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arXiv:quant-ph/0009114 · Brazilian Journal of Physics

New method for obtaining complex roots in the semiclassical coherent-state propagator formula

Ademir Luix Xavier Jr.

Published Sep 1, 2001• 1 citation
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Abstract

A semiclassical formula for the coherent-state propagator requires the determination of specific classical paths inhabiting a complex phase-space and governed by a Hamiltonian flux. Such trajectories are constrained to special boundary conditions which render their determination difficult by common methods. In this paper we present a new method based on Runge-Kutta integrator for a quick, easy and accurate determination of these trajectories. Using nonlinear one dimensional systems we show that the semiclassical formula is highly accurate as compared to its exact counterpart. Further, we clarify how the phase of the semiclassical approximation is correctly retrieved during the time evolution.

DOI
10.1590/s0103-97332001000300018

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A semiclassical formula for the coherent-state propagator requires the determination of specific classical paths inhabiting a complex phase-space and governed by a Hamiltonian flux. Such trajectories are constrained to special boundary conditions which render their determination difficult by common methods. In this paper we present a new method based on Runge-Kutta integrator for a quick, easy and accurate determination of these trajectories. Using nonlinear one dimensional systems we show that the semiclassical formula is highly accurate as compared to its exact counterpart. Further, we clarify how the phase of the semiclassical approximation is correctly retrieved during the time evolution.
A semiclassical formula for the coherent-state propagator requires the determination of specific classical paths inhabiting a complex phase-space and governed by a Hamiltonian flux. Such trajectories are constrained to special boundary conditions which render their determination difficult by common methods. In this paper we present a new method based on Runge-Kutta integrator for a quick, easy and accurate determination of these trajectories. Using nonlinear one dimensional systems we show that the semiclassical formula is highly accurate as compared to its exact counterpart. Further, we clarify how the phase of the semiclassical approximation is correctly retrieved during the time evolution.
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.
We study the application of the semiclassical coherent-state path-integral technique to the problem of scattering. A numerical method based on initial-value trajectories and their stability information is proposed to solve the root search for the complex trajectories that determine the semiclassical propagator. Very good agreement between the numerically exact quantum and semiclassical results for the correlation function can be achieved for scattering from one-dimensional barrier as well as well-type potentials by using just one single complex trajectory at each time.
The semiclassical approximation to the coherent state propagator requires complex classical trajectories in order to satisfy the associated boundary conditions, but finding these trajectories in practice is a difficult task that may compromise the applicability of the approximation. In this work several approximations to the coherent state propagator are derived that make use only of real trajectories, which are easier to handle and have a more direct physical interpretation. It is verified in a particular example that these real trajectories approximations may have excellent accuracy.
The semiclassical limit of the coherent state propagator $⟨{\mathbf{z}}^{\ensuremath{''}}\ensuremath{\mid}{e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}T∕\ensuremath{\hbar}}\ensuremath{\mid}{\mathbf{z}}^{\ensuremath{'}}⟩$ involves complex classical trajectories of the Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}(\mathbf{u},\mathbf{v})=⟨\mathbf{v}\ensuremath{\mid}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\mid}\mathbf{u}⟩$ satisfying $\mathbf{u}(0)={\mathbf{z}}^{\ensuremath{'}}$ and $\mathbf{v}(T)={\mathbf{z}}^{\ensuremath{''}}*$. In this work we study mostly the case ${\mathbf{z}}^{\ensuremath{'}}={\mathbf{z}}^{\ensuremath{''}}$. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as $\ensuremath{\hbar}$ goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.