Acceleration-assisted entanglement harvesting and rangefinding

The details of the setup closely follow Ref. (VerSteeg2009, ) with any modifications clearly described. The background field is a massless scalar field in ${(3+1)}$-dimensional Minkowski spacetime Birrell1982 , initially in the Minkowski vacuum state. When comparing to the results of an expanding universe VerSteeg2009 , we will assume conformal coupling Birrell1982 because minimal coupling has a very different effect Nambu2011 ; Nambu2013 , but both are identical for Minkowski spacetime (with Ricci scalar $R=0$). We consider two local detectors (defined below), labeled $a$ and $b$, on the following spacetime trajectories:

where $b$ claims the prime, $\tau$ and $\tau^{\prime}$ are the proper times for observers $a$ and $b$, respectively, and $\bm{x}_{i}$ is the spatial trajectory of observer $i$. The 4-vector index $\mu$ is henceforth dropped.

We model the detectors as point-like, two-level quantum systems with an energy gap $\Omega$, each of which interacts with the field via an Unruh-DeWitt interaction DeWitt:1979tl ; Unruh1976 , whose interaction-picture Hamiltonian is

Here $\eta(\tau)$ is a window function (i.e., time-dependent coupling strength) that governs the interaction with the ambient scalar field, $x(\tau)$ is either of the 4-vectors $x_{a}$ or $x_{b}$, $\hat{\phi}[x(\tau)]$ is the interaction-picture field amplitude at spacetime point $x$, and $\hat{\sigma}_{x}(\tau)$ is the interaction-picture Pauli-$x$ operator for the two-level system. The detectors are identical aside from their trajectories, and we formally delay readout (i.e., projective measurement) of the detectors indefinitely because we wish to analyze the quantum features of the resulting state of the detectors. The detectors start in their respective ground states ($\mathinner{\lvert g\rangle}\otimes\mathinner{\lvert g\rangle}$), so if the interaction events are sufficiently far separated, then any entanglement appearing in the final state of the detectors must have been harvested from the field through the local interactions.

We choose $\eta$ at all times to be very small (i.e., $0<\eta(\tau)\ll 1$), with nontrivial support only over a small range in $\tau$ (as compared to the light-crossing time between them). This choice gives a short, weak interaction with the field. Its shortness enforces spacelike separation of the detection events, and its weakness allows us to treat these interactions perturbatively.

For calculational simplicity, we take $\eta(\tau)$ to be a Gaussian with variance $\sigma^{2}$:

with $\sigma\ll L$ and $0<\eta_{0}\ll 1$. Note that this function is analytic and therefore cannot be strictly compact in $\tau$. This means that, in general, the detection events are only approximately spacelike separated. Analyticity of the Gaussian will allow us to evaluate integrals using complex analysis, but the fact that it has vanishing but nonzero support for $\tau\to\pm\infty$ will have interesting consequences that we will explore later on.

The fact that the field is in the (Minkowski) vacuum does not necessarily imply that a detector will fail to become excited when interacting with it MartinMartinez:2012gh ; Birrell1982 . This is due to counter-rotating terms (such as $\hat{\sigma}_{-}\hat{a}$ and $\hat{\sigma}_{+}\hat{a}^{\dagger}$) in the expansion of Eq. (3). In fact, most window functions and trajectories will produce a nonzero response (excitation probability) in an Unruh-DeWitt detector due to the presence of these terms. Their contribution—and thus the probability of excitation—will tend to 0, however, when the energy gap and coupling strength (during detection) are constant, the trajectory is inertial, and the detection time tends to infinity Birrell1982 .

