Semiclassical propagation of coherent states and wave packets: Hidden saddles

Semiclassical methods are extremely important in the subjects of wave-packet and coherent-state dynamics. Unfortunately, these essentially saddle-point approximations are considered nearly impossible to carry out in detail for systems with multiple degrees of freedom due to the difficulties of solving the resulting two-point boundary-value problems. However, recent developments have extended the applicability to a broader range of systems and circumstances. The most important advances are first to generate a set of real reference trajectories using appropriately reduced dimensional spaces of initial conditions, and second to feed that set into a Newton-Raphson search scheme to locate the exposed complex saddle trajectories. The arguments for this approach were based mostly on intuition and numerical verification. In this paper, the methods are put on a firmer theoretical foundation and then extended to incorporate saddles hidden from Newton-Raphson searches initiated with real trajectories. This hidden class of saddles is relevant to tunneling-type processes, but a hidden saddle can sometimes contribute just as much as or more than an exposed one. The distinctions between hidden and exposed saddles clarifies the interpretation of what constitutes tunneling for wave packets and coherent states in the time domain.

Figure 1

Initial conditions of exposed saddles satisfying Eq. (11) for a wave packet defined by Eqs. (5) and (6), and propagated with the Hamiltonian, Eq. (13). The complex momentum of a saddle's initial conditions follows from the complex position using Eq. (8). In the upper panel, the propagation time is the period of the central trajectory with initial condition $(q_0,p_0)=(0,20)$ and in the lower panel twice this period. The density of exposed saddles increases with increasing time, but even in the limit of small times, the total number of just the exposed saddles is infinite. The only exception is for $t=0$.

Figure 2

Illustration of the limitations of LWPD. The vertically oriented ellipses are contours of the initial wave packet's Wigner transform using the parameters of Eq. (6). This density is classically propagated for one period of the central trajectory's motion ($\tau$), forming curvy foliations. The LWPD produced wave packet generates stretched and rotated ellipses. Only phase points within $1\sigma$ are still approximately linearly related to the central trajectory by this propagation time

Figure 3

Illustration of distinct classical transport pathways. The vertically oriented ellipses are contours of the initial wave packet's Wigner transform using the parameters of Eq. (6). The left panel is propagated classically for one period of the motion ($\tau$) and the right panel three periods. They generate the distorted spirals. The insets show expanded views of localized regions. In the $t=3\tau$ figure, the $5\sigma$ contour can be divided into nine foliations (enumerated in later figures), a trajectory member of which can be used to generate initial conditions for a Newton-Raphson scheme to locate unique saddles. The blackened dots show the locations of the real parts of the nine associated saddles' complex final phase-space points contributing to $\varphi (x=0,3\tau )$ (the initial conditions follow by back propagating a time $3\tau$). An analogous statement applies to the $t=1\tau$ figure on the left.

Figure 4

Contours of the final position $x$'s real part as a function of the trajectory initial conditions on the initial wave packet's Lagrangian manifold. Real and imaginary parts of ${\mathcal{p}}_0$ follow using Eq. (8) with the parameters of Eq. (6); a similar figure was shown as Fig. 3 of [10]. The black dot near $\mathrm{⑦}$ indicates the real initial condition of the Wigner density centroid. The initial conditions of exposed saddles associated with foliations $\mathrm{⑥}, \mathrm{⑦}, \mathrm{⑧}$ as a function of the propagated wave function's position $x$ appear as nearly vertical solid lines perpendicular to the contour lines of $\text{Re}x$; they are located on $\text{Im}x=0$ contours which are not shown. The transition from exposed to hidden occurs at the avoided crossing of two saddles from different foliations, which occurs near the classical turning point. The solid line turns to dashed or dotted where it transitions to hidden. The dashes are for saddles that crossed a Stokes line and must be excluded.

Figure 5

Comparison of the quantum propagation of a wave packet and the GGWPD approximation. The particular wave packet illustrated is the one defined by Eqs. (5) and (6) propagated for $3\tau$. The lower two panels show the imaginary and real parts of the classical actions, respectively. The dashed and dotted lines indicate where saddles become hidden. The (avoided) crossing marks the transition in how quickly the real and imaginary parts of the classical action are changing as a function of $x$, respectively. The dashed line indicates a saddle has crossed a Stokes line, which is indicated here by a crossing of the real parts of two classical actions, and must be excluded. The dotted lines indicate hidden saddles, which can be included, and may be physically relevant if the imaginary parts of their actions are not too large.

Figure 6

Expanded view of the left side of Fig. 5. The propagated wave function decreases quite a bit to the left of where foliations $\mathrm{⑧}$ and $\mathrm{⑨}$ switch from exposed to hidden. Just to the left of this switch, the hidden saddle $\mathrm{⑧}$'s contribution is just as significant as those of the exposed saddles $\mathrm{⑩}$ and $\mathrm{⑪}$, whereas $\mathrm{⑨}$ has crossed a Stokes line and ceases to contribute. Foliations $\mathrm{⑩}$ and $\mathrm{⑪}$ are from larger-than-$5\sigma$ contours, not plotted in Figs. 3 and 5. The semiclassical inaccuracies seen where $\mathrm{⑨}$ and $\mathrm{⑪}$ cross Stokes lines can be healed (uniformized) following Berry's prescription [34].