Quantization and coherent states for a time-dependent Landau problem

TL;DR

This paper extends the Landau problem to include time-dependent electric fields, mass, and harmonic frequency, deriving exact solutions and constructing coherent states using invariant operators and algebraic methods.

Contribution

It introduces a method to solve the time-dependent Landau problem with exact Hamiltonian solutions and constructs related coherent states, expanding understanding of quantum symmetries and dynamics.

Findings

Exact spectrum obtained for the time-dependent Hamiltonian

Wave functions expressed via generalized Laguerre polynomials

Construction of coherent and nonlinear states based on system symmetries

Abstract

The ordinary Landau problem consists of describing a charged particle in time-independent magnetic field. In the present case the problem is generalized onto time-dependent uniform electric fields with time-dependent mass and harmonic frequency [1].The spectrum of a Hamiltonian describing this system is obtained. The configuration space wave functions of the model is expressed in terms of the generalised Laguerre polynomials. To diagonalize the time-dependent Hamiltonian we employ the Lewis-Riesenfeld method of invariants. To this end, we introduce an unitary transformation in the framework of the algebraic formalism to construct the invariant operator of the system and then to obtain the exact solution of the Hamiltonian. We recover the solutions of the ordinary Landau problem in the absence of the electric and harmonic fields, for a constant particle mass. The quantization of this system exhibits many symmetries such as $U(1), SU(2), SU(1,1)$. We therefore construct the coresponding coherent states and the associated photon added and nonlinear coherent states.