Birth and death of entanglement between two accelerating Unruh-DeWitt detectors coupled with a scalar field

We study two accelerating Unruh-DeWitt detectors coupled linearly or quadratically with a scalar field, by using an approach different from previous ones. We use Rindler frame, and integrate over the spacetime coordinates before over the momenta. In this way, the divergent problem previously encountered is avoided. We show that entanglement between the detectors can be created by coupling with the field, and is divergent only in the limit that the two detectors coincide in spacetime. For linear coupling, entanglement increases monotonically with the decrease of acceleration, as previously found, while for quadratic coupling, entanglement behaves non-monotonically, as noted for the first time.


Introduction
Quantum field theory in curved spacetime, in which the curvedness of the spacetime is considered but the gravity is not quantized [1], has made many remarkable predictions, such as the Unruh effect [2] and Hawking radiation [3].An approach to studying these phenomena is to consider a particle detector, which provides a good illustration of the notion of particle in quantum field theory.As the simplest model, the Unruh-DeWitt detector [4,5] describes a two-level system coupled linearly with a scalar field, similar to an atomic dipole coupled with an electromagnetic field [6].It is natural to extend the consideration to other kinds of interaction, for instance, the Φ2 coupling, the coupling with a Dirac field [7], etc.However, these interactions run into severe divergence problems in their correlation features [8], which reveals the entanglement of the quantum fields.
The Reeh-Schlieder theorem [9] shows that in quantum field theory, all field variables in any spacetime region are entangled with those in the other regions, thus the vacuum state is entangled, as can be explicitly described by the thermofield double [10], which is closely related to the Unruh effect.Given that the field is entangled, people are interested in the so-called entanglement harvesting [11], i.e. swapping the correlation in the quantum field to that between particle detectors.There are a few different scenarios, in which two detectors may be static, uniformly accelerating, or freely moving in a gravitational field.Because of the divergences mentioned above, however, few studies consider particle detector models with nonlinear coupling with the quantum fields.
In this paper we focus on the entanglement harvesting of two uniformly accelerating detectors, both of which coupled quadratically with a scalar field.Other than working in the inertial frame, as in most of the previous studies, we tackle the problem from the Rindler observers' perspective, which is more natural for our purpose, and the divergences appear only when these two detectors conincide in spacetime.Furthermore, for Φ 2 coupling, there appears a striking feature that the entanglement does not decrease monotonically with the acceleration, in contrast to the usual situation that the acceleration enhances entanglement harvesting.This feature was noticed only recently [12], and is quite generic in our case, although for large enough acceleration, the entanglement decreases in the end till the sudden death [13].We will also discuss how the entanglement depends on the distance between the detectors.
This paper is organized as the following.In section 2 we will give a brief introduction to the Rindler modes, which are crucial in our calculation, and we will derive the reduced density matrix of our model which determines the entanglement properties.In section 3 and 4 we will study detectors that are linearly and quadratically coupled with the scalar field respectively.It can be seen that in these two different circumstances, the entanglement exhibits different features.In section 5, we will summarize our results and give an outlook.

