Squeezed Light

Squeezed light is an electromagnetic field whose fluctuations in one declared quadrature are smaller than the fluctuations of the corresponding vacuum mode. For a normalized mode with annihilation operator $\hat a$, define

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

The orthogonal quadrature is $\hat X_{\theta+\pi/2}$, and

$$\left[ \hat X_\theta, \hat X_{\theta+\pi/2} \right] = i.$$

Vacuum and coherent states have

$$\operatorname{Var}_0(X_\theta) = \frac12$$

for every $\theta$. The operational squeezing criterion in this convention is

$$\min_\theta \operatorname{Var}(X_\theta) \lt \frac12.$$

The word declared matters. Squeezing is always relative to a mode, a quadrature angle, a bandwidth, and a calibrated vacuum reference. It does not mean that every field observable is quieter, that the uncertainty principle has been evaded, or that every photon-counting statistic is sub-Poissonian.

For an ideal single-mode squeezed vacuum with squeezing strength $r$,

$$V_{\min} = \frac12e^{-2r}, \qquad V_{\max} = \frac12e^{2r},$$

so the uncertainty product remains

$$\sqrt{V_{\min}V_{\max}} = \frac12.$$

Squeezing redistributes quantum noise. The narrow quadrature is accompanied by an anti-squeezed quadrature. Angle error mixes some of that larger noise into the measurement; ordinary optical loss instead replaces part of the field with vacuum noise.

Required background. Quantized Electromagnetic Modes defines normalized optical modes, Coherent Light establishes the vacuum noise reference, Squeezed States introduces the oscillator squeeze operator, and Density Operators explains mixed states. This page connects those states to optical preparation, mode-selective measurement, and loss.

Optical Field Quadratures

A single optical mode behaves algebraically like one harmonic oscillator, but its quadratures are physical field amplitudes rather than particle coordinates. Choose

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

Then

$$\hat X_\theta = \hat X\cos\theta + \hat P\sin\theta,$$

and

$$\hat X_{\theta+\pi/2} = -\hat X\sin\theta + \hat P\cos\theta.$$

For a narrowband traveling mode, the positive-frequency electric field has the schematic form

$$\hat E^{(+)}(t) = \mathcal E \hat a e^{-i\omega t},$$

where $\mathcal E$ contains the mode normalization. Interference with a phase reference converts a chosen linear combination of $\hat E^{(+)}$ and $\hat E^{(-)}$ into $\hat X_\theta$. A quadrature is therefore not an independent polarization or a separate beam. It is a phase-selected component of one complex mode amplitude.

Equivalent normalizations

Three conventions are common:

$$\hat X = \frac{ \hat a+\hat a^\dagger }{ \sqrt2 }, \qquad \operatorname{Var}_0(X) = \frac12;$$ $$\hat x = \frac{ \hat a+\hat a^\dagger }{ 2 }, \qquad \operatorname{Var}_0(x) = \frac14;$$

and a noise-normalized convention with

$$v_\theta \equiv \frac{ \operatorname{Var}(X_\theta) }{ \operatorname{Var}_0(X_\theta) }, \qquad v_{\mathrm{vac}}=1.$$

All three describe the same physics. Problems arise only when a formula or plot is compared without converting its reference level. This page uses $\operatorname{Var}_0(X_\theta)=1/2$ for operators and $v_{\mathrm{vac}}=1$ for measured noise ratios.

Means versus fluctuations

Write the fluctuation operator as

$$\delta\hat X_\theta = \hat X_\theta - \langle\hat X_\theta\rangle.$$

Squeezing concerns

$$\operatorname{Var}(X_\theta) = \left\langle (\delta\hat X_\theta)^2 \right\rangle,$$

not the mean field. A squeezed vacuum has

$$\langle\hat a\rangle=0,$$

whereas a displaced squeezed state can be bright:

$$\langle\hat a\rangle=\alpha.$$

Both can have the same covariance ellipse. Conversely, a highly stable bright coherent beam has a large mean amplitude but vacuum-level quadrature fluctuations rather than squeezing.

Vacuum Reference and Decibels

The normalized quadrature noise is

$$v_\theta = 2\operatorname{Var}(X_\theta)$$

in the convention used here. Vacuum gives $v_\theta=1$, squeezing gives $v_\theta\lt1$, and anti-squeezing gives $v_\theta\gt1$.

Noise power relative to vacuum is often plotted as

$$L_\theta = 10\log_{10}v_\theta \ \mathrm{dB}.$$

Thus a squeezed trace lies at a negative value. Experimental prose often quotes a positive squeezing magnitude,

$$R_{\mathrm{sq}} = -10\log_{10}v_{\min} \ \mathrm{dB}.$$

For an ideal squeezed vacuum,

$$v_{\min}=e^{-2r}, \qquad v_{\max}=e^{2r},$$

so

$$R_{\mathrm{sq}} = \frac{ 20r }{ \ln 10 } \ \mathrm{dB} \approx 8.686r \ \mathrm{dB}.$$

Three decibels of squeezing means approximately half the variance:

$$10^{-3/10} \approx 0.501.$$

The corresponding standard deviation is smaller by approximately $1/\sqrt2$. Using $20\log_{10}$ directly on a variance double-counts the conversion.

Uncertainty Ellipse

For $\hat{\boldsymbol R}=(\hat X,\hat P)^{\mathsf T}$, define the covariance matrix

$$V_{jk} = \frac12 \left\langle \delta\hat R_j\delta\hat R_k + \delta\hat R_k\delta\hat R_j \right\rangle.$$

The variance along the unit vector

$$\boldsymbol u_\theta = \begin{pmatrix} \cos\theta\\ \sin\theta \end{pmatrix}$$

is

$$\operatorname{Var}(X_\theta) = \boldsymbol u_\theta^{\mathsf T} V \boldsymbol u_\theta.$$

The eigenvectors of $V$ are the principal quadrature axes. Its eigenvalues are $V_{\min}$ and $V_{\max}$. For one canonical mode, the uncertainty principle requires

$$\det V \ge \frac14.$$

An ideal pure squeezed vacuum saturates the bound. A mixed squeezed state can still satisfy $V_{\min}\lt1/2$ while having

$$\det V \gt \frac14.$$

Its ellipse has more area than the minimum-uncertainty ellipse.

