Harvesting Correlations from Thermal and Squeezed Coherent States

Harvesting correlations from thermal and squeezed coherent states

Petar Simidzija1, 2, ∗ and Eduardo Mart´ın-Mart´ınez1, 2, 3, † 1Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 2Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada We study the harvesting of entanglement and mutual information by Unruh-DeWitt particle detectors from thermal and squeezed coherent field states. We prove (for arbitrary spatial dimensions, switching profiles and detector smearings) that while the entanglement harvesting ability of detectors decreases monotonically with the field temperature T , harvested mutual information grows linearly with T . We also show that entanglement harvesting from a general squeezed coherent state is independent of the coherent amplitude, but depends strongly on the squeezing amplitude. More- over, we find that highly squeezed states i) allow for detectors to harvest much more entanglement than from the vacuum, and ii) ensure that the entanglement harvested does not decay with their spatial separation. Finally we analyze the spatial inhomogeneity of squeezed states and its influence on harvesting, and investigate how much entanglement one can actually extract from squeezed states when the squeezing is bandlimited.

I. INTRODUCTION while most of these entanglement harvesting studies have focused on conventional linear Unruh-DeWitt (UDW) The entanglement structure of a quantum field has particle detectors [31] coupled to real scalar fields [22], been an important area of research over the last few there have also been several interesting results coming decades. Besides being an interesting focus of study in from other variations of the setup. Some examples in- its own right, the presence of entanglement between local clude: hydrogenoid atomic detectors coupled to the full degrees of freedom in general field states (and in particu- electromagnetic field [24], non-linear couplings of UDW lar the vacuum [1, 2]) has been used as a means to better detectors to neutral [27] and charged [29] scalar fields, tri- understand important fundamental questions, from the partite entanglement in flat spacetime [21], and multiple black hole information loss problem [3–8], to the dynam- detector harvesting in curved spacetimes [26]. Entan- ics of quantum phase transitions in statistical mechan- glement harvesting using infinite dimensional harmonic ics [9, 10]. Moreover, operational approaches which har- oscillator detectors has been looked at in several works ness this entanglement to perform useful tasks have also as well. An example which is very relevant to this paper been studied, leading to, for example, the development is an article by Brown where the issue of harvesting from of protocols for quantum energy teleportation [11–13]. thermal states is considered [18]. Another widely studied protocol making use of the en- While some of the above mentioned parameters affect- tanglement present in a quantum field is concerned with ing entanglement harvesting are difficult to control in a the extraction of field entanglement onto a pair of initially lab setting (such as the geometry and topology of space- uncorrelated first-quantized systems (detectors). These time), other parameters, such as the energy gap of the so called entanglement harvesting protocols were initially detectors or the state of the field, are more easily tunable. studied in the 90s by Valentini [14], then later by Reznik A major motivation for studying the sensitivity of entan- et al. [15, 16], and have in the last decade or so experi- glement harvesting to these types of parameters is that enced a great deal of attention from many different per- it may lead to experimental realizations of entanglement spectives [17–30]. harvesting protocols. This would not only be an impor- Many of these recent lines of research into entangle- tant achievement from a fundamental perspective, but it ment harvesting are related to the fact that the amount could also potentially be a method of obtaining entanglement that could then be used for quantum information arXiv:1809.05547v1 [quant-ph] 14 Sep 2018 of harvestable entanglement is generally sensitive to the many variable parameters of the setup. For instance, purposes [19]. the sensitivity of entanglement harvesting on the position With this ultimate motivation in mind, it has been and motion of the detectors has resulted in harvesting- shown that a non-zero detector energy gap is crucial based proposals in metrology — from rangefiding [23] to in protecting an entanglement harvesting UDW pair precise vibration detection [20] — while, on the more against fluctuation induced, entanglement harming, lo- fundamental side, it has also been shown that entan- cal noise [32, 33]. Furthermore, for harmonic oscil- glement harvesting is sensitive to the geometry [17] and lator detectors, this noise has been found to increase topology [25] of the background spacetime. Furthermore, with field temperature, leading to detrimental effects on the amount of entanglement harvested [18] by oscillator pairs. Meanwhile, and perhaps surprisingly, for UDW detectors interacting with coherent states of the field, the ∗ [email protected] presence of leading order local noise does not end up af- † [email protected] fecting the amount of entanglement that can be harvested 2 from the field [34, 35]. the amount obtainable from the vacuum. We will also an- In this paper, we fill in significant gaps in the study of alyze whether this advantage carries over to more exper- entanglement harvesting sensitivity on thermal and gen- imentally attainable field configurations where states are eral squeezed coherent field states. While, to our knowl- squeezed across a narrow frequency range of field modes. edge, this is the first study of squeezed state entangle- This paper is structured as follows: We begin in Sec. II ment harvesting, we would also like to point out that our by reviewing the setup of entanglement harvesting by study of thermal state harvesting differs in several crucial UDW detectors from arbitrary states of a scalar field. regards to the previous work in [18]. In [18] it was shown In Sec. III we particularize to the case of thermal field that for a pair of pointlike oscillator detectors interact- states, and study the harvesting of entanglement and mu- ing with a massless field in a one-dimensional cavity, the tual information in this setting. Then, in Sec. IV we look amount of entanglement extracted decays rapidly with at entanglement harvesting from squeezed field states, the temperature. In contrast, i) we consider spatially both those with uniform and bandlimited squeezing am- smeared qubit detectors interacting with a field of any plitudes. Finally, Sec. V is left for the conclusions. Units mass in a spacetime of any dimensionality, rather than of ~ = c = kb = 1 are used throughout. pointlike oscillator detectors interacting with a massless field in (1+1)-dimensions, ii) we look at the continuum free space case rather than being in a cavity, and hence II. CORRELATION HARVESTING SETUP we are not forced to introduce any UV cutoffs to handle numerical sums, and iii) we directly compute the evolved Before studying the harvesting of correlations from detectors’ density matrix from the field’s one and two- thermal and squeezed coherent field states, let us re- point functions, rather than using the significantly dif- view the general correlation harvesting setup that can be ferent formalism of Gaussian quantum mechanics (see, found in extensive literature (see, e.g. [38] and references e.g. [36]). therein) and that is applicable to any field state. We start Despite these significant differences between our ap- with a free Klein-Gordon field φˆ in (n + 1)-dimensional proach and that in [18], we will find that, for thermal Minkowski spacetime, which can be expressed in a basis states, our results are in qualitative agreement with their of plane wave modes as general conclusions, i.e. that temperature is detrimental Z dnk h i φˆ(x, t) = aˆ† ei(ωkt−k·x) + H.c. , (1) to entanglement harvesting. However, since we obtain p n k analytical expressions for entanglement measures, rather 2(2π) ωk than being restricted to numerical calculations, we are p 2 2 where ωk := |k| + m , and the creation and annihila- able to provide an explicit proof that the amount of en- tion operators,a ˆ† anda ˆ , satisfy the canonical commu- tanglement that (qubit) detectors can harvest from the k k tation relations field rapidly decays with its temperature. In particular, † † † (n) 0 we will show that the optimal thermal state for harvest- 0 [âk, aˆk ] = [âk, aˆk0 ] = 0, [âk, aˆk0 ] = δ (k − k ). (2) ing entanglement from the field is the vacuum. On the other hand, we will see that this is not the case for the We denote by |0i the ground state of the field, by which harvesting of mutual information, which is a measure of we mean the state annihilated by all thea ˆk operators. the total (quantum and classical) correlations of the de- For now, let us suppose that the field is in an arbitrary tector pair. In fact we will see that for high field tem- (potentially mixed) stateρ ˆφ. We will later particularize peratures T (while still in the perturbative regime) the to the case of thermal and squeezed coherent states. mutual information harvested by the detectors increases Next we consider the pair of first-quantized particle de- proportionally with T . tectors that couple to the field with the aim of extracting (i.e. harvesting) entanglement. We will model the detec- We will then consider the case of squeezed coherent tors (labeled ν ∈ {A, B}) as two-level quantum systems, states [37], where, to the authors’ knowledge, no previous with ground states |g i, excited states |g i, and proper literature exists. We will first prove that the statement ν ν energy gaps Ω . We assume that the detectors are at rest “entanglement harvesting is independent of the field’s co- ν at positions x , that they have spatial profiles given by herent amplitude” is true not only for non-squeezed co- ν the smearing functions F (x), and that they are initially herent states, as was shown in [34], but also for arbitrarily ν (i.e. prior to interacting with the field) in the separable squeezed coherent states. On the other hand we will show stateρ ˆ ⊗ ρˆ . Then, we describe the interaction of the that, unlike the coherent amplitude, the choice of field’s a b detectors and the field using the Unruh-DeWitt (UDW) squeezing amplitude ζ(k) does in fact affect the ability model [31], which is a successful model of the light-matter of UDW detectors to become entangled, and moreover interaction when angular momentum exchange can be the Fourier transform of ζ(k) directly gives the locations neglected [24, 39]. In this model the coupling of detec- in space near which entanglement harvesting is optimal. tors to field is given by the interaction picture interaction Perhaps surprisingly, we will also find that for highly and Hamiltonian, Hˆ (t) = Hˆ (t) + Hˆ (t), where uniformly squeezed field states, the amount of entangle- i i,a i,b Z ment that the detectors can harvest is independent of ˆ n ˆ their spatial separation, and is often much higher than Hi,ν (t) := λν χν (t)ˆµν (t) d x Fν (x − xν )φ(x, t). (3) 3

