Entanglement harvesting in cosmic string spacetime

We study the entanglement harvesting phenomenon for static detectors locally interacting with massless scalar fields in the cosmic string spacetime which is locally flat but with a conical structure characterized by a deficit angle. Specifically, three alignments of the detectors with respect to the string, i.e., parallel and vertical alignments with the detectors on the same side of the string, and vertical alignment with the detectors on two different sides, are examined. For the alignments on the same side of the string, we find that the presence of a cosmic string may either assist or inhibit entanglement harvesting both in the sense of the entanglement harvested and the harvesting-achievable range of interdetector separation depending on the detector-to-string distance, and this is remarkably different from the case of a locally flat spacetime with a reflecting boundary where the boundary always enlarges the harvesting-achievable range. For the alignment with detectors on two different sides of the string, the detectors notably can always harvest more entanglement than those in flat spacetime without a cosmic string, which is in sharp contrast to those on the same side. Interestingly, the presence of a cosmic string enlarges the harvesting-achievable range for the detectors in vertical alignment only in the vicinity of the string, while it always reduces the harvesting-achievable range for the detectors in parallel alignment.


I. INTRODUCTION
In the framework of formal algebraic quantum field theory, it has been recognized for a long time that the vacuum state of a free quantum field can maximally violate Bell's inequalities [1,2], and there exists quantum entanglement between both timelike and spacelike separated regions in the vacuum state of a free quantum field, suggesting that the vacuum is a potential resource for quantum entanglement.After the pioneering work of Valentini [3] (and later of Reznik [4]), it was realized that such vacuum entanglement could be extracted by multiple detectors through local interactions with the field.The process that two initially uncorrelated Unruh-DeWitt (UDW) detectors interact locally with a quantum field in some state (typically the vacuum state) to extract entanglement has now become recognized as the so-called entanglement harvesting protocol [5,6].The entanglement harvesting phenomenon has been examined in a wide range of scenarios .It has been demonstrated that entanglement harvesting is sensitive to the topology [12,23,27] and curvature [11,[13][14][15][16] of spacetime, and the superpositions of temporal order [21], the intrinsic motion [18,20,23,24] and the energy gaps of detectors [25][26][27].It was argued that the sensitivity of entanglement harvesting phenomenon to nontrivial topology can be used to distinguish locally flat spacetimes that differ only in their topologies [12].Recently, it was found that the presence of a reflecting boundary in flat spacetime, which renders the spacetime topologically nontrivial, could play a quite interesting role in entanglement harvesting, i.e., inhibiting entanglement harvesting in the near zone of the boundary while assisting it in the far zone [23,27].
In addition to the locally flat spacetime with a reflecting boundary, there is yet another interesting locally flat but topologically nontrivial spacetime characterized by a conical structure with a deficit angle, which is physically interesting in that it describes the spacetime of a cosmic string [29].Cosmic strings as a topological defect can either occur as a result of the phase transitions in the early universe [30,31] or in some gauge extensions of the standard model of particle physics [32].The conical structure around the string, which gives rise to interesting cosmological, astrophysical and gravitational phenomena [32][33][34], also modifies quantum fluctuations of fields, leading to interesting modifications to a lot of quantum phenomena, such as the Casimir-Polder effect [35,36], atomic transitions [37][38][39][40], resonance interaction [41], lightcone fluctuations [42] and entanglement dynamics [43].Noteworthily, a proposal for experimental detection of analogue spacetime metric fluctuations through the observable flight time variance near fabricated analogue cosmic string has been made in Ref. [44].
The simplest cosmic string spacetime is the one with a deficit planar angle for a static, straight and infinitely thin cosmic string [29].It has been found that the atomic transition rate and resonance interaction of atoms in the cosmic string spacetime would exhibit some behaviors similar to those in a flat spacetime with a perfectly reflecting boundary [39][40][41].
