23 Jul 2026

Research Overview: Exceptional-Hermite Angular Quenches

An overview of a seven-part research sequence on stable magnetic oscillators, exact exceptional-Hermite state preparation, angular quenches, covariance response, survival probability, spectral dynamics, synchronization, and non-Gaussian mutual information.

dissertation quantum mechanics exceptional Hermite quantum quench non-Gaussian dynamics quantum information

This research sequence studies how an exactly prepared non-Gaussian quantum state responds when an angular-momentum coupling is switched on suddenly. The system is a two-dimensional anisotropic oscillator, motivated by a charged particle in a magnetic field, whose one-dimensional preparation potentials are rational extensions of the harmonic oscillator. Their exact eigenstates are expressed through exceptional-Hermite polynomials.

The seven posts form one connected argument. They begin with the physical Hamiltonian and its stability, construct the non-Gaussian initial state, define a controlled quench, derive exact short-time responses, propagate the state at finite time, and finally determine which correlations require the complete quantum state rather than its covariance matrix.

Central physical question

The preparation Hamiltonian is separable in two rotated coordinates, $q_+$ and $q_-$. At the quench, the coordinate confinement is kept fixed and the signed angular generator

\[L_z=q_+p_- -q_-p_+\]

is activated:

\[H_f=H_{\mathrm{sep}}^{\mathrm{RE}}-\omega_cL_z.\]

Here $\omega_c$ is signed, while $H_{\mathrm{sep}}^{\mathrm{RE}}$ contains the two exactly solvable rational oscillators. The problem is then to determine how the prepared product state departs from stationarity, develops cross-mode correlations, exhibits relative-fluctuation synchronization, and generates mutual information.

The difficulty is that exceptional-Hermite states are non-Gaussian. Their first and second moments do not determine the complete state, and the covariance equations do not generally close. The research therefore combines exact operator results with controlled finite-basis propagation instead of assuming Gaussian dynamics.

Main findings and new results

Coordinate reduction is not dynamical diagonalization

An orthogonal rotation removes the mixed coordinate term from the confining potential, but it does not remove the magnetic angular-momentum coupling. The rotated confinement scales $\Omega_\pm$ are therefore not the true phase-space normal-mode frequencies. The analysis derives the distinct dynamical frequencies and shows that strict confinement is governed by the bare-trap positivity conditions, equivalently

\[\Omega_\pm^2>\omega_c^2.\]

This separation prevents a coordinate rotation from being mistaken for a complete solution of the magnetic dynamics.

The non-Gaussian initial state is exact

A Darboux factorization produces regular rational oscillator potentials from even pseudo-Hermite seeds. The construction gives normalized exceptional-Hermite eigenfunctions, their shifted spectrum, parity, exact moments, and coordinate and momentum matrix elements. The initial two-mode state is therefore an exact product eigenstate of the preparation Hamiltonian, not a fitted wavefunction or a Gaussian approximation.

The quench changes one physical interaction

The protocol holds the coordinate potentials and their scales fixed while switching on only $-\omega_cL_z$. This distinguishes the intended angular quench from a literal change of magnetic field, which would generally alter both the signed angular term and the quadratic diamagnetic contribution. The energy variance provides an exact state-specific test of whether the prepared state must evolve after the switch.

The initial covariance response is exact without Gaussian closure

For a real, definite-parity product state, the complete first-order cross-covariance response follows directly from the Heisenberg equations:

\[\left.\dot{\sigma}_{q_+q_-}\right|_0 =\omega_c\left(V_{q,-}-V_{q,+}\right),\] \[\left.\dot{\sigma}_{p_+p_-}\right|_0 =\omega_c\left(V_{p,-}-V_{p,+}\right),\]

while the two mixed position-momentum cross rates vanish initially. These relations require no Gaussian approximation. They show that the signed linear response is controlled by the mismatch of the corresponding local variances.

Vanishing covariance slopes do not imply stationarity

When the two factors have equal local variances, every first-order cross covariance rate can vanish. The state can nevertheless be nonstationary. The survival probability detects this through the exact short-time law

\[\mathcal S(t) =1-\omega_c^2 \left( V_{q,+}V_{p,-}+V_{q,-}V_{p,+}-\frac12 \right)t^2+O(t^4).\]

This supplies a concrete counterexample to the idea that an unchanged covariance matrix proves that the complete state is stationary.

Finite-time dynamics can be obtained without closing the moments

The exact exceptional-product basis converts the Schrödinger equation into an infinite system for the expansion coefficients. Parity separates the dynamics into invariant sectors, and finite matrices are propagated spectrally. Increasing quadrature orders and basis cutoffs, together with boundary-population and observable-level comparisons, distinguishes a converged finite-time result from a merely norm-preserving truncation.

