particle detectors, cavities, and the weak …particle detectors, cavities, and the weak equivalence...

Particle Detectors, Cavities, and the Weak Equivalence Principle

Erickson Tjoa,1, 2, ∗ Robert B. Mann,1, 2, † and Eduardo Martın-Martınez3, 2, 4, ‡

1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada2Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

3Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada4Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, N2L 2Y5, Canada

We analyze a quantum version of the weak equivalence principle, in which we compare the responseof a static particle detector crossed by an accelerated cavity with the response of an accelerateddetector crossing a static cavity in (1+1)-dimensional flat spacetime. We show, for both massiveand massless scalar fields, that the non-locality of the field is enough for the detector to distinguishthe two scenarios. We find this result holds for vacuum and excited field states of different kindsand we clarify the role of field mass in this setup.

I. INTRODUCTION

The weak equivalence principle (WEP) has been oneof the central tenets of gravitational physics. It has a va-riety of formulations, but it asserts that the local effectsof motion in a curved spacetime cannot be distinguishedfrom those of an accelerated observer in flat spacetime.The proviso of locality eliminates measurable tidal forces(that would originate, for example, from a radially con-vergent gravitational field) acting upon finite sized physi-cal bodies. It implies that the trajectories of bodies withnegligible gravitational binding energy are independentof their composition and structure, and depend only theirinitial positions and velocities.

With the development of the Unruh-DeWitt (UDW)model in quantum field theory, WEP can be analyzed inthe presence of quantum fields in contrast to the origi-nal classical formulation. While not equivalent to a fullquantum version of the WEP, this approach provides anoperational means of understanding some important as-pects of the WEP in a quantum context. In particular,since UDW detectors capture fundamental features of thelight-matter interaction for atomic systems [1], one canoperationally study the WEP by asking if a free-fallingdetector in a stationary cavity in uniform gravitationalfield has a different response from that of a stationarydetector surrounded by a free-falling cavity. This prob-lem has recently been revisited in the context of movingmirrors [2].

Renewed effort has been expended in recent years to-wards reanalyzing the role of atomic detector models cou-pled to a real scalar field with regards to the connectionto gravitational phenomena. It has been argued thatnon-inertiality can be distinguished locally by exploitingnonlocal correlations of the field [3–5], effectively provid-ing an accelerometer. An analysis of the behaviour of aUDW detector in a static cavity indicated that QFT mayprovide a way of distinguishing between flat-space accel-

∗ [email protected]† [email protected]‡ [email protected]

eration and free-fall in the near-horizon regime [4]. Morerecently [6] atoms falling through a cavity near an eventhorizon, together with short-wavelength approximations,led to radiation that is Hawking-like as seen by observersat spatial infinity. Even more recently, an analysis of amoving mirror in cavity [2] has been used to argue onceand for all that a “qualitative WEP” should hold in aquantum-field theoretic setting, and emphasized the im-portance of the initial state of the field in determiningradiation from a moving mirror. This investigation fo-cused on mirrors lacking internal degrees of freedom, butnonetheless had the advantage of providing informationabout the stress-energy tensor in the cavity, which wasapparently missed in the past.

Here we will complement these recent studies by show-ing that atomic UDW detectors also exhibit a qualitativeWEP. In particular, we revisit the old problem of com-puting the response of a static detector surrounded byan accelerating cavity, and the response of an accelerat-ing detector that is surrounded by a static cavity. Thekey issue here is not the composition of the detector (thebody), but rather of its quantum field (vacuum) environ-ment. We consider the response for various field states,including the (scalar) vacuum, excited Fock states, andalso single-mode coherent field states. We find that themass of the quantum scalar field does not enter into theresponse of the detector apart from providing a degrada-tion in the transition amplitude and larger mode frequen-cies in the mode decomposition. This is a consequenceof the fact that the conformal invariance of the masslessKlein-Gordon equation is not a physical effect, and is tobe distinguished from the conformal flatness of the space-time under consideration. For non-vacuum field states,we show how resonance can be used to amplify the tran-sition probability via co-rotating terms and demonstratethe irrelevance of the mass of the field in the physics un-derlying WEP.

Our paper is organized as follows. In Section II, werevisit the formulation of WEP and clarify the contextsin which this work and others, in particular [2, 6], areperformed. In Section III we provide the standard setupand generic expressions for a UDW detector coupled toa Klein-Gordon field, without restriction to the vacuum

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state of the field. In Section IV we consider an accel-erating detector traversing the entire static cavity, not-ing the necessary changes if detector starts acceleratingsomewhere within the cavity. In Section V we considera static UDW detector that encounters an acceleratingcavity, entering one end and leaving the other due to themotion of the cavity; we also note the necessary modi-fications if the trajectory of the detector is changed. InSection VI we consider various non-vacuum field excita-tions and the role of resonance between atomic gap andexcited cavity modes. In Section VII we compute tran-sition rate to better understand if the difference betweentwo scenarios are not averaged out by the transition prob-ability calculations.

Throughout we adopt c = ~ = 1 so that the massparameter m has units of inverse length.

II. WEAK EQUIVALENCE PRINCIPLEREVISITED

It is a remarkable property of gravity, in contrast toother non-gravitational forces, that every test particleequally and universally experiences the influence of grav-itational fields. This underlies the WEP, which statesthat phenomenology of bodies observed from frames inuniform gravitational fields is equivalent to that of framesthat accelerate uniformly relative to inertial [free-falling]frames [7]. In other words, WEP states that a mass free-falling in a stationary cavity with uniform gravitationalfield g = −gez is completely equivalent to a stationarymass with uniformly accelerating cavity1 with a = gez,as shown in Figure 1. This principle has been verifiedto great accuracy through various experiments and ef-fectively sets inertial mass and gravitational mass to beequal.

The purely classical version of the WEP, while assert-ing that free-fall is independent of a body’s composition,does not consider internal quantum degrees of freedom ofa body (unlike a qubit). For example, the body is con-sidered to be uncharged and the space inside the cavityto be free of electromagnetic fields. The quantum versionof the WEP essentially requires us to consider an atomicdetector coupled to some field prepared in some state,as shown in Figure 2. The state most closely resemblingthe classical environment of the WEP is the quantumvacuum. Other field states can of course be considered,but they will in general produce environments analogousto those of air or some other fluid that produces dragon the body. For example [2], having an electromagneticfield makes the argument less trivial: a classical electriccharge in uniform accelerated motion radiates and it is

1 Alternatively, the normal force experienced by a test mass on thefloor of a closed cavity cannot be attributed to cavity accelera-tion or uniform gravitational field without additional non-localinformation (e.g. by looking out of the cavity).

nontrivial to ask whether a free-falling charge radiates.The reason is because in general relativity, free-fallingis an inertial motion (geodesic motion) and accelerationcorresponds to non-geodesics in spacetime. According tothe equivalence principle, however, we should be able tospeak of uniform acceleration and constant gravitationalfield intercontrovertibly. Where is the problem?

2

cavity, entering one end and leaving the other due to themotion of the cavity; we also note the necessary modi-fications if the trajectory of the detector is changed. InSection VI we consider various non-vacuum field excita-tions and the role of resonance between atomic gap andexcited cavity modes. In Section VII we compute tran-sition rate to better understand if the dierence betweentwo scenarios are not averaged out by the transition prob-ability calculations.



It is a remarkable property of gravity, in contrast toother non-gravitational forces, that every test particleequally and universally experiences the influence of grav-itational fields. This underlies the WEP, which statesthat phenomenology of bodies observed from frames inuniform gravitational fields is equivalent to that of framesthat accelerate uniformly relative to inertial [free-falling]frames [7]. In other words, WEP states that a mass free-falling in a stationary cavity with uniform gravitationalfield g = ≠gez is completely equivalent to a stationarymass with uniformly accelerating cavity1 with a = gez,as shown in Figure 1. This principle has been verifiedto great accuracy through various experiments and ef-fectively sets inertial mass and gravitational mass to beequal.

The purely classical version of the WEP, while assert-ing that free-fall is independent of a body’s composition,does not consider internal quantum degrees of freedom ofa body (unlike a qubit). For example, the body is con-sidered to be uncharged and the space inside the cavityto be free of electromagnetic fields. The quantum versionof the WEP essentially requires us to consider an atomicdetector coupled to some field prepared in some state,as shown in Figure 2. The state most closely resemblingthe classical environment of the WEP is the quantumvacuum. Other field states can of course be considered,but they will in general produce environments analogousto those of air or some other fluid that produces dragon the body. For example [2], having an electromagneticfield makes the argument less trivial: a classical electriccharge in uniform accelerated motion radiates and it isnontrivial to ask whether a free-falling charge radiates.The reason is because in general relativity, free-fallingis an inertial motion (geodesic motion) and accelerationcorresponds to non-geodesics in spacetime. According tothe equivalence principle, however, we should be able to


speak of uniform acceleration and constant gravitationalfield intercontrovertibly. Where is the problem?

mm

a

g

FIG. 1. Classical WEP setup: test mass m in a closed cavity.WEP claims that these two are kinematically indistinguish-able in absence of second-order eects such as tidal forces ornon-uniform acceleration. The space inside the cavity is atrue vacuum in the classical sense, apart from existence ofgravitational field to mimic acceleration.

a

g

FIG. 2. Quantum WEP setup: a two-level atomic ‘detector’with gap (with respect to its proper frame) in a cavity.The cavity contains a quantum field whose modes may leadto atomic excitations (even in the vacuum state), and hencedetector responses. The atom serves as a detector in the sensethat a particle is detected when the atom is excited and thenemit radiation [8].