Inspired by these studies, a question naturally arises as to what role the cosmic string will play in entanglement harvesting.Since both the effects of a reflecting plane boundary and a cosmic string on the vacuum fields can be studied by considering the contributions from the "images" due to the boundary and the conical structure of the string in the Wightman functions of the fields, it is also quite of interest to compare the phenomenon of entanglement harvesting in the cosmic string spacetime with that in flat spacetime with a reflecting boundary, which may provide a useful way to distinguish locally flat spacetimes that differ only in topology caused by cosmic strings and reflecting boundaries.These are what we are planning to explore the present paper.This paper is organized as follows.In the following section, we review the entanglement harvesting protocol, including the UDW detector model and the basic formula of the detector-field coupling.In Sec.III, we derive the expressions for the detectors' transition probability and their nonlocal correlations in the cosmic string spacetime, and investigate the entanglement harvesting phenomenon for two static detectors in three different alignments with respect to the cosmic string in detail.Numerical evaluations are employed when necessary to clearly exhibit the behaviors of entanglement harvesting and comparisons are made between the entanglement harvesting phenomena in the cosmic string spacetime and flat spacetime with a reflecting boundary.Finally, we end up with summaries in Sec.IV.
For convenience, the natural units ℏ = c = k B = 1 are adopted throughout this paper.

II. THE BASIC FORMULAS
In the standard entanglement harvesting protocol, one considers two UDW detectors A and B which locally interact with a massless quantum scalar field ϕ[x D (τ )] (D ∈ {A, B}) along their worldlines.The classical trajectory of the detector, x D (τ ), is parameterized in terms of its proper time τ .Suppose that the UDW detector has an energy gap Ω D between its ground state |0⟩ D and excited state |1⟩ D .Then the interaction Hamiltonian for such a detector locally coupling with the scalar field in the interaction picture is given by where constant λ ≪ 1 denotes the weak coupling strength, χ(τ Assume that two UDW detectors A and B are prepared in their ground state and the scalar field is in the vacuum state |0⟩.The initial state of the detector-field system then can be written as |Ψ i ⟩ = |0⟩ A |0⟩ B |0⟩ and the final state can be shown to be given by where T denotes the time ordering operator and t is the coordinate time with respect to which the vacuum state of the field is defined.For simplicity, we presume that the two detectors have the same energy gap Ω (Ω A = Ω B ) and parameter σ (σ A = σ B ) characterizing the interaction duration .Tracing out the field degrees of freedom, one can obtain, with the perturbation theory, that the density matrix for the final state of the detectors in the basis B is, to the leading order in the coupling strength, given by [12,14,16] where with W (x, x ′ ) := ⟨0|ϕ(x)ϕ(x ′ )|0⟩ being the Wightman function of the scalar field in vacuum, and θ(x) denoting the Heaviside step function.In particular, if the two detectors are at rest, we have t = τ and t ′ = τ ′ .Here, the matrix element P D represents the detector's transition probability from the ground state to the excited state due to the interaction between the detector and the field, and the quantities C and X represent the nonlocal correlations between the two detectors.
To quantify the entanglement acquired by the two detectors, we employ the concurrence as a measure of entanglement [45].For the density matrix (3), the concurrence can be evaluated straightforwardly [12,14,16], This indicates that the concurrence C(ρ AB ) is determined by the competition between the nonlocal correlation term X and the geometric mean of the transition probabilities P A and P B , which both crucially depend on the Wightman function of the scalar field.

III. HARVESTING ENTANGLEMENT IN THE COSMIC STRING SPACETIME
We now begin to study the entanglement harvesting phenomena for two UDW detectors near a long straight cosmic string which is situated along the z-axis in flat spacetime.The line element of the cosmic spacetime can be written in cylindrical coordinates as [29] ds where θ ∈ [0, 2π/ν] and ν := (1 − 4Gµ) −1 with G and µ being the Newton's gravitational constant and the cosmic string linear energy density, respectively.The dimensionless quantity Gµ measures the strength of the gravitational effects of the string by a deficit angle δθ := 2π(ν − 1)/ν = 8πGµ with respect to the usual flat spacetime.Notice that for the cosmic string spacetime one always has the deficit-angle parameter ν > 1 .
