.$$

The first beam splitter rotates the two-mode Schwinger pseudospin, the arms accumulate a $J_z$ phase, and the second beam splitter rotates the phase information into a number difference. Up to convention-dependent axes and global phases, the complete Mach–Zehnder acts as one SU(2) rotation.

This representation is valuable because a fixed total photon-number sector $N$ is a spin-$j=N/2$ space. Different optical input states correspond to different pseudospin states and therefore have different phase response and noise.

Top: the stated $BP(\phi)B$ convention, including both arm phases. Bottom: ideal single-photon probabilities $p_c=(1+\cos\phi)/2$ and $p_d=(1-\cos\phi)/2$. At $\phi=\pi/2$ both outputs have probability $1/2$ and the difference has maximum slope. At a dark fringe the count response is quadratic and cannot determine the sign of a small phase without another setting or reference.

Single-Photon Interference

Output state

For one photon in input $a$ and vacuum in $b$, the Schrödinger output state in the chosen convention is

$$|\psi_{\mathrm{out}}(\phi)\rangle ={} \cos\left(\frac\phi2\right) |1,0\rangle - i\sin\left(\frac\phi2\right) |0,1\rangle.$$

Ideal output counting gives

$$\begin{aligned} p_c(\phi) &= \cos^2\left(\frac\phi2\right) = \frac{1+\cos\phi}{2}, \\ p_d(\phi) &= \sin^2\left(\frac\phi2\right) = \frac{1-\cos\phi}{2}. \end{aligned}$$

The probabilities sum to one. At $\phi=0$, output $c$ is bright and $d$ is dark; at $\phi=\pi$, their roles reverse. Each run produces one localized click, while repeated runs reconstruct the fringe.

Open and closed arrangements

After the first beam splitter, the photon occupies a coherent superposition of arm modes. Removing the second splitter measures in the arm basis and does not convert the relative phase into output populations. Inserting the second splitter measures in a phase-sensitive superposition basis.

This distinction is a change of measurement, not a retroactive change in what happened earlier. Delayed-choice interpretation is developed at Delayed-Choice Experiments.

Visibility

A common empirical model is

$$\begin{aligned} p_c(\phi) &= \frac12 \left[ 1+\mathcal V \cos(\phi+\phi_0) \right], \\ p_d(\phi) &= \frac12 \left[ 1-\mathcal V \cos(\phi+\phi_0) \right], \end{aligned}$$

where $0\leq\mathcal V\leq1$. The offset $\phi_0$ fixes the instrument’s reference phase. Reduced visibility can result from amplitude imbalance, mode mismatch, bandwidth averaging, unresolved path markers, or random phase noise. A fitted $\mathcal V$ alone does not identify which mechanism is responsible.

From Fringes to Phase Estimates

Local error propagation

Suppose an observable $\hat M$ has mean $m(\phi)$ and standard deviation $\Delta M$. Near a calibrated operating point $\phi_0$, the local error-propagation estimate is

$$\Delta\phi_{\mathrm{ep}} = \frac{ \Delta M }{ \left| \partial_\phi m \right| }_{\phi=\phi_0}.$$

This formula is a linearized property of a chosen observable. It is not a universal lower bound, and it can fail at a fringe extremum where the slope vanishes, for a biased estimator, or when the likelihood is multimodal.

An interferometer is commonly locked near a point with a large, stable slope. Feedback keeps the residual phase inside the locally invertible region, and calibration converts the error signal into physical phase.

Binary-count Fisher information

For one detected photon with the ideal probabilities above, the classical Fisher information is

$$\mathcal I_1(\phi) = \sum_{j=c,d} \frac{ \left[ \partial_\phi p_j(\phi) \right]^2 }{ p_j(\phi) }.$$

Away from the exact zero-probability endpoints, direct substitution gives

$$\mathcal I_1(\phi)=1.$$

The neighboring-phase limit is also one at the endpoints, but this must not be identified with an ordinary regular Fisher information at the exact dark fringe. There one outcome has zero probability and signed phase is locally ambiguous: the usual regularity and local-unbiasedness conditions fail. For $\nu$ independent detected photons at a regular operating point,

$$\mathcal I_\nu=\nu,$$

the local Cramér–Rao bound for an unbiased estimator is

$$\Delta\phi \geq \frac1{\sqrt\nu}.$$

This calculation makes the resource explicit: $\nu$ is the number of independent detected one-photon trials. It does not include uncounted losses, state-preparation overhead, or a cost for finding the correct fringe.

