Quantum metrology for relativistic quantum fields

Mehdi Ahmadi,1,* David Edward Bruschi,2,† and Ivette Fuentes1,‡1School of Mathematical Sciences, University of Nottingham, University Park,

Nottingham NG7 2RD, United Kingdom2School of Electronic and Electrical Engineering, University of Leeds,

Woodhouse Lane, Leeds LS2 9JT, United Kingdom(Received 21 January 2014; published 20 March 2014)

In quantum metrology quantum properties such as squeezing and entanglement are exploited in thedesign of a new generation of clocks, sensors and other measurement devices that can outperform theirclassical counterparts. Applications of great technological relevance lie in the precise measurement ofparameters which play a central role in relativity, such as proper accelerations, relative distances, time andgravitational field strengths. In this paper we generalize recently introduced techniques to estimate physicalquantities within quantum field theory in flat and curved space-time. We consider a bosonic quantum fieldthat undergoes a generic transformation, which encodes the parameter to be estimated. We presentanalytical formulas for optimal precision bounds on the estimation of small parameters in terms ofBogoliubov coefficients for single-mode and two-mode Gaussian channels.

DOI: 10.1103/PhysRevD.89.065028 PACS numbers: 04.62.+v, 06.20.-f

I. INTRODUCTION

Quantum metrology provides techniques to enhancethe precision of measurements of physical quantities byexploiting quantum properties such as squeezing andentanglement. Such techniques are being employed todesign a new generation of quantum measurement tech-nologies such as quantum clocks and sensors. Impressively,the quantum era is now reaching relativistic regimes.Tabletop experiments demonstrate relativistic effects inquantum fields [1] and long-range quantum experiments[2] will soon reach regimes where relativity kicks in [3,4].It is therefore of great interest to develop quantummetrology techniques to measure physical quantities thatplay a role in relativity. These can lead to the measurementof gravitational waves and applications in relativisticgeodesy, positioning, sensing and navigation.Recently, techniques that apply quantum metrology to

quantum field theory in curved and flat space-time havebeen developed [5–7]. Quantum field theory allows one toincorporate relativistic effects at the regimes that are beingreached by cutting-edge quantum experiments. The appli-cation of quantummetrology to quantum field theory can inprinciple produce technologies that outperform (nonrela-tivistic) quantum estimation of Newtonian gravitationalparameters [7]. Indeed, it was shown that relativistic effectscan be exploited to improve measurement technologies [7].In this paper we generalize metrology techniques to

estimate physical quantities that play a role in relativistic

quantum field theory. Previous work aligned with this spiritshowed that entanglement could be used to determine theexpansion rate of the Universe [8] and that phase estimationtechniques could be employed to measure the Unruh effectat accelerations that were within experimental reach [5,9].The limitations imposed by the quantum uncertaintyprinciple in the measurement of space-time parameterswere investigated within locally covariant quantum fieldtheory [6] and nonrelativistic quantum mechanics [10].Reference [7] shows that quantum metrology techniquesemploying the covariance matrix formalism are ideallysuited to estimate parameters in quantum field theory. Thetechniques were applied to the estimation of parameters of afield contained in a moving cavity undergoing arbitrarymotion. This led to the design of an accelerometer that canimprove by 2 orders of magnitude the state-of-the-artoptimal measurement precision. It is important to pointout that the equations presented in [7] are only directlyapplicable to the example of the moving cavity.In this paper we generalize the techniques introduced in

[7] by considering a bosonic quantum field that undergoes ageneric transformation, which encodes the parameter to beestimated. Such transformation can involve, for example,space-time dynamics such as the expansion of the Universe,a general change of an observer’s reference frame, as wellas the example of a cavity undergoing nonuniform motion.In the case that the transformation admits a series expansionaround the parameter to be estimated, we present generalexpressions for optimal precision bounds in terms ofBogoliubov coefficients. Our analysis is restricted tosingle-mode and two-mode Gaussian states for whichelegant and simple formulas can be provided.The structure of this paper is as follows: In the next

section we review notions from quantum field theory

*mehdi.ahmadi@nottingham.ac.uk†Present address: Racah Institute of Physics and Quantum

Information Science Centre, the Hebrew University of Jerusalem,91904 Jerusalem, Israel.

‡Previously known as Fuentes-Guridi and Fuentes-Schuller.

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showing how the state of the field and its transformationscan be described in the covariance matrix formalism.In Sec. III, we will review quantum metrology techniqueswhere the quantum Fisher information provides optimalbounds on measurement precisions. In Sec. IV we general-ize techniques for relativistic quantum metrology byproviding analytical expressions for the quantum Fisherinformation in terms of the Bogoliubov coefficients ofgeneral transformations. In the last section, we comparesingle-mode and two-mode channels showing, in particu-lar, that two-mode channels improve measurement preci-sion in the example of a nonuniformly moving cavity.

II. QUANTUM FIELD THEORY IN CURVEDSPACE-TIME AND THE COVARIANCE

MATRIX FORMALISM

We start by describing the states of a bosonic quantum fieldthat undergoes a transformation which encodes the parameterwe want to estimate. We consider a real massless scalarquantum fieldΦ in a space-timewithmetric gμν. The signatureof the metric is ð−;þ;þ;þÞ. The field obeys the Klein-Gordon equation □Φ ¼ 0 where the d’Alambertian takesthe form □ ≔ ð ffiffiffiffiffiffi−gp Þ−1∂μ

ffiffiffiffiffiffi−gpgμν∂ν. Here g ¼ detðgμνÞ is

the determinant of the metric. The field can be expanded interms of a discrete set of modes [11],

