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Differential atom interferometry beyond the standard quantum limit K. Eckert1, P. Hyllus1,2, D. Bruß1,3, U.V. Poulsen1,4,5, M. Lewenstein1,6, , ∗ C. Jentsch7, T. Mu¨ller7, E.M. Rasel7, and W. Ertmer7 1Institut fu¨r Theoretische Physik, Universität Hannover, D-30167 Hannover, Germany. 2 The Blackett Lab – QOLS, Imperial College London, London SW7 2BW, United Kingdom. 3Institut fu¨r Theoretische Physik III, Universität Du¨sseldorf, D-40225 Du¨sseldorf, Germany. 4Dipartimento di Fisica, Università di Trento, I-38050 Povo (TN), ∗ Italy; and ECT , I-38050 Villazzano (TN), Italy. 5Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C., Denmark. 6ICFO - Institut de Ciències Fotòniques, 08034 Barcelona, Spain. 7Institut fu¨r Quantenoptik, Universität Hannover, D-30167 Hannover, Germany. (Dated: February 1, 2008) 6 Weanalyzemethodstogobeyondthestandardquantumlimitforaclassofatomicinterferometers, 0 where the quantity of interest is the difference of phase shifts obtained by two independentatomic 0 ensembles. An example is given by an atomic Sagnac interferometer, where for two ensembles 2 propagating in opposite directions in the interferometer this phase difference encodes the angular n velocityoftheexperimentalsetup. Wediscussmethodsofsqueezingseparatelyorjointlyobservables a of the two atomic ensembles, and compare in detail advantages and drawbacks of such schemes. J In particular we show that the method of joint squeezing may improve the variance by up to a 9 factor of 2. We take into account fluctuations of the number of atoms in both the preparation 1 and the measurement stage, and obtain bounds on the difference of the numbers of atoms in the two ensembles, as well as on the detection efficiency, which have to be fulfilled in order to surpass 2 the standard quantum limit. Under realistic conditions, the performance of both schemes can be v improved significantly by reading out the phase difference via a quantum non-demolition (QND) 1 measurement. Finally, we discuss a scheme using macroscopically entangled ensembles. 6 0 PACSnumbers: 42.50.St, 42.50.Ct,95.75.Kk,42.81.Pa 7 0 5 I. INTRODUCTION is to measure the phase shift which occurs when the 0 laser setup (laboratory frame) is rotating relative to the / h Comparing the phase shifts obtained in two inde- frame of the freely flying atomic ensembles. This phase p shift is given by φat = 4πA Ω m /h as compared to pendent interferometric setups has several applications, at - φlight = 4πA Ω/(λc) for laser interferometers, where t prominent examples being the comparison of atomic n m is the mass of the atoms, A is the oriented enclosed clocks [1], and Sagnacinterferometryto discriminate be- at a area of the interferometer, Ω is the vector of angular u tween rotations and accelerations [2]. In addition, the q comparison of gravitational forces at different points in velocity, and λ is the wavelength of the light. For an : atomic gyroscope working with 87Rb, the phase φat is v spaceorfordifferentatomicspecies[3]allowstotestpre- 1011 times larger than the corresponding phase φlight of i dictions of possible violations of Einstein’s general rela- X tivity [4, 5]. For such differential interferometers, the alightinterferometerenclosingthesameareaandoperat- r ing atλ=103 nm. Hence, atominterferometerspromise a quantityofinterestisencodedineitherthedifference,or anenormouslyimprovedresolutionforrotationmeasure- the sum of the individual phase shifts. ments as compared to “classical” photonic devices. Especiallyforthemeasurementofinertialforces,atom The Sagnac phase can be measured by letting two en- interferometerspromisehighresolution. Hereatomopti- sembles of atoms pass through the interferometer from cal elements like beam splitters and mirrors can be real- opposite sides. They obtain then phase shifts φ due to izedusingRamantransitionsbetweentwoatomicground ± the rotation of the laboratory frame, where the sign de- state levels [6]. The accumulated phase difference be- pends on the direction of propagation of the ensembles, tweenthe interferometerpathsis encodedinthe number anda commonphase shift θ due to effects suchasanac- difference of atoms in the exit ports labeled by differ- celerationofthesetup. Subtractingthephasesofthetwo ent internal states. For example, this can be measured ensemblesyieldsthedesiredphase2φ,whichencodesthe by state selective fluorescence detection. Such schemes rotationofthesetuparoundanaxisperpendiculartothe have already been successfully implemented to measure plane of the interferometer. Collisions between the two inertial forces and the earth’s gravity with a high ac- ensembles in the interferometer can safely be neglected curacy [6, 7]. In Sagnac atom interferometry, the goal due to the low atomic densities of the ensembles. In the standard quantum limit for phase measurements in atomic interference experiments, the variance ∗alsoatInstitucioĆatalanaderecercaiestudisavançats. ofthe phase due to quantumprojectionnoiseis givenby 2 (∆φ)2 = 1/N, where N is the number of atoms in a populationmeasurements ofthe levels 1 and 2 by flu- SQL | i | i sample. By feeding the interferometer with non-classical orescencetechniques [18]. Hence, only the z components states of atoms, the so-called squeezed states, this limit of the collective spin vectors can be measured directly. can be surpassed, with a fundamental bound given by (∆φ)2 = 1/N2, the so-called Heisenberg limit [8, 9, 10]. H Squeezed atomic states can be produced by a quantum II.A. Description of the interferometer non-demolition (QND) interaction of the atoms with a light beam [11], or by absorption of non-classical states Initially, all the atoms are assumed to be prepared in of the light [12]. Squeezed states of atomic ensembles the state 1 , leading to the following expectation value | i might also be useful as quantum memory for states of and variance of the collective spin vector: light [13]. It has been shownthat for suchsqueezedstates atoms Jˆ = NJˆz, (∆Jˆ)2 =0, (∆Jˆ )2 = NJ, (2) z x,y within an ensemble are entangled with each other [14]. h i 2 4 In addition to this entanglement on a microscopic level, AnalogousvaluesareobtainedfortheensembleLˆ. Corre- itisalsopossibletoentanglemacroscopicdegreesoffree- dom of two atomic ensembles with a similar interaction sponding to the definitions in Eq.(1), the z axis denotes [15, 16]. This in principle enables teleportation of the the difference of the number of atoms in states 1 and | i 2 ,whereasthephasedifferencebetweenthesetwostates macroscopic state of an ensemble. Furthermore, it has | i beenshownrecentlythatsuchmacroscopicallyentangled is encoded in the x y plane. The uncertainties stem − from the single particle uncertainties, and correspondto ensembles can improvethe efficiency of measurementsof a state with a fixed number of atoms. the components of a magnetic field [17]. Here, we try to exploit microscopic as well as macroscopic entangle- A typical interferometer sequence used for the mea- ment between two atomic ensembles in the context of surement of inertial forces consists of three atom–light differential interferometry, where we focus especially on interactions as shown in Fig. 1. The first beam splitting an atomic Sagnac interferometer setup. Raman pulse transfers all the atoms from the ground In Section II we calculate the phase variance for a state 1 tothesuperposition 1 (1 + 2 ). Atomstrans- | i √2 | i | i differential interferometer using non-squeezed coherent feredtostate 2 obtainamomentumkickoftwophoton | i states in order to introduce our methodology. We show recoil if the two Raman lasers are counter-propagating, also how to include number fluctuations into the calcu- sothatthepartialwavesdelocalize,asdepictedinFig.1. lations. These will turn out to be important later. In The second pulse exchanges the populations, 1 2 , | i ↔ | i Section III we show that the phase uncertainty can be and deflects the partial waves. Finally, they are recom- reduced by feeding the interferometer with individually binedinthelastinteractionzone,actingasabeamsplit- squeezedensembles. InSectionIVweconsidersqueezing ter. These pulses can be represented as rotations of the of a joint observable of both ensembles. In Section V we discuss how decoherence affects the interferometer, and in Section VI we compare the variances of the schemes discussed so far for realistic parameters. Finally, in Sec- tion VII the use of macroscopically entangled states in the interferometer is discussed. II. COHERENT INPUT STATES Fora singleatomwe define apseudo spin-1/2through twogroundstateatomichyperfinelevels 1 and 2 [2,5] | i | i FIG. 1: Scheme of the atom interferometer. π and π label by introducing the relevant spin operators as 2 the beamsplitting and the mirror pulses, respectively. The 1 1 populationsinthetwoexitportsaredetectedstate-selectively σˆz = (1 1 2 2) , σˆx = (1 2 + 2 1) , viafluorescence measurements. 2 | ih |−| ih | 2 | ih | | ih | 1 σˆ = (1 2 2 1). (1) collective spin vectors Jˆ and Lˆ around an axis in the y 2i | ih |−| ih | h i h i x y plane, the angle being givenbyπ/2for beamsplit- − We will subsequently only consider the collective spin ters and by π for mirrors. For a fixed coordinate system Jˆ = NJ σˆ(i) for the first, and in analogy Lˆ for the theanglebetweenthexaxisandtherotationaxisisgiven i=1 second ensemble; N are the number of atoms in the by the laser phase [19]. The laser phases change if the J,L P ensemblesJ andL,respectively. Themeasurementsthat setup(laboratoryframe)rotateswithrespecttothepath can typically be performed in atom interferometry are of the freely flying ensembles, which causes the Sagnac 3 phaseshift. This changecorrespondsto arotationofthe Lˆ, because the Sagnac phase φ takes a different sign, collective spin vectors in the x y plane around