For $r=\ln2$, the squeezed vacuum has $V_X=1/8$ and $V_P=2$. The ellipse’s semiaxes are standard deviations $\sqrt{V_X}$ and $\sqrt{V_P}$, not variances or diameters; the dashed vacuum circle has radius $1/\sqrt2$. The lower panel plots $v_\theta=2V_\theta$. Pure loss with $\eta=3/4$ raises its minimum from $1/4$ to $7/16$ and lowers its maximum from $4$ to $13/4$, moving both toward vacuum.

Rotated ellipse

For a general symmetric covariance matrix,

$$V = \begin{pmatrix} V_{XX} & V_{XP}\\ V_{XP} & V_{PP} \end{pmatrix},$$

the stationary axes satisfy

$$\tan(2\theta)=\frac{2V_{XP}}{V_{XX}-V_{PP}}.$$

This tangent equation alone does not distinguish the minimum from the orthogonal maximum. A branch-safe minimum is

$$\theta_{\min}=\frac12\operatorname{atan2}(-2V_{XP},V_{PP}-V_{XX}) \pmod\pi,$$

unless $V$ is proportional to the identity, when every angle is equivalent. The factor of two reflects the ellipse’s invariance under a rotation by $\pi$. The squeeze-operator phase below is twice the quiet quadrature angle.

Squeeze Operator and State Family

Let

$$\zeta = r e^{i\phi}, \qquad r\ge0.$$

One standard single-mode squeeze operator is

$$\hat S(\zeta) = \exp\left[ \frac12 \left( \zeta^*\hat a^2 - \zeta(\hat a^\dagger)^2 \right) \right].$$

It generates the squeezed vacuum

$$\lvert0;\zeta\rangle = \hat S(\zeta)\lvert0\rangle.$$

The corresponding Bogoliubov transformation is

$$\hat S^\dagger(\zeta) \hat a \hat S(\zeta) = \hat a\cosh r - e^{i\phi} \hat a^\dagger\sinh r.$$

The commutator is preserved because

$$\cosh^2r-\sinh^2r=1.$$

For the squeezed vacuum,

$$\operatorname{Var}(X_\theta) = \frac12\cosh(2r) - \frac12\sinh(2r) \cos(2\theta-\phi) .$$

The minimum occurs at

$$\theta_{\mathrm{sq}} = \frac{\phi}{2} \pmod{\pi},$$

and the orthogonal quadrature is anti-squeezed:

$$\begin{aligned} \operatorname{Var}(X_{\theta_{\mathrm{sq}}}) &= \frac12e^{-2r}, \\ \operatorname{Var}(X_{\theta_{\mathrm{sq}}+\pi/2}) &= \frac12e^{2r}. \end{aligned}$$

Changing the sign convention in $\hat S$ shifts the quoted squeezing phase. The invariant content is the measured principal axis and its variance.

Displaced squeezed states

The displacement operator

$$\hat D(\alpha) = \exp\left( \alpha\hat a^\dagger - \alpha^*\hat a \right)$$

produces

$$\lvert\alpha,\zeta\rangle = \hat D(\alpha) \hat S(\zeta) \lvert0\rangle.$$

With this ordering,

$$\langle\hat a\rangle=\alpha,$$

while the covariance is that of the squeezed vacuum. The alternative order $\hat S(\zeta)\hat D(\alpha)$ represents the same family with a transformed displacement parameter, but it does not use the same numerical $\alpha$.

The relative angle between displacement and squeezing controls direct intensity fluctuations. An amplitude-squeezed bright state can be sub-Poissonian, while a phase-squeezed bright state can be super-Poissonian. Quadrature squeezing and sub-Poissonian counting are different witnesses of nonclassicality.

Squeezed thermal states

A useful mixed Gaussian model is

$$\rho_{\mathrm{sth}} = \hat S(\zeta) \rho_{\mathrm{th}} \hat S^\dagger(\zeta),$$

where the unsqueezed thermal mode has mean occupation $n_{\mathrm{th}}$. Along the principal axes,

$$\begin{aligned} V_{\min} &= \left( n_{\mathrm{th}}+\frac12 \right) e^{-2r}, \\ V_{\max} &= \left( n_{\mathrm{th}}+\frac12 \right) e^{2r}. \end{aligned}$$

The state is squeezed below vacuum only if

$$(2n_{\mathrm{th}}+1)e^{-2r} \lt 1.$$

Its covariance determinant is

$$\det V = \left( n_{\mathrm{th}}+\frac12 \right)^2,$$

and its purity is

$$\operatorname{Tr}(\rho_{\mathrm{sth}}^2) = \frac1{2n_{\mathrm{th}}+1}.$$

Parametric gain can create a visibly elongated ellipse without pushing the narrow axis below vacuum when the input or environment is too noisy. “Phase-sensitive amplification” is therefore broader than “generation of nonclassical squeezed light.”

Number-state signature

The squeezed vacuum contains only even photon numbers. Its mean occupation is

$$\bar n = \sinh^2r,$$

and its number variance is

$$\operatorname{Var}(N) = 2\bar n(\bar n+1).$$

For $\bar n>0$,

$$g^{(2)}(0) = 3+\frac1{\bar n}.$$

The state is strongly super-Poissonian even though one quadrature is quieter than vacuum. This is not contradictory: photon number is not the squeezed quadrature. Photon Number States develops number-resolved diagnostics, while the oscillator page gives the canonical even-number expansion.

