A regular semiclassical approximation for the propagation of wave packets with complex trajectories

Abstract

The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V (x) = 1/x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.

Complex trajectories in chaotic dynamical tunneling

“…Indeed, E vanishes at g → 0 implying that the oscillations become indiscernible in the semiclassical limit. One obtains a quantity which is well behaved in this 16 This is due to the limitations of the numerical approach: one cannot obtain solutions to the Schrödinger equation at arbitrarily small g, see Appendix B. limit by averaging the reflection probability over several periods of oscillations. To be more precise, we consider the smoothed probability…”

Off-Center Coherent-State Representation and an Application to Semiclassics

“…In fact, we obtain a family of semiclassical propagators parameterized by a real number λ, of which the van Vleck formula is a particular case (λ = 1). While the original result involves real classical paths that begin at x ′ and end at x ′′ , we show that complex trajectories may appear in the semiclassical evaluation of a position-position propagator and not only in problems involving Gaussian initial states, as has been reported so far [5][6][7][8]. Yet, within the realm of semiclassical approximations, since λ is a continuous parameter, it may be used in variational and optimization processes, although we do not address these issues in the present work.…”

Communication: Systematic elimination of Stokes divergences emanating from complex phase space caustics

“…4,5 However, simply removing trajectories with σ(ξ) > 0 leads to inaccurate numerical results, due to the rapidly fluctuating phase e i σ(ξ) at the boundary σ(ξ) = 0. A number of criteria have been proposed to remove divergences based on the value of σ(ξ) or components thereof, 11,12,15,20 but these are empirical at best and do not lead to satisfactory results for long time propagation.…”

“…In continuous time dynamics, such a root search is quite challenging and has been successful so far only for relatively short times. 5,[11][12][13][14] Long time dynamics where the trajectory manifold is plagued by a multitude of caustics, remains elusive.…”