(1) (2) Here, λν is the coupling strength of detector ν to the ward to show thatρ âb andρ âb take the forms field, χ (t) is the time-dependent switching function ν Z ∞ which models the duration of the interaction and how (1) X (0) ρâb = i λν dtχν (t)[ˆρab , µˆν (t)]V (xν , t), (12) the detector ν is turned on and off, and theµ ˆν (t) are ν∈{A,B} −∞ operators on the two detector Hilbert space given by " Z ∞ Z ∞ µâ(t) :=m ˆ a(t) ⊗ 1b, andµ ˆb(t) := 1a ⊗ mˆ b(t), where (2) X 0 0 ρâb = λν λη dt dt χν (t )χη(t) mˆ ν (t) is the interaction picture monopole moment of de- −∞ −∞ tector ν: ν,η∈{A,B} 0 (0) 0 × µˆν (t )ˆρab µˆη(t)W (xη, t, xν , t ) mˆ (t) = |e ihg |eiΩν t + |g ihe |e−iΩν t. (4) ν ν ν ν ν Z ∞ Z t 0 0 − dt dt χν (t)χη(t ) −∞ −∞ To determine how entangled (if at all) the detectors are 0 (0) 0 × µˆν (t)ˆµη(t )ˆρ W (xν , t, xη, t ) following their interactions with the field, we calculate ab the time-evolved two-detector stateρ ˆ as ab Z ∞ Z t 0 0 − dt dt χν (t)χη(t ) −∞ −∞ h ˆ ˆ †i ρâb := Trφ U ρâ ⊗ ρˆb ⊗ ρˆφ U , (5) # (0) 0 0 × ρâb µˆη(t )ˆµν (t)W (xη, t , xν , t) . (13) where the time-evolution unitary Uˆ is formally given by 0 Here, V (xν , t) and W (xη, t, xν , t ) are given by Z " Z ∞ # n ˆ ˆ V (xν , t) := d x Fν (x − xν )v(x, t), (14) U = T exp −i dt Hi(t) , (6) −∞ Z Z 0 n n 0 0 W (xη, t, xν , t ) := d x d x Fη(x − xη)Fν (x − xν ) with T denoting the time-ordering operation. By as- × w(x, t, x0, t0), (15) suming that the detector-field coupling constants λν — which have units of (length)(n−3)/2 in (n+1)-dimensional while the one- and two-point correlation functions, v(x, t) 0 0 spacetime — are small compared to other scales with the and w(x, t, x , t ), of the field in the stateρ ˆφ, are defined same units in the setup, we can expand Uˆ in powers of as λν , obtaining h ˆ i v(x, t) := Trφ φ(x, t)ˆρφ , (16)

0 0 h ˆ ˆ 0 0 i Z ∞ Z ∞ Z t w(x, t, x , t ) := Trφ φ(x, t)φ(x , t )ˆρφ . (17) ˆ 1 ˆ 0 ˆ ˆ 0 3 U = −i dtHi(t) − dt dt Hi(t)Hi,a(t ) +O(λν ). −∞ −∞ −∞ | {z } | {z } After computing the evolved two-detector stateρ âb us- Uˆ (1) Uˆ (2) ing Eq. (8), we can use it to compute the amount of (7) correlations present between the detectors A and B fol- Then, the final two-detector stateρ âb in Eq. (5) can be lowing their interactions with the field. In this paper we perturbatively expressed as will focus on two types of correlations: entanglement and mutual information.

(0) (1) (2) 3 More precisely, we will quantify the entanglement that ρâb =ρ âb +ρ âb +ρ âb + O(λν ), (8) the detectors A and B harvest from the field by computing the negativity N , which, for a stateρ âb on the where Hilbert space Ha ⊗ Hb, is defined as [9] X ta N [ˆρab] := max 0, −Eab,i , (18) (0) ρâb :=ρ â ⊗ ρˆb ⊗ ρˆφ, (9) i (1) ˆ (1) ˆ (1)† ta ρâb := Trφ U ρˆ0 +ρ ˆ0U , (10) where the Eab,i are the eigenvalues of the partially transposed matrixρ ˆta . It is well known that the negativ- (2) ˆ (2) ˆ (1) ˆ (1)† ˆ (2)† ab ρâb := Trφ U ρˆ0 + U ρˆ0U +ρ ˆ0U . (11) ity of a two-qubit system is an entanglement monotone that vanishes if and only if the two-qubit state is separable [40, 41]. Hence the negativity is often used as a By using the definitions of Uˆ (1) and Uˆ (2) in Eq. (7) and measure of entanglement in harvesting scenarios, and it ˆ the expression for Hi given by Eq. (3), it is straightfor- is the measure that we will use. 4

