Unruh deWitt probe of late time revival of quantum correlations in Friedmann spacetimes Ankit Dhanukaand Kinjalk Lochany

2025-05-06 0 0 721.17KB 29 页 10玖币

侵权投诉

Unruh deWitt probe of late time revival of quantum correlations in Friedmann

spacetimes

Ankit Dhanuka∗and Kinjalk Lochan†

Department of Physical Sciences, IISER Mohali,

Sector 81, SAS Nagar, Manauli PO 140306, Punjab, India

Unruh deWitt detectors are important constructs in studying the dynamics of quantum ﬁelds in

any geometric background. Curvature also plays an important role in setting up the correlations

of a quantum ﬁeld in a given spacetime. For instance, massless ﬁelds are known to have large

correlations in de Sitter space as well as in certain class of Friedmann-Robertson-Walker (FRW)

universes. However, some of the correlations are secular in nature while some are dynamic and

spacetime dependent. An Unruh deWitt detector responds to such divergences diﬀerently in diﬀerent

spacetimes. In this work, we study the response rate of Unruh deWitt detectors which interact with

quantum ﬁelds in FRW spacetimes. We consider both conventionally as well as derivatively coupled

Unruh deWitt detectors. Particularly, we consider their interaction with massless scalar ﬁelds in

FRW spacetimes and nearly massless scalar ﬁelds in de Sitter spacetime. We discuss how the term

which gives rise to the infrared divergence in the massless limit in de Sitter spacetime manifests

itself at the level of the response rate of these Unruh deWitt detectors in a wide class of Friedmann

spacetimes. In order to carry out this study, we make use of an equivalence that exists between

massless scalar ﬁelds in FRW spacetimes with massive scalar ﬁelds in de Sitter spacetime. Further,

we show that while the derivative coupling regulates the divergence appearing in de Sitter spacetime,

it does not completely remove them in matter dominated universe. This gives rise to large transitions

in the detector which can be used as a probe of setting up of large correlations in late time era of

the universe as well. We also apply the results of these otherwise formal analyses to the coupling

of hydrogen atoms with gravitational waves. We show that the coupling of hydrogen atoms with

gravitational waves takes a form that is similar to derivatively coupled UdW detectors and hence

has signiﬁcant observational implications as a probe of late time revival of quantum correlators.

∗ankitdhanuka555@gmail.com, ph17006@iisermohali.ac.in

†kinjalk@iisermohali.ac.in

arXiv:2210.11186v1 [gr-qc] 20 Oct 2022

I. INTRODUCTION

Unruh deWitt (UdW) detectors are quantum probes which follow classical trajectories in spacetime while

measuring eﬀects of quantum correlations of a background ﬁeld on quantum systems [1]. Other than their

classical motion in spacetime, they have an internal quantum structure with discrete quantum levels. These

detectors couple to quantum ﬁelds and the coupling of a detector with a quantum ﬁeld can cause the detector

to make transitions between its internal quantum levels. The probability amplitude for a detector to undergo

these transitions depends crucially upon the state of the quantum ﬁeld and the trajectory of the detector

in spacetime. The coupling of particle detectors with quantum ﬁelds (and hence the probability amplitude)

senses the correlations of quantum ﬁeld in the state in which it is placed and the response of the detector

along any particular trajectory also encapsulates in it the fact that the particle content of a quantum ﬁeld in

any state is an observer dependent quantity [2, 3]. Diﬀerent particle detectors diﬀer from each other by their

internal quantum structure (i.e., whether it is a two-level system [3, 4], a quantum harmonic oscillator [5, 6],

etc.), their interaction with the ﬁeld (i.e., whether it employs a monopole coupling, a derivative coupling

[7–12], etc.) and other things like whether they are operative for a ﬁnite time [13] or indeﬁnite time or has

some other form for the switching function [14] or whether they are taken to be point sized or they have

some ﬁnite spatial size [15–18], etc. Though particle detectors have been traditionally used to study quan-

tum ﬁeld theory content in non-inertial settings or classical gravitational settings [2, 3], etc, they have also

been employed to investigate quantum eﬀects of gravity on sensitive observables such as the entanglement

between two entangled UdW detectors with diﬀerent types of relative motion between them [19, 20] (for

more on observer dependent entanglement, refer to [21] and references therein).

