Method for finding the exact effective Hamiltonian of time driven quantum systems

TL;DR

This paper introduces a general method to determine the exact effective Hamiltonian of time-driven quantum systems using Lie group structures, enabling solutions for complex systems previously only approximated or unsolved.

Contribution

The paper presents a novel, systematic approach to find exact effective Hamiltonians in time-dependent quantum systems based on Lie group decompositions.

Findings

Reproduces known solutions for optical lattice and Kapitza pendulum.

Provides the first exact solution for the Paul trap.

Contains approximate solutions consistent with previous studies.

Abstract

Time-driven quantum systems are important in many different fields of physics like cold atoms, solid state, optics, etc. Many of their properties are encoded in the time evolution operator which is calculated by using a time-ordered product of actions. The solution to this problem is equivalent to find an effective Hamiltonian. This task is usually very complex and either requires approximations, or in very particular and rare cases, a system-dependent method can be found. Here we provide a general scheme that allows to find such effective Hamiltonian. The method is based in using the structure of the associated Lie group and a decomposition of the evolution on each group generator. The time evolution is thus always transformed in a system of ordinary non-linear differential equations for a set of coefficients. In many cases this system can be solved by symbolic computational algorithms. As an example, an exact solution to three well known problems is provided. For two of them, the modulated optical lattice and Kapitza pendulum, the exact solutions, which were already known, are reproduced. For the other example, the Paul trap, no exact solutions were known. Here we find such exact solution, and as expected, contain the approximate solutions found by other authors.