Harvesting correlations from vacuum quantum fields in the presence of a reflecting boundary

When the detectors system is very close the boundary (i.e., $\Delta{z}/\sigma\ll 1$), the transition probabilities can be approximated as

and the correlation term $|X|$ as

The concurrence, denoted here by $\mathcal{C}_{P}(\rho_{AB})$ can be approximately expressed as

Eq. (20) shows that the concurrence vanishes in the limit of $\Delta{z}\rightarrow 0$, i.e, the two detectors are located at the boundary, as well as when $L/\sigma\gg 1$, i.e., the interdetector separation is very large. This means that entanglement harvesting cannot occur in these cases. While for small interdetector separation ($L/\sigma\ll 1$), the harvested entanglement is an increasing function of the energy gap ($\Omega$) and the detector-to-boundary distance ($\Delta{z}$). In the far zone to the boundary ($\Delta{z}/\sigma\gg 1$), one can also approximately estimate the concurrence

In the case of $L/\sigma\ll 1$, the concurrence is now a decreasing function of $\Delta{z}$, which is in contrast to the case of the near zone to the boundary where the concurrence is a growing function of $\Delta{z}$ (see Eq. (20)). Therefore, one can infer that the amount of harvested entanglement ought to peak at a certain distance between the detectors and the boundary. This is what has been demonstrated in the numerical evaluations in Ref. Zhjl:2021 .

When the detectors’ energy gap is much larger than the Heisenberg energy ($\Omega\gg 1/\sigma$), one can easily deduce from Eq. (15) that the concurrence should be vanishingly small. For such identical detectors near the boundary with not too large separation ($\Delta{z}/\sigma\ll 1,L/\sigma\ll 1$) the concurrence quantifying the harvested entanglement reads approximately,

As a result, the harvested entanglement degrade to zero as the energy gap increases.

In order to gain a better understanding of the entanglement harvesting for nonidentical detectors with different energy gaps in the presence of the reflecting boundary, we now resort to numerical calculation since analytical approximations are hard to obtain in generic cases. We first show the results from our numerical evaluation on the role played by the reflecting boundary on entanglement harvesting in Figs. (2) and (3).

In Fig. (2), the amount of acquired entanglement is plotted as a function of interdetector separation for various detectors’ energy gaps. Obviously, the harvested entanglement monotonically degrade with the increasing interdetector separation no matter whether the two detectors are identical or not. For a fixed distance $\Delta{z}$ from the boundary, the identical detectors usually have an advantage in acquiring more entanglement than nonidentical detectors with unequal energy gaps (e.g., see Fig. (2b)). In contrast, when the energy gaps are small and the interdetector separation is large enough with respect to the duration time parameter ($\Omega_{A}\sigma<\Omega_{B}\sigma\ll 1$ and $L>\sigma$), the nonidentical detectors are instead more likely to harvest more entanglement [see Fig. (2a)]. Notice that there is only a finite harvesting-achievable range for the interdetector separation for entanglement harvesting, which can be enlarged by a large energy gap (or energy gap difference).

In Fig. (3), we show how the presence of the boundary influences entanglement harvesting by plotting the concurrence as a function of $\Delta{z}/\sigma$ for various detectors’ energy gaps. It is easy to see that the amount of harvested entanglement is zero in the limit of $\Delta{z}\rightarrow 0$ and approaches to their corresponding values in flat spacetime without any boundaries as $\Delta{z}\rightarrow\infty$ as expected. No matter whether the two detectors are nonidentical with different energy gaps or identical, the harvested entanglement measure by concurrence has a peak value when the distance $\Delta{z}$ is comparable to the duration time $\sigma$ [e.g., see Fig. (3a)]. The exact position of detectors where the harvested entanglement peaks is not sensitive to the energy gap difference. Moreover, the reflecting boundary plays a double-edged role in entanglement harvesting (inhibiting the entanglement harvesting near the boundary or enhancing it when $\Delta{z}/\sigma>1$ as compared to the case without any boundaries), regardless of whether the detectors are identical or not. Furthermore, both the amount of harvested entanglement and the influence of the boundary can be suppressed by increasing detectors’ energy gaps or the energy gap difference between the two detectors.