Φ ¼Xn

½ϕnan þ H:c:�; (1)

where the creationandannihilationoperatorsa†n andan satisfythe canonical bosonic commutation relations ½am; a†n� ¼ δmn.The modes fϕnjn ¼ 1; 2; 3;…g are solutions to the Klein-Gordon equation and form a complete set of orthonormalmodes with respect to the inner product denoted by ð·; ·Þ [12].The vacuum state j0i of the field is defined as the state that isannihilated by the operators an for all n, i.e., anj0i ¼ 0.It is important to note that the mode decomposition is

not unique, the field can be decomposed in terms of anew set of modes which we denote ~ϕn. For example, acoordinate transformation between different observersresults in a Bogoliubov transformation between the modesϕn and mode solutions ~ϕn in the new coordinate system.Indeed a Bogoliubov transformation is the most general

linear transformation between two sets of field modes ϕnand ~ϕn. The transformation between the correspondingoperators am and ~an is given by

~am ¼Xn

ðα�mnan − β�mna†nÞ; (2)

where αmn ≔ ð ~ϕm;ϕnÞ and βmn ≔ ð ~ϕn;ϕ�mÞ are the

Bogoliubov coefficients. The operators ~an define thevacuum state j~0i in the new basis through the condition~anj~0i ¼ 0. Paradigmatic examples are the Bogoliubovtransformations between inertial and uniformly accelerated

observers in flat space-time, between Kruskal andSchwarzschild observers in a black-hole space-time andbetween observers at past and future infinity in an expand-ing universe [12].In quantum field theory different observers do not

always agree on the particle content of a state [12]. Notethat j~0i is annihilated by the initial field operators an onlyif all coefficients βmn are zero. Therefore, βmn ≠ 0 is anindication of the particle creation. This occurs, for instance,in the Unruh effect where the inertial vacuum state is seenas a thermal state by uniformly accelerated observers [12].An example of particular interest in this work is that of acavity undergoing noninertial motion [13–15]. The vacuumstate of an inertial cavity becomes populated after the cavityundergoes nonuniformly accelerated motion [1]. As wementioned before, the Bogoliubov coefficients dependon the parameter we want to estimate. Examples of suchparameters are proper time, the expansion rate of theUniverse, the mass of a black hole, and the accelerationof a cavity, among others.Let us nowdescribe the state of the field and its Bogoliubov

transformations in the covariance matrix formalism. Thisformalism has been used to investigate entanglement inquantum field theory [15–17] since it is applicable to systemsconsisting of a (discrete) infinite number of bosonic modes.The covariance matrix formalism enables elegant and sim-plified calculations to quantify bipartite and multipartiteentanglement forGaussianstates [14,17]. Inorder to introducethis formalism we define the quadrature operators X2n−1 ¼1ffiffi2

p ðan þ a†nÞ and X2n ¼ 1ffiffi2

piðan − a†nÞ, which correspond to

thegeneralizedposition andmomentumoperators of the field.Wewill restrict ouranalysis toGaussian states since they takeasimple form in this formalism. These states are completelydescribed by the expectation values of the quadratures hXii(also known as the first moments of the field) and thecovariance matrix Σij ¼ hXiXj þ XjXii − 2hXiihXji. Theunitary transformations in theHilbert space that are generatedby a quadraticHamiltonian can be represented as a symplecticmatrix S in phase space. These transformations form the realsymplectic group Spð2n;RÞ, the group of real (2n × 2n)matrices that leave the symplectic form Ω invariant, i.e.,SΩST ¼ Ω, where Ω ¼ ⨁n

k¼1Ωk and Ωk ¼ −iσy and σy isone of the Pauli matrices. The time evolution of the field, aswell as the Bogoliubov transformations, can be encoded inthis structure. The symplectic matrix corresponding to theBogoliubov transformation in Eq. (2) can be written in termsof the Bogoliubov coefficients as

S ¼

0BBBBBB@

M11 M12 M13 � � �M21 M22 M23 � � �M31 M32 M33 � � �... ..

. ... . .

.

1CCCCCCA; (3)

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where theMmn are the 2 × 2 matrices

Mmn ¼�

ℜðαmn − βmnÞ ℑðαmn þ βmnÞ−ℑðαmn − βmnÞ ℜðαmn þ βmnÞ

�: (4)

Hereℜandℑdenote therealandimaginaryparts, respectively.The first moments and covariance matrix after a Bogoliubovtransformation are given by h ~Xi ¼ ShX0i and ~Σ ¼ SΣ0ST

respectively, where hX0i andΣ0 encode the initial state of thefield. In this formalism it is relatively easy to applymetrologytechniques to estimate physical quantities which are encodedin the Bogoliubov coefficients.

III. QUANTUM METROLOGY

In this section we briefly review metrology techniquesfor Gaussian states. Quantum metrology deals with theestimation of quantities that do not correspond to anobservable of the system [18]. Examples are temperature,time, acceleration, and coupling strengths. The techniquesassume that the state of a quantum system undergoes atransformation which encodes the parameter to be esti-mated. The main aim is to find the optimal estimationstrategy, i.e. finding an optimal initial state and the set ofmeasurements on the final state that will allow us toestimate the parameter with the highest possible precision.We are interested in estimating the precision in the casewhere one measures only one or two modes in a Gaussianstate. Note that symplectic transformations take Gaussianstates into Gaussian states.A parameter θ can be estimated with high accuracy when

the states ρθ and ρθþdθ, which differ from each other by aninfinitesimal changedθ, canbedistinguished.Theoperationalmeasure that quantifies the distinguishability of these twostates is the Fisher information [19]. Let us suppose that anagent performs N independent measurements to obtain anunbiased estimator θ for the parameter θ. The Fisher informa-tionFðθÞ tellsushowmuchinformationcanbeextractedaboutthe unknown parameter θ. In other words it gives us a lowerbound to the mean-square error in the estimation of θ via theclassical Cramér-Rao inequality [20], i.e., hðΔθÞ2i ≥ 1

NFðθÞ,where FðθÞ ¼ R dλpðλjθÞðd ln½pðλjθÞ�=dλÞ2 and pðλjθÞ isthe likelihood function with respect to a chosen positiveoperator valued measurement fOλg with

PλOλ ¼ 1.