Here we revisit this problem using an Unruh-DeWitt(UDW) detector to replace the mirror in [2]. Specifically,we consider two dierent “Experiments”:

(1) Stationary cavity, accelerating detector: In thisscenario, we let an atomic detector undergo uniformacceleration as it crosses the cavity containing thequantum field. This mimics the scenario of free-falling atom in gravitational field e.g. outside a blackhole especially near the horizon, when the metric isapproximately Rindler-like [6].

(2) Stationary detector, accelerating cavity: Inthis scenario, a rigid cavity is accelerating such that astationary detector traverses across the cavity. Thismimics a free-falling cavity in gravitational field un-der appropriate quasilocal approximations.

If WEP holds, then we should expect that Experiment1 and Experiment 2 should be qualitatively symmetricin some regimes for the scalar field vacuum. More gener-ally., we might expect that for other field states that themotion of the body is independent of the field mass. In

FIG. 1. Classical WEP setup: test mass m in a closed cavity.WEP claims that these two are kinematically indistinguish-able in absence of second-order effects such as tidal forces ornon-uniform acceleration. The space inside the cavity is atrue vacuum in the classical sense, apart from existence ofgravitational field to mimic acceleration.

2

cavity, entering one end and leaving the other due to themotion of the cavity; we also note the necessary modi-fications if the trajectory of the detector is changed. InSection VI we consider various non-vacuum field excita-tions and the role of resonance between atomic gap andexcited cavity modes. In Section VII we compute tran-sition rate to better understand if the dierence betweentwo scenarios are not averaged out by the transition prob-ability calculations.



It is a remarkable property of gravity, in contrast toother non-gravitational forces, that every test particleequally and universally experiences the influence of grav-itational fields. This underlies the WEP, which statesthat phenomenology of bodies observed from frames inuniform gravitational fields is equivalent to that of framesthat accelerate uniformly relative to inertial [free-falling]frames [7]. In other words, WEP states that a mass free-falling in a stationary cavity with uniform gravitationalfield g = ≠gez is completely equivalent to a stationarymass with uniformly accelerating cavity1 with a = gez,as shown in Figure 1. This principle has been verifiedto great accuracy through various experiments and ef-fectively sets inertial mass and gravitational mass to beequal.

The purely classical version of the WEP, while assert-ing that free-fall is independent of a body’s composition,does not consider internal quantum degrees of freedom ofa body (unlike a qubit). For example, the body is con-sidered to be uncharged and the space inside the cavityto be free of electromagnetic fields. The quantum versionof the WEP essentially requires us to consider an atomicdetector coupled to some field prepared in some state,as shown in Figure 2. The state most closely resemblingthe classical environment of the WEP is the quantumvacuum. Other field states can of course be considered,but they will in general produce environments analogousto those of air or some other fluid that produces dragon the body. For example [2], having an electromagneticfield makes the argument less trivial: a classical electriccharge in uniform accelerated motion radiates and it isnontrivial to ask whether a free-falling charge radiates.The reason is because in general relativity, free-fallingis an inertial motion (geodesic motion) and accelerationcorresponds to non-geodesics in spacetime. According tothe equivalence principle, however, we should be able to


speak of uniform acceleration and constant gravitationalfield intercontrovertibly. Where is the problem?

mm

a

g

FIG. 1. Classical WEP setup: test mass m in a closed cavity.WEP claims that these two are kinematically indistinguish-able in absence of second-order eects such as tidal forces ornon-uniform acceleration. The space inside the cavity is atrue vacuum in the classical sense, apart from existence ofgravitational field to mimic acceleration.

a

g

FIG. 2. Quantum WEP setup: a two-level atomic ‘detector’with gap (with respect to its proper frame) in a cavity.The cavity contains a quantum field whose modes may leadto atomic excitations (even in the vacuum state), and hencedetector responses. The atom serves as a detector in the sensethat a particle is detected when the atom is excited and thenemit radiation [8].

Here we revisit this problem using an Unruh-DeWitt(UDW) detector to replace the mirror in [2]. Specifically,we consider two dierent “Experiments”:



If WEP holds, then we should expect that Experiment1 and Experiment 2 should be qualitatively symmetricin some regimes for the scalar field vacuum. More gener-ally., we might expect that for other field states that themotion of the body is independent of the field mass. In

FIG. 2. Quantum WEP setup: a two-level atomic ‘detector’with gap Ω (with respect to its proper frame) in a cavity.The cavity contains a quantum field whose modes may leadto atomic excitations (even in the vacuum state), and hencedetector responses. The atom serves as a detector in the sensethat a particle is detected when the atom is excited and thenemit radiation [8].

Here we revisit this problem using an Unruh-DeWitt(UDW) detector to replace the mirror in [2]. Specifically,we consider two different “Experiments”:



3

If WEP holds, then we should expect that Experiment1 and Experiment 2 should be qualitatively symmetricin some regimes for the scalar field vacuum. More gener-ally., we might expect that for other field states that themotion of the body is independent of the field mass. Inorder to make useful and valid comparisons in the con-text of WEP, we generally need to ensure two additional‘requirements’ on the setup in question.

First of all, we will need to be able to set up a kindof cavity undergoing constant acceleration across its fullspatial extent. This is forbidden in special relativitywithout abandoning the rigidity condition [9–11] (staticboundary condition in Rindler coordinates). Therefore,a rigid accelerating cavity suitable for WEP is necessar-ily in a quasilocal regime in the sense of aL 1 where Lis the proper length of the cavity at rest as measured inthe lab frame. Outside this regime, we see that the ac-celerating cavity will have detectable non-uniform properaccelerations across the cavity and hence in the comovingframe of the cavity, the stationary detector (with respectto lab frame) does not undergo uniform acceleration sincethe worldline of the detector crosses hypersurfaces of con-stant but different accelerations. Therefore, Experiment1 and Experiment 2 are only equivalent in quasilocal ap-proximations.

Secondly, we will need to show that the distinction be-tween detector responses in Experiments 1 and 2 shouldbe qualitatively independent of the mass of the quantumfield and the initial state of the field within the quasilocalregime. In other words, quantitative differences betweenExperiments 1 and 2 would then be attributed to non-local correlations: the atom is sensitive to the inequiva-lent setups in the two experiments and also the fact thatmoving-boundary/stationary-atom is not the same as amoving-atom/stationary-boundary from a physical pointof view.

Furthermore, the role of the field mass should onlyserve to degrade non-local correlations of the field andhence diminish transition amplitudes, all else beingequal. This requirement, however, is in slight tensionwith previous results [3] claiming (in the non-relativisticregime) that the field mass term can enhance the tran-sition probability of a detector, making it a better ac-celerometer in the case of highly excited field states. Thiswould mean that the mass of a scalar field leads to ad-ditional physical effects beyond suppressing correlations.For the WEP in particular, one could imagine increasingthe mass more and more to detect increasingly small lo-cal accelerations. We will recover consistency with WEPby showing that this discrepancy is in part due to mix-ing conformal flatness with conformal invariance of theKlein-Gordon equation. We also note that the idea thatmassive excitations should be ‘harder’ to detect than themassless ones, all things being equal, is not new — it hasbeen investigated e.g. in [12]. A more complete discus-sion of these issues is given in Appendix A and B.

In light of these two requirements, in the next few sec-tions we will consider the setup and demonstrate that the

qualitative WEP is indeed observed. In particular, we re-cover the expectation that massless field should be able todetect relative acceleration (non-inertiality) as well as themassive field, if not better, in quasilocal regime. The ideathat detection of massive excitations should be ‘harder’than the corresponding massless ones is not new (see, forexample, [12]). This entails the clarification that con-formal invariance in the massless case has nothing to dowith the physics of uniform acceleration and hence WEP;it is a computational convenience that one can invoke (cf.Appendix A), to be distinguished from the fact that alltwo-dimensional spacetimes are conformally flat. We willstrengthen this claim by considering an arbitrary Fockstate and a single-mode coherent state, and check thatno essential differences arise even in the transition rate(which is a differential version of the detector response).

III. SETUP

Our starting point is the Klein-Gordon equation fora real scalar field: the covariant formulation of Klein-Gordon equation which governs the dynamics of a realscalar field reads

1√−g ∂µ(gµν√−g∂νφ

)+m2φ = 0 . (1)

For global Minkowski spacetime, the solutions are givenby plane waves. Recall that all (1+1) dimensional space-times are conformally equivalent to Minkowski space-time: by this we mean that there exists a coordinatesystem in which the metric is conformally flat, i.e. withmetric that takes the form

gµν(x) = Ω2(x)ηµν . (2)

This conformal flatness can be exploited in the case ofm = 0 to map the solutions of the Klein-Gordon equa-tion to the plane-wave solutions in Minkowski spacetimebecause the massless Klein-Gordon equation is confor-mally invariant in (1 + 1) dimensions. This allows usto obtain an exact closed form for the spectrum of thefield modes2. For m 6= 0, the conformal invariance ofthe wave equation is lost and hence conformal flatnessprovides no particular advantage. Therefore, even for auniformly accelerating frame the field modes can be writ-ten in closed form; however neither the normalization northe spectrum can.

To probe the field, we consider a pointlike Unruh-DeWitt (UDW) detector whose interacting Hamiltonianis given by

HI(τ) = λχ(τ)µ(τ)φ(τ, x(τ)) ,µ = eiΩτ σ+ + e−iΩτ σ− ,

(3)

2 An important point here is that conformal invariance is conve-nient but not necessary. We show this in Appendix A.

4

where τ is the proper time of the detector, λ is the cou-pling strength of the detector and the field, µ(τ) is themonopole moment of the detector, σ± are su(2) ladderoperators characterizing the two-level atomic detector,and χ(τ) is the switching function of the detector. Hereσ+ |g〉 = |e〉 and σ− |e〉 = |g〉 where |g〉 , |e〉 refer to theground and excited states of the atom respectively, sepa-rated by energy gap Ω. Note that the interacting Hamil-tonian is given in the Dirac picture.