In general, then, each detector (initially in the ground state) is prone to excitation through its traveling interaction with the field. To lowest nontrivial order (i.e., second order) in the detector coupling strength $\eta_{0}$, the probability $A$ that exactly one of the detectors will be excited is given by Birrell1982

where $\Omega$ is the detector energy gap, and $D^{+}(x;x^{\prime})=\langle\hat{\phi}(x)\hat{\phi}(x^{\prime})\rangle$ is the Wightman function for the field $\hat{\phi}$, which, for the case of the Minkowski vacuum, is given by Eq. (3.59) of Ref. Birrell1982 :

Note that, due to symmetry, we could equally well make the trajectory substitution $x_{a}(\cdot)\mapsto x_{b}(\cdot)$ and get the same value for $A$. Also note that all quantities are evaluated with $\epsilon\to 0^{+}$ at the end.

To quantify the harvested entanglement, we follow the standard procedure first proposed by Reznik (Reznik2003, ) and used also by later authors Reznik2005 ; VerSteeg2009 ; Nambu2011 ; Nambu2013 : We calculate the elements of the reduced density matrix of the detectors and then use the partial transpose criterion Peres1996 , which is necessary and sufficient for two qubits Horodecki1996 , to detect the presence or absence of entanglement after interaction with the field.

As in previous work Reznik2003 ; Reznik2005 ; VerSteeg2009 ; Nambu2011 ; Nambu2013 , the partial transpose criterion for our setup reduces to a simple comparison between two quantities. Entanglement exists between the detectors if and only if

where $A$ is defined in Eq. (5), and $X$ can be interpreted as an amplitude for (virtual) particle exchange between the detectors (to second order in $\eta_{0}$):

where the Wightman functions are taken with respect to the two trajectories $x_{a}$ and $x_{b}$ (cf. Eq. (5), which only involves one trajectory at a time), and $\Omega$ is the energy gap of the detector. The amount of entanglement harvested can be quantified by the negativity (Vidal2002, ) of the final state of the detectors, which in this case is simply

A two-qubit state has nonzero negativity iff it is entangled Horodecki1996 .

Entanglement is therefore seen as a competition between the probability of actual detector excitation ($A$) and the magnitude of an off-diagonal term in the density matrix representing the amplitude for particle exchange ($X$) Reznik2005 . When the detection events are entirely spacelike separated, the exchange term can only be interpreted as swapping preexisting entanglement out of the field and into the detectors—rather than simply using the field as a transmission medium by which to exchange real excitations. When the detection events are only approximately spacelike separated, as is the case for some choices of parameters here (since $\eta(\tau)$ is Gaussian), there are additional subtleties, which we address later.

The canonical formalism for studying accelerated observers in Minkowski spacetime Unruh1976 ; Birrell1982 ; Crispino2008 involves defining a Rindler wedge corresponding to an accelerating trajectory and using that wedge to define three additional wedges that separate Minkowski spacetime into left, right, future, and past regions with lightsheet boundaries. A key feature of this partitioning is that all four wedges share a common apex. This suggests a natural set of accelerating trajectories: those hyperbolas that share the bounding lightsheets as asymptotes.

We take a more operational approach to the problem by focusing on the physical objects: two detectors accelerating uniformly through space. There is no physical reason why the Rindler wedges defined by each observer should be constrained to meet at the origin, and in fact we argue that this approach is restrictive. Forcing the wedges to meet at the origin is tantamount to forcing the distance between the detectors at closest approach, $L$, to be constrained by the acceleration parameter $\kappa$: specifically, $L=2\kappa^{-1}$. We relax the condition of wedges with overlapping apexes and study two uniformly accelerated detectors with an arbitrary distance at closest approach. We also consider both anti-parallel and parallel trajectories, both of which are described by the following trajectories:

where symmetry allows us to set $\tau^{\prime}=\tau$ in Eqs. (II.1), the plus/minus sign in $x_{b}$ refers to parallel/anti-parallel trajectories, respectively illustrated in left/right panels of Fig. 1, and the other spatial coordinates are taken to be 0. The quantity $L$ represents the distance of closest approach as measured by an inertial observer along a trajectory of constant $x$. For such an observer, the detectors are separated by this minimum distance when $t=t^{\prime}=0$.

