source_sections/09_quantum.tex

\section{Classical and quantum amplification in the same theory}\label{sec:quantum}
The equivalence in this section concerns the \emph{same} constrained material
programs, not a separate quantum optimization ansatz. The underlying
Bogoliubov correspondence and hyperbolic photon-number formula are known
\cite{echave,kim,bae}; the new consequences within this release are their
combination with the resource extremizer, exact structural defect, and
finite-observation rigidity estimate.

\subsection{Canonical quantization of the Maxwell mode}
For a real standing-wave component of a transverse vector potential
$\mathcal A$, use a consistent unit-volume mode normalization and its
canonical momentum $\Pi=\varepsilon\dot{\mathcal A}$. The mode Hamiltonian is
\begin{equation}\label{eq:MaxwellHamiltonian}
\mathscr H(t)=\frac{\Pi^2}{2\varepsilon(t)}+
\frac{k^2\mathcal A^2}{2\mu(t)},\qquad
\{\mathcal A,\Pi\}=1.
\end{equation}
Its Hamilton equations reproduce the transverse wave mode; no derivatives
of material coefficients are needed. For constant $\mu$, the constant
canonical transformation
\[
Q=-\frac{\sqrt\mu}{|k|}\Pi,\qquad
P=\frac{|k|}{\sqrt\mu}\mathcal A
\]
has $\{Q,P\}=1$ and gives
$\mathscr H=(P^2+\omega(t)^2Q^2)/2$, with
$\omega(t)^2=k^2/(\mu\varepsilon(t))$.
Thus $Q$ is a constant multiple of the electric induction variable used
in the classical reduction. With
$q=\sqrt{\omega_-}Q$, $p=P/\sqrt{\omega_-}$, and $s=\omega_-t$, the
Hamiltonian generating $s$-evolution is $(p^2+u(s)q^2)/2$.
After quantization, $[q,p]=i\hbar$ and the Heisenberg equations are
exactly the classical linear system. No semiclassical closure is involved
for these quadratic observables.

For both time-dependent material parameters one instead keeps
$q=\mathcal A$, $p=\Pi$, which gives
$\mathscr H=(a(t)p^2+b(t)q^2)/2$ with $a=\varepsilon^{-1}$,
$b=k^2\mu^{-1}$. The rectangle theorem in Section~\ref{sec:rectangle}
therefore enters the identical quantum construction below.

\subsection{A fixed vacuum convention and exact Bogoliubov coefficients}
Fix a reference $\Omega>0$ independently of the chosen material profile and
of its translated temporal cut. Define
\[
\hat a=\frac{\sqrt\Omega q+ip/\sqrt\Omega}{\sqrt{2\hbar}},\qquad
W_\Omega=\diag(\sqrt\Omega,1/\sqrt\Omega).
\]
This reference specifies the initial vacuum and the number measurement.
It is not automatically the instantaneous ground state of each competing
profile. Input/output matching stages, when included physically, must be
incorporated in the transfer before applying the formulas.
For a real canonical monodromy $M=(M_{ij})$,
\begin{align}
\alpha&=\tfrac12\{M_{11}+M_{22}+i(M_{21}/\Omega-\Omega M_{12})\},\nonumber\\
\beta&=\tfrac12\{M_{11}-M_{22}+i(M_{21}/\Omega+\Omega M_{12})\}.
\label{eq:bogoliubov}
\end{align}
Substitution of $q'=M_{11}q+M_{12}p$ and $p'=M_{21}q+M_{22}p$
into $\hat a$ proves $\hat a'=\alpha\hat a+\beta\hat a^\dagger$.
The determinant-one identity gives $|\alpha|^2-|\beta|^2=1$ and
$\Re\alpha=D=\tfrac12\tr M$. Hence
\begin{equation}\label{eq:SU11}
\mathcal B=\begin{pmatrix}\alpha&\beta\\\beta^*&\alpha^*\end{pmatrix}
\in SU(1,1),\qquad \det\mathcal B=1.
\end{equation}
This is a fixed similarity transformation of $M$, so the two matrices have
the same multipliers, discriminant, and Floquet exponent.

For clarity about implementation, write
$\alpha=e^{i\phi}\cosh r$, $\beta=e^{i\psi}\sinh r$.
Phase rotations and the exponential of the quadratic squeezing generator
implement this transformation on one mode. On finitely many standing
modes these Gaussian unitaries also implement the continuous linear
propagator, up to a phase that has no effect on number or covariance.
Approximating a bounded measurable coefficient by temporal step functions
converges in the integral linear equation; the resulting finite Gaussian
transformations define the same limiting covariance and particle counts.
Thus none of the formulas requires differentiating a discontinuous
coefficient or assuming a finite initial classical field.