Visibility reduces information

For the visibility model, the Fisher information per detected event is

$$\mathcal I_1(\phi) = \frac{ \mathcal V^2 \sin^2(\phi+\phi_0) }{ 1- \mathcal V^2 \cos^2(\phi+\phi_0) }.$$

At quadrature,

$$\phi+\phi_0 = \frac{\pi}{2},$$

so

$$\mathcal I_1=\mathcal V^2.$$

With $N_{\mathrm{det}}$ independent detected photons, the corresponding local bound is

$$\Delta\phi \geq \frac{ 1 }{ \mathcal V\sqrt{N_{\mathrm{det}}} }.$$

Unequal efficiencies and background counts require the full detector likelihood; they cannot always be represented by one visibility.

Quantum Fisher information

For a pure state that acquires phase through $e^{-i\phi\hat J_z}$, the quantum Fisher information is

$$F_Q = 4 \left( \Delta J_z \right)^2.$$

This is computed in the state inside the interferometer, after the first splitter and before the phase. It bounds the Fisher information of every possible output measurement:

$$\mathcal I(\phi)\leq F_Q.$$

For one photon split equally between the arms, $(\Delta J_z)^2=1/4$ and $F_Q=1$. Output photon counting therefore attains the available local information at regular phase points. The nonregular exact endpoint still needs the limiting qualification above. General proofs and multiparameter cautions are in Quantum Sensing.

Coherent Light and Shot Noise

Output state and mean counts

Send a coherent state $|\alpha\rangle$ into port $a$ and vacuum into port $b$. Let

$$\bar n=|\alpha|^2$$

be the mean input photon number in the selected measurement interval. Passive linear optics preserves the product-coherent form:

$$|\alpha,0\rangle \longmapsto{} \left| \alpha\cos\left(\frac\phi2\right) \right\rangle_c \otimes \left| -i\alpha\sin\left(\frac\phi2\right) \right\rangle_d.$$

The output means are

$$\begin{aligned} \overline n_c &= \bar n \cos^2\left(\frac\phi2\right), \\ \overline n_d &= \bar n \sin^2\left(\frac\phi2\right). \end{aligned}$$

Each output has Poisson statistics, and the two ideal coherent outputs are independent.

Difference readout

Define

$$\hat M = \hat n_c-\hat n_d.$$

Then

$$\langle\hat M\rangle = \bar n\cos\phi.$$

For independent Poisson outputs,

$$\begin{aligned} (\Delta M)^2 &= (\Delta n_c)^2 + (\Delta n_d)^2 \\ &= \overline n_c+\overline n_d = \bar n. \end{aligned}$$

At quadrature the mean common intensity cancels from the difference, but photon shot noise remains. Away from quadrature, fluctuations in the input intensity also multiply the nonzero fringe signal. At quadrature, $|\sin\phi|=1$, error propagation gives

$$\Delta\phi_{\mathrm{coh}} = \frac1{\sqrt{\bar n}}.$$

This is the coherent-state shot-noise scaling for one measurement interval. After many equal intervals, replace $\bar n$ by the total detected coherent photon number when the trials are independent and the phase is stable.

Full Poisson likelihood

The Fisher information in both output counts is

$$\mathcal I_{\mathrm{P}} = \sum_{j=c,d} \frac{ \left( \partial_\phi\overline n_j \right)^2 }{ \overline n_j }.$$

For the ideal coherent input,

$$\mathcal I_{\mathrm{P}}=\bar n.$$

The neighboring-phase limit remains finite at a dark fringe even though difference-signal error propagation has zero slope there. The exact point is nonregular, so this limit alone does not guarantee the usual unbiased estimator bound. Near a dark port, the small mean count is quadratic in phase, and a full likelihood can estimate $|\phi|$. It cannot determine the sign without an offset, phase modulation, homodyne reference, or prior information.

What shot noise means

For an ideal coherent state, photon counts in a fixed mode and interval obey

$$\operatorname{Var}(N) = \langle N\rangle.$$

This Poisson variance is called optical shot noise. It remains when classical intensity noise, electronic noise, and environmental drift have been removed. “Shot-noise limited” means that the measured uncertainty agrees with a declared quantum-statistical model over a stated frequency and power range. It does not mean zero noise.

