Spatially extended Unruh-DeWitt detectors for relativistic quantuminformation

Antony R. Lee and Ivette Fuentes*

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD,United Kingdom

(Received 5 December 2012; published 29 April 2014)

Unruh-DeWitt detectors interacting locally with a quantum field are systems under consideration forrelativistic quantum information processing. In most works, the detectors are assumed to be pointlike and,therefore, couple with the same strength to all modes of the field spectrum. We propose the use of a morerealistic detector model where the detector has a finite size conveniently tailored by a spatial profile. Wedesign a spatial profile such that the detector, when inertial, naturally couples to a peaked distribution ofMinkowski modes. In the uniformly accelerated case, the detector couples to a peaked distribution ofRindler modes. Such distributions are of special interest in the analysis of entanglement in noninetialframes. We use our detector model to show the noise detected in the Minkowski vacuum and in singleparticle states is a function of the detector’s acceleration.

DOI: 10.1103/PhysRevD.89.085041 PACS numbers: 03.70.+k, 11.10.-z

I. INTRODUCTION

Relativistic quantum information is a multifaceted areathat incorporates key principles from quantum field theory,quantum information, and quantum optics to answer,primarily, questions of an information theoretic nature.In order to implement quantum information tasks inrelativistic settings it is necessary to find suitable localizedsystems to store information. Moving pointlike detectorscoupled to quantum fields have been considered to carryquantum information in spacetime [1–3], perform telepor-tation [4], and extract entanglement from the Minkowskivacuum [2,5,6]. For a review see [7]. Our research programaims at developing new detector models which are morerealistic and simpler to treat mathematically so they can beused in relativistic quantum information processing.In this paper we utilize finite-size detectors [8–10], i.e.

detectors with a position-dependent coupling strength,which are not only more realistic but also have theadvantage of coupling to peaked distributions of modes.We design Gaussian-type spatial profiles such that auniformly accelerated detector naturally couples to peakeddistributions of Rindler modes. By expanding the field interms of Unruh modes, we show the accelerated detectorcouples simultaneously to two peaked distributions ofmodes corresponding to left and right Unruh modes. Asexpected, the same detector interacts with a Gaussiandistribution of Minkowski modes when it follows aninertial trajectory. In the Minkowski vacuum, the responseof the detector has a thermal signature when it is uniformlyaccelerated and the temperature depends on the properacceleration of the detector.

In the prototypical studies of quantum entanglement innoninertial frames, observers are assumed to analyze statesinvolving sharp frequency modes [11,12]. In particular,recent works analyzing the entanglement degradationbetween global modes seen by uniformly acceleratedobservers consider states of modes labeled by Unruhfrequencies [13–15]. Our analysis provides further insightinto the physical interpretation of the particle states whichwere analyzed in these works. Finite-size detectors aresuitable to discuss such results from an operational per-spective since sharp frequency modes are an idealization ofthe peaked distributions finite-size detectors couple to. It isshown that a finite-size detector observes the usual thermalspectrum plus an additional noise term, which depends onthe detector’s acceleration, when the field contains a singleUnruh particle. Therefore, a degradation of global modeentanglement in noninertial frames as a function of accel-eration should be detected by uniformly accelerated finite-size detectors. Although global mode entanglement cannotbe detected directly it can be extracted by Unruh-DeWitttype detectors becoming useful for quantum informationtasks. Throughout our work we use natural units and themetric has signature ð−;þ;þ;þÞ.

II. DETECTOR MODEL

The action of a detector interacting with a quantum fieldin the interaction picture is given by [8,9]

SI ¼Z

dVM · ϕ; (1)

where dV is the volume element for the spacetime and Mis the monopole moment of the detector parametrized byarbitrary coordinates in a (3þ 1)-dimensional spacetime.*Previously known as Fuentes-Guridi and Fuentes-Schuller.

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We will assume a flat spacetime and the field ϕ to be a realmassive scalar field which satisfies the Klein-Gordonequation,

∇μ∇μϕ −m2ϕ ¼ 0; (2)

where m ≥ 0 is the free field mass and ∇μ denotes thecovariant derivative on the spacetime. We require thedetector to have, in a comoving reference frame withcoordinates ðτ; ~ζÞ, a constant spatial profile. In this casethe monopole moment of the detector factorizes into atemporal operator valued function and a spatial functionMðτ; ~ζÞ ¼ MðτÞfð~ζÞ [16–18]. The temporal operator val-ued function describes the internal structure of the detectorwhich we model by a two-level system of characteristicenergy Δ; therefore, MðτÞ ≔ σ−e−iτΔ þ σþeiτΔ. The oper-ators σþ and σ− create and annihilate, respectively, exci-tations in the internal structure of the detector. In this case,the action can be written in terms of the detector’s propertime τ,

