A Gaussian density matrix under decoherence and friction

TL;DR

This paper analyzes the time evolution of a Gaussian density matrix for a particle in a harmonic potential under decoherence and friction, showing convergence to a thermal state and exploring parameter dependencies.

Contribution

It provides an analytical and numerical study of the asymptotic behavior of a Gaussian density matrix with decoherence and friction, including fixed point solutions.

Findings

Density matrix converges to a thermal state with (5) temperature.

Analytical expression for the asymptotic density matrix.

Dependence of the fixed point on oscillator frequency, friction, and decoherence strength.

Abstract

The time evolution of a Gaussian density matrix of a one dimensional particle, generated by a quadratic, ${\cal O}(\partial_t^2)$ effective Lagrangian, describing a harmonic potential, a friction force and decoherence, is studied within the Closed Time Path formalism. The density matrix converges to an asymptotic form, given by a completely decohered thermal state with an ${\cal O}(\hbar)$ temperature in the translation invariant case. The time evolution of the state of a harmonic oscillator is followed numerically. The asymptotic density matrix, the fixed point of the master equation, is found analytically and its dependence on the oscillator frequency, the friction constant and the decoherence strength is explored.