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A bosonic state with Gaussian phase-space characteristic function, specified by its first and second optical quadrature moments. Vacuum, coherent states, optical squeezed vacuum and thermal states are examples. Gaussian here describes phase-space statistics, not a Gaussian optical pulse envelope.

Raw vectors $a^{†}|\psi ⟩$ and $a|\psi ⟩$ have squared norms $\mu +1$ and $\mu$. Matrix elements of the number operator in these vectors must be divided by their squared norms before calling them means. For an optical squeezed vacuum with mean $s$, the raw moments are $(3s+1)(s+1)$ and $s(3s+1)$, while both normalized means equal $3s+1$ when subtraction is possible.

Sub-Poissonian number variance in a specified mode or detection gate. It concerns photon counting, not an optical quadrature. An optical squeezed vacuum has quadrature squeezing but fails this number-variance condition.

A degenerate optical parametric amplifier sends a strong pump at $2\omega$ through a phase-matched second-order nonlinear medium, creating pairs in a selected mode near $\omega$. With the pump treated classically and undepleted, the effective interaction is quadratic in the mode's creation and annihilation operators. Vacuum input evolves into an optical squeezed vacuum; the pump phase fixes the quiet optical quadrature. A cavity enhances the interaction below oscillation threshold. balanced homodyne detection compares this optical quadrature's noise with the blocked-signal vacuum reference; phase locking, detector balance, electronic-noise measurement and loss calibration distinguish squeezing from an instrumental noise reduction. A primary cavity experiment observed below-vacuum noise by degenerate down conversion and balanced homodyne readout.^[ref]

Use $S(r)=\exp [\frac{r}{2}(a^2-a^{†2})]$, so $S^{†}aS=a\cosh r-a^{†}\sinh r$. Write $s={\sinh }^{2}r$, $c=\cosh r$ and $N=a^{†}a$. Vacuum expectations then givePair creation makes the mean field zero while the mean photon number is positive. Expanding the transformed ladder operators in the vacuum gives the squeezed-vacuum photon-number moments: $⟨N(N-1)⟩=3s^2+s$ and $⟨N^2⟩=3s^2+2s$; equivalently $\mathrm{Var}N=2s(s+1)$.

The vectors produced by photon addition and photon subtraction in the paper are not normalized. Their squared norms are $s+1$ and $s$, respectively. A particularly direct evaluation isBoth successful operations therefore prepare the same squeezed one-photon state, up to an overall phase, whenever subtraction has nonzero probability. Since $⟨1|S^{†}NS|1⟩=c^2+2s=1+3s$, the unnormalized moments areThe printed photon-added formula equals $M_{+}$, not a normalized mean. With its literal unnormalized convention, the corresponding subtraction result is $M_{-}$ andThis difference tends to one as $r\to 0$ and grows as $\frac{3}{4}{e}^{2r}$ for large positive $r$. Keeping this convention recovers the paper's requested expression without misidentifying it as a physical conditional mean.

For the normalization of photon-added and photon-subtracted states, divide by the squared norms to obtain actual conditional means:These equalities for subtraction require $s>0$. As $r\to 0$ the normalized successful subtraction tends to $|1⟩$, but its success probability vanishes; exactly at $r=0$, $a|0⟩=0$ gives no conditional state. At large $r$, both normalized means grow as $3e^{2r}/4$, while their difference remains zero.

For the photon statistics of a squeezed vacuum, the even-number probabilities areAfter subtraction the normalized distribution is $p_n^{-}=(n+1)p_{n+1}/s$. It has only odd occupation and no vacuum component. photon addition gives $p_n^{+}=n{p}_{n-1}/(s+1)$, identical here to $p_n^{-}$. The diagram uses normalized distributions with an explicitly chosen $r$, rather than treating the unnormalized moments as means.

A subtraction event samples initial occupation with weight $n$, preferentially selecting the larger components before removing one photon. In general ${\bar{n}}_{-}=⟨N(N-1)⟩/⟨N⟩=⟨N⟩-1+\mathrm{Var}N/⟨N⟩$. Here the large variance gives ${\bar{n}}_{-}-⟨N⟩=1+2s>0$. This conditional photon-number bias does not create energy by removing a photon: the unconditional ensemble also includes the runs without a heralded event.

For $X=(a+a^{†})/2$ and $Y=(a-a^{†})/(2i)$, $[X,Y]=i/2$, so $\Delta X\Delta Y\ge 1/4$. Vacuum and coherent states have variance $1/4$ in either optical quadrature. quadrature squeezing means that some rotated optical quadrature has variance below this reference, with compensating noise elsewhere. A reduced second moment about zero is not the correct test when the field's mean is displaced.

For $S(\zeta )=\exp [({\zeta }^{\ast }{a}^2-\zeta a^{†2})/2]$ and $\zeta =r{e}^{i\theta }$, the quiet principal axis of the optical squeezed vacuum is at angle $\theta /2$ under the squeeze-operator convention used above, andThe pure Gaussian quantum state saturates the uncertainty product. Displacement moves its centre without changing its covariance ellipse; squeezed thermal states instead have a larger uncertainty product. The optical squeezed vacuum has zero mean field, even photon-number parity and mean occupation ${\sinh }^{2}r$; it need not have sub-Poissonian photon statistics. number squeezing and quadrature squeezing are different conditions.

Degenerate parametric amplification generates single-mode squeezing; a two-mode squeezed vacuum instead has correlated occupations and joint quadrature squeezing. Either individual mode of an ideal two-mode state is thermal after the other is ignored, not individually squeezed. balanced homodyne detection measures the noise versus local oscillator phase. Reference phase drift admixes the anti-squeezed optical quadrature, while electronic noise and detector imbalance need separate checks.

