Distinguishing quantum features in classical propagation

Abstract

The strictly classical propagation of an initial Wigner function, referred to as TWA or LSC-IVR, is considered to provide approximate averages, despite not being a true Wigner function: it does not represent a positive operator. We here show that its symplectic Fourier transform, the truncated chord approximation (TCA), coincides with the full semiclassical approximation to the evolved quantum characteristic function (or chord function) in a narrow neighbourhood of the origin of the dual chord phase space. Surprisingly, this small region accounts for purely quantum features, such as blind spots and local wave function correlations, as well as the expectation of observables with a close classical correspondence. Direct numerical comparison of the TCA with exact quantum results verifies the semiclassical predictions for an initial coherent state evolving under the Kerr Hamiltonian. The resulting clear criterion for any further features, which may be estimated by classical propagation, is that, within the chord representation, they are concentrated near the origin.

Symmetry-induced many-body quantum interference in chaotic bosonic systems: an augmented truncated Wigner method

“…This particular derivation, which was already provided in various ways [31,[33][34][35], is complementary to the usual justification of the TWA based on the truncation of the time evolution equation of the system's Wigner function [28][29][30] and essentially explains why, in practice, the TWA can, for sufficiently simple one-body observables, yield reliable predictions even for long evolution times [31,35]. While using the Vleck-Gutzwiller propagator is not the unique path to derive the TWA (see in this context also [36]), it intrinsically offers a formalization in terms of trajectory pairs and thus serves here as a useful tool, well-suited to later ease the incorporation of symmetries in terms of symmetry-related partner trajectories. Section 3 is devoted to a discussion of discrete symmetries and explains the notion of symmetric and nonsymmetric trajectory families.…”

“…An approach that is particularly convenient for our purpose is based on the semiclassical van Vleck-Gutzwiller propagator [32] in combination with the diagonal approximation [38]. While the derivation of the TWA based on this approach was provided in various previous studies [33,34] (see also [36] for a related derivation via the truncated chord approximation), we nevertheless present it here in extensive detail, for pedagogical purpose and to lay out proper foundations for the subsequent sections devoted to the effect of symmetries.…”

“…The enumeration of different solutions γ is what defines the set of trajectory families γ, whereas the individual trajectories within one family relate to each other by smooth deformation with variation of the boundary values q i,f and t without crossing a point for which the stability amplitude (6) diverges. For example, two trajectories where one passes 5 We note that in alternative derivations that do not employ a semiclassical approximation at the early level of individual propagators can lead to a formulation where a pair of phase-space trajectories emerges as companions of a single classical midpoint trajectory [30,36,[39][40][41]. These should be distinguished from the two individually classical trajectories γ, γ in (7).…”

“…One can formally derive (see appendix C) a variant of the Dirac-delta identity (19) that selects only symmetric families γ, namely by using the approximate evolution q(R i , P i , t) instead of the true time evolution. Note that the explicit linear approximation (36) is not even necessary for that purpose and can be replaced by a less restrictive set of sufficient properties of q as detailed in appendix C. We find…”

Semiclassical energy transition of driven chaotic systems: phase coherence on scar disks

“…In general, the SC approximation of a driven eigenstate W l (x|τ ) does not equal its simple classical evolution, which is known as the truncated Wigner approximation [15,16], but it will be justified in the following section that it is identified with the Wigner function of the lth eigenstate of the driven Hamiltonian:…”