Phonon creation by gravitational waves

Carlos Sabın,1 David Edward Bruschi,2 Mehdi Ahmadi,1 and Ivette Fuentes1

1School of Mathematical Sciences, University of Nottingham,University Park, Nottingham NG7 2RD, United Kingdom

2Racah Institute of Physics and Quantum Information Science,The Hebrew University of Jerusalem, 91904, Givat Ram, Jerusalem, Israel

We show that gravitational waves create phonons in a Bose-Einstein condensate (BEC). A trav-eling spacetime distortion produces particle creation resonances that correspond to the dynamicalCasimir effect in a BEC phononic field contained in a cavity-type trap. We propose to use thiseffect to detect gravitational waves. The amplitude of the wave can be estimated applying recentlydeveloped relativistic quantum metrology techniques. We provide the optimal precision bound onthe estimation of the wave’s amplitude. Finally, we show that the parameter regime required todetect gravitational waves with this technique is within experimental reach.

Introduction. Einstein’s theory of general relativity [1]predicts the existence of gravitational waves [2]. Gravita-tional waves are perturbations of the spacetime generatedby accelerated mass distributions. The theory predictsthat the amplitude of gravitational waves is extremelysmall and thus, finding experimental evidence of theirexistence is a difficult task. Indeed the quest for the de-tection of these spacetime distortions [3] has been one ofthe biggest enterprises of modern science and the focus ofa great amount of work, both in theory and experiment.

In this paper we show that small spacetime distortionsproduce phononic excitations in a BEC. We propose ascheme that exploits this effect to detect gravitationalwaves. Phononic excitations of a trapped BEC satisfy aKlein-Gordon equation on a curved background metric.The metric has two terms [4–6], one corresponding to thereal spacetime metric and a second term, correspondingto the analogue gravity metric, which depends on BECparameters such as velocity flows and energy density. Inthe field of analogue gravity, the real spacetime metric isconsidered to be flat while the BEC parameters are mod-ified by the experimentalist in order to mimic spacetimedynamics. In this way, sonic black holes and expandinguniverses are simulated in a BEC. Interestingly, the ef-fects of the real spacetime metric on the phononic fieldhave been ignored. In this paper, we show that changes inthe real spacetime metric produce phononic excitationsthat can be in principle detected. This generalises thework presented in reference [7] to the curved backgroundcase. In particular we consider the case of gravitationalwaves such as those created by supernovas and gamma-ray bursts. We propose to use this effect, which is both aquantum and a relativistic effect, to design a novel grav-itational wave detector. By applying recently developedtechniques in relativistic quantum metrology [7, 8], weshow that the regime of functionality of the phononicgravitational wave detector is within experimental reach.

Quantum metrology [9] exploits quantum properties toimprove the performance of measurement technologies.Indeed, quantum metrology promises to provide useful

k

n

k

n

k

n

0

FIG. 1: Sketch of the setup: A BEC in a box-like poten-tial that acts as a cavity for the phononic excitations. Twophononic modes k and n are initially prepared in a two-modesqueezed state represented by the initial covariance matrix σ0.A gravitational wave of amplitude ε transforms the state pro-ducing excitations. The new state σε depends on ε. Measure-ments on the modes can be used to estimate the amplitude ofthe spacetime distortion.

techniques for gravitational wave detection [10] and arecurrently being used in one of the most ambitious pro-grams of gravitational-wave astronomy, the Laser Inter-ferometry Gravitational-Wave Observatory (LIGO) [11].LIGO is a laser interferometer that attempts to detectgravitational waves by means of the changes they producein the optical paths of the laser in the interferometer’sarms. Small path changes should produce phase shiftsthat can be measured at the output of the interferometer.Since the gravitational waves are very small, the interfer-ometer arms must be 4 km long and the mirrors’ weightis 10 kg. LIGO’s sensitivity can be enhanced employinglaser quantum states, such as squeezed states. By thismethod, the noise can be in principle reduced to shot-noise quantum limit [12]. Unfortunately, the regimes atwhich these techniques would provide important advan-tages, have not been reached yet in LIGO. Sources oferror such as the thermal noise of the test masses, arestill challenges to overcome before one can hope to reachthe quantum regime. An alternative method that hasbeen proposed to detect gravitational waves using quan-tum states involves an atom interferometer [13, 14]. Inan atom interferometer, the wavefunction of a BEC is

arX

iv:1

402.

