Time-dependent conformal transformations and the propagator for quadratic systems

TL;DR

This paper develops a method using conformal transformations to simplify quadratic quantum systems with time-dependent parameters, enabling explicit calculation of propagators and extension of phase corrections, exemplified by the Mathieu profile.

Contribution

It introduces a generalized conformal transformation approach to solve and analyze time-dependent quadratic quantum systems, extending previous methods and including phase correction generalizations.

Findings

Explicit quantum propagator for time-dependent quadratic systems.

Extension of Maslov phase correction to arbitrary time-dependent frequencies.

Application to Mathieu profile demonstrates method effectiveness.

Abstract

The method proposed by Inomata and his collaborators allows us to transform a damped Caldiroli-Kanai oscillator with time-dependent frequency to one with constant frequency and no friction by redefining the time variable, obtained by solving a Ermakov-Milne-Pinney equation. Their mapping ``Eisenhart-Duval'' lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.