It is also possible for Alice and Bob to be classically To see this concretely, from the definition (20) of correlated via their interactions with the field. We will ρˆβ and the canonical commutation relations (CCRs) in quantify the total amount of correlations (quantum and Eq. (2), we can straightforwardly calculate the one- and classical) between them by computing the mutual infor- two-point correlation functions defined in (16) and (17). mation, I, which is defined as Because the field is composed of a linear superposition † ofa ˆk anda ˆk operators, we first compute the following I[ˆρab] := S[ˆρa] + S[ˆρb] − S[ˆρab], (19) useful expression: 1 where S[ˆρ] := −Tr(ˆρ logρ ˆ) is the von Neumann entropy −βHˆφ Trφ ρˆβ aˆk = Trφ e aˆk (22) of the stateρ ˆ, whileρ â := Trb(ˆρab) andρ ˆb := Tra(ˆρab) Z are the reduced states of detectors A and B following the 1 ˆ ˆ ˆ = Tr e−βHφ aˆ eβHφ e−βHφ detector-field interactions. In particular, if entanglement Z φ k βω is zero and the mutual information is not, the correlations e k ˆ = Tr aˆ e−βHφ have to be either classical correlations or discord [42, 43]. Z φ k βωk = e Trφ ρˆβ aˆk , III. THERMAL FIELD STATE where in the third line we made use of the identity −βHˆφ βHˆφ βωk Let us suppose now that the two Unruh-DeWitt detec- e aˆke = e aˆk, which can be easily proved using the Zassenhaus formula and the CCRs. Then, com- tors are initially in their ground states,ρ ˆν = |gν ihgν |, and that the field is in a thermal stateρ ˆ of inverse temper- paring the first and last lines of Eq. (22), we conclude that β † ature β. It will be sufficient for our purposes to formally Trφ ρˆβ aˆk = 0. Hence Trφ(ˆρβ aˆk) = 0, and therefore the defineρ ˆβ as a Gibbs state in the usual way. Namely we one-point function v(x, t) = 0. Then, from Eqs. (12) and write (1) (14), we conclude that the first order contributionρ âb to ˆ ρâb is identically zero for a thermal field state. exp(−βHφ) 0 0 ρˆβ := , (20) To calculate the two-point function w(x, t, x , t ) we Z first compute: ˆ where Z := Tr[exp(−βHφ)] is the partition function of 1 † −βHˆφ † ˆ Trφ ρˆβ aˆkaˆ 0 = Trφ e aˆkaˆ 0 (23) the free field. Here Hφ is the Shrödingerpicture free k Z k field Hamiltonian, which, after subtracting off an infinite † 1 −βHˆ βHˆ −βHˆ aˆ = Tr e φ aˆ e φ e φ k0 zero-point energy (which does not affect any observable Z φ k dynamics), takes the form βωk e −βHˆ aˆ† = Tr aˆ e φ k0 Z Z φ k ˆ n † Hφ = d k ωkaˆkaˆk. (21) βωk † = e Trφ ρˆβ aˆk0 aˆk We would like to emphasize that, strictly speaking, βωk † 0 = e Trφ ρˆβ aˆkaˆk0 + δ(k − k ) , the Gibbs definition ofρ ˆβ in Eq. (20) is not well de- ˆ fined when Hφ is the Hamiltonian of a field in free space, where in the last step we again made use of the CCRs. ˆ since then Hφ is an operator acting on a Hilbert space Comparing the first and last lines of this expression gives of uncountably many dimensions, and certain technical the result issues arise in with performing its exponentiation and βωk trace. We could proceed rigorously by instead consider- † e 3 0 Tr(ˆρβ aˆkaˆk0 ) = δ (k − k ). (24) ing our field to be in a large box of length L, such that its eβωk − 1 Hilbert space is of countable dimension, and then in the end taking the limit L → ∞. Alternatively we could Similarly we obtain the identities formalize our treatment by making use of the Kubo- † 1 3 0 0 Martin-Schwinger (KMS) definition of a thermal state, Tr(ˆρβ aˆkaˆk ) = δ (k − k ), (25) eβωk − 1 which is rigorously defined even for continuous variable 0 Tr(ˆρβ aˆkaˆk ) = 0, (26) systems [44, 45]. In this case the definition ofρ ˆβ would correspond to a KMS state of KMS parameter β with † † Tr(ˆρβ aˆkaˆk0 ) = 0. (27) respect to the time t proper to both detectors. However we will shortly see that, for our limited purposes, these Notice that, as alluded to above, the calculations in more rigorous definitions ofρ ˆβ are unnecessary in the Eqs. (22) and (23) would turn out the same if we rig- sense that formal calculations using the Gibbs definition orously considered the field in a box and then took the in Eq. (20) yield the same results. This can be checked L → ∞ limit in the end. In particular the only differ- by comparing the results we will obtain with, e.g., [46]. ence would be that the CCRs contain a Kronecker delta, 5 which in the limit of free space becomes a Dirac delta, to second order in the coupling strength λ, and where thus recovering our results in a more rigorous fashion. we work in the basis {|gai|gbi, |gai|ebi, |eai|gbi, |eai|ebi}. Furthermore, our final expressions in Eqs. (24)-(27) are The terms Lνη(β) and M(β) are defined to be equal to those obtained using the KMS definition ofρ ˆβ (see equation 14.3 in [46]). Hence our formal use of the Z dnk F¯∗(k)F¯ (k)eik·(xη −xν ) L (β) = Lvac + 2πλ λ ν η Gibbs definition ofρ ˆβ in Eq. (20) is justified. νη νη ν η 2ω eβωk − 1 We can now use the identities in Eqs. (24)-(27) to k h write the two-point function of the field, defined by ∗ × χ¯ν (ωk − Ων )¯χη(ωk − Ωη) 0 0 ˆ ˆ 0 0 w(x, t, x , t ) := Tr[ˆρβ φ(x, t)φ(x , t )], as ∗ i +χ ¯ν (ωk + Ων )¯χη(ωk + Ωη) , (33) w(x, t, x0, t0) = wvac(x, t, x0, t0) + wth(x, t, x0, t0). (28) β Z n ¯ ¯∗ ik·(xa−xb) vac d k Fa(k)Fb (k)e M(β) = M − 2πλaλb vac 0 0 th 0 0 2ω eβωk − 1 Here w (x, t, x , t ) and wβ (x, t, x , t ) are the vacuum k h (β-independent) two-point function and the thermal (β- × χ¯∗(ω − Ω )¯χ (ω + Ω ) dependent) contribution, respectively, and are explicitly a k a b k b given by ∗ i +χ ¯a(ωk + Ωa)¯χb(ωk − Ωb) . (34) n Z d k 0 0 wvac(x, t, x0, t0) = e−iωk(t−t )eik·(x−x ), n : m 2(2π) ωk Here we define the Fourier transformg ¯ R → C of a m (29) function g : R → R as

h 0 0 i Z dnk eiωk(t−t )e−ik·(x−x ) + c.c 1 Z th 0 0 g¯(k) := dmx g(x)eik·x, (35) wβ (x, t, x , t ) = . p m n βωk (2π) 2(2π) ωk e − 1 (30) and as always we use the superscript “vac” to denote Before we proceed to use the two-point function to quantities that do not depend on the inverse temperature calculate the time-evolved two-detector density matrix β, i.e. those terms which arise from the “vacuum” part vac vac ρâb, it should be noted that in the literature one often w of the two-point function. The vacuum terms Lνη finds a very different looking expression for the two-point and Mvac are explicitly given by function of a thermal field state. For instance, in [47], the thermal two-point function for a massless field in (3 + 1)- Z dnk vac ¯∗ ¯ −ik·(xν −xη ) Lνη = 2πλν λη Fν (k)Fη(k)e (36) dimensions is shown to be 2ωk " # × χ¯ (ω + Ω )¯χ∗(ω + Ω ), 1 π(r + t) π(r − t) ν k ν η k η w(x, t, 0, 0) = coth + coth n ∞ t Z d k Z Z 0 8πrβ β β Mvac = −λ λ dt dt0e−iωk(t−t ) (37) a b 2ω i h i k −∞ −∞ + δ(3)(r + t) − δ(3)(r − t)] , (31) h ¯ ¯∗ ik·(xa−xb) 0 8πr × Fa(k)Fb (k)e χ¯a(t)¯χb(t )

0 i where r := |x|. The advantage of this expression over × ei(Ωat+Ωbt ) + (A ↔ B) . the one in Eq. (28) is that there are no integrals over momentum space that have to be evaluated. The disad- vantage is that it is restrictive to the massless (3 + 1)- dimensional case. Furthermore the method used in [47] A. Harvesting entanglement to obtain Eq. (31) is much less direct than the method we employed in obtaining Eq. (28). In any case, as a Having computed the time-evolved density matrixρ ˆ consistency check in Appendix A we show that the ex- ab of the Unruh-DeWitt detector pair, we can now compute pression in Eq. (31) is indeed a specific case of Eq. (28) the negativity of this state and thus quantify the amount when m = 0, n = 3, and x0 = t0 = 0. of entanglement the detectors harvest from the thermal We now come back to our main objective: use the two- field state. Using the expression (32) forρ ˆ , we find that point function w(x, t, x0, t0) in Eq. (28) to compute the ab in the same computational basis, to O(λ2) the partially density matrixρ ˆ in (5). Substituting (28) into (13) we ab transposed matrixρ ˆta takes the form obtain ab

 ∗   ∗  1 − Laa(β) − Lbb(β) 0 0 M (β) 1 − Laa(β) − Lbb(β) 0 0 Lab(β) ∗ ∗  0 Lbb(β) Lab(β) 0  ta  0 Lbb(β) M (β) 0  ρˆab =   , ρˆab =   .  0 Lab(β) Laa(β) 0   0 M(β) Laa(β) 0  M(β) 0 0 0 Lab(β) 0 0 0 (32) (38) 6

2 −1 −1 As discussed in [22], at O(λ ) a matrix of this form has Now, let us consider two temperatures, β1 < β2 . only one potentially negative eigenvalue: Then, defining 1 ta 1 1 Eab,1 = Laa(β) + Lbb(β) (39) h(k) := − , (44) 2 eβ2ωk − 1 eβ1ωk − 1 p 2 2 − (Laa(β) − Lbb(β)) + 4|M(β)| . which is strictly greater than zero, we can rewrite Lνν (β) and M(β) to read Hence we find that the negativity N , defined in Eq. (18), can be written as Z n ¯ 2 2 d k h(k)|F (k)| Lνν (β2) = Lνν (β1) + πλ ta ωk N [ˆρab] = max 0, −Eab,1 . (40) 2 2 × |χ¯(ωk − Ω)| + |χ¯(ωk + Ω)| , (45) Now suppose that the detectors A and B are identi- Z dnk h(k)|F¯(k)|2 cal. That is, they have the same shapes F (x) = Fν (x), 2 M(β2) = M(β1) − 2πλ the same proper energy gaps Ω = Ων , the same cou- ωk pling constants λ = λ , and the same switching profiles iΩ(ta+tb) ik·(xa−xb) ν × e e cos[ωk(ta − tb)] χ(t − t ) = χ (t). Note that we are still allowing for the ν ν × χ¯∗(ω − Ω)¯χ(ω + Ω). (46) detectors to couple to the field at potentially different k k spacetime locations (ta, xa) and (tb, xb). However, since Taking the magnitude of the latter expression we obtain the local terms Lνν are translationally invariant, we find that L (β) = L (β), and the negativity can be written aa bb Z dnk h(k)|F¯(k)|2 more simply as 2 |M(β2)| ≤ |M(β1)| + 2πλ ωk N = max 0, |M(β)| − Lνν (β) . (41) iΩ(ta+tb) ik·(xa−xb) × e e cos[ωk(ta − tb)]

