Investigating quantum transport with an initial value representation of the semiclassical propagator

Abstract

Quantized systems whose underlying classical dynamics possess an elaborate mixture of regular and chaotic motion can exhibit rather subtle long-time quantum transport phenomena. In a short wavelength regime where semiclassical theories are most relevant, such transport phenomena, being quintessentially interference based, are difficult to understand with the system's specific long-time classical dynamics. Fortunately, semiclassical methods applied to wave packet propagation can provide a natural approach to understanding the connections, even though they are known to break down progressively as time increases. This is due to the fact that some long-time transport properties can be deduced from intermediate-time behavior. Thus, these methods need only retain validity and be carried out on much shorter time scales than the transport phenomena themselves in order to be valuable. The initial value representation of the semiclassical propagator of Herman and Kluk [Chem. Phys. 91, 27 (1984)] is heavily used in a number of molecular and atomic physics contexts, and is of interest here. It is known to be increasingly challenging to implement as the underlying classical chaos strengthens, and we ask whether it is possible to implement it well enough to extract the kind of intermediate-time information that reflects wave packet localization at long times. Using a system of two coupled quartic oscillators, we focus on the localizing effects of transport barriers formed by stable and unstable manifolds in the chaotic sea and show that these effects can be captured with the Herman-Kluk propagator.

Spectra of harmonium in a magnetic field using an initial value representation of the semiclassical propagator

“…For four phase space degrees of freedom typically 10 6 trajectories are needed for converged results. In cases of strongly chaotic dynamics (not considered here) and if long-time information is needed this number may, however, increase dramatically [31]. We note in passing that in contrast to the multi-trajectory Herman-Kluk method [32], the single trajectory Thawed Gaussian Wavepacket method [33] uses just a single trajectory to express the final wavefunction.…”

Interference nature of quantum breather oscillation

“…We note that there are 'many faces of tunneling' [36], one of the more prominent ones, besides barrier tunneling, being the so-called dynamical tunneling. In one special case it has been shown (see, e.g., figure 3 in [37]) that a converged Herman-Kluk calculation shows no traces of dynamical tunneling. Dynamical tunneling [10], as well as above barrier reflection [15] has been discussed in the literature as sources for the energy splitting in the breather (local mode) case.…”

An analytical approach to high harmonic generation

“…This fact has been used in [10] to explain the plateau formation in HHG. It has been shown that 10 6 trajectories are needed to converge the results [30] and that a trajectory removal procedure [25,31,32] is not appropriate for the HHG problem.…”