The phrase “standard quantum limit” is context dependent:

• for independent optical probes, it often means $1/\sqrt N$ phase scaling;
• for continuous displacement sensing, it can mean an optimum between measurement imprecision and radiation-pressure backaction;
• for clocks or spins, it often refers to projection noise of uncorrelated particles.

These baselines should not be conflated. The relevant detector, generator, bandwidth, and resource count must be named.

Standard Quantum Limit derives the independent-probe baseline and explains which resources enter a comparison. Here the Poisson likelihood makes that baseline concrete.

Classical and technical noise

Real count or photocurrent variance can be written schematically as

$$V_{\mathrm{meas}} = V_{\mathrm{shot}} + V_{\mathrm{technical}} + V_{\mathrm{electronic}},$$

provided the contributions are independent and expressed in compatible units. Technical terms can include laser intensity noise, frequency noise converted by arm mismatch, vibration, thermal drift, beam pointing, scattered-light interference, and detector gain fluctuations.

Power scaling helps diagnose them. Coherent shot-noise amplitude grows as the square root of optical power, while many classical amplitude-noise terms grow linearly with power. That heuristic is not a substitute for calibrated noise injection and transfer-function measurements.

Phase Noise and Imperfect Contrast

Random phase

Let the actual phase be

$$\phi_{\mathrm{act}} = \phi+\delta\phi.$$

For zero-mean Gaussian phase noise with variance $\sigma_\phi^2$,

$$\left\langle e^{i\delta\phi} \right\rangle = e^{-\sigma_\phi^2/2}.$$

The averaged fringe is therefore

$$\langle\cos\phi_{\mathrm{act}}\rangle = e^{-\sigma_\phi^2/2} \cos\phi.$$

This average depends on the sampled angle distribution, not solely on whether the noise is called fast or slow. If every photon samples an independent phase, the binary visibility model describes the resulting independent trials. If all photons in one pulse share a random phase, their outcomes are correlated after averaging: the same mean fringe does not imply the same Fisher information or count variance. At quadrature, coherent pulses with mean $\bar n$ and independent pulse-to-pulse Gaussian jitter obey

$$\operatorname{Var}M=\bar n+\frac{\bar n^2}{2}(1-e^{-2\sigma_\phi^2}).$$

The second term is classical phase noise shared within a pulse. Slow drift between successive pulses can additionally correlate different measurement intervals. Specify these correlations before multiplying single-event information by the number of detected photons.

Mode mismatch

Only overlapping output modes interfere. If one contribution decomposes into a component parallel to the other and an orthogonal remainder, the parallel component contributes to the fringe while the orthogonal part adds incoherent power. Spatial, temporal, spectral, and polarization overlap all matter.

A visibility loss is not automatically decoherence of the source. It may be ordinary mismatch in the measurement basis. Conversely, excellent average spatial overlap does not rule out hidden spectral or polarization markers.

Loss

Equal phase-independent loss after the interferometer reduces the detected photon number:

$$N_{\mathrm{det}} = \eta N_{\mathrm{in}}.$$

For a coherent input, the detected state remains coherent and the local phase uncertainty becomes

$$\Delta\phi_{\mathrm{loss}} = \frac1{\sqrt{\eta N_{\mathrm{in}}}}.$$

Unequal arm loss reduces visibility. For balanced mixers, coherent input, and ideal counting at both outputs, quadrature remains optimal; additional asymmetries or background can shift the best operating point. Loss inside the arms can also carry path information into environmental modes. A trustworthy model places each loss channel at its physical location rather than attaching one efficiency to the final formula.

