A novel derivation of quantum propagator useful for time-dependent trapping and control

TL;DR

This paper introduces a new derivation method for quantum propagators of quadratic systems using Heisenberg equations, enabling explicit calculations of energy, heat, and state probabilities for trapped particles.

Contribution

It presents a novel Heisenberg-based derivation of quantum propagators applicable to time-dependent trapping potentials and control scenarios.

Findings

Explicit formulas for energy, work, and heat in quantum traps

Derivation of the density matrix for harmonic oscillators and Paul traps

Method generalizable to multiple interacting harmonic oscillators

Abstract

A novel derivation of quantum propagator of a system described by a general quadratic Lagrangian is presented in the framework of Heisenberg equations of motion. The general corresponding density matrix is obtained for a derived quantum harmonic oscillator and a particle confined in a one dimensional Paul trap. Total mean energy, work and absorbed heat, Wigner function and excitation probabilities are found explicitly. The method presented here is based on the Heisenberg representation of position and momentum operators and can be generalized to a system consisting of a set of linearly interacting harmonic oscillators straightforwardly.