We consider the initial state to be a separable state|g〉 ⊗ |ψ〉 = |g, ψ〉 where |ψ〉 is some initial pure state ofthe field. If the field is in some |out〉 state after the inter-action and the detector is in excited state |e〉, then thetransition probability of the detector is given by Born’srule after tracing out the field state:

P (Ω) =∑

out

∣∣∣〈e, out|U |g, ψ〉∣∣∣2

(4)

where the time evolution operator in the Dirac picture is

U = T exp(− i~

∫ ∞

−∞dτHI(τ)

). (5)

Employing the Dyson expansion

U = 11 + U (1) +O(λ2) ,

U (1) = − i~

∫ ∞

−∞dτHI(τ)

(6)

we obtain the leading order contribution to the transitionprobability

P (Ω) = λ2∫

dτ∫

dτ ′χ(τ)χ(τ ′)×

e−iΩ(τ−τ ′)W (τ, τ ′) +O(λ4) ,W (τ, τ ′) = 〈ψ|φ[x(τ)]φ[x(τ ′)]|ψ〉 ,

(7)

where W (τ, τ ′) is the pullback of the Wightman functionon the detector’s trajectory x(τ). The remaining task isto compute the Wightman function for different scenar-ios and choose an appropriate switching function of thedetector.

We would like to study further the situation when onespeaks of the weak equivalence principle in the presenceof a quantum field subject to a boundary condition (aDirichlet cavity) in (1+1) dimensions. We are interestedin two types of scenarios (“Experiments”) that can besummarized as follows:

(1) A cavity is static relative to some laboratory frame(t, x) and the detector is accelerating with constantproper acceleration. In the language of the equiv-alence principle, this should also describe a staticcavity in a constant gravitational field (like on thesurface of the Earth), with a free-falling detector.

(2) The detector is static relative to the lab frame and arigid cavity uniformly accelerates, mimicking a free-falling cavity in a uniform gravitational field.

4

operators characterizing the two-level atomic detector,and ‰(·) is the switching function of the detector. Here‡+ |gÍ = |eÍ and ‡≠ |eÍ = |gÍ where |gÍ , |eÍ refer to theground and excited states of the atom respectively, sepa-rated by energy gap . Note that the interacting Hamil-tonian is given in the Dirac picture.

We consider the initial state to be a separable state|gÍ ¢ |ÂÍ = |g,ÂÍ where |ÂÍ is some initial pure state ofthe field. If the field is in some |outÍ state after the inter-action and the detector is in excited state |eÍ, then thetransition probability of the detector is given by Born’srule after tracing out the field state:

P () =ÿ

out

---Èe, out|U |g,ÂÍ---2

(4)

where the time evolution operator in the Dirac picture is

U = T exp3

≠ i

~

⁄ Œ

≠Œd·HI(·)

4. (5)

Employing the Dyson expansion

U = 11 + U (1) +O(⁄2) ,

U (1) = ≠ i

~

⁄ Œ

≠Œd·HI(·)

(6)

we obtain the leading order contribution to the transitionprobability

P () = ⁄2⁄

d·

⁄d· Õ‰(·)‰(· Õ)◊

e≠i(·≠· Õ)W (·, · Õ) +O(⁄4) ,W (·, · Õ) = ÈÂ|„[x(·)]„[x(· Õ)]|ÂÍ ,

(7)

where W (·, · Õ) is the pullback of the Wightman functionon the detector’s trajectory x(·). The remaining task isto compute the Wightman function for dierent scenar-ios and choose an appropriate switching function of thedetector.

We would like to study further the situation when onespeaks of the weak equivalence principle in the presenceof a quantum field subject to a boundary condition (aDirichlet cavity) in (1+1) dimensions. We are interestedin two types of scenarios (“Experiments”) that can besummarized as follows:

(1) A cavity is static relative to some laboratory frame(t, x) and the detector is accelerating with constantproper acceleration. In the language of the equiv-alence principle, this should also describe a staticcavity in a constant gravitational field (like on thesurface of the Earth), with a free-falling detector.

(2) The detector is static relative to the lab frame and arigid cavity uniformly accelerates, mimicking a free-falling cavity in a uniform gravitational field.

x

t

tmax

x1 x2

L

›1 ›2

FIG. 3. Spacetime diagram for the setup under considera-tion. The accelerating detector-static cavity scenario corre-sponds to the trajectories in blue, with accelerating detectordenoted by the solid curved arrow. The accelerating (rigid)cavity-static detector scenario corresponds to the trajectoriesin black, with the static detector denoted by the vertical solidvertical curve. In both scenarios the trajectories of the end-points of the cavity are given by dashed lines.

The spacetime diagram for these two setups are shown inFigure 3. Case (1) corresponds to the trajectories in blue,with accelerating detector denoted by solid curved arrow.Case (2) corresponds to trajectories in black, with staticdetector denoted vertical solid vertical curve. In bothscenarios the cavity trajectories are in dashed lines.

We will see that this result continues to hold evenfor excited field states, including coherent states [13].Therefore, this eectively shows that massive scalar fieldsdo not generally provide additional insight over masslessscalar fields, apart from introducing an additional degra-dation factor (for example in the studies of entanglementgeneration or degradation [14]).

IV. ACCELERATING DETECTOR

Let us first consider the case of a static cavity relativeto the lab frame (t, x). In this case, the cavity is equiva-lent to a Dirichlet boundary condition „(x1) = „(x2) = 0where xj are the locations of the boundary and the lengthof the cavity as measured by the lab frame is L = x2≠x1.The equation of motion for the quantized scalar field re-duces to

(ˆµˆµ +m2)„ = 0 (8)

FIG. 3. Spacetime diagram for the setup under considera-tion. The accelerating detector-static cavity scenario corre-sponds to the trajectories in blue, with accelerating detectordenoted by the solid curved arrow. The accelerating (rigid)cavity-static detector scenario corresponds to the trajectoriesin black, with the static detector denoted by the vertical solidvertical curve. In both scenarios the trajectories of the end-points of the cavity are given by dashed lines.

The spacetime diagram for these two setups are shown inFigure 3. Case (1) corresponds to the trajectories in blue,with accelerating detector denoted by solid curved arrow.Case (2) corresponds to trajectories in black, with staticdetector denoted vertical solid vertical curve. In bothscenarios the cavity trajectories are in dashed lines.

We will see that this result continues to hold evenfor excited field states, including coherent states [13].Therefore, this effectively shows that massive scalar fieldsdo not generally provide additional insight over masslessscalar fields, apart from introducing an additional degra-dation factor (for example in the studies of entanglementgeneration or degradation [14]).

IV. ACCELERATING DETECTOR

Let us first consider the case of a static cavity relativeto the lab frame (t, x). In this case, the cavity is equiva-lent to a Dirichlet boundary condition φ(x1) = φ(x2) = 0where xj are the locations of the boundary and the lengthof the cavity as measured by the lab frame is L = x2−x1.The equation of motion for the quantized scalar field re-duces to

(∂µ∂µ +m2)φ = 0 (8)

5

and under a Dirichlet boundary condition the modes arestanding waves:

φ(t, x) =∞∑

n=1un(x)

(e−iωntan + eiωnta†n

),

un(t, x) = 1√Lωn

sin nπ(x− x1)L

,

(9)

where ω2n =

(nπL

)2 +m2. The normalization of un canbe found using the Klein-Gordon inner product [8]

(φ1, φ2) = −i∫

ΣdΣµ (φ1∂µφ

∗2 − (∂µφ1)φ∗2) , (10)

where Σ is the spacelike hypersurface foliated by theglobal time function defining the timelike Killing vectorof the spacetime.

For the detector/cavity configuration (see figure 3), ifthe detector is accelerating from left wall to the rightwall of the cavity then x1 = 0; if the detector starts frommidpoint, then x1 = −L/2. Starting at the midpoint(as in [3]) is useful if we wish to consider the a = 0limit, since the Dirichlet boundary renders the limit ill-defined for a detector starting from the left where thefield vanishes (i.e. the detector ‘merges’ with the wall).We will consider trajectories in which the detector travels

from one wall to the other as well as from the midpoint asappropriate; we shall refer to the latter kind of trajectoryas a ‘midpoint trajectory’.

If the initial state of the field is the Minkowski vacuumstate |0M 〉, then it is straightforward to show that thepullback of the Wightman function along the trajectoryγ(τ) is

W0(τ, τ ′) =∞∑

n=1un(x(τ))u∗n(x(τ ′)) . (11)

For a uniformly accelerating detector, this trajectory isgiven by

x(τ) = 1a

(sinh aτ, cosh aτ − 1) (12)

where the integration constant is chosen so thatx(γ(0)) = 0. Solving for the time taken to traverse thecavity, we obtain

τmax = cosh−1(1 + aL)a

. (13)

If the detector starts from the midpoint of the cavity,then the expression for the time to exit the cavity is givenby Eq. (13) with L → L/2. Finally, putting everythingtogether, we obtain

PD0 (Ω) = λ2∞∑

n=1

1Lωn

∣∣∣∣∫ ∞

−∞dτχ(τ) sin nπ

L

(cosh aτ − 1

a− x1

)e−iΩτ exp

(−iωn

sinh aτa

)∣∣∣∣2

(14)

for the detector transition probability for the field in thevacuum state. Note that the limits of integration are ef-fectively governed by the switching function. We shallgenerally choose χ(τ) = 1 for the interval [0, τmax] andzero otherwise (the so-called top-hat switching). We usethe superscript D to denote an accelerating detector in acavity that is static with respect to the lab frame (t, x);otherwise we use a superscript C. Note that the cavityforces the field to be compactly supported in the interval[x1, x2], beyond which the detector experiences no inter-action with the field.