As acknowledged in [22], this form for the negativity ∗ makes evident the competition between the non-local × χ¯ (ωk − Ω)¯χ(ωk + Ω) term |M(β)|, which increases the negativity, and the lo- Z n ¯ 2 cal term Lνν (β), which decreases it. We note however, 2 d k h(k)|F (k)| that although this interpretation of Eq. (41) is pleasantly ≤ |M(β1)| + 2πλ ωk consistent with the intuition that entanglement is a non- × |χ¯∗(ω − Ω)||χ¯(ω + Ω)| (47) local phenomenon, it should not be taken too literally. k k For instance, in [34, 35] it was shown that a detector Finally, combining Eqs. (45) and (47) we find pair interacting with a coherent field state extracts the exact same amount of entanglement as it would from a |M(β2)| − Lνν (β2) vacuum state, despite the fact that inherently local terms Z n 2 d k h(k)D(k) of O(λ) appear inρ âb for the former but not the latter ≤ |M(β1)| − Lνν (β1) − πλ case. ωk Having obtained an expression in (41) for the negativ- ≤ |M(β1)| − Lνν (β1), (48) ity N of two identical Unruh-DeWitt detectors following 2 2 their interactions with a thermal field state, we would where D(k) := |F¯(k)| (|χ¯(ωk − Ω)| − |χ¯(ωk + Ω)|) is now like to determine the temperature dependence of N . a non-negative function characterized by the switching, In other words, we want to answer the question, “what is smearing, and energy gap of the detectors. Hence, using the optimal field temperature for Unruh-DeWitt detec- the definition (39) of the negativity, Eq. (48) proves our tors to harvest entanglement?” first result: the amount of entanglement that two iden- To answer this question, let us first particularize the tical UDW detectors can harvest from a thermal field −1 terms Lνη(β) and M(β) in Eqs. (33) and (34) for iden- state decreases with the temperature β . This is true tical detectors. We obtain regardless of the dimensionality of spacetime, the mass of the field, and the properties (spatial smearing, temporal Z dnk |F¯(k)|2 L (β) = Lvac + πλ2 eik·(xη −xν ) switching, energy gap) of the detectors. νη νη βω ωk e k − 1 In fact, we can obtain a somewhat stronger statement about the negativity of a pair of detectors interacting × |χ¯(ω − Ω)|2ei(ωk−Ω)tη e−i(ωk−Ω)tν k with a thermal field state. First, notice from Eq. (44) that for given values of β and k, the value of the func- + |χ¯(ω + Ω)|2e−i(ωk+Ω)tη ei(ωk+Ω)tν , (42) 1 k tion h(k) can be increased arbitrarily by choosing a small n 2 iΩ(t +t ) ik·(x −x ) Z d k |F¯(k)| e a b e a b enough value of β2. Therefore, from Eq. (48), as long as M(β) = Mvac − 2πλ2 βω D(k) is not identically equal to zero, we find that the ωk e k − 1 ∗ value of |M(β2)|−Lνν (β2) can be made negative by tak- × χ¯ (ωk − Ω)¯χ(ωk + Ω) cos[ωk(ta − tb)]. (43) −1 ing a large enough temperature β2 . Hence, not only 7 does the amount of entanglement harvested by a UDW 4 detector pair decreases monotonically with the tempera- d=1τ ture, but also by increasing the temperature of the field ) - 4 3 d=2τ

to a high enough value we can always (as long as D(k) is 10 not identically zero) ensure that the thermal noise pre- 2 d=3τ vents the detectors from becoming entangled at all. This 2 is true regardless the mass of the field, spacetime dimensionality and the detector properties. 1 Knowing that the negativity N of a detector pair de- Negativity (× λ creases with the temperature of the field, we can ask what is the rate of this decrease. We can straightforwardly ob- 0 tain a bound on dN / dβ from Eq. (48). First, writing 0.0 0.1 0.2 0.3 0.4 ta T(unitsτ -1) Eab,1(β) = Lνν (β) − |M(β)| for identical detectors, the second line of Eq. (48) can be expressed as FIG. 1. Negativity of identical detectors as a function of field Z dnk h(k)D(k) temperature, for different spatial separations d of their centers Eta (β ) − Eta (β ) ≤ −πλ2 . (49) ab,1 1 ab,1 2 ω of mass. The detectors are coupled to the field at the same k time according to a Gaussian switching function of width τ, Dividing both sides of this expression by β − β , taking their spatial profiles are Gaussians of width σ = τ, and their 1 2 energy gap is Ω = 3/τ. the limit β1 → β2, and using the fact that

h(k) d 1 ω eβ1ωk lim = − = k , Then it is straightforward to show that the terms Mth, β1ωk 2 β1→β2 β1 − β2 dβ1 e − 1 β1ωk th th e − 1 Lab and Lνν , which make up the thermal contributions (50) to the density matrixρ âb, evaluate to ta we find the rate of change of the eigenvalue Eab,1(β) with 2 − 1 Ω˜ 2 iΩ˜∆˜ + ∞ − 1 k˜2(1+˜σ2) respect to the inverse temperature β to be bounded from λ˜ e 2 e Z e 2 Mth = − dk˜ (55) β˜k˜ below according to 4πd˜ 0 e − 1 × sin(d˜k˜) cos(∆˜ −k˜), d Z eβωk ta 2 n E (β) ≤ −πλ d k D(k) . (51) 1 ˜ 2 ˜ ˜ − ∞ 1 ˜2 2 ab,1 2 ˜2 − 2 Ω −iΩ∆ Z − 2 k (1+˜σ ) dβ βωk λ e e e e − 1 Lth = dk˜ (56) ab β˜k˜ 2πd˜ 0 e − 1 Therefore in regions where the negativity N (β) is non- × sin(d˜k˜) cosh[(Ω˜ + i∆˜ −)k˜], zero, we have that 2 − 1 Ω˜ 2 ∞ − 1 k˜2(1+˜σ2) λ˜ e 2 Z ke˜ 2 Lth = dk˜ cosh(Ω˜k˜). (57) Z βωk νν β˜k˜ dN 2 n e 2π 0 e − 1 ≥ πλ d k D(k) 2 . (52) dβ βωk e − 1 Here, every quantity with a tilde is a dimensionless expression of the scales of the problem in units of τ (e.g. This puts a lower bound on how fast N must grow with Ω˜ := Ωτ, β˜ := β/τ), and we have defined d˜:= |x −x |/τ the inverse temperature β, in regions where N is non- a b and ∆˜ ± := (t ± t )/τ. Meanwhile the terms Mvac and zero. Of course if N is zero, then increasing β will only b a Lvac, which give the vacuum (β independent) contribu- result in N remaining zero. νη tions toρ ˆ , can be found in equations 29-31 in [22]. Having proven the general result that temperature is ab Assuming these detector spatial profiles and switching always detrimental to entanglement harvesting (at least functions, in Fig. 1 we show the dependence of the neg- for identical detectors), let us now consider some partic- ativity of the detector pair on the temperature T = β−1 ular parameters for the detectors A and B, so that we of the field. We see that, in accordance with our general may explicitly see the manifestation of this phenomenon. discussion above, the negativity is a monotonically de- To that end, let us suppose that the two detectors are creasing function of T , and that it is identically zero after located in (3 + 1) dimensional spacetime, that they have a certain finite temperature. These findings are qualita- Gaussian spatial profiles of width σ, tively the same as what was found in [18], namely that

2 harmonic oscillator detectors in a (1+1)D cavity har- 1 − |x| F (x) = √ e σ2 , (53) vest less entanglement as the ﬁeld temperature increases. ( πσ)3 This is, of course, all in agreement with our intuition that and that their temporal switching functions are also “thermal noise” is detrimental to the detectors obtaining Gaussians (of width τ), non-local correlations. We will soon see however, that this seemingly reasonable intuition does not apply when 2 − t we quantify the correlations using the mutual informa- χ(t) = e τ2 . (54) tion rather than the negativity. In particular we will 8

) d=1τ T=0 2 0.100 ) 1.5 1 d=3τ - 3 T= 12τ

10 0.010 d=5τ

2 1 T= 6τ d 10 1.0 0.001 = τ

10-4

0.5 10-5 Negativity (× λ λ (× Information Mutual 10-6 0.0 0.001 0.010 0.100 1 10 100 0 1 2 3 4 5 T(unitsτ -1) Ω(unitsτ -1) FIG. 3. Mutual information of identical detectors as a func- FIG. 2. Negativity of identical detectors as a function of their tion of field temperature, for different spatial separations d of energy gap, for different field temperatures T . The detectors their centers of mass. The detectors are coupled to the field are coupled to the field at the same time according to a Gaus- at the same time according to a Gaussian switching function sian switching function of width τ, and they have Gaussian of width τ, their spatial profiles are Gaussians of width σ = τ, spatial profiles of width σ = τ, the centers of which are sepa- and their energy gap is Ω = 3/τ. rated in space by d = 2τ.