After analytically solving the Klein-Gordon equation for a massless scalar field, the Wight-man function of the field in the cosmic string spacetime can be found to be given by [42] W where with ∆t = t − t ′ − iϵ, ∆z = z − z ′ , and ∆θ = θ − θ ′ .Here, [ν/2] denotes the integer part of ν/2 and the prime in the summation means that when ν is an even integer the term with m = ν/2 should be multiplied by an additional factor 1/2. Obviously, the summation has no contribution when ν < 2, and the Wightman function (9) usually is not a continuous function of ν due to the truncating-integer operation [ν/2].Furthermore, if ν is an integer number, the third term in Eq. ( 9) will vanish and the Wightman function takes a simple form The first term in the above equation is just the Wightman function in a trivial flat spacetime and the second term is a summation of the Wightman functions corresponding to ν − 1 "images" due to the conical topology with a planar deficit angle [see Fig.  two detectors in this case can be expressed as In order to calculate the concurrence, we first need to calculate the transition probabilities of the detectors.Substituting the trajectory ( 14) into Eq.( 9) yields After some manipulations [see Appendix A], the transition probability (4) can be given by with where Erfc(x) := 1 − Erf(x) with the error function being Erf(x) : Note that the first term P 0 is just the transition probability for a static detector in flat spacetime without a cosmic string, which approaches to zero in the limit of Ωσ → ∞ as expected [12].Obviously, the transition probability P D will reduce to P 0 in the limit of l → ∞ or ν = 1, i.e., when the detectors are located infinitely far from the string, or when the deficit angle vanishes.Moreover, for small l/σ, i.e., when the detectors are very close to the string, the transition probability P D can be further approximated as So, the transition probability of the detector is ν times that in flat spacetime without a cosmic string, suggesting that the presence of the deficit angle increases the detector's transition probability.Especially, when the detector is located on the string, the transition probability can be exactly written in a simple form where in the last step we have considered the fact that for non-integer ν Let us now turn to calculate the correlation term X.For convenience, we denote it by X P for the case of the parallel alignment.Substituting the trajectories ( 14) and the Wightman function (9) into Eq.( 5) , we have (see Appendix B) with FIG.3: The transition probability is plotted as a function of l/σ in (a) and as a function of ν in (b) with fixed parameter Ωσ = 0.10.Here, the black dashed line in the left plot represents the corresponding results in flat spacetime without a cosmic string (i.e., ν = 1).For convenience, all relevant physical quantities are expressed in the unit of the interaction duration parameter σ. and where the auxiliary function f is defined as From above equations, one may see that the correlation term X P will be exponentially suppressed when the detector energy gap Ω is increased.So, a large energy gap still plays a strongly inhibiting role in entanglement harvesting.
Notice that the first term of the correlation term X P , is just the result for flat spacetime without a cosmic string.It is worth noting that the value of |X 0 | will diverge in the limit of d → 0 due to the ill-defined point-like approximation of the UDW detector model for d/σ ≪ λ which results in that the concurrence is mathematically divergent.However, it has been shown that a finite-size detector model with a spatial smearing function can resolve this issue of divergence in the entanglement harvesting protocol [6].The second term X P 1 and the third term X P 2 are dependent on the detectorto-string distance l, and both of them vanish in the limit of l → ∞, i.e., when the detectors are very far away from the string.As a result, X P reduces to that in flat spacetime without a cosmic string X 0 and so does the concurrence.While, when the detectors are very close to the string, i.e., when l/σ ≪ 1, the correlation term can be approximated as Then using Eqs.( 7), ( 20) and ( 28), one finds, for the concurrence quantifying the entanglement harvested by the detectors, with representing the concurrence in the case of flat spacetime without a cosmic string.In particular, when the detectors are located on the string (l = 0), we have This means that the presence of the cosmic string amplifies the amount of entanglement harvested by the detectors in the vicinity, and the bigger the deficit-angle parameter ν, the larger the concurrence C P (ρ AB ).However, the analytical approximations of X P and C P (ρ AB ) are only obtainable in the asymptotic regions, i.e., when detectors are very close or very far away from the string.For general locations, numerical evaluations will be needed and be performed latter after an analytical analysis of the vertical alignment case which follows.