Interaction of electromagnetic waves with atoms can be modelled by UdW type coupling [22, 23] and hence

quantum optical setups can be used to test the predictions obtained by analysing UdW type couplings. For

example, one can try to test the prediction that detectors moving along inertial trajectories in ﬂat spacetime

have a vanishing response rate whereas a UdW detector sees a thermal response rate when it follows a

uniformly accelerating trajectory in ﬂat spacetime [2, 3]. In addition to the investigation of UdW detectors

in ﬂat spacetime, it is also important as well as interesting to analyse how curvature contributes to the

response rate of UdW detectors in curved spacetimes [24]. One natural arena where curvature is present

in the analysis is that of the evolution of our own universe where diﬀerent epochs can be approximated by

Friedmann-Robertson-Walker (FRW) spacetimes. In fact, the quantum dynamics of metric ﬂuctuations over

these spatially homogeneous and isotropic background FRW spacetimes play an important role in shaping

the present day universe the way we observe it [25]. Quantum ﬁeld theories in FRW spacetimes have been

studied extensively [1, 26–32] and they provide important insights into the formal aspects of QFT as well as

into our universe. A number of past studies, [33–38], have also analysed the behaviour of quantum ﬁelds in

FRW spacetimes by coupling them to UdW type particle detectors. For example, [33] studies the response

rate of UdW detector which are coupled with real quantum scalar ﬁelds in the conformal as well as massless

case in de Sitter spacetime, [37] studies the response rate for quadratically coupled complex scalar ﬁelds again

in both conformal as well as massless cases in de Sitter spacetime, [35] studies the transition probability for

conformal vacua in FRW spacetimes. One important and well known feature of quantum ﬁelds in a class of

FRW spacetimes is that their Wightman functions have infrared divergences [32, 39–44]. Such a divergence

has a root in the fact that massless ﬁelds in power law FRW universes are conformally equivalent to massful

ﬁelds in de Sitter spacetime, which may harbour such divergences in correlations. It has been argued recently

[45] that potential divergences in correlation functions strongly enhance the UdW responses to reveal small

acceleration dependence. Therefore, we expect that the infrared divergences in FRW spacetimes should also

lead to enhancement of the UdW response rates. With this motivation, we seek to investigate the coupling

of UdW detectors with quantum ﬁelds in FRW spacetimes.

First, we consider the conventional Unruh deWitt (UdW) type coupling where there is a monopole coupling

between the ﬁeld and the operator which causes the transitions in the internal quantum space of the de-

tector [3]. In this case, the response rate of transition between quantum states of the detector is related

to the Wightman function of the quantum ﬁeld in the considered spacetime. Therefore, the behaviour of

the correlations of quantum ﬁeld between spacetime points along the trajectory of the detector is imprinted

in the expression of the transition response rate. In the case of de Sitter spacetime, we consider nearly

massless scalar ﬁelds and place them in the Bunch Davies vacuum [40]. The Wightman function of a scalar

ﬁeld in de Sitter spacetime has an infrared divergence [42–44, 46] in the mass going to zero limit. The term

corresponding to the infrared divergence has no spacetime dependence and it provides a dominant secular

contribution to the response rate of the detector.

We also consider massless scalar ﬁelds in radiation dominated spacetime and matter dominated spacetimes

to study detector response in other epochs of cosmological expansion. For these cases, we make use of an

equivalence between massless scalar ﬁelds in FRW spacetimes with that of massive scalar ﬁelds in de Sitter

spacetime [32, 47, 48]. Using this equivalence, we place massless scalar ﬁelds in FRW spacetimes in the Bunch

Davies like vacuum of the corresponding massive scalar ﬁeld in de Sitter spacetime. The Wightman function

of massless ﬁelds in matter dominated cases inherit the infrared divergence of the de Sitter spacetime but

now with a time dependent conformal factor multiplying the divergent term [32, 47]. Thus, we ﬁnd that the

term corresponding to the infrared divergence provides the dominant contribution to the transition response

rate. For radiation dominated case, the massless scalar ﬁeld correlator does not possess any such infrared

divergent term and hence provides ﬁnite detector response.