SI ¼Z

dτMðτÞ ~ϕðτÞ; (3)

where the field the detector couples to is given by

~ϕðτÞ ≔Z

d3 ~ζffiffiffiffiffiffiffiffiffiffiffiffi−gð~ζÞ

qfð~ζÞϕðτ; ~ζÞ; (4)

and g ¼ detðgμνÞ is the determinant of the metric tensor. Weassume the center of the detector follows a classicaltrajectory in spacetime and the spatial profile fð~ζÞ deter-mines how the detector couples to the field along thetrajectory. This function, which must be real to ensure theaction is Hermitian, can be interpreted as a position-dependent coupling strength. Consider u~νð~ζðτÞÞ to be fieldsolutions to the Klein-Gordon equation evaluated along apointlike worldline parametrized by τ corresponding to thecenter point of the detector. The frequencies ~ν of the modesare determined by an observer comoving with the center ofthe detector. The Hamiltonian in terms of these modes takesthe form

HI ¼ MðτÞ ·Z

d3 ~ν ~fð~νÞðu~νðτÞa~ν þ H:c:Þ; (5)

where a†~ν and a~ν are creation and annihilation operatorsassociated with the field modes of frequency ~ν. Thefrequency distribution ~fð~νÞ corresponds to a transformationof fð~ζÞ into frequency space. In the ideal case where thedetector is considered to be pointlike, the spatial profileis fð~ζÞ ¼ δ3ð~ζ − ~ζ0Þ [here δ3ð~ζ− ~ζ0Þ≔ δðζ1−ζ1

0Þδðζ2−ζ2

0Þδðζ3−ζ30Þ is the three-dimensional Dirac delta distri-

bution]. The detector couples locally to the field and thecoupling strength is uniformly equal for all frequencymodes. When we model a finite-size detector, whichcorresponds to a more realistic situation, the detector

couples naturally to a distribution of field modes. Thefrequency distribution will be determined by the spatialprofile. In this sense the field ~ϕðτÞ corresponds to a windowof frequencies.

III. INERTIAL TRAJECTORY

Now we specify a trajectory for the detector. When thecenter of the detector follows an inertial trajectory it isconvenient to use Minkowski coordinates ðt; ~xÞ where~x ≔ ðx; y; zÞ. In this case, the proper time of a comovingobserver is τ ¼ t and we can also write the comovingspatial coordinates as ~ζ ¼ x!. The solutions to the Klein-Gordon equation correspond to plane waves,

u~kðt; ~xÞ ≔1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ð2πÞ3ωp e−iωtþi ~k· ~x; (6)

where the frequency of the mode ω≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~k · ~kþm2

qis

strictly positive and ~k ∈ R3 denotes the momentum~k ≔ ðkx; ky; kzÞ. The inner product satisfies ðu~k; u~k0 Þ ¼δ3ð~k − ~k0Þ [19]. In this case the creation and annihilationoperators associated with the Minkowski field modes

satisfy ½a~k; a†~k0� ¼ δ3ð~k − ~k0Þ. The field can be expanded

in Minkowski modes as

ϕðt; ~xÞ ¼Z

d3 ~kða~ku~kðt; ~xÞ þ H:c:Þ: (7)

From this, the frequency distribution expressed inMinkowski modes is

~fð~kÞ ¼Z

d3 ~xfð~xÞeþi ~k· ~x; (8)

which is the Fourier transform of the spatial profile fð~xÞ.We now design a spatial profile tailored so that the

corresponding frequency detection window of the detectoris a Gaussian distribution of modes peaked around aMinkowski frequency ~λ. This choice is motivated by earlyworks on relativistic entanglement where the states ana-lyzed involved sharp frequencies Ω and Ω0. For example,the Bell state,

jϕi ¼ 1ffiffiffi2

p ðj0iΩj0iΩ0 þ j1iΩj1iΩ0 Þ; (9)

which was analyzed in a flat (1þ 1)-dimensional space inthe uncharged massless bosonic [13] and charged [14] case.Entanglement for Bell states in noninertial frames was alsodiscussed for Dirac fields [15,20,21]. Sharp frequencystates, j1iΩ ¼ a†Ωj0i, are an idealization of Gaussian wavepackets of the form

j1λi ¼Z

dkΦðλ; kÞa†kj0i; (10)