Nonclassical without Wigner negativity

The Wigner function of an ideal squeezed vacuum is a positive Gaussian. Yet a quadrature variance below vacuum is incompatible with a nonnegative Glauber–Sudarshan $P$ probability measure, including delta measures. Squeezing is therefore a nonclassicality witness even when the Wigner function never becomes negative.

Wigner negativity is sufficient for nonclassicality, not necessary. This distinction is central when Gaussian resources are compared with photon-added, photon-subtracted, or number-state resources.

Which Optical Mode Is Squeezed?

The operator $\hat a$ must represent a normalized field mode. For a temporal or spectral wave packet,

$$\hat A_f = \int d\omega\, f^*(\omega) \hat a(\omega),$$

with

$$\int d\omega\, \lvert f(\omega)\rvert^2 = 1.$$

Then

$$[\hat A_f,\hat A_f^\dagger]=1.$$

The statement “the pulse is squeezed” means that a specified wave-packet mode $f$ has a sub-vacuum quadrature. A detector or local oscillator selecting a different mode $g$ observes only their overlap,

$$\mu = \int d\omega\, g^*(\omega)f(\omega),$$

with the orthogonal component entering as loss or added noise according to its state.

Continuous-wave sidebands

Continuous-wave squeezing is commonly reported at an analysis frequency $\Omega$ around a carrier $\omega_0$. The electronic photocurrent at $\Omega$ combines optical sidebands near

$$\omega_0+\Omega \qquad\text{and}\qquad \omega_0-\Omega.$$

A spectral quadrature is therefore generally a joint observable of the upper and lower sideband modes. Pair generation correlates those sidebands, and homodyne detection converts their correlation into sub-vacuum noise at the electronic analysis frequency.

Calling this “one monochromatic mode squeezed at $\omega_0$” hides important frequency structure. A continuous-wave measurement specifies:

• the optical carrier;
• the analysis sideband frequency;
• the resolution and video bandwidths;
• whether the noise spectrum is single-sided or double-sided;
• the temporal integration used to define the detected mode.

Multimode squeezing

A pulsed nonlinear interaction can have the schematic pair-creation kernel

$$\begin{aligned} \hat H_I &= \frac{i\hbar}{2} \iint d\omega\,d\omega'\, \hat{\mathcal K}(\omega,\omega'), \\ \hat{\mathcal K}(\omega,\omega') &= \kappa(\omega,\omega') \hat a^\dagger(\omega) \hat a^\dagger(\omega') - \mathrm{h.c.} \end{aligned}$$

Pump bandwidth, dispersion, and phase matching determine the symmetric kernel $\kappa(\omega,\omega')$; the factor $1/2$ avoids double counting identical-mode pairs. Its Takagi decomposition identifies orthogonal squeezed modes for a fixed pair Hamiltonian. More generally, a Bloch–Messiah decomposition of the full Gaussian transformation identifies passive mode rotations and independent squeeze factors. The output is not generally one uniformly squeezed pulse mode.

This matters experimentally. A local oscillator matched to the leading supermode can reveal strong squeezing, while an unmatched ultrashort pulse averages squeezed and unsqueezed modes. Mode Decompositions explains how the selected modes define the subsystem decomposition.

Generation Mechanisms

Deterministic Gaussian preparation from coherent states and vacuum requires an active Gaussian interaction that mixes annihilation and creation operators. Passive phase shifts and beam splitters can redistribute existing squeezing but cannot create it from these inputs. Non-Gaussian nonlinear dynamics, including Kerr shearing below, can also produce squeezing. The material response, mode overlap, and validity of the undepleted-pump reduction are developed in Nonlinear Quantum Optics.

Degenerate parametric interaction

In a second-order nonlinear medium, a strong classical pump near $2\omega$ can drive the effective Hamiltonian

$$\hat H_I = \frac{i\hbar\kappa}{2} \left( e^{-i\phi_p}\hat a^2 - e^{i\phi_p}(\hat a^\dagger)^2 \right).$$

For interaction time $t$, the evolution is a squeeze operator with

$$r=\kappa t$$

in this ideal undepleted-pump model. The pump supplies energy and fixes the squeezing phase. At the operator level, the output has the general phase-sensitive form

$$\hat a_{\mathrm{out}} = \mu\hat a_{\mathrm{in}} + \nu\hat a_{\mathrm{in}}^\dagger,$$

where

$$\lvert\mu\rvert^2 - \lvert\nu\rvert^2 = 1.$$

One principal quadrature is deamplified while its conjugate is amplified. The vacuum seed is transformed into squeezed vacuum; a coherent seed becomes a displaced squeezed state.

Energy and momentum matching

For nondegenerate parametric down-conversion,

$$\omega_p = \omega_s+\omega_i.$$

Efficient conversion also requires phase matching, schematically

$$\boldsymbol k_p \approx \boldsymbol k_s+\boldsymbol k_i,$$

with a reciprocal-lattice contribution for quasi-phase matching. Degenerate squeezing sets $\omega_s=\omega_i=\omega_p/2$ but can still involve distinct spatial, polarization, or sideband modes.

The phrases degenerate, single-mode, and collinear answer different questions. Degenerate frequencies do not guarantee a single spatiotemporal mode, and a single beam can contain many independently squeezed supermodes.

Optical parametric oscillators

Placing the nonlinear medium in a cavity enhances the interaction and selects a frequency-dependent output mode. Below threshold, an optical parametric oscillator can emit squeezed vacuum with no coherent carrier in the squeezed port. Its useful description requires:

• cavity linewidth and escape efficiency;
• pump fraction relative to threshold;
• internal absorption and propagation loss;
• detuning and phase-lock errors;
• squeezing spectrum versus analysis frequency.