7009

v2 [

quan

t-ph

] 2

8 Fe

b 20

14

2

split and recombined by means of laser pulses, giving riseto a phase shift proportional to the acceleration inducedspacetime distortion. Interestingly, although both laserand atom interferometer schemes operate in the overlapof quantum mechanics and relativity, they do not con-sider relativistic quantum field theoretical effects, suchas particle creation [15, 16] or mode-mixing [17]. Theschemes mentioned above use non-relativistic quantummechanics on one hand, and general relativity on theother. However, these theories are known to be incom-patible.

Quantum field theory in curved spacetime allows oneto properly incorporate quantum and relativistic effectsat low energies. The energy regime in which the the-ory is applicable includes the natural energy regime ofgravitational waves that are produced by, for example,supernovas. Interestingly, the application of metrologytechniques to quantum field theory in curved spacetimeremains practically unexplored. Only recently, a frame-work for relativistic quantum metrology has been devel-oped applying metrology techniques to estimate param-eters of quantum fields that undergo relativistic trans-formations [7, 8, 18, 19]. The techniques can be appliedto estimate spacetime parameters, proper times, grav-itational field strenghts and accelerations, among otherquantities of great interest to science and technology. Us-ing this framework, it was shown that the non-uniformacceleration of a BEC cavity trap produces phononic ex-citations that can be detected with cutting-edge technol-ogy. This effect occurs in a flat spacetime metric withmoving boundary conditions, i.e. corresponding to thedynamical Casimir effect. Phonon creation provides thebasis of a quantum accelerometer that exploits relativis-tic effects to improve the state of the art in quantumaccelerometers [7].

In this paper, we apply relativistic quantum metrol-ogy techniques to calculate the optimal precision boundachieved for detecting gravitational waves using thephononic creation effect we described above. We charac-terise the transformation that a gravitational wave gen-erates on the phononic excitations in a BEC (see Fig. 1).The final state depends on the amplitude of the gravi-tational wave. We compute the quantum Fisher infor-mation associated to this state and -via the quantumCramer-Rao bound- we obtain the fundamental boundto the error in the measurement of the spacetime ripple.We find the experimental regime of parameters for whichthis bound is low enough to enable the detection of agravitational wave.

Modelling the gravitational wave spacetime. The metricof a gravitational wave spacetime is commonly modelledby a small perturbation hµν to the flat Minkowski metricηµν , i.e. [2],

gµν = ηµν + hµν (1)

where

ηµν =

−c2 0 0 0

0 1 0 00 0 1 00 0 0 1

. (2)

and c is the speed of light in the vacuum. We con-sider Minkowski coordinates (t, x, y, z). In the transversetraceless (TT) gauge [2], the perturbation correspondingto a gravitational wave moving in the z-direction can bewritten as,

hµν =

0 0 0 00 h+(t) h×(t) 00 h×(t) −h+(t) 00 0 0 0

, (3)

where h+(t), h×(t) correspond to time-dependent pertur-bations in two different polarisations. Later on we willrestrict the analysis to 1-dimensional fields, where theline element takes a simple form,

ds2 = −c2 dt2 + (1 + h+(t)) dx2. (4)

The null geodesics of the above line element yield thespeed of propagation of photons in the spacetime, whichis c(t) = dx

dt =' c(1− 12 h+(t)) up to the first order in h.

Bose-Einstein condensates on a curved and flat space-time. We are interested in describing a BEC in this space-time and exploring the possibility that the gravitationalwaves produce observable effects in the system. The sys-tem is extremely small, we will consider the BEC to be 1µm long. Thus, it is natural to think that the effects aretoo small to be observable. Surprisingly, amplificationeffects produced by the slow propagation of excitationson the BEC make the effects, in principle, observable.To show this we describe the BEC on a general space-time metric following references [4–6]. In the superfluidregime, a BEC is described by a mean field classical back-ground Ψ plus quantum fluctuations Π. These fluctua-tions, for length scales larger than the so-called healinglength, behave like a phononic quantum field on a curvedmetric. Indeed, in a homogenous condensate, the fieldobeys a massless Klein-Gordon equation Π = 0 wherethe d’ Alembertian operator = 1/

√−g ∂a(

√−ggab∂b)

depends on an effective spacetime metric gab -with deter-minant g- given by [4–6]

gab =

(n20 c

−1s

ρ0 + p0

)[gab +

(1− c2s

c2

)VaVb

]. (5)

The effective metric is a function of the real spacetimemetric gab (that in general may be curved) and back-ground mean field properties of the BEC such as thenumber density n0, the energy density ρ0, the pressurep0 and the speed of sound cs = c