In what follows, we will use time variables corresponding to the sum and difference of the two proper times:

Using the trajectories, we define

where the second line follows from the symmetry of the trajectories—both for Eqs. (II.4) and for the ones from Ref. VerSteeg2009 , which will be used for comparison. Furthermore,

or equivalently using $x_{b}(\cdot)$ instead. In fact, for the particular trajectories in question, these formulas are all invariant under the exchange $x_{a}(\cdot)\leftrightarrow x_{b}(\cdot)$.

Using trajectories (II.4), we compute the quantity $X$ from Eq. (II.3). We find this integral to be very difficult to compute analytically and to suffer from convergence issues when evaluated using naive numerical methods. Fortunately, we can calculate the integral by completing the square in the exponential and shifting the contour of integration to convert the complex Gaussian into a real Gaussian. By making these transformations, we remove the highly oscillatory prefactor and allow for evaluation of the integral using Cauchy’s residue theorem. In particular, we obtain

The inverse-Gaussian prefactor comes from completing the square, and we have shifted the contour of integration up by $2i\sigma^{2}\Omega$ and changed variables (see Appendix A). Shifting the contour of integration will, in general, cross poles, which give rise to the residue contributions noted explicitly in Eq. (II.4). The pole locations are functions of the input parameters $L$ and $\Omega$, as well as functions of $x$ and $y$. As such, computing the residue contributions proves challenging, and details are presented in Appendix A.

We will evaluate $X$ in several different scenarios. The ones that are the focus of this work are parallel and anti-parallel acceleration, as shown in Fig. 1. For the parallel case we use $D(x,y)=D_{\text{{\tiny((}}}(x,y)$, with

Similarly, for the anti-parallel case we use $D(x,y)=D_{\text{{\tiny)(}}}(x,y)$, with

The subscripts on $D_{\text{{\tiny((}}}$ and $D_{\text{{\tiny)(}}}$ are used to visually indicate the respective trajectories in Fig. 1.

For comparison with known results in the literature VerSteeg2009 ; Nambu2013 , we will also need the following additional Wightman functions. The first is for two comoving detectors in de Sitter spacetime with expansion rate $\kappa$, separated by constant comoving distance $L$ Birrell1982 ; Nambu2013 :

A second Wightman function is for two inertial detectors with fixed proper distance $L$ in the case where the field has been heated to a finite temperature $T=(2\pi k_{B})^{-1}\kappa$ in the rest frame of the detectors Weldon2000 ; Nambu2013 :

Any of the four Wightman functions above can substitute for $D(x,y)$ when calculating $X$ in Eq. (II.4).

We use a similar method to calculate the excitation probability $A$, although in this case we can avoid crossing any poles by restricting our analysis to $\kappa\sigma^{2}\Omega<\pi$ (see Appendix A for details). As such, the residue contributions vanish here, giving

In this expression, $D_{\text{detect}}(y)$ from Eq. (13) is the detector response function for a single detector in a thermal bath at temperature $T=(2\pi k_{B})^{-1}\kappa$ Birrell1982 :

In order to evaluate $A$ and $X$, we employ a saddle point approximation, following Ref. Nambu2013 . In particular,

Note that there are no residue contributions to $A$ due to the aforementioned restriction $\kappa\sigma^{2}\Omega<\pi$. The detector response is identical in all cases considered:

Plugging in for each of the possible choices for $D$ gives the following for $X$ in each case:

where ‘r.t.’ stands for residue terms, which will be evaluated later.