For an optical loss channel with efficiency $\eta$ and vacuum entering the unused port, optical squeezing degradation by loss givesThe linear-loss limit follows directly by writing the output field as $\sqrt{\eta }a+\sqrt{1-\eta }v$ with independent vacuum mode $v$. Loss therefore places a floor under measured squeezing. Squeezed input can improve phase-sensitive measurements when aligned with the measured optical quadrature; it does not reduce every noise source or both conjugate uncertainties at once.

An optical squeezed vacuum has zero mean field, nonzero mean photon number ${\sinh }^{2}r$, and unequal noise in two principal optical quadratures. With $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$, the vacuum variance is $1/4$ and $[X_1,X_2]=i/2$. In axes aligned with the squeezing,The ideal pure state saturates $\Delta X_1\Delta X_2=1/4$. A below-threshold degenerate optical parametric amplifier or oscillator can generate it: a coherent pump drives pair creation into a mode, and the pump phase selects the squeezed quadrature. Loss mixes in vacuum noise, while technical phase fluctuations mix the squeezed and antisqueezed axes.

For the stated vacuum–one-photon superposition, normalization requires $|ϵ|\le 1$. Direct Fock-basis matrix elements giveUsing $a{a}^{†}=a^{†}a+1$,Consequently there is no squeezing of the specified quadrature:It equals the vacuum uncertainty only at $ϵ=0$ and reaches $\sqrt{3}/2$ at $ϵ=\pm 1$. The curve is even in $ϵ$. The orientation matters: the same real superposition has $(\Delta X_1{)}^2=1/4-{ϵ}^2/2+{ϵ}^4$, which is below $1/4$ for $0<{ϵ}^2<1/2$. This does not turn it into a Gaussian squeezed vacuum; it simply illustrates quadrature squeezing of a vacuum–one-photon superposition.

For the printed beam splitter convention, expansion gives the operator identitiesThuswhere different input modes commute. The source's equality mixing an expectation on one side with an operator on the other is understood as this operator identity, or with expectation brackets on both sides. The mean difference reveals phase-sensitive mutual coherence between inputs; it is not by itself a photon-number correlation measurement of the usual Hanbury Brown–Twiss type. Repeated fluctuations, rather than one mean value, are needed to infer quadrature uncertainty.

For the independent coherent local oscillator, replace $b,b^{†}$ by $\beta ,{\beta }^{\ast }$ in the mean. The effective signal operator isTherefore $\varphi =0$ measures $X_2$, and $\varphi =-\pi /2$ measures $X_1$, each multiplied by $2|\beta |$. Choosing $\varphi =\pi /2$ measures $-X_1$. These phase choices follow the specified complex beam-splitter matrix; importing a different beam-splitter convention would shift them.

This is balanced homodyne detection. A stable strong local oscillator supplies the phase reference, the signal and oscillator meet at a 50:50 beam splitter, and matched photodetector currents are subtracted. Varying the reference phase selects different quadrature projections. Recording their distributions supplies more information than their means and supports optical homodyne tomography.

The strong-reference limit also needs a noise qualification. Define $X_{\vartheta }=X_1\cos \vartheta +X_2\sin \vartheta$, with $\vartheta =\varphi +\pi /2$. For independent signal and coherent reference, an exact ideal-detector calculation givesOne derivation writes $b=\beta +\delta b$, where $\delta b$ is in vacuum. The reference-amplitude term gives the first contribution; the remaining vacuum term contributes $⟨a^{†}a⟩$. Thus the normalized variance approaches the signal quadrature variance when the reference is sufficiently strong relative to the signal. A finite reference is not exactly a classical noiseless field.

A single detector can also carry quadrature information, but its mean alone still does not determine the variance. It measures $n_C=(n_a+n_b+I_{CD})/2$, retaining the large reference background and its fluctuations. For fixed signal moments and a large coherent reference,The additional vacuum-level term and the uncancelled reference power must be calibrated. In practice the reference may also have much larger classical intensity noise. Balanced subtraction cancels the common total-power term and strongly rejects common-mode technical noise while retaining the phase-sensitive difference. Detector imbalance, unequal efficiencies and reference phase noise limit that advantage. It is therefore the more direct and usually more sensitive quadrature-noise measurement, rather than a claim that a single-port measurement contains no useful information.

Even and odd occupation sectors have eigenvalues $+1$ and $-1$. Pair creation preserves parity, whereas one photon addition or photon subtraction reverses it. An optical squeezed vacuum is even and a squeezed one-photon state is odd.

The single-mode squeeze operator changes occupation in pairs. Consequently an optical squeezed vacuum has only even photon numbers, with mean $s={\sinh }^{2}r$ and variance $2s(s+1)$. Its large number fluctuations are compatible with quadrature squeezing: they concern different observables.

For real $ϵ$ with $|ϵ|\le 1$, let $|\psi ⟩=\sqrt{1-{ϵ}^2}|0⟩+ϵ|1⟩$. The Fock state matrix elements give $⟨a⟩=ϵ\sqrt{1-{ϵ}^2}$, $⟨a^2⟩=0$ and $⟨a^{†}a⟩={ϵ}^2$. ThereforeOnly the first of these optical quadratures is squeezed for this real superposition, with minimum variance $3/16$ at ${ϵ}^2=1/4$. The second is never below vacuum noise. This non-Gaussian superposition is distinct from an optical squeezed vacuum.

photon addition to an optical squeezed vacuum gives $a^{†}S|0⟩=\cosh rS|1⟩$; subtraction gives $aS|0⟩=-\sinh rS|1⟩$. After successful-event normalization these are the same state up to phase, with mean occupation $1+3{\sinh }^{2}r$ and odd photon parity. At zero squeezing subtraction never succeeds; the normalized limiting state is not an output at that exact parameter value.