Testing the (t, t') method for wave packet evolution in the presence of time varying fields: Gaussian well dynamics

Abstract

Solving the Time-dependent Schrodinger equation (TDSE) beyond the perturbative regime in the presence of time-dependent potentials is not an easy task. Several methods are developed over time that is computationally efficient and provide the best possible solution. The floquet theorem and the (t,t’) formalism are employed with an objective to replace the time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The (t,t’) method uses an expanded Hilbert space to convert the time-dependent Hamiltonian into an effective time-averaged Hamiltonian. However, the floquet method for solving the TDSE involves very heavy diagonalization of the floquet matrix at each step. The evolution operator requires a vast storage space with a large operation count for the propagation. To get rid of these, we will use the Split operator method and the Zassenhauss formula. By doing so, we can separate the number matrix from the full floquet matrix and analytically block diagonalize the floquet matrix. By using the Zassenhauss formula, an analytical expression for the matrix transformation can be derived. Hence the only storage to be taken care of is the matrix of the order of the number of points in the spatial grid. Taking Xenon model potential as a test case, we are checking the stabilization in a high-intensity, high-frequency laser field.