07 Jun 2026
IV: Exact First-Order Covariance Response Without Gaussian Closure
Covariances are useful observables in a non-Gaussian quench, but they do not generally form a closed dynamical system. Here the initial cross-covariance rates can nevertheless be derived exactly, directly from the Heisenberg equations and the symmetry of the prepared state.
Setting and conventions
We use $\hbar=2M=1$ and a signed half-cyclotron frequency
\[\omega_c=\frac{q_{\mathrm{el}}B}{2Mc}.\]The preparation bipartition is fixed as
\[A\equiv +,\qquad B\equiv -.\]An exact product eigenstate of the separable exceptional-Hermite Hamiltonian is prepared first,
\[\lvert\Psi_0\rangle =\lvert m_+,\ell_+\rangle_A \otimes\lvert m_-,\ell_-\rangle_B,\]and then evolved with
\[H_f=p_+^2+p_-^2+U_+(q_+)+U_-(q_-) -\omega_c(q_+p_--q_-p_+).\]The functions $U_\pm$ contain the regular rational extensions and the same diamagnetically dressed confinement used during preparation. The quench switches on the angular term only.
Throughout the first-order theorem below, each initial factor is assumed to be real and of definite parity. These hypotheses are essential; the stated selection rules need not hold for a displaced, current-carrying, or indefinite-parity input.
Why the moment hierarchy remains open
The exact Heisenberg equations are
\[\begin{aligned} \dot q_+&=2p_++\omega_cq_-, &\dot q_-&=2p_--\omega_cq_+,\\ \dot p_+&=-U_+'(q_+)+\omega_cp_-, &\dot p_-&=-U_-'(q_-)-\omega_cp_+. \end{aligned}\]For a quadratic potential, $U_\alpha’$ is linear and the second moments close. A rational extension changes that structure. For example,
\[\frac{d}{dt}\langle q_+^2\rangle =2\langle q_+p_++p_+q_+\rangle +2\omega_c\langle q_+q_-\rangle,\]while
\[\begin{aligned} \frac{d}{dt}\langle q_+p_++p_+q_+\rangle ={}&4\langle p_+^2\rangle -\langle q_+U_+'+U_+'q_+\rangle\\ &+2\omega_c\langle q_+p_-+q_-p_+\rangle. \end{aligned}\]The rational-force expectation is not fixed by the covariance matrix. Higher equations generate further force-weighted moments, so the exact dynamics do not admit a Gaussian second-moment closure.
This does not prevent local analytic results. For any observable $\mathcal O$, its derivatives at the quench are
\[\left.\frac{d^n}{dt^n}\langle\mathcal O(t)\rangle\right|_{t=0} =i^n\left\langle\operatorname{ad}_{H_f}^n(\mathcal O)\right\rangle_0, \qquad \operatorname{ad}_{H_f}(X)=[H_f,X].\]This nested-commutator identity is exact. Keeping finitely many Taylor terms is, separately, a short-time approximation.
Complete first-order cross-covariance theorem
Define the initial local variances
\[V_{q,\alpha}=\langle q_\alpha^2\rangle_0, \qquad V_{p,\alpha}=\langle p_\alpha^2\rangle_0.\]For the real definite-parity product input, all first moments vanish. Initial cross correlations and local symmetrized $q_\alpha p_\alpha$ covariances also vanish. Therefore cross moments and cross covariances agree to first order at $t=0$.
Direct differentiation gives the complete response:
\[\boxed{ \left.\frac{d}{dt}\operatorname{Cov}(q_+,q_-)\right|_0 =\omega_c(V_{q,-}-V_{q,+}) }\]and
\[\boxed{ \left.\frac{d}{dt}\operatorname{Cov}(p_+,p_-)\right|_0 =\omega_c(V_{p,-}-V_{p,+}). }\]Both symmetrized mixed blocks vanish:
\[\left.\frac{d}{dt}\frac12 \langle q_+p_-+p_-q_+\rangle\right|_0=0,\] \[\left.\frac{d}{dt}\frac12 \langle q_-p_++p_+q_-\rangle\right|_0=0.\]The force terms that might enter the momentum cross rate factorize into products such as $\langle U_+’\rangle_0\langle p_-\rangle_0$, and parity makes them vanish. No assumption of Gaussianity is used.
