Stationary entanglement in $N$-atom subradiant degenerate cascade systems

We address ultracold $N$-atom degenerate cascade systems and show that stationary subradiant states, already observed in the semiclassical regime, also exist in a fully quantum regime and for a small number of atoms. We explicitly evaluate the amount of stationary entanglement for the two-atom configuration and show full inseparability for the three-atom case. We also show that a continuous variable description of the systems is not suitable to detect entanglement due to the non-Gaussianity of subradiant states.

Figure 1

Time evolution of $\langle N_i\rangle$ for $m=0$ (red), $m=1$ (blue dashed), $m=2$ (green dotted), and $\langle N_{\text{ph}}\rangle$ (black dot-dashed) with $t$ in units of $1/g$. Left: $N=2$, $\kappa =0.2g,\epsilon =0.3$; right: $N=3$, $\kappa =0.3g,\epsilon =0.5$.

Figure 2

Time evolution of $P_0$ (blue dotted) and $P_1$ (red) with $t$ in units of $1/g$. Left: $N=2$, $\kappa =0.2g,\epsilon =0.3$; right: $N=3$, $\kappa =0.3g,\epsilon =0.5$.

Figure 3

Steady-state probability $P_1$ as a function of $\epsilon$ for different values of $\kappa$ (red dashed: $\kappa =0.002g$; green dashed: $\kappa =0.01g$; blue dashed: $\kappa =0.04g$; brown dashed: $\kappa =0.08g$) and as given by Eq. (9) (continuous black).

Figure 4

Left: steady-state probability as a function of $\epsilon$ for $N=2$ (red) and $N=3$ (blue dashed). Right: negative eigenvalue of the partially transposed state as a function of $\epsilon$ for $N=2$ (red) and $N=3$ (blue dashed). In the $N=3$ case, we have set $\kappa =0.8g$.

Figure 5

Left: $\delta (\varrho )$ for $N=50$ vs $\epsilon$ accounting for relations (11) and (12). Right: $\delta (\varrho )$ for $N=2$ (red) and $N=3$ (blue dashed); in this second case we have set $\kappa =0.8g$.