The analysis shows that the infrared divergences of the de Sitter and matter dominated spacetimes man-

ifest themselves in the detector response. However, in de Sitter spacetime the divergence of correlators is

sometimes argued to be originated from breaking of de Sitter symmetry [40, 42] and any physically sensible

result should be free of any divergences. A line of argument to that end is to regard only those operators

as physical which are infrared ﬁnite. For example, [49] argues that the shift invariant operators like the

diﬀerences of the ﬁeld operators, derivatives of ﬁelds etc., are to be regarded as true physical observables

as they are infrared ﬁnite for massless scalar ﬁelds in de Sitter spacetime. Similarly, the derivatives present

in the stress energy operator also renders it infrared ﬁnite for the de Sitter spacetime [50]. Keeping these

arguments in mind, we look at the response of more ’physical’ derivatively coupled UdW detectors.

In the derivative coupling case, the detector couples to the derivative of the ﬁeld with respect to the proper

time along the trajectory [7]. For this case, the response rate of transition between quantum states of the

detector depends upon the double derivative of the Wightman function of the ﬁeld with respect to the proper

time at diﬀerent points along the detector’s trajectory [7, 51]. In the case of de Sitter spacetime, under the

action of the derivatives, this term goes away as the infrared divergent term in the Wightman function for

nearly massless scalar ﬁelds does not have any spacetime dependence. Thus the transition response rate

of derivatively coupled UdW detector for nearly massless scalar ﬁelds in de Sitter spacetime remains ﬁnite.

However, in the case of massless scalar ﬁelds in matter dominated spacetimes, the infrared divergent term

has time dependence and it does not vanish under the action of derivatives with respect to the detector’s

proper time and provides the dominant contribution to the response rate. Thus, even though the derivative

coupling could cure the infrared divergence of the de Sitter spacetime, this does not happen for the matter

dominated spacetimes. Using this we argue that the realistic physical systems, e.g. the derivatively couple

UdW detectors are expected to capture the revival of quantum correlations in the matter dominaated era of

the universe.

In addition to these formal analyses of UdW detectors and derivatively coupled UdW detectors for quantum

scalar ﬁelds in the considered spacetimes, we investigate the scenario of the coupling of hydrogen atoms with

gravitational waves where a derivatively coupled UdW like coupling occurs. Following the treatment given

in [52, 53], we consider the interaction of a non-relativistic hydrogen atom (whose center of mass is moving

along some time-like classical trajectory) with the curvature of the spacetime. Considering gravitational

wave perturbations over the homogeneous and isotropic FRW backgrounds, one can ﬁnd the form of the

above interaction term upto to leading order in gravitational perturbations. The interaction between gravi-

tational waves and hydrogen atom has the form of a generalized derivatively coupled UdW detector. For this

setting, the above analysis of the response rate of derivatively coupled UdW detector in matter dominated

spacetimes can be carried over and the implications of the dominant infrared term on the transition of the

electron of the atom between its diﬀerent atomic states can be investigated. Such an analysis also provides

a potential avenue to look for observational signatures of quantized gravitational waves.

The rest of the paper is divided in four sections. In section II, we consider conventional UdW detectors for

scalar ﬁelds in de Sitter, radiation-dominated and matter-dominated spacetimes and calculate the response

rate for them. In section III, we perform a similar analysis as is done in section II but for derivatively coupled

UdW detectors. In section IV, we consider a speciﬁc UdW coupling where the detector couples with the

stress energy tensor of the ﬁeld. In this section, we also look at the dynamics of a hydrogen atom in FRW

spacetimes with and without gravitational wave perturbations which harbours a derivatively coupled UdW

detector like structure. In section V, we summarize the results obtained in this paper and their implications.

