Semiclassical propagation of coherent states using complex and real trajectories

A semiclassical approximation to the time evolution of coherent states may be derived from a saddle point approximation to the exact quantum propagator, and in general it involves complex classical dynamics. We generalize previous one-dimensional results to $d$ dimensions, and for the case $d=2$ we present several applications. We also consider other simple approximations that depend only on real classical trajectories, but are not initial value representations. These approximations are able to reproduce interference and tunneling effects and involve propagating a few classical initial conditions compatible with the quantum uncertainties.

Figure 1

(Color online) Exact (up) and semiclassical (down) probability densities (times ${10}^2$) at $T=4$, with $\mathbf{q}=(-10,1)$ and $\mathbf{p}=(3,0)$, in the case of an attractive Gaussian potential. Except for the main peak (notice the change in scale), the wave function is accurately reproduced, and ${\mid ⟨\psi \mid {\psi }_{\mathrm{sc}}⟩\mid }^2\approx 92\%$.

Figure 2

(Color online) Probability density at $T=2.4$ for the quartic oscillator, with $\mathbf{q}=(0,0)$ and $\mathbf{p}=(2,0)$. The upper panel shows the exact calculation, the middle one is ${\mid {\psi }_{\mathbf{q}}(x,y)\mid }^2$, and the lower one is ${\mid {\psi }_{\mathrm{sc}}(x,y)\mid }^2$. Using only real trajectories the result is very poor, but it becomes excellent when complex ones are used: the overlap (48) in this case is around 95%.

Figure 3

Cut of the probability densities in Fig. 2 along the line $y=0$. The solid line is the exact result, the dashed line is ${\mid {\psi }_{\mathrm{sc}}(x,y)\mid }^2$, and the dotted line is ${\mid {\psi }_{\mathbf{q}}(x,y)\mid }^2$. Notice that the latter must be cut because of the presence of a caustic.

Figure 4

(Color online) Probability density at $T=0.5$ in the case of a circular billiard, with $\mathbf{q}=(0,0)$ and $\mathbf{p}=(4,0)$. The upper panel shows the exact calculation and the lower one is ${\mid {\psi }_{\mathbf{q}}(x,y)\mid }^2$. Using only real trajectories we have a very good result $({\mid ⟨\psi \mid {\psi }_{\mathbf{q}}⟩\mid }^2\approx 97\%)$, including effects due to curvature and interference.

Figure 5

Cut of the probability densities in Fig. 4 along the line $y=0$, displaying the exact (solid) and the semiclassical (dashed) results. The latter is obtained from the interference of a direct and a reflected trajectory.

Figure 6

Probability density at $T=2.5$ for the ridge potential, with $\mathbf{q}=(-10,0)$ and $\mathbf{p}=(4,0)$. The upper panel shows the exact calculation and the lower one is ${\mid {\psi }_{\mathbf{q}}(x,y)\mid }^2$ (times ${10}^2$). Using only real trajectories it is possible to accurately reproduce tunneling effects $({\mid ⟨\psi \mid {\psi }_{\mathbf{q}}⟩\mid }^2\approx 94\%)$.