09 Jun 2026
VI: Exceptional-Basis Spectral Propagation and Synchronization
The short-time commutator expansion tells us how an observable first responds to the quench. It does not, by itself, produce a finite-time trajectory. Here we use the complete exceptional-Hermite product basis to propagate the same non-Gaussian state beyond short times, without imposing a Gaussian closure.
The result is a numerical approximation to the infinite-dimensional Schrödinger evolution, not a new finite-dimensional exact model. Its reliability therefore rests on observable-level cutoff tests.
The fixed-confinement quench
We use
\[\hbar=2M=1,\]and retain the sign of the half-cyclotron frequency
\[\omega_c=\frac{qB}{2Mc}.\]After the coordinate potential has been diagonalized, the post-quench Hamiltonian is
\[H_{\mathrm{RE},B} =H_{\mathrm{sep}}^{\mathrm{RE}}-\omega_c L_z, \qquad L_z=q_+p_- -q_-p_+.\]The preparation Hamiltonian $H_{\mathrm{sep}}^{\mathrm{RE}}$ already contains the coordinate scales $\Omega_\pm$, including the target diamagnetic dressing. At $t=0$, only the angular generator $-\omega_cL_z$ is activated. This is a fixed-confinement angular-momentum quench. It is not a literal switch of the complete magnetic field, which would also alter the diamagnetic coordinate term.
The signed $\omega_c$ matters. Reversing it reverses every first-order angular-response coefficient, although even-in-time quantities such as the leading survival loss depend on $\omega_c^2$.
Expansion in the exceptional basis
Fix the even exceptional-Hermite codimensions
\[\boldsymbol m=(m_+,m_-).\]The separable one-mode eigenstates form a complete product basis, so the evolving state can be written as
\[|\Psi(t)\rangle =\sum_{r,s=0}^{\infty}c_{rs}(t) |m_+,r\rangle\otimes|m_-,s\rangle.\]Projecting the Schrödinger equation gives
\[i\dot c_{rs} =E^{\mathrm{sep}}_{\boldsymbol m,(r,s)}c_{rs} -\omega_c\sum_{u,v=0}^{\infty} \mathsf L_{rs,uv}c_{uv},\]where the angular-momentum matrix elements factor into one-dimensional integrals:
\[\begin{aligned} \mathsf L_{rs,uv} ={}&\langle r|q_+|u\rangle \langle s|p_-|v\rangle\\ &-\langle r|p_+|u\rangle \langle s|q_-|v\rangle. \end{aligned}\]The mode labels $m_+$ and $m_-$ are suppressed inside these one-mode matrix elements. For momentum elements, the derivative acts on the exceptional polynomial ratio as well as on its Gaussian envelope. Reweighting a Gaussian probability density is therefore insufficient for the momentum sector.
Both $q_\alpha$ and $p_\alpha$ reverse one-mode parity. Each term in $L_z$ consequently flips both mode parities and preserves their product. The propagation separates into fixed total-parity sectors, a useful structural check on the matrix assembly.
What the finite cutoff means
For computation we retain
\[0\le r,s\le N\]and Hermitize the projected Hamiltonian before diagonalizing it. If
\[H_N=V_ND_NV_N^\dagger,\]then the coefficient vector within the retained space is propagated as
\[\boldsymbol c_N(t) =V_Ne^{-iD_Nt}V_N^\dagger\boldsymbol c_N(0).\]This spectral exponential is unitary within the truncated space, so its norm is conserved to floating-point accuracy. That fact does not establish convergence to the infinite-basis dynamics: a truncated Hermitian matrix can conserve its norm while omitting physically important high-level amplitudes.
We therefore monitor two more demanding diagnostics:
- the maximum change in each observable curve when $N$ is increased; and
- the population on the cutoff boundary,
There is no claim that any finite $N$ closes the rational dynamics exactly. The cutoff must be reconsidered for stronger quenches, more highly excited inputs, or longer time windows.
Synchronization as a relative-fluctuation contrast
For the mass convention above, define dimensionless quadratures
\[Q_j=\sqrt{\frac{\omega_0}{2}}\,x_j, \qquad P_j=\sqrt{\frac{2}{\omega_0}}\,p_j.\]The complete-synchronization contrast is
\[S_c(t)=\langle\mathcal D_c(t)\rangle^{-1},\]with
\[\mathcal D_c =\frac12\left[ (\delta Q_x-\delta Q_y)^2 +(\delta P_x-\delta P_y)^2 \right].\]This is an equal-time relative-fluctuation measure. A large value, an extremum, or an oscillatory interval does not by itself prove phase locking or stationary synchronization.
In the benchmark below, $g=0$, so the rotated $+,-$ axes coincide with the laboratory axes up to a permutation. For $g\ne0$, one must keep track of which quadratures define the synchronization observable.
Exact local slope for the codimension-two input
For the $m=2$, $\ell=0$ exceptional-Hermite factor, let
\[\mathcal M =\sqrt{\frac{\pi}{2}}e^{1/2} \operatorname{erfc}\!\left(\frac{1}{\sqrt2}\right).\]Its physical variances are
\[V_{q,\alpha} =\frac{2(\mathcal M-1/2)}{\Omega_\alpha}, \qquad V_{p,\alpha} =\frac{\Omega_\alpha}{2} \left(\frac32+\frac{\mathcal M}{3}\right).\]At $\omega_0=1$, define the dimensionless variance sum
\[A_\alpha^{(2)} =\frac{\mathcal M-1/2}{\Omega_\alpha} +\Omega_\alpha\left(\frac32+\frac{\mathcal M}{3}\right).\]This is $V_{Q,\alpha}+V_{P,\alpha}$, not $V_{q,\alpha}+V_{p,\alpha}$.
