Evolution of squeezed states under the Fock-Darwin Hamiltonian

TL;DR

This paper provides a comprehensive analytical framework for understanding how squeezed states of a charged particle evolve over time under the Fock-Darwin Hamiltonian, including effects of a time-dependent electric field, with applications to cold-ion experiments.

Contribution

It generalizes previous relations for harmonic oscillator states to the Fock-Darwin system, linking squeezed and unsqueezed state evolution and their Wigner functions.

Findings

Derived analytical expressions for state evolution under Fock-Darwin Hamiltonian.

Established relations between squeezed and unsqueezed state dynamics.

Computed response functions to external electric fields.

Abstract

We develop a complete analytical description of the time evolution of squeezed states of a charged particle under the Fock-Darwin Hamiltonian and a time-dependent electric field. This result generalises a relation obtained by Infeld and Pleba\'nski for states of the one-dimensional harmonic oscillator. We relate the evolution of a state-vector subjected to squeezing to that of state which is not subjected to squeezing and for which the time-evolution under the simple harmonic oscillator dynamics is known (e.g. an eigenstate of the Hamiltonian). A corresponding relation is also established for the Wigner functions of the states, in view of their utility in the analysis of cold-ion experiments. In an appendix, we compute the response functions of the FD Hamiltonian to an external electric field, using the same techniques as in the main text.