Optimizing over all the possible quantum measurementsprovides an even stronger lower bound [21], i.e.,

NhðΔθÞ2i ≥ 1

FðθÞ ≥1

HðθÞ ; (5)

where HðθÞ is the quantum Fisher information (QFI).This quantity is obtained by determining the eigenstates ofthe symmetric logarithmic derivative Λρθ defined by 2 dρθ

dθ ¼Λρθρθ þ ρθΛρθ . Alternatively, the QFI can be related to theUhlmann fidelity F of the two states ρθ and ρθþdθ through

HðθÞ ¼8�1 −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF ðρθ; ρθþdθÞ

p �dθ2

; (6)

whereF ðρ1; ρ2Þ ¼ ðtr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ1

pρ2

ffiffiffiffiffiρ1

pp Þ2.Theoptimalmeasure-ments forwhich thequantumCramér-Raobound (5) becomesasymptotically tight can be computed from Λρθ [22].Unfortunately, these optimal measurements are usually noteasilyimplementableinthelaboratory.Nevertheless, intypicalproblems involving optimal implementations one can devisesuboptimal strategies involving feasible measurements suchas homodyne or heterodyne detection, see, e.g., [18]. In thispaperweareinterestedincomparingthequantumCramér-Raobound for different initial states.Expressions for the fidelity for one- and two-mode states

have been previously obtained. The fidelity between twogenerally mixed single-mode Gaussian states σ and σ0 isgiven by [23]

Fðσ; σ0Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiΛþ Δ

p−

ffiffiffiffiΛ

p exp½−hΔXiTA−1hΔXi�; (7)

where

A ¼ σ þ σ0;

hΔXi ¼ hXiσ0 − hXiσ;

Δ ¼ 1

4detðAÞ;

Λ ¼ 1

4detðσ þ iΩÞ detðσ0 þ iΩÞ:

Note that we follow the conventions used in [14,15] for thenormalization of the covariance matrix, which differ fromother conventions [23].Now let σ and σ0 be two-mode Gaussian states with zero

initial first moments. The fidelity between them is givenby [23]

F ðσ; σ0Þ;¼ exp½−hΔXiTA−1hΔXi�ffiffiffiffiΛ

p þ ffiffiffiΓ

p−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið ffiffiffiffi

Λp þ ffiffiffi

Γp Þ2 − Δ

q ; (8)

where

Γ ¼ 1

16detðiΩσiΩσ0 þ 1Þ;

Λ ¼ 1

16detðiΩσ þ 1Þ detðiΩσ0 þ 1Þ;

Δ ¼ 1

16detðAÞ;

A ¼ σ þ σ0; (9)

and 1 is the identity matrix.

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IV. RELATIVISTIC QUANTUM METROLOGY

We now develop the main formalism contained in thiswork. We consider a bosonic quantum field which under-goes a general Bogoliubov transformation that depends ona dimensionless parameter θ, where θ is the parameter wewant to estimate. We assume that the Bogoliubov coef-ficients which relate the initial and final states of the fieldhave a series expansion in terms of θ,

αmn ¼ αð0Þmn þ αð1Þmnθ þ αð2Þmnθ2 þOðθ3Þ; (10a)

βmn ¼ βð1Þmnθ þ βð2Þmnθ2 þOðθ3Þ: (10b)

This implies that our formalism applies only to smallparameters θ. Such assumption enables us to provideanalytical expressions for the quantum Fisher informationin terms of general Bogoliubov coefficients. A commentabout the small parameter θ is in order. Note that θ willgenerally be a function of a dimensionful parameter weultimately want to estimate which needs not to be small inabsolute terms. As an example consider θ ¼ v=c, where vis the velocity of the system and c is the speed of light.In this case, our perturbative treatment allows for theestimation of velocities as large as v ∼ 0.1c.To find the leading order term in θ of the quantum Fisher

information we need to compute the second order term inthe expansion of the fidelity, i.e. the coefficient of dθ2.Equations (10) lead to an expansion for the symplectictransformation

SðθÞ ¼ Sð0Þ þ Sð1Þθ þ Sð2Þθ2 þOðθ3Þ: (11)

Therefore, the first moments and the covariance matrix canbe written as

hXiθ ¼ SðθÞhX0i¼ hXið0Þ þ hXið1Þθ þ hXið2Þθ2 þOðθ3Þ (12)

and

σðθÞ ¼ σð0Þ þ σð1Þθ þ σð2Þθ2 þOðθ3Þ: (13)

In general, it is of interest to analyze the evolution of a finiteset of modes of the field. UsingWillamson’s decomposition[24], we can write the marginal covariance matrix of the Nmodes as σðθÞ ¼ sðθÞσ⊕ðθÞs†ðθÞ, where s is a 2N × 2Nsymplectic transformation with the expansion

sðθÞ ¼ sð0Þ þ sð1Þθ þ sð2Þθ2 þOðθ3Þ; (14)

and σ⊕ðθÞ ¼ diagðν1ðθÞ; ν2ðθÞ; ν1ðθÞ; ν2ðθÞÞ is known asthe symplectic form of sðθÞ. The functions νiðθÞ are calledthe “symplectic eigenvalues” of sðθÞ. They are simply the

eigenvalues of the matrix jiΩσðθÞj. In general νi ≥ 1, wherethe equality holds for pure states.In this paper we consider states that are initially pure and

that remain pure to lowest order in the parameter θ i.e.,det½σðθÞ� ¼ 1þOðθÞ. This is means that the Bogoliubovtransformations do not entangle modes to zero order. Thisrequirement, together with the unitarity condition of theBogoliubov transformations, implies that the α coefficientsto zero order are αð0Þmn ¼ Gnδmn, where Gn ¼ expðiϕnÞ.This also implies that the expansion of the symplecticeigenvalues νiðθÞ in terms of θ is

