Entanglement trapping in a nonstationary structured reservoir

We study a single two-level atom interacting with a reservoir of modes defined by a reservoir structure function with a frequency gap. Using the pseudomodes technique, we derive the main features of a trapping state formed in the weak coupling regime. Utilizing different entanglement measures we show that strong correlations and entanglement between the atom and the modes are in existence when this state is formed. Furthermore, an unexpected feature for the reservoir is revealed. In the long time limit and for weak coupling the reservoir spectrum is not constant in time.

Figure 1

Diagrammatic representation of the atom-pseudomodes system. The band-gap reservoir is represented by two interacting pseudomodes PM${}_1$ and PM${}_2$. The two pseudomodes are also coupled to two independent Markovian reservoirs that induce the decay of the pseudomodes at rates ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$, respectively. The atom couples only to one of the two pseudomodes. For a perfect gap, i.e., $D\left({\omega }_c\right)=0$, the decay rate for the first pseudomode ${\Gamma }_1^{\prime }$ is zero.

Figure 2

(a) The populations $|c_a{\left(t\right)|}^2$ (black), $|a_1{\left(t\right)|}^2$ (red short-dashed), $|a_2{\left(t\right)|}^2$ (blue dashed), and ${\Pi }_j\left(t\right)$ (green long-dashed), for ${\Gamma }_1=10{\Omega }_0$, ${\Gamma }_2=0.2{\Omega }_0$, and $W_1=50W_2$. (b) The final populations $|c_a{\left(\infty \right)|}^2$ (black), $|a_1{\left(\infty \right)|}^2$ (red short-dashed), and ${\Pi }_j\left(\infty \right)$ (green-dashed) as functions of the dimensionless parameter $\eta =2{\Omega }_0/\sqrt{{\Gamma }_1{\Gamma }_2}$.

Figure 3

The reservoir spectrum $S({\omega }_{\lambda },t)$ as a function of time, (a) for ${\Gamma }_1=10{\Omega }_0$, ${\Gamma }_2=0.2{\Omega }_0$, and $W_1=50W_2$, and (b) for ${\Gamma }_1=0.5{\Omega }_0$, ${\Gamma }_2=0.01{\Omega }_0$, and $W_1=50W_2$. Panels (c) and (d) are snapshots for the reservoir spectrum for ${\Omega }_{0}t=50$ and for the parameters of panels (a) and (b), respectively. The red dashed line in panels (c) and (d) is the spectrum for a reservoir with a Lorentzian structure function with ${\Gamma }_2=W_2=0$, $W_1=1$ and ${\Gamma }_1=10{\Omega }_0$ and ${\Gamma }_1=0.5{\Omega }_0$, respectively.

Figure 4

(a) The probability current $J_{\lambda ,a}\left(t\right)$ [Eq. (14)], for ${\Gamma }_1=10{\Omega }_0$, ${\Gamma }_2=0.2{\Omega }_0$, and $W_1=50W_2$, and (b) the total probability current $Q\left(t\right)$ [Eq. (15)] for the same parameters.

Figure 5

The concurrences $C_{a,1}\left(t\right)$ (black solid), $C_{a,2}\left(t\right)$ (red short-dashed), and $C_{a,\left(12\right)}\left(t\right)$ (blue dashed), for (a) ${\Gamma }_1=10{\Omega }_0$, ${\Gamma }_2=0.2{\Omega }_0$, and for (b) ${\Gamma }_1=0.5{\Omega }_0$, ${\Gamma }_2=0.01{\Omega }_0$. For both panels $W_1=50W_2$.

Figure 6

(a) The atom-modes density of entanglement ${\mathcal{E}}_A({\omega }_{\lambda },t)$ as a function of time and the mode frequency ${\omega }_{\lambda }$, and (b) the density of entanglement ${\mathcal{E}}_R({\omega }_{\lambda },{\omega }_{\mu },t)$ between a mode ${\omega }_{\mu }={\omega }_c+0.1{\Omega }_0$ and the rest of the reservoir modes as a function of time. Panels (c) and (d) are snapshots for the mode-mode density of entanglement ${\mathcal{E}}_R({\omega }_{\lambda },{\omega }_{\mu },t)$ for ${\Omega }_{0}t=10$ and ${\Omega }_{0}t=30$, respectively. For all panels ${\Gamma }_1=10{\Omega }_0$, ${\Gamma }_2=0.2{\Omega }_0$, and $W_1=50W_2$.

Figure 7

The concurrence $C_A^2\left(t\right)$ (black solid), $C_R^2\left(t\right)$ (red short-dashed) and the total concurrence $C^2\left(t\right)$ (blue dashed), for ${\Gamma }_1=10{\Omega }_0$, ${\Gamma }_2=0.2{\Omega }_0$, and $W_1=50W_2$.