Vertical alignment
For the vertical alignment with both the detectors on the same side of the string [see Fig. ( )], the spacetime trajectories can be written as with l representing the distance to the string of the detector which is closer.It is easy to see that the transition probability for detector A is just given by Eq. ( 16), and P B can be obtained directly from Eq. ( 16) by replacing l with l + d.Similarly, the correlation term X, denoted now by X V for the vertical alignment, can be written in the form: with and Analogous to the correlation term in the parallel alignment, X V also approaches X 0 in the limit of l → ∞, and as a result, the concurrence becomes that in the case of flat spacetime without a cosmic string.Moreover, for small l/σ and d/σ, the correlation term X V agrees with Eq. ( 28), so that the concurrence can be approximately written as C V (ρ AB ) ≈ νC 0 (ρ AB ).
After presenting the analytical analysis for the asymptotic regions, we now begin our numerical analysis for the general locations of the detectors.Here, it is worth pointing out that when the energy gap of the detectors is much larger than the Heisenberg energy (Ωσ ≫ 1), both the transition probability and the correlation term are vanishingly small so that entanglement can hardly be harvested.So, we will only consider small energy gap in the numerical evaluations.
In Fig. (4), we demonstrate how the concurrence depends upon interdetector separation for fixed detector-to-string distance (i.e., fixed l/σ) and deficit angle (characterized by parameter ν).Obviously, the concurrence is a monotonically decreasing function of interdetector separation no matter how the two detectors are aligned with respect to the cosmic string.Interestingly, for small interdetector separation (d/σ ≪ 1), the two detectors aligned parallel to the string will harvest more entanglement than those in vertical alignment for a fixed small l/σ, while for not too small interdetector separation, the detectors aligned vertically to the string will instead harvest more entanglement.One may understand this property as follows.When both the interdetector separation and detector-to-string distance are small (d/σ ≪ 1 and l/σ ≪ 1), the transition probabilities of the detectors in these two alignments, given by Eq. ( 16), are almost the same.However, comparing Eq. ( 22) with Eq. (33), one can easily infer that the correlation term |X P | is larger than |X V | because the auxiliary function f (•) is an exponentially decreasing function of its argument.Therefore, one can see from Eq. ( 7) that C P (ρ AB ) is larger than C V (ρ AB ) for small d/σ.When interdetector separation is not too small (d/σ > 1 and d ≫ l), the correlation term |X P | becomes approximately equal to |X V |.However, the geometric mean of the detectors' transition probabilities in the case of the vertical alignment is much smaller than that in the parallel alignment.This is because the transition probability is a decreasing function of detector-to-string distance and the vertical alignment has a comparatively longer effective distance from the string than the parallel alignment.As a result, C V (ρ AB ) is greater than C P (ρ AB ) for not too small interdetector separation.
According to the afore-analysis, the harvested entanglement will approach to the result in flat spacetime without a cosmic string as the detector-to-string distance grows to infinity, which is analogous to the entanglement harvesting phenomenon for such detectors far way from the boundary in flat spacetime with a reflecting boundary.To gain a better under- standing on how entanglement harvesting depends upon the detector-to-string distance, we plot the concurrence as a function of detector-to-string distance for various ν in Fig. (5).In addition, the dependence of concurrence on detector-to-boundary distance in flat spacetime with a reflecting plane boundary is simultaneously depicted in these plots for comparison.