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where Φðλ; kÞ is a Gaussian distribution centred around λ[13]. Our detector model will be useful for investigatingquestions of entanglement in noninertial frames from anoperational perspective and extracting entanglement forrelativistic quantum information processing. An interestingquestion, which we intend to address in future work, is howmuch entanglement can be extracted by our detectors fromfield mode entangled states as a function of the detector’sacceleration.We find that a Gaussian frequency window of width σ

centered around frequency ~λ, as shown in Fig. 1, can beengineered by choosing the following spatial profile:

fð~xÞ ¼ nσe−12σ−2 ~x· ~xðe−i~λ·~x þ eþi~λ·~xÞ; (11)

which corresponds to a Gaussian distribution multiplied bya superposition of plane waves of opposite momentum ~λ. nσis a normalization constant. This spatial profile is of greatmathematical convenience and, moreover, similar couplingsnaturally arise in the interaction of harmonic oscillators andquantum fields. For example, consider the coupling betweenthe two-level system and the field is given by [22,23],

fðxÞ ¼ φ�−ðxÞ∂xφþðxÞ; (12)

where φ−ðxÞ and φþðxÞ are the ground and excited wavefuntions of the detector. One can consider the two energylevels to be two eigenstates of a quantum harmonicoscillator. In this case, the interaction strength will be anoscillatory Gaussian-type function that can be approximatedby the coupling strength we propose in Eq. (11). The wavefunctions of the two-level system in this case can be takento be Hermite functions of the form,

φn ¼ Nne−12x2HnðxÞ; (13a)

φm ¼ Nme−12x2HmðxÞ; (13b)

where Nk are normalization constants, Hk are Hermitepoynomials, and we impose n < m. Inserting these wavefunctions into the coupling strength Eq. (12), and using thewell-known recursion relations of the Hermite functions,we find,

fðxÞ ¼ φ�n

� ffiffiffiffim2

rφm−1 −

ffiffiffiffiffiffiffiffiffiffiffiffimþ 1

2

rφmþ1

�: (14)

From the properties of the Hermite polynomials, we knowthat the quantum numbers of the quantum harmonicoscillator wave functions need to take particular values.To be precise, we need n ¼ 2k (an even integer) andm ¼ 2pþ 1 (an odd integer). This is to ensure our examplecoupling strength accurately approximates the quantumharmonic oscillator coupling. A different choice of pairingwould either result in a coupling strength that is overall anodd function or would necessarily be zero at the origin, bothcontradicting the form of Eq. (11). Assuming the groundstate to be given by n ¼ 0, it can be easily shown that thepairs ðn;mÞ ¼ ð0; 1Þ and (0, 3) can be accurately approxi-mated by our coupling strength. The two pairs correspondto a bound state between the lowest energy eigenstate of thequantum harmonic oscillator and its first and third energystates. As an illustrative example, we consider the systemwhich is limited to interactions between the ground andthird excited state. In this case the normalized wavefunctions are

φ0 ¼ N0e−12x2H0ðxÞ; (15a)

φ3 ¼ N3e−12x2H3ðxÞ: (15b)

Consequently, it can be seen that the correspondingcoupling strength, FðxÞ ∼ ð2x4 − 9x2 þ 3Þe−x2 , is approxi-mated by a coupling of the form (11) with the parameterchoices ðλ; σÞ ¼ ð2.5; 1Þ. Our coupling strength wouldtherefore be a close approximation for a bound quantumharmonic oscillator. See Figs. 2 and 3 for plots showing theðn;mÞ ¼ ð0; 1Þ and (0, 3) systems, respectively. We bringto the reader’s attention Ref. [24] where a two-dimensionalversion of the coupling function (11) has been experimen-tally demonstrated in a Bose-Einstein condensate-cavitysystem. It should be noted, however, if the bound system isbetween high energy level wave functions then ourassumed coupling strength would no longer be a goodapproximation to the physical coupling strength. This isdue to the asymptotic form of the harmonic oscillator wavefunctions which approach either a purely cosine or sineform and lose their Gaussian nature. As a final comment,we would like to point out that a physically realized spatial

10 5 5 10

0.2

0.4

0.6

0.8

1.0

f

FIG. 1 (color online). ð1þ 1Þ-dimensional example of a fre-quency distribution peaked around �λ given by Eq. (16) forσ ¼ 1 and λ ¼ 5. This frequency distribution peaks around thedesired frequency λ but has a double peaking due to the twoexponential terms. In the ð1þ 1Þ massless case, the field isexpanded as an integral over ω > 0 and so the peak in the ω < 0region does not contribute.

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profile will generally be given by an effective coupling ofthe internal degrees of freedom of the detector and the field.In practice, it will have contributions from more than justthe wave function of the detector. To continue, usingEq. (8), the spatial profile is transformed into the momen-tum distribution,

~fð~kÞ ¼ e−12σ2ð~k−~λÞ·ð~k−~λÞ þ e−

12σ2ð~kþ~λÞ·ð~kþ~λÞ: (16)

