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

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.

Figure 1

$\left(q_1,p_1\right)$-Poincare sections illustrating the chaotic domain for the coupling coefficients $\lambda =0.0$, $\lambda =-0.35$, and $\lambda =-0.6$ from top to bottom, respectively.

Figure 2

(a) $\left(q_2,p_2\right)$ and (b) $\left(q_1,p_1\right)$-Poincare sections for $\lambda =-0.35$ showing the different subregions in the chaotic sea and the regular islands; the large regular island inside region $5^{\prime }$ is denoted as island $\mathcal{A}$; modified from [7] and reprinted with permission.

Figure 3

Classical (solid horizontal line), semiclassical (long dashes), and quantum mechanical (dotted line) cross-correlation functions for the regular island with a probe Gaussian in the chaotic region for different numbers of trajectories (tr.) in the HK calculations: $2\cdot {10}^6\text{tr.}$ in (a), $64\cdot {10}^6\text{tr.}$ in (b). The classical calculations were performed with ${10}^4$ trajectories. The quantum result was obtained with the split-operator fast Fourier transform technique [77].

Figure 4

Quantum mechanical transition probabilities in the zero-barrier case (a) and for one barrier (b) at long propagation times. For the zero-barrier case, both wave packets are in region $5^{\prime }$ and for the one-barrier case, the final wave packet is placed in region 6 (the regions are shown in Fig. 2). The initial wave packet width is ${\gamma }_{5^{\prime }}=3.3$.

Figure 5

Locally time-smoothed quantum and classical transition probabilities for zero $\left(j=5^{\prime },k=5^{\prime }\right)$ and one barrier $\left(j=5^{\prime },k=6\right)$. The quantum zero-barrier case is represented by the solid line, the classical zero-barrier case by the dashed line, the quantum one-barrier case by dash-dots, and finally the classical one-barrier case by the dotted line. The smoothing parameter $\zeta =0.35$ in (a) and 0.8 in (b) and the results shown are obtained with ${10}^6$ trajectories. Note that the two classical cases are very close to each other at longer times whereas the two quantum cases are not. Also note the logarithmic scale in (a).

Figure 6

Locally time-smoothed quantum and classical transition probabilities for one $\left(j=5^{\prime },k=6\right)$ and three barriers $\left(j=5^{\prime },k=2\right)$. The quantum one-barrier case is represented by the solid line, the classical one-barrier case by the dashed line, the quantum three-barrier case by dash-dots, and finally the classical three-barrier case by the dotted line. The smoothing parameters and numbers of trajectories are identical to Fig. 5. Here the two classical cases take much longer to approach each other, but eventually do. On the other hand, the ratio of the values of the two quantum cases are far from unity indicating the extra localization due to three barriers rather than one.

Figure 7

Comparison of the semiclassically and quantum mechanically calculated transition probability. Shown is the one-barrier case in the previous two figures: quantum result (solid line) versus semiclassical result with ${10}^6$ trajectories (dashed line) and semiclassical result with ${10}^7$ trajectories (dotted line). The convergence for longer times with greater numbers of trajectories is clear.

Figure 8

Comparison of the semiclassical and quantum mechanically calculated transition probability for the three-barrier case: quantum result (solid line) versus the semiclassical result with ${10}^6$ trajectories (dashed line) and the semiclassical result with ${10}^7$ trajectories (dotted line).