νiðθÞ ¼ 1þ νð1Þi θ þ νð2Þi θ2 þOðθ3Þ: (15)

Imposing the necessary constraints F ðσðθÞ, σðθÞÞ ¼ 1and [25]

∂F ðσðθÞ; σðθ þ dθÞÞ∂dθ

����dθ¼0

¼ 0 (16)

allows us to write the expansion of fidelity as

F ðσðθÞ; σðθ þ dθÞÞ ¼ 1 − F ð2Þ dθ2

2þOðθdθ2 þ θ2dθÞ:

(17)

We find that the second order term in the fidelity has twocontributions F ð2Þ ¼ Eð2Þ þ Cð2Þ. The first contributioncomes from the expansion of the exponential term in thefidelity formulas (7) and (8) which is given by

e−hΔXiTA−1hΔXi ¼ 1 − Eð2Þdθ2

¼ 1 − hXið1ÞTA−1ð0ÞhXið1Þdθ2; (18)

where A−1ð0Þ ¼ Að0Þ−1. The second contribution Cð2Þ comesfrom the denominator in Eqs. (7) and (8). Having found thesecond order contributions to the fidelity we can write theQFI as

H ¼ 4ðEð2Þ þ Cð2ÞÞ: (19)

In the following sections we compute Eð2Þ and Cð2Þ forsingle-mode and two-mode detection schemes.

A. Single-mode detection

We start by considering the scenario in which a singlemode k of the quantum field is detected to estimate theparameter θ. Therefore, we assume that all of the modes ofthe quantum field are initially in the vacuum state except formode k which is in a squeezed displaced vacuum, i.e.jΨ0i ¼ SkðrÞDkðδÞj0i, whereDkðδÞ ¼ exp½δða†k − akÞ� andSkðrÞ ¼ exp½r

2ða†2k − a2kÞ� are the displacement and squeez-

ing operators, respectively. The parameters δ and r determinethe amount of displacement and squeezing in the state.

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The corresponding reduced 2 × 2 covariance matrix of modek takes the form σ0 ¼ diagðer; e−rÞ. The reduced covariancematrix of mode k after the Bogoliubov transformation can bewritten as

σkðθÞ ¼ MkkðθÞσ0MTkkðθÞ þ

Xn≠k

MknðθÞMTknðθÞ; (20)

where the matrices MijðθÞ can be found using Eqs. (4)and (10).Let us first compute the contribution Eð2Þ to the fidelity.

In order to do so we first need to compute the first momentsof the quantum field after the Bogoliubov transformation.The initial first moments of the mode k of the quantum fieldcan be written as hXki ¼ ð ffiffiffi

2p

δ; 0ÞT . Now using Eq. (12) wecan write the first contribution to the final first moments as

hXið1Þ ¼ ffiffiffi

2p

δℜ½αð1Þkk − βð1Þkk �− ffiffiffi

2p

δℑ½αð1Þkk − βð1Þkk �

!: (21)

Using (18) we find that the second order contribution Eð2Þto the fidelity is given by

Eð2Þ1 ¼ 2δ2ðjαð1Þkk − βð1Þkk j2 cosh r

þℜ½ðαð1Þkk − βð1Þkk Þ2G�k2� sinh rÞ: (22)

Note that this term depends both on the amount ofsqueezing r and the amount of displacement δ.In order to obtain the contribution Cð2Þ to the fidelity,

we need to find the perturbative expansions of Λ and Δ.Using Eq. (7) we find

Δðθ;θþdθÞ¼ 1þΔð2Þdθ2

2þΔð2ÞðθþdθÞdθþOðθ2dθ2Þ;

Λðθ;θþdθÞ¼ 1

4ðΔð2ÞθðθþdθÞÞ2þOðθ3ðθþdθÞ3Þ;

(23)

where

Δð2Þ ¼ 2�σð0Þ11 σ

ð2Þ22 þ σð2Þ11 σ

ð0Þ22 − 2σð0Þ12 σ

ð2Þ12

�þ 1

2

�σð1Þ11 σ

ð1Þ22 − 2ðσð1Þ12 Þ2

�; (24)

and σðnÞij ’s are the elements of nth order matrices in (13).After some algebra we find the second contribution Cð2Þ tothe fidelity formula (7) as

Cð2Þ1 ¼ Δð2Þ

4

¼hðfkα þ fkβÞ coshðrÞ −

Xn≠k

ℜ½αð1Þnk βð1Þ�nk � sinhðrÞ

−1

4ðℜ½αð1Þkk β

ð1Þ�kk � sinhð2rÞ þ jαð1Þkk j2cosh2ðrÞ

þ 1

2

�jβð1Þkk j2sin2ðϕkÞ þ

1

2ℑ½βð1Þkk �ℜ½βð1Þkk � sinð2ϕkÞ

−ℜ½ðαð1Þkk Þ2� cosð2ϕkÞÞisinh2r

�: (25)

B. Two-mode detection

In this section we analyze the precision in the measure-ment of θ when two modes of the quantum field aredetected. Our aim is again to find the second ordercontribution to the fidelity which will allow us to findthe leading order term in the QFI and hence the lowerbound on the error in estimation of the parameter θ.Let us start analyzing the expansion of Λ, Γ and Δ in the

fidelity formula (8). We first notice that

Λðθ; θ þ dθÞ ¼Yi

ð1 − ν2i ðθÞÞð1 − ν2i ðθ þ dθÞÞ

¼ ðνð1Þ1 νð1Þ2 θðθ þ dθÞÞ2 þOðθ3ðθ þ dθÞ3ÞÞ;(26)

which implies that Λ does not contribute to the term Cð2Þ inthe fidelity expansion. Using the multiplicativity propertyof the determinants, we can write

Γ ¼ detð1þ σ⊕ðθÞPðθ; θ þ dθÞÞ;Δ ¼ detðσ⊕ðθÞ þ Pðθ; θ þ dθÞÞ; (27)

where Pðθ; θ þ dθÞ ≔ s−1ðθÞσðθ þ dθÞðs−1ðθÞÞ†.For a given a matrix M with perturbative expansion