√∂p/∂ρ. Here p is the

3

total pressure, ρ the total density and Va is the 4-velocityflow on the BEC. In the field of analogue gravity, the realspacetime metric is considered to be flat and, therefore,it’s effects are neglected. Analogue spacetimes are simu-lated through the artificial manipulation of what we callthe analogue gravity metric ,

Gab =

(n20 c

−1s

ρ0 + p0

)[(1− c2s

c2

)VaVb

]. (6)

Experimentalists change the background parameters ofthe analogue metric Gab to simulate sonic black holes orexpanding universes [20]. Here we are interested solely onthe effects of the real spacetime metric. To ensure this,we consider that in the comoving frame Va = (c, 0, 0, 0)and obtain,

gab =

(n20 c

−1s

ρ0 + p0

)gab +

(c2 − c2s) 0 0 0

0 0 0 00 0 0 00 0 0 0

. (7)

In the absence of a gravitational wave, the real spacetimemetric is gab = ηab, where ηab has been defined in Eq.(2). Therefore, the effective metric of the BEC phononicexcitations on the flat spacetime metric is given by,

gab =

(n20 c

−1s

ρ0 + p0

)−c2s 0 0 0

0 1 0 00 0 1 00 0 0 1

. (8)

Ignoring the conformal factor, we notice that the metricis the flat Minkowski metric with the speed of light be-ing replaced by the speed of sound cs. By considering arescaled time coordinate τ = (c/cs)t we recover the stan-dard Minkowski metric ds2 = −cdt2 + dx2. This meansthat the phonons live on a spacetime that is Minkowskihowever, due to the BEC ground state properties, timeflows slower and excitations propagate accordingly. Asa result of this, we will show that changes in the realspacetime metric are amplified, becoming observable.

The solutions of the Klein-Gordon equation with themetric in Eq. (8) describe massless excitations propa-gating with the speed of sound cs. Therefore, the fre-quency of the mode ωk is given by the dispersion relationωk = cs |k|, where k is the mode’s momentum. We con-sider that the BEC is contained in a 1-dimensional cavitytrap. Therefore, we impose close to hard-wall boundaryconditions [21–23] that give rise to the spectrum,

ωn =nπ csL

, (9)

where L is the cavity length and n ∈ 1, 2.... The modesolutions to the Klein-Gordon equation are given by,

φn =1√nπ

sinnπ(x− xL)

Le−i ωn t, (10)

where xL and xR are the positions of the left and rightwalls, respectively.The phononic field Π(t, x) is then quantised by associat-

ing creation and annihilation operators a†k and ak to themode solutions [24],

Π(t, x) =∑k

(φk(t, x) ak + φ∗k(t, x) a†k

). (11)

The operators ak and a†k obey the canonical commu-tation relations. In what follows we will describe thephonons in the spacetime of a gravitational wave.

Phonon creation by a spacetime distortion. Consider thatinitially the spacetime is flat and that the phonons are ina given initial state σ0. We are interested in computingthe state of the phonons after a gravitational wave haspassed by. The real spacetime metric of a gravitationalwave is given by Eq. (1) and thus, the effective metricfor the phonons is,

gab =

(n20 c

−1s

ρ0 + p0

)−c2s 0 0 0

0 1 + h+(t) h×(t) 00 h×(t) 1− h+(t) 00 0 0 1

.

(12)For simplicity, we considered a quasi one-dimensionalBEC. The line element is conformal to,

ds2 = −c2s dt2 + (1 + h+(t)) dx2. (13)

The effects of the spacetime distortion on the phononscan be computed using a Bogoliubov transformation [25].The flat field operators ak in Eq. (11) are transformedinto,

am =∑n

(α∗mnan + β∗mna

†n

), (14)

where a†k and ak are creation and annihilation oper-ators associated to the perturbed mode solutions andαmn(h+(t)) and βmn(h+(t)) are Bogoliubov coefficientsthat depend on the wave’s spacetime parameters. In or-der to find the perturbed mode solutions and the Bo-goliubov coefficients induced by the wave, we apply atechnique developed in reference [17]. The technique en-ables the computation of Bogoliubov coefficients associ-ated to continuous variations of spacetime by integratingdiscrete changes. Following this procedure, we assumethat spacetime is initially flat and instantaneously un-dergoes a discrete spacetime perturbation ε. The newmetric will be given by,

ds2 = −c2s dt2 + (1 + ε) dx2. (15)

Considering the change of coordinates,

t = t′, x = x′ (1− ε

2), (16)