Interestingly, this negativity formula and entanglement criterion exactly match those for comoving (inertial) observers, at a fixed comoving distance $L$, in a de Sitter universe with expansion rate $\kappa$ Nambu2013 ; VerSteeg2009 . Indeed, using $X_{\text{dS}}$ from Eq. (26) and verifying that we again cross no poles, we see that $\left\lvert{X_{\text{{\tiny((}}}}\right\rvert=\left\lvert{X_{\text{dS}}}\right\rvert$, which means that the negativity is the same, $N_{\text{dS}}=N_{\text{{\tiny((}}}$, and so is the entanglement criterion, Eq. (28). The only differences are in their interpretation: $\kappa$ now refers to the expansion rate instead of the parallel acceleration rate, and $L$ now refers to the comoving distance instead of to the distance as measured by an inertial observer on the trajectory $x(t)=\text{(constant)}$. Notice that the motion in these two cases seem very different. In the case of parallel acceleration, the detectors are moving in the same direction, whereas in the de Sitter case the detectors are moving in opposite directions. However, this intuition is misleading because the similarity between the two is the constancy of $L$. For parallel acceleration, the distance at closest approach as measured by a stationary observer is constant, while in de Sitter space the detectors are separated by a constant comoving distance.

Due to the mathematical connection between Rindler and de Sitter geometries and the properties of conformal fields (see Chapter 5 of Ref. Birrell1982 ), one might expect a priori such a connection between the parallel-acceleration case and the case of de Sitter expansion. We point out a few reasons why this intuition is not enough to prove that they must be the same. First, while $A$ is the same in both cases, $X_{\text{{\tiny((}}}$ and $X_{\text{dS}}$ actually differ by a phase, Eqs. (24) and (26). As such, while the negativity (which depends only on $\left\lvert{X}\right\rvert$) is exactly the same in both cases, the actual density matrix in each case is different, resulting in different correlations. Second, the calculations leading to the equivalence involve two approximations—second-order perturbation theory and a saddle-point approximation—that limit the equivalence to the cases where these approximations are valid. Finally, notice that $D_{\text{{\tiny((}}}$ and $D_{\text{dS}}$ are not the same, Eqs. (II.4) and (17). This leads us to suspect that when the aforementioned approximations break down, the equivalence may be broken as well. On the other hand, our work does not rule out the existence of other coordinates and/or modified trajectories (perhaps parallel acceleration but displaced in a different spatial direction) for which there is exact equivalence between the two cases. This is left as an open problem.

From Ref. VerSteeg2009 , we already know that the entanglement profile for two inertial detectors with fixed proper distance $L$ in a thermal bath in the detectors’ rest frame will differ from the de Sitter case. From the above results, this means it must also differ from the parallel-acceleration case, despite $A$ being the same in all three cases. Solving the analogous inequalities using $X_{\text{th}}$ from Eq. (27) and noting that no poles have been crossed, we find entanglement whenever

with the temperature $T$ of the thermal state of the field chosen to be the Gibbons-Hawking temperature associated with $\kappa$ Gibbons1977 —i.e., $T=(2\pi k_{B})^{-1}\kappa$. Thus, we have reproduced the results first reported in Ref. VerSteeg2009 (and later confirmed using this saddle-point method in Ref. Nambu2013 ), which show that two detectors can distinguish between Gibbons-Hawking and thermal radiation by whether they are able to harvest entanglement or not. If entangled, the negativity in this case is

For comparison, the entanglement boundary for inertial detectors in the Minkowski vacuum is given by

which can be obtained by taking the $\kappa\to 0$ limit of either Eq. (28) or Eq. (30) while fixing the other physical parameters $L$, $\sigma$, and $\Omega$. For entangled states, the negativity in this case is Nambu2013

All of these results are shown in Fig. 2 (left). Notice that the negativity is generally exceptionally small since, for all cases considered here, $N\sim e^{-(\sigma\Omega)^{2}}$.

Perhaps one of the most interesting features of entanglement harvested with anti-parallel acceleration is that the acceleration allows for entanglement harvesting where inertial detectors could not. In particular, certain detectors with a distance of closest approach $L>2\sigma^{2}\Omega$ can harvest entanglement. This is surprising because it is outside the allowed region for inertial entanglement harvesting, given by Eq. (32). The allowed region for inertial detectors is represented by the blue triangular region in Fig. 2 (left), the boundary of which is given by the blue line in Fig. 2 (right), assuming the same resonant frequency and window function VerSteeg2009 . This line should be interpreted as the $\kappa\to 0$ limit of non-inertial motion.