Squeezed-Input Improvement

Dark-port linearization

Take $\alpha$ real and bright in input $a$, and inject a zero-mean state into input $b$. Near the dark operating point $\phi=0$,

$$\begin{aligned} \hat d_{\mathrm{out}} &= -i\sin\left(\frac\phi2\right) \hat a + \cos\left(\frac\phi2\right) \hat b \\ &\approx \hat b-\frac{i\phi}{2}\hat a. \end{aligned}$$

Define

$$\hat P_d = \frac{ \hat d-\hat d^\dagger }{ i\sqrt2 }.$$

Replacing the bright carrier by its mean to leading order gives

$$\langle\hat P_d\rangle \approx -\frac{\alpha\phi}{\sqrt2}.$$

Vacuum in port $b$ has

$$\operatorname{Var}(P_b)=\frac12,$$

which recovers

$$\Delta\phi_{\mathrm{vac}} = \frac1{|\alpha|} = \frac1{\sqrt{\bar n}}.$$

If the measured quadrature of port $b$ is squeezed by $r$,

$$\operatorname{Var}(P_b) = \frac12e^{-2r},$$

then the ideal local uncertainty is

$$\Delta\phi_{\mathrm{sq}} = \frac{ e^{-r} }{ \sqrt{\bar n} }.$$

This is the Caves dark-port mechanism in its simplest linearized form. Here $\bar n=|\alpha|^2$ counts carrier photons; a squeezed-vacuum input adds $\sinh^2r$ photons to the total probe mean. The homodyne reference is also an explicit part of the measurement setup. Direct photon counting exactly at a dark fringe has a quadratic response; homodyne readout, DC offset, RF modulation, or another phase reference provides a signed linear error signal.

Squeezing-angle error

If the squeezed ellipse is misaligned from the measured quadrature by $\epsilon$, the injected variance is

$$\begin{aligned} V(\epsilon) = \frac12 \left( e^{-2r}\cos^2\epsilon + e^{2r}\sin^2\epsilon \right). \end{aligned}$$

The anti-squeezed term grows rapidly with $r$. Increasing source squeezing without improving phase control can therefore worsen the measured noise.

Loss mixes in vacuum

For net efficiency $\eta$, a pure-loss channel gives

$$V_{\mathrm{det}} = \frac12 \left[ \eta e^{-2r} + 1-\eta \right].$$

If the same efficiency attenuates the signal slope, the local phase uncertainty becomes

$$\Delta\phi_{\mathrm{sq,loss}} = \frac{ \sqrt{ 1-\eta+\eta e^{-2r} } }{ \sqrt{\eta\bar n} }.$$

As $r\to\infty$, loss leaves a vacuum floor. Quoting source squeezing before propagation is therefore insufficient; the relevant quantity is the quadrature variance in the detected mode at the readout frequency.

Backaction and frequency dependence

In a static phase model, more carrier photons and stronger phase squeezing both improve imprecision. A movable mirror changes the problem. Carrier amplitude fluctuations exert radiation-pressure force, and squeezing the phase quadrature anti-squeezes the conjugate amplitude quadrature. At high power or low frequency, increased backaction can offset the imprecision gain.

Filter cavities and frequency-dependent readout can rotate the squeezing ellipse so that different quadratures are useful in different spectral regions. The full force-measurement tradeoff is developed in Optomechanics.

Other nonclassical inputs

Twin-Fock, Holland–Burnett, NOON, and other entangled number states can have larger phase-generator variance than independent photons. In an ideal fixed-$N$ model, suitably chosen states and measurements can approach $1/N$ local scaling.

That statement has strict qualifications:

• phase may be ambiguous over multiple fringes without prior information;
• preparation and detection resources must be counted;
• photon loss can destroy high-order coherence;
• the measurement that attains quantum Fisher information may be difficult;
• asymptotic scaling does not guarantee an advantage at finite $N$;
• indefinite-number resource constraints require special care.

This page does not assign one universal “Heisenberg limit.” The operational comparison is between complete protocols under the same declared resources and noise.

Readout Strategies

Two-port photon counting

Counting both outputs uses energy conservation and rejects some common-mode fluctuations. Number-resolving detectors provide the full joint count distribution; threshold detectors generally lose information at higher occupation. Detector efficiency, dark counts, dead time, saturation, and afterpulsing belong in the likelihood.

The dedicated Photon Counting page develops continuous records and jumps. Optical Photon Counting derives laboratory counting POVMs and detector corrections.

Balanced homodyne

Homodyne detection mixes the signal output with a strong phase reference and measures a selected quadrature. It is natural for dark-port phase signals and squeezed inputs because it can choose the low-noise quadrature directly. Local-oscillator mode mismatch is equivalent to vacuum loss only when the additional selected modes are empty; otherwise their noise enters. Phase error can mix in anti-squeezing.