4

we find that the mode solutions to the Klein-Gordonequation in the new metric after imposing hard-wallboundary conditions at x′L and x′R are

φn(t′.x′) =1√nπ

sinnπ(x′ − x′L)

L′e−i ω

′n t, (17)

where

L′ = x′L − x′R = L , ω′n = ωn . (18)

We assume that the ε-perturbation doesn’t change therigidity of the trap, that is the proper length [10]:

L(ε) =

∫ xR(ε)

xL(ε)

dx√

1 + ε (19)

remains constant L(ε) = L, up to order ε. Thismeans that in coordinates (t, x), the boundary conditionschange in the following way:

xL(ε) = xL(1− ε

2), xR(ε) = xR(1− ε

2). (20)

Thus, in coordinates (t′, x′):

x′L = xL; x′R = xR (21)

The Bogoliubov coefficients αmn(ε) = (φm, φn) and

βmn(ε) = −(φn, φ∗m) can be calculated using the Klein-

Gordon inner product [25] between the flat and ε-perturbed mode solutions in Eqs. (10) and (17). Tofirst order in ε and for xL = 0 we find that,

βmn = − (−1)m+n√mn

2(m+ n)ε (m 6= n)

βnn = 0

αmn =(−1)m+n

√mn

2(m− n)ε (m 6= n)

αnn = 1 (22)

[36]. After the instantaneous perturbation the fieldmodes undergo a period of free evolution before under-going the next instantaneous perturbation. During freeevolution the modes pick up a time-dependent phase.Following [17], the Bogoliubov coefficients correspond-ing to the continuous perturbation produced by a grav-itational wave h+(t) = ε sin Ω t, can be computed byapproximating the perturbation h+(t) by instantaneousperturbations in ε followed by infinitesimal intervals offree evolution. Taking the continuos limit, we obtain theBogoliubov coefficients corresponding to a sinusoidal per-turbation,

βmn(t) = (ωm + ωn)βmn

∫ t

0

e−i (ωm+ωn) t′sin (Ω t′) dt′

αmn(t) = (ωm − ωn)αmn

∫ t

0

e−i (ωm−ωn) t′sin (Ω t′)dt′.

In quantum field theory, β 6= 0 is associated with particlecreation. Therefore, we conclude that the gravitationalwave has produced phonons in the system. The numberof particles produced is n =

∑m |βmn|2 [26], therefore,

proportional to the amplitude, frequency and period ofthe wave. We note that there is a particle creation reso-nance at

Ω = ωm + ωn. (23)

At resonance, assuming that the duration of the wave tis long enough,

ω1 t >> 1, (24)

and that m + n = odd, the first order terms of the Bo-goliubov coefficients are given by,

βjk(t) =ε

2

√n

mωm t δjm δkn +O(ε2)

αjk(t) = 0 +O(ε2), (25)

that is, the only non-negligible coefficients are βnm. TheBogoliubov coefficients we obtained coincide with thoseof a cavity in flat spacetime with sinusoidally varyinglength,

L(t) = L(0)(1 + ε sin(Ω t)), (26)

which where computed in reference [27] using an alter-native method.

Introducing the covariance matrix formalism. It is con-venient to use the covariance matrix formalism to de-scribe the Bogoliubov transformation that the wave in-duces on the field states. Gaussian states of bosonicfields and their transformations take a very simple formin this framework. This has enabled the fast develop-ment of quantum information and quantum metrologytechniques for Gaussian states and is a natural formal-ism for the application of quantum metrology to rel-ativistic quantum fields [7, 8]. We start by definingthe quadrature operators X2n−1 = 1√

2(an + a†n) and

X2n = 1√2 i

(an − a†n), which correspond to the gener-

alised position and momentum operators of the field, re-spectively. In the covariance matrix formalism Gaussianstates are completely defined by the field’s first and sec-ond moments. The first moment correspond to 〈Xi〉 andthe second moments are encoded in the covariance ma-trix Σij = 〈XiXj +XjXi〉 − 2〈Xi〉〈Xj〉. We restrict ouranalysis to initial Gaussian states with vanishing first mo-ments 〈Xi〉 = 0. In this case, the state of the field aftera Bogoliubov transformation is given by

σε = Sεσ0STε , (27)

where σ0 encodes the initial state of the field and theBogoliubov transformation in Eq. (25) is encoded in the

5

symplectic matrix

S =

M11 M12 M13 · · ·M21 M22 M23 · · ·M31 M32 M33 · · ·

......

