Semiclassical coherent-states propagator

In this work, we derived a semiclassical approximation for the matrix elements of a quantum propagator in coherent states (CS) basis that avoids complex trajectories; it only involves real ones. For that purpose, we used the symplectically invariant semiclassical Weyl propagator obtained by performing a stationary phase approximation (SPA) for the path integral in the Weyl representation. After that, for the transformation to CS representation SPA is avoided; instead a quadratic expansion of the complex exponent is used. This procedure also allows us to express the semiclassical CS propagator uniquely in terms of the classical evolution of the initial point without the need of any root search typical of van Vleck–Gutzwiller-based propagators. For the case of chaotic Hamiltonian systems, the explicit time dependence of the CS propagator has been obtained. The comparison with a realistic chaotic system that derives from a quadratic Hamiltonian, the cat map, reveals that the expression here derived is exact up to quadratic Hamiltonian systems.

Figure 1

The points $X_2$ and $X_1$ have their midpoint in $X$ and ${\xi }_0$ denotes the chord that joins them. The orbit $\gamma$, has also its center point in $X$ while its chord ${\xi }_{\gamma }^t$ joins the points $x_i$ and $x_f$. The orbit $\gamma$, starts in $x_i$ that is shifted from $X_2$ by the point shift ${\delta }_{\gamma }^t$. Meanwhile, orbit ${\gamma }_2$ starts in $X_2$ and evolves after a time $t$ up to $x_{2f}$, the drift from $X_1$ is denoted by $d^t$. For this last orbit ${\gamma }_2$, the chord is denoted by ${\xi }_{{\gamma }_2}^t$, while the midpoint is $X_{{\gamma }_2}$. For simplicity, in the figure we have dropped the time superscripts $t$.

Figure 2

Relative error of the amplitude of the semiclassical expressions $\langle X_1|{\widehat{U}}^t{|X_2\rangle }_{\mathrm{SC}1}$ and $\langle X_1|{\widehat{U}}^t{|X_2\rangle }_{\mathrm{SC}2}$ as a function of $N$. In full line the error of $\langle X_1|{\widehat{U}}^t{|X_2\rangle }_{\mathrm{SC}1}$ and in dotted line the error of $\langle X_1|{\widehat{U}}^t{|X_2\rangle }_{\mathrm{SC}2}$.