The power of this enhancement can be understood physically. Consider two inertial detectors in Minkowski spacetime, configured with $L~\gtrsim~2\sigma^{2}\Omega$—i.e., just below the blue diagonal line in Fig. 2 (right). These two detectors would not be able to harvest entanglement and would remain in a separable state. However, if we instead consider two detectors with the same $\Omega$, accelerating oppositely and reaching a distance at closest approach equal to $L$, in some cases they would become entangled. Notice that we do not have to physically bring the detectors closer together than the original $L$ in order to entangle them. It is worth pointing out that this feature arises in the portion of $X$ that does not come from residue terms. Therefore, we conjecture that the enhancement is not an artifact of the infinite tails in the Gaussian window functions but would persist even if the Gaussian were cut off smoothly for $\left\lvert{\tau}\right\rvert\gg\sigma$ (see Sec. A.3).

For small accelerations and finite interaction times, the two accelerating detectors would look almost like inertial detectors, yet they could in principle harvest entanglement that truly inertial detectors could not. This seems to present a paradox: the behavior as $\kappa\to 0$ should match up with that of actually achieving the limit $\kappa=0$, but it appears not to do so. This apparent paradox is resolved by noting that for small accelerations, both $L$ and $\Omega$ must be very large to see this effect (since the terms $L\kappa$ and $\kappa\sigma^{2}\Omega$ both become small), and at the limit point $\kappa=0$, $L$ and $\Omega$ must become infinite, which is impossible. Therefore, for all practical purposes, the inertial bound, Eq. (32), becomes the relevant one in the limit $\kappa\to 0$.

As mentioned in Sec. II, the usual picture employed in the literature for studying accelerated observers Unruh1976 ; Birrell1982 ; Crispino2008 suggests a natural restriction to trajectories embedded within Rindler wedges sharing a common apex, which corresponds to $L=2\kappa^{-1}$. We do not impose this restriction in this work because there is no physical reason why the separation between two independent objects should necessarily be related to their acceleration.

This restriction is not entirely without physical motivation, however. The Unruh effect is usually discussed in terms of the Minkowski vacuum being describable as a collection of two-mode squeezed states between pairs of Rindler modes, with one mode from each pair localized to the left and right Rindler wedges, respectively Unruh1976 ; Birrell1982 ; Crispino2008 .²²2Recent results, however, bring the naturalness of this picture into question Rovelli:2012jm . Due to reflection symmetry between the two wedges, the modes that are entangled (i.e., two-mode squeezed) happen to be those that are naturally detected by anti-parallel-accelerating observers, one within each wedge. For such detectors, $L=2\kappa^{-1}$. One might therefore wonder if having detectors on these trajectories might somehow be the best way to see the effects of anti-parallel acceleration due to resonance with these two-mode-squeezed modes. In fact, perhaps this configuration might be necessary for making use of such a resonant effect to harvest a significant amount of entanglement using anti-parallel-accelerating detectors. Surprisingly, this conjecture will turn out to be false.

Before we say why this is false, it is worth asking what such a resonant effect would look like in our calculations. We do not expect to see any effect in evaluating $A$ since this is just the response of our detector to the perceived thermal noise, and this does not depend on the relative positions of the trajectories. Since the detector is only coupled (nontrivially) to the field for a finite time $\sigma$, this response is expected to be finite.³³3In the case of constant and always-on coupling, $A$ would diverge, which is why it is common to talk about transition rates in that case, rather than the total probability of excitation Birrell1982 . It is still conceivable that we might find a divergence in $X$, however. Such a divergence would signal a situation in which higher-order terms—representing multiple real or virtual transitions—would be necessary to capture the true dynamics. Since we do not go beyond second order in this analysis, we cannot be sure of what exactly is happening in such a case, but we can still consider such a divergence to be evidence of a resonance condition, which should be detectable in the final state of the detectors. In what follows, we take this interpretation of such divergences.