Approaching threshold increases ideal low-frequency squeezing and anti-squeezing, but it also increases sensitivity to phase noise, pump noise, and technical drift. Above threshold the device becomes an oscillator with a macroscopic signal, and the linearized state description must be rebuilt around that operating point.

Four-wave mixing

A third-order nonlinearity can drive

$$2\omega_p = \omega_s+\omega_i$$

through annihilation of two pump photons and correlated creation of signal and idler photons. Atomic vapors, optical fibers, integrated resonators, and other $\chi^{(3)}$ platforms can generate single-mode quadrature squeezing, twin-beam intensity-difference squeezing, or two-mode squeezing depending on the selected modes and pumps.

The first experimental observation of optical squeezing used four-wave mixing in an atomic cavity. The mechanism also introduces platform-specific noise: spontaneous emission, Raman scattering, pump leakage, absorption, and excess phase noise can obscure the ideal pair Hamiltonian.

Kerr shearing and emitter-based squeezing

A Kerr interaction proportional to

$$\hat H_{\mathrm K} \propto (\hat a^\dagger\hat a)^2$$

shears a coherent-state distribution in phase space. For suitable evolution and displacement, part of the sheared state can have reduced quadrature variance. At stronger evolution the state is non-Gaussian, so one ellipse no longer captures its full structure.

Resonance fluorescence and related driven-emitter fields can also display quadrature squeezing. Such sources are not equivalent to an ideal squeezed vacuum: their bandwidth, photon statistics, and non-Gaussian correlations are set by the emitter dynamics.

How Squeezing Is Measured

Quadrature squeezing is a phase-sensitive statement, so it requires a phase-sensitive measurement. Balanced homodyne detection is the standard reference method.

Balanced homodyne detection

Let $\hat a$ be the signal mode and $\hat b$ a strong coherent local-oscillator mode with

$$\beta = \lvert\beta\rvert e^{i\theta}.$$

A balanced beam splitter produces output modes

$$\hat c = \frac{ \hat a+\hat b }{ \sqrt2 }, \qquad \hat d = \frac{ \hat b-\hat a }{ \sqrt2 }.$$

The photon-number difference is

$$\hat N_c-\hat N_d = \hat a^\dagger\hat b + \hat b^\dagger\hat a.$$

For a strong local oscillator, replace $\hat b$ by $\beta$ in the leading signal term:

$$\hat N_c-\hat N_d \approx \sqrt2 \lvert\beta\rvert \hat X_\theta.$$

For an independent coherent local oscillator, the exact finite-power variance is

$$\operatorname{Var}(N_c-N_d) =2|\beta|^2\operatorname{Var}(X_\theta)+\langle N_a\rangle.$$

It follows by writing $b=\beta+\delta b$ with vacuum fluctuations in $\delta b$. The vacuum-signal reference is $|\beta|^2$, so the extra normalized noise is $\langle N_a\rangle/|\beta|^2$. It must be small relative to the squeezed variance being resolved, not merely small relative to one. This formula assumes a balanced, ideal, mode-matched receiver.

Changing the local-oscillator phase scans the quadrature angle. Balanced subtraction removes the large common local-oscillator intensity to first order and leaves a difference noise proportional to the signal quadrature variance.

The local oscillator does more than supply gain and phase. Its spatial, temporal, spectral, and polarization profile defines the measured mode. Excellent photodiodes cannot recover squeezing in a mode orthogonal to the local oscillator.

Establishing the vacuum level

A standard vacuum reference is obtained by blocking the signal before the homodyne beam splitter while keeping the local oscillator, detectors, electronic gain, filters, and analysis settings unchanged. The unused signal input then supplies vacuum fluctuations.

The measured difference-noise power should:

• scale linearly with local-oscillator optical power over the working range;
• remain well above electronic dark noise;
• stay below detector saturation and digitizer clipping;
• be stable under repeated vacuum–signal switching;
• be insensitive to classical common-mode local-oscillator noise after balancing.

A line labelled “shot noise” is not automatically a trustworthy vacuum calibration. Electronic noise subtraction, detector imbalance, resolution bandwidth, windowing, and spectral averaging all affect the displayed trace.

Spectral squeezing

For a stationary continuous-wave field, one measures a quadrature-noise power spectral density

$$S_{X_\theta X_\theta}(\Omega)$$

relative to the vacuum spectrum at analysis frequency $\Omega$. The result is often reported as

$$v_\theta(\Omega) = \frac{ S_{X_\theta X_\theta}(\Omega) }{ S_{\mathrm{vac}}(\Omega) }.$$

Squeezing can exist in one frequency band and disappear in another. A cavity source may be quantum-noise limited near its resonance yet dominated by pump, control, or acoustic noise at low analysis frequencies.

Comparisons require the same convention for:

• single-sided versus double-sided spectra;
• angular frequency versus ordinary frequency;
• power spectral density versus amplitude spectral density;
• raw spectrum-analyzer bins versus integrated mode variance.

A level in $\mathrm{dB}/\mathrm{Hz}$ is not the same object as a dimensionless single-mode variance until a temporal or spectral mode and integration rule have been supplied.

Direct detection answers a different question

Direct photon counting measures intensity or number statistics, not a phase-selected quadrature. An undisplaced squeezed vacuum has zero mean field, even-number support, and super-Poissonian number fluctuations. Its squeezing is therefore not revealed by asking whether its raw count variance is below a Poisson value.

A bright amplitude-squeezed beam can display sub-shot-noise intensity fluctuations, but that is a special alignment of displacement and covariance. Phase squeezing generally requires interferometric or homodyne conversion before it appears as an intensity signal.