Set up
Before we introduce the model, it is worthwhile to briefly review quantum field theory in the Rindler wedge since we will be working in the Rindler perspective.
The right Rindler coordinate (τ, ξ, x, y) is related to the Minkowski coordinate (t, x, y, z) as t = a −1 e aξ sinh aτ, z = a −1 e aξ cosh aτ, while (x, y) ≡ x ⊥ is the same as in the Rindler coordinate.A massive scalar field can be expanded in the right Rindler wedge as [14] Φ where the lower bound of the integration over ω should be 0, with κ ≡ k 2 ⊥ + m 2 and K ν (x) is the modified Bessel function.Since the annihilation operator for a certain mode âR ωk ⊥ in the Rindler wedge is different from that in the Minkowski spacetime, the vacuum state of the scalar field is different in the two coordinate systems.
The Minkowski vacuum can be expressed in the right Rindler wedge as [14] where C ω = 1 − exp(−2πω/a), the product over ω and k ⊥ should be understood as the limit of the discrete case.The mathematically rigorous formulation is based on the S matrix between the two Fock spaces [1].Now we consider two Unruh-DeWitt detectors which accelerate in +z direction with the same acceleration a, but are separated in the x direction with distance x 0 .The Hamiltonian in the interaction picture is where, with j = A, B, where λ j is the coupling constant, χ j is the switch function, O is the field operator, which is Φ or Φ 2 , f (x) represents the shape of the detector; σ + j and σ − j are the raising and lowering operators of the two-level system, Ω j is the energy gap of the system, g is the determinant of the metric tensor.Suppose the initial state of the system is |ψ 0 = |G |G |0 M , where |G represents the ground state of each detector.The evolution operator is where T is the time-ordering operator.According to the perturbation theory, it can be written as The following steps are straigtfoward.We obtain the final state and subsequently the reduced density matrix of the detectors by tracing out the field state.Here we adopt the notations in [8] and the form of the reduced density matrix is where with j = A, B, is just the transition probability for A and B detectors respectively, (2.12) Our task now is to calculate these matrix elements.We shall start with linear coupling as an illustration of the method, then we switch to quadratic coupling.Our method is to express the vacuum in terms of the states in the Fock space of Rindler modes.This means that we work in the reference frame of the accelerating observers.We will see that with this method, divergence of the elements appears only when these two detectors concide in spacetime.

Linear coupling
We start with the generic case and then focus on the special scenario in which the two detectors are identical.The main purpose is to calculate the Wightman function.Normally one expands the field operator in Minkowski spacetime and integrates over the momentum variables to obtain the Wightman function as a function of the spacetime coordinates.
Then one could substitute the detectors' trajectories into the coordinates, over which one integrates in the end to obtain the matrix elements.
Here we use a different method.By using the field expansion (2.2) and the expression of the Minkowski vaccum (2.4) in terms of Rindler modes, we obtain Note that our detectors reside in the same Rindler wedge.This result can also be seen directly from the expectation value of the Rindler number operator on the Minkowski vacuum.
Subsequently we do not integrate in the momentum space to obtain an explicit form of the Wightman function.It should also be pointed out that expanding the field with Rindler modes gives the same Wightman function as Minkowski modes [14].Instead, To calculate the matrix elements we integrate over the spacetime coordinates, and thus express the elements as integrals over momentum variables.In some sense this method is more natural since we write quantities like transition probabilities as the sum of all different modes.The details are as follows.

Calculation of P
We can substitute the above equation into the transition probability and integrate over the spacetime variables, obtaining first where where dτ χ(τ )e iντ is the Fourier transformation of the switch function.
We are specifically interested in the case where the scalar field is massless and the detectors are point-like ones with Gaussian switch functions, which means Then the Fourier transformation takes the form Consequently, Substitute the above equations into (3.2),we obtain The details are given in the appendix.

Calculation of L AB
This is quite similar to the calculation of P , as can be easily seen from their expressions.We substitute the correlation function (3.1) into the expression (2.11) of L AB , obtaining where and (3.12) Now we assume that these two detectors seperate in x diretion with x 0 and there is a time delay τ 0 between their switching functions, that is, Then we have (details are given in the appendix) . (3.15)

Calculation of M
This is where things can get quite messy.We first calculate the time-ordered Minkowski vacuum expectation value in which θ(τ ) is the standard Heaviside function.Substituting the above equation into (2.12),we obtain with where ǫ is a positive infinitesimal real number.Similarly we can obtain To further simplify the expressions, we assume Ω A = Ω B = Ω, σ A = σ B = σ and τ 0 = 0. Then we obtain The details are given in the Appendix.The final results are Substitute these into M , and we obtain a . (3.24) where in the equality we've taken the field to be massless.
Here is where the divergence may occur.If and only if two detectors concide in spacetime, i.e. x 0 = 0, M is divergent: (3.25)

.28)
The reduced density matrix has already been given in Sec. 2, The partial transpose of it with respect to qubit A is with the four eigenvalues We obtained the numerical result as in Fig. 1.It can be seen that the negativity decreases with the acceleration and comes to zero at a certain point.In addition, the entanglement decreases with the increase of the distance between the detectors, as anticipated.