We remark that for a trajectory where the detectortraverses the entire cavity (from one wall to another),the divergences associated with sudden switching do notoccur because the Dirichlet boundary condition causesthe field to vanish there (see for instance [15]). Effec-tively, the detector does not see the discontinuity in theswitching. Furthermore, while divergences due to sud-den switching arise in quite general contexts [16], it isalso now known that the spurious divergence due to sud-den switching in Minkowski space is in fact dimension-dependent [17] and the setup (1+1)D does not suffer this

problem due to logarithmic nature of the singularity inthe Wightman function. Since the mode sum is conver-gent even without a UV regulator, imposing UV cutoffis a computational convenience (cf. Appendix C). An IRcutoff naturally arises from the Dirichlet boundary con-dition; thus the usual divergence of a massless scalar fieldin (1+1) dimensions does not appear either.

V. ACCELERATING CAVITY

Now suppose we consider a rigid cavity of length L asmeasured in the lab frame at t = 0. The cavity is uni-formly accelerating in the positive x-direction, and thereis an inertial UDW detector at rest at (t, xd) where xdis constant. This corresponds to the detector passingthrough a cavity with moving boundary conditions. In(1+1) dimensions, there is an analytic solution to thisseemingly difficult problem m = 0: we perform a coordi-nate transformation

t = eaζ

asinh aς , x = eaζ

acosh aς (15)

6

where (ς, ζ) are sometimes known as the Lass or radarRindler coordinates [18] — we will refer to these as con-formal Rindler coordinates. This coordinate system cov-ers the usual Rindler wedge and has the special propertythat the metric is conformal to the Minkowski metric:

ds2 = dt2 − dx2 = e2aζ (dς2 − dζ2) . (16)

Each line of constant ζ describes a uniformly acceleratingtrajectory with proper acceleration |aµaµ|1/2 = ae−aζ .Consequently the kinematical parameter a for the lineζ = 0 corresponds to the proper acceleration of the testparticle along this trajectory. In these coordinates, thecavity walls correspond to Dirichlet boundary conditionsat ζ = ζ1, ζ2. Since we are comparing the scenarios inwhich the detector traverses the entire cavity, we will alsochoose ζ1 = 0 so that the proper acceleration of the leftwall matches the acceleration of the detector3. Invertingthe coordinates, the trajectory of the static detector is

ς = 1a

tanh−1 t

xd, ζ = 1

alog a

√x2d − t2 . (17)

If we define the left wall to be at ζ = ζ1 = 0, then theproper length of the cavity in conformal coordinates is

L = x2 − x1

∣∣∣t=0

=∫ ζ2:=L′

ζ1=0dζ eaζ = eaL

′ − 1a

, (18)

which can be inverted to give L′ = a−1 log(1 + aL). Cru-cially, x2−x1 6= ζ2−ζ1. If the detector starts at the rightwall and the cavity accelerates in the positive x-direction,then we have xd = a−1 + L. The maximum interactiontime is obtained by solving for

a√x2d − t2 = 1 =⇒ tmax =

√2La

+ L2 . (19)

For the massless field, the Klein-Gordon equation is con-formally invariant under the above transformation andhence the modes in this coordinate system read

φ(ς, ζ) =∞∑

n=1vn(ζ)

(e−iωnς bn + eiωnς b†n

), (20)

vn = 1√nπ

sin nπ(ζ − ζ1)L′

, (21)

where we have used the fact that the normalization sim-plifies due to

√L′ωn =

√nπ. Note that L′ 6= L since

conformal transformation does not preserve length, i.e.the comoving length of the cavity in radar coordinates isζ2 − ζ1 6= L.

Since t = τ is the proper time, the full transition prob-ability for traversing the entire cavity is

PC0 (Ω) = λ2∞∑

n=1

∣∣∣∣∣

∫ ∞

−∞dτχ(τ) sin nπ log

√(1 + aL)2 − a2τ2

log 1 + aLe−iΩτ exp

(−inπ tanh−1 aτ

1+aLlog(1 + aL)

)∣∣∣∣∣

2

, (22)

with the top-hat switching in the interval [0, τmax], notingthat here tmax = τmax.

If the detector trajectory is such that at t = 0 it is atmidpoint of the cavity (‘midpoint detector’), then someparts of these expressions will need to be changed if wewant the kinematical parameter a to be the proper accel-eration at the centre of the cavity (such as is done in [3]).

3 Note that if we consider the midpoint of the cavity to have ac-celeration a, then it is not true that the walls are located atζj = ±L′/2: conformal transformations do not preserve dis-tances between two points. In particular, it can be shown that

xj = 1a±L

2=⇒ ζj = log

(1± aL

2

)

which is manifestly not symmetric with respect to the detectorposition ζd = 0.

Both tmax and L′ will change for the midpoint detector

L′ = log 2 + aL

2− aL , tmax =√L

a− L2

4(23)

and there will be a slight modification of Eq. (22). Also,clearly ζ1 would not be zero in this case.

If the field is massive, the Klein-Gordon equation is nolonger invariant under a conformal transformation, andit is more advantageous to use the manifestly simplerstandard Rindler coordinates

t = ξ sinh η , x = ξ cosh η . (24)Let us work this out explicitly from the Klein-Gordonequation: since √−g = ξ, the covariant Klein-Gordonequation gives

1ξ2∂2φ

∂η2 −(

1ξ

∂φ

∂ξ+ ∂2φ

∂ξ2

)+m2φ = 0 . (25)

7

Separating variables φ = v(ξ)T (η), we can show thatT (η) ∝ exp±iωη and hence we obtain the modifiedBessel differential equation of imaginary order for thespatial mode v(ξ):

ξ2 ∂2v

∂ξ2 + ξ∂v

∂ξ+ (ω2 −m2ξ2)v = 0 . (26)

Implementing the Dirichlet boundary condition v(ξ1) =v(ξ2) = 0 as before, the modes will have discrete spec-trum labelled by n ∈ Z and the spatial mode can beexpressed in terms of modified Bessel functions of imag-inary order [19]:

vn(ξ) = |An| (Re (Iiωn(mξ1))Kiωn

(mξ)−Re (Iiωn(mξ))Kiωn(mξ1)) ,

1 = 2|An|2ωn∫ ξ2

ξ1

dξξ|vn(ξ)|2 .

(27)

where the normalization follows from Klein-Gordon in-ner product in Eq. (10). The discrete spectrum and thenormalization must be solved numerically. Similar to themassless case, we can then do the pullback of the Wight-man function onto the trajectory of the detector whichis given by

ξ(τ) =√x2d − t2 η(τ) = tanh−1 t

xd(28)

where the constant xd describes the static detector tra-jectory with respect to the lab coordinates.

We pause to comment about rigid body motion in theRindler frame. Note that even if the leftmost wall accel-eration gets arbitrarily large, the centre of mass acceler-ation is bounded above by the rigidity condition: a rigidcavity of length L in the lab frame must have a differ-ent proper acceleration at each point in order to remainrigid. The proper acceleration at any point x within thecavity is given by

a(x) = a11 + a1(x− x1) , (29)

where x1 = a−11 is the location of left wall and a1 is the

proper acceleration of the left wall. We see that at thecentre xc = a−1

1 + L/2 of the cavity we have the limit

lima1→∞

ac = lima1→∞

2a12 + a1L

= 2L. (30)

If the centre of the cavity attains an acceleration largerthan this, the rear wall will cross the future Rindler hori-zon, which is an unphysical cavity setup.

Another way to see this geometrically is by looking atthe spacetime diagram (cf. Figure 3). For a uniformly ac-celerating rigid cavity, the two walls must both be on twodifferent hypersurfaces of constant ξ in order for them tobe a Dirichlet boundary i.e. ξ = ξ1 and ξ = ξ2. Differ-ent values of ξ correspond to trajectories with differentproper accelerations, and the lab observer does not see

this cavity as rigid because the the cavity shrinks acrossplane of simultaneity of constant t. The rigidity con-dition essentially means that cavity has constant lengthwhen measured in the plane of simultaneity of constantη.

In Figure 4 we plot the absolute probability differencebetween the accelerating cavity and the accelerating de-tector scenarios as a function of the proper acceleration a.A larger energy gap generally suppresses the transitionprobability in massless scenario as shown in Figure 5, andsimilar qualitative suppression is observed in the massivecase. For comparison of the convergence of the modesums, we considered ranging both N = 15 and N = 100.The larger value of N is required for larger accelerationparameters a (see also Appendix C for separate conver-gence checks).

In Figure 6 we compare the absolute probability dif-ference for massless and massive fields. Here our resultsagree with previous work [3] in that if the initial fieldstate is the vacuum, then for aL 1 the differencein responses between inertial and non-inertial detectorsquickly vanishes. For completeness, we plot in Figure 7the transition probability of the accelerating cavity sce-nario for very small mass. We see that indeed it providesthe correct massless limit despite the rather complicatedmode functions involving modified Bessel functions. Theaccelerating detector case will trivially have the correctlimit since the functional form of the Wightman functionsis the same.

Furthermore, we do see considerable distinction be-tween the massive and massless cases once a becomessufficiently large (cf. Figure 6). The larger differencein response for the two setups at large a should not betaken to be a fundamental violation of the WEP: forlarge a, the non-uniformity of the cavity acceleration atdifferent points is more pronounced, similar to how thenon-uniformity of Earth’s gravitational field is detectableif we consider a large enough region in space.

VI. EXCITED FIELD STATES

After considering the vacuum state of the field, a natu-ral question then arises: can sensitivity to non-inertialitybe enhanced if the field state is not a vacuum state? Theadditional terms in the Wightman function due to the ex-cited field states may have co-rotating term of the formΩ−ωn which may produce resonant-like behaviour, whilefor the vacuum state this cannot occur for a ground stateatom. We will consider both single-mode excited Fockstates and single-mode coherent states.