For the real product eigenstate, both local variance rates vanish at the instant of the quench:
\[\left.\frac{dV_{Q,\alpha}}{dt}\right|_{0}=0, \qquad \left.\frac{dV_{P,\alpha}}{dt}\right|_{0}=0.\]The cross-covariance rates remain
\[\left.\frac{d}{dt}\langle q_+q_-\rangle\right|_0 =\omega_c(V_{q,-}-V_{q,+}),\] \[\left.\frac{d}{dt}\langle p_+p_-\rangle\right|_0 =\omega_c(V_{p,-}-V_{p,+}),\]while both mixed position-momentum cross rates vanish. Combining these facts gives the exact first-order expansion
\[\boxed{ S_c^{(2,2)}(t) =\frac{2}{A_+^{(2)}+A_-^{(2)}} -\frac{4\omega_c\bigl(A_+^{(2)}-A_-^{(2)}\bigr)} {\bigl(A_+^{(2)}+A_-^{(2)}\bigr)^2}\,t +O(t^2) }.\]This statement is local to $t=0$. It is exact for the stated real, definite-parity product preparation, but the displayed Taylor truncation is only a first-order short-time approximation.
If $A_+^{(2)}=A_-^{(2)}$, the linear slope vanishes. That cancellation does not freeze the state: the survival probability can still have nonzero quadratic curvature. It also does not establish stationary synchronization.
A cutoff-tested finite-time benchmark
The representative quench uses
\[g=0, \qquad m_+=m_-=2, \qquad \ell_+=\ell_-=0,\]and
\[\Omega_+=\frac54, \qquad \Omega_-=\frac{17}{20}, \qquad \omega_c=\frac34, \qquad \omega_0=1.\]The bare-confinement stability margins are positive:
\[\Omega_+^2-\omega_c^2=1, \qquad \Omega_-^2-\omega_c^2=\frac4{25}.\]One-mode $q$ and $p$ matrices were evaluated with 360-point Gauss-Hermite quadrature. We propagated $0\le t\le12$ at square cutoffs $N=30,34,38,42$. At $N=42$, the maximum one-mode overlap error was $1.71\times10^{-12}$, and the maximum propagated norm error was $5.77\times10^{-15}$.
The more informative observable-level comparison is:
| $N$ | $\max | \Delta\mathcal S | $ | $\max | \Delta S_c | $ | $\max | \Delta I | $ | $\max P_\partial$ |
|---|---|---|---|---|---|---|---|---|---|---|
| 30 | $1.69\times10^{-5}$ | $9.60\times10^{-4}$ | $3.49\times10^{-4}$ | $1.01\times10^{-6}$ | ||||||
| 34 | $1.25\times10^{-5}$ | $7.25\times10^{-4}$ | $2.39\times10^{-4}$ | $5.15\times10^{-7}$ | ||||||
| 38 | $9.83\times10^{-6}$ | $3.70\times10^{-4}$ | $1.58\times10^{-4}$ | $1.59\times10^{-7}$ | ||||||
| 42 | reference | reference | reference | $1.04\times10^{-7}$ |
Every $\Delta$ is the largest absolute difference from the $N=42$ curve over the stated time window. Thus $N=42$ is the numerical reference, not a mathematically certified infinite-cutoff answer.
Short-time and finite-time checks
The analytic synchronization values are
\[S_c(0)=0.5106376395009\ldots,\] \[\dot S_c(0)=-0.1229730434570\ldots.\]At the first numerical step, $t=10^{-3}$, the propagated values give
\[S_c(0)=0.5106376395002\ldots, \qquad \dot S_c(0)\big|_{\mathrm{one\,step}} =-0.1229306242038\ldots.\]The slope difference is consistent with the remainder of a one-sided finite difference. The analytic coefficient, together with the stability margins and the codimension-two moment formulas, was independently evaluated in Maxima 5.49.0.
At the $N=42$ reference cutoff:
- the survival probability reaches a minimum $0.996300824942$ near $t=9.188$, so the state is not stationary;
- $S_c(t)$ ranges from $0.4935354$ to $0.5232898$ over $0\le t\le12$; and
- the exact-within-cutoff rotated-mode mutual information reaches $0.0433166$ nats.
These are bounded-window dynamics. They do not establish equilibration, an asymptotic synchronized state, or uniform long-time convergence.
What this calculation establishes
The exceptional basis turns the formal coefficient equations into a reproducible finite-time method for one stable quench. It also separates four logically different claims:
- the exceptional-Hermite preparation is analytic and exact;
- the synchronization slope is an exact coefficient at $t=0$;
- truncating its Taylor series is a short-time approximation;
- spectral propagation at finite $N$ is a cutoff-tested numerical approximation.
No Gaussian closure is used. The rational force continues to generate an open moment hierarchy, while the basis calculation evolves the state amplitudes directly.
Discussion