The two real standing modes can be combined as
$\hat a_k=(\hat a_c-i\hat a_s)/\sqrt2$ and
$\hat a_{-k}=(\hat a_c+i\hat a_s)/\sqrt2$.
Their identical real dynamics then give
$\hat a_k'=\alpha\hat a_k+\beta\hat a_{-k}^\dagger$.
For an initial two-mode vacuum, the mean photon count \emph{per traveling
mode}, equal to the mean pair count, is $\nu=|\beta|^2$; the mean total
photon count in the pair is $2\nu$. The pure output has geometric pair
probabilities
\[
\Pr(j\text{ pairs})=\frac{\nu^j}{(1+\nu)^{j+1}},\qquad j=0,1,\ldots.
\]
Indeed annihilation by the transformed vacuum operator gives the coefficient
recurrence $c_{j+1}=-(\beta/\alpha^*)c_j$ up to a phase convention, and
normalization and summing the geometric series give the displayed
probabilities and mean. Single standing-mode photons have an even-photon
squeezed-vacuum distribution instead; the geometric statement is for pairs.

\begin{theorem}[Exact finite-cycle and asymptotic quantum laws]\label{thm:quantum}
Let $M\in SL(2,\R)$ be the one-period monodromy of any admissible program,
with the fixed reference above. Denote its $N$-period Bogoliubov coefficient
by $\beta_N$, its mean pair count by $\nu_N=|\beta_N|^2$, and its
squeezing parameter by $r_N=\operatorname{arsinh}\sqrt{\nu_N}$.
Then, with $U_j$ the Chebyshev polynomial of the second kind,
\begin{equation}\label{eq:quantumCheb}
\beta_N=U_{N-1}(D)\beta,\qquad
r_N=\log\|W_\Omega M^N W_\Omega^{-1}\|_2.
\end{equation}
If $|D|>1$ and $L=\arcosh|D|$, then
\begin{align}
\nu_N&=\chi\sinh^2(NL),\qquad
\chi=\frac{|\beta|^2}{\sinh^2L}
=1+\frac{(\Im\alpha)^2}{\sinh^2L}\ge1,\label{eq:quantumhyper}\\
r_N&=NL+\tfrac12\log\chi+o(1).\label{eq:quantumr}
\end{align}
If $|D|<1$, put $\phi=\arccos D$; then
\begin{equation}\label{eq:quantumstable}
\nu_N=|\beta|^2\frac{\sin^2(N\phi)}{\sin^2\phi},
\end{equation}
which is bounded for the fixed monodromy. If $|D|=1$, then
\begin{equation}\label{eq:quantumparabolic}
\nu_N=N^2|\beta|^2.
\end{equation}
At $M=\pm I$ this is identically zero; otherwise this is polynomial photon
growth, with $r_N=\log N+O(1)$, not a positive Floquet exponent.
In every case,
\begin{equation}\label{eq:quantumlimits}
\lim_{N\to\infty}\frac{r_N}{N}=L,\qquad
\lim_{N\to\infty}\frac{\log(1+\nu_N)}N=2L.
\end{equation}
\end{theorem}
\begin{proof}
Cayley--Hamilton gives
$M^N=U_{N-1}(D)M-U_{N-2}(D)I$, with $U_{-1}=0$.
The off-diagonal Bogoliubov coefficient is linear in matrix entries and
vanishes on $I$, proving the first formula. The identities
\[
U_{N-1}(\cos\phi)=\frac{\sin N\phi}{\sin\phi},\quad
|U_{N-1}(\pm\cosh L)|=\frac{\sinh NL}{\sinh L},\quad
|U_{N-1}(\pm1)|=N
\]
follow either by the recurrence or by finite geometric sums. They give
all three count formulas. Since $|\alpha|^2-|\beta|^2=1$ and
$(\Re\alpha)^2=D^2$, one has
$|\beta|^2=D^2-1+(\Im\alpha)^2$, proving $\chi\ge1$.

For $K=W_\Omega M^N W_\Omega^{-1}$, let its larger singular value be
$\sigma\ge1$. Its determinant is one, so the other is $1/\sigma$.
Direct substitution in \eqref{eq:bogoliubov} gives
$|\beta_N|^2=(\tr K^{\mathsf T}K-2)/4
=(\sigma-\sigma^{-1})^2/4$.
Thus $r_N=\log\sigma=\log\|K\|_2$.
In the hyperbolic case, expanding
$\operatorname{arsinh}(\sqrt\chi\sinh NL)$ gives \eqref{eq:quantumr}.
The stable and parabolic cases yield zero limits in
\eqref{eq:quantumlimits} directly. These arguments retain both signs of
real Floquet multipliers and do not assume a branch of a complex logarithm.
\end{proof}
The prefactor $\chi$ in \eqref{eq:quantumhyper} cannot generally be dropped.
The statement $\nu_N\sim\sinh^2(NL)$ with ratio tending to one is false
unless $\chi=1$. The known Petermann-factor representation in
\cite{kim} makes the same distinction between exponential rate and
photon-number prefactor.