Tomography versus witnessing

Observing one quadrature below vacuum witnesses nonclassicality for the measured mode under a trusted calibration. It does not reconstruct the full density operator.

Scanning many local-oscillator phases samples quadrature marginals and can support optical homodyne tomography. Reconstruction then requires a model for efficiency, phase sampling, finite statistics, and regularization. The general phase-space inversion belongs with Gaussian States and Wigner Functions and Phase-Space Distributions.

Loss and Added Noise

Loss is especially important for squeezing because vacuum entering an unobserved port fills in the narrow quadrature. Model pure loss of transmissivity $\eta$ as

$$\hat a_{\mathrm{out}} = \sqrt\eta\, \hat a_{\mathrm{in}} + \sqrt{1-\eta}\, \hat v,$$

where $\hat v$ is an independent vacuum mode. In vacuum-normalized units,

$$v_{\mathrm{out}} = \eta v_{\mathrm{in}} + (1-\eta).$$

For an ideal input squeezed vacuum,

$$\begin{aligned} v_{\min,\mathrm{out}} &= 1-\eta+\eta e^{-2r}, \\ v_{\max,\mathrm{out}} &= 1-\eta+\eta e^{2r}. \end{aligned}$$

Even infinite input squeezing cannot overcome the loss floor:

$$\lim_{r\to\infty} v_{\min,\mathrm{out}} = 1-\eta.$$

The corresponding best possible observed squeezing is

$$R_{\max} = -10\log_{10}(1-\eta) \ \mathrm{dB}.$$

For example, $\eta=0.90$ imposes a $10\ \mathrm{dB}$ ceiling even before phase noise or technical noise is included.

What belongs in the efficiency

A simple effective efficiency can combine independent vacuum-loss channels:

$$\eta_{\mathrm{tot}} = \eta_{\mathrm{esc}} \times \eta_{\mathrm{prop}} \times \eta_{\mathrm{vis}} \times \eta_{\mathrm{det}},$$

where the symbols may represent:

• source escape efficiency;
• propagation transmission;
• homodyne mode-overlap efficiency;
• photodiode quantum efficiency.

The mode-overlap factor is often the squared visibility after correcting for power imbalance. Multiplying efficiencies is valid only when each imperfection is well represented by independent mixing with vacuum. Stray coherent light, thermal background, electronic noise, or nonlinear detector response requires a different model.

Thermal or noisy environments

For an independent thermal environment mode of mean occupation $n_{\mathrm{env}}$, the normalized quadrature variance becomes

$$v_{\mathrm{out}} = \eta v_{\mathrm{in}} + (1-\eta) \left( 2n_{\mathrm{env}}+1 \right).$$

Mean occupation alone does not fix an arbitrary environment’s quadrature noise. For a squeezed environment use its angle-dependent variance; initial signal–environment correlations also add cross terms. The displayed formula is a thermal-loss Gaussian channel. At optical frequencies, a laboratory environment is often effectively vacuum at the carrier frequency, but technical sidebands and control-field leakage need not be. At microwave frequencies, thermal occupation and amplifier-added noise are frequently central.

Gaussian Channels gives the canonical channel description. Calling every degradation “loss” can substantially overestimate the recoverable source squeezing.

Inferring source squeezing

Under a trusted pure-loss model,

$$v_{\mathrm{in}} = \frac{ v_{\mathrm{out}}-(1-\eta) }{ \eta }.$$

This inversion becomes ill-conditioned when the observed variance is near the loss floor or $\eta$ is uncertain. A mature result quotes the directly observed value first, then any inferred source value with propagated uncertainty and a stated channel model.

Phase Noise

Consider an undisplaced squeezed vacuum. Suppose the intended measurement axis is its squeezed principal quadrature, but the local-oscillator angle has a fixed error $\delta$. The measured normalized variance is

$$v(\delta) = e^{-2r}\cos^2\delta + e^{2r}\sin^2\delta.$$

For small $\delta$,

$$v(\delta) \approx e^{-2r} + \delta^2 \left( e^{2r}-e^{-2r} \right).$$

Large anti-squeezing makes even a small phase error costly. Increasing parametric gain can therefore make the observed minimum worse when phase control is not improved.

For statistically independent zero-mean Gaussian angle jitter of variance $\sigma_\theta^2$, constant during each sampled mode but varying between repetitions, averaging over the angle gives

$$\overline v = \cosh(2r) - e^{-2\sigma_\theta^2} \sinh(2r).$$

With pure loss after the phase error,

$$v_{\mathrm{obs}} = 1-\eta + \eta\overline v.$$

For a displaced field, the measured mean also changes with angle. The law of total variance adds the variance of those conditional means; averaging only the noise ellipse is then incomplete. Rapid within-mode phase noise can mix temporal modes and requires its noise spectrum. Moreover, a random mixture of rotated Gaussian states is generally non-Gaussian, so its covariance determinant alone does not determine its purity.

Phase noise can arise from pump-phase fluctuations, local-oscillator drift, cavity-length noise, control sidebands, or a frequency-dependent rotation between source and detector.

Frequency-dependent angle

Dispersion and cavities can rotate the squeezing ellipse by an angle that depends on $\Omega$. This can be unwanted, or it can be engineered. A filter cavity deliberately rotates squeezed vacuum so that different quadratures are used in different frequency bands.

One homodyne phase cannot reveal the minimum at every analysis frequency when the ellipse rotates across the spectrum. Reporting only the best point can hide poor broadband performance.