straightforward to derive the asymptotic behaviour as show that the mutual information between the detector −1 β → ∞. Defining L± := βL± and Lνη := βLνη, we pair can increase with the field temperature. notice from Eq. (33) that L± and Lνη are independent To conclude this section, let us briefly investigate how of β in the limit β−1 → ∞. Then from Eq. (58) it is the negativity of the detectors varies with their energy straightforward to show that in the β−1 → ∞ limit the gap Ω. These results are summarized in Fig. 2. Notice mutual information goes as that, for a given field temperature T , the detectors can- not become entangled if their energy gap is below some 1 I[ˆρ ] ∼ L+ log L+ + L− log L− finite value Ωmin(T ). We also notice that Ωmin(T ) is a ab β monotonically increasing function of temperature. This − log − log . (60) tells us that if we have a way to control the energy gap of Laa Laa Lbb Lbb the detectors, then by measuring the amount of entan- Combining this with the fact that the mutual informa- glement that this detector pair harvests from the field tion is always non-negative, we conclude that in the large we have, in principle, a quantum thermometer capable temperature limit (of course with a coupling constant of measuring the field temperature. small enough so that we are still within the perturbative regime) the total correlations that the detectors harvest from the field grow proportionally to the temperature B. Harvesting mutual information β−1. To see explicitly the dependence of I[ˆρab] on the tem- Having shown that the amount of entanglement har- perature, let us once again particularize to the case of vested by two Unruh-DeWitt detectors decreases with identical detectors with Gaussian spatial smearings (53) the temperature of the field with which they interact, we and Gaussian switching functions (54). These results are can ask what happens to other types of correlations. As plotted in Fig. 3. We see that for low T = β−1 the mentioned above, the mutual information I[ˆρab], defined mutual information approaches a constant finite value, in Eq. (19), quantifies the total correlations (quantum which corresponds to the correlations that the detectors and classical) present between the two detectors. Using would obtain if they interacted with the field vacuum. the time-evolved density matrixρ âb in Eq. (32) for the For intermediate field temperatures, we find that the mu- two detectors, we find that I[ˆρab] takes the form tual information has a non-trivial dependence on T , and in fact, unlike the negativity, I[ˆρ ] does not always in- I[ˆρ ] =L log(L ) + L log(L ) (58) ab ab + + − − crease with T . However, as we showed for the case of ar- 4 − Laa log(Laa) − Lbb log(Lbb) + O(λ ), bitrary detectors above, in the asymptotic limit T → ∞ the mutual information is proportional to T . It should be where L± is defined as emphasized that in a full, non-perturbative calculation, this upwards trend of I[ˆρab] with temperature would not 1 p 2 2 L± = Laa + Lbb ± (Laa − Lbb) + 4|Lab| . (59) continue indefinitely for the simple reason that for a two 2 qubit system the mutual information is bounded from Although the general dependence of I[ˆρab] on the tem- above by 2 log 2. Nevertheless it is interesting that, at perature β−1 is highly non-trivial, from Eq. (58) it is least in the perturbative regime (i.e. if for a given tem- 9 perature we consider a small enough coupling strength), implies that the amount of entanglement harvested from the field by 0 0 an Unruh-DeWitt detector pair is hindered by high field hα(k), ζ(k, k )|aˆk00 |α(k), ζ(k, k )i temperatures, whereas the total correlations in fact grow † 00 = h0|Sˆ [â 00 + α(k )]Sˆ |0i with T . ζ k ζ = α(k00), (65)

and hence, using the mode expansion (1) of the field op- IV. SQUEEZED COHERENT FIELD STATE erator, the one-point function (16) of the field in the state |α(k), ζ(k, k0)i is Again let us suppose that each Unruh-DeWitt detector is in its ground state, and that now the field Z dnk v(x, t) = α(k)e−i(ωkt−k·x) + c.c . is in an arbitrary, multimode, squeezed coherent state. p n 2(2π) ωk The physical relevance of squeezed coherent states is (66) that they are the most general set of states that sat- urate the Heisenberg uncertainty principle. The most Thus we see that the one-point function is indepen- general multimode squeezed coherent state is given by dent of the squeezing amplitude ζ(k, k0). Similarly we 0 ˆ ˆ |α(k), ζ(k, k )i = DαSζ |0i, where the displacement oper- can show that the two-point function (17) in the state ˆ ˆ ato Dα and the squeezing operator Sζ are unitary oper- |α(k), ζ(k, k0)i is of the form ators defined by [37] w(x, t, x0, t0) = wind(x, t, x0, t0) + wcoh(x, t, x0, t0), (67) Z Dˆ := exp d3k α(k)â† − H.c. , (61) α k where wind is independent of the coherent amplitude Z Z α(k), where wcoh is given by a product of one-point func- ˆ 1 3 3 0 ∗ 0 Sζ := exp d k d k ζ (k, k )âkaˆk0 − H.c. , tions: 2 (62) wcoh(x, t, x0, t0) = v(x, t)v(x0, t0), (68)

We call the complex valued distributions α(k) and and vanishes if α(k) = 0 for all k. 0 ζ(k, k ) respectively the coherent amplitude and squeez- Even without calculating the α(k)-independent con- 0 ing amplitude of the state |α(k), ζ(k, k )i. Through the tribution wind to the two-point function, we can see that ˆ ˆ integrals in the definitions of Dα and Sζ , these distribu- it is the product of two one-point functions. In [35] it tions generalize the familiar notion of a squeezed coherent was shown that when this is the case, then the α(k)- state of a single harmonic oscillator to the case where we dependent contributions ofρ âb arising from the one-point have an uncountably infinite number of field mode oscil- function exactly cancel the contributions from the two- ta lators that can be pairwise two-mode squeezed with each point function, so that the eigenvalues ofρ âb andρ âb — other. and therefore the negativity N [ˆρab] as well — are com- In order to calculate the one and two-point functions of pletely independent of α(k). This result was used in [35] the field in a squeezed coherent state, we will make use to prove that the entanglement harvested by an Unruh- ˆ ˆ of the identities governing the action of Dα and Sζ on DeWitt detector pair is independent of the coherent am- the creation and annihilation operators. Namely, by us- plitude of a (non-squeezed) coherent state. Since this is ing the canonical commutation relations and the Baker- a general consequence of the special relationship between Campbell-Hausdorff lemma it is straightforward to show the α(k)-dependent parts of the one and two-point func- that tions, we conclude that this result is true even in the presence of squeezing. Namely, to O(λ2), the negativ- ˆ † ˆ DαaˆkDα =a ˆk + α(k)11. (63) ity of a detector pair interacting with a general squeezed coherent state |α(k), ζ(k, k0)i is independent of the co- On the other hand, we are not aware of a similarly con- herent amplitude distribution α(k). In other words, en- ˆ† ˆ venient closed-form expression for Sζ aˆkSζ in the case of tanglement harvesting from a squeezed coherent state is an arbitrary, continuous, multimode squeezing. However, insensitive to the coherent amplitude. since Sˆζ is the exponential of terms quadratic ina ˆk and Therefore, since we are interested in studying the en- † ˆ† ˆ tanglement harvested by the detector pair from a gen- aˆk, by expanding out the exponentials in Sζ aˆkSζ it is not difficult to prove that this expression takes the form of a eral squeezed coherent state, we can, without loss of gen- † erality, restrict our attention only to squeezed vacuum linear superposition ofa ˆk anda ˆk operators, i.e. states (i.e. we can make α(k) identically zero). Addi- Z tionally, for mathematical simplicity—i.e. in order to ob- ˆ† ˆ 3 0 h 0 0 † i Sζ aˆkSζ = d k K1(k, k )âk0 + K2(k, k )âk0 , (64) ˆ† ˆ tain an explicit expression for Sζ aˆkSζ in Eq. (64)—from here on we will consider only squeezed coherent states for some bi-distributions K1 and K2. In particular this in which the squeezing is not “mixed” between modes, 10 i.e. such that the squeezing amplitude is of the form invariant with respect to spacetime translations. As we 0 0 ζ(k, k ) = ζ(k)δ(k − k ). In this case we find that Sˆζ will see, a physical consequence of this is that the neg- simplifies to ativity harvested by a pair of UDW detectors from a squeezed coherent state depends not only on the space- 1 Z Sˆ = exp d3k ζ∗(k)â2 − H.c. , (69) time interval between the detectors, but also on where in ζ 2 k the spacetime they are centered. With the expression (71) for the two-point function of and that Sˆ†aˆ Sˆ can be conveniently expressed as ζ k ζ a squeezed vacuum field state, and with the vanishing