The result has three immediate consequences:
- Field reversal changes the sign of each nonzero rate because $\omega_c$ is signed.
- Equality $V_{q,+}=V_{q,-}$ removes only the coordinate-coordinate rate.
- Every linear cross-covariance rate vanishes only when both $V_{q,+}=V_{q,-}$ and $V_{p,+}=V_{p,-}$.
These are statements about derivatives at the quench. They do not imply that the same pattern persists at later times, when the nonlinear hierarchy becomes active.
The synchronization numerator
When the rotated and laboratory axes coincide up to permutation—as for $g=0$—introduce dimensionless quadratures
\[Q_j=\sqrt{\frac{\omega_0}{2}}x_j, \qquad P_j=\sqrt{\frac{2}{\omega_0}}p_j.\]The complete-synchronization measure is
\[S_c=\langle\mathcal D_c\rangle^{-1}, \qquad \mathcal D_c=\frac12\left[ (\delta Q_x-\delta Q_y)^2+(\delta P_x-\delta P_y)^2 \right].\]An equal-time value of $S_c$ measures relative fluctuations; by itself it is not evidence of phase locking or a stationary synchronized regime.
For two codimension-two ground factors, set $\omega_0=1$ and define
\[\mathcal M=\sqrt{\frac{\pi}{2}}e^{1/2} \operatorname{erfc}\!\left(\frac{1}{\sqrt2}\right),\] \[A_\alpha^{(2)} =V_{Q,\alpha}+V_{P,\alpha} =\frac{\mathcal M-1/2}{\Omega_\alpha} +\Omega_\alpha\left(\frac32+\frac{\mathcal M}{3}\right).\]The local variance rates vanish at $t=0$ for these real stationary one-mode factors. Hence
\[\left.\frac{d}{dt}\langle\mathcal D_c\rangle\right|_0 =\omega_c\underbrace{\left(A_+^{(2)}-A_-^{(2)}\right)}_{ \mathcal N_{\mathrm{sync}}},\]where $\mathcal N_{\mathrm{sync}}$ is the synchronization numerator. It yields
\[\boxed{ S_c(t)= \frac{2}{A_+^{(2)}+A_-^{(2)}} -\frac{4\omega_c\left(A_+^{(2)}-A_-^{(2)}\right)} {\left(A_+^{(2)}+A_-^{(2)}\right)^2}t +O(t^2). }\]The slope is odd under field reversal and disappears when the two dimensionless second-moment sums coincide. This formula is local in time and specific to the stated axes, parity, factor choice, and scaling. It is not a claim of persistent or stationary synchronization.
For the manuscript benchmark
\[\Omega_+=\frac54, \qquad \Omega_-=\frac{17}{20}, \qquad \omega_c=\frac34,\]Maxima gives
\[S_c(0)=0.5106376395008639\ldots, \qquad \dot S_c(0)=-0.1229730434570357\ldots.\]The symbolic residuals used in the publication check vanish. A separate converged basis propagation reproduces these coefficients numerically, but that propagation uses tested finite cutoffs—it is not an exact finite-dimensional reduction.
What this result does and does not establish
The first-order response is exact for the stated initial state. It provides a sharp diagnostic for analytic work and numerical propagation. It does not close the rational moment hierarchy, determine later-time covariances from initial second moments alone, or turn a non-Gaussian state into a covariance-complete one.
In particular, vanishing linear covariance rates do not prove stationarity. The survival probability supplies an independent second-order test, developed in the next article.
Discussion