Fast Converging Path Integrals for Time-Dependent Potentials I: Recursive Calculation of Short-Time Expansion of the Propagator

TL;DR

This paper introduces a recursive method to efficiently compute the short-time expansion of the quantum propagator in systems with time-dependent potentials, enabling faster simulations and analytical studies.

Contribution

It develops a novel recursive approach for calculating high-order short-time expansions of the propagator in time-dependent quantum systems.

Findings

Recursion relations for the effective potential are derived and solved analytically.

Numerical verification confirms the accuracy of the recursive method.

The approach accelerates Monte Carlo simulations and analytical approximations.

Abstract

In this and subsequent paper arXiv:1011.5185 we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not yet been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte Carlo simulations for path integrals, or to set up various analytic approximation techniques to study properties of quantum systems in time-dependent potentials. The analytically derived results are numerically verified by treating several simple models.