Dynamics of the Bloch vector in the thermal Jaynes-Cummings model

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 $\mathbit{S}(t)$ 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 $\mathbit{S}(t)$ oscillate hard and their periods seem infinite. We observe that the trajectory of the time evolution of $\mathbit{S}(t)$ 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 $\mathbit{S}(t)$. We examine the histogram of ${\phantom{|}{S}_{z}(n\ensuremath{\Delta}t)|n=0,1,\dots{},N}$ for small $\ensuremath{\Delta}t$ 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 multivalued 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.