Temporal and spatial dependence of quantum entanglement from a field theory perspective

We consider the entanglement dynamics between two Unruh-DeWitt detectors at rest separated at a distance $d$. This simple model when analyzed properly in quantum field theory shows many interesting facets and helps to dispel some misunderstandings of entanglement dynamics. We find that there is spatial dependence of quantum entanglement in the stable regime due to the phase difference of vacuum fluctuations the two detectors experience, together with the interference of the mutual influences from the backreaction of one detector on the other. When two initially entangled detectors are still outside each other’s light cone, the entanglement oscillates in time with an amplitude dependent on spatial separation $d$. When the two detectors begin to have causal contact, an interference pattern of the relative degree of entanglement (compared to those at spatial infinity) develops a parametric dependence on $d$. The detectors separated at those $d$ with a stronger relative degree of entanglement enjoy longer disentanglement times. In the cases with weak coupling and large separation, the detectors always disentangle at late times. For sufficiently small $d$, the two detectors can have residual entanglement even if they initially were in a separable state, while for $d$ a little larger, there could be transient entanglement created by mutual influences. However, we see no evidence of entanglement creation outside the light cone for initially separable states.

Figure 1

The zeroth-order results, no mutual influence is included here. $\gamma ={10}^{-5}$, $\Omega =2.3$, ${\Lambda }_0={\Lambda }_1=20$, and $(\alpha ,\beta )=(1.1,4.5)$. The two plots on the left are for the zeroth-order $E_{\mathcal{N}}^{(0)}$ which is seen to decrease and disentangle in time. (The behavior of ${\Sigma }^{(0)}$ is similar to $-E_{\mathcal{N}}^{(0)}$ but the amplitude of oscillation in time is smaller.) The two plots on the right are for the relative values of $E_{\mathcal{N}}^{(0)}$ at spatial separation $d$ to the value at infinite spatial separation, as given in (35). In the upper-right plot, the brighter color corresponds to the higher value of $E_{\mathcal{N}\mathrm{rel}}$.

Figure 2

The oscillating curve represents the value of ${{\rm Y}}^{(0)}$ [defined in (45)] as a function of $d$. The bottom curve represents its lower bound [Eq. (46)]. It becomes negative when $d<616$, which signifies the violation of uncertainty relation. To rectify this, one needs to add on the mutual influences, as shown in Fig. 3. Here $\gamma ={10}^{-4}$, $\Omega =2.3$, ${\Lambda }_0={\Lambda }_1=25$.

Figure 3

Plots for $\Sigma$ (solid curve) and ${\rm Y}$ (dashed curve) at late times as a function of $d$, with parameters the same as those in Fig. 2. Two detectors are separable when $\Sigma \ge 0$ (shaded zone). One can see that $\Sigma$ becomes negative when $d<0.025$. With the mutual influences included, the uncertainty relation [see Eq. (9) and below] now holds for all $d$.

Figure 4

The plot of $\Sigma$ as a function of $d$ and $t$, up to the first-order correction. $\Sigma$ is negative in the dark region and positive in the bright region. For a fixed $d$, the disentanglement time $t_{dE}$ is at the border of the lowest dark region or the earliest time that the detectors become separable. The interference pattern in Fig. 1 for $\Sigma$ at early times signifies that the disentanglement time $t_{dE}$ is longer or shorter than those at $d\to \infty$ [Eqs. (65, 66)]. The gridded profile in the left plot shows that after $t_{dE}$ there could be some short-time revivals of entanglement. Here the parameters are the same as those in Fig. 1 except $(\alpha ,\beta )=(1.5,0.2)$ in the left plot and (1.1, 4.5) in the right (cf. Fig. 3 in Ref. 1).

