= 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.<Generation of Squeezed States by Parametric Down Conversion by Ling-An Wu and others>[1986, abstract: cavity down conversion and homodyne noise reduction]{ref}
Use $S(r)=\exp[\tfrac r2(a^2-a^{\dagger2})]$, so $S^\dagger aS=a\cosh r-a^\dagger\sinh r$. Write $s=\sinh^2r$, $c=\cosh r$ and $N=a^\dagger a$. Vacuum expectations then give
$$
\boxed{\langle N\rangle=s},\qquad
\langle a^2\rangle=-\sinh r\cosh r.
$$
Pair 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>: $\langle N(N-1)\rangle=3s^2+s$ and $\langle N^2\rangle=3s^2+2s$; equivalently $\operatorname{Var}N=2s(s+1)$.
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 is
$$
a^\dagger S(r)|0\rangle=cS(r)|1\rangle,\qquad
aS(r)|0\rangle=-\sinh r\,S(r)|1\rangle.
$$
Both successful operations therefore prepare the same <squeezed one-photon state>, up to an overall phase, whenever subtraction has nonzero <probability>. Since $\langle1|S^\dagger NS|1\rangle=c^2+2s=1+3s$, the unnormalized moments are
$$
M_+=\langle\Psi_+|N|\Psi_+\rangle=(c^2+2s)c^2=(3s+1)(s+1),
\qquad M_-=\langle\Psi_-|N|\Psi_-\rangle=s(3s+1).
$$
\b[The printed photon-added formula equals $M_+$, not a normalized mean.] With its literal unnormalized convention, the corresponding subtraction result is $M_-$ and
$$
\boxed{M_+-M_-=1+3\sinh^2r>0.}
$$
This difference tends to one as $r\to0$ and grows as $\tfrac34e^{2r}$ 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:
$$
\boxed{\bar n_+=\frac{M_+}{s+1}=1+3s,
\qquad \bar n_-=\frac{M_-}{s}=1+3s,
\qquad \bar n_+-\bar n_-=0.}
$$
These equalities for subtraction require $s>0$. As $r\to0$ the normalized successful subtraction tends to $|1\rangle$, but its success <probability> vanishes; exactly at $r=0$, $a|0\rangle=0$ 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> are
$$
p_{2m}=\frac{(2m)!}{2^{2m}(m!)^2}\frac{\tanh^{2m}r}{\cosh r},\qquad p_{2m+1}=0.
$$
After 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 $p^+_n=np_{n-1}/(s+1)$, identical here to $p^-_n$. The diagram uses normalized distributions with an explicitly chosen $r$, rather than treating the unnormalized moments as means.
\Image[/past-exam-of-the-nonlinear-optics-and-quantum-states-of-light-course-of-the-university-of-cambridge/2014/paper-1-conditioned-photon-statistics.png]
{title=Original normalized even squeezed-vacuum and odd conditional photon-number distributions, with raw-moment and mean conventions distinguished}
A subtraction event samples initial occupation with weight $n$, preferentially selecting the larger components before removing one <photon>. In general $\bar n_-=\langle N(N-1)\rangle/\langle N\rangle=\langle N\rangle-1+\operatorname{Var}N/\langle N\rangle$. Here the large <variance> gives \b[$\bar n_--\langle N\rangle=1+2s>0$]. This <conditional photon-number bias> does not create <energy> by removing a <photon>: the unconditional ensemble also includes the runs without a heralded event.
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