A. Single-mode excited Fock state

The simplest excited field state we can consider is asingle-mode non-vacuum Fock state, i.e. when the k-th momentum has nk excitations. This is a straight-

8

0.2 0.4 0.6 0.8 1.0a

0.0002

0.0004

0.0006

0.0008

0.0010

|δProb|/λ2Absolute difference in Probability, L = 1, Ω = π /L

N = 100 N = 15

FIG. 4. Absolute difference in probability |PC0 −PD

0 |/λ2 as afunction of acceleration for Ω = π/L for M = 0. Here and forsubsequent plots we set L = 1 for convenience. For small ac-celerations, the mode sums quickly converge for small N andthe difference in transition probability of the two scenarios isvanishingly small in low acceleration limit.

0.2 0.4 0.6 0.8 1.0a

0.00005

0.00010

0.00015

0.00020

0.00025

|δProb|/λ2Absolute difference in Probability, L = 1, Ω = 2π /L

N = 100 N = 15

FIG. 5. Absolute difference in probability |PC0 − PD

0 |/λ2 asa function of acceleration for larger gap Ω = 2π/L,L = 1 form = 0.

forward generalization from the expression found in [3].We denote this by |nk〉 which formally reads |nk〉 ∼|000...0 nk 000...〉, where the enumeration is formallyvalid because of the countably infinite spectrum. Thecorresponding Wightman function is formally given by

W (x, x′) = 〈nk|φ(x)φ(x′)|nk〉 (31)

Employing the result

φ(x′) |nk〉 =∑

l 6=ku∗l (x′) |1l, nk〉+

√nk + 1u∗l (x′) |nk + 1〉+√nkul(x′) |nk − 1〉

(32)

0.2 0.4 0.6 0.8 1.0a

0.0002

0.0004

0.0006

0.0008

0.0010

|δProb/λ2|Absolute probability difference (N = 100)

0.05 0.10 0.15 0.20a0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

M = 2 M = 0

FIG. 6. Comparing the absolute difference in probability|PC

0 −PD0 |/λ2 as a function of acceleration for L = 1,Ω = π/L

when the field is initially in the vacuum state. Here L = 1 forconvenience. The difference between an accelerating cavityand an accelerating detector vanishes quickly at low acceler-ations.

0.2 0.4 0.6 0.8 1.0a

0.001

0.002

0.003

0.004

Prob/λ2Massless limit for accelerating cavity, Ω = π /L, N = 100

Cavity (M = 0.00001) Cavity (M = 0)

FIG. 7. Transition probability (divided by λ2) as a functionof acceleration for Ω = π/L,L = 1, showing that in the smallmass limit the results agree with massless case. We chooseN = 100 instead of the value N = 15 as in previous plots.Note that a value of M = 0.0001 is small enough to be indis-tinguishable from the M = 0 case, with relative difference inprobability of 2 parts in a billion (10−9) at a ≈ 0.01.

where the uj are the eigenmodes of the Klein-Gordonequation (not just the spatial part), we obtain

W (x, x′) =∑

j,l 6=kuj(x)u∗l (x′) + (nk + 1)uk(x)u∗k(x′)

+ nku∗k(x)uk(x′)

=∑

j

uj(x)u∗j (x′) + nkuk(x)u∗k(x′)

+ nku∗k(x)uk(x′)

= W0(x, x′) +Wexc(x, x′)

(33)

9

for the full expression for the Wightman function. There-fore, for an excited field state given by a single-mode Fockstate, the Wightman function is the sum of the vacuumWightman functionW0 and an additional pieceWexc thatis explicitly dependent on which mode it is excited. Sincethe transition probability is linear in W (x, x′), we see thatthe transition probability for this state reads

Ptot(Ω) = P0(Ω) + nk

∣∣∣∣∫

dτχ(τ)e−iΩτuk(τ)∣∣∣∣2+

nk

∣∣∣∣∫

dτχ(τ)e−iΩτu∗k(τ)∣∣∣∣2.

(34)

We are interested in Wexc since we found W0 in the pre-vious section and we can always subtract off the vacuumcontribution. Note that

e−iΩτuk(τ) ∼ e−i(ωkT (τ)+Ωτ) ,

e−iΩτu∗k(τ) ∼ e−i(ωkT (τ)−Ωτ) ,(35)

where T (τ) is the time function (which in our case iseither η(τ) or t(τ)) along the trajectory of the detector.The third term in Ptot is the ‘co-rotating term’ whichwill tend to dominate over the second (‘counter-rotating’)term.

The above results teach us that there are two ways inwhich one can “neglect” the vacuum contribution. Oneis when we have an approximate resonance (up to someDoppler shifts) i.e. when Ω ∼ ωk. In this case, theresonance will amplify the transition rate and the vacuumcontribution can be rendered negligible compared to therest. The other is if there is a sufficiently higher numberof excitations nk: in this case the transition probabilityscales as

Pexcited ∼nkk

(36)

where the denominator 1/k comes from the normaliza-tion of uk. This means for a given energy gap Ω, thehigher momentum mode will need an excitation of ordernk ∼ k to achieve a given probability amplitude. Whenit is off-resonance, a larger gap tends to diminish thetransition probability, which simply reflects the fact thatatoms with larger energy gaps are harder to excite.

Some of these results are shown in Figure 8. A no-table result upon comparison of the two figures is thatone can indeed amplify transition probability by consid-ering gaps that are ‘close’ to the excited field state fre-quency. In Figure 8, by considering ‘off-resonant’ gapat Ω = 3π/L ± ε, there are regimes of accelerations inwhich the massive fields have better transition probabil-ities for both accelerating detector/cavity scenarios thando their massless counterparts, and vice versa depend-ing whether Ω = ωn − ε or Ω = ωn + ε (in the plots,ε = 0.5π/L). However, for each mass the distinction be-tween an accelerating detector and an accelerating cavityquickly vanishes for small a.

Here we make a parenthetical comment that the rela-tive magnitude of Ω−ωk or Ω/ωk does matter : for a given

fixed atomic gap Ω, one can engineer a situation in whichmassive fields can have larger transition probability thanthe massless counterpart using resonance and vice versa.This is already apparent in Figure 8 for small a, wheretransition probability for massive case can be lower orhigher than the massless case depending on choice of gapΩ. This is, however, a separate problem from fundamen-tally distinguishing local accelerations.

0.00 0.05 0.10 0.15 0.200

20

40

60

80k = 3, nk = 3, Ω = 2.5π /L

Cavity (M = 0) Detector (M = 0) Cavity (M = 2)

Detector (M = 2)

0.00 0.05 0.10 0.15 0.20

20

40

60

80

100k = 3, nk = 3, Ω = 3.5π /L


Detector (M = 2)

FIG. 8. Transition probability (divided by λ2) as a functionof acceleration for two different gaps, comparing massless andmassive cases. The field is in the third excited state i.e. k = 3and we chose n3 = 3. Top: Ω = 2.5π/L. Bottom: Ω =3.5π/L. The plots are for L = 1.

The relative magnitude matters less as one moves awayfrom resonance, e.g. when Ω/ωk 1. We check thisfor the case of highly populated field state nk 1 asshown in Figure 9, where we choose k = 1 and nk = 1000to match the setup in [3] for convenience. In the topfigure, the massive field seems to outperform the masslesscase for distinguishing local accelerations. However, thiscan be attributed to resonant effect, since for our choiceof fixed Ω, the magnitude of Ω − ωk is smaller for the

10

massive case than the massless case. A possibly fairercomparison would be to use the same |Ω−ωk| or Ω/ωk, asshown in the middle and bottom plot of Figure 9. Whenthis is done, we see that the apparent advantage of themassive field over massless one disappears and masslessfield seems to perform equally well if not better4.

We conclude from these that massive fields do not seemto offer any obvious fundamental advantages at low ac-celerations as compared to their massless counterpart. InAppendix B we suggest a possible reason for the disparitywith the results found in [3].

B. Coherent field state

An interesting case to consider is when the field is ina coherent state, analogous to that of a laser field inquantum optics scenarios. It is defined as the continuumlimit of a quantum-mechanical coherent state for a quan-tum harmonic oscillator using the displacement operatorDα(k) (see, for instance [13]):

|α(k)〉 := Dα(k) |0〉

= exp[∫

dk[α(k)a†k − α∗(k)ak

]]|0〉 .

(37)

Here α(k) is the coherent amplitude distribution defininga coherent amplitude for every mode k. As a coherentstate, it satisfies the ‘eigenvalue’ equation

ak′ |α(k)〉 = α(k′) |α(k)〉 , (38)

noting that |α(k)〉 does not mean an explicit dependenceon k but rather on coherent amplitude distribution α(k).In a cavity, the spectrum becomes discrete and so welabel the modes with integers n instead (for example, thecontinuous variable k becomes discrete: kn = nπ/L instatic cavity scenario). The coherent state has a simplerform

|α(n)〉 = exp[ ∞∑

n=1(αna†n − α∗nan)

]|0〉 . (39)

Note that in this case we can formally write

|α(n)〉 ∼ |α1α2...αj ...〉 ∼∞⊗

n=1|αn〉 (40)

which denotes tensor product of coherent states eachwith complex coherent amplitude αj . For single-mode

4 This issue is somewhat tricky since it may arguable which com-parison is fairer. However, this ‘fairness’ is necessary for WEPsince fair comparison is analogous to “not being able to lookout of the window of a rocket” to decide the asymmetry of theproblem.

