Constrained Adiabatic Trajectory Method (CATM): a global integrator for explicitly time-dependent Hamiltonians

TL;DR

The paper reexamines the Constrained Adiabatic Trajectory Method (CATM) as a global integrator for explicitly time-dependent Hamiltonians, highlighting its spectrum dilation and capacity to handle complex time dependencies in quantum systems.

Contribution

It provides a detailed analysis of CATM's relation to other integrators and demonstrates its effectiveness in complex, time-dependent quantum problems through numerical examples.

Findings

CATM can dilate the Hamiltonian spectrum for perturbative treatment.

CATM effectively handles complex time dependencies in quantum systems.

Numerical results on H2+ ion under laser pulse validate CATM's capabilities.

Abstract

The Constrained Adiabatic Trajectory Method (CATM) is reexamined as an integrator for the Schr\"odinger equation. An initial discussion places the CATM in the context of the different integrators used in the literature for time-independent or explicitly time-dependent Hamiltonians. The emphasis is put on adiabatic processes and within this adiabatic framework the interdependence between the CATM, the wave operator, the Floquet and the (t,t') theories is presented in detail. Two points are then more particularly analysed and illustrated by a numerical calculation describing the $H_2^+$ ion submitted to a laser pulse. The first point is the ability of the CATM to dilate the Hamiltonian spectrum and thus to make the perturbative treatment of the equations defining the wave function possible, possibly by using a Krylov subspace approach as a complement. The second point is the ability of the CATM to handle extremely complex time-dependencies, such as those which appear when interaction representations are used to integrate the system.