M ¼ 1þMð1;0Þθ þMð0;1Þdθ þMð1;1Þθdθ

þMð2;0Þθ2 þMð0;2Þdθ2;

and eigenvalues

λi ¼ 1þ λð1;0Þi θ þ λð0;1Þi dθ þ λð1;1Þi θdθ þ λð2;0Þi θ2

þ λð0;2Þi dθ2;

we prove that

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detð1þMÞ ¼ 16þ 8Tr½Mð1;0Þ�θ þ 8Tr½Mð0;1Þ�dθ

þ�4Xi<j

λð1;0Þi λð0;1Þi þ 8Tr½Mð1;1Þ��θdθ

þ�4Xi<j

λð1;0Þi λð1;0Þi þ 8Tr½Mð2;0Þ��θ2

þ�4Xi<j

λð0;1Þi λð0;1Þi þ 8Tr½Mð0;2Þ��dθ2:

(28)

Using (27) and (28) we find that, up to third ordercorrections, Γ ¼ Δþ Δc, where Δc is of the orderθðθ þ dθÞ. Considering the perturbative expansions of Λ,Γ,Δ and the relation between Γ andΔ, we find that the termffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið ffiffiffiffi

Λp þ ffiffiffi

Γp Þ2 − Δ

qin (8) does not have any term propor-

tional to dθ2. Therefore, we conclude that the only termwhich contributes to the order dθ2 in fidelity is Γ. We canthen obtain the second contribution to the fidelity for ageneral two-mode Gaussian state as

Cð2Þ2 ¼ 1

16½ðTr½ðσð0ÞÞ−1σð1Þ�Þ2 − Tr½ððσð0ÞÞ−1σð1ÞÞ2�

þ 4Tr½ðσð0ÞÞ−1σð2Þ��: (29)

Note that in order to find the second contribution Cð2Þ forany initial state of the quantum field we need to computethe covariance matrix elements up to second order.

Consider that the initial state of the quantum field is ageneric Gaussian state for modes k and k0, and that all othermodes are in their vacuum state. The reduced covariancematrix for modes k and k0 is given by

σkk0 ¼�

ψk ϕkk0

ϕTkk0 ψ 0

k

�; (30)

where ψk and ψk0 are the reduced covariance matrices of themodes k and k0 respectively. The 2 × 2 matrix ϕkk0 containsthe correlations between the two modes and it vanishes forproduct states. The transformed covariance matrix is thengiven by

~σkk0 ¼�Ckk Ckk0

Ck0k Ck0k0

�; (31)

where

Cij ¼ MTkiψkMkj þMT

k0iϕTkk0Mkj þMT

kiϕkk0Mk0j

þMTk0iψk0Mk0j þ

Xn≠k;k0

MTniMnj: (32)

We now assume that both modes k and k0 are in a displacedsqueezed state, which is a product state. The 2 × 2 blocksin (30) are ψk ¼ ψk0 ¼ diagðer; e−rÞ while ϕkk0 vanishes.Using Eqs. (29) and (32) we find the contribution Cð2Þ to thefidelity as

Cð2Þ2 ¼ 1

4

Xi;j

ℜh4 coshðrÞðfiα þ fiβÞ þ 4 sinh rG�2

j Gαβjj þ 2cosh2rðjβð1Þij j2 − fiα þ fiβÞ − 2cosh4rjβð1Þij j2

− 2sinh2rðG�2j αð1Þij

2 þ G2jβ

ð1Þij

2 þ 2Giαð2Þii

�Þ − 4 sinh 2rðαð1Þij βð1Þij Þ þ 2 sinhð2rÞcosh2ðrÞαð1Þij β

ð1Þij

þ sinh4rðjαð1Þij j2 − jβð1Þij j2 −G�2j αð1Þij

2 −G2jβ

ð1Þij

2Þ − 1

4sinh22rðjαð1Þij j2 − 3jβð1Þij j2 −G�2

j αð1Þij2 −G2

jβð1Þij

2Þi; (33)

where we have defined

fiα ≔1

2

Xn≠k;k0

jαð1Þni j2;

fiβ ≔1

2

Xn≠k;k0

jβð1Þni j2;

Gαβij ≔

Xn≠k;k0

αð1Þni βð1Þnj

�:

The first moments of the initial state in this case can bewritten as hX0i ¼ ð ffiffiffi

2p

δ; 0;ffiffiffi2

pδ; 0ÞT . Again after comput-

ing the first order contribution to the first moments of thequantum field after the Bogoliubov transformation we findthe second order contribution Eð2Þ as

Eð2Þ2 ¼ 2δ2½cosh rðjAkk0 j2 þ jAk0kj2Þ

þ sinh rðcosð2ϕkÞjAkk0 j2 þ sinð2ϕkÞℑ½A2kk0 �

þ cosð2ϕk0 ÞjAk0kj2 þ sinð2ϕk0 Þℑ½A2k0k��; (34)

where Aij ¼ αð1Þii þ αð1Þij − βð1Þii − βð1Þij . Note that this termdepends on both the amount of initial displacement δ andthe amount of initial squeezing r in each mode. Also it isworth pointing out that having nonzero initial displacementis necessary for the term Eð2Þ to contribute to the QFI.

V. EXAMPLE: CAVITY IN NONUNIFORMMOTION

In this section we show how to apply our techniques toestimate acceleration using a massless scalar field Φ

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confined within rigid moving boundary conditions. In thiscase, a massless scalar field is a good approximation to oneof the polarization modes of the electromagnetic field [26].We consider the scenario where a cavity of length L isinitially inertial in a (1þ 1)-dimensional flat space-timewith coordinates ðt; xÞ and then moves noninertially for afinite period of time. Initially, we will consider generaltrajectories and towards the end of the example we willpresent results for a specific trajectory. Independent of thetrajectory details, we consider that at all times the fieldvanishes at the cavity walls, i.e. Φnðt; xLÞ ¼ Φnðt; xRÞ ¼ 0.The field Φ satisfies the Klein-Gordon equation □Φ ¼ 0,where the d’Alambertian takes the simple form □ ¼ ∂2

t −c2∂2

x in Minkowski coordinates. The solutions to theKlein-Gordon equation are plane waves. After imposingthe corresponding boundary conditions we find the modesolutions ϕn. We assume that, after the cavity has under-gone noninertial motion, the cavity will once more moveinertially and it is then possible to find a new set of fieldmodes ~ϕn. The two sets of modes ϕn and ~ϕn are related by aBogoliubov transformation, details of this transformationcan be found in [13]. The transformation can be expandedin the form (10) where the small parameter in this case ish ¼ aL

c2 . Here a is the proper acceleration at the center ofthe cavity and c is the speed of light. In this particular setup,the diagonal first order Bogoliubov coefficients are zero,

i.e. αð1Þii ¼ βð1Þii ¼ 0. Our aim is to employ the formulaspresented in the sections above to estimate the properacceleration a. To do so we consider different initial statesof the cavity and we compare the quantum Fisher infor-mation between these different cases. In [7] this examplewas studied only in the case that the initial state of the fieldwas a state with two modes, each one of them squeezed.

Now we will be able to analyze what initial states provide abetter estimation of the acceleration.

A. Single-mode detection

Let us start with the case where all the modes of thequantum field inside the cavity are in their vacuum stateexcept mode k which is in a squeezed displaced vacuumstate. Since in the case of a moving cavity the diagonal firstorder contribution to the Bogoliubov coefficients vanish,the term Eð2Þ

1 is zero, which implies that there is noadvantage in measuring h by initially displacing the state.Using Eqs. (19) and (25) we find that the leading order termof the QFI is

H1 ¼ 4½ðfkα þ fkβÞ coshðrÞ −ℜ½αð1Þnk βð1Þnk

�� sinhðrÞ�; (35)

which only depends on the first order Bogoliubov coeffi-cients and the squeezing parameter. The Bogoliubov coef-ficients here correspond to any trajectory followed by thecavity. We note that when r ¼ 0 then H1 is very small. Thiswill give rise to a very large error which can be decreased byincreasing the squeezing in the state. As already noted, initialdisplacement will not affect the precision.

B. Two modes in a product form

Let us consider a different initial state for the field insidethe cavity. We assume that two modes k and k0 are initiallyprepared in a separable state where each mode is in adisplaced squeezed vacuum. To simplify calculations weassume the same amount of squeezing r and displacement δin each mode. Using Eqs. (33) and (37) and considering

that αð1Þnn ¼ βð1Þnn ¼ 0 we find

Cð2Þ2 ¼ 1

4ℜh4 cosh rðfkα þ fkβ þ fk

0α þ fk

0β Þ − 4cosh4rjβð1Þkk0 j2 þ 4cosh2rð2jβð1Þkk0 j2 þ fkβ − fkα þ fk

0β − fk

0α Þ

− 4sinh2rðG�2k0 α

ð1Þkk0

2 þ G2k0β

ð1Þkk0

2 þ G�kα

ð2Þkk þG�

k0αð2Þk0k0 Þ − 4 sinh 2rðαð1Þkk0β

ð1Þkk0 þ αð1Þk0kβ

ð1Þk0kÞ

þ 4 sinh rðG�2k Gαβ

kk þG�2k0 G

αβk0k0 Þ þ 4 sinh 2rcosh2rðαð1Þkk0β

ð1Þkk0 þ αð1Þk0kβ

ð1Þk0kÞ þ 2sinh4rðjαð1Þkk0 j2 − jβð1Þkk0 j2

−G�k02αð1Þkk0

2 − G2k0β

ð1Þkk0

2Þ − 1

2sinh22rðjαð1Þkk0 j2 − 3jβð1Þkk0 j2 −G�

k02αð1Þkk0

2 −G2k0β

ð1Þkk0

2Þi

(36)

and

Eð2Þ2 ¼ 2δ2½cosh rðjαð1Þkk0 − βð1Þkk0 j2 þ jαð1Þk0k − βð1Þk0kj2Þ þ sinh rðcosð2ϕkÞjαð1Þkk0 − βð1Þkk0 j2 þ sinð2ϕkÞℑ½ðαð1Þkk0 − βð1Þkk0 Þ2�

þ cosð2ϕk0 Þjαð1Þk0k − βð1Þk0kj2 þ sinð2ϕk0 Þℑ½ðαð1Þk0k − βð1Þk0kÞ2��: (37)

The QFI in this case is given by H2 ¼ 4ðCð2Þ2 þ Eð2Þ2 Þ.

C. Two-mode squeezed state

Finally we consider an initial state for two modesk; k0 containing entanglement. The state is known asthe two-mode squeezed state and is given by

ϕkk ¼ ϕk0k0 ¼ coshðrÞ1 and ϕkk0 ¼ sinhðrÞσz where σz isthe Pauli zmatrix. A two-mode squeezed state has zero firstmoments, therefore, the contribution to the fidelity comingfrom the exponential term Eð2Þ

3 vanishes. Using once moreEq. (33) we find that H3 ¼ 4Cð2Þ3 which in terms ofBogoliubov coefficients yields

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H3 ¼ ℜh4 cosh rðfkα þ fkβ þ fk

0α þ fk

0β Þ − 4cosh4rjβð1Þkk0 j2 þ 4cosh2rð2jβð1Þkk0 j2 þ fkβ − fkα þ fk

0β − fk

0α Þ

− 4sinh2rð−cos2ðϕk þ ϕk0Þαð1Þkk0

2 þG2k0β

ð1Þkk0

2 þG�kα

ð2Þkk þ G�

k0αð2Þk0k0 − ℑ½GkGk0α

ð1Þkk0 �ℜ½αð1Þkk0 � sinðϕk þ ϕk0 ÞÞ

þ 4 sinh rðℜ½Gαβkk0 þ Gαβ

k0k� cos½ϕk þ ϕk0 � − ℑ½Gαβk0k þ Gαα

kk0 � sinðϕk þ ϕk0 ÞÞ

−1

2sinh22rð2jαð1Þkk0 j2 − 3jβð1Þkk0 j2 −Gk0

2βð1Þkk02i:

Note that in the limit of zero squeezing for the one-modedetection scheme the QFI reduces to

H1jr¼0 ¼ 8fkβ; (38)

whereas for the two-mode detection scheme we find theQFI as

H2jr¼0 ¼ H3jr¼0 ¼ 4Cð2Þvac ¼ 8fkβ þ 8fk0β þ 4jβð1Þkk0 j2:

We note that the term jβð1Þkk0 j2 is directly related to theentanglement generated between the modes k and k0 due tothe nonuniform motion of the cavity. It was shown that thenegativity N (which quantifies entanglement) generatedbetween the modes after the Bogoliubov transformation

has the simple expression N ¼ jβð1Þkk0 j [27]. From Eqs. (38)and (39) one can see that H2jr¼0 ¼ H3jr¼0 ∼ 2H1jr¼0þ4N 2 ≥ H1jr¼0. Therefore, employing two modes providesa higher precision compared to the single-mode case due tothe entanglement generated by the transformation.So far we have presented our results considering that the

cavity follows an arbitrary trajectory. In order to make acomparison on the estimation of acceleration using thestates analyzed above, we specify our example further. Wewill consider that the cavity is initially inertial and thenundergoes a single period of uniform acceleration a andduration τ after which the cavity returns to inertial motion.In this case the Bogoliubov coefficients α; β are given interms of simple expressions of the inertial-to-Rindlercoefficients ∘α; ∘β. These are the coefficients that relatethe solutions of the Klein-Gordon equation in the inertialframe to the solutions in the uniformly accelerated frame.For the particular travel scenario that we consider here, thefinal Bogoliubov coefficients are given by

αð1Þij ¼ ∘αð1Þij ðGi−GjÞ;

βð1Þij ¼ ∘βð1Þij ðGi−G�

jÞ;αð2Þij ¼Gi∘α

ð2Þij −Gj∘α

ð2Þji þ

Xk

½Gk∘αð1Þki ∘α

ð1Þkj −G�

k∘βð1Þki ∘β

ð1Þkj �;

αð2Þij ¼Gi∘βð2Þij −G�

j ∘βð2Þji þ

Xk

½Gk∘αð1Þki ∘β

ð1Þkj −G�

k∘βð1Þki ∘α

ð1Þkj �;(39)

where ∘αðnÞij and ∘β

ðnÞij are the nth order contributions of the

Bogoliubov coefficients ∘α; ∘β which are given in full detailin [13]. The coefficients Gn ¼ exp½i ~ωnτ� are phase factors

that each mode n accumulates due to free evolution duringthe acceleration period. The angular frequencies with respectto the proper time at the center of the cavity are given by~ωn ¼ nπh

2Larctanhðh=2Þ. We choose to plot our results in terms of

the dimensionless parameter u ¼ ~ωnτ2πn ¼ hτ

4Larctanhðh=2Þ.In order to make a fair comparison between the schemes

considering different initial states, we assume that theenergy of the initial state is the same in each case, i.e.E0 ¼ ℏNkðωk þ ωk0 Þ, where ωn ¼ nπc

L is the frequency ofmode n and Nn ¼ sinh2rþ δ2 is the average photonnumber of mode n before the Bogoliubov transformation.We define the parameter x to denote the fraction of initialenergy associated to squeezing i.e., sinh2r ¼ xNn andtherefore the fraction of energy associated to displacementis given by δ2 ¼ ð1 − xÞNn. In Fig. 1 we plot the QFI H2

as a function of u for x ¼ 0, 0.5 and 1.In order to understand in more detail the role of squeezing

in the analysis, we plot our results in the case that the initialdisplacements are zero. In Fig. 2 we compare the quantumFisher information as a function of u for the three differentinitial states: Single-mode squeezed state (solid black), aproduct state of two single-mode squeezed states (dotted red)and an entangled two-mode squeezed state (dashed blue).We have used different squeezing parameters for differentinitial states in such a way that the total energy of the statesis the same. The two single-mode squeezed state providesus with a better precision in the estimation of proper

x 1

x 0.5

x 0

0.2 0.4 0.6 0.8 1.0u

2

4

6

8

10

12

H

FIG. 1 (color online). Quantum Fisher information vs u ¼hτ

4Larctanhðh=2Þ for modes k ¼ 1 and k0 ¼ 2 when the initial state is

only squeezed x ¼ 0, only displaced x ¼ 1 and with an equalamountofenergyassociated tosqueezinganddisplacementx ¼ 0.5.

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acceleration for every value of u. This improvement is due tothe entanglement generated between the modes k and k0.Furthermore, the entanglement generated due to the motionof the cavity reaches a local minimum as shown in [15]. Thishas a direct consequence on the QFI, which also has aminimum for u ¼ 0.5, as can be seen in Fig. 2. It is worthpointing out that the comparison between the performanceof a single-mode squeezed state and the performance of atwo-mode squeezed state, i.e.H1 andH3, needs to be done ata particular proper time τ. In other words, for some observersthe single-mode squeezed state is a better choice of initialstate while for the others it is more convenient to prepare thetwo modes k and k0 in a two-mode squeezed state.

