Semiclassical treatment of quantum propagation with nonlinear classical dynamics: A third-order thawed Gaussian approximation

The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of $\hslash$ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.

Figure 1

The first recurrence of the wave packet with its initial coherent state in the absolute value cubic potential described inline. At $t=340$ the wave packet has just reflected off of the right wall and is fairly Gaussian in appearance. As it proceeds back across the potential at $t=360$ to $t=400$, the absolute value of its wave function becomes far more asymmetric and oscillatory. The cubic approximation to this evolution, expanded around the classical guiding trajectory at the center of the wave packet, appears to be able to capture this latter behavior far better than the quadratic approximation and GGWPD.

Figure 2

The first recurrence in the autocorrelation according to (a) full quantum propagation, (b) the semiclassical quadratic and (c) cubic approximations, (d) the normalization of the cubic approximation, and (e) $\epsilon$, which is defined in the text. These are for the coherent state and cubic potential described in text with time step $\mathrm{\Delta }t=8\times {10}^{-5}$. The cubic autocorrelation diverges from unit normalization immediately before the recurrence when the wave packet bounces against the near wall.

Figure 3

The $4.9\text{−}\sigma$ confidence intervals of the initial and classically propagated final states in phase space around the time of the middle of the recurrence. Along with the full time evolution, also shown is the quadratically and cubically approximated evolution of the state around the guiding trajectory marked by “x.”

Figure 4

The error $|C_{\beta }^{\text{c.s.c./h.s.c}}\left(t\right)-C_{\beta }\left(t\right)|/|C_{\beta }\left(t\right)|$ averaged over all absolute values greater than $0.01$ in the time interval $t=350$ to $t=500$ with $\mathrm{\Delta }t=1$ for different values of $\hslash$. These are plotted on a logarithmic scale with $\sigma /\hslash$ kept fixed. When $\hslash =1$, $\sigma =1/\sqrt{8}$, which are the parameters used in the evolution under the absolute value cubic potential discussed in the text.

Figure 5

(a) The cubically approximated wave function from Fig. 1 and (b) the cubically approximated dynamics of the classical density that are real from Fig. 3. As described in the text, the abrupt terminations of the purely real parts of the classical density at time steps 340 and 360 produce the two lobes seen in the wave packet.

Figure 6

(a) The $4.9\text{−}\sigma$ confidence intervals of the classical densities alongside their (b) corresponding exact and GGWPD-generated wave packets at four time points. Also shown are the real parts of the complex guiding trajectories $(p_t,q_t)$ (marked as black dots, with their area of coalescence magnified in the inset) used to produce the wave packet at every $q$ value. A pair of guiding trajectory contributions, each from one of the two branches, was included for the part of the quadratic wave packet to the left of the hairpin. The part to the right was constructed by only including the upper branch.

Figure 7

The (a) $4.9\text{−}\sigma$ confidence intervals from Fig. 6 and the final real “off-center” guiding trajectories $(p_t,q_t)$ (marked as black dots, with their area of coalescence magnified in the inset). (b) The quadratic approximation to the wave packets at every $q$ value produced from these guiding trajectories. Like in the GGWPD case, a pair of guiding trajectory contributions, each from one of the two branches, was included for the part of the quadratic wave packet to the left of the hairpin. The part to the right was constructed by including only the upper branch. The initial values of the real trajectories were chosen to be the real parts of the initial complex saddle-point trajectories used in GGWPD in Fig. 6.

Figure 8

(a) The real part of the GGWPD autocorrelation is compared to the exact first recurrence. Notice the underestimation of the exact amplitude by GGWPD in the middle of the recurrence. (b) The $4.9\text{−}\sigma$ confidence intervals are shown along with the two saddle-point trajectories drawn as black dots for the coherent state representation of the GGWPD. These saddle points appear to coalesce together around $t=380$.

Figure 9

Error, $\frac{|C_{\beta }^{\mathrm{hsc}/\csc }\left(t\right)-C_{\beta }\left(t\right)|}{|C_{\beta }\left(t\right)|}$, for guiding trajectories $(p_t,q_t)$ ending at the phase-space point indicated by the axes for the time interval shown in Fig. 2. Darker regions correspond to guiding trajectories that lead to the most accurate approximation of the overlap, while lighter regions correspond to poorer trajectories. As can be seen in this side-by-side comparison between the quadratic and cubic approximations, while the area of acceptable phase space from which the quadratic approximation's guiding center can be selected is quite consistent and appears almost Gaussian shaped itself, it is significantly more variable for the cubic case. In particular, the cubic approximation's valid guiding center area is smallest when it is least applicable immediately after a turning point and is largest when the third-order action derivatives dominate over all others.

Figure 10

The (a) absolute value and (b) real part of the autocorrelation of an initial coherent state with $\sigma =0.057$, $q_{\beta }=0$, $p_{\beta }=200$ in the Morse potential $H=\frac{p^2}{2m}+D{[1-exp(-\alpha q)]}^2$ with $m=1$, $D=50000$, $\alpha =8$ and with time step $\mathrm{\Delta }t=0.00005$. The cubic expansion is able to capture the local nonlinearity in the autocorrelation. (Inset: The collection of periodic orbits shown in phase space which additively contribute to the semiclassical wave packet [the sums over paths in Eq. (31) at $t=350$.)