This means that in order to couple our detector to a peakedGaussian distribution of modes centered around ~λ it isnecessary to engineer a field-detector coupling strengthwhich not only is peaked around the atom’s trajectory butalso oscillates with position. Sharp frequency modes~fð~kÞ ¼ δ3ð~k − ~λÞ þ δ3ð~kþ ~λÞ are obtained when fð~xÞ∼expð−i~λ · ~xÞ þ expðþi~λ · ~xÞ. In the massless ð1þ 1Þ-dimensional case the frequency distribution obtained froma given spatial profile is defined as a function ofω ≥ 0 only.Therefore, given the window profile peaks are sufficientlynarrow and separated, the second peaking corresponding tothe ω < 0 region does not contribute to the frequencywindow in this case. The field to which the detectorcouples, given by Eq. (4), is therefore

ϕðτÞ ¼Z

dωNωe−12σ2ðλ−ωÞ2 ½e−iωtaω þ eþiωta†ω�: (17)

In the general case, the frequency window is peaked aroundtwo modes corresponding to negative and positive momen-tum. Also see [23] which, complementary to this work,looks in detail at the coupling of Unruh-DeWitt detectors toMinkwoski modes. Equation (16) shows there is a trade-offbetween the width of the frequency window and the spatialprofile of the detector.We now proceed to calculate the instantaneous transition

rate of the detector model we have introduced. Thetransition rate is a function of the detector’s energy gapΔ given by [10]

_FτðΔÞ ≔ 2

Z∞

0

ds × Re½e−iΔsWðτ; τ − sÞ�; (18)

where Wðτ; τ0Þ ≔ hψ jϕðτÞϕðτ0Þjψi is the so-calledWightman function and jψi denotes the state of the field.Note we have assumed that the detector is turned on in thedistant past. Expanding the field in terms of Minkowskimodes we find that the vacuum transition rate for astationary detector is

_FτðΔÞ ¼ Θð−Δ −mÞΞðΔÞ; (19)

where

ΞðΔÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið−ΔÞ2 −m2

qj ~fð−ΔÞj2; (20)

and ΘðxÞ is the Heavisde theta function defined as

ΘðxÞ ¼�0∶ x < 0

1∶ x ≥ 0. (21)

We have explicitly written −Δ in Eq. (20) to emphasize thatin the limit m → 0 we recover the standard literature result.Note that in the above result we have explicitly assumed~fð~kÞ ¼ ~fðj~kjÞ i.e. the Fourier transform of the spatialprofile fð~xÞ depends on the magnitude of ~k only. Theresult is explicitly independent of the time parameter τ. Thiscan be traced back to the fact that an inertial trajectory is astationary orbit in flat spacetime. The interesting result hereis that the transition rate of the detector is tempered by thesquare of the frequency distribution ~f. As examples, wehave plotted the transition rate Eq. (19) for a Dirac deltapeaked profile [fðxÞ ∼ δðxÞ] and a double peaked Gaussianprofile Eq. (16) in Figs. 4 and (5, respectively. In Fig. 4, wecan see that pointlike excited detectors always have apossibility of undergoing spontaneous emission for−Δ > m. Outside of this region, the detector will eitherremain excited or in its ground state. In other words,spontaneous emission (or absorption) is not possible. InFig. 5, we observe how a Gaussian type spatial profile

3 2 1 1 2 3 x

0.4

0.2

0.2

0.4

0.6

0.8

1.0

f x

Approx

Exact

FIG. 3 (color online). Comparison of exact coupling strength(as modeled with a quantum harmonic oscillator) and ourapproximate coupling strength. The choices are quantumnumbers ðn;mÞ ¼ ð0; 3Þ and parameters ðλ; σÞ ¼ ð2.5; 1Þ.

3 2 1 1 2 3 x

0.2

0.4

0.6

0.8

1.0

f

Approx

Exact

FIG. 2 (color online). Comparison of exact coupling strength(as modeled with a quantum harmonic oscillator) and ourapproximate coupling strength. The choices are quantum num-bers ðn;mÞ ¼ ð0; 1Þ and parameters ðλ; σÞ ¼ ð1.66; 1= ffiffiffiffiffiffiffiffiffi

0.89p Þ.

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modifies the transition rate of an inertial detector. In starkcontrast to the pointlike detector, for a detector in its excitedstate with a large energy gap, the transition rate is negligibleoutside a “resonance” region. These regions effectively tellus the detector is sensitive to modes with energy ∼jΔj.We are also interested in analyzing the response of the

detector when the field has a single Minkowski excitation,

j1Φi ≔Z

d3 ~kΦð~kÞa†~kj0i; (22)

where we define a delta normalized state to have theproperty

Rd3 ~kjΦð~kÞj2 ¼ δ3ð0Þ and a properly normalized

state to have the propertyRd3 ~kjΦð~kÞj2 ¼ 1. This state is

the generalization of Eq. (10) to three spatial dimensions.The Wightman function in this case is

Wðτ; τ0Þ ¼ h1ΦjϕðτÞϕðτ0Þj1Φi: (23)

Writing the states and the field in terms of the Minkowskimodes and normal ordering the associated operators wefind that