Quadratic coupling
Now we consider two detectors which are quadratically coupled with a scalar field.The Hamiltonian in the interaction picture is where j=A,B.Here we take the field operator to be Wick ordered to avoid potential divergences.
There are two equivalent ways to realize Wick ordering, one being putting the creation operators to the left of the annihilation operators, the other one being taking the field to be the original one minus its vacuum expectation value.Both ways may cause ambiguity since the notion of mode expansion and the vacuum state is not uniquely defined in quantum field theory in curved spacetime.In this paper, we use the latter way.In this way, the Wightman function for the quadratic coupling is related to the that for the linear coupling, which has been calculated in the last section.
The reduced density matrix shares the same structure with the case of Φ coupling, where the matrix elements are It should be clear now that we are to calculate the vacuum expectation value V Φ 2 .It is closely related to the Wightman function, and we use (3.1) to obtain Now it is a good time to make a digression to say a few words about the general structure of the interaction between detectors and quantum fields.The structure is well studied in [15], which gives the Feynman rules for the amplitude.In our case some clues of the Feynman rules can be observed.For example, for the quadratic coupling there are two vertices that give rise to double integrals over the momentum variables.Besides, in our case, when the detectors start unentangled, the correlation term M represents a field propagator.In genric cases (for instance the detectors star entangled), however, there are terms that represent detector propagators .T e −iΩ(τ 1 −τ 2 ) .We wish to have a deeper insight to such terms in future works.
We follow the same procedure as in section 3 to calculate the matrix elements.The approach is the same but the calculation could be tedious so we leave the details in the Appendix and only describe the main steps and results.

Calculation of P j and L AB
Let's start with P j and L AB , since they are quite similar, It is straightforward to write down the form of the η's.For instance, (4.10) The final expression for the parameter values m = 0, σ, τ 0 = 0, x 0 = 0 will be given later along with M .

Calculation of M
Now we calculate M , which was divergent in previous calculations by other other authors.We need to calculate the time-ordered vacuum expectation value Substituting this expression into M , we obtain These terms looks complicated but can be calculated straightforwardly.Here we take W 0 ω 1 k 1⊥ ,ω 2 k 2⊥ as an example.
As before, keep in mind that in the end we will integrate over k 1⊥ and k 2⊥ .The integral over u can be calculated by using the same technique as in the linear case, The other W 's can be treated the similar way.Substituting the W s into M and taking the field to be massless, one obtains As in the linear case, M is divergent when x 0 = 0: )

.17)
As before, the negativity for the reduced density matrix of the detectors is and the numerical result is depicted in  It can be seen that now the negativity can be enhanced by the acceleration, though it still vanishes when the acceleration is large enough.The entanglement also decreases with the increase of the distance.

Conclusion
We have studied two Unruh-DeWitt detectors coupled with a scalar field linearly or quadratically.Other than working in the inertial frame, as people usually do, we use Rindler coordinates, as the Hamiltonian itself depends on the proper time of the detectors.In our calculations, the divergence people usually encounter in calculating correlations only appears when the detectors coincide in spacetime.Moreover, the dependence of the entanglement on the acceleration is more complex in the quadratic-coupling case, and the dependence is not monotonic.In all cases studied, the entanglement decreases with the increase of the distance between the detectors, in consistency with our intuition.
To deal with the Wightman function, which is divergent at certain points in spacetime, our approach is different from previous ones, and is based on the field expansion in the Rindler coordinates.Moreover,in calculating the matrix elements of the reduced density matrix, rather than making integration over the momentum variables immediately, we first integrate over the spacetime coordinates.This may be the reason why we do not encounter divergence except when the detectors coincide in spacetime.
Given the equivalence between the Rindler spacetime and the Schwarzchild spacetime near the horizon, our results can be applied to the case of black hole.We hope that our method can extended to more general cases.In our model, it is supposed that the two detectors reside in the same Rindler wedge to avoid ambiguities.Moreover, it is quite promising that we could apply our method to the case of Dirac interaction.Further extension can be made to consider the case that the detectors are entangled initially.We list the ηs as follows.

Figure 1 .
Figure 1.The graph for the negativity of the two detectors linearly coupled with the scalar field versus the acceleration (σ = 1, Ω = 1)

Figure 2 .
Figure 2. Negativity of the two detectors quadratically coupled with the scalar field, as a function of the acceleration (σ = 1, τ = 1)