And to facilitate the comparison, we recall here the results for the case with a reflecting boundary in flat spacetime [23,27].The transition probability of the detector satisfies [see Eq. (3.4) in Ref. [23]] where l represents the distance between the detector and the boundary.Quite different from the transition probability (16) in the cosmic string spacetime, Eq. ( 36) does not take the maximum value but vanishes when the detector is located at the boundary (l = 0).
The correlation term X in the case of the parallel-to-boundary alignment satisfies, for an interdetector separation d, while in the case of the orthogonal-to-boundary (vertical-to-boundary) alignment it reads with l now being the distance to the boundary of the detector which is closer [see Ref. [27] for more details].Accordingly, the concurrence in flat spacetime with a boundary can be straightforwardly obtained by substituting Eqs.(36) ( 37) and ( 38) into Eq.( 7).In comparing Eqs.(37) (38) with Eqs. ( 22) (33), one may find that the contributions of the "images" to the correlation term in flat spacetime with a boundary are subtracted from X 0 rather than added in the cosmic string spacetime.Now let us discuss what conclusions we can draw from Fig. (5).First, the presence of a cosmic string may either assist or inhibit entanglement harvesting, depending on the detector-to-string distance.For a small detector-to-string distance (l/σ ≪ 1), the presence of the cosmic string will assist entanglement harvesting in both parallel and vertical alignments, which is in sharp contrast with the inhibiting role played by the presence of a reflecting plane boundary in entanglement harvesting in the vicinity of the boundary [23,27].However, for a sufficiently large detector-to-string distance, the harvested entanglement falls below the corresponding result in flat spacetime without a cosmic string, i.e., the presence of the cosmic string now plays an inhibiting role in entanglement harvesting, which is contrary to the assisting role played by the boundary in entanglement harvesting for sufficiently large detector-toboundary distance.Remarkably, the harvested entanglement with a cosmic string/boundary possesses a dip/peak at the certain large detector-to-string/detector-to-boundary distance.
Second, Fig. ( 5) also shows that the detectors in the vertical-to-string/parallel-to-boundary alignment still have opportunities to harvest relatively more entanglement than the parallelto-string/vertical-to-boundary ones if the detector-to-string/detector-to-boundary distance is sufficiently large.
In order to clearly reveal the influence of the deficit angle on entanglement harvesting, we further plot the concurrence as a function of the deficit-angle parameter in Fig. (6).Obviously, the dependence of the concurrence on the deficit-angle parameter is also significantly impacted by the detector-to-string distance.When the detector-to-string distance is small   with respect to the duration time (l/σ ≪ 1), the concurrence is a monotonically increasing function of ν, which is in accordance with our afore-analysis.However, when the detectorto-string distance is not too small, the concurrence initially decreases as ν increases, and then becomes a continuously increasing function of ν.In order to understand this property, we plot the behaviors of P D and |X 0 | versus ν in the case of the parallel alignment in Fig. (7).
As we can see, both the transition probability P D and the correlation term |X 0 | increase as ν grows, but the increasing degree is different from each other.At the beginning, P D grows more rapidly than |X 0 | as ν increases; however, when ν becomes large enough, P D ceases increasing more rapidly than |X 0 |.Recalling the expression of concurrence (7), one can conclude that the concurrence would initially decrease and then increase as ν increases when ν is large enough.
In what follows, we analyze how the presence of a cosmic string impacts the harvestingachievable range of interdetector separation.We introduce d max to denote the maximum harvesting-achievable separation, beyond which entanglement harvesting cannot occur any more, and plot it as a function of detector-to-string distance in Fig. (8).As we can see from Fig. (8) that the presence of a cosmic string can, in comparison with the case of a trivial flat spacetime, either reduce or enlarge the harvesting-achievable range, depending on the detectors' alignment and detector-to-string distance.Concretely, when the two de- tectors are aligned parallel to the string, the presence of the cosmic string always reduces the harvesting-achievable range, which is in sharp contrast with the fact that a reflecting boundary always enlarges the harvesting-achievable range [23]; however, when the detectors are aligned vertically to the string, the presence of the cosmic string would instead enlarge the harvesting-achievable range in the vicinity of the string, but would reduce the range as the detector-to-string distance becomes comparable to the duration parameter (l > σ).So, in the sense of the harvesting-achievable range, the cosmic string also plays a double-edged role in entanglement harvesting.

B.
Vertical alignment with two detectors on two different sides of the string For this vertical alignment [see Fig. (2c)], the trajectories of the detectors can be written as where l still represents the distance to the string of the detector which is closer and d denotes the interdetector separation.Notice that now d ⩾ 2l > 0. The transition probability P A is again Eq. ( 16), and P B can now be obtained by replacing l with d − l in Eq. ( 16).Similarly, the nonlocal correlation term X, denoted here by X T for the vertical intersecting alignment, reads with It is easy to find out that both X T 1 and X T 2 vanish in the limit of l → ∞, resulting in X T = X 0 .For small l/σ (i.e., 1 ≫ l/σ and d ≫ 2l), the correlation term then can be approximated as And, for small d/σ and integer ν, one may further have, by taking Eqs.(7) ( 20) and ( 43) In particular, when the two detectors are in symmetric alignment with respect to the cosmic string (i.e., d = 2l), we have Notice that when ν is an even integer, X T 2 vanishes and the last term (m = ν/2) in the summation in Eq. ( 44) reads which accordingly results in divergence of X T and thus of C T (ρ AB ) as well.Physically, this is a result of that there exists one "image" of detector B which is angularly identical In Fig. (10), we illustrate how the concurrence depends upon the deficit-angle parameter ν.In contrast to the results in the alignments with two detectors on the same side of the string, the concurrence now is no longer continuous function at points of integer ν, and the ν-dependence of concurrence is not qualitatively impacted by the concrete value of l/σ.
In particular, if the two detectors are aligned symmetrically to the string (d = 2l), the concurrence becomes divergent at any even integer ν due to the divergent correlation term X T caused by the spatial overlapping of the image and the detector that makes the pointlike detector model ill-defined [e.g., see plots (10a) and (10b)].Interestingly, the harvested entanglement overall will exhibit a degradation between every neighboring even integer pairs of ν [e.g., see plots (10a) and (10c)].Moreover, if ν is just an odd integer (or even integer in the asymmetrical alignment case), one can see that the bigger the value of ν, the more the entanglement harvested.The presence of the cosmic string indeed assists the entanglement harvesting.
In addition, we plot d max /σ as a function of l/σ in Fig. (11), which shows that for not too large detector-to-string distance l1 , the harvesting-achievable range of interdetector separation with the detectors on two different sides of the string in the cosmic string spacetime is always larger than that in a trivial flat spacetime.For this alignment, the role played by the cosmic string on the harvesting-achievable range is similar to that by the reflecting boundary.

IV. CONCLUSION
We have performed, in the framework of the entanglement harvesting protocol, a detailed study of the entanglement harvesting phenomenon of two UDW detectors locally interacting with a massless scalar field near a cosmic string.Specifically, three alignments of the detectors with respect to the string, i.e, the interdetector separation is parallel to the string, and vertical to the string with two detectors on the same side and two detectors respectively on two sides, are studied.We find that the presence of the cosmic string in general enhances the transition probability of the static detectors for a finite interaction duration.For the alignments on the same side of the string, we find the amount of entanglement harvested is always a monotonically decreasing function of interdetector separation regardless of whether the two detectors are aligned parallel or vertically to the string.Meanwhile, the harvested entanglement does not always decrease as the detector-to-string distance grows, but displays a dip that falls below the corresponding result in flat spacetime without a cosmic string at a certain large detector-to-string distance.This is quite different from that in the case with the presence of a reflecting boundary where the harvested entanglement will show a peak as the detector-to-boundary distance grows to be comparable to the duration time parameter.In other words, the presence of a cosmic string may assist entanglement harvesting in the vicinity of the string, but would inhibit it in the far zone, which is contrary to that the boundary inhibits entanglement harvesting in the vicinity of the boundary, but assists it in the far zone.
