Geometric phase for nonlinear coherent and squeezed state

TL;DR

This paper formulates and analyzes the geometric phases for nonlinear coherent and squeezed states, revealing how non-linear functions influence phase evolution and potential applications in quantum control.

Contribution

It introduces the non-unitary, non-cyclic geometric phases for nonlinear coherent and squeezed states, highlighting their properties and effects of non-linear functions on phase dynamics.

Findings

Non-linear functions significantly affect geometric phase evolution.

The geometric phase approaches a limit as the squeezing parameter r -> infinity.

Cyclic geometric phases are derived under specific periodic conditions.

Abstract

The geometric phases for standard coherent states which are widely used in quantum optics have attracted a large amount of attention. Nevertheless, few physicists consider about the counterparts of non-linear coherent states, which are useful in the description of the motion of a trapped ion. In this paper, the non-unitary and non-cyclic geometric phases for two nonlinear coherent and one squeezed states are formulated respectively. Moreover, some of their common properties are discussed respectively, such as gauge invariance, non-locality and non-linear effects. The non-linear functions have dramatic impacts on the evolution of the corresponding geometric phases. They speed the evolution up or down. So this property may have application in controlling or measuring geometric phase. For the squeezed case, when the squeezed parameter r -> \infinity, the limiting value of the geometric phase is also determined by non-linear function at a given time and angular velocity. In addition, the geometric phases for standard coherent and squeezed states are obtained under a particular condition. When the time evolution undergoes a period, their corresponding cyclic geometric phases are achieved as well. And the distinction between the geometric phases of the two coherent states maybe regarded as a geometric criterion.