0.2 0.4 0.6 0.8 1.0a

0.5

1.0

1.5

|δProb|/λ2k = 1, nk = 1000, Ω = 4π /L

0.05 0.10 0.15

0.02

0.04

0.06

0.08

0.10

0.12

0.14

M = 0 M = 2

0.2 0.4 0.6 0.8 1.0a

0.5

1.0

1.5

|δProb|/λ2k = 1, nk = 1000, Ω - ωk = 8.84

0.05 0.10 0.15

0.02

0.04

0.06

0.08

0.10

0.12

0.14

M = 0 M = 2

0.0 0.2 0.4 0.6 0.8a0.0

0.5

1.0

1.5

2.0|δProb|/λ2

k = 1, nk = 1000, Ω/ωk = 3.37

0.05 0.10 0.15

0.05

0.10

0.15

0.20

0.25

0.30

0.35

M = 0 M = 2

FIG. 9. Absolute probability difference (divided by λ2) asa function of acceleration for large nk. The field is in thethird excited state i.e. k = 3 and we chose n3 = 3. Top:Ω = 4π/L. Middle: Ω − ωk = 8.84 where the reference ωk

is chosen to be the angular frequency for the massive case.Bottom: Ω/ωk = 3.37. The plots are for L = 1.

11

coherent state, say for the j-th momentum, we have (cf.Eq. (38))

aj |α(k)〉 = αjδjk |α(k)〉 , αj ∈ C . (41)

For a countably infinite multimode coherent state above,we require that

∞∑

n=1|αn|2 <∞ (42)

which means that modes with higher momenta have sup-pressed coherent amplitude. Here we will not employthe infinite multimode coherent state, and instead fo-cus specifically on the more realistic single-mode coherentstate as is used in quantum optics.

The Wightman function for the coherent state reads

W (x, x′) = 〈0|D†α(n)φ(x)φ(x′)Dα(n)|0〉

=∞∑

n=1un(x)u∗n(x′) +

∞∑

n=1

∞∑

j=1α∗nαjuj(x)u∗n(x′)

+∞∑

n=1

∞∑

j=1αnα

∗ju∗j (x)un(x′)

+∞∑

n=1

∞∑

j=1αnαjuj(x)un(x′)

+∞∑

n=1

∞∑

j=1α∗nα

∗ju∗j (x)u∗n(x′) .

(43)Note that similar to the single-mode excited Fock state,the vacuum contribution to the Wightman function doesnot vanish. If we define the one-point function of thecoherent state as

J(x) := 〈α(n)|φ(x)|α(n)〉 =∑

n

αnun(x) , (44)

we can compactly write the full Wightman function as

W (x, x′) = W0(x, x′) + J(x)J(x′) + J(x)J∗(x′)+ J∗(x)J(x′) + J∗(x)J∗(x′)

= W0(x, x′) +Wc(x, x′) ,Wc(x, x′) = 4Re[J(x)]Re[J(x′)] .

(45)

The fact that Wc(x, x′) factorizes into product of one-point functions allow us to simplify the expression forthe transition probability. The transition probability dueto the purely coherent part (i.e. modulo the vacuumcontribution W0(x, x′)) then reads

Pc(Ω) = λ2∫

dτ dτ ′χ(τ)χ(τ ′)e−iΩ(τ−τ ′)Wc(τ, τ ′)

= 4λ2∣∣∣∣∫

dτχ(τ)e−iΩτRe[J(x(τ))]∣∣∣∣2.

(46)

With a judicious choice of αn, it may be possible toperform the infinite sum in J(τ) exactly. Before we pro-ceed, it is worth noting that resonant behaviour similarto that of the previous section is expected, since the realpart of J(τ) contains cosωnt(τ) term which produces co-rotating term when combined with the exponential of thegap e−iΩτ .

For single-mode coherent state, there is no real restric-tion on the coherent amplitude; we obtain

J(x) = δmn∑

n

αnun(x) , (47)

where m-th mode is to be the coherent state and the restare all vacuum modes. For simplicity we can consider,for example, m = 2 and restrict α ∈ R (though α can bearbitrary complex number).

We illustrate the case when the second mode k = 2 isin a coherent state with coherent amplitude α2 = 1 whileothers are in the vacuum state, shown in Figure 10. Wealso intentionally adjust the energy gap of the detector sothat Ω = 1.9ωn, which is different for massless and mas-sive fields. This comparison can be thought of as mak-ing the comparison somewhat fairer since the amountby which the atom is off-resonant from the mode fre-quency is of the same weight. We see that even withmassive fields, the overall behaviour remains unchangedand as expected, the transition amplitude degrades withlarger mass. This contrast is even more apparent whenwe compute absolute probability difference between theaccelerating cavity and accelerating detector in massiveand massless fields, as shown in Figure 11. While we donot probe extremely non-relativistic regimes due to com-putational resources, it is clear that the role of mass isvanishingly small for smaller acceleration. We have ig-nored the vacuum contribution because we have chosenthe value of Ω such that the vacuum contribution is neg-ligible compared to the contribution due to the excitedfield state. Furthermore, we have shown that vacuumstates are not sensitive to local accelerations.

We pause to comment that the response of an ac-celerating cavity response ‘underperforms’ relative toan accelerating detector for a fixed mass m for largeaccelerations. We see in Figure 8 that this un-der/overperformance is reversed for a . 0.35. This canpresumably be attributed to non-linearities introducedby non-uniform acceleration across the accelerating cav-ity, though we do not yet have a full understanding ofthis effect.

C. Resonance

The resonance phenomenon, while not very exact dueto accelerated motion of the detector or cavity, can bemade manifest if we study the “resonance peak” of thedetector. The resonance peak for the case of the field in aFock state |nk〉 is shown in Figure 12. Recall that in thisnotation, it is the k-th momentum having nk excitations:

12

0.2 0.4 0.6 0.8 1.0a

0.01

0.02

0.03

0.04

0.05

0.06

Prob/λ2M = 0, Ω = 1.9ωn

Cavity Detector

0.2 0.4 0.6 0.8 1.0a

0.01

0.02

0.03

0.04

0.05

Prob/λ2M = 2, Ω = 1.9ωn

Cavity Detector

FIG. 10. Transition probability (modulo the vacuum contri-bution) for an accelerating detector and an accelerating cavityfor two different masses when the second mode (k = 2) is incoherent state and other modes are in vacuum state. In theseplots L = 1.

if field is in the seventh excited state with 20 excitations,then we write |207〉.

From Figure 12 we observe that for large accelerationthere is a larger Doppler shift, which smears out reso-nance and damps out transition probability. The numberof resonance peaks matches the mode number k that de-fines the excited state of the field. As the accelerationdecreases, the resonance peaks becomes narrower andhigher, indicating that we approach resonance in staticinertial scenario. Figure 12 also shows that resonancedominates when

Ωτ ≈ lima→0

ωkt(τ) , (48)

where t(τ) is the pullback of the coordinate time in termsof proper time τ of the detector. Crucially, the roughestimate of the right hand side gives

ωkt(τ) ≈ kπ

aL

(aτ +O(a3τ3)

)∼ kπτ

L+O(a3τ3) , (49)

which is to first order the same as the case for staticdetector and static cavity.

0.2 0.4 0.6 0.8 1.0a

0.005

0.010

0.015

0.020

|δProb|/λ2

M = 0 M = 2

0.2 0.4 0.6 0.8 1.0a

1.0

1.2

1.4

1.6

Prob M=0

Prob M=2

Ratio M=0

M=2, Ω = 1.9ωn, second mode coherent

Cavity Detector

FIG. 11. Top: absolute probability difference |δP |/λ2 =|PC(Ω) − PD(Ω)|/λ2 (modulo the vacuum contribution) formassless and massive fields. The difference is vanishing forsmall a regardless of mass. Bottom: the ratio of probabili-ties between massless and massive ones. We see that in lowacceleration regimes the ratio approaches 1.

We remark that near resonance Ω− ωk ≈ 0, Figure 12seem to indicate that the probability amplitude may bedivergent if a is small enough, since λ2 may not be smallenough to make the probability amplitude less than 1(e.g. set λ = 0.1). We expect this to be an artifact ofthe approximations in the whole setup, including pertur-bative calculations of transition probability P (Ω). As anexample of such artifacts, note that in Eq. (34) the transi-tion probability scales linearly with nk (this also appearsin [3]). Clearly, this cannot be valid for arbitrary nksince for large enough excitations, the probability can bemade greater than 1. These may be cancelled by higherorder terms which would also contain co-rotating terms.Also, recall that since our detector starts from one endof the cavity, in the limit where a = 0 we should expectno excitation at all due to Dirichlet boundary conditiongiven the choice of coupling. This suggests that for com-putations involving non-vacuum contributions and co-rotating terms, one should be careful in extrapolatingresults.

Nonetheless, our results so far do not change even if we

13

6 7 8 9kπ /L

1

2

3

4

Pr/λ2Accelerating Cavity, k = 7

a = 0.02, nk = 7 a = 0.02, nk = 14

a = 0.1, nk = 7 a = 0.1, nk = 14

6 7 8 9kπ /L

200

400

600

800

1000

1200

Pr/λ2Accelerating Cavity, k = 7

a = 0.0001, nk = 14

a = 0.0001, nk = 28

FIG. 12. Transition probability for fixed acceleration as afunction of energy gap Ω = kπ/L where n is real and nk isthe number of excitations of the massless scalar field in modek, with L = 1.

stay away from the Ω ≈ ωk limit (cf. Figure 10), sinceall that the resonance condition and large nk limit do isallow us to ignore vacuum contributions from W0(τ, τ ′)by amplifying the non-vacuum contributions. Even ifthe excited parts Wexc,Wc of the Wightman function aresmaller than W0, we could simply subtract off the W0part since we find a negligible difference between the re-sponses in Experiment 1 and Experiment 2.

VII. TRANSITION RATE

Computation of a transition probability — also knownas a response function F (Ω) of a particle detector withenergy gap Ω — as a detector traverses through a quan-tum field coupled to it has a physical interpretation: itprovides an operational way of defining the particle con-tent of the field without invoking a high degree of space-time symmetry [20, 21]. However, the fact that it is a

double integral may obscure information about the atom-field interaction. This prompts us to consider whetherthe transition rate, essentially the time-derivative of theresponse function along the detector trajectory, can pro-vide further insights into the WEP.

