A uniform approximation for the coherent state propagator, valid in the vicinity of phase space caustics, was recently obtained using the Maslov method combined with a dual representation for coherent states. In this paper we review the derivation of this formula and apply it to two model systems: the one-dimensional quartic oscillator and a two-dimensional chaotic system.
Using phase-space complexification, an Initial Value Representation (IVR) for the semiclassical propagator in position space is obtained as a composition of inverse Segal-Bargmann (S-B) transforms of the semiclassical coherent state propagator. The result is shown to be free of caustic singularities and identical to the Herman-Kluk (H-K) propagator, found ubiquitously in physical and chemical applications. We contrast the theoretical aspects of this particular IVR with the van Vleck-Gutzwiller (vV-G) propagator and one of its IVRs, often employed in order to evade the non-linear "root-search" …
Recent experimental progress using ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methods, with propagators expressed as integrals over phase space using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that…
As a model for the semiclassical analysis of quantum-mechanical systems with both potentials and boundary conditions, we construct the WKB propagator for a linear potential sloping away from an impenetrable boundary. First, we find all classical paths from point $y$ to point $x$ in time $t$ and calculate the corresponding action and amplitude functions. A large part of space-time turns out to be classically inaccessible, and the boundary of this region is a caustic of an unusual type, where the amplitude vanishes instead of diverging. We show that this curve is the limit of caustics in the usu…
Investigating quantum transport with an initial value representation of the semiclassical propagator
量子系统的基本经典动力学具有规则和混沌运动的复杂混合，可以表现出相当微妙的长时间量子输运现象。在半经典理论最相关的短波长范围内，这种传输现象是典型的基于干涉的，很难用系统特定的长时间经典动力学来理解。幸运的是，应用于波包传播的半经典方法可以提供一种自然的方法来理解连接，即使它们已知会随着时间的增加而逐渐分解。这是因为一些长时间的输运性质可以从中时行为推导出来。因此，这些方法只需要保持有效性，并在比传输现象本身短得多的时间尺度上进行，才有价值。Herman和Kluk [M. F. Herman和E. Kluk，Chem.Phys.91，27（1984）]大量用于许多分子和原子物理学背景中，并且在此是感兴趣的。众所周知，随着潜在的经典混沌的加强，实现这一点越来越具有挑战性，我们问是否有可能很好地实现它，以提取反映长时间波包本地化的中间时间信息。使用一个系统的两个耦合的四次振子，我们专注于在混沌海中的稳定和不稳定的流形形成的运输障碍的局部化效果，并表明这些效果可以捕获的赫尔曼-克鲁克传播。
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…
This paper deals with the critical issue of approximating the pre-exponential factor in semiclassical molecular dynamics. The pre-exponential factor is important because it accounts for the quantum contribution to the semiclassical propagator of the classical Feynman path fluctuations. Pre-exponential factor approximations are necessary when chaotic or complex systems are simulated. We introduced pre-exponential factor approximations based either on analytical considerations or numerical regularization. The approximations are tested for power spectrum calculations of more and more chaotic mode…
We study the phase-space representation of dynamics of bosons in the semiclassical regime where the occupation number of the modes is large. To this end, we employ the van Vleck-Gutzwiller propagator to obtain an approximation for the Green's function of the Wigner distribution. The semiclassical analysis incorporates interference of classical paths and reduces to the truncated Wigner approximation (TWA) when the interference is ignored. Furthermore, we identify the Ehrenfest time after which the TWA fails. As a case study, we consider a single-mode quantum nonlinear oscillator, which displays…
A general semiclassical method in phase space based on the final value representation of the Wigner function is considered that bypasses caustics and the need to root-search for classical trajectories. We demonstrate its potential by applying the method to the Kerr Hamiltonian, for which the exact quantum evolution is punctuated by a sequence of intricate revival patterns. The structure of such revival patterns, lying far beyond the Ehrenfest time, is semiclassically reproduced and revealed as a consequence of constructive and destructive interferences of classical trajectories.