Interestingly, when the detector-to-string distance is small with respect to the interaction duration parameter, the larger the deficit angle, the more the entanglement harvested; however, if the detector-to-string distance is not too small, the entanglement harvested may initially appear to decrease slightly as the defect angle increases, and later becomes a monotonically increasing function as the deficit angle further increases.Therefore, one may conclude that the amount of entanglement harvested is generally amplified because of the deficit angle regardless of whether the two detectors are aligned parallel or vertically to the string, when the deficit angle is large enough.Notably, in the vicinity of the string, the detectors in parallel alignment with a small interdetector separation, can harvest more entanglement than those in vertical alignment.However, the detectors in vertical alignment can instead harvest comparatively more entanglement when the interdetector separation or the detector-to-string distance becomes sufficiently large.
As far as the harvesting-achievable range of interdetector separation is concerned, when the two detectors are aligned parallel to the cosmic string, the presence of the string always reduces the harvesting-achievable range in comparison with the result in a trivial flat spacetime; however, when the detectors are aligned vertically to the string, the harvestingachievable range would be enlarged in the vicinity of the string, but reduced in the far zone.Therefore, the presence of the cosmic string can either assist or inhibit entanglement harvesting in the sense of harvesting-achievable range, which is quite different from the fact that a reflecting boundary always enlarges the harvesting-achievable range.
Regarding the vertical alignment with two detectors on two different sides of the string, the amount of entanglement harvested always decreases as the detector-to-string distance grows due to the positive relation between the detector-to-string distance and the interdetector separation.Remarkably, different from the entanglement harvesting for the two detectors on the same side of the string, the detectors on two different sides always harvest more entanglement than those in a trivial flat spacetime, and the presence of the cosmic string enlarges the harvesting-achievable range in the vicinity of the string.Another interesting feature is that the harvested entanglement now is not a continuous function of the deficit-angle parameter ν, which is quite different from the case with the two detectors on the same side.In particular, when the two detectors are aligned symmetrically to the string (d = 2l), the harvested entanglement diverges at an even integer ν due to the spatial overlapping of the image and the detector as a result of the conical topology of the cosmic string spacetime.
Finally, it is worth noting that the nontrivial topologies due to both the presence of a cosmic string and a reflecting boundary in locally flat spacetime, although all impact the entanglement harvesting phenomenon significantly, show distinctive influences in the fact that the cosmic string usually assists, in the sense of the amount of entanglement harvested, the entanglement harvesting for two detectors on the same side in the vicinity and inhibits it in far zone while the boundary inhibits the entanglement harvesting in the vicinity and assists it in the far zone.However, from the perspective of the harvestingachievable range, the cosmic string plays a double-edged role, in contrast to the fact that a reflecting boundary always plays a single role of enlarging the harvesting-achievable range.
These sharp contrasting properties may provide a potential way to distinguish locally flat spacetimes with different topologies due to cosmic strings and reflecting boundaries.
based upon the requirement of the principal value of the logarithmic function.From the l-dependence in P 1 and P 2 , one can also infer that the transition probability is a monotonically decreasing function of the detector-to-string distance l, with the maximum value attained at l = 0 for a fixed ν.In order to clearly show how the transition probability depends on the detector-to-string distance and the deficit angle, we resort to numerical calculation with the results shown in Fig.(3).Obviously, the transition probability in the cosmic string spacetime generally decreases as the detector-to-string distance grows, ultimately converging to the corresponding result in flat spacetime without a cosmic string when the detector-to-string distance approaches to infinity [see Fig.(3a)].Moreover, the transition probability indeed attains its maximum value when the detector is located on the string, which is in accordance with our afore analytical analysis.It is also easy to see that the transition probability is an increasing function of the deficit-angle parameter ν, i.e., the deficit angle enhances the transition probability [see Fig.(3b)].

