08 Jun 2026
V: Survival Curvature Beyond the Linear Covariance Response
When all first-order cross-covariance rates vanish, it is tempting to call the post-quench state stationary. That inference is false. The survival probability probes the state vector directly and reveals an exact quadratic response that covariance symmetry can miss.
Fixed-confinement quench and domain assumption
We retain the conventions
\[\hbar=2M=1, \qquad \omega_c=\frac{q_{\mathrm{el}}B}{2Mc},\]with signed $\omega_c$. The preparation coordinates define the bipartition
\[A\equiv +,\qquad B\equiv -.\]A real, definite-parity product eigenstate is prepared exactly,
\[\lvert\Psi_0\rangle =\lvert m_+,\ell_+\rangle_A \otimes\lvert m_-,\ell_-\rangle_B,\]under the additive Hamiltonian $H_{\mathrm{sep}}^{\mathrm{RE}}$. At $t=0$, only the angular coupling is activated:
\[H_f=H_{\mathrm{sep}}^{\mathrm{RE}}-\omega_cL_z, \qquad L_z=q_+p_- -q_-p_+.\]The diamagnetically dressed coordinate confinement is unchanged. This is not a literal quench of the complete electromagnetic field.
For the fourth-order remainder below, assume explicitly
\[\lvert\Psi_0\rangle\in D(H_f^2).\]The regular even-seed exceptional-Hermite inputs satisfy this condition: their rational-polynomial derivatives remain Gaussian-decaying, so the required fourth energy moment is finite.
Survival probability and energy variance
Define
\[\mathcal S(t)= \left|\langle\Psi_0|e^{-iH_ft}|\Psi_0\rangle\right|^2.\]Because survival probability is even in time and the domain assumption supplies the needed moments,
\[\mathcal S(t)=1-\operatorname{Var}_0(H_f)t^2+O(t^4).\]The initial state is an eigenstate of $H_{\mathrm{sep}}^{\mathrm{RE}}$, while reality and parity give
\[\langle L_z\rangle_0=0.\]Therefore the entire energy variance comes from the quenched angular term:
\[\operatorname{Var}_0(H_f) =\omega_c^2\langle L_z^2\rangle_0.\]Introduce the local second moments
\[V_{q,\alpha}=\langle q_\alpha^2\rangle_0, \qquad V_{p,\alpha}=\langle p_\alpha^2\rangle_0.\]Expanding $L_z^2$, factorizing expectations between $A=+$ and $B=-$, and using
\[\langle q_\alpha p_\alpha\rangle_0=\frac{i}{2}, \qquad \langle p_\alpha q_\alpha\rangle_0=-\frac{i}{2},\]gives
\[\langle L_z^2\rangle_0 =V_{q,+}V_{p,-}+V_{q,-}V_{p,+}-\frac12.\]The $-1/2$ is an ordering contribution: the two crossed products in $L_z^2$ each contribute $1/4$ beneath a minus sign. Defining
\[\boxed{ \Lambda_0 =V_{q,+}V_{p,-}+V_{q,-}V_{p,+}-\frac12, }\]we obtain the exact local expansion
\[\boxed{ \mathcal S(t)=1-\omega_c^2\Lambda_0t^2+O(t^4). }\]The coefficient $\Lambda_0$ is independent of the sign of the field, while first-order covariance flows are odd in $\omega_c$. Thus field reversal reverses the direction of the linear covariance response but not the initial survival loss.
A mode-exchange-symmetric counterexample
Suppose the two factors have the same codimension, excitation, and coordinate scale. Their second moments are then mode-exchange symmetric:
\[V_{q,+}=V_{q,-}=V_q, \qquad V_{p,+}=V_{p,-}=V_p.\]Every first-order cross-covariance rate vanishes, because those rates are proportional to the corresponding variance differences. Define the one-mode uncertainty product
\[U_m=\Delta q\,\Delta p=\sqrt{V_qV_p}.\]The survival coefficient reduces to
\[\boxed{ \Lambda_0=2U_m^2-\frac12. }\]For two equal ordinary oscillator vacua, $U_0=1/2$, so $\Lambda_0=0$. This is consistent with rotational invariance. For two equal codimension-two added states, however,
\[U_2=0.5172471466\ldots\]and hence
\[\Lambda_0 =2(0.5172471466\ldots)^2-\frac12 =0.03508922135\ldots>0.\]The state therefore leaves its initial ray quadratically in time whenever $\omega_c\neq0$, even though all linear cross-covariance rates vanish. This is a direct counterexample to the implication
\[\text{zero linear covariance response} \quad\Longrightarrow\quad \text{stationary state}.\]The correct conclusion is narrower: mode-exchange-symmetric second moments suppress the linear covariance channel. They do not suppress higher moments or the full non-Gaussian state dynamics.
Maxima-verified benchmark
For the unequal-scale codimension-two benchmark
\[\Omega_+=\frac54, \qquad \Omega_-=\frac{17}{20}, \qquad \omega_c=\frac34,\]the bare-confinement stability margins are
\[\Omega_+^2-\omega_c^2=1, \qquad \Omega_-^2-\omega_c^2=\frac4{25},\]and Maxima evaluates
\[\Lambda_0=0.07537829213164765\ldots.\]The corresponding finite-time propagation gives a nonzero survival loss and reproduces the analytic curvature near $t=0$. This numerical evolution is performed in a systematically enlarged exceptional-product basis. It does not rely on Gaussian closure, and no finite cutoff is claimed to be exact.
Scope of the result
The formula for $\Lambda_0$ is exact for the stated real definite-parity product input and fixed-confinement angular quench. The displayed Taylor polynomial remains a short-time expansion, not a global solution. Later-time dynamics require propagation of the full state or an independently controlled approximation.
Survival curvature and covariance answer different questions. Covariance detects selected quadratic flows; survival detects departure from the complete initial state. Their disagreement in the symmetric codimension-two example is therefore not a paradox—it is a concrete signature of information outside the linear covariance response.
Discussion