VI. CONCLUSIONS

In this paper we have provided techniques for theoptimal estimation of parameters which appear in quantum

field theory in curved space-time. This enables the esti-mation of parameters such as proper accelerations, propertimes, relative distances, gravitational field strengths, aswell as space-time parameters of interest such as theexpansion rate of the Universe or the mass of a blackhole. We have combined techniques from quantum metrol-ogy, continuous variable quantum information, symplecticgeometry and quantum field theory in order to develop aframework which is applicable in a generic scenario inwhich the parameter to be estimated is encoded in aBogoliubov transformation. By restricting the analysis tosmall parameters and Gaussian states, we have been able toprovide analytical formulas for the QFI in terms ofBogoliubov coefficients.We have applied our results to a scenario of current

physical interest, namely the estimation of the acceleration ofa cavity which undergoes nonuniform motion. This examplewas studied before in [7] considering an initial stateconsisting of two single-mode squeezed states in a productform. Here we have extended the analysis to one-mode andtwo-mode Gaussian states. We have shown that the gen-eration of entanglement in two-mode detection schemesincreases the precision, therefore, improving single-modeschemes.Our techniques are applicable to analogue gravity

systems where the effects of space-time on quantumfields can be investigated in realizable experimentalsetups [28].

ACKNOWLEDGMENTS

We thank Carlos Sabín, Antony Lee, Jandu Dradoumaand Nicolai Friis for useful discussions and comments.M. A. and I. F. acknowledge support from EPSRC (CAFGrant No. EP/G00496X/2 to I. F.). D. E. B. was supportedby the UK Engineering and Physical Science ResearchCouncil Grant No. EP/J005762/1. D. E. B. also acknowl-edges hospitality from the University of Nottingham.

[1] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen,J. R. Johansson, T. Duty, F. Nori, and P. Delsing, Nature(London) 479, 376 (2011).

[2] X.-S. Ma, T. Herbst, T. Scheidl, D. Wang, S. Kropatschek,W. Naylor, A. Mech, B. Wittmann, J. Kofler, E. Anisimova,V. Makarov, T. Jennewein, R. Ursin, and A. Zeilinger,Nature (London) 489, 269 (2012).

[3] S. Schiller, A. Görlitz, A. Nevsky, S. Alighanbari,S. Vasilyev, C. Abou-Jaoudeh, G. Mura, T. Franzen,U. Sterr, S. Falke et al., arXiv:1206.3765.

[4] D. Rideout, T. Jennewein, G. Amelino-Camelia, T. F.Demarie, B. L. Higgins, A. Kempf, A. Kent, R. Laflamme,

X. Ma, R. B. Mann, E. Martín-Martínez, N. C. Menicucci,J. Moffat, C. Simon, R. Sorkin, L. Smolin, and D. R. Terno,Classical Quantum Gravity 29, 224011 (2012).

[5] M. Aspachs, G. Adesso, and I. Fuentes, Phys. Rev. Lett.105, 151301 (2010).

[6] T. G. Downes, G. J. Milburn, and C. M. Caves,arXiv:1108.5220.

[7] M. Ahmadi, D. E. Bruschi, N. Friis, C. Sabn, G. Adesso,and I. Fuentes, arXiv:1307.7082.

[8] J. L. Ball, I. Fuentes-Schuller, and F. P. Schuller, Phys. Lett.A 359, 550 (2006).

[9] D. J. Hosler and P. Kok, Phys. Rev. A 88, 052112 (2013).

H1

H2

H3

0.2 0.4 0.6 0.8 1.0u

2

4

6

8

10

12

H

FIG. 2 (color online). Quantum Fisher information vs u ¼hτ

4Larctanhðh=2Þ for three different initial states of the cavity with

the same average energy. A single-mode squeezed state in modek ¼ 1 (solid black line), two single-mode squeezed states in aproduct form inmodes k ¼ 1 and k0 ¼ 2 (dotted red line) and a two-mode squeezed state in modes k ¼ 1 and k0 ¼ 2 (dashed blue line).

QUANTUM METROLOGY FOR RELATIVISTIC QUANTUM FIELDS PHYSICAL REVIEW D 89, 065028 (2014)

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[10] H. Salecker and E. P. Wigner, Phys. Rev. 109, 571 (1958).[11] The expansion is only possible when positive and negative

modes solutions can be distinguished, i.e. when the space-time admits a timelike Killing vector field. In the case of acontinuous mode decomposition, it is possible to construct adiscrete basis formed of wave packets of continuous modes.

[12] N. D. Birrell and P. C. W. Davies, Quantum Fields inCurved Space (Cambridge University Press, Cambridge,England, 1982).

[13] D. E. Bruschi, I. Fuentes, and J. Louko, Phys. Rev. D 85,061701(R) (2012).

[14] G. Adesso and F. Illuminati, J. Phys. A 40, 7821 (2007).[15] N. Friis and I. Fuentes, J. Mod. Opt. 60, 22 (2013).[16] G. Adesso, I. Fuentes-Schuller, and M. Ericsson, Phys. Rev.

A 76, 062112 (2007).[17] G. Adesso, S. Ragy, and D. Girolami, Classical Quantum

Gravity 29, 224002 (2012).[18] V. Giovanetti, S. Lloyd, and L. Maccone, Nat. Photonics 5,

222 (2011).

[19] M. G. A. Paris, Int. J. Quantum. Inform. 07, 125 (2009).[20] H. Cramér, Mathematical Methods of Statistics (Princeton

University, Princeton, NJ, 1946).[21] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439

(1994).[22] A. Monras, arXiv:1303.3682.[23] P. Marian and T. A. Marian, Phys. Rev. A 86, 022340

(2012).[24] C. Weedbrook, S. Pirandola, N. J. Cerf, T. C. Ralph,

J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621(2012).

[25] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439(1994).

[26] N. Friis, A. R. Lee, and J. Louko, Phys. Rev. D 88, 064028(2013).

[27] N. Friis, D. E. Bruschi, J. Louko, and I. Fuentes, Phys. Rev.D 85, 081701(R) (2012).

[28] C. Barceló, S. Liberati, and M. Visser, Living Rev.Relativity 8, 12 (2005).

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