Wðτ; τ0Þ ¼Z

d3 ~kjΦð~kÞj2 · h0jϕðτÞϕðτ0Þj0i

þ 2Re½I�ΦðτÞIΦðτ0Þ�; (24)

where

IΦðτÞ ¼Z

d3 ~kΦð~kÞ ~fð~kÞ 1ffiffiffiffiω

p e−iωτ: (25)

We notice there are two terms in the Wightman function.The first one corresponds to the vacuum state and thesecond is the contribution from the particle present inthe field.The single particle contribution factorizes into two

independent functions of τ and τ0. This allows us to analyzethe transition rate with relative ease. Substituting Eq. (24)into Eq. (18) we obtain an expression which depends on

ιτðΔÞ ≔Z

∞

0

dse−isΔIΦðτ − sÞ: (26)

This integral is essentially a Fourier transform of IΦðτ − sÞin the s variable and can be computed, either analytically ornumerically, for specifically chosen ~f and g. Employing theRiemann-Lebesgue lemma, which can only be used forfunctions which are integrable on the real line, one showsthat IΦðτÞ → 0 as τ → �∞ as long as ~f and g are wellbehaved. IΦðτÞ vanishes in the distant past and future wherethe detector is responding only to vacuum fluctuations. Inother words, the detector only observes a constant spectrumin these asymptomatic regions. In the intermediate regionsthe oscillatory response is due to the presence of theparticle.

IV. ACCELERATED TRAJECTORY

We now consider a detector following a uniformlyaccelerated trajectory. Conformally flat Rinder coordinatesξ ¼ ðτ; ~ξÞ ¼ ðρ; ξ; y; zÞ are a convenient choice in this case.The transformation between Rindler and Minkowski coor-dinates is given by

t ¼ a−1eaξ sinhðaρÞx ¼ a−1eaξ sinhðaρÞ; (27)

where a is a postive parameter and the other spatialcoordinates do not change, i.e., y ¼ y, z ¼ z. This trans-formation holds for the spacetime region jtj > x whichis called the right Rindler wedge. The coordinate

4 3 2 1 1

1

2

3

4

F

m 1.5

m 1

m 0

FIG. 4 (color online). The transition rate for an inertial, Diracdelta spatial profile (i.e. pointlike) detector probing a ð3þ 1Þmassive field. We have plotted the transition rate for mass valuesm ¼ 0, 1, 1.5. We observe that if the detector is in its excited state(−Δ > 0), its energy gap needs to be larger than the mass of thefield to undergo spontaneous emission. Also, for all energy gaps−Δ < m, the detector cannot undergo spontaneous emission, i.e.,it is stable in its excited state.

4 3 2 1 1

0.5

1.0

1.5

2.0

2.5

F

m 1.5

m 1

m 0

FIG. 5 (color online). The transition rate for an inertial, doublepeaked Gaussian spatial profile detector [see Eq. (11)] probing að3þ 1Þmassive field. We have plotted the transition rate for massvalues m ¼ 0, 1, 1.5. We observe that if the detector is in itsexcited state (−Δ > 0), for high energy gaps the detector’stransition rate is negligibly small. This implies that for largeenergy gaps, a nonpointlike detector is stable, i.e., it will notundergo spontaneous emission.

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transformation for jtj > −x (left region) differs fromEq. (27) only by an overall sign in the x and t coordinates.The coordinates are tailored specifically to the trajectory~ξ ¼ ~0 so that an observer traveling along this worldline willmeasure a proper acceleration

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi−AμAμ

p ¼ a and the propertime is parametrized by the coordinate time ρ. The Klein-Gordon equation for a massive bosonic field in a (3þ 1)-dimensional flat spacetime in this case takes the form

∂ρρϕ − ½∂ξξ þ e2aξð∂yy þ ∂zzÞ −m2e2aξ�ϕ ¼ 0; (28)

and the solutions are the Rindler modes [13,19]

uΩ; ~k⊥;αðρ; ~ξÞ ≔ NΩ=aKiΩ=aðMa−1eaξÞe−iΛαðρ; ~ξÞ

Λαðρ; ~ξÞ ≔ αρΩ − ~k⊥ · ~x⊥; (29)

with M ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~k⊥ · ~k⊥ þm2

qand NΩ=a is the mode normali-

zation constant. The functions KiΩ=aðRÞ are modifiedBessel functions of the second kind. Here ~x⊥ ≔ ðy; zÞand ~k⊥ ≔ ðky; kzÞ are position and momentum vectorsperpendicular to the direction of acceleration. Ω isstrictly positive and denotes the Rindler frequency andα ¼ þ1ð−1Þ corresponds to right (left) Rindler regions,respectively. The canonical orthonormality relation for theð3þ 1Þ massive field is