11 FIG. 4 :
FIG. 4: The concurrence versus d/σ for ν = 2 in plot (a) and ν = 11 in plot (b) with Ωσ = 0.10 and l/σ = 0.10.Here, the black dashed lines in all plots correspond to the results in flat spacetime without a cosmic string.

11 FIG. 5 :
FIG. 5: The concurrence is plotted as a function of l/σ for ν = 2 in plot (a) and ν = 11 in plot (b) with fixed Ωσ = 0.10 and d/σ = 0.50.The red dashed and green dashed lines respectively describe the corresponding cases of parallel and vertical alignments in the presence of a reflecting boundary.The black dashed lines indicate the corresponding results in flat spacetime without any cosmic strings and boundaries.

FIG. 7 :
FIG. 7: The correlation term |X P | and the transition probability P D are plotted as a function of parameter ν for l/σ = 2.00, Ωσ = 0.10 and d/σ = 0.10.The vertical dashed line (ν ≈ 5.867) indicates where the minimum difference between |X P | and P D occurs.

6 FIG. 8 :
FIG. 8: The maximum harvesting-achievable separation between detectors, d max /σ, is plotted as a function of l/σ for the alignments with two detectors on the same side of the string with ν = 3 and Ωσ = 0.10.Here, the red dashed and green dashed lines respectively describe the corresponding cases of parallel and vertical alignments in the presence of a reflecting boundary.The black dashed line indicates the result in flat spacetime without any strings or boundaries (trivial flat spacetime).
with detector A due to the conical topology of the cosmic string spacetime and consequently detector A and the "image" of detector B spatially overlap.As mentioned before, a finite-size detector model with a spatial smearing function is needed to resolve this issue of divergence, details of which are however not relevant to our main interests of this paper.In order to demonstrate the properties of entanglement harvesting clearly, in what follows we will plot the concurrence as a function of the detector-to-string distance and the deficit-angle parameter in Figs.(9) and(10), respectively.As shown in Fig.(9), the concurrence is a monotonically decreasing function of the detector-to-string distance.This is because the interdetector separation is positively related to the detector-to-string distance in the vertical alignment with two detectors on two different sides of the string and the concurrence usually is a monotonically decreasing function of the interdetector separation.Quite different from the entanglement harvesting in the alignments with detectors on the same side of the string, one can find that the detectors aligned on two different sides of the string always harvest more entanglement than those in flat spacetime without a cosmic string.

FIG. 9 :
FIG. 9: The concurrence is plotted as a function of l/σ with Ωσ = 0.10 for different interdetector separations d/l = {2.00,2.50} in left-to-right order.The dashed lines represent the results in flat spacetime without a cosmic string (i.e., ν = 1).
FIG. 10: The concurrence is plotted as a function of ν for detectors in symmetrical alignment with l/σ = 0.10 in plot (a) and l/σ = 1.00 in plot (b), and for detectors in non-symmetrical alignment with d/l = 2.50, l/σ = 0.10 in plot (c) and d/l = 2.50, l/σ = 1.00 in plot (d).Here, we set Ωσ = 0.10 for all plots.The circle point exactly indicates the corresponding value of concurrence at a certain integer ν.

4 FIG. 11 :
FIG.11:The maximum harvesting-achievable interdetector separation versus the detector-to-string distance for the alignment with two detectors on two different sides of the string.Here, we set ν = 3 and Ωσ = 0.10.The black dashed line represents the corresponding result in flat spacetime without a cosmic string.There is a certain value of the detector-to-string distance, i.e., l = l 0 ≈ 2.219σ, beyond which the detectors cannot harvest entanglement any more due to the positive relation between the interdetector separation and the detector-to-string distance (d ≥ 2l).