It turns out that for any detector resonance frequency $\Omega$, we can create a situation in which such a resonant condition occurs for any value of $L$ up to $4\kappa^{-1}$. To find this critical distance $L_{\text{crit}}$, we evaluate where the saddle-point approximation of $X$, Eq. (22) diverges. Using $D=D_{\text{{\tiny)(}}}$, we find

Interestingly, while this method reveals the location of the divergence in $X$, the saddle-point approximation gets the sign wrong. Direct numerical integration reveals that $X\to\infty$ as $L\to L_{\text{crit}}$, but the saddle point approximation predicts $X\to-\infty$ instead. The behavior of $X$ with respect to $L$ around $L\sim L_{\text{crit}}$ is shown in Fig. 3. Notice that $X$ gets more negative before turning around and shooting up to $+\infty$. The saddle-point approximation does not reveal this turnaround to positive values and instead predicts that $X$ will continue to decrease as the critical distance is approached.

The critical distance is shown as the orange S-shaped curve in Fig. 2 (right). Given a value of $L<4\kappa^{-1}$, we can tune the detectors to reveal this effect by setting $\Omega=\Omega_{\text{res}}$, where

This divergence appears in the residue-free portion of $X$, which leads us to believe it to be robust to smooth cutoffs in the window functions (see Sec. A.3). An application of this resonance to a rangefinding thought experiment is discussed in Sec. V.1. Note that we have assumed point-like detectors and that finite size effects could be important for more realistic detectors.

Notice that $L=2\kappa^{-1}$ is not necessary to see this resonance effect. For the special case $\kappa\sigma^{2}\Omega=\frac{\pi}{2}$, of course, we get resonance at that distance, but this is an additional requirement. Therefore, $L=2\kappa^{-1}$ alone is neither a necessary nor a sufficient condition for resonance. This is contrary to the intuition provided by the Rindler-wedge picture often used to derive and discuss the Unruh effect Unruh1976 ; Birrell1982 ; Crispino2008 .

The entanglement harvested by detectors in different regions of parameter space seems to arise from distinct physical mechanisms. For $L\kappa<2$, the detectors are timelike separated, and real particles can, in principle, be exchanged. There is a residue contribution to $X$ that is largest for small $L\kappa$ and vanishes when $L\kappa>2$. Due to this behavior, we hypothesize that the residue contribution in this region may arise from causal dynamics (see Sec. A.2).

For detectors on the hyperbolic trajectories shown in the right panel of Fig. 1, the Gaussian window functions cause the detectors to spend a very long time traveling near the speed of light and interacting with the underlying field. Although the interaction is very weak, we conjecture that the long interaction time gives rise to a nontrivial contribution to $X$ comparable to (or even greater than) the residue-free portion.

Specifically, we hypothesize that the first detector’s interaction with the field for a long time in the infinite past may produce a large effect concentrated near the lightlike past asymptote of the first detector’s trajectory that is felt by the other detector (in the coherence term $X$) as the second detector crosses the future extension of that asymptote. (The same effect also happens from the second to the first detector.) Similarly, the second detector will spend a long time near the lightlike future asymptote of its own trajectory, thus amplifying any effect due to the presence of the first detector near the past extension of that asymptote (and vice versa). The net result of these four (symmetric) processes is a nontrivial contribution to the coherence term $X$, which indicates that the detectors “feel each other’s presence” due to the long tails of the Gaussian and overlapping wedges, as shown in Fig. 4.

Even as the causal residue contribution discussed in Sec. IV.3 vanishes as $L\kappa>2$, a second residue contribution takes over in that region when, in addition, $\kappa\sigma^{2}\Omega>\tfrac{\pi}{2}$. This means that there exists a nontrivial residue contribution for spacelike separated detectors ($L\kappa>2$) but only for certain choices of the other parameters. Because the detectors are spacelike separated for the entirety of their trajectories in this case, this contribution is more mysterious. There are some clues that we can use to hypothesize about its physical origin, however.

