Entanglement creation between two causally disconnected objects

We study the full entanglement dynamics of two uniformly accelerated Unruh-DeWitt detectors with no direct interaction in between but each coupled to a common quantum field and moving back-to-back in the field vacuum. For two detectors initially prepared in a separable state our exact results show that quantum entanglement between the detectors can be created by the quantum field under some specific circumstances, though each detector never enters the other’s light cone in this setup. In the weak coupling limit, this entanglement creation can occur only if the initial moment is placed early enough and the proper acceleration of the detectors is not too large or too small compared to the natural frequency of the detectors. Once entanglement is created it lasts only a finite duration, and always disappears at late times. Prior result by Reznik [B. Reznik, Found. Phys. 33, 167 (2003).] derived using the time-dependent perturbation theory with extended integration domain is shown to be a limiting case of our exact solutions at some specific moment. In the strong coupling and high acceleration regime, vacuum fluctuations experienced by each detector locally always dominate over the cross correlations between the detectors, so entanglement between the detectors will never be generated.

Figure 1

Evolution of $⟨Q_A,Q_B⟩$ for the detectors initially in their ground state, with $\gamma =0.01$, $\Omega =1.3$, $a=2$, $\hslash =1$, and starting at ${\tau }_0=-60$ (left) and ${\tau }_0=0$ (right). In the left plot one can see that when $0\lesssim \tau \lesssim -{\tau }_0=60$, $⟨Q_A,Q_B⟩$ behaves according to (5), while after $\tau >60$ it behaves according to (8). In the right plot one sees that for ${\tau }_0=0$, $⟨Q_A,Q_B⟩$ behaves like (9) and its amplitude never grows.

Figure 2

Evolution of $c_{-}$ of the detectors initially in their ground states ($\alpha =\beta =\sqrt{{\Omega }_r}$). (Left) The parameters in the left plot are the same as those in Fig. 1 (left) (i.e., ${\tau }_0=-60$, etc.) to facilitate direct comparison. One can see that the three-stage profile of $⟨Q_A,Q_B⟩$ emerges in the evolution of $c_{-}$, and creates transient entanglement during $19\lesssim \tau \lesssim 80$, which is much longer than the natural period of the detectors. (Right) The right plot assumes the same conditions as the left plot except that the initial moment is ${\tau }_0=-10$. Now $c_{-}$ is always greater than 0.5 because the cross correlators have not had sufficient time to become strong enough to create quantum entanglement. This shows clearly that the entanglement creation process is intrinsically non-Markovian in nature.

Figure 3

Evolution of $c_{-}$ of the detectors initially in their ground state ($\alpha =\beta =\sqrt{{\Omega }_r}$) in the ultraweak coupling limit, with $\gamma ={10}^{-5}$, $\Omega =2.3$, $\hslash =a=1$, ${\Lambda }_0={\Lambda }_1=20$, ${\tau }_0=-2/\gamma$ (left) and ${\tau }_0=-1/4\gamma$ (right). Quantum entanglement is created ($c_{-}<\hslash /2$) after $\tau \gtrsim 0$, but disappears ($c_{-}\ge \hslash /2$) at late times.

Figure 4

Evolution of $c_{-}$ of the detectors initially in their ground states in the strong coupling limit, with $\gamma =0.1$, $\Omega =1.3$, $\hslash =a=1$, ${\Lambda }_0={\Lambda }_1=20$, and ${\tau }_0=-60$. Owing to the strong coupling the self correlators always dominate over the cross correlators, so quantum entanglement is never created.

Figure 5

Evolution of $\Sigma$ with each detector initially in a squeezed state with minimum uncertainty. The parameters are the same as those in Fig. 1 (left) except $\alpha$ and $\beta$, which are indicated in the plots. In the dark spots of the above plots, $\Sigma <0$ and the detectors are entangled. One can see that the parameters in $(\alpha ,\beta )$ space far away from the ones for the ground state (at $(\alpha ,\beta )=(\sqrt{{\Omega }_r},\sqrt{{\Omega }_r})$, in the third plot from the left in the upper row) tend to increase the value of $\Sigma$, namely, decrease the degree of entanglement.

Figure 6

Evolution of $|{\rho }_{11,00}^R|$ (dotted curve) and $|{\rho }_{10,10}^R|$ (solid curve) of the truncated RDM under the same condition as in Fig. 2 (left). The duration the detectors are entangled determined by the condition $|{\rho }_{11,00}^R|>|{\rho }_{10,10}^R|$ matches exactly with the result determined by $c_{-}<\hslash /2$.

Figure 7

Early-time evolutions of the ${\rho }_{10,10}^R$ in TDPT with the original positive frequency Wightman function, $D^{+}(z_A^{\mu }(s),z_A^{\nu }(s^{\prime }))$ in Eq. (a7) (dashed curve) with $ϵ=e^{-14-{\gamma }_e}/\Omega$ (${\gamma }_e$ is the Euler’s constant), the ${\rho }_{10,10}^R$ with the modified one, $D^{\prime +}(z_A^{\mu }(s),z_A^{\nu }(s^{\prime }))$ in Eq. (a8) (solid curve) with ${ϵ}^{\prime }=e^{-14-{\gamma }_e}/\Omega$, and the ${\rho }_{10,10}^R$ in (a10) with the original $D^{+}(z_A^{\mu }(s),z_A^{\nu }(s^{\prime }))$ and $ϵ=e^{-30}/\Omega$, together with the physical cutoffs ${ϵ}_0={ϵ}_1=e^{-14-{\gamma }_e}/\Omega \gg ϵ$ (dots). The left and the right plots are the same results but shown in different scales to stress that the early-time behavior of the dotted curve really matches with the climbing solid curve. Here $\gamma ={10}^{-5}$ and $\Omega =2.3$. The slope of the dotted curve in its linear-growing phase agrees quite well with $2\gamma /(e^{2\pi \Omega /a}-1)$ in (a9).

Figure 8

(Left) An example of $|D^{+}(z_A^{\mu }(s),z_B^{\nu }(s^{\prime }))|$ in Eq. (a11). Here $ϵ=e^{-8}$ and $a=1$. One can see that the correlation of the massless scalar field at $z_A^{\mu }(s)$ and $z_B^{\nu }(s^{\prime })$ is enhanced around $s^{\prime }=-s$. Such an enhancement will be suppressed if the trajectories of the detectors are off the $z_A^{\mu }(s)$ and $z_B^{\nu }(s^{\prime })$ in this paper or the scalar field is massive. (Right) Contour plot of the same $|D^{+}(z_A^{\mu }(s),z_B^{\nu }(s^{\prime }))|$. The long-dashed lines, solid lines, short-dashed lines, and dotted lines are borders of the integration domain at $\tau =-10$, 0, $1/a{sinh}^{-1}{ϵ}^{-1}$ ($\approx 8.69$), and 15, respectively.