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A normalized nonnegative probability measure over coherent states produces a classical mixture. The measure may contain Dirac delta distributions, so a pure coherent state is included. Conditioning on $\alpha$ gives Poisson distribution photon statistics; hence $\mathrm{Var}n=⟨n⟩+\mathrm{Var}(|\alpha {|}^2)\ge ⟨n⟩$. Each optical quadrature has variance at least its vacuum value because mixing adds nonnegative variance of conditional means. These inequalities provide simple nonclassicality witnesses.

For one optical mode, $E=E_0(a+a^{†})$ and $[a,a^{†}]=1$. A coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$, so normal ordering gives $⟨E^2⟩=E_0^2[(\alpha +{\alpha }^{\ast }{)}^2+1]$. Subtracting the squared mean leaves $E_0^2$, independent of the coherent displacement. The extra one is the canonical commutation relation, not noise from a random coherent amplitude. A rotated optical quadrature has the same coherent-state variance with the same amplitude normalization.

An experimental optical physicist studying quantum-enhanced measurements. His research includes the generation and use of squeezed light for precision measurement, including magnetometry and gravitational-wave detection. Reducing fluctuations in a chosen optical quadrature connects the quantum state of the probe field to the sensitivity of an instrument.^[ref]

For the convention of balanced homodyne detection, assume independent signal and coherent reference states and ideal detectors. Write $b=\beta +\delta b$, with $\delta b$ in vacuum. The current difference splits into $2|\beta |X_{\vartheta }$ and $i(a^{†}\delta b-a\delta b^{†})$. Cross terms have zero expectation, and the second term has variance $⟨a^{†}a⟩$. Consequently the exact variance is $4|\beta {|}^2\mathrm{Var}{X}_{\vartheta }+⟨a^{†}a⟩$. Division by $4|\beta {|}^2$ approaches the signal optical quadrature variance when the photon-number correction is negligible.

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.

Ling-An Wu, H. J. Kimble, John L. Hall and Huifa Wu reported cavity degenerate parametric down conversion and below-vacuum optical quadrature noise measured by balanced homodyne detection. The publisher abstract also emphasizes attenuation when comparing measured and inferred squeezing. doi.org/10.1103/PhysRevLett.57.2520

An experimental optical physicist who coauthored a 1986 generation of optical squeezed vacuum. A cavity enhanced spontaneous parametric down-conversion, while balanced homodyne detection measured reduced optical quadrature noise. The comparison with attenuation models linked a directly observed noise reduction to the source's underlying quantum field.^[ref]

An experimental physicist who coauthored an early optical squeezed vacuum experiment. Its balanced homodyne detection compared the field's optical quadrature fluctuations with vacuum noise. The paper distinguishes observed noise reduction from the larger inferred reduction before optical attenuation, showing why propagation losses matter when quantifying squeezing.^[ref]

Jeffrey Shapiro develops the quantum optics of optical information and communication. His teaching treatment of optical quadratures explains why a squeezed coherent state can reduce fluctuations of one measured field component while increasing those of its conjugate component. The Heisenberg uncertainty principle constrains their product, rather than requiring equal fluctuations in both components. Comparing these states with coherent states connects the mathematics of a quantum harmonic oscillator with the statistics of optical measurements.^[ref]

Lecturer named on the cover of Quantum Optical Communication lecture 6 by Jeffrey H. Shapiro.

An experimental optical physicist who coauthored a 1986 demonstration of optical squeezed vacuum. Degenerate spontaneous parametric down-conversion inside a cavity produced reduced fluctuations of an optical quadrature, measured with balanced homodyne detection. The result made a noise level below the vacuum reference a directly measured optical property.^[ref]

A coherent reference field that selects the measured optical quadrature through its relative phase. Mode matching and phase stability determine how faithfully the desired signal optical quadrature is sampled.

Sufficient single-mode witnesses include a Fano factor $\mathrm{Var}n/⟨n⟩<1$, an optical quadrature variance below $1/4$ for $[X,P]=i/2$, or photon antibunching with $g^{(2)}(0)<1$. Classical mixtures obey $\mathrm{Var}n\ge ⟨n⟩$, and $g^{(2)}(0)=\mathbb{E}(|\alpha {|}^4)/\mathbb{E}(|\alpha {|}^2{)}^2\ge 1$ by nonnegative variance. The tests are sufficient rather than necessary and exclude the vacuum case when dividing by zero mean occupation.

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.

Interference of a signal with a coherent reference at the same optical carrier frequency to measure phase-sensitive field information. The reference fixes an optical quadrature. The inferred noise depends on reference strength, detector efficiency and the detection arrangement.