This calculation produces survival, covariance, synchronization, and entanglement-related observables from the propagated amplitudes themselves. No assumed Gaussian covariance model replaces the state.

Exact mutual information and a covariance bound are different

For a propagated pure state with coefficient matrix $C(t)$, the reduced density operator is

\[\rho_A(t)=C(t)C^\dagger(t).\]

Its eigenvalues determine the exact mutual information within the converged finite basis. The local covariance instead determines a Gaussian maximum-entropy upper bound. For a globally pure state, their difference is

\[2h(\nu_A)-I(A:B) =2\delta_{\mathrm{NG}}(\rho_A).\]

Thus the excess of the covariance bound is exactly twice the relative-entropy non-Gaussianity of the reduced state. In particular, the covariance bound can be positive even when the exact mutual information is zero for an initial product state.

Methodology

The research uses four complementary levels of analysis.

First-principles analytical construction

The Hamiltonian is derived from minimal coupling in the symmetric gauge. Canonical coordinate and momentum rotations are performed together, the stability conditions are obtained from positivity of the bare trap, and the true dynamical frequencies follow from the coupled Hamilton or Heisenberg equations.

The exceptional-Hermite preparation is derived from the one-dimensional Schrödinger equation using Darboux factorization. This fixes the potential, energy shift, normalization, parity, and low moments before any time evolution is attempted.

Operator and short-time methods

Commutators with the post-quench Hamiltonian generate the exact Heisenberg equations. Product structure, parity, reality, and local stationarity are then used to derive the first covariance rates. A Taylor expansion of the survival amplitude relates its curvature to the post-quench energy variance. These calculations remain exact even though the full non-Gaussian moment hierarchy is open.

Symbolic verification with Maxima

Compact Maxima calculations verify the coordinate reduction, exceptional Schrödinger residuals, Gaussian control cases, covariance coefficients, survival curvature, stability margins, and entropy identities. The symbolic checks accompany the derivations and return exact zero residuals or the stated numerical values.

Controlled numerical propagation

One-mode matrix elements are evaluated with Gauss-Hermite quadrature and assembled into the exceptional-product representation of $H_f$. The projected Hermitian Hamiltonian is diagonalized, and its spectral phases propagate the coefficient vector. Reliability is assessed through orthogonality and Hermiticity errors, parity leakage, boundary population, norm conservation, analytic short-time targets, and convergence of each reported observable as the cutoff increases.

The calculation carefully separates three kinds of statement:

• exact identities in the infinite-dimensional theory;
• exact evolution inside a chosen projected matrix; and
• numerical evidence that a finite-window observable has converged toward its infinite-dimensional value.

Overall contribution

The sequence provides a complete route from a physical Hamiltonian to non-Gaussian information dynamics. Its central conceptual result is that covariance can give exact local response coefficients and useful bounds, but it cannot generally determine the state, close the dynamics, or supply the exact mutual information. Survival probability and the reduced density spectrum recover information that second moments omit.

The work also supplies a reproducible research framework: exact exceptional-Hermite preparation, a precisely defined fixed-confinement quench, Maxima-verified analytical benchmarks, and observable-level convergence tests for the finite-time calculation.

The seven research posts

I: Magnetic oscillator stability and rotated-coordinate reduction

Derives the charged anisotropic oscillator from minimal coupling, rotates the coordinate quadratic form, distinguishes the rotated scales from the true dynamical frequencies, and establishes the strict confinement conditions.

II: Exact exceptional-Hermite preparation before the angular quench

Builds regular rational oscillator potentials and normalized exceptional-Hermite eigenstates from Darboux factorization, including their spectrum, parity, exact moments, and operator matrix elements.

III: A fixed-confinement angular-momentum quench

Defines the sudden quench that activates only the signed angular-momentum coupling, proves how it couples the preparation coordinates, and derives an exact energy-variance test for post-quench stationarity.

IV: Exact first-order covariance response without Gaussian closure

Derives the initial means and complete covariance response from the Heisenberg equations, identifies the open non-Gaussian moment hierarchy, and obtains the exact short-time relative-noise synchronization response.

V: Survival curvature beyond the linear covariance response

Uses energy variance to derive the survival-probability curvature and shows why vanishing first-order covariance rates do not prove that the quantum state is stationary.

VI: Exceptional-basis spectral propagation and synchronization

Constructs parity-resolved finite-basis propagation, explains the physical meaning of the cutoff, verifies analytic short-time targets, and studies finite-time survival, covariance, synchronization, and convergence.

VII: Non-Gaussian mutual information: exact entropy and covariance bounds

Computes mutual information from the reduced density spectrum, distinguishes it from globally matched Gaussian information and local covariance bounds, and identifies the bound gap with reduced-state non-Gaussianity.