\begin{corollary}[Resource-constrained classical--quantum extremality]\label{cor:quantumargmax}
For the fixed $(R,S,m)$ admissible class,
\[
\operatorname*{arg\,max}_u L(u)
=\operatorname*{arg\,max}_u\lim_{N\to\infty}r_N(u)/N
=\operatorname*{arg\,max}_u\lim_{N\to\infty}\log(1+\nu_N(u))/N.
\]
When $F(R,S,m)>0$, this is exactly the union of the canonical binary
translation orbits in Theorem~\ref{thm:resource}. A deficit $\delta$ in the
asymptotic squeezing rate per cycle gives
$\dist_1(u,\mathcal O_*)\le C\sqrt\delta$; a deficit $\delta$ in the
logarithmic pair-count rate gives $C\sqrt{\delta/2}$.
\end{corollary}
\begin{proof}
The limits in \eqref{eq:quantumlimits} hold for every individual admissible
profile. Maximizing their identical functions gives the equality of sets;
Theorem~\ref{thm:globalstability} gives the distance estimates.
\end{proof}
These are rates of mean pair production, not optimization of the probability
of exactly one pair. That probability is $\nu/(1+\nu)^2$ and need not
increase with mean amplification.

\begin{proposition}[Finite-cycle photon optimization is a different problem]\label{prop:quantumfinite}
Even within a single classical maximizing translation orbit, one-period
squeezing and photon production need not agree. Thus the finite-$N$ quantum
argmax is not universally the full classical argmax.
\end{proposition}
\begin{proof}
For $R=2$, $m=2$, $S=3\pi/4$, use reference $\Omega=1$. Starting at the
high-state switch gives
\[
M_0=\diag(-2,-1/2),\qquad |\beta_0|^2=9/16.
\]
Translate the marked start by $\pi/8$ inside the high segment. The new
monodromy is the conjugate
\[
M_t=E(2,\pi/8)M_0E(2,\pi/8)^{-1}
=\begin{pmatrix}-5/4&3/8\\3/2&-5/4\end{pmatrix},
\qquad |\beta_t|^2=225/256.
\]
Both have the same period, material resources, and maximal $L=\log2$.
Their different $|\beta|^2$ give different finite-$N$ counts by
\eqref{eq:quantumCheb}. At least the smaller-count member of the classical
argmax cannot maximize the finite-cycle count. Fixing the reference and
marked cut is essential in making that comparison meaningful.
\end{proof}

\begin{theorem}[Finite-observation quantum rigidity]\label{thm:quantumstability}
For a class with $\|W_\Omega M W_\Omega^{-1}\|_2\le K_0$ uniformly,
\begin{equation}\label{eq:finitequantumerror}
0\le r_N-NL\le\log\{N(K_0+1)\}.
\end{equation}
For the scalar bounds, one may take
$K_0=\max(\Omega,\Omega^{-1})e^{BS}$.
Let $e_N=\log\{N(K_0+1)\}/N$. For the fixed-resource class with $F>0$,
either certificate
\[
r_N(u)/N\ge F-\delta
\quad\text{or}\quad
\sup_v r_N(v)/N-r_N(u)/N\le\delta
\]
implies
\begin{equation}\label{eq:finitequantumrigidity}
\boxed{\dist_1(u,\mathcal O_*)\le C\sqrt{\delta+e_N}.}
\end{equation}
\end{theorem}
\begin{proof}
Spectral radius is bounded by operator norm, so $r_N\ge NL$.
The Chebyshev identities give
$|U_{N-1}(D)|\le N e^{(N-1)L}$ in every regime: in the stable regime use
$|U_{N-1}|\le N$, and in the hyperbolic regime use its geometric sum.
Cayley--Hamilton and the triangle inequality yield
$\|W M^N W^{-1}\|_2\le N(K_0+1)e^{NL}$, proving
\eqref{eq:finitequantumerror}. Since $\|A(u)\|_2\le B$, the integral ODE
estimate gives $\|M\|_2\le e^{BS}$ and the stated $K_0$.
The first certificate implies $L\ge F-\delta-e_N$.
For the second, a classical extremizer has $r_N/N\ge F$, so it implies
the first certificate. Apply Theorem~\ref{thm:globalstability}.
\end{proof}
This finite-cycle consequence is a structural bound, not a claim that
finite-cycle quantum optimizers are exactly classical optimizers. The
$O(\log N/N)$ allowance explicitly retains possible transient amplification,
parabolic growth, and the vacuum-normalization prefactor.