II. CONVENTIONAL UDW DETECTORS

In this section, we consider the response rate of a conventional Unruh deWitt detector which couples with

massless scalar ﬁelds in FRW spacetimes. A conventional Unruh deWitt detector couples with a quantum

ﬁeld by the following type of interaction term [1, 3]

Hint =cˆµ(τ)χ(τ)ˆ

φ(x(τ)) ,(1)

where ˆµ(τ) is the detector term which governs the transitions within the internal quantum structure of the

detector and ˆ

φ(x(τ)) is a quantum ﬁeld which the detector is coupled to. Here τrepresents the proper time

of the detector along its classical timelike trajectory, x(τ), and χ(τ) is a real-valued switching function which

decides how the detector is turned on and oﬀ [13, 54] .

Let us consider the case when detector makes a transition from some state |0iDto another state |ΩiD

which have energies 0 and Ω, respectively and the ﬁeld starts in some state, |ψi, while it is allowed to go to

any ﬁnal state which are traced over. The transition probability for this case, in ﬁrst order time-dependent

perturbation theory [1], is given by

P0→Ω=c2|DhΩ|ˆµ(0)|0iD|2ZZ dτ1dτ2χ(τ1)χ(τ2)e−iΩ(τ1−τ2)G(x(τ1), x(τ2)),(2)

where G(x(τ1), x(τ2)) = hψ|ˆ

φ(x(τ1)) ˆ

φ(x(τ2))|ψiis the two point function of the quantum ﬁeld in the state |ψi.

For a switching function operating uniformly over τito τf, the transition probability is given by

P0→Ω=c2|DhΩ|ˆµ(0)|0iD|2Zτf

τiZτf

τi

dτ1dτ2e−iΩ(τ1−τ2)G(x(τ1), x(τ2)) .(3)

As motivated above that we want to investigate the role of the curvature of cosmological space and the

divergent structure of correlations of quantum ﬁelds in a class of these spacetimes on the response rate of

UdW detectors, we specialize to Friedmann spacetimes. We consider the case in which the UdW detector

moves along comoving trajectories for which the spatial coordinates are ﬁxed and the comoving time is the

proper time. Thus, we can go to the conformal coordinates in which dτ =a(η)dη where a(η) denotes the

scaling factor of the FRW spacetime under consideration i.e., ds2=a2(η)(−dη2+d~x2). The probability

amplitude, expressed in conformal coordinates, is given by

P0→Ω=c2|DhΩ|ˆµ(0)|0iD|2Zηf

ηiZηf

ηi

dη1dη2e−iΩ(τ(η1)−τ(η2))a(η1)a(η2)G(x(η1), x(η2)) ,(4)

where ηiand ηfare the values for the conformal coordinate corresponding to the τiand τf, respectively.

We now make a change of variables and introduce new coordinates ˜η≡(η1+η2)/2 and ∆η≡η1−η2.For

any ﬁxed ˜η∈(ηi,(ηi+ηf)/2), we have η2∈(ηi,2˜η−ηi) and ∆η∈(−2(˜η−ηi),2(˜η−ηi)).Similarly, for any

ﬁxed value of ˜η∈((ηi+ηf)/2, ηf), we have η2∈(2˜η−ηf, ηf) and ∆η∈(−2(ηf−˜η),2(ηf−˜η)).

Using Eq. (4) and going to the (˜η, ∆η) coordinates, we ﬁnd that the rate of transition probability with

respect to ˜η, for ˜η∈(ηi,(ηi+ηf)/2)), is given by

c2|DhΩ|ˆµ(0)|0iD|2

dP0→Ω

d˜η=Z2(˜η−ηi)

−2(˜η−ηi)

d(∆η)e−iΩ(τ(˜η+(∆η)/2)−τ(˜η−(∆η)/2))

Gx˜η+ (∆η)/2, x˜η−(∆η)/2a˜η+ (∆η)/2a˜η−(∆η)/2,(5)

whereas for ˜η∈((ηi+ηf)/2, ηf), the rate of transition probability is given by

c2|DhΩ|ˆµ(0)|0iD|2

dP0→Ω

d˜η=Z2(ηf−˜η)

−2(ηf−˜η)

d(∆η)e−iΩ(τ(˜η+(∆η)/2)−τ(˜η−(∆η)/2))