Worked Example: Loss and Phase Error

Suppose a source produces ideal $10\ \mathrm{dB}$ squeezing:

$$v_{\min,\mathrm{in}}=0.10, \qquad v_{\max,\mathrm{in}}=10.$$

With total efficiency $\eta=0.80$ and perfect phase lock,

$$v_{\min,\mathrm{out}} = 0.20+0.80(0.10) = 0.28.$$

The directly observed squeezing is

$$-10\log_{10}(0.28) \approx 5.53\ \mathrm{dB}.$$

The observed anti-squeezing is

$$v_{\max,\mathrm{out}} = 0.20+0.80(10) = 8.20,$$

or about $9.14\ \mathrm{dB}$ above vacuum.

Now add a fixed phase error $\delta=2^\circ$. Before loss,

$$\begin{aligned} v(\delta) &= 0.10\cos^2(2^\circ) + 10\sin^2(2^\circ) \\ &\approx 0.112. \end{aligned}$$

After loss,

$$v_{\mathrm{obs}} \approx 0.20+0.80(0.112) \approx 0.290.$$

The observed squeezing falls to approximately $5.38\ \mathrm{dB}$. The phase penalty looks modest here, but it grows rapidly with anti-squeezing or phase jitter. This is why source gain, efficiency, and phase control must be optimized together.

Squeezing and Precision Measurement

Squeezing helps when a signal is encoded in a quadrature whose uncertainty has been reduced and when the anti-squeezed quadrature does not couple back into the estimator.

For a bright field with real displacement $\alpha$ and a small phase shift $\varphi$ generated by $U=e^{-i\varphi N}$,

$$\langle\hat P\rangle \approx -\sqrt2 \alpha\varphi.$$

Error propagation gives

$$\Delta\varphi \approx \frac{ \Delta P }{ \sqrt2\lvert\alpha\rvert }.$$

For coherent light,

$$\Delta P = \frac1{\sqrt2},$$

so

$$\Delta\varphi_{\mathrm{coh}} \approx \frac1{2\sqrt{\bar n}}, \qquad \bar n=\lvert\alpha\rvert^2.$$

If the phase quadrature is ideally squeezed by $r$,

$$\Delta\varphi_{\mathrm{sq}} \approx \frac{ e^{-r} }{ 2\sqrt{\bar n} }.$$

This local formula shows the noise advantage, but it is not a universal metrological bound. Here $\bar n$ counts displacement photons; the total mean for this pure squeezed state is $|\alpha|^2+\sinh^2r$. A fixed-resource comparison must also include losses, bandwidth, and the relevant preparation and readout resources.

Interferometer dark-port injection

Near a dark operating port, vacuum fluctuations entering the nominally unused input contribute to the output noise. Injecting squeezed vacuum into that port can reduce the measured quadrature noise without placing a bright carrier there. The squeezing angle must be aligned with the signal quadrature at the readout.

This strategy does not make all interferometer noise disappear:

• phase squeezing can reduce high-frequency photon-counting noise;
• the conjugate amplitude fluctuations can increase radiation-pressure backaction;
• optical loss replaces part of the injected state with ordinary vacuum;
• detuning and optomechanical response rotate the relevant quadrature with frequency.

Carlton Caves’s interferometer analysis made this tradeoff explicit and motivated squeezed-vacuum injection as a measurement resource.

Frequency-dependent squeezing in gravitational-wave detectors

Large interferometers need different quadrature orientations where photon-counting noise and radiation-pressure noise dominate. Filter cavities can rotate the injected squeezing angle across frequency so the same source reduces both contributions over a broad band.

This is an engineering achievement as much as a state-preparation result. Useful operation depends on low-loss optics, long-term phase and alignment control, stable filter-cavity detuning, and a detector-wide noise budget. Modern gravitational-wave observatories have demonstrated squeezed-light operation and frequency-dependent quantum-noise reduction, but the achieved astrophysical sensitivity is never determined by a squeezing number alone.

Single-Mode and Two-Mode Squeezing

Single-mode squeezing reduces one quadrature of one declared mode. It is not by itself bipartite entanglement because no subsystem split has been named.

Two-mode squeezing correlates modes $A$ and $B$. In one convention, collective quadratures such as

$$\hat X_A-\hat X_B \qquad\text{and}\qquad \hat P_A+\hat P_B$$

are squeezed. Each mode separately can have a thermal reduced state even though the joint state is pure and entangled.

A balanced beam splitter can convert two single-mode squeezed inputs with orthogonal squeezing angles into two-mode entanglement. Conversely, changing the mode basis can reveal single-mode squeezing hidden inside a multimode Gaussian state. The relevant statements depend on the physically declared mode decomposition, not only on a drawing of two beams.

The derivation and entanglement criteria are in Squeezed States as Entangled Modes.

Exercises

1. Rotated squeezed-vacuum variance

For $\zeta=re^{i\phi}$, use

$$\hat S^\dagger(\zeta) \hat a \hat S(\zeta) = \hat a\cosh r - e^{i\phi} \hat a^\dagger\sinh r$$

to derive

$$\operatorname{Var}(X_\theta) = \frac12\cosh(2r) - \frac12\sinh(2r) \cos(2\theta-\phi) .$$

Find the squeezed angle, the two principal variances, and their product.

Solution

Transform the quadrature:

$$\begin{aligned} \hat B_\theta &= e^{-i\theta} \hat a\cosh r - e^{i(\phi-\theta)} \hat a^\dagger\sinh r \\ \hat S^\dagger \hat X_\theta \hat S &= \frac{ \hat B_\theta+\hat B_\theta^\dagger }{ \sqrt2 }. \end{aligned}$$

The vacuum mean vanishes. Using

$$\langle0\lvert \hat a\hat a^\dagger \rvert0\rangle = 1$$

and the vanishing of the other normally unordered quadratic vacuum moments gives

$$\operatorname{Var}(X_\theta) = \frac12 \left( \cosh^2r+\sinh^2r \right) - \cosh r\,\sinh r \cos(2\theta-\phi).$$

Now use

$$\cosh^2r+\sinh^2r = \cosh(2r)$$

and

$$2\cosh r\,\sinh r = \sinh(2r)$$

to obtain the stated expression.

The minimum requires

$$\cos(2\theta-\phi)=1,$$

so

$$\theta_{\mathrm{sq}} = \frac{\phi}{2} \pmod{\pi}.$$

At that angle,

$$\begin{aligned} V_{\min} &= \frac12 \left[ \cosh(2r)-\sinh(2r) \right] \\ &= \frac12e^{-2r}. \end{aligned}$$

The orthogonal angle changes the cosine to $-1$:

$$V_{\max} = \frac12e^{2r}.$$

Therefore

$$V_{\min}V_{\max} = \frac14,$$

or

$$\Delta X_{\min} \Delta X_{\max} = \frac12.$$

The ideal squeezed vacuum remains a minimum-uncertainty state.

2. Convert squeezing between decibels and state parameters

An ideal squeezed vacuum is quoted as having $6.0\ \mathrm{dB}$ of squeezing. Find:

1. its normalized minimum variance;
2. its minimum standard deviation relative to vacuum;
3. the squeezing parameter $r$;
4. the mean photon number $\bar n=\sinh^2r$;
5. its ideal anti-squeezing in decibels.

Solution

The normalized variance is

$$v_{\min} = 10^{-6.0/10} \approx 0.251.$$

The standard-deviation ratio is

$$\frac{ \Delta X_{\min} }{ \Delta X_{\mathrm{vac}} } = \sqrt{v_{\min}} \approx 0.501.$$

For an ideal squeezed vacuum,

$$v_{\min}=e^{-2r},$$

so

$$r = -\frac12\ln v_{\min} \approx 0.691.$$

The mean occupation is

$$\bar n = \sinh^2(0.691) \approx 0.56.$$

Ideal squeezing has reciprocal principal variances:

$$v_{\max} = e^{2r} = \frac1{v_{\min}} \approx 3.98.$$

Thus the anti-squeezing is

$$10\log_{10}(v_{\max}) \approx +6.0\ \mathrm{dB}.$$

Real data need not have symmetric squeezing and anti-squeezing in decibels, because loss and excess noise enlarge the covariance determinant.

3. Loss floor and observed squeezing

A source produces $8.0\ \mathrm{dB}$ of ideal squeezing and is measured with total efficiency $\eta=0.75$.

1. Find the detector-plane normalized minimum variance.
2. Convert it to observed squeezing in decibels.
3. Find the maximum observable squeezing allowed by this efficiency even for an infinitely squeezed source.

Solution

The source variance is

$$v_{\mathrm{in}} = 10^{-8.0/10} \approx 0.1585.$$

Pure loss gives

$$\begin{aligned} v_{\mathrm{out}} &= 1-\eta+\eta v_{\mathrm{in}} \\ &= 0.25+0.75(0.1585) \\ &\approx 0.3689. \end{aligned}$$

The observed squeezing magnitude is

$$-10\log_{10}(0.3689) \approx 4.33\ \mathrm{dB}.$$

For infinite source squeezing,

$$v_{\mathrm{out}} \longrightarrow 1-\eta = 0.25.$$

The efficiency ceiling is therefore

$$R_{\max} = -10\log_{10}(0.25) \approx 6.02\ \mathrm{dB}.$$

No increase in source gain can move the detector below this floor under the assumed channel.

4. Leakage from the anti-squeezed quadrature

An ideal source has

$$v_{\min}=0.10, \qquad v_{\max}=10.$$

The homodyne angle is offset from the squeezed axis by $3^\circ$.

1. Find the variance before loss.
2. Find the variance and squeezing in decibels after efficiency $\eta=0.90$.
3. Explain why phase stability becomes more demanding as anti-squeezing increases.

Solution

At angle error $\delta$,

$$v(\delta) = v_{\min}\cos^2\delta + v_{\max}\sin^2\delta.$$

With $\delta=3^\circ$,

$$\begin{aligned} v(3^\circ) &= 0.10\cos^2(3^\circ) + 10\sin^2(3^\circ) \\ &\approx 0.127. \end{aligned}$$

After pure loss,

$$\begin{aligned} v_{\mathrm{obs}} &= 0.10+0.90(0.127) \\ &\approx 0.214. \end{aligned}$$

The observed squeezing is

$$-10\log_{10}(0.214) \approx 6.69\ \mathrm{dB}.$$

For small error,

$$v(\delta) \approx v_{\min} + \delta^2 \left( v_{\max}-v_{\min} \right).$$

The error coefficient grows approximately as $v_{\max}$. Raising the parametric gain narrows one axis but enlarges the noise available to leak from the orthogonal axis. Better intrinsic squeezing can therefore produce worse observed squeezing if the phase lock is not improved.

5. When is a squeezed thermal state below vacuum?

A mode begins in a thermal state with

$$n_{\mathrm{th}}=0.25$$

and is squeezed by $r=0.50$.

1. Find the minimum $r$ needed to cross below vacuum.
2. Find the normalized principal variances for $r=0.50$.
3. Find the squeezing in decibels and the state purity.

Solution

The normalized narrow-axis variance is

$$v_{\min} = (2n_{\mathrm{th}}+1)e^{-2r}.$$

Sub-vacuum noise requires

$$r \gt \frac12 \ln(2n_{\mathrm{th}}+1).$$

Here,

$$r_{\mathrm{threshold}} = \frac12\ln(1.5) \approx 0.203.$$

At $r=0.50$,

$$v_{\min} = 1.5e^{-1} \approx 0.552,$$

and

$$v_{\max} = 1.5e \approx 4.08.$$

The squeezed noise is

$$-10\log_{10}(0.552) \approx 2.58\ \mathrm{dB}$$

below vacuum. The purity is unchanged by the unitary squeeze operation:

$$\operatorname{Tr}(\rho^2) = \frac1{2n_{\mathrm{th}}+1} = \frac23.$$

The product

$$v_{\min}v_{\max} = (2n_{\mathrm{th}}+1)^2 = 2.25$$

also shows that the state is mixed rather than a minimum-uncertainty squeezed vacuum.