ðuΩ; ~k⊥;α; uΩ0; ~k0⊥;α0Þ ¼ δðΩ −Ω0Þδ2ð~k⊥ − ~k⊥

0Þδαα0 ; (30)

and commutation relations satisfy

½aΩ; ~k⊥;α; a†Ω0 ~k0⊥;α0

� ¼ δðΩ −Ω0Þδ2ð~k⊥ − ~k⊥0Þδαα0 : (31)

From our coordinate definitions Eq. (27), we choose thedetector to be traveling in the right Rindler wedge. Thisimplies that our comoving coordinates can be parametrizedas τ ¼ ρ and ~ζ ¼ ~ξ. The field expansion in terms of theparametrized Rindler modes is

ϕðτ; ~ξÞ ¼Z

dΩd2 ~k⊥½uΩ; ~k⊥;þðρ; ~ξÞaΩ; ~k⊥;þ þ H:c:�: (32)

Note the left Rindler modes do not appear in Eq. (32) as thedetector is assumed to be moving in the right Rindlerwedge. The explicit form of the accelerated detectorsfrequency distribution is

~fðΩ; ~k⊥Þ ¼Z

d3 ~ξe2aξfð~ξÞKiΩ=aðMa−1eaξÞeþi ~k⊥· ~x⊥ : (33)

The most significant difference between the inertial and theaccelerated frequency distributions is the appearance of anontrivialmetric factor and theBessel function.Note also thatfor both massless and massive fields, the Rindler modes aredefined as an integral over Ω ∈ Rþ, unlike the Minkwoski

mode case. Equation (33) is a Fourier transform in the y and zcoordinates; however, it is a nonstandard integral trans-formation in the ξ coordinate. Reminiscent of a Hankeltransformation, we expect our desired properties of arbitrarymode peaking to still hold. Using the integral representationof the modified Bessel function for the second kind

KiΩ=aðRÞ ¼ffiffiffiπ

p ð12RÞiΩ=a

ΓðiΩ=aþ 1=2ÞZ

∞

0

dtðsinhðtÞÞ2iΩ=aþ1

eR coshðtÞ ; (34)

valid for Ω=a > 0 and R > 0, we can write the frequencydistribution as a Fourier type integral that takes the form

~fð~kÞ ¼Z

d3 ~ξβð~ξÞei~ξ· ~k; (35)

where now ~k ¼ ðΩþ δ; ky; kzÞ and

βð~ξÞ ¼ffiffiffiπ

p ð12Ma ÞiΩ=a

ΓðiΩ=aþ 1=2Þ fð~ξÞZ

∞

0

drðsinhðrÞÞ2iΩ=aeMae

aξ coshðrÞ ; (36)

δ is a phase that is acquired from the integral representation ofthe modified Bessel function. This shows that, in principle,the standard properties of the Fourier transformation can beused to design a detector profile such that we obtain a peakeddistribution in momentum space. For a concrete example, weshall consider the massless ð1þ 1Þ field case. The appro-priate transformation, in terms of Rindler modes, is given by

~fðΩÞ ¼Z

dξe2aξfðξÞeiΩξ; (37)

and the spatial profile we propose in this case is

fðξÞ ¼ NðσÞe−2aξe−12σ−2ξ2ðe−i~λξ þ eþi~λξÞ; (38)

which includes the conformal metric factor that arises fromthe Rindler coordinate transformation. Here NðσÞ is anormalization constant and ~λ dictates a preferred modefrequency. This profile reduces the integral transformationEq. (37) to a standard Fourier transformation and the resultingfrequency distribution is

~fðΩÞ ¼ e−12σ2ð~λ−ΩÞ2 þ e−

12σ2ð~λþΩÞ2 : (39)

Substituting this frequency distribution intoEq. (4), weobtain

ϕðτÞ ¼Z

dΩNΩ=ae−12σ2ð~λ−ΩÞ2 ½e−iΩτaΩ;I þ H:c:�: (40)

Therefore, our detector couples to a Gaussian distributioncentered around a Rindler frequency ~λ. In the limiting casewhere the acceleration goes to zero, the Rindler frequencygoes to theMickowski frequency, i.e.,Ω → ω, and the spatialprofile reduces to the Minkwoski profile given by Eq. (11).

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Wewould now like to expand the field in terms of Unruhmodes which pay an important role in the literature. Thesemodes are given by [13,19]

uΩ; ~k⊥;R ¼ coshðrΩ=aÞuΩ; ~k⊥;þ þ sinhðrΩ=aÞu�Ω; ~k⊥;−;uΩ; ~k⊥;L ¼ coshðrΩ=aÞuΩ; ~k⊥;− þ sinhðrΩ=aÞu�Ω; ~k⊥;þ; (41)

where tanhðrΩ=aÞ ¼ e−πΩ=a. Upon parametrizing themodes with our accelerated comoving coordinates, i.e.,ðρ; ~ξÞ ¼ ðτ; ~ξÞ, and noting that the left Rindler modes haveno support in the right Rindler wedge, we find the Unruhmodes reduce to

uΩ; ~k⊥;Rðτ; ~ξÞ ¼ coshðrΩ=aÞuΩ; ~k⊥;þðτ; ~ξÞ;uΩ; ~k⊥;Lðτ; ~ξÞ ¼ sinhðrΩ=aÞu�Ω; ~k⊥;þðτ;

