Dynamics of Bloch vector in thermal Jaynes-Cummings model

TL;DR

This paper studies the complex dynamics of a two-level atom's Bloch vector in the Jaynes-Cummings model with a thermal field, analyzing its time evolution, histogram distributions, and entanglement bounds.

Contribution

It introduces a method to analyze the Bloch vector dynamics in a thermal field, including time-averaging and histogram analysis, and estimates atom-field entanglement bounds.

Findings

Bloch vector exhibits oscillatory and non-closed trajectories.

Time-averaged Bloch vector smooths out rapid oscillations.

Histogram analysis reveals dependence of variance on temperature.

Abstract

In this paper, we investigate the dynamics of the Bloch vector of a single two-level atom which interacts with a single quantized electromagnetic field mode according to the Jaynes-Cummings model, where the field is initially prepared in a thermal state. The time evolution of the Bloch vector S(t) seems to be in complete disorder because of the thermal distribution of the initial state of the field. Both the norm and the direction of S(t) oscillate hard and their periods seem infinite. We observe that the trajectory of the time evolution of S(t) in the two- or three-dimensional space does not form a closed path. To remove the fast frequency oscillation from the trajectory, we take the time-average of the Bloch vector S(t). We examine the histogram of {S_{z}(n\Delta t)|n=0,1,...,N} for small \Delta t and large N. It represents an absolute value of a derivative of the inverse function of S_{z}(t). (When the inverse function of y=S_{z}(t) is a multi-valued function, the histogram represents a summation of the absolute values of its derivatives at points whose real parts are equal to y on the Riemann surface.) We examine the dependence of the variance of the histogram on the temperature of the field. We estimate the lower bound of the entanglement between the atom and the field.