Semiclassical evolution in phase space for a softly chaotic system

Abstract

An initial coherent state is propagated exactly by a kicked quantum Hamiltonian and its associated classical stroboscopic map. The classical trajectories within the initial state are regular for low kicking strengths, then bifurcate and become mainly chaotic as the kicking parameter is increased. Time-evolution is tracked using classical, quantum and semiclassical Wigner functions, obtained via the Herman-Kluk propagator. Quantitative comparisons are also included and carried out from probability marginals and autocorrelation functions. Sub-Planckian classical structure such as small stability islands and thin/folded classical filaments do impact semiclassical accuracy, but the approximation is seen to be accurate for multiple Ehrenfest times.

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

“…Since in this case they both have diverging pre-factors, this might be hopeless. Despite these problems, the H-K has been found to perform unexpectedly well for situations of soft chaos, in which phase space is populated by both regular and chaotic dynamics [12,14,66]. The reason for this might be that the main contributions come from the regular trajectories, since, as states earlier, for chaotic trajectories the pre-factors diverge very fast.…”

“…The only difference is that, whilst the angular velocity is the same for all orbits in the SHO, in the Kerr system it is conserved per orbit, but monotonically increasing as a function of the distance from the origin. Using (15), it is also easy to show that the classical action for the Kerr system as obtained from the flow above is given by S Kerr (q (q, p), q, t) = 1 4 ω(q, p) t + 2 p q(cos [ω(q, p) t] − 1) + (p 2 − q 2 ) sin [ω(q, p) t] , (66) while the symmetric action in ( 27) is just…”

Quantum-Chaotic Evolution Reproduced from Effective Integrable Trajectories

“…There is strong evidence, however, that quantum mechanics can be accurately reproduced by employing classical trajectories even when they are chaotic, despite the " -area rule" [3,8,9]. We here shift direction by investigating the extent to which the trajectories of a specifically tailored integrable system supply a semiclassical approximation for the exact quantum evolution corresponding to a chaotic system -and for how long.…”

“…We apply our methods to the propagation of an initial coherent state under the dynamics of the recently introduced "coserf map" [9], which is exactly quantizable. The short, long and very long time-regimes are examined for a kicking strength that renders the system strongly chaotic.…”

“…The effective integrable system is devised using the Baker-Hausdorff-Campbell formula and its trajectories are obtained using a recently proposed numerical algorithm able to deal with Hamiltonians that are not sums of kinetic and potential terms [12]. The semiclassical approximations are calculated using the Herman-Kluk propagator, which is very accurate and easily modified to deal with discrete systems [9,13,14].…”