Optical homodyne tomography reconstructs an optical quantum state from repeated optical quadrature measurements at different phases. A strong coherent local oscillator interferes with the signal on a beam splitter; balanced homodyne detection subtracts the two detector outputs to estimate the quadrature selected by the reference phase.

Many identically prepared signals give a distribution for that quadrature. Repeating at several phases supplies complementary distributions from which a density matrix or optical Wigner function can be estimated. One quadrature distribution alone generally does not determine the state, just as a single projection of a three-dimensional object does not reveal its full shape. Detector efficiency, mode matching and phase noise must be included in the measurement model; otherwise apparent loss or excess noise can be assigned incorrectly to the signal state.

A single optical mode in a squeezed vacuum state. In principal optical quadratures, its variances are $e^{-2r}/4$ and $e^{2r}/4$, while its mean photon number is ${\sinh }^{2}r$. Its mean field vanishes. A degenerate optical parametric amplifier generates the state by pair creation with phase fixed by the pump. Loss admixes unsqueezed vacuum and limits the measured variance reduction.

Mixing a mode with independent vacuum on a beam splitter of intensity transmission $\eta$ gives the displayed optical quadrature variance when vacuum variance is $1/4$. It approaches vacuum noise as transmission falls. Phase noise instead mixes squeezed and anti-squeezed axes, and cannot in general be represented by the same efficiency alone.

The coherent state satisfies $⟨a⟩=\alpha$, $⟨a^{†}⟩={\alpha }^{\ast }$ and $⟨a^{†}a⟩=|\alpha {|}^2$. The canonical commutation relation adds the vacuum term $⟨a{a}^{†}⟩=|\alpha {|}^2+1$. Thereforeand the coherent-state electric-field variance isHere $E_0$ is taken positive as the field-amplitude scale. The coherent state displaces the mean field without changing this vacuum optical quadrature noise.

A single-mode coherent state is an annihilation operator eigenstate:Its photon statistics are Poisson distribution: $⟨n⟩=\mathrm{Var}n=|\alpha {|}^2$. For optical quadratures $X=(a+a^{†})/2$, $P=(a-a^{†})/(2i)$, one has $[X,P]=i/2$ and $\mathrm{Var}X=\mathrm{Var}P=1/4$. Its field mean oscillates classically while retaining vacuum fluctuations.^[ref][ref]

A classical mixture of coherent states has a nonnegative Glauber-Sudarshan representation. A nonclassical state of light cannot be represented that way. Examples are a squeezed vacuum state, with quadrature variances $e^{-2r}/4$ and $e^{2r}/4$, and a nonvacuum number state, with zero photon number variance. Noise below the vacuum level in one quadrature, sub-Poisson number noise, or photon antibunching can witness nonclassicality. For $|n⟩$, $g^{(2)}(0)=1-1/n<1$; coherent light has $g^{(2)}(0)=1$. Thermal light has excess noise but is a classical mixture: noncoherent does not by itself mean nonclassical.

Choose the normalization constant real and positive; its otherwise arbitrary phase is physically immaterial. Orthogonality of the Fock states givesActing with the annihilation operator and shifting the summation index givesThus a coherent state is an annihilation-operator eigenket. It is not also an eigenket of the creation operator: a common eigenket would make $[\hat{a},{\hat{a}}^{†}]$ annihilate it, contradicting $[\hat{a},{\hat{a}}^{†}]=I$. Taking the adjoint of the annihilation-operator equation instead gives the distinct, valid bra relationWith $\hat{n}={\hat{a}}^{†}\hat{a}$, the photon number expected value is $\mu =|\alpha {|}^2$. Normal ordering ${\hat{n}}^2=({\hat{a}}^{†}{)}^2{\hat{a}}^2+{\hat{a}}^{†}\hat{a}$ gives $⟨n^2⟩={\mu }^2+\mu$. ThereforeAt large mean photon number the relative fluctuations vanish, giving a classical-field limit even though quantum fluctuations remain. For the vacuum $\mu =0$, the relative ratio itself is undefined.

The Born rule applied to the number-state coefficient gives the Poisson distributionIt depends on the single parameter $\mu =⟨n⟩$, and has equal expected value and variance, consistent with the preceding moments.

For a Fock state $|n⟩$, raising or lowering the number makes an orthogonal state, so the optical quadrature has $⟨\hat{X}⟩=0$. Expanding its square givesThe first two terms have zero diagonal matrix elements, while the last two contribute $n+1$ and $n$. In particular, the vacuum retains quadrature variance $1/2$.