Homodyne Detection describes stochastic records and conditional states. Homodyne and Heterodyne Detection explains the optical receiver and its calibration.

Modulation and lock-in readout

A small known phase modulation can move the signal away from low-frequency technical noise and provide a signed error signal around a dark fringe. Demodulation estimates the response at the modulation frequency. The modulation depth, sideband transfer, detector bandwidth, and demodulation phase become part of the calibration.

Adaptive phase estimation

When the phase is not already localized to one fringe, a fixed operating point can be inefficient or ambiguous. Adaptive protocols update a controllable reference phase using previous outcomes. Bayesian or maximum likelihood estimators can then combine information across settings.

Adaptive performance depends on the prior, control latency, estimator, and loss. A local Cramér–Rao bound does not by itself certify globally reliable phase acquisition.

Applications to Metrology

Displacement

For one-way propagation in vacuum,

$$\phi = \frac{2\pi}{\lambda} \Delta L.$$

In a simple Michelson round trip, a mirror displacement $\Delta x$ changes the optical path by $2\Delta x$, giving

$$\phi = \frac{4\pi}{\lambda} \Delta x.$$

The inferred displacement uncertainty is

$$\Delta x = \frac{\lambda}{4\pi} \Delta\phi$$

for that ideal double-pass geometry. Cavities, folded paths, incidence angle, dispersion, and control loops modify the transfer function.

Refractive index and spectroscopy

A sample of length $L$ and refractive index $n(\omega)$ contributes

$$\phi(\omega) = \frac{\omega}{c} n(\omega)L.$$

Comparing it with a reference arm can reveal index changes, dispersion, absorption-induced contrast loss, or resonant phase shifts. Near a resonance, the full complex transfer function is required because amplitude and phase responses are linked and detector saturation or technical laser noise may dominate.

Rotation

Counterpropagating waves in a Sagnac loop enclose area $A$. For rotation component $\Omega$ normal to the loop, the ideal vacuum phase is

$$\phi_{\mathrm{Sag}} = \frac{ 8\pi A\Omega }{ \lambda c }.$$

Fiber coils increase effective area through multiple turns. Real gyroscopes must model polarization, backscatter, thermal gradients, scale-factor drift, and nonreciprocal phase shifts.

Gravitational-wave detectors

Kilometer-scale detectors use power-recycled Michelson interferometers with Fabry–Perot arm cavities, multiple control degrees of freedom, and frequency-dependent response. A gravitational wave produces a tiny differential arm-length signal. Quantum shot noise, radiation-pressure backaction, optical loss, mirror thermal noise, coating noise, seismic motion, and control noise contribute in different frequency bands.

Tse and collaborators demonstrated squeezed-vacuum enhancement in Advanced LIGO in 2019. Its effect was measured in the detector strain-noise spectrum; a source squeezing parameter alone does not specify an observatory’s sensitivity or observing duty cycle.

General transduction

If a physical parameter $\theta$ produces phase $\phi(\theta)$, then local uncertainties obey

$$\Delta\theta = \frac{ \Delta\phi }{ \left| \partial_\theta\phi \right| }.$$

Improvement can come from a steeper transduction coefficient, lower phase noise, more independent resources, or a better estimator. Calling every improvement “quantum enhanced” obscures which part of the measurement actually changed.

Bandwidth and Calibration

Dynamic response

A real interferometer has a frequency-dependent transfer function. In the Fourier domain, write

$$y(\Omega) = H_{\phi}(\Omega) \phi(\Omega) + n_y(\Omega),$$

where $y$ is the detector record, $H_\phi$ converts phase to output units, and $n_y$ is additive output noise. Calibration estimates $H_\phi$ and its uncertainty.

A static slope is sufficient only when the signal frequencies lie well inside the flat, linear response region. Cavity storage time, detector bandwidth, digital filters, feedback loops, and propagation delay can all alter magnitude and phase.

