Phase-space gaussian ensemble quantum camouflage

TL;DR

Extends the Weyl-Wigner phase-space formulation to nonlinear, separable Hamiltonians $H^W(q,p)=K^W(p)+V^W(q)$ with the constraint $\partial^2 H^W/\partial q\partial p=0$, enabling analytic Wigner-flow descriptions. The approach uses the Wigner continuity equation $\partial_t W+\partial_q J_q+\partial_p J_p=0$ with a hierarchical $\hbar$-expansion for the currents, recovering classical Liouville flow at leading order and incorporating quantum distortions via higher-order terms. Gaussian ensembles $\mathcal{G}_\alpha$ (and squeezed variants $\mathcal{G}_\zeta$) are shown to act as quantum ground states, yielding convergent series that quantify quantum distortions around classical trajectories and enabling a quantum camouflage where stationary classical ensembles are mirrored by stationary Gaussian quantum ensembles. For a class of non-standard Hamiltonians $\tilde{\mathcal{H}}(x,k)$, the Gaussian state appears as a zero-mode ($\tilde{\mathcal{H}}\mathcal{G}_\zeta=0$), producing stationary Wigner flows that emulate classical stationary patterns and highlighting a path toward a full spectrum of exotic Hamiltonians.

Abstract

Extending the phase-space description of the Weyl-Wigner quantum mechanics to a subset of non-linear Hamiltonians in position and momentum, gaussian functions are identified as the quantum ground state. Once a Hamiltonian, $H^{W}(q,\,p)$, is constrained by the $\partial ^2 H^{W} / \partial q \partial p = 0$ condition, flow properties for generic $1$-dim systems can be analytically obtained in terms of Wigner functions and Wigner currents. For gaussian statistical ensembles, the exact phase-space profile of the quantum fluctuations over the classical trajectories are found, so to interpret them as a suitable Hilbert space state configuration for confronting quantum and classical regimes. In particular, a sort of {\em quantum camouflage} where the stationarity of classical statistical ensembles can be camouflaged by the stationarity of gaussian quantum ensembles is identified. Besides the broadness of the framework worked out in some previous examples, our results provide an encompassing picture of quantum effects on non-linear dynamical systems which can be interpreted as a first step for finding the complete spectrum of non-standard Hamiltonians.