the z depending ondirection in which the ensemble passes the − direction. interferometer. Wedefinethephaseoperatorforthecase Inorderto makethe scheme applicabletoamoregen- of coherent states (cs) as eral scenario for differential interferometers, we model it as follows for each of the ensembles labeled by Jˆ and Jôut Lôut Lˆ, respectively: the first beam splitter rotates each col- φˆcs = z + z (5) − N N J L lective spin vector by π/2 around the y axis, then we collect all the phase shifts occurring in the interferom- and obtain to the first order in φ and θ eter in rotations around z of Jˆ and Lˆ by Φ and Φ , J L respectively. Finally, a π and a π/2 pulse, both around φˆ = φ (6) cs x, implement the mirror and the final beam splitter, re- h i 1 1 spectively. These last two pulses can be combined to a (∆φˆ )2 = + . (7) cs 4N 4N single rotation around the x axis by π/2. All relevant J L − The variance of φˆ corresponds to the standard quan- cs tum limit. Note that θ can be obtained in an analogous way. From the definition it is obvious that determining the number of atoms is important for the calculation of the phase shift. A major source of error will come from fluctuationsinthenumberofatomsoftheensemblesand hencewewilldiscusshowtoincludethisprocessintothe FIG. 2: (Color online) Interferometric scheme for a coherent calculations in the next section. input state: (a) collective spin after the first beam splitter, (b)thetotalphaseφaccumulatedintheinterferometerresults in a rotation around z, before (c) a final rotation around x by −π/2 implements the mirror and the final beam splitter, II.B. Number fluctuations encoding thephase in thez direction. interferometric steps are described in Fig. 2, where the There are two sources of deviations of the number of atomicspinvectorisdepictedtogetherwithadiscrepre- atoms: thepreparationprocessandthenumbermeasure- sentingthecorrespondinguncertaintiesofthespinopera- ment process. The atomic ensembles produced from the tors. Tosimplifynotationwewilltakethestateafterthe sourcearebestdescribedasastatisticalmixtureofstates first beam splitter as the initial state: Jîn = (π/2)Jˆ, withdifferentatomnumbers,butwewillassumethatthe y R see Fig. 2(a). Here (α), i x,y,z , is the matrix final number measurement projects onto a number state i rotating a vector by tRhe angleα∈a{round}the direction eˆ . with N and N atoms in the two ensembles. Defining i J L In the Heisenberg picture, and neglecting collisions be- N¯ = (N +N )/2, we assume that N N = γ√N¯, J L J L tween the atoms within the ensemble, the spin operator whichreflectsthevariancesofthenum|ber−opera|torsNˆ J,L changes according to Jôut = ( π/2) (Φ )Jîn. This ofthe ensemblesafterthe production. Thus,γ is the pa- x z J R − R leads to rameter which describes how well the atom numbers in the two ensembles match. JîncosΦ JînsinΦ x J − y J Wetreatthenumbermeasurementsbyintroducingop- Jôut = Jˆzin . (3) erators δNˆ with JînsinΦ JîncosΦ − x J − y J   δNˆ =0, [∆(δNˆ)]2 =αN, (8) It is also possible to consider a balanced atom interfer- h i ometer in which all rotations are around the x axis. In this case, an extra π/2 shift around z leads to the same where α describes the quality of the number measure- result. The advantage of the extra pulse is that for the ment. For fluorescence measurements, α−1 is given by initial state of Eq. (2) the meannumberoftimes anatomgoesthroughthe flu- orescencecycleandscattersaphotonwhichsubsequently N N Jôut = J sinΦ JΦ (4) isregisteredinthedetectors[20]. Typicalvaluesinnowa- h z i − 2 J ≈− 2 J daysexperimentsareα 1 50...100. WereplaceN by − J ≈ for small angles Φ , while with the unmodified balanced N0+δNˆ(1)+δNˆ(2)andJôutbyJôut,0+(δNˆ(1) δNˆ(2))/2. J J J J z z J − J scheme Jôut wouldbeproportionaltocosΦ ,andhence N0 refers to the actual number that would have been h z i J J sensitive to Φ only in second order around Φ =0. measured in perfect number projection measurements, J J As explained in the introduction, the phase will be andinδNˆ(i) the index icorrespondsto the atomiclevels J given by Φ = φ + θ for Jˆ and by Φ = φ + θ for i . J L − | i 4 With these substitutions, we obtain the Hamiltonian describing the interaction of the light with the first atomic ensemble can be brought to the 1 1 φˆ = φ α + (9) form [11] h csi − N0 N0 J L (cid:16) (cid:17) (∆φˆ )2 = 1 + 1 +α 1 + 1 , (10) Hˆ =~ΩJSˆzJˆz, (13) cs 4N0 4N0 N0 N0 J L J L (cid:16) (cid:17) with a frequency Ω . Due to the interaction, the collec- J where terms of higher order have been neglected. The tivespinvectoroftheatomsisrotatedaroundthez axis contribution from the number fluctuations is in agree- by χ Sˆ , with the atom–photon coupling χ =Ω t and ment with Ref. [20]. The expectation value of φˆ is J z J J cs the effective interaction time t. The Stokes vector un- shifted for α=0. As this shift is of the order of the sec- dergoes the same variationwith χ Sˆ replacedby χ Jˆ. ondtermin(6∆φˆ )2,itisofsecondorderofthestandard J z J z cs This rotation of the Stokes vector is due to the Faraday deviation only, and can thus safely be neglected. From effectofthelightpassingtheatoms,whiletherotationof the expressionsitis clearthatα 1 isrequiredin order the atomic spin vector is due to an AC Stark shift origi- ≪ to reach the fundamental limit for the phase resolution nating from the light field. The coupling χ is given by J using coherent input states. [11] In the following, we will drop the superscript 0 on the atomnumber, as long as there is no dangerof confusion. L ∆ ωd2 χ =2g2 , g = , (14) J c 14Γ2+∆2 s2~ǫ0AL III. SEPARATELY SQUEEZED ENSEMBLES whereAandLarecrosssectionandlengthoftheatomic sample,Γanddareline-widthanddipolemomentofthe It is known that by taking squeezed input states it is atomic transition, respectively, and ∆ is the detuning possible to surpass the standard quantum limit in inter- from the atomic resonance frequency ω. ferometry[8]. Inthis section,weconsiderthe casewhere After the interaction, bothensemblesaresqueezedseparatelywiththe method introducedin[11],i.e.,byaQNDinteractionwithalaser Sôut =sin(χ Jˆ)Sîn+cos(χ Jˆ)Sîn, (15) beam shortly after the first beam splitter. y J z x J z y and if initially III.A. Squeezing a single ensemble n n Sîn = Jxˆ, (∆Sîn)2 =0, (∆Sîn )2 = J, (16) h i 2 x y,z 4 We assume to have the situation of Fig. 3, i.e., the electromagnetic field mode a couples states 1 and 3 , where n is the number of photons in the ensemble Jˆ, 1 J and a couples states 2 and 4 . Here transi|tioins 1| i then we can effectively replace Sîn by its macroscopic 2 | i | i | i↔ x 4 and 2 3 haveto be suppressedto the firstorder expectation value n /2. Furthermore, developing the J | i | i↔| i in the coupling constant ([11], see also the discussion in trigonometricexpressionsandassumingthatN χ2 1, J J ≪ Section V). For the light, an effective spin vector can be we obtain in leading order n χ Sôut J Jîn+Sîn. (17) y ≈ 2 z y Henceameasurementoftheycomponentoftheoutgoing light vector gives information about Jˆ , while Jˆ itself z z h i is not affected by the rotation around the z axis. FIG. 3: For the QND interaction, the levels |1i and |2i are coupled off-resonantly to states |3i and |4i. IfsuchaQNDmeasurementisperformedafterthefirst beam splitter of the interferometer, then the operator defined, the so-called Stokes vector. Its components are 2 given by Jˆ Jôut Sôut (18) z′ ≡ z − n χ y J J 1 1 Sˆz = 2 aˆ†1aˆ1−aˆ†2aˆ2 , Sˆx = 2 aˆ†1aˆ2+aˆ†2aˆ1 ,(11) measures the difference between the z component of the 1(cid:16) (cid:17) (cid:16) (cid:17) ensemble’s atomic spin vector after the second beam Sˆy = 2i aˆ†1aˆ2−aˆ†2aˆ1 , (12) splitter and the estimated value after the first one. The (cid:16) (cid:17) fluctuations of this operator are reduced as compared to where aˆ†1 and aˆ†2 create a photon in mode a1 and a2, Jˆzout [11],whilethefluctuationsofJˆy areenlarged,which respectively. Sˆ measuresthedifferenceofphotonsinthe is depicted in Fig. 4. Hence, the state of the atomic spin z two modes. By an appropriate choice of the parameters, vector is squeezed in the z direction. 5 III.B. Modified interferometric scheme provided that α is small enough, the variance is dom- inated by the first two terms scaling as NJ−/2L. They We modify the scheme introduced in Section II by in- originate from the projection noise of the light, and in serting the QND interaction shortly after the first beam principle allow to improvethe resolutionbelow the stan- splitter, followed by an extra rotation around x by π/2, dard quantum limit. However, nχ2 equals, except for whichrotatesthe uncertaintyellipse suchthatthe phase a factor of order unity, the fraction of atoms which are uncertainty is reduced as desired, cf. Fig. 4. In the lost due to spontaneous processes during the squeezing experiment, the latter pulse must not transfer momen- process, cf. Section V. Thus, nχ2 1 is necessary, ≪ tum to the particles, which can be achieved by using and even though we obtained a Heisenberg-like scaling co-propagatingRamanlasersforthisstep, providedthat (∆φˆss)2 ∼ 1/NJ2,L, we are far from reaching the Heisen- the two transition frequencies are approximately equal, berg limit. so that the two recoilmomenta cancelinthe transitions. Furthermore,asitbecomesclearfromthesecondterm, the resolution is limited by the accuracy of the fluorescence number measurements. These measurements are necessary in any case to determine the phase, because Jôut and Lôut haveto be rescaledproperly. However, hJˆzanid Lˆ hitszelfican be measured using another QND z z interaction. As will be shown in the next section, this reduces the dependence on the quality of the number measurements. III.C. QND output measurement Let us consider now a modification of the scheme us- FIG. 4: (Color online) The interferometric scheme for ing squeezed states, where a second QND laser beam is squeezed input states: (a) collective spin after thefirst beam sent through each ensemble shortly after the last beam splitter, (b) theQND measurement prepares a spin squeezed splitter. We define state which (c) is rotated around x by π/2; step (d)→(e) as step (b)→(c)in Fig. 2. 1 2 φˆ = Sôut Sôut ss+ −N nχ y,r − y J The outgoing spin vector now is calculated as Jôut = 1 2 (cid:16) (cid:17) x( π/2) z(ΦJ) x(π/2) z(χSˆz)Jîn, and in analogy +N nχ Tˆyo,urt−Tˆyout , (22) R − R R R L for the second ensemble Lˆ, with corresponding Stokes (cid:16) (cid:17) vector Tˆ. where the extra index + is supposed to indicate the ad- IncomparisontoEq.(5),Jˆzout nowhastobecorrected ditional QND measurement. Furthermore, Sˆr and Tˆr toincorporatethespinsqueezing(ss)asdescribedabove: correspond to individually prepared light pulses used to read out Jˆ and Lˆ , respectively. The resulting expecta- z z 1 2 φˆ = Jôut Sôut tion value and variance in leading order are ss −N z − n χ y J J J +N1L(cid:16)Lôzut− nL2χLTˆyout(cid:17). (19) hφˆss+i = φ+αθ N1J + N1L (23) (cid:16) (cid:17) 2 (cid:16)1 1 (cid:17) Calculating the expectationvalue andvarianceas before (∆φˆss+)2 = nχ2 N2 + N2 yields (cid:16) J L(cid:17) 1 1 +α(θ2+φ2) + . (24) φˆ = φ α 1 + 1 (20) 4NJ 4NL h ssi − N N (cid:16) (cid:17) J L 1 (cid:16)1 1 (cid:17) 1 1 Let us compare the modified scheme including an (∆φˆss)2 = nχ2 N2 + N2 +α N + N .(21) additional QND measurement with the scheme using (cid:16) J L(cid:17) (cid:16) J L(cid:17) squeezed states. The magnitude of α in the variance of Here it has been assumedn =n =:n andχ =χ =: themodifiedschemeiseffectivelyreducedforsmallθand J L J L χ. Deviations fromthese assumptions enter the variance φ, α α(θ2 +φ2)/4, as compared to Eq. (21). Hence, → onlyinhigherordertermsaslongas χ χ /(χ + the dependence on the number measurement is reduced. J L J || |−| || | | χ ) 1, and similar for n . Further assumptions Furthermore, the leading term shifting the expectation L J l|ead|ing≪to these expressions are N¯χ2 1 as mentioned valuefromthe desiredresultφis smallerbyafactorθ as before, as well as nχ2 √8 [√N¯(θ≪2+φ2)] 1. Now, compared to Eq. (20). − ≪ · 6 As a further possible advantage, the effect of a non- Wedefinethephaseoperatorfortheschemeemploying symmetric atom-light interaction is compensated if the jointly squeezed (js) ensembles as couplingoftheread-outQNDpulseissimilartothecou- 1 1 2 plingofthesqueezingpulse[21]. However,thisadvantage φˆ = Jôut+ Lôut Sôut . (27) is probably not very relevant for the Sagnac interferom- js − NJ z NL z − nχN¯ y (cid:16) (cid:17) eter considered here due to the long time-of-flight of the Note that in this definition Jôut and Lôut are divided ensembles in the interferometer between the two QND z z by the atom numbers of the respective ensembles as be- pulses. fore, while the QND measurement yields an estimate of A disadvantage of this scheme is that the first term of Jôut +Lôut without such a correction, cf. Eq. (26). As the varianceofEq.(24)comeswitha factorof2 because z z aconsequencewe expectthat weloose the advantagesof ofthetwoprojectionmeasurementsofthelightnecessary squeezing if the numbers of atoms in the two ensembles per atomic ensemble. It is possible, although technically differ strongly, i.e., if γ 1. We find in leading order demanding, to reduce this contribution by re-using the ≫ light from the first QND interaction for the read-out. In 2α φˆ = φ (28) this case, χ χ is needed in the second interaction h jsi − N¯ ainnaolrodgeyrftoorot→bhtea−isnectohneddeinffseermenbclee.JˆzoTuhte−sJiˆgzinn [c2a1n],aalnsodbine (∆φˆjs)2 = nχ21N¯2 + 2N¯α + 8γN¯22. (29) achieved by a π rotation of the atom spin vector around the x axis in between the final beam splitter and the Theimportanceofthenumberfluctuationsattheprepa- QND read-out pulse. rationstageisreflectedinthefactthatinordertoarrive at these equations, the assumption IV. JOINTLY SQUEEZED ENSEMBLES γ N¯(nχ2)2(θ2+φ2) 1 (30) ≪ p is necessary in addition to the assumptions leading to In the preceding section we have seen that the 1/(nχ2N2)terminthevariancecomeswithafactorgiven Eq. (21). Furthermore, now 1/(nχ2N¯2) is the leading bythenuJmberofQNDinteractionswithdifferent ensem- term only if nχ2γ2/8 1. ≪ A QND measurement could also be used after the in- bles of light, cf. Eq. (24). For this reason let us consider terferometer to directly read out the joint observable the case of preparing the initial state of the two atomic Jôut+Lôut by defining ensembles with only a single QND pulse that interacts z z with both ensembles consecutively. The first interaction 2 1 (with ensemble Jˆ) transforms Jîn → Rz(χSˆzin)Jîn, the φˆjs+ =−nχN¯ Sˆyo,urt−Sˆyout , (31) second interaction (with ensemble Lˆ) transforms Lîn (cid:16) (cid:17) → (χSîn)Lîn, because Sîn itself remains unchanged dur- where again the indices r refers to the read-out QND Rz z z ingtheQNDinteraction. TheStokesvectorSˆ transforms measurement. Notice that in this way it is not asSôut = (χLîn) (χJîn)Sîn,andthey componentof possible to measure the correct rescaled observable the outgoiRngzlighzt isRgzivenzby Jˆzout/NJ +Lôzut/NL,andconsequentlythereisanimpor- tant contribution from the difference of the atom num- Sôut = cos(χLîn) cos(χJîn)Sîn+sin(χJîn)Sîn + bers in the expectation value already: y z z y z x h i N N αφ sin(χLîn) cos(χJîn)Sîn sin(Jînχ)Sîn ,(25) φˆ =φ+ L− Jθ+ . (32) z z x − z y h js+i N +N 2N¯ L J h i such that for Nχ2 1, and Sˆ initially prepared as in Compared to the variance of the scheme using jointly ≪ Eq. (16), we have squeezedensembleswithout the QND read-outmeasure- ment, Eq. (29), the dependence of the variance on the nχ Sôut (Jîn+Lîn)+Sîn. (26) number measurements and on the atom number differ- y ≈ 2 z z y ence in the two ensembles is reduced for small θ, φ: MeasuringSôut thus revealsinformationabout Jîn+Lîn 2 α y z z (∆φˆ )2 = + (θ2+φ2)+ and performs a squeezing operation on this joint opera- js+ nχ2N¯2 2N¯ tor. Now we apply the same operations as before to the γ2 ensemble Jˆ, but for the ensemble Lˆ we perform a rota- + α(θ2+φ2). (33) 8N¯2 tionbyπaroundthexaxisbeforethefinalmeasurement. Aftertheextrapulse,the Sagnacshiftφiseffectivelyen- However, the γ-dependent correction to the expectation codedinthesumofthe z componentsinsteadofintothe value φˆ is only negligible compared to the standard js+ difference. deviathion ∆iφ if γ2N¯nχ2θ2/8 1. This is generally a js+ ≪ 7 stronger criterion than the limit on γ encountered with- for the Stokes vector. The leading order corrections to out the QND read-out, cf.Eq. (30). the variance (∆φˆ )2 given in Eq. (21) for the case of ss The offset can be compensated by using an estimate separately squeezed ensembles reads for θ from the final fluorescence measurement nχΓ 1 2Γ (∆φˆ )2 (∆φˆ )2+ + , (42) θˆ= 1 Jôut+ 1 Lôut, (34) ss → ss ∆ N¯ n2χ∆ −N z N z J L and similarly for the other schemes. We will use this to define a corrected phase operator estimateinthefollowingdiscussionandleaveanin-depth analysisofdecoherenceprocesses,e.g.,followingthelines N N φˆc =φˆ L− Jθˆ (35) of [22], to further investigations. Generally, according js+ js+− NL+NJ to Eq. (38) also the expectation value changes due to decoherence. Thiscanbeaccountedforbyrescaling,and which takesinto accountthe bias of φˆjs+ . We find that doing so gives only higher-order corrections to Eq. (42). h i to leading order For usual choices of parameters, the last term in Eq. (42) is negligible, while the contribution proportional to φˆc = φ+ αφ γ2α (36) N¯−1 is comparableinsize to the terms in Eq.(21). This h js+i 2N¯ − 2N¯2 limits the coupling χ and thus the achievable squeezing. ∆φˆc 2 = 2 + αφ2 + γ2 . (37) For all other experimental parameters fixed, there exists js+ nχ2N¯2 2N¯ 8N¯2 an optimalchoice for the detuning ∆ and thus for χ [see (cid:16) (cid:17) Eq. (14)] which minimizes the variance. Taking into Hence, the γ-dependent bias in the expectation value is account only the first term in Eq. (21), and in the limit reducedbyafactorαγ/(N¯3/2θ) 1,whilethe1/N¯ term ∆ Γ,thisoptimalchoiceof∆leadstoaminimalvalue ≪ in the variance still has a factor αφ2. for≫(∆φˆ )2: ss These are the same advantages that we also found in the case of separately squeezed ensembles with QND 2 2ǫ ~cAΓ rteeramd-opurot,pcofr.tiEoqn.al(2to4)1./Hnχow2Ne¯v2eirs,rtehdeuccoendtbriybuatfiaocntoorfotfh2e m∆in (∆φss)2 = N32 dr 0ω . (43) (cid:2) (cid:3) in (∆φˆc )2, because only two instead of four projective The scaling is thus no longer as in the Heisenberg limit, js+ measurements of the light are necessary in this case. but it is still better than in shot-noise limited measurements (see also [23]). In addition, the decay of the states 1 and 2 dur- | i | i V. DECOHERENCE ing the interferometer step has to be takeninto account. Choosing long-lived hyperfine ground-state levels to implement 1 and 2 minimizes this decay. Also, spin The attainable squeezing is limited by the absorp- | i | i squeezed states have been shown to be robust with re- tion of photons during the interaction between light and spect to both, particle loss and dephasing [24], in con- atoms [22]. Each atom which absorbs and subsequently trast to, e.g., GHZ states, which are maximally fragile spontaneously emits a photon is no longer correlated to under particle losses. the rest of the atoms, but still adds to the variance. We estimate the number of atoms contributing to such an uncorrelatedbackgroundasthenumberofscatteredpho- VI. COMPARISON OF THE SCHEMES tonsnκ,whereκ=N χΓ/∆istheopticaldensity. Then, J in the limit of κ 1 and for just a single ensemble, one ≪ To analyze the performance of the schemes discussed finds [22] in the preceding sections we will fix the number of pho- nχΓ tons as n = 1011 and take N¯ = 1010 as a reasonable hJzouti → hJzouti 1− ∆ , (38) parameter for the mean atom number per ensemble. We (cid:18) (cid:19) will first consider a close-to-ideal