Spectral sensitivity

Fix the Fourier convention before integrating a noise spectrum. For a stationary real phase fluctuation, define the two-sided angular-frequency PSD by

$$C_\phi(\tau)=\int_{-\infty}^{\infty}\frac{d\Omega}{2\pi} S_\phi^{(2)}(\Omega)e^{-i\Omega\tau}.$$

The variance after a filter $F$ is

$$\sigma_\phi^2=\int_{-\infty}^{\infty}\frac{d\Omega}{2\pi} |F(\Omega)|^2S_\phi^{(2)}(\Omega).$$

For a one-sided PSD per hertz, use $f>0$ and $S_{\phi,f}^{(1)}(f)=2S_\phi^{(2)}(2\pi f)$; the same variance is $\int_0^\infty df\,|F(2\pi f)|^2S_{\phi,f}^{(1)}(f)$. Its amplitude spectral density has units $\mathrm{rad}/\sqrt{\mathrm{Hz}}$. Thus the positive-frequency factor of two and the $d\Omega/(2\pi)$ measure are separate conventions. A spectrum also needs the measurement bandwidth and averaging filter before it becomes an rms uncertainty.

Calibration chain

A defensible calibration records:

1. a traceable actuator or reference phase;
2. the optical response from actuator to detected signal;
3. detector gain, linearity, and saturation;
4. digital filtering and timing;
5. uncertainty and drift in every conversion;
6. injections that test the model over the signal band.

The final physical uncertainty combines statistical noise with calibration and systematic uncertainty. More photons reduce only the terms that truly average with photon number.

Exercises

1. Multiply the Mach–Zehnder matrices

Using

$$B = \frac1{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}$$

and

$$P(\phi) = \begin{pmatrix} e^{-i\phi/2}&0\\ 0&e^{i\phi/2} \end{pmatrix},$$

derive $S_{\mathrm{MZI}}=BPB$. Check $\phi=0$ and $\phi=\pi$.

Solution

First multiply on the right:

$$P B = \frac1{\sqrt2} \begin{pmatrix} e^{-i\phi/2}&e^{-i\phi/2}\\ e^{i\phi/2}&-e^{i\phi/2} \end{pmatrix}.$$

Let

$$z=e^{-i\phi/2}.$$

Then

$$\begin{aligned} B P B &= \frac12 \begin{pmatrix} z+z^* & z-z^* \\ z-z^* & z+z^* \end{pmatrix} \\ &= \begin{pmatrix} \cos(\phi/2)&-i\sin(\phi/2)\\ -i\sin(\phi/2)&\cos(\phi/2) \end{pmatrix}. \end{aligned}$$

At $\phi=0$, this is the identity. At $\phi=\pi$,

$$S_{\mathrm{MZI}}(\pi) = \begin{pmatrix} 0&-i\\ -i&0 \end{pmatrix},$$

which swaps the ports and attaches a convention-dependent phase $-i$.

2. Single-photon Fisher information

For

$$p_c = \cos^2\left(\frac\phi2\right), \qquad p_d = \sin^2\left(\frac\phi2\right),$$

calculate the Fisher information at regular phase points. Why does its limit not justify a regular signed-phase bound at an exact dark fringe?

Solution

The derivatives are

$$\partial_\phi p_c = -\frac12\sin\phi, \qquad \partial_\phi p_d = \frac12\sin\phi.$$

Therefore

$$\mathcal I_1 ={} \frac{ \sin^2\phi }{ 4\cos^2(\phi/2) } + \frac{ \sin^2\phi }{ 4\sin^2(\phi/2) }.$$

Using

$$\sin^2\phi = 4 \sin^2\left(\frac\phi2\right) \cos^2\left(\frac\phi2\right)$$

gives

$$\mathcal I_1 = \sin^2\left(\frac\phi2\right) + \cos^2\left(\frac\phi2\right) =1.$$

At $\phi=0$ or $\pi$, one displayed denominator and its numerator both vanish. The likelihood is well defined but changes support at the endpoint; the displayed value one is its neighboring-phase limit. At the exact point, identical probabilities for opposite small phase shifts prevent a locally unbiased signed-phase estimator on a two-sided neighborhood. An offset, phase stepping, or a suitable one-sided prior changes the estimation problem.

3. Coherent difference-count sensitivity

A coherent state with mean input number $\bar n$ enters port $a$. Derive the mean and variance of $\hat M=\hat n_c-\hat n_d$. Find the error-propagation phase uncertainty and identify its best operating point.