The most important clue is that the effect is strongest near $L\kappa\gtrsim 2$. This suggests that the detectors may be feeling the effects of entangled Rindler modes Unruh1976 ; Birrell1982 ; Crispino2008 . To understand this reasoning, consider dividing spacetime into four wedges (using the usual Rindler prescription Unruh1976 ) whose common origin is at ${(x=0,t=0)}$. If the actual detector trajectories have asymptotes that are close to the boundaries of these origin-centered Rindler wedges (i.e., ${L\kappa\gtrsim 2}$), then the detectors will be nearly resonant with the Rindler modes that are two-mode squeezed (see Sec. IV.2). This effect only increases as $L\kappa\to 2^{+}$. However, the fact that it cuts off sharply as soon as $L\kappa<2$ also suggests that the infinite tails may play a role in this effect. Perhaps confinement of the infinite tails to the left and right origin-centered wedges is required to see the effect. If the detectors are too close, then the long tails of each detector’s trajectory will leak out into the origin-centered future and past wedges. This effect would then be traded out for the causal effect described in Sec. IV.3, which is tiny for $L\kappa\lesssim 2$. Further work is needed to explore the effect of using switching functions with compact support, but this requires new calculational methods and is beyond the scope of this work. We therefore leave the above as a working hypothesis. Further justification may be found in Sec. A.2.

Having found the critical distance in Eq. (34), we explore the properties of the harvested entanglement near $L_{\text{crit}}$. We will study the dependence of $\operatorname{Re}(X)$ on $L$. For convenience, we define

to be the distance away from the critical value. Fig. 3 shows $\operatorname{Re}(X)$ as a function of $\delta L$ for $\kappa=0.001$, $\sigma=1$, and $\Omega=1250$ (i.e., $\kappa\sigma^{2}\Omega=1.25$). These parameter values were chosen to lie in a region of parameter space in which residue contributions to $X$ are negligible; specifically, we choose $1.2<\kappa\sigma^{2}\Omega<\tfrac{\pi}{2}$ and $1.1<L\kappa<2$. In this region of parameter space $X$ is heavily dominated by the residue-free contribution (i.e., the residue terms are negligible), and the shape of $\operatorname{Re}(X)$ is generic for other values of $\Omega$.

Through explicit numerical evaluation we find that the coherence term, $X$, flips sign in a narrow corridor around $L=L_{\text{crit}}$. (Interestingly, the saddle-point approximation of Eq. (25) misses this behavior.) This sign flip is detectable by performing local measurements on the detectors and collecting statistics. To see how this is possible, consider the bipartite state of the two-detector system in the basis defined by the tensor product of their respective ground and first excited states, up to second order in the detector coupling. The density matrix is of the form

By measuring $\sigma_{x}\otimes\sigma_{x}$ or $\sigma_{y}\otimes\sigma_{y}$, we have that

Therefore,

Operationally, this means that we can make measurements of either ${\sigma_{x}\otimes\sigma_{x}}$ or ${\sigma_{y}\otimes\sigma_{y}}$ and send the results of the measurements, along with a timestamp and information about the choice of measurement, back to a neutral third party at $x=0$. If the third party finds that the measurements were made in the same basis, then the results of the measurements are kept. With access to an ensemble of detectors and measurements, one can collect statistics of the outcomes of the measurements for comparison. Using this scheme, it is possible to use entanglement to measure distance between the detectors. It is in this sense that we say information about the distance between the detectors is stored nonlocally in the phase of the joint state.

Suppose for a moment that we had access to such an ensemble of detectors with $1.2<\kappa\sigma^{2}\Omega<\tfrac{\pi}{2}$, and we performed the measurements described above. In this case, $\operatorname{Re}(X)$ can only be positive when $L$ is close to $L_{\text{crit}}$ (i.e., $\delta L\simeq 0$). As such, a measurement outcome of $\operatorname{Re}(X)>0$ necessarily implies that the detectors must have been separated by $L_{\text{crit}}$ up to $\mathcal{O}(\delta L)$.

