Semiclassical description of wave packet revival

Abstract

We test the ability of semiclassical theory to describe quantitatively the revival of quantum wavepackets -a long time phenomena-in the one dimensional quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are considered: time-dependent WKB and Van Vleck propagation. We show that both approaches describe with impressive accuracy the autocorrelation function and wavefunction up to times longer than the revival time. Moreover, in the Van Vleck approach, we can show analytically that the range of agreement extends to arbitrary long times.

Complexified phase spaces, initial value representations, and the accuracy of semiclassical propagation

“…The Kerr system has been already investigated in depth [24,27,69,72], but its dynamics in phase space is worthwhile revisiting due to its fascinating geometry. We start by noticing that a classical phase space distribution under the action of the Kerr flow (65) will simultaneously revolve around the origin and be deformed into a filament, since outer points move faster than inner points.…”

“…1 inspires pessimism with regard to a semiclassical approximation being able to reproduce quantum evolution, especially considering that its filamentary classical backbone gets thinner and thinner as time evolves (although its area obviously remains constant, by Liouville's theorem). This expectation was proven wrong in at least three occasions: In [72] it was was shown that a careful application of the vV-G propagator was successful in reproducing the evolved wave functions for more than one revival time; in [77] the H-K propagator was used to model a 0-dimensional Bose-Hubbard chain, for which the hamiltonian is given by the slightly different (yet dynamically identical) expression; and in [27] a value representation using final instead of initial values, proposed first in [64], was able to reproduce quantum dynamics with calculations performed directly in phase space. Since we know that these semiclassical analyses of the Kerr system were successful, we can be sure that semiclassical methods are supposed to work for this system.…”

“…It is fundamental to keep in mind that the previous analysis [72] of the Kerr system using the vV-G propagator used a series of approximations to obtain accurate results, such as filtering trajectories and approximating the action up to second order. Here, however, we are not interested in obtaining accurate results, but in providing fair comparisons between all semiclassical propagators, which are calculated in the same grids and with the same number of trajectories unless explicitly stated.…”

Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems

“…Multiple attempts to improve the TWA [19,33,34] suggest that more sophisticated methods [35][36][37][38][39][40][41][42][43] should be applied in order to describe the quantum evolution in terms of continuously distributed classical phase-space trajectories. Alternatively, different types of discrete phase-space sampling were proposed [44][45][46][47][48][49][50] in order to emulate evolution of average values using the main idea of the TWA.…”

Metaplectic sheets and caustic traversals in the Weyl representation

“…There is no need for the classical Hamiltonian to be derived from a Lagrangian, so that the path integral may just as easily be constructed for e.g. the Kerr Hamiltonian [3,4,5], H(x) = (ap 2 + bq 2 ) 2 , as for the usual form, H(x) = p 2 /2m + V (q). A classical trajectory coincides with a stationary phase of the Weyl path integrand, just as with Feynman path integrals in the position representation.…”