To obtain the response rate, we need to rewrite theresponse function in such a way that it can be easilydifferentiated. This is done by changing variables [22]

F (Ω) = 2Re∫ τ

τ0

du∫ u−τ0

0dse−iΩsW (u, u− s) , (50)

where τ0 denotes the time in which the detector isswitched on. Instead of the usual response function whichgives transition probability of exciting the atom from itsground state, we can now compute the instantaneoustransition rate of a detector turned on at time τ0 andread at time τ , given by [22]

F (Ω) = dF (Ω)dτ = 2Re

∫ τ−τ0

0ds e−iΩsW (τ, τ − s) . (51)

Despite some subtleties in handling this observable forfree space involving regularization, we expect that cavitysetup removes these difficulties since the field is com-pactly supported and there is an infrared cutoff. Inour scenario it is convenient to compute the case whereτ0 = 0. If different field states have a chance of causingdifferent responses to the detector, transition rate maybe able to pick this up5. Conversely, if transition rate isidentical, then the response of the detector should be thesame under integration.

Since the response rate is linear in W (τ, τ ′), we willsplit them into two parts:

F (Ω) = F0(Ω) + F1(Ω) (52)

where F0 is the vacuum contribution and F1 is the re-maining contribution due to the field in excited state.The vacuum state transition rate is shown in Figure 13.The crucial thing to note here is that the vacuum contri-bution for both cases have negligible differences in tran-sition rate — therefore the transition probability mustbe the same as well after integrating across the full tra-jectory. For computational time convenience, we chosea = 0.02 to represent massive case and the same con-clusion holds. This justifies our earlier results (also in[3]) that vacuum contributions are not sensitive to localaccelerations.

Two examples for a highly populated first excited state(k = 1, nk = 100) for the massless case are shown in Fig-ure 14. We see that while the rate appears qualitativelydifferent at different read-out times, the difference be-tween an accelerating cavity and an accelerating detector

5 On the other hand, it is possible that response function washesout differential differences due to mean value theorem.

14

is very small (of course it is only exactly zero for a staticsetup). As far as differences go, massive fields generi-cally do not perform better than massless ones, which isconsistent with the idea that the role of mass tends to‘kill off’ correlations at large distances and diminish theamplitudes.

From Figure 14, one might be led to think that a mas-sive field seems to have a very large response rate com-pared to a massless field, but this is not the right com-parison. Note that the co-rotating frequency Ω− ωn de-termines quite directly the magnitude of these rates, andgiven the same gap Ω, one of the two fields will be closerto ‘resonant frequency’ than the other. In Figure 15, weadjust the gap so that both massless and massive fieldsthe atomic gap is Ω = 1.012ω1, where ω1 is the frequencyof the first mode. As expected, the absolute value of thetransition rate for massless fields dominate the massivecase. The difference in response rates ∆F = FD − FC ,where C,D denotes accelerating cavity and detector re-spectively, are of approximately the same order as seenin Figure 15.

With hindsight we should not be surprised by these re-sults, since they are basically an Unruh-type setup con-fined to cavity. As clarified in [2], what is important inthese WEP considerations is really the fact that thereis relative acceleration between the atom and the cavity.In the slow acceleration limit, every point in the cavitycan be approximated to have the same constant properacceleration (hence the same clock ticking rates) and soan accelerating cavity-static detector and an acceleratingdetector-static cavity should lead to the same physical re-sults. The mass parameter of the scalar field enters thequantum field via the mode frequency and amplitude,which generally degrades response since the integral overWightman function is more oscillatory and the normal-ization for each mode is smaller than those for a mass-less field. In this respect, if a ‘fair’ comparison is madebetween the massless and massive cases (e.g. adjustingΩ/ωk or |Ω−ωk| instead of fixing Ωl, cf. Section VI), themassless field should lead to larger detector responses be-cause mass suppresses nonlocal correlations. Note thatthis suppression is independent of WEP.

Why would the responses be different at large a? As ar-gued in the context of mirrors [2], the accelerating cavity-static detector and the accelerating detector-static cav-ity setups are also not mathematically equivalent: if ourexperiments are sensitive enough to non-uniformity ofacceleration across the cavity, then the notion of “rela-tive acceleration” becomes blurred. For an acceleratingdetector, in the cavity frame one observes that the de-tector has a constant-acceleration trajectory; for an ac-celerating cavity, in the cavity frame observes that thedetector is not uniformly accelerating because its world-line crosses all the hypersurfaces of constant ξ betweenone cavity wall ξ = ξ2 to another ξ = ξ1. In the slowacceleration limit, these constant-ξ surfaces describe ap-proximately the same acceleration and hence the detec-tor is observed to be approximately uniformly acceler-

ating. We can think of the correlation functions of thefield as capturing this non-local difference and the in-equivalent setups lead to unequal responses. It is in thisspirit that WEP makes sense — the responses betweenthe free-falling cavity-stationary detector and free-fallingdetector-stationary cavity will be different once the non-uniformity of the gravitational field is detectable.

Finally, a small qualification about the comparison be-tween the two different scenarios (accelerating detectorand accelerating cavity) is in order. There are a coupleof ways in which the two scenarios can be argued not tobe on equal footing, First, we note that in relativity thereis no absolute rigidity [9–11]; it is impossible to maintainfixed coordinate distance between two cavity walls in allframes. Accelerating the cavity whilst keeping it rigidin the cavity rest frame (Fermi-Walker rigidity) is thesimplest and most natural setup. The fact that for accel-erating cavities the detector is seen to be non-uniformlyaccelerating from the cavity’s frame, is sufficient to showthat the detector response should be different from theconstantly accelerating detector scenario.

However, there is also a perhaps more fundamental andeasier argument for the lack of equivalence between thetwo scenarios: The accelerating cavity is a setup of accel-erating mirrors, which are perfectly reflecting boundaryconditions, whereas an accelerating detector is a quan-tum object that can absorb, transmit and reflect parts ofan illuminating plane wave. Consequently, they consti-tute rather distinct field configurations (e.g., dynamicalCasimir effect and Unruh radiation respectively) and thetwo setups are not identical beyond the ‘non-uniformity’of the acceleration either. We can estimate the deviationbetween the two scenarios e.g. from Eq. (49), which canbe seen to be third order in the dimensionless parameterthat depends on acceleration and duration of the inter-action.

VIII. CONCLUSION

We have investigated a quantum version of the WEPin which we consider the response of a particle detectorin two scenarios: a) a detector accelerating in a staticcavity and b) a static detector in an accelerating cav-ity. We found that the qualitative WEP is indeed satis-fied insofar as quasilocal approximations are valid. Wedo this by investigating the transition probability of atwo-level atomic detector on various field states, namelyvacuum state (Minkowski-like and Rindler-like vacuum),arbitrary Fock state, and single-mode coherent state. Wealso check the effect of bringing the atomic gap closer tothe resonant frequency when we have co-rotating termsand clarify the validity of some approximations such aslarge nk limit for Fock state of the field. Importantly,the results support the idea that a ‘quantum accelerom-eter’ in non-relativistic regime would work equally for amassless field and for a massive field. We strengthen theresults by computing the transition rates to ensure that

15

0.2 0.4 0.6 0.8 1.0Nτmax

-0.0010

-0.0005

0.0005

Rate/λ2M = 0, vacuum state, Ω = π /L

Cavity Detector

0.2 0.4 0.6 0.8 1.0N (units of τmax)

-0.005

-0.004

-0.003

-0.002

-0.001

0.001

0.002Rate/λ2

a = 0.02, M = 2, vacuum state, Ω = 1.2π /L

Cavity Detector

FIG. 13. Transition rate as a function of time τ . Note thatin both cases the transition rates for accelerating cavity andaccelerating detector scenarios are practically indistinguish-able regardless of mass. We chose different parameters forvariations.

no fundamental physical differences are averaged out byintegration when we compute transition probabilities. Inthis sense, our results complement those of [2, 6].

ACKNOWLEDGEMENT

This work was supported in part by the Natural Sci-ences and Engineering Research Council of Canada. E.Tjoa thanks Jorma Louko, Robie Hennigar and RichardLopp for useful discussions and A. Dragan for helpful cor-respondence. E. M-M also acknowledges funding form hisOntario Early Researcher Award.

Appendix A: Solving massless Klein-Gordonequation without conformal transformation

In this section we solve for the solution for the masslessKlein-Gordon field equation without invoking conformaltransformation of any sort. We quote again the standardRindler coordinates for convenience:

t = ξ sinh η , x = ξ cosh η .

From the general Klein-Gordon field equation (cf.Eq. (1)) in this coordinate system, which gives the mod-

0.2 0.4 0.6 0.8 1.0Nτmax

20

40

60

Rate/λ2M = 0, first excited state, Ω = 1.2π /L

Cavity Detector

0.2 0.4 0.6 0.8 1.0N (units of τmax)

50

100

150

200

Rate/λ2a = 0.02, M = 2, first excited state, Ω = 1.2π /L

Cavity Detector

0.2 0.4 0.6 0.8 1.0N (units of τmax)

-3

-2

-1

1

2

ΔRate/λ2a = 0.02, first excited state, Ω = 1.2π /L

M = 0 M = 2

FIG. 14. Transition rate as a function of time τ for the firstexcited state of the field. Top: massless case. Middle: mas-sive case. Bottom: Difference in transition rate for both sce-narios. It appears that transition rate and hence transitionamplitude is slightly more advantageous for massless case for agiven acceleration. Here “∆Rate” is simply ∆F = FD = FC .

ified Bessel differential equation for the spatial modesv(η, ξ):

ξ2 d2v

dξ2 + ξdvdξ + (ω2 −m2ξ2)v = 0 . (A1)

The solution basis for m 6= 0 is given by Re(Iiω) andKiω which are both real and linearly independent dueto nontrivial Wronskian [19]. Now let us set m = 0 on

16

0.2 0.4 0.6 0.8 1.0N (units of τmax)

50

100

150

200

Rate/λ2a = 0.02, M = 0, Ω = 1.012ω1


Detector (M = 2)

FIG. 15. Transition rate as a function of time τ for the firstexcited state of the field. Top: massless case. Middle: mas-sive case. Bottom: Difference in transition rate for bothscenarios. It appears that transition rate and hence transi-tion amplitude is slightly more advantageous for massless casefor a given acceleration.

