 Solution

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.