In order to further study how the entanglement harvesting phenomenon for two nonidentical detectors depends on the energy gap difference, we plot concurrence as a function of the energy gap difference in Fig. (4). When the interdetector separation $L$ is small relative to the duration time $\sigma$, the energy gap difference would generally hinder harvesting of entanglement, i.e., the harvested entanglement would rapidly degrade to zero with the increasing energy gap difference between two nonidentical detectors [see Fig. (4a)]. However, when the interdetector separation grows to comparable to or larger than the interaction duration time parameter, i.e., $L\gtrsim\sigma$, the concurrence is no longer a monotonically decreasing function of the energy gap difference but may initially increase with the increasing energy gap difference, and then reach a maximum value at certain nonzero energy gap difference before degrade to zero as the energy gap difference further increases [see Fig. (4b)]. This suggests that there exist an optimal energy gap difference that maximizes the amount of entanglement harvested, which we denote by $\Delta\Omega_{\mathcal{C}}$. It seems that the value of $\Delta\Omega_{\mathcal{C}}$ depends upon both the interdetector separation and the detector-to-boundary distance. In order to analyze this clearly, we demonstrate how the value of the optimal energy gap difference is influenced by the presence of the boundary for various interdetector separations in Fig. (5). Obviously, the optimal energy gap difference is an increasing function of $\Delta{z}/\sigma$, i.e., the presence of the boundary reduces the optimal energy gap difference for entanglement harvesting. It is also easy to find the larger the interdetector separation, the larger the optimal value of the energy gap difference, which is consistent with the conclusion in Ref. Zhjl:2022.4 .

When the detectors system is placed near the boundary (i.e., $\Delta{z}/\sigma\ll 1$), the correlation term $C$ is approximately given by

then the harvested mutual information can be approximated as

As can be seen from Eq. (24), the mutual information harvested by identical detectors, in the limit of $\Delta{z}\rightarrow 0$, i.e., the detectors are located at the boundary, must vanish. While, as the detector-to-boundary distance $\Delta{z}$ increases, the mutual information grows like $\Delta{z}^{2}$ in the near zone of the boundary. It is also interesting to note that the harvested mutual information is a decreasing function of the detectors’ energy gap $\Omega$ in the case of $L/\sigma\ll 1$, while it is an increasing function of the energy gap in the case of $L/\sigma\gg 1$. This is similar to the behavior for the mutual information harvesting in flat spacetime without any boundaries Pozas-Kerstjens:2015 .

For $\Omega\sigma\ll 1$ and $\Delta{z}/\sigma\gg 1$, the harvested mutual information can be approximated as,

Eq. (25) shows that the amount of harvested mutual information is an increasing function of $\Delta{z}$, and its boundary dependence is of minus $\Delta{z}^{-2}$. The harvested mutual information is also a decreasing function of $\Omega$ in the case of $L/\sigma\ll 1$, but becomes an increasing function of $\Omega$ in the case of $L/\sigma\gg 1$. In the limit of $\Delta{z}\rightarrow\infty$, the Wightman function (III) reduces to that in flat spacetime, and so the harvested correlations (mutual information or entanglement) approach to the corresponding results in flat spacetime without any boundaries as expected.

When the detectors’ energy gap is much larger than the Heisenberg energy ($\Omega\gg 1/\sigma$), the mutual information for two identical detectors near the boundary with not too large separation ($\Delta{z}/\sigma\ll 1,L/\sigma\ll 1$) reads approximately

One can see that the harvested mutual information degrades to zero as the energy gap increases.

In more general cases, the mutual information harvesting behavior can be captured by resorting to numerical estimations. In Figs. (6) and (7), the amount of mutual information is plotted as a function of interdetector separation and detectors-to-boundary distance for various detectors’ energy gaps, respectively. Obviously, similar to the entanglement, the harvested mutual information is also a monotonically decreasing function of the interdetector separation no matter whether the two detectors are identical or not. Interestingly, when the energy gaps are small and the interdetector separation is large enough with respect to the duration time parameter ($\Omega_{A}\sigma<\Omega_{B}\sigma\ll 1$ and $L>\sigma$), the nonidentical detectors with unequal energy gaps are likely to harvest more mutual information than the identical detectors [see Fig. (6a)], contrary to the usual wisdom that the identical detectors may be advantageous in mutual information harvesting. It should be pointed out that, unlike entanglement harvesting, mutual information can be harvested by two static detectors with an arbitrarily large interdetector separation, and physically, this means that the mutual information harvested outside the harvesting-achievable range of entanglement is either classical correlation or nondistillable entanglement Pozas-Kerstjens:2015 .

Fig. (7) demonstrates that in the limit of $\Delta{z}\rightarrow\infty$ the mutual information approaches to its corresponding values in flat spacetime without any boundaries, while on the boundary the mutual information vanishes as expected. In sharp contrast to the entanglement harvesting phenomenon which displays a peak-value behaviour when the distance $\Delta{z}$ is comparable to the duration time $\sigma$ [see Fig. (3)], the harvested mutual information is always a monotonically increasing function of $\Delta{z}/\sigma$, approaching asymptotically to the value without any boundaries. So, this means that the reflecting boundary always plays an inhibiting role in mutual information harvesting in contrast to its double-edged role in entanglement harvesting (inhibiting the entanglement harvesting near the boundary and enhancing it when $\Delta{z}/\sigma>1$). Moreover, both the amount of harvested mutual information and the influence of the boundary can be suppressed by increasing detectors’ energy gaps or the energy gap difference between the two detectors.