~ξÞ: (42)

In the case the field is massless and in (1þ 1) dimensions,the detector interacts with two peaked distributions corre-sponding to right and left Unruh modes, respectively,

ϕðτÞ ¼ ϕRðτÞ þ ϕLðτÞ; (43)

where

ϕRðτÞ ¼Z

dΩNΩae−

12ð~λ−ΩÞ2chΩe−iΩτaΩ;R þ H.c.; (44a)

ϕLðτÞ ¼Z

dΩNΩae−

12ð~λ−ΩÞ2shΩeþiΩτaΩ;L þ H.c.; (44b)

and to shorten notation we have defined chΩ ≔ coshðrΩ=aÞand shΩ ≔ sinhðrΩ=aÞ. Note that the effective interactionstrength is now modulated by hyperbolic trigonometricfunctions. We are currently using this detector model toanalyze entanglement extraction of sharp frequency Unruhstates. As uniform acceleration is also a stationary orbit offlat spacetime, we expect a time-independent vacuumtransition rate [25]. Using the parametrized Unruh modes,we can calculate the transition rate of the accelerateddetector. We find for the field in its vacuum state,

_FτðΔÞ ¼1

e2πΔ=a − 1ΞðΔÞ; (45)

where

ΞðΔÞ ≔ Rd2 ~k⊥½N2

~Δj ~fð ~ΔÞj2ΘðΔÞ − N2

− ~Δj ~fð− ~ΔÞj2Θð−ΔÞ�;

(46)

with � ~Δ ≔ ð�Δ; ky; kzÞ and N ~Δ denotes the appropriatenormalization for the Rindler modes. We can see immedi-ately the transition rate of the detector is the expectedthermal distribution, where the temperature is inversely

proportional to the acceleration parameter a, but againmodified by the frequency of the field operator. Again asexamples, we plot the transition rate (45) for the (1þ 1)massless field for different accelerations. In Fig. (6), weplot the transition rate for a Dirac delta profile. In Fig. (7),we plot the transition rate for a double Gaussian peakingprofile. We note the qualitative and quantitative differencesbetween the standard pointlike detector and our modi-fied model.We also note that Eq. (45) satisfies the Kubo-Martin-

Schwinger condition [26,27]

_FτðΔÞ ¼ e−2πaΔ _Fτð−ΔÞ: (47)

The transition rate is, as expected, independent of time dueto the stationarity of the trajectory and the invariance of thevacuum state. Now we shall analyze the response of ouraccelerated detector model when the field contains a singleUnruh particle. In the literature, well analyzed states of thefield correspond to maximally entangled Bell states; see forexample, [11,28,29]. These states contain both the vacuumand a single Unruh particle. Our starting point will again bethe Wightman function which, for the one particle state,takes the form

Wðτ; τ0Þ ≔ h1pjϕðτÞϕðτ0Þj1pi; (48)

where j1pi is a one Unruh particle state defined as

j1pi ≔RdΩd2 ~k⊥ΦðΩ; ~k⊥Þa†Ω; ~k⊥;pj0i. Continuing in the

exact same fashion as the Minkowski one particle statewe find

Wðτ; τ0Þ ¼Z

dΩd2 ~k⊥jΦð~kÞj2 · h0jϕðτÞϕðτ0Þj0i

þ 2Re½I�pðτÞIpðτ0Þ�; (49)

2 1 0 1 2

1

3

5

7

F

a 1.5

a 1

a 0.1

FIG. 6 (color online). The transition rate for an accelerated,Dirac delta spatial profile detector probing a (1þ 1) masslessfield. We have plotted the transition rate for acceleration valuesa ¼ 0.1, 1, 1.5. We see that the transition rate is zero only forasymptotically large values of Δ. The divergent region Δ ¼ 0 isdue to the nature of the massless (1þ 1) field.