Eq. (A1). The eigenbasis6 of the solution space is givenby sin (ω log ξ) and cos (ω log ξ). Note that we could alsoobtain this by doing a series expansion for small m →0+ on the mode solutions in Eq. (27) which satisfies theDirichlet boundary condition at ξ = ξ1 [19]. Since ηis dimensionless, so is ω here. If we let the boundaryconditions to be at ξ1 = a−1 and ξ2 = a−1 + L, we get

vn ∝ sin (ωn log ξ)− tan(ωn log 1

a

)cos (ωn log ξ) ,

(A2)where ωn is now a discrete spectrum due to the secondboundary condition ξ = ξ2. The normalization can befound by standard Klein-Gordon inner product [8]. Re-markably, even after imposing the second boundary con-dition, the spectrum is still exact, which reads

ωn = nπ

log(1 + aL) , n ∈ N , (A3)

which is precisely what we got from the conformal trans-formation where we identify the denominator as aL′, theconformally transformed length of the cavity multipiledby the kinematical parameter a. In some sense this isperhaps not surprising, since the same physical situa-tion should be described by the same differential oper-ator with the same set of spectrum (which is invariantunder coordinate transformations).

Some representative plots of the modes for small andlarge accelerations are given in Figure 16. Now it is very

6 This is not the ones used in e.g. [3, 23], but for our purposeseither one will work. Roughly speaking, one can check from theseries expansion at small m that this is analogous to the choiceof writing solutions to harmonic oscillator equation in terms ofcosine/sine functions or plane waves.

clear that the spatial modes approach Minkowski staticcavity scenario very quickly for not too small a ∼ 0.01,while for large acceleration (of the left wall) the modesare “deformed sine functions”. These deformed modesare in fact very similar in form as the modes for massivecase described in terms of modified Bessel functions ofimaginary order Re(Iiω) and Kiω.

100.2 100.4 100.6 100.8 101.0ξ

-2

-1

1

2

vn(ξ)Accelerating cavity, a = 0.01, M = 0

n = 1 n = 2 n = 5

0.4 0.6 0.8 1.0 1.2ξ

-1

1

2

vn(ξ)Accelerating cavity, a = 4.00, M = 0

n = 1 n = 2 n = 5

FIG. 16. Sample plots of mode functions for the second moden = 2 for small and large accelerations. This makes clearthat large acceleration limit is “Bessel-like”, in that the modefunction is a deformed sine function, squashed in the directionof acceleration. These plots are not normalized since we areconcerned with their forms rather than their amplitudes.

This clearly demonstrates that the differential equationgoverning the form of the spatial modes is solvable di-rectly even if the metric is not the one conformally equiv-alent to the Minkowski metric. In this standard Rindlercoordinates, the Klein-Gordon equation would also notbe conformally invariant under the change of coordinates.However, the standard Rindler coordinates and confor-mal Rindler coordinates both cover the Rindler wedgeportion of Minkowski spacetime and each hypersurfaceof constant ξ in either coordinates describe the trajec-tory of uniformly accelerating test particles. One wouldnot conclude that massless fields cannot distinguish thetwo scenarios on grounds of conformal invariance, whilemassive fields can; instead, one would conclude that bothshould have qualitatively similar behaviour up to some

17

degradation factor due to mass of the field that enters thenormalization constant and phase factor in the integralof transition probability.

Here we make a short remark on the distinction be-tween conformal flatness and conformal invariance of fieldequations via a conformal transformation. A spacetimeM is said to be conformally flat if there exists a coordi-nate system in which the metric can be rewritten as

gµν(x) = Ω(x)2ηµν , (A4)

and in (1+1) dimensions all Lorentzian manifolds areconformally flat. The massless KG field is conformallyinvariant because under conformal transformation, theKG equation takes the same form as the wave equation inglobal Minkowski coordinates. However, performing con-formal transformation is a calculational advantage thatdoes not change the physics, since we could equally dophysics using non-conformally equivalent metric that de-scribes the same spacetime. Alternatively, we say thatthe physics is contained in Ω(x) and so the physics willstill be different from static Minkowski spacetime [2]. Agood example is the de Sitter expanding universe, whichcan be written in coordinates such that it is conformallyflat — the mode functions inherit the form in flat space,but static detector in conformal vacuum of the de Sit-ter spacetime detects particles while static detector inMinkowski vacuum does not.

0.5 1.0 1.5 2.0

5

10

15

20M = 0

Cavity Detector Detector (discrepancy)

0.5 1.0 1.5 2.0

2

4

6

8

10

M = 2

Cavity Detector Detector (discrepancy)

FIG. 17. The transition probability plot simulating the plotsfound in [3]. The discrepancy is possibly related to incorrectnormalization for massive field-accelerating detector scenario(labelled ‘discrepancy’ here), thus producing the result thatmassive fields can better distinguish local acceleration.

Appendix B: Discrepancy with past results

Based on the argument above, there is a slight dispar-ity in a result we obtained here and the results obtainedin [3]. Since the exact parameters used previously [3] areunknown, we attempt to emulate the construction andthe result is shown in Figure 17. From what we can dis-cern, this discrepancy arises from making the same (inap-propriate) normalization choice for both the massive andmassless cases. For a detector accelerating in a staticcavity with a massive scalar field [3], this leads to theconclusion that (in the non-relativistic regime) massivefields can distinguish local acceleration whereas masslessfields cannot.

Despite the discrepancy, the results here and in [3]nonetheless show that detector responses can indeed de-tect non-uniformity of accelerations in cavity which leadto distinguishability between the two scenarios. Essen-tially, it boils down to the fact that in the accelerat-ing cavity scenario, the static detector is only approx-imately uniformly accelerating from the perspective ofcavity frame, since the vertical worldlines cross hypersur-faces of constant but different ξ, which is approximatelyconstant for very short cavity or very small accelerations.On the other hand, an accelerating detector is an ex-actly uniformly accelerating test body; thus the setupis not mathematically equivalent — hence “qualitativeweak equivalence principle” [2].

To summarize, we first note that both accelerating cav-ity and accelerating detector setups are kinematically in-equivalent for any nonzero aL, as illustrated in Section Vand Appendix A. What conformal invariance in (1 + 1)dimensions gives us is convenience, a point made alsoin [2]. It boils down to the fact that in the rest frameof an accelerating cavity the detector does not undergouniform acceleration. Therefore, for any value of aL,there exists a finite difference in transition probability∆ Pr = |Prcav −Prdet| between the two setups regardlessof the mass of the field. This difference quickly van-ishes as aL → 0: in this ‘quasilocal regime’, we can ap-proximate the whole cavity as accelerating with a singleproper acceleration, recalling that the acceleration alongthe length of the cavity a(x) is related to the accelerationof the rear wall a1 by

a(x) = a11 + a1(x− x1) ≈ a1 (B1)

if a1(x − x1) < a1L 1. For this reason, ∆ Pr fallsquickly as a → 0, becoming exactly zero when a = 0(entirely static detector and cavity setups). So long asaL 6= 0, in principle we can always distinguish local accel-erations using nonlocal correlations of the field regardlessof mass. Choosing the detector gap to be closer to theresonant frequency of the field (e.g. excited Fock state)will help in amplifying very small transition probabili-ties, noting that the resonant frequencies between mass-less and massive cases would be different.

An alternative interpretation would be to require that

18

if ∆ Pr is below certain threshold we lose the capacityto distinguish local accelerations in the non-relativisticregime. All things being equal (taking into account res-onant effects etc.), this would mean that generically nei-ther massless nor massive fields can do the job of framedistinction if the threshold is not exceeded. While oper-ationally sensible, we prefer the previous interpretationsince ∆ Pr generally never actually vanishes except whenboth the cavity and the detector are at rest relative toone another. Neither massless nor massive fields are ‘pre-ferred’ in their capacity to distinguish local relative ac-celerations; any quantitative difference is purely due toquantum-theoretic aspects of nonlocal field correlationsand their dependence on mass.

Appendix C: Convergence of mode sums

We show some plots demonstrating how quickly themode sums converge for certain choices of parameters. InFigure 18 we plot the transition probabilities as a func-tion of mode sum for the field initiated as vacuum statefor two different accelerations a.

We see that the convergence is attained for relativelysmall N ∼ 100, and even if we sum N = 15 (the smallestN in these plots), the values do not stray far from theconverged value, thus for practical purposes we choose toperform calculations involving vacuum state for N = 15.Note that for fields initiated in excited states, the Wight-man function has vacuum and excited state contributionsbut the latter does not occur as sums over modes and

hence convergence issue does not appear.

100 200 300 400 500 600No. of modes

0.00007730

0.00007735

0.00007740

0.00007745

0.00007750

0.00007755Prob/λ2

Convergence, a = 0.01, Ω = π /L

Cavity Detector

100 200 300 400 500 600No. of modes

0.0046

0.0048

0.0050

0.0052

0.0054

0.0056

Prob/λ2Convergence, a = 1, Ω = π /L

Cavity Detector

FIG. 18. Probability as a function of mode sumN for a = 0.01and a = 1.0 with M = 0.

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