Solution

The output means are

$$\begin{aligned} \overline n_c &= \bar n\cos^2\left(\frac\phi2\right), \\ \overline n_d &= \bar n\sin^2\left(\frac\phi2\right). \end{aligned}$$

Hence

$$\langle M\rangle = \overline n_c-\overline n_d = \bar n\cos\phi.$$

The two coherent outputs have independent Poisson counts, so

$$\begin{aligned} (\Delta M)^2 &= (\Delta n_c)^2 + (\Delta n_d)^2 \\ &= \overline n_c+\overline n_d = \bar n. \end{aligned}$$

Because

$$\left| \partial_\phi\langle M\rangle \right| = \bar n|\sin\phi|,$$

error propagation gives

$$\Delta\phi_{\mathrm{ep}} = \frac{ 1 }{ \sqrt{\bar n}|\sin\phi| }.$$

It is minimized at quadrature, $\phi=\pi/2$ or $3\pi/2$, where

$$\Delta\phi_{\mathrm{ep}} = \frac1{\sqrt{\bar n}}.$$

The divergence at a fringe extremum is a failure of this signed difference observable’s linear slope, not proof that the full count distribution has zero information at nearby regular phases. At the exact endpoint, the regular Fisher bound itself needs the qualification in Exercise 2.

4. Contrast and detected resources

For

$$p_\pm = \frac12 \left[ 1\pm\mathcal V\cos\phi \right],$$

derive the Fisher information per detected photon. Evaluate it at quadrature and include a phase-independent detection efficiency $\eta$ for $N_{\mathrm{in}}$ incident independent photons.

Solution

The derivatives are

$$\partial_\phi p_\pm = \mp\frac{\mathcal V}{2}\sin\phi.$$

Since

$$p_+p_- = \frac14 \left( 1-\mathcal V^2\cos^2\phi \right),$$

the binary Fisher information is

$$\mathcal I_1(\phi) = \frac{ \mathcal V^2\sin^2\phi }{ 1-\mathcal V^2\cos^2\phi }.$$

At quadrature,

$$\mathcal I_1=\mathcal V^2.$$

If loss only removes events independently, the expected detected resource is

$$N_{\mathrm{det}} = \eta N_{\mathrm{in}}.$$

The local Cramér–Rao bound becomes

$$\Delta\phi \geq \frac{ 1 }{ \mathcal V \sqrt{\eta N_{\mathrm{in}}} }.$$

This simple factorization fails when loss is arm dependent, phase dependent, or correlated with another degree of freedom.

5. Gaussian phase jitter

Let $\delta\phi$ be a zero-mean Gaussian random variable of variance $\sigma_\phi^2$. Show how averaging over it changes an ideal $\cos\phi$ fringe. What is the small-noise visibility loss?

Solution

The Gaussian characteristic function is

$$\left\langle e^{i\delta\phi} \right\rangle = e^{-\sigma_\phi^2/2}.$$

Therefore

$$\begin{aligned} \left\langle \cos(\phi+\delta\phi) \right\rangle &= \operatorname{Re} \left[ e^{i\phi} \left\langle e^{i\delta\phi} \right\rangle \right] \\ &= e^{-\sigma_\phi^2/2} \cos\phi. \end{aligned}$$

The effective visibility is

$$\mathcal V_{\mathrm{eff}} = e^{-\sigma_\phi^2/2}.$$

For $\sigma_\phi\ll1$,

$$\mathcal V_{\mathrm{eff}} \approx 1-\frac{\sigma_\phi^2}{2}.$$

This is an ensemble average. Pulse-shared jitter and independent photon-to-photon jitter can produce this same mean visibility while giving different counting likelihoods. Slow drift also correlates distinct pulses.

6. Unequal losses in the arms

Put intensity transmissions $\eta_u,\eta_\ell$ in the two arms before the final balanced mixer. For coherent input mean $\bar n$, derive the output means and the counting Fisher information at quadrature.

Solution

Replace the phase matrix by $\operatorname{diag}(\sqrt{\eta_u}e^{-i\phi/2}, \sqrt{\eta_\ell}e^{i\phi/2})$. Vacuum environmental inputs preserve the commutators, and coherent output means are

$$\mu_{c,d}=\frac{\bar n}{4} \left(\eta_u+\eta_\ell\pm2\sqrt{\eta_u\eta_\ell}\cos\phi\right).$$

The expected total count is $N_{\rm det}=\bar n(\eta_u+\eta_\ell)/2$, and the visibility is $\mathcal V=2\sqrt{\eta_u\eta_\ell}/(\eta_u+\eta_\ell)$. The independent Poisson likelihood at quadrature therefore has

$$\mathcal I=N_{\rm det}\mathcal V^2 =\frac{2\bar n\eta_u\eta_\ell}{\eta_u+\eta_\ell}.$$

For $(\eta_u,\eta_\ell)=(1/4,1)$, the detected mean is $5\bar n/8$ but the information is only $2\bar n/5$. Treating the same mean loss as equal attenuation of both outputs misses the additional contrast loss.