Figure 5

(Upper left) The solid curve and the long-dashed curve represent the values of $\Sigma$ and ${\rm Y}$, respectively, while the dotted line is for the value of $\Sigma$ with all $⟨\cdots {⟩}_{\mathrm{v}}$ set to zero. The detectors are separable initially (the parameters here are the same as those in Figs. 2, 3 except $d=0.01$ and $(\alpha ,\beta )=(1,1)$). Quantum entanglement has been generated after $t\approx 0.15$, and $\Sigma$ oscillates in frequency ${\Omega }_{+}$ at early times. Around the time scale $t\sim 1/{\gamma }_{+}$ ($\approx 5000$ here) (upper right), $\Sigma$ has an oscillation with long period $\pi /({\Omega }_{-}-{\Omega }_{+})$ ($\approx 361.4$). $\Sigma$ appears to be settling down at a value (about $-0.046$ here) depending in this case only on the value $\alpha$ in the initial data. However, in a much longer time scale $t\sim 1/{\gamma }_{-}$ ($\approx 1.13\times {10}^8$) (lower left), one sees that the value of $|\Sigma |$ is actually decaying exponentially to the late-time value ($\approx -6.8\times {10}^{-7}$) consistent with the results in Fig. 3 with $d=0.01$ and independent of the initial data of the detectors.

Figure 6

(Upper left) The solutions to (a3) for the complex frequency $K_j=x_j+i{y}_j$ of $q_{+}^{(A)}$ [defined in (a1)] are located at the intersections of the dashed and solid curves, which represent the solutions to Eqs. (a4, a5), respectively. Here $\gamma =0.25$, $\Omega =0.9000202$, $d=1$. (Upper right) The same case, but here “$+$” denotes complex solutions and “$\times$” denotes purely imaginary solutions for $K$. There are two purely imaginary solutions for $K$ in this case. (Lower left) There are three purely imaginary solutions for $K$ when $\gamma =0.25$, $\Omega =0.8$, $d=1$. (Lower right) Solutions for $K$ when $\gamma =0.25$, $\Omega =0.3$, $d=1$. There is only one purely imaginary solution, which is located in the lower half of the complex $K$ plane.

Figure 7

The early-time evolution of $E_{\mathcal{N}}$ with the first-order mutual influence included for different initial states of the detectors with $1/8<d<15$. (Upper row) All parameters are the same as those in Fig. 1 where $(\alpha ,\beta )=(1.1,4.5)$ and the initial state of the detectors is entangled. Compared with Fig. 1, one can see that the distortion of the interference pattern due to the mutual influences is tiny. (Lower row) $(\alpha ,\beta )=(1.5,0.2)$, the detectors also initially entangled. The distortion by the mutual influences is also tiny. As indicated by Eq. (35), the complicated structure of $E_{\mathcal{N}\mathrm{rel}}$ outside the light cone is reducing to simple oscillations as time goes larger.

Figure 8

The early-time evolution of $E_{\mathcal{N}}$ for an initially separable detector pair with the first-order mutual influence included. The parameters are the same as those in Fig. 1 except $(\alpha ,\beta )=(1,1)$ and $1/15<d<15$ here. From the left plot, one finds that quantum entanglement is created at small $d$ due to the mutual influences. In the middle plot, one can see that there is no clear interference pattern in $d$ similar to those in Fig. 1 for $E_{\mathcal{N}\mathrm{rel}}$ inside the light cone. Note that here $E_{\mathcal{N}\mathrm{rel}}\equiv -{log}_{2}2c_{-}(t,d)+{log}_{2}2c_{-}(t,\infty )$ instead of $E_{\mathcal{N}}(t,d)-E_{\mathcal{N}}(t,\infty )$. The detectors with smaller separation $d$ always get a greater value of $-{log}_{2}2c_{-}$. In the right plot, while there are small oscillations outside the light cone, the smaller separation $d$ always associates the greater value of $E_{\mathcal{N}\mathrm{rel}}$. The amplitude of the small oscillation is the same order of ${\gamma }^2{\Lambda }_0$ and ${\gamma }^2{\Lambda }_1$.