† iθ(k) † one-point function (66), we can proceed to calculate the Sˆ aˆkSˆζ = cosh[r(k)]âk − e sinh[r(k)]â , (70) ζ k evolved stateρ âb of the two UDW detectors following their interactions with this field. From (13) we obtain where we have written ζ(k) = r(k)eiθ(k) in polar form. The two-point function (17) of the state Sˆζ |0i, with Sˆζ  ∗  in the above form, can be written as 1 − Laa[ζ] − Lbb[ζ] 0 0 M [ζ] ∗  0 Lbb[ζ] Lab[ζ] 0  w(x, t, x0, t0) = wvac(x, t, x0, t0) + wsq(x, t, x0, t0), (71) ρâb =   ,  0 Lab[ζ] Laa[ζ] 0  M[ζ] 0 0 0 where wvac is the vacuum two-point function given in (74) Eq. (29), while wsq is the contribution that depends to second order in the coupling strength λ, and where on ζ(k) and vanishes if ζ(k) = 0 for all k. Explicitly we work in the basis {|g i|g i, |g i|e i, |e i|g i, |e i|e i}. wsq(x, t, x0, t0) is given by a b a b a b a b The matrix terms Lνη[ζ] and M[ζ] are now functionals of Z n the squeezing distribution ζ(k), and they take the forms sq 0 0 d k w (x, t, x , t ) = n sinh[r(k)] (72) 2(2π) ωk iθ(k) −iω (t+t0) ik·(x+x0 vac sq × − e cosh[r(k)]e k e Lνη[ζ] = Lνη + Lνη[ζ], (75) vac sq 0 0 M[ζ] = M + M [ζ]. (76) + sinh[r(k)]e−iωk(t−t )eik·(x−x + c.c, (73)

vac vac Notice that, unlike Eq. (28) for a thermal ﬁeld state, the As before, the vacuum terms Lνη and M are given by two-point function for a squeezed coherent state is not Eqs. (36) and (37), while the ζ(k) dependent terms read

Z d3k sq 2 ¯ ¯∗ ∗ ik·(xν −xη ) Lνη[ζ] = πλν λη sinh [r(k)]Fν (k)Fη (k)¯χν (ωk − Ων )¯χη(ωk − Ωη)e (77) ωk 2 ¯∗ ¯ ∗ −ik·(xν −xη ) + sinh [r(k)]Fν (k)Fη(k)¯χν (ωk + Ων )¯χη(ωk + Ωη)e −iθ(k) ¯∗ ¯∗ −ik·(xν +xη ) − e sinh[r(k)] cosh[r(k)]Fν (k)Fη (k)¯χν (ωk + Ων )¯χη(ωk − Ωη)e iθ(k) ¯ ¯ ∗ ∗ ik·(xν +xη ) − e sinh[r(k)] cosh[r(k)]Fν (k)Fη(k)¯χν (ωk − Ων )¯χη(ωk + Ωη)e , Z d3k sq −iθ(k) ¯∗ ¯∗ −ik·(xa+xb) M [ζ] = 2πλaλb e sinh[r(k)] cosh[r(k)]Fa (k)Fb (k)¯χa(ωk + Ωa)¯χb(ωk + Ωb)e (78) ωk iθ(k) ¯ ¯ ∗ ∗ ik·(xa+xb) + e sinh[r(k)] cosh[r(k)]Fa(k)Fb(k)¯χa(ωk − Ωa)¯χb(ωk − Ωb)e 2 ¯∗ ¯ ∗ −ik·(xa−xb) − sinh [r(k)]Fa (k)Fb(k)¯χa(ωk + Ωa)¯χb(ωk − Ωb)e 2 ¯ ¯∗ ∗ ik·(xa−xb) − sinh [r(k)]Fa(k)Fb (k)¯χa(ωk − Ωa)¯χb(ωk + Ωb)e .

A. Harvesting entanglement identical UDW detectors with Gaussian spatial profiles of width σ, given by Eq. (53), and Gaussian temporal In order to study the dependence of field squeezing on switching functions of width τ, as in Eq. (54). Then the sq sq the ability of detectors to harvest entanglement, let us matrix elements Lνη[ζ] and M [ζ] given by Eqs. (77) once again particularize to the case of a massless field and and (78) become 11

˜2 − 1 Ω˜ 2 Z 3˜ sq λ e 2 d k − 1 |k˜|2(1+˜σ2) 2 |k˜|Ω˜ −i|k˜|(t˜ −t˜ ) ik˜·(x˜ −x˜ ) L [ζ] = e 2 sinh [r(k)]e e ν η e ν η (79) νη 16π2 |k˜| ˜ ˜ ˜ ˜ ˜ ˜ + sinh2[r(k)]e−|k|Ωei|k|(tν −tη )e−ik·(x˜ν −x˜η ) ˜ ˜ ˜ ˜ − e−iθ(k) sinh[r(k)] cosh[r(k)]ei|k|(tν +tη )e−ik·(x˜ν +x˜η ) ˜ ˜ ˜ ˜ − eiθ(k) sinh[r(k)] cosh[r(k)]e−i|k|(tν +tη )eik·(x˜ν +x˜η ) ,

˜2 − 1 Ω˜ 2 Z 3˜ sq λ e 2 d k − 1 |k˜|2(1+˜σ2) 2 −i|k˜|(t˜ −t˜ ) ik˜·(x˜ −x˜ ) M [ζ] = − e 2 sinh [r(k)]e a b e a b (80) 16π2 |k˜| ˜ ˜ ˜ ˜ + sinh2[r(k)]ei|k|(ta−tb)e−ik·(xã−x˜b) ˜ ˜ ˜ ˜ ˜ ˜ − e−iθ(k) sinh[r(k)] cosh[r(k)]e−|k|Ωei|k|(ta+tb)e−ik·(xã+x˜b) ˜ ˜ ˜ ˜ ˜ ˜ − eiθ(k) sinh[r(k)] cosh[r(k)]e|k|Ωe−i|k|(ta+tb)eik·(xã+x˜b) ,

where, as before, we denote by a tilde any quantity re- 2.5 ferred to the scale τ (e.g., Ω˜ = Ωτ,σ ˜ = σ/τ, etc.). r=0

With these explicit expressions for the matrix elements ) 2.0 r= 0.5 - 2 r=1 ofρ ˆab at hand, we can now readily compute the negativ- 10 2 1.5 ity N = max 0, |M[ζ]| − Lνν [ζ] , and thus quantify the r=1.5 amount of entanglement that the two detectors harvest from the ﬁeld. 1.0

λ (× Negativity 0.5

1. Uniform squeezing 0.0 -1.0 -0.5 0.0 0.5 1.0

Let us begin by considering the simplest possible type xCoM (unitsτ) of squeezing: that in which all field modes are squeezed equally. To that end we take ζ(k) = r, where we also FIG. 4. Negativity of identical detectors as a function of their assume that r is real and positive. (We will shortly see center of mass position, for different values of the squeezing what the effect is of r having a complex phase.) parameter r = |ζ(k)|. Here the squeezing is uniform across all field modes. The detectors are coupled to the field through In Fig. 4, for different values of r, we plot the negativity Gaussian switching functions of width τ centered at t = 0, and of the detectors following their interactions with the field their energy gaps are Ω = τ−1. The detectors are centered as a function of their joint center of mass. We see that— at (xcom ± τ, 0, 0) and have Gaussian spatial profiles of width as we anticipated already from the two-point function—a σ = τ. squeezed field state is in general not translationally invariant, and as such the entanglement harvesting ability of a pair of detectors from such a state is not transla- Let us now attempt to better understand the non- tionally invariant either. In particular we find that if the translation-invariance of squeezed field states in general, detectors’ center of mass is near the spatial origin of the and in particular the consequences of this for entangle- coordinate system, then the detectors can harvest more ment harvesting from these states. Concretely, with re- entanglement from a uniformly squeezed field state than gards to the plots in Fig. 4, it is natural to ask why is from the vacuum. On the other hand if the detectors are the spatial origin of our chosen coordinate system the far enough away from the origin, then, regardless of the preferred location of UDW detectors that hope to har- amount of squeezing, they are unable to extract entan- vest entanglement? First, let us note once again that, glement. The proximity to the origin that is necessary as can be seen in Fig 4, in the absence of squeezing the for squeezing to be beneficial for entanglement harvest- translation-invariance of entanglement harvesting is re- ing is dictated by the amount of squeezing r: for a highly stored. Therefore, the picking out of a preferred point in squeezed field state the detectors can harvest a lot more space near which entanglement harvesting is maximized entanglement, but they have to be highly centered near (in this case the origin of the coordinate system) must the origin; for a less squeezed state the improvement in be a direct consequence of the squeezing amplitude ζ(k) harvesting is not as noticeable, but the detectors do not that we choose for the field. In fact, we notice that the need to be so precisely centered. Fourier transform of the uniform amplitude ζ(k) = r is 12 proportional to δ(x), and therefore the origin x = 0 is Ω= 0.5τ -1 3.5 clearly a special point in this case. As we will now show, this relationship between the Fourier transform of the 3.0 squeezing amplitude and the preferred location of detec- )