scenario and assume Jout 2 1 nχΓ 2 Jout 2 + nNJχΓ(39) that the noise from the fluorescence measurements can h z i → − ∆ h z i 2∆ be neglected by setting α = 2 10 7. Also, we will (cid:0) (cid:1) (cid:18) (cid:19) (cid:0) (cid:1) set γ = 10, which for N¯ = 1010×ato−ms corresponds to for the collective atomic spin vector and N N = 10 4N¯, and we will initially not include J L − |deco−herenc|e. Fig. 5 (a) shows the scaling with N¯ of N χΓ Sout Sout 1 J , (40) (∆φ)2/(∆φ )2, i.e., of the various variances normalized h z i → h z i − ∆ cs (cid:18) (cid:19) to the case of coherent ensembles. For all the methods Sout 2 1 NJχΓ 2 Sout 2 + NJ2χΓ(41) involving squeezing of some observable a Heisenberg like h z i → − ∆ h z i 4∆ scaling is visible. The offsets of these curves are given (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) 8 0 10 0 Separately squeezed (((aaa))) Separately squeezed + (b) QND readout ) Sum squeezed B d -5 ( S Separately squeezed + n 2 C QND readout o ) ti c Sum squeezed +joint QND u 2 (10-1 Sum squeezed readout + eliminatio n of -10 ed ) 10-9 e r ( Separately squeezed al ois Sum squeezed + joint QND readout ptim -15 n o + elimination of Sum squeezed + joint QND readout -10 10-2 (no correction of ) 10 108 109 1010 1011 -20 8 9 10 11 8 9 10 11 10 Mean atom1 n0umber N 10 10 10 Mean atom1 n0umber N 10 10 FIG. 5: Double-logarithmic plot of the variances (∆φ)2 for the methods discussed in the text normalized to the variance for coherentstates(∆φcs)2,asafunctionofthemeannumberofatomsN¯. Thescaleontherighthandsitegivesthenoisereduction in dB. In both figures n = 1011, and we fixed θ = φ = 0.01 in order to operate close to the point of maximal sensitivity of the interferometer. In (a) we consider the close-to-ideal scenario with α=2×10−7, γ =10, and decoherence is not included. χ=3.23×10−10 isused,correspondingtoadetuning∆=2.28×1010 s−1 fortheRbD2 lineandacrosssection A=0.3mm2 of the laser beam. In this close-to-ideal scenario, the scaling as N¯−2 is visible for all the methods employing squeezing. The graphsforseparately squeezedensembles(without QNDread-out)andfor jointly squeezedensembleswith QNDread-outand corrected expectation value lie on top of each other. In the case of a joint QND read-out, failing to correct the expectation value for the contribution from θ results in a scaling as 1/N¯ for N¯ & 109. (b) Realistic scenario with α=2×10−2, γ =104, and including decoherence. In each case and for each value of N¯ the detuning has been adjusted to minimize the variance [25]. Theinset shows thecorresponding optimal values of thecoupling parameter χ. Forlarge atom numbersit is clearly seen thattheschemeswhichdonotemployafinalQNDmeasurementarestronglyaffectedbythelimitationsfrom thefluorescence detection. The curvefor jointly squeezed ensembles using a joint QNDreadout lies outside therange of thefigure. by the numbers of QND measurements performed, i.e., cases the variance scales as 1/N¯ due to decoherence and by the factor multiplying the 1/(nχ2N¯2) term in the thenoisefromthefluorescencemeasurements. Forallthe variance of each of the considered schemes that involve proceduresnot involvinga QND read-out,the strongin- squeezing. For the measurement of φ via reading out fluenceofthelattercontributioncanbeobserved,though Jˆ +Lˆ through a QND interaction, the term propor- for N¯ = 1010 atoms the total noise still is reduced by z z tional to θ shifting the expectation value [see Eq. (32)] around 7 dB with respect to the limit set by quantum hasbeenincludedintothevariance,inordertoallowfora projection noise. On the other hand, the read-out via faircomparison. Itisthistermwhichmakesthevariance QND measurements allows to reduce the noise by more scale only proportional to 1/N¯ for N¯ & 109. Obviously, than 10 dB compared to this limit, while this method correcting this contribution of θ as described in Section does not require an additional experimental setup as IVavoidsthistermandmaintainsthe improvementbya compared to the scheme where QND measurements are factor of 2 compared to the case of squeezing and QND performed only on the incoming atomic ensembles. measuring both ensembles separately. In Fig. 5 (b), the relative variances are plotted for realistic experimental parameters and including decoherence. α=2 10 2 correspondsto 50 fluorescencecycles − × per atom, and γ = 104 is equivalent to a difference of the number of atoms in the two ensembles of 10% of the Whileintheclose-to-idealcasesqueezingofajointob- mean number N¯ at N¯ = 1010. With all other parame- servablegivesanadvantageofafactorof2inthevariance ters fixed, for each value of N¯ the interactionstrength χ compared to individual squeezing and read-out (corre- is determined by choosing the detuning ∆ from atomic sponding to a 3 dB noise reduction), for an experimen- resonancesuchthatthevariance(∆φ)2 isminimized,see tal reasonable scenario this advantage reduces to about Eq. (43) [25]. 1.5 dB. Of course the method of squeezing a joint ob- As can be seen from Fig. 5 (b), in such a realistic sce- servable of both ensembles has a basic advantage since nario the noise reduction obtained from squeezing de- itonlyneeds asinglesqueezingoperationinsteadoftwo, creases for all methods. For large atom numbers, in all and thus less technical effort is necessary. 9 VII. ENTANGLED ENSEMBLES These measurements yield both angles: N N αφ Julsgaard et al. demonstrated experimentally in φˆ = φ+ L− Jθ+ (48) h EEi N +N 2N¯ Ref.[16]the generationofmacroscopicentanglementbe- L J N N αθ tween two atomic ensembles. The scheme to generate θˆ = θ+ L− Jφ+ , (49) h EEi N +N 2N¯ suchamacroscopicallyentangledstate,describedfirstin L J Ref. [15], is motivated by the fact that under the ideal wheretheoffsetscanbecorrectedasabove. Forparame- condition of γ = 0 two commuting joint observables can tersasinSectionV.theleadingtermsofthecorrespond- be constructed from Jˆ and Lˆ: ing variances read (at θ =0 and φ=0) h[Jˆy −Lˆy,Jˆz+Lˆz]i∝(NJ −NL)=0, (44) (∆φˆ )2 = 1 2 + γ2 nχ2. (50) EE cos2ϕN¯2nχ2 8N¯ i.e., Jˆ Lˆ can be measured without affecting Jˆ +Lˆ , y y z z − 1 2 γ2 nχ2 and vice versa. This can be seen directly from (∆θˆ )2 = + . (51) EE sin2ϕN¯2nχ2 8N¯ ( (χSˆ )Jîn (χSˆ )Lîn) =Jîn Lîn, (45) Rz z −Rz z y y − y Changing ϕ allows to trade in a lower variance of one i.e., the first QND interaction leaves the difference of component for a higher variance of the other. But these the y components unaffected. Thus after squeezing the variancesareonly scalingwith 1/N¯2 ifγ is closeto zero, sumJˆ +Lˆ , alsothe difference Jˆ Lˆ canbe squeezed otherwisethelastterm γ2nχ2/N¯ inEqs. (50)and(51) z z y y − ∝ without loosing the information gained in the first mea- is dominating the scaling. However, γ 0 is an obvious ≈ surement. To realize this experimentally in the interfer- requirement in this case, as otherwise the commutator ometer, after the first squeezing interaction Jˆ is rotated does not vanish in Eq. (44), and thus the two squeezing byaclassicalπ/2pulsearoundthexaxissothatJˆ Jˆ operations are not compatible. y z → while Lˆ is rotated by π/2 around x giving Lˆ Lˆ . y z − − → Thenasecondlaserpulse,preparedagainasinEq. (16), VIII. CONCLUSION AND OUTLOOK interacts consecutively with both ensembles and thus finally carries information about Jˆ Lˆ . The outgoing y y state corresponds to a macroscopic−ally entangled EPR We have presented and compared in detail several state [15]. It is now a natural question to ask whether methods to improve the detection of a differential phase entangled atomic ensembles are of use in Sagnac atom shift of two atomic interferometers beyond the standard interferometry. quantumlimit,havinginmindespeciallythe application For the schemes discussed so far, the collective spin to Sagnac interferometry. For this purpose, we have an- vectors lie in the x z plane after the last step of the alyzed the squeezing of individual and joint observables interferometer. Ther−efore always Jˆ Lˆ = 0, and an and,inbothcases,the read-outofthe interferometervia y y additionaloperationisnecessaryinhord−ertoiencodephase fluorescence detection of the atoms only or by an addi- information in the y components as well. This can be tional QND interaction. achieved by rotating both ensemble vectors Jˆ and Lˆ by Ifdecoherence andmeasurementimperfections arene- an angle ϕ and ϕ around the x axis before and after glected, all the methods of squeezing improve the be- the interferometr−ic phase is applied, respectively. In this haviorofthe varianceofthe differentialphaseto a 1/N¯2 way the plane of rotation of the phase shift is effectively scaling,modifiedbyafactork/(nχ2) 1,whichisdeter- ≫ tilted by ϕ around the x axis. minedbythenumberk ofQNDinteractionsinvolved,by Themeasurementprocessnowconsistsoffirstrotating thenumberofphotonsn,andbythecouplingχbetween Jˆ andLˆ by π/2 and using another QND interaction to atoms and photons. In the case of jointly squeezed ob- measure the±sum of the z components. This measure- servables, we found that this limit can only be attained ment, scaled correctly by a ϕ-dependent factor, reveals if some constraints on the difference of the number of φ. To be more explicit, the corresponding operator for atoms in both ensembles can be fulfilled. In all cases, entangled ensembles (EE) reads the achievable squeezing is limited by decoherence due to the absorption of photons during the QND measure- φˆ = 1 2 Sôut Sôut . (46) ment. EE cosϕN¯nχ2 y,r − y Using fluorescence measurements to read out the (cid:16) (cid:17) atomic spins after the interferometer always produces A measurement of the sum of the y components can be additional noise scaling as 1/N¯ due to the photon shot realizedafteranotherrotationaroundxbyeitheraQND noise. As an alternative method, a QND measurement or a projection measurement. 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