Long-lived spin entanglement induced by a spatially correlated thermal bath

We investigate how two spatially separated qubits coupled to a common heat bath can be entangled by purely dissipative dynamics. We identify a dynamical time scale associated with the lifetime of the dissipatively generated entanglement and show that it can be much longer than either the typical single-qubit decoherence time or the time scale on which a direct exchange interaction can entangle the qubits. We give an approximate analytical expression for the long-time evolution of the qubit concurrence and propose an ion trap scheme in which such dynamics should be observable.

Figure 1

(Color online) Concurrence of the initially separable state $|\uparrow \downarrow ⟩$ in $x$, $y$, or $z$ $\left(\Lambda =-1\right)$ as a function of time (scaled by ${\lambda }_1$) and $R=\text{tanh}\left(\Delta /2k_B{T}_B\right)$ calculated from the full Liouvillian, although ignoring the Lamb shift.

Figure 2

(Color online) Main: concurrence as a function of (scaled) time calculated analytically (dashed lines, valid for $t>{\gamma }_0^{-1}$) and numerically (solid lines). We consider four initial states: the Bell singlet $\left(1/\sqrt{2}\right)\left(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩\right)$ (blue, $\Lambda =-3$), the pure state $|\uparrow \downarrow ⟩$ (red, $\Lambda =-1$), the maximally mixed state $\rho =\left(1/4\right)I$ (green, $\Lambda =0$), and the Bell state $\left(1/\sqrt{2}\right)\left(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩\right)$ (black, $\Lambda =1$). Parameters: $\delta =0.05$ and $R=0.9$. Inset: behavior of the initial state $|\uparrow \downarrow ⟩$ where, in the numerical calculations, the Lamb shift and the exchange interactions have been included at an arbitrarily chosen strength $B=\xi =1/\left(2\left|{\lambda }_1\right|\right)$.