6. Photon bunching of squeezed vacuum

Use

$$\bar n = \langle N\rangle$$

and

$$\operatorname{Var}(N) = 2\bar n(\bar n+1)$$

to derive the equal-time intensity correlation

$$g^{(2)}(0) = \frac{ \langle N(N-1)\rangle }{ \bar n^2 }.$$

Interpret its low-occupation limit.

Solution

The factorial moment is related to the variance by

$$\langle N(N-1)\rangle = \operatorname{Var}(N) + \bar n^2 - \bar n.$$

Substitution gives

$$\begin{aligned} \langle N(N-1)\rangle &= 2\bar n(\bar n+1) + \bar n^2 - \bar n \\ &= 3\bar n^2+\bar n. \end{aligned}$$

Therefore

$$g^{(2)}(0) = 3+\frac1{\bar n}.$$

As $\bar n\to0^+$, the normalized correlation diverges; at exactly vacuum it is undefined. The squeezed vacuum is created in pairs, so conditioning on there being light strongly favors a two-photon contribution over two unrelated single-photon events. The absolute pair rate still tends to zero with the source brightness; a divergent normalized $g^{(2)}$ does not mean a large count rate.

This strong bunching coexists with quadrature squeezing because number and field quadrature are different observables.

7. Homodyne difference current and mode mismatch

For

$$\hat c = \frac{ \hat a+\hat b }{ \sqrt2 }, \qquad \hat d = \frac{ \hat b-\hat a }{ \sqrt2 },$$

derive $\hat N_c-\hat N_d$. Then take

$$\hat b\longrightarrow \lvert\beta\rvert e^{i\theta}$$

and identify the measured quadrature.

Finally, suppose the local oscillator overlaps the squeezed mode with efficiency $\eta_{\mathrm{mm}}$ and the orthogonal component is vacuum. Show that the normalized observed variance is

$$v_{\mathrm{obs}} = \eta_{\mathrm{mm}}v_{\mathrm{sq}} + (1-\eta_{\mathrm{mm}}).$$

Solution

Expanding the two output number operators gives

$$\begin{aligned} \hat N_c &= \frac12 \left( \hat a^\dagger\hat a + \hat a^\dagger\hat b + \hat b^\dagger\hat a + \hat b^\dagger\hat b \right), \\ \hat N_d &= \frac12 \left( \hat b^\dagger\hat b - \hat b^\dagger\hat a - \hat a^\dagger\hat b + \hat a^\dagger\hat a \right). \end{aligned}$$

Subtracting cancels the direct intensities:

$$\hat N_c-\hat N_d = \hat a^\dagger\hat b + \hat b^\dagger\hat a.$$

For a strong coherent local oscillator,

$$\begin{aligned} \hat N_c-\hat N_d &\approx \lvert\beta\rvert \left( \hat a^\dagger e^{i\theta} + \hat a e^{-i\theta} \right) \\ &= \sqrt2 \lvert\beta\rvert \hat X_\theta. \end{aligned}$$

Thus the local-oscillator phase selects the measured quadrature.

Represent the detected mode as a superposition of the desired squeezed mode and an orthogonal vacuum mode:

$$\hat A_{\mathrm{det}} = \sqrt{\eta_{\mathrm{mm}}} \hat A_{\mathrm{sq}} + \sqrt{1-\eta_{\mathrm{mm}}} \hat A_{\mathrm{vac}}.$$

The two modes are independent, so their quadrature variances add:

$$v_{\mathrm{obs}} = \eta_{\mathrm{mm}}v_{\mathrm{sq}} + (1-\eta_{\mathrm{mm}})v_{\mathrm{vac}}.$$

With $v_{\mathrm{vac}}=1$, this is the pure-loss formula. Mode mismatch is therefore equivalent to loss for this selected-mode measurement.

8. Optimal squeezing with uncertain phase

For undisplaced squeezed vacuum with independent Gaussian angle jitter, minimize $\overline v=\cosh(2r)-c\sinh(2r)$ over $r\ge0$, where $c=e^{-2\sigma_\theta^2}$. Then include efficiency $\eta$.

Solution

For nonzero finite jitter, differentiation gives $\tanh(2r_{\rm opt})=c$. Thus

$$r_{\rm opt}=\frac12\operatorname{artanh}c,\qquad \overline v_{\min}=\sqrt{1-c^2},\qquad v_{\rm obs,min}=1-\eta+\eta\sqrt{1-e^{-4\sigma_\theta^2}}.$$

For $\sigma_\theta\ll1$ the phase-noise floor is approximately $2\sigma_\theta$ before loss. Infinite source squeezing is optimal only in the ideal zero-jitter limit; at fixed nonzero jitter it eventually amplifies the unwanted quadrature too strongly.

9. A finite local oscillator

For a pure squeezed vacuum with $r=\ln2$, find its mean occupation and the ideal squeezed variance relative to vacuum. How large must the local oscillator occupation be to keep its finite-power excess below $1\%$ of that squeezed variance?

Solution

$\bar n=\sinh^2(\ln2)=9/16$ and $v_{\min}=1/4$. The exact ideal receiver noise is $v_{\min}+\bar n/|\beta|^2$. Requiring the second term to be smaller than $0.01/4$ gives $|\beta|^2>225$. Thus an oscillator that is only ten times brighter than the signal would not suffice for this relative precision. Real receivers also need adequate electronic-noise clearance and linearity.

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