7. Generator variance for independent photons

After the first balanced splitter, one photon occupies

$$|\psi\rangle = \frac{ |1,0\rangle+|0,1\rangle }{ \sqrt2 }$$

inside the arms. Compute $(\Delta J_z)^2$ for $\hat J_z=(\hat n_u-\hat n_\ell)/2$. Extend the result to $N$ independent photons and obtain the quantum Fisher information.

Solution

For one photon, the two possible $J_z$ eigenvalues are $+1/2$ and $-1/2$, each with probability $1/2$. Thus

$$\langle J_z\rangle=0$$

and

$$\left( \Delta J_z \right)^2 = \frac12 \left( \frac14+\frac14 \right) = \frac14.$$

The pure-state quantum Fisher information is

$$F_Q = 4 \left( \Delta J_z \right)^2 =1.$$

For $N$ independent photons, variances add:

$$\left( \Delta J_z^{(N)} \right)^2 = \frac N4.$$

Therefore

$$F_Q^{(N)}=N, \qquad \Delta\phi\geq\frac1{\sqrt N}.$$

Correlations can change the total generator variance, but whether that variance yields an operational advantage depends on preparation, measurement, loss, and global phase ambiguity.

8. Displacement sensitivity and noise budget

A simple Michelson interferometer uses wavelength $\lambda=1064\,\mathrm{nm}$ and detects $N_{\mathrm{det}}=10^{12}$ coherent photons in one integration interval. Ignoring all other noise, estimate the shot-noise-limited mirror displacement uncertainty. Then state three reasons this is not yet a detector sensitivity claim.

Solution

The coherent phase uncertainty is

$$\Delta\phi = \frac1{\sqrt{N_{\mathrm{det}}}} = 10^{-6}\,\mathrm{rad}.$$

For a double-pass Michelson,

$$\phi = \frac{4\pi}{\lambda}x,$$

so

$$\begin{aligned} \Delta x &= \frac{\lambda}{4\pi} \Delta\phi \\ &= \frac{ 1064\times10^{-9}\,\mathrm m }{ 4\pi } 10^{-6} \\ &\approx 8.5\times10^{-14}\,\mathrm m. \end{aligned}$$

This is not yet a complete sensitivity claim because, among other issues:

• the integration time and equivalent bandwidth were not specified;
• contrast, optical loss, detector efficiency, and calibration uncertainty were assumed ideal;
• vibration, laser frequency and intensity noise, thermal noise, scattered light, and radiation-pressure backaction were omitted.

It is a useful quantum-statistical baseline for the stated detected photon resource, not a prediction for a full instrument.

9. Shared phase noise within a coherent pulse

At quadrature, every photon in a coherent pulse shares one Gaussian phase offset of variance $\sigma^2$. The offset is independent between pulses. Derive the variance of the count difference and its error-propagation phase uncertainty. Compare increasing pulse brightness with averaging independent pulses.

Solution

Conditioned on the angle, $\langle M\rangle_\delta=-\bar n\sin\delta$ and $\operatorname{Var}(M|\delta)=\bar n$. Total variance gives

$$\operatorname{Var}M=\bar n+\bar n^2\langle\sin^2\delta\rangle =\bar n+\frac{\bar n^2}{2}(1-e^{-2\sigma^2}).$$

The averaged slope has magnitude $\bar n e^{-\sigma^2/2}$, hence

$$(\Delta\phi_{\rm ep})^2=\frac{e^{\sigma^2}}{\bar n}+\sinh(\sigma^2).$$

At high brightness, one pulse retains phase variance approximately $\sigma^2$ for small jitter. Averaging $m$ independent pulses divides both terms by $m$; a common drift shared across all pulses would not average down this way. Visibility alone does not specify that correlation.

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