In order to further investigate the influence of the energy gap difference on mutual information harvesting, we plot mutual information as a function of the energy gap difference in Fig. (8). Similar to entanglement harvesting, the harvested mutual information is also a monotonically decreasing function of the energy gap difference when the interdetector separation is timelike and very small relative to the duration time $\sigma$ [see Fig. (8a)]. However, when the interdetector separation is spacelike ($L\gg\sigma$), the mutual information is no longer a monotonically decreasing functions of the energy gap difference but may peak at a certain nonzero energy gap difference and then degrade to zero with the increase of the energy gap difference [see Fig. (8b)], i.e., there also exists an optimal energy gap difference that maximizes the mutual information. We demonstrate how the optimal energy gap difference, denoted here by $\Delta{\Omega_{\mathcal{I}}}$, depends upon the presence of the boundary in Fig. (9). Analogous to that in the case of entanglement harvesting, the optimal energy gap difference $\Delta{\Omega_{\mathcal{I}}}$ generally increases as the interdetector separation increases. However, the presence of the boundary can either increase or decrease the optimum energy gap difference for mutual information harvesting. Specifically, when the interdetector separation is large with respect to the duration time, the optimum energy gap difference for mutual information harvesting is an increasing function of the detector-to-boundary distance; however, when the interdetector separation grows to too large, the optimum energy gap difference for mutual information harvesting becomes a decreasing function rather than an increasing function of the detector-to-boundary distance, which is quite different from that in entanglement harvesting.

It is well known that once there is a reflecting plane boundary in flat spacetime, the isotropy of spacetime would be lost. Therefore, the orientation of two detectors system in various angular alignments with respect to the boundary may have a non-negligible impact on correlation harvesting. So, in the next section, we will consider the situation in which the two detectors are orthogonally aligned to the boundary plane.

For small energy gap ($\Omega\sigma\ll 1$), the concurrence, denoted here by $\mathcal{C}_{V}(\rho_{AB})$, can be shown after some long algebraic manipulations to take the following approximate form

Similar to the case of parallel-to-boundary alignment, Eq. (30) demonstrates that the entanglement cannot be extracted at extremely large interdetector separation $L$ or vanishing distance $\Delta{z}$ from the boundary. Comparing Eq. (30) with Eqs. (20) and (21), we can find for $\Delta{z}/\sigma\ll 1$ the harvested entanglement in the orthogonal-to-boundary scenario is also an increasing function of $\Delta{z}$, but the boundary dependence behaves like $\Delta{z}$ rather than $\Delta{z}^{2}$, while for $\Delta{z}/\sigma\gg 1$ the approximate form of concurrence in both the orthogonal-to-boundary and parallel-to-boundary alignment looks the same and behaves as a decreasing function of $\Delta{z}$. This non-monotonicity of a function with respect to $\Delta{z}$ implies that the harvested entanglement in the orthogonal-to-boundary case also possesses a peak behavior. Beside these, one may find that the detectors orthogonally aligned to boundary can harvest comparatively more entanglement than these in parallel-to-boundary alignment for the case of $\Delta{z}\ll L\ll\sigma$. This is physically in accordance with our finding the boundary inhibits the entanglement harvesting for the detectors close to it.

For large energy gap ($\Omega\gg 1/\sigma$), the concurrence in the case of $\Delta{z}/\sigma\ll 1$ and $L/\sigma\ll 1$ can be approximated as

Similar to the case of the parallel-to-boundary alignment, the harvested entanglement in the orthogonal-to-boundary alignment also degrade with the increasing detectors’ energy gap.

From above approximations, it is easy to infer that the influence of the boundary on the entanglement harvesting phenomenon for two detectors orthogonally aligned to the boundary should be similar to that for two detectors aligned parallel to the boundary, i.e., qualitatively, the reflecting boundary should play a double-edge role (inhibiting entanglement harvesting for $\Delta{z}\ll\sigma$ and assisting entanglement harvesting for $\Delta{z}>\sigma$), and moreover, entanglement harvesting possesses a finite harvesting-achievable range regardless of the presence of the boundary. However, the quantitative details should be slightly different. To better understand it, we define the difference of concurrence between the scenarios of orthogonal-to-boundary and parallel-to-boundary alignments: $\Delta\mathcal{C}:=\mathcal{C}_{V}-\mathcal{C}_{P}$.

The difference in the amount of harvested entanglement is plotted as a function of the distance from the boundary in Fig. (11). As we can see from Fig. (11), the detectors in orthogonal-to-boundary alignment near the boundary ($\Delta{z}/\sigma\ll 1$) could harvest comparatively more entanglement than that in parallel-to-boundary alignment. However, when the detectors are located far from the boundary ($\Delta{z}/\sigma\gg 1$) with not too large interdetector separation, this would be reversed, i.e., the parallel-to-boundary alignment turns out to be favorable for entanglement harvesting. This is because the boundary would play a strongly inhibiting role in entanglement harvesting for two detectors placed near the boundary, and the detectors system in orthogonal-to-boundary alignment has a relatively longer effective distance from the boundary and thus less inhibiting effect, resulting in more entanglement harvested in comparison with those in parallel-to-boundary alignment. While when two detectors are far away from the boundary, the assisting role played by the boundary leads to more entanglement harvested by detectors in parallel-to-boundary alignment in comparison to orthogonal-to-boundary alignment.

In Fig (12), we demonstrate how the optimal energy gap difference that maximizes the amount of harvested entanglement depends upon the detector-to-boundary distance. It is easy to find that the optimal energy gap difference $\Delta{\Omega_{\mathcal{C}}}$ in orthogonal-to-boundary alignment is also an increasing function of the detector-to-boundary distance, but the quantitative detail is slightly different from that in the scenario of parallel-to-boundary alignment [comparing Fig (12) and Fig. (5)].

For small energy gap ($\Omega\sigma\ll 1$), the mutual information denoted here by $\mathcal{I}_{V}(\rho_{AB})$ in orthogonal-to-boundary alignment can be approximated as

In comparison with Eq. (24), the boundary dependence of mutual information, in the case $\Delta{z}/\sigma\ll 1$, is $\Delta{z}^{2}\ln{(\sigma/\Delta{z})}$ in orthogonal-to-boundary alignment, rather than $\Delta{z}^{2}$ in parallel-to-boundary alignment. Hence, the detectors orthogonally aligned to the boundary would harvest more mutual information than those aligned parallel to the boundary in the near zone of the boundary. Physically, we can understand it as follows. Since the reflecting boundary always inhibits the harvesting of mutual information as previously discovered, the orthogonal-to-boundary alignment has a comparatively longer effective detector-to-boundary distance than the parallel-to-boundary alignment, so the detectors orthogonally aligned to the boundary could harvest more mutual information. When compared with Eq. (25) for $\Delta{z}/\sigma\gg 1$, the approximated expressions of Eq. (32) have the same form as that of Eq. (25), which in the limit of $\Delta{z}\rightarrow\infty$ reduce to the result in flat Minkowski spacetime, which is isotropic, as expected.

For large energy gap ($\Omega\gg 1/\sigma$), the mutual information in the case of $\Delta{z}/\sigma\ll 1$ and $L/\sigma\ll 1$ can be approximated as

Obviously, the harvested mutual information in orthogonal-to-boundary alignment will degrade to zero as the energy gap grows.

With the above discussions, it is easy to infer that the influence of the boundary on the mutual information harvesting phenomenon for two detectors orthogonally aligned to the boundary should be similar to that for two detectors aligned parallel to the boundary, i.e., qualitatively, the reflecting boundary should always play an inhibiting role in mutual information harvesting, and moreover mutual information can be harvested at arbitrarily large interdetector separation. However, the quantitative details may in general be slightly different. To better understand it, we define the difference of mutual information between the scenarios of orthogonal-to-boundary and parallel-to-boundary alignments: $\Delta\mathcal{I}:=\mathcal{I}_{V}-\mathcal{I}_{P}$.

The difference in the amount of harvested mutual information is plotted, in Fig. (13), as a function of the distance between the boundary and the detector which is closer. As we can see from Fig. (13), the case of orthogonal-to-boundary alignment in general is more favorable for mutual information harvesting than that of parallel-to-boundary alignment. This is because the boundary always plays an inhibiting role in mutual information harvesting and the case of orthogonal-to-boundary alignment has relatively longer effective distance from boundary.

In addition, the boundary dependence for the optimal energy gap difference between two nonidentical detectors that maximizes the amount of mutual information is displayed in Fig. (14). Similar to the case of parallel-to-boundary alignment, the optimal energy gap difference $\Delta{\Omega_{\mathcal{I}}}$ in the case of orthogonal-to-boundary alignment can be either an increasing or a decreasing function of detector-to-boundary distance $\Delta{z}$. Specifically, when the interdetector separation is large with respect to the duration time, the optimum energy gap difference $\Delta{\Omega_{\mathcal{I}}}$ is an increasing function of $\Delta{z}$; however, when the interdetector separation is too large, $\Delta{\Omega_{\mathcal{I}}}$ becomes a decreasing function of $\Delta{z}$.