- 2 2.5

tors trying to harvest entanglement is valid in general. 10 To that end, let us consider an arbitrary squeezing 2 2.0 amplitude ζ(k). With this choice of squeezing, there will 1.5 be some preferred points in space near which it is easier d=0 for detectors to harvest entanglement, and others near 1.0 d=1τ λ (× Negativity which it is more difficult. Suppose now that we change 0.5 d=3τ 0 ik·x0 the squeezing by a local phase ζ(k) → ζ (k) = e ζ(k). d= 20τ How do the positions of the preferred points change? 0.0 To answer this question, let us recall from Eq. (5) that 0.0 0.5 1.0 1.5 2.0 2.5 3.0 the stateρ âb of the two detectors following their inter- r actions with a squeezed field state with amplitude ζ0 is -1 given by Ω=1τ

2.5 ˆ 0 ˆ† ˆ ˆ 0† ρˆ = Tr U ρˆ ⊗ ρˆ ⊗ S 0 |0ih0|S 0 U , (81) ab φ a b ζ ζ )

- 2 2.0

10 d=0 ˆ 0 2 where U is the time-evolution unitary 1.5 d=1τ d=3τ h Z 0 X 1.0 Uˆ = T exp − i dt λν χν (t)ˆµν (t) (82) d= 20τ ν Z λ (× Negativity 0.5 n ˆ i × d x Fν (x − xν )φ(x, t) . 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Now let us deﬁne the ﬁeld momentum operator to be ˆ R 3 † r P := d k kaˆkaˆk. Then, using the fact that

iPˆ·x0 −iPˆ·x0 ik·x0 FIG. 5. Negativity of identical detectors as a function of the e aˆke =a ˆke , (83) squeezing parameter r = |ζ(k)|, for different values of their ˆ ˆ spatial separation d and energy gaps Ω. Here the squeezing ˆ −iP ·x0/2 ˆ iP ·x0/2 we find that we can write Sζ0 = e Sζ e . is uniform across all field modes. The detectors are coupled Making use of the cyclicity of the partial trace with re- to the field through Gaussian switching functions of width τ spect to the subsystem being traced over, we find that centered at t = 0; they are centered at (±d/2, 0, 0) and have ρâb can be expressed as Gaussian spatial profiles of width σ = τ. ρˆ = Tr Uˆ ρˆ ⊗ ρˆ ⊗ Sˆ†|0ih0|Sˆ Uˆ † , (84) ab φ a b ζ ζ ¯ that the Fourier transform ζ is shifted by (x0) can be removed by choosing a different convention for the expo- ˆ ˆ where Uˆ := eiP ·x0/2Uˆ 0e−iP ·x0/2. Using (83) we readily nent in the definition (35) of a Fourier transform. There- obtain fore we conclude that (up to a potential re-scaling) the Fourier transform of the field’s squeezing amplitude ζ di- h Z X Uˆ = T exp − i dt λν χν (t)ˆµν (t) (85) rectly tells us where in space the UDW detectors should ν be centered if they want to harvest more entanglement Z x i from the squeezed field state. These preferred locations × dnx F x − x − 0 φˆ(x, t) . ν ν 2 are commensurate with where the fluctuations of the field amplitude, and the stress energy density, are localized in Hence changing the field’s squeezing amplitude by a local space. phase ζ → eik·x0 ζ is equivalent to shifting the detectors Having expounded the dependence of the detectors’ in space by an amount x0/2. In other words, a local center of mass on their ability to harvest entanglement phase change of the squeezing amplitude effects a trans- from a squeezed field state, and having related this to lation of the points in space near which it is easier for the the local phase of the squeezing amplitude, let us now detectors to harvest entanglement. However, such a local turn to the question of how the magnitude of the squeez- phase change of ζ also effects a translation of its Fourier ing amplitude affects the detector’s abilities to harvest ¯ ¯ transform: namely ζ(x) → ζ(x − x0). Note that the dis- entanglement. crepancy by a factor of 2 between the amount that the In Fig. 5 we plot the negativity of a UDW detector pair preferred points are translated (x0/2) and the amount as a function of ζ(k) = r, which we once again assume 13 to be uniform across all field modes. We notice several squeezed, while all other modes are in their vacuum interesting features from these plots. states. More precisely, we set Interestingly, high squeezing can remove the depen- ( dence of entanglement harvesting on the distance be- r if |k0 − k | < for i ∈ {x, y, z} ζ(k0) = i i 2 , (86) tween the detectors. Indeed, we find that while at low 0 otherwise squeezing amplitude the amount of entanglement that the detectors can harvest depends on their spatial separa- where k0 = (k0 , k0 , k0 ), k = (k , k , k ), and : x y z x y z tion d = |xa −xb|, at high squeezing this is not the case. parametrizes the bandwidth of the squeezing. With this In other words, in the limit of large uniform squeezing of choice of squeezing amplitude, and assuming again that the field, a detector pair separated by a large spatial dis- the spatial and temporal profiles of the detectors are tance will harvest the same amount of entanglement as if Gaussians given by Eqs. (53) and (54), the matrix el- they were at the same location in space. A similar effect ements Lsq [ζ] and Msq[ζ] of the evolved two detector of removal of the distance scale in a setup where vacuum νη density matrixρ âb are again given by the expressions in entanglement is relevant was seen in [48] where quantum Eqs. (79) and (80), except that now the limits of momen- energy teleportation could be made independent of sep- 0 tum space integration are such that |ki −ki| < /2. With aration between sender and receiver if one uses squeezed the use of these expressions we can compute the negativ- field states. ity N = max 0, |M[ζ]| − Lνν [ζ] , and thus observe how Furthermore, from Fig. 5, we find that the amount of the amount of entanglement that the detectors can har- entanglement that the detectors harvest is also indepen- vest depends on the bandwidth of the field’s squeezing dent of the squeezing parameter ζ(k) = r in the limit as amplitude. r → ∞. Hence although squeezing the field modes often However before showing plots of N versus , since we increases the amount of harvestable entanglement from are in this section trying to upgrade our theoretical find- that allowed by the field vacuum, this trend of increas- ings to the realm of what is experimentally feasible, it is ing negativity does not continue indefinitely, but rather important that we also discuss what values of squeezing plateaus to a constant asymptotic value at large r. amplitude r we can expect to obtain in our bandlimited frequency range. As far as we are aware, the highest experimentally attained squeezed state of the electromag- 2. Bandlimited squeezing netic field resulted in a squeezed quadrature noise reduction of 15 dB below the vacuum level [51]. Using the To an experimentalist looking to make an entangle- conversion formula [52] ment harvesting measurement in the lab, perhaps the most interesting results of the previous section are that i) ˆ 2 ∆Noise (in dB) = 10 log10 2h∆X i , (87) the amount of entanglement harvested by a pair of UDW detectors from a highly (uniformly) squeezed field state between the reduction in noise of the squeezed quadra- is independent of the spatial separation of the detectors, ture Xˆ and the variance h∆Xˆ 2i := hXˆ 2i − hXî2 of that and ii) if the detectors are centered near the “preferred” quadrature in the squeezed state |ζ(k)i, as well as the locations in space (as determined by the Fourier trans- expression form of the squeezing function ζ(k)), then the amount of entanglement that they harvest could be much higher 1 h∆Xˆ 2i = e−2r, (88) than in the case of a vacuum field state. 2 However such an experimentalist would be quick to note that there is an obvious difficulty with attempting between h∆Xˆ 2i and r, we find that to translate the theoretical results of the previous section into an actual experiment in the lab. Namely, in the ∆Noise (in dB) = −20 log10(e)r. (89) previous section we assumed the field to be uniformly squeezed across all field modes, while squeezed states in Hence a noise reduction of 15 dB corresponds to a squeez- experimental quantum optics [49] and superconducting ing amplitude of r ≈ 1.7. To be on the safe side with setups [50] are generally bandlimited to a very narrow respect to experimental feasibility, we will for the below range of field modes. We expect that in this case, where discussion set r = 1 (corresponding to ∼ 8.7 dB). only a narrow frequency range of modes are squeezed, In Fig. 6 we plot the dependence of the negativity that the field state will behave more similarly to the vacuum two UDW detectors can harvest from the field, as a func- state, in which case squeezing might not give much of tion of the bandwidth of field modes that are squeezed an advantage in terms of entanglement harvesting. The (we assume the squeezed modes to be centered around key question is then: what range of field modes must be some wavevector k). In the top plot of this figure, we squeezed in order to produce a significant entanglement suppose that the detectors are near enough in space such harvesting advantage over the vacuum state? that they are able to harvest entanglement from the field To answer this question, let us now assume that only vacuum ( = 0). Perhaps unintuitively, we find that as the field modes near some momentum k are uniformly we start squeezing around the mode k (i.e. we increase 14

d=2τ modes is squeezed, i.e. if the bandwidth is larger than 2.0 some critical value c. We notice from the plots in Fig. 6 that the critical )

