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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]

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 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.