Gx˜η+ (∆η)/2, x˜η−(∆η)/2a˜η+ (∆η)/2a˜η−(∆η)/2.(6)

In order to analyse the case of interest i.e., massless scalar ﬁelds in power-law type FRW spacetimes, we

make use of an equivalence [32, 47, 48] according to which a massless scalar ﬁeld in an FRW spacetime

with scaling factor, a(η) = (Hη)−q, can be mapped to a massive scalar ﬁeld in de Sitter spacetime with

m2=H2(1−q)(2+q). One also ﬁnds that the Wightman functions in the two equivalent settings are related

by the following relation

GF RW (x1, x2)=(Hη1)q−1(Hη2)q−1GdS (x1, x2),(7)

for more details, refer Appendix A.2 of [32].

As for the state in the corresponding de Sitter spacetime is concerned, we take that to be the Bunch-

Davies vacuum [40] for which the Wightman function1is given by

GdS (x1, x2) = H2

16π2Γ3

2+νΓ3

2−ν2F13

2+ν, 3

2−ν, 2,1−y

4,(8)

where

y(x1, x2) = −(η1−η2−i)2+ (~x1−~x2)2

η1η2

,(9)

and ν=q9

4−m2

H2.

From the formula that, m2=H2(1 −q)(2 + q), we see that the square of the mass is positive only for the

cases in which q∈[−2,1). We consider only those FRW spacetimes which have qbelonging to this range.

Since a(η)=(Hη)−q, we see that η∈(0,∞) corresponds to expanding spacetimes for q∈[−2,0], whereas

for q∈[0,1), η∈(−∞,0) corresponds to expanding spacetimes. Let us brieﬂy look at the response rate for

UdW detectors which remain operative for the full time range of these spacetimes. We argue in Appendix

A that the inﬁnite time response rate with respect to ˜ηhas the following dependence on Ω and H

c2|DhΩ|ˆµ(0)|0iD|2

dP0→Ω

d˜η∝(ΩH−q)1

1−q.(10)

From this expression, we see that, for q∈(−2,0), the exponent of His positive and hence the response

rate increases with increasing H. While for q∈(0,1), the exponent of His negative and the response rate

decreases with increasing H. In fact, the Ricci scalar for FRW spacetimes is given by R∝H2qand we

conclude that, for q∈(−2,0), it increases with decreasing Hwhile, for q∈(0,1), it increases with increasing

H. Hence, the behaviour of the response rate and the Ricci scalar with respect to H(for an FRW spacetime)

are opposite of each other. Other important point to notice is that the response rate gets most enhanced

with H(>1) for q=−2 whereas for H(<1), it is for q= 1 case that the response rate is most signiﬁcantly

enhanced with H. As we see below that the infrared divergent factors in de Sitter and matter dominated

spacetimes cause very fast transitions, the above conclusion implies that for these cases, the Hdependence

also contributes maximally to the response rate compared to the other considered spacetimes.

In standard cosmological setting universe remains in a given phase only for a ﬁnite time, thus it will be

worthwhile to consider the ﬁnite time response rates for diﬀerent cosmological eras as we do next.

1Though we are considering Bunch Davies vacuum here, one could also consider other physically well-behaved normalizable

states [55]. Such well-behaved states also share the divergent Bunch Davies correlator structure in addition to their own

characteristic features which, however, do not signiﬁcantly alter the characteristics used in our analysis [48, 55, 56].

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

UnruhdeWittprobeoflatetimerevivalofquantumcorrelationsinFriedmannspacetimesAnkitDhanukaandKinjalkLochanyDepartmentofPhysicalSciences,IISERMohali,Sector81,SASNagar,ManauliPO140306,Punjab,IndiaUnruhdeWittdetectorsareimportantconstructsinstudyingthedynamicsofquantumeldsinanygeometricbackground.Curvat...

展开>> 收起<<

Unruh deWitt probe of late time revival of quantum correlations in Friedmann spacetimes Ankit Dhanukaand Kinjalk Lochany.pdf

共29页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Unruh deWitt probe of late time revival of quantum correlations in Friedmann spacetimes Ankit Dhanukaand Kinjalk Lochany

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: