Dressed-state master equation for two strongly coupled two-level atoms with long-lived entanglement

Abstract

We derive a dressed-state master equation in Lindblad form for two strongly coupled two-level atoms. The resulting decay dynamics are governed by Lindblad operators that couple different dressed states. We show that the eigenvalues and eigenvectors of the Liouvillian can be obtained in a compact form, since each off-diagonal element in the dressed-state basis constitutes an eigenvector. Depending on the interatomic distance and the atomic transition frequency, two distinct time scales emerge. On a short time scale, the system relaxes toward two states, one of which corresponds to a transient, maximally entangled configuration. On a longer time scale, this entangled state gradually decays to the steady state.

Figures (4)

• Figure 1: Schematic of two atoms are positioned symmetrically along the $z$-axis at $\vec{r}_1=-R\hat{ z}$ and $\vec{r}_2=+R\hat{z}$ . The dipole moments $\vec{d}_1$ and $\vec{d}_2$, can be oriented in any direction. The coupling strength between the atoms is denoted by $g$.
• Figure 2: Energy levels of the dressed states, Eq. \ref{['eq:dressedbasis']}, or eigenstates of $H_S$. We also present the relevant frequency differences $\Omega_i$, Eq. \ref{['eq:dressedOmegas']} or \ref{['eq:OmegasChi']}, and the decay rate between dressed states $\gamma^\pm_i$ given in \ref{['eq:gammas']}, $i\in\{1,2\}$. Note that no decay mechanism occurs between states $\ket2$ and $\ket1$ as both present the same number of excitations. The dissipative process connecting to $\ket1$ corresponding to $\gamma_i^-$ are depicted with dashed lines as they can be greatly suppressed leading to a transient protection of the maximally entangled state $\ket1$.
• Figure 3: Decay rates between dressed states, as shown in Fig. \ref{['fig:levels']}, for two different values of the ratio $\gamma/\Omega$ between the single atom spontaneous decay rate $\gamma$, given in Eq. \ref{['eq:gammasi']}, and the atomic frequency $\Omega$. For the left (right) panel $\gamma/\Omega=10^{-5}$ ($\gamma/\Omega=10^{-8}$). The interval between the vertical lines at $\chi_-=(39\gamma/\Omega)^{1/3}$ and $\chi_+=(39\gamma/\Omega)^{1/5}$ presents a clear separation between a fast time scale marked by $\gamma_i^+$ involving the dressed state $\ket2$, and a transient time where $\ket1$ is protected slowly decaying at rates $\gamma_i^-$. Furthermore, at least in this interval, the performed rotating wave approximation is valid as the eigenvalues of $H_S$ do not present values close to any degeneracy.
• Figure 4: Concurrence as a function of time, presented in solid curves, for the initial state $\ket{eg}$ and different values of the ratio $\Omega/\gamma$, the atomic frequency and the free atomic decay rate. In colored-dashed lines we have included the prediction of the concurrence given in Eq. \ref{['eq:conctemp']}. We have chosen an interatomic distance of $1\mu$m. This also fixes the value of the mean photon numbers for the three cases. Top: $n_1=n_2\propto 10^{-5}$. Middle: $n_1=0.127$, $n_2=0.132$. Bottom: $n_1=0.363$, $n_2=0.367$. The dashed-black curve corresponds to the approximation for zero mean photon number given in Eq. \ref{['eq:conczero']}. In the case $\Omega=10^4\gamma$, one achieves small occupation numbers and the prediction for zero temperature in Eq. \ref{['eq:conczero']} and shown with a dashed black line is an excellent prediction.