.... . .

, (28)

where the 2× 2 matrices Mmn are given by

Mmn =

(<(αmn − βmn) =(αmn + βmn)−=(αmn − βmn) <(αmn + βmn)

). (29)

< and = denote the real and imaginary parts, respec-tively. The Bogoliubov coefficients and thus, the finalstate of the field, depend on the amplitude of thegravitational wave ε. Our main aim is now to determineunder what circumstances these changes are observableand moreover, use specialised techniques in relativisticquantum metrology to estimate the amplitude of thegravitational wave by making measurements on the BECphononic field.

Relativistic Quantum Metrology: estimating the ampli-tude of the wave. Quantum metrology provides tech-niques to estimate parameters associated to the trans-formation of a quantum state. The formalism providesstrategies, that include finding both, initial states andmeasurements basis, that enable one to estimate the pa-rameter with optimal precision. Recently a formalismto estimate parameters in relativistic quantum fields hasbeen developed in the covariance matrix formalism [7, 8].In this section we apply these techniques to provide abound on the optimal precision that can be achievedwhen estimating the wave amplitude ε through mea-surements on σε. The quantum Cramer-Rao theoremstates that the error in the estimating the parameter ε isbounded by [28],

〈(∆ε)2〉 ≥ 1

MHε. (30)

where Hε is the Quantum Fisher Information (QFI) andM the number of measurements. The QFI can be com-puted using the Uhlmann fidelity F between the state σεand a state with an infinitesimal increment in the param-eter, i.e. σε+dε,

Hε =8(1−

√F(σε, σε+dε)

)dε2

. (31)

The precision in the estimation of the parameter will beincreased when σε and σε+dε are more distinguishable.Now let σε be a two-mode Gaussian state with zero initialfirst moments. The Fidelity between them is given by [29]

F(σε, σε+dε),=1

√Λ +√

Γ−√

(√

Λ +√

Γ)2 −∆, (32)

where

Γ =1

16det(iΩσεiΩσε+dε + 11)

Λ =1

16det(iΩσε + 11)det(iΩσε+dε + 11)

∆ =1

16det(σε + σε+dε)

(33)

where 11 is the identity matrix and the symplectic formΩ is given by Ω =

⊕nk=1 Ωk, Ωk = −iσy and σy is one of

the Pauli matrices..In reference [7, 8] analytical formulas are provided for

the computation of the QFI in terms of Bogoliubov coef-ficients that admit a perturbative expansion in terms ofthe parameter to be estimated. We consider the state ofthe field to be a two-mode squeezed state for two modes nand m with squeezing parameter r -the rest of the modesare initially in the vacuum state. In the covariance ma-trix formalism, the reduced state of a particular set ofmodes is obtained by simply deleting the correspondingrows and columns of the covariance matrix. The QFI forthis particular initial state was computed in [8] in termsof general Bogoliubov coefficients,

H = ε−2<[4 cosh r(fnα + fnβ + fmα + fmβ )

+ 4 cosh2 r(2|βnm(t)|2 − fnα + fnβ − fmα + fmβ )

+ 4 sinh2 r(fnα − fnβ + fmα − fmβ − 2βnm(t)2 + 2αnm(t)2)

+ 4 sinh r<[Gαβnm + Gαβnm]− 4 cosh4 r|βnm(t)|2

− 1

2sinh2 2r(2|αnm(t)|2 − 3|βnm(t)|2 − βnm(t)2

].

(34)

where

f iα =1

2

∑j 6=n,m

|αji|2

f iβ =1

2

∑j 6=n,m

|βji|2

Gαβij =∑k 6=n,m

αkiβ∗kj

. (35)

Substituting Eq. (25), we obtain,

Hε =n

4mω2m t

2 (8− 4 cosh4(r) + 2 sinh2(2r)). (36)

Is the effect observable? The fact that gravitationalwaves can generate photons in fields confined in certainregions of spacetime was pointed out in [30, 31]. How-ever, the effect is negligible in optical cavities since grav-itational waves typically have frequencies several orders

6

of magnitude below the optical regime. The conditionfor a particle creation resonance given by Eq. (23) inthe case of an optical cavity, would require waves in thePHz regime. Fortunately, the situation is very differentfor the phononic excitations of a BEC. The slow propa-gation of the excitations acts as an amplification effectmaking particle creation observable at regimes that canbe in principle reached in the experiment. Consideringa cavity with length L = 1 µm and typical values forthe speed of sound in such systems cs = 10 mm/s re-sults in a fundamental frequency of ω1 = 2π × 5000Hz,which contrasts with the typical fundamental frequenciesin optical cavities, which are in the PHz regime. Exper-imental timescales between 0 and 2000 seconds, wouldallow to detect persistent sources such as the stochas-tic cosmic gravitational wave background [32], as well asshort and long-lived [33] gravitational waves transientssuch as gamma-ray bursts. Typical phononic states on aBEC last approximately a few seconds, however, for verylow densities the state’s lifetime can be between 100-1000seconds. We consider values of the initial squeezing pa-rameter between r = 2 and r = 10 that seem are in prin-ciple achievable in time-dependent potential traps [34].Each pair of mode numbers in turn provides a differentvalue of the resonant frequency of the gravitational wave.This fact can be exploited to detect waves in different fre-quencies. Using these parameters, we plot in Figs. (2a)and (2b) our bound to a typical figure of merit that isused in the literature of gravitational wave astronomy,i. e. strain sensitivity [3]. This figure of merit is givenby ∆ε/

√Ω. The range of frequencies considered are be-

tween 104 − 105 Hz, which partially overlaps with thehigh frequency band of LIGO and also enters into thehigh-frequency realm [35]. The number of measurementsis assumed to be M = 1014. This number of measure-ments is assumed in non-relativistic proposals to detectgravitational waves with BECs in space [13]. As can beseen in Figs. (2a) and (2b), we find that the strain sen-

sitivity can be as low as 10−27 Hz−1/2 for the optimalparameters. This is well below the threshold for the de-tection of gravitational waves. State-of-the-art in LIGOis 7× 10−23 Hz−1/2.

Our intention in this paper is to suggest that gravi-tational waves can be detected through the excitation ofthe phononic modes of a BEC and to show, using realisticparameters, that the effects are in principle observable.A detailed experimental proposal will be provided else-where.Conclusions. We have proposed a method forgravitational-wave astronomy based on a relativisticquantum field theoretical approach. We have shownthat spacetime distortions produce phonons in a Bose-Einstein condensate and we have suggested to use thiseffect to detect gravitational waves. The spacetime wavegives rise to particle creation resonances when the BEC isin a box-like potential, similar to the Dynamical Casimir

a)

b)

0 500 1000 1500 20000

2.¥ 10-26

4.¥ 10-26

6.¥ 10-26

8.¥ 10-26

1.¥ 10-25

tHsL

De WHHz

-1 2L

0 500 1000 1500 20000

1.¥ 10-20

2.¥ 10-20

3.¥ 10-20

4.¥ 10-20

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FIG. 2: Optimal bound of the strain sensitivity ∆εΩ

versus du-ration of the gravitational wave for L = 1 µm, cs = 10 mm/s,M = 1014. a) r = 10 and mode numbers k = 1, m = 2(blue, solid), m = 1, n = 6 (red,dashed), m = 10, n = 11(yellow,dotted). b)m = 10, n = 11 and squeezing parameterr = 2 (blue, solid), r = 3 (red,dashed) and r = 4 (yellow,dotted).

Effect. Particle creation through the motion of bound-ary conditions has been extensively analysed in the con-text of the Dynamical Casimir Effect [15, 16]. However,to the best of our knowledge, this is the first time it isshown that the real dynamics of spacetime can generatephonons in a BEC. Since the final state of the phononsdepends on the amplitude of the wave, the amplitudecan be estimated. We calculated a bound on the opti-mal precision that can be achieved with this method andstudied the regime of experimental parameters in whichthe sensitivity is low enough to detect the spacetime rip-ple. In the best scenario, the predicted strain sensitivityis several orders of magnitude beyond the performance ofhighly sophisticated programmes for gravitational wavedetection such as LIGO. The experiment clearly presentschallenges, however, introducing new methods for the de-tection of spacetime effects might not only lead to the de-tection of gravity waves but also deepen our understand-ing of the overlap of quantum theory and relativity. Thetechniques presented in this paper can be extended toexplore the effects of other spacetimes including expand-ing metrics. We hope our scheme for relativistic quantummetrology will not only play a role in the upcoming era ofgravitational wave astronomy but also show how the in-

7

terplay between quantum and relativistic effects can giverise to a new generation of quantum technologies.Acknowledgements We thank Tupac Bravo, Jorma Loukoand Chris Westbrook for useful discussions and com-ments. C. S, M. A. and I. F. acknowledge support fromEPSRC (CAF Grant No. EP/G00496X/2 to I. F.).

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