The negativity contours [Fig. 2(Right)] for anti-parallel acceleration show a narrow region for large $\kappa\sigma^{2}\Omega$ where the negativity is zero. We have affectionately dubbed this oddly shaped region the “necktie”, for obvious reasons. Just to the right of the necktie (e.g., $L\kappa\gtrsim 2$ and $\kappa\sigma^{2}\Omega\gtrsim 2.4$) the negativity suddenly drops to zero as we cross $L\kappa=2$ from the right. This sudden death of entanglement (in parameter space) is curious in its own right and has been discovered in a number of other systems(Yu2009, ; Ostapchuk:2011ud, ; Doukas:2013noa, ). In addition, we can use this phenomenon for rangefinding by identifying the presence or absense of entanglement as a signal for crossing a particular distance.

To see how entanglement sudden death can be used for rangefinding, we shall construct a simple toy protocol that uses entanglement as a trigger for signalling that a particular distance ($L=2\kappa^{-1}$) has been crossed. For the remainder of this section, we will always require $\kappa\sigma^{2}\Omega\gtrsim~2.4$ in order to focus on the necktie.

Consider two sets of detectors separated by $L={2\kappa^{-1}\pm\delta}$ for $0<\delta\ll 1$. The detectors separated by $L={2\kappa^{-1}+\delta}$ will be able to harvest entanglement and will evolve into an entangled state, whereas the detectors separated by $L=2\kappa^{-1}-\delta$ will not be able to harvest any entanglement and will remain in a separable state. Using an ensemble of detectors with varying $L$, it is possible to use entanglement to identify those detectors lying on either side of the $L\kappa=2$ boundary. This identification marks the desired $L=2\kappa^{-1}$ distance. In order to verify that the detectors have evolved into an entangled state, it suffices to show violation of a Bell inequality, which can always be done if and only if the state is entangled (with sufficient extra resources) Masanes:2008ib .

In addition to the sudden death of entanglement, there is another feature of the necktie region that we can use for rangefinding. To reiterate an earlier point, our aim here is simply to highlight the relationship between entanglement and distance. From Fig. 2(Right), we see that the negativity has a steep gradient in the region just to the left of the necktie. In this region, the negativity changes rapidly over very small changes in distance $L$. As such, the amount of entanglement harvested in this region of parameter space is highly sensitive to the distance between the detectors; very small changes in the separation distance $L$ manifest as (proportionally) large changes in the negativity. We can use this feature to determine changes in distance very precisely by measuring relatively coarse changes in the entanglement.

To make this argument precise, suppose we have two pairs of detectors with a given acceleration $\kappa$ and energy gap $\Omega$. Suppose further that the pairs are both separated by a known distance at closest approach $L$. If the separation between one pair of detectors is perturbed slightly, the negativity for that pair will be vastly different from the negativity of the unperturbed, reference pair. If we measure $A$ and $X$ for the two pairs of detectors [the latter through coincidence measurements as in Eq. (40)] and compare them, we can map the change in negativity to a change in distance. The steep negativity gradient allows even relatively imprecise knowledge of the negativities to correspond to fairly precise values of distance. Note that this technique does not allow us to measure arbitrary absolute distances, but rather fluctuations in distance around some target value.

Obviously, much more practical methods exist to measure distance between objects. Furthermore, the amount of entanglement harvested using these methods is extremely small due to the Gaussian prefactor $e^{-(\sigma\Omega)^{2}}$, so even the proportionally large changes in entanglement predicted in, e.g., Sec. V.3 are between values that are exceedingly small.

As such, we would like to remind the reader that the purpose of this section is simply to outline several thought experiments to show—in principle—how the particular correlations in the harvested entanglement reveal details about the distance between the traveling detectors. We expect that this effect—of the harvested entanglement showing a critical dependence on distance—is generic and will eventually lead to realistic proposals for rangefinding using entanglement harvesting. One example of such a proposal is Ref. Brown:2014en , which was motivated directly by the results reported here.