- 2 bandwidth necessary to achieve an improvement in en- 1.5 c

10 tanglement harvesting over the vacuum is at least of the 2 order |k|, where k is the wavevector of the mode around 1.0  which we squeeze. Hence for instance if we wanted to k = (0,0,0) use a 300 THz squeezed laser source to entangle a pair  -1 0.5 k = (τ ,0,0) of atomic detectors, we would need to squeeze all the λ (× Negativity  k = (2τ-1,0,0) modes up to 600 THz with wavevectors pointing in the direction of the laser, as well a wide range of field modes 0.0 pointing in other directions. As far as we are aware, 0 2 4 6 8 10 current experimental setups featuring squeezed electro- ϵ(unitsτ -1) magnetic field states do not squeeze such large band- widths of field modes. Hence, in order to make use of d= 20τ the benefits of squeezed field states with respect to entanglement harvesting, it may be necessary to increase 2.0 the experimentally achievable squeezing bandwidth. Al- )

- 2 ternatively, it might still be possible to obtain high levels

10 1.5 2 of harvestable entanglement with narrowly bandlimited squeezed states, but for which the squeezing amplitude 1.0  ζ(k) is non-uniform in the bandlimited range. This re- k = (0,0,0)  mains to be investigated in future work. k = (τ-1,0,0)

λ (× Negativity 0.5  k = (2τ-1,0,0)

0.0 V. CONCLUSIONS 0 2 4 6 8 10 ϵ(unitsτ -1) We studied the ability of a pair of Unruh DeWitt particle detectors to harvest quantum and classical correla- FIG. 6. Negativity of identical detectors as a function of tions from thermal and squeezed states of a scalar field the bandwidth of of modes squeezed, centered around a with which they interact. We find several interesting re- mode k. The squeezing inside the bandlimited range is of sults: uniform amplitude r = 1, and outside is zero. The detectors First, we prove that the amount of entanglement that are coupled to the field through Gaussian switching functions a pair of identical detectors (with arbitrary spatial proof width τ centered at t = 0, they are centered at (±d/2, 0, 0) files and time-dependent switching functions) can harvest and have Gaussian spatial profiles of width σ = τ, and their from a thermal state of the field decreases monotonically energy gaps are Ω = τ−1. with temperature. Additionally, we obtain a lower bound on this rate of decrease, and hence show that for temperatures higher than a certain threshold the detectors ), the negativity of the detectors initially begins to de- are unable to harvest any entanglement from the field. crease. That is, for a small bandwidth of field squeez- With these findings we also extend the main results in ing, regardless of the mode k around which the squeez- [18], where it was numerically shown (using the very dif- ing is being performed, the amount of entanglement that ferent formalism of Gaussian quantum mechanics) that the detectors can harvest from the field is actually less temperature is detrimental to entanglement harvesting than what they could harvest from the vacuum. Eventu- by harmonic oscillator detectors from a massless field in ally however, as the bandwidth is increased further, the 1+1 dimensional spacetime. Indeed, we prove that this amount of entanglement that the detectors can harvest is also the case for qubit detectors of arbitrary shape and from the field becomes higher than in the vacuum case. switching interacting with a field of any mass in any di- Meanwhile, detectors with a large spatial separation mensionality of spacetime. (bottom plot of Fig. 6) are unable to harvest entangle- On the other hand, we find that unlike the negativ- ment from the vacuum ( = 0), as was already shown ity, the mutual information — which is a measure of the in Ref. [22]. In this case increasing the squeezing band- total (quantum and classical) correlations — that the de- width allows the detectors to harvest some entanglement, tectors harvest from the field actually increases linearly but this only occurs for larger than some critical value with the field temperature (again extending the numer- c. Hence, regardless of separation, the ability of a pair ical findings of [18] to qubit detectors). Hence, while of UDW detectors to harvest more entanglement from thermal noise hinders the ability of UDW detectors to a squeezed field state than from the vacuum is depen- harvest entanglement, it is beneficial in the harvesting of dent on whether a large enough frequency interval of field non-entanglement correlations. 15

Moving on to squeezed field states, we start by proving Appendix A: Thermal two-point function that, at least to leading perturbative order, the amount of entanglement that a UDW detector pair can harvest We will show that our expression for the thermal two- from a squeezed coherent state is independent of its co- point function in Eq. (28) reduces to the special case in herent amplitude. This greatly generalizes the result Eq. (31) when m = 0, n = 3, and x0 = t0 = 0. of Ref. [34], which considered only unsqueezed coherent Let us first evaluate the second term in Eq. (28), states, to hold for all general squeezed coherent states. wβ(x, t, 0, 0), which is given in Eq. (30). Working in polar We also show that, unlike the coherent amplitude, the coordinates, with k := |k| and r := |x|, we straightfor- field’s squeezing amplitude ζ(k) does affect the amount wardly obtain of entanglement that the detectors can harvest from the Z ∞ field. In particular, we find that the amount of entan- 1 dk wβ(x, t, 0, 0) = 2 βk sin(kr) cos(kt) glement that detectors centered at a spatial point x0 2π r 0 e − 1 can harvest is directly related to the amplitude of the 1 Fourier transform of ζ(k) evaluated at x . Hence, con- = P − (A1) 0 4π2(r2 − t2) trary to vacuum [22], coherent [34], and thermal states, " #! harvesting entanglement from general squeezed states is 1 π(r + t) π(r − t) + coth + coth , generally not a translationally invariant process. 8πrβ β β However, and perhaps surprisingly, we find that for detectors centered at a particular location x0, the amount where P denotes the principal value of the integral (this of entanglement harvested from a highly and uniformly expression only has meaning as a distribution). Interest- squeezed state is independent of the spatial separation ingly, notice that the last term does not depend on the of the detectors. Moreover, this amount of entanglement temperature. is often much larger than detectors at the same separa- We can similarly calculate the first term in Eq. (28), tion would be able to harvest from the vacuum, raising w0(x, t, 0, 0), which is given in Eq. (29). We obtain the idea of the possibility of using squeezed states to experimentally test entanglement harvesting. This result is 1 Z ∞ w (x, t, 0, 0) = dk e−ik(t−r) − e−ik(t+r) commensurate with the finding that squeezed states can 0 2 8π ir 0 remove the distance decay of protocols that rely on field Z s 1 −ik(t−r) −ik(t+r) entanglement such as quantum energy teleportation [48]. = 2 lim dk e − e 8π ir s→∞ 0 Finally, we have also studied how entanglement har- 1 1 vesting is modified when we allow for squeezing only in a = P + lim (A2) 4π2(r2 − t2) s→∞ 8π2r finite frequency bandwidth of field modes. We find that " if we restrict the modes of the field that are squeezed to a i sin s(r + t) i sin s(r − t) × − narrow bandwidth (namely, when the bandwidth is below r + t r − t the order of the frequency being squeezed), then squeez- # ing states give no noticeable advantage over vacuum en- cos s(r + t) cos s(r − t) − − . tanglement harvesting, at least for uniform squeezing. It r + t r − t remains to be seen whether a more general squeezing amplitude (e.g. with continuously varying magnitude and Notice that although these limits do not converge as real phase) can provide the necessary advantages in entan- functions, they do converge as distributions on test func- glement harvesting that we have found here for uniform tions. Namely we have squeezing, while at the same time being implementable in a lab setting. This is an important direction for future sin(sx) lim = δ(x), (A3) research, since such a squeezed field state could over- s→∞ πx come the main experimental limitation of entanglement cos(sx) harvesting: the fast decay with detector separation. lim = 0 = the zero distribution. (A4) s→∞ πx Hence Eq. (A2) simplifies to

1 w0(x, t, 0, 0) =P 2 2 2 ACKNOWLEDGMENTS 4π (r − t ) i h i + δ(3)(r + t) − δ(3)(r − t)] , (A5) 8πr P.S. gratefully acknowledges the support of the NSERC CGS-M and Ontario Graduate Scholarships. where it should again be emphasized that the principal E.M.-M. acknowledges the funding from the NSERC Dis- value and the delta functions only make sense as distri- covery program and his Ontario Early Research Award. butions. Finally, combining Eqs. (A1) and (A5), we find 16 that for a massless field in (3+1)-dimensions our expres- distribution in Eq. (31), which was obtained in [47] by a sion for the two-point function, Eq. (28), reduces to the completely different method.

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