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where IpðτÞ ≔RdΩd2 ~k⊥ ~fðΩ; ~k⊥ÞΦðΩ; ~k⊥ÞUΩ;pðτÞ with

UΩ;pðτÞ ¼ NΩ=a

�coshðrΩ=aÞe−iΩτ∶ p ¼ R

sinhðrΩ=aÞeþiΩτ∶ p ¼ L. (50)

An Unruh particle has with it an associated wave function,

defined asRdΩd2 ~k⊥ΦðΩ; ~k⊥ÞUΩ;pðt; xÞ. The accelerated

detector will probe the Unruh particles’ wave function as itapproaches. As the particles’ wave function oscillates as afunction of τ, the detector will observe different phases atdifferent times. It is these oscillations that contribute to theundulatory behavior of the detectors’ transition rate. For theUnruh state, the corresponding accelerated expression forEq. (26) again has a time-dependent oscillatory integral. Itis clear, for appropriately behaved functions ~f and g, thesame analysis can be applied here as for the Minkwoskiparticle. The Riemann-Lebesgue lemma can be used toshow in the asymptotic past and future, the response of thedetector is the same as the vacuum and hence has a thermalsignature. As with the Minkowski particle, the intermediateregions between past and future asymptotic times give riseto an oscillatory response function. In the limit of highacceleration, the second term in Eq. (49) becomes negli-gible and the state tends to the maximally mixed state.Thus, the single particle state of the field is dominated bythe vacuum fluctuations.

V. CONCLUSIONS

We introduce a detector model which naturally couplesto peaked frequency distributions of Minkowski, Unruh,and Rindler modes. This detector model is suitable forstudies of entanglement extraction in noninertial frames. Inthe (3þ 1)-dimensional case, the frequency window of the

detector peaks around positive and negative momentuminducing a double peaking. In the (1þ 1)-dimensionalcase, frequency distributions naturally peak around a singlefrequency. We obtain analytical results for the instanta-neous transition rates of the detectors undergoing inertialand uniformly accelerated motion. In particular, the tran-sition rate of the accelerated detector is the expectedthermal distribution modified by a smearing function thatarises from the detectors’ spatial profile. We have alsoshown the well-studied single Unruh particle states producean oscillatory response that is only thermal in the asymp-totic past and future. Since for accelerated detectors thermalnoise is observed in both the vacuum and the single Unruhparticle state, entanglement is expected to be degraded bythe Unruh effect for global Unruh modes. We also see thedegradation effects occur for properly normalized wavepackets.As shown in Figs. 2–5, the response of the Unruh-DeWitt

detector depends strongly on the profile of its interactionand, thus, on the physical system that implements thedetector. Contributing factors to this profile will be theeigenstates of the detector, external components such asdriving laser fields, and the geometry of the system.We haveshown the response of the detector can vary significantly andthis will have observable consequences in the experimentalverification of results in quantum field theory and relativisticquantum information. To this end, we comment on thephysical feasibility and impact on constructing the spatiallydependent coupling profile we have described.A spatially dependent coupling strength can be engi-

neered by placing the quantum system in an externalpotential which is time and space dependent. These tunableinteractions have been produced in ion traps [30,31], cavityQED [32], and superconducting circuits [33–36]. In an iontrap, the interaction of the ion with its vibrational modescan be modulated by a time- and spatial-dependent classicaldriving field, such as a laser [37]. In cavity QED, time- andspace-dependent coupling strengths are used to engineer aneffective coupling between two cavity modes [38,39].Within this setting, our work is particularly relevant wherecurrent investigations [40–42] study the effect of artificialatoms, with large spatial profiles, interacting with modesof an electromagnetic field. Our model can also be, inprinciple, realized in a Bose-Einstein condensate where aharmonic oscillator detector corresponds to an impuritycreated by a potential well within the condensate[24,43,44]. The spatial profile is engineered by choosingthe right shape of the trapping potential. The harmonicoscillator couples to the phononic field of the condensate,which obeys a Klein-Gordon equation in an effectivecurved metric. Therefore, our model can be used to studyeffects in analogue gravity.Unruh-DeWitt-type detectors have allowed us to

explore different coupling scenarios in quantum fieldtheory. In tandem, our results have paved the way for

2 1 0 1 2

0.5

1.0

1.5

2.0

2.5F

a 1.5

a 1

a 0.1

FIG. 7 (color online). The transition rate for an accelerated,double peaked Gaussian spatial profile detector probing a (1þ 1)massless field. We have plotted the transition rate for accelerationvalues a ¼ 0.1, 1, 1.5. We see that there are resonant values for Δwhere probability of the detector undergoing spontaneous emis-sion (or absorption) is increased. This can be traced back to theGaussian spatial profile which causes the detector to interact morestrongly with modes with energy ∼jΔj.

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the development of novel mathematical techniques thatallow a nonperturbative treatment of Unruh-DeWitt detec-tors [45,46] and more advanced phenomena, such asdetector-field backreaction [47]. With these considerations,we hope that Unruh-DeWitt detectors can be used to gainfurther insight into quantum information within quantumfield theory.

ACKNOWLEDGMENTS

The authors would like to thank N. Friis, D. E. Bruschi,J. Louko, A. Dragan, L. Hodgkinson, G. Adesso,S. Tavares, S. Y. Lin, and T. Tufarelli for useful commentsand discussions. I. F. was supported by EPSRC (CAF GrantNo. EP/G00496X/2).

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