Atomic homodyne detection of continuous variable entangled twin-atom states
C. Gross, H. Strobel, E. Nicklas, T. Zibold, N. Bar-Gill, G. Kurizki, M.K. Oberthaler
AAtomic homodyne detection of continuous variable entangled twin-atom states.
C. Gross, H. Strobel, E. Nicklas, T. Zibold, and M.K. Oberthaler
Kirchhoff-Institut f¨ur Physik, Universit¨at Heidelberg,Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
N. Bar-Gill ∗ and G. Kurizki Weizmann Institute of Science, Rehovot 76100, Israel
Historically, the completeness of quantum theory has been questioned using the concept of bipar-tite continuous variable entanglement [1]. The non-classical correlations (entanglement) between thetwo subsystems imply that the observables of one subsystem are determined by the measurementchoice on the other, regardless of their distance. Nowadays, continuous variable entanglement isregarded as an essential resource allowing for quantum enhanced measurement resolution [2], therealization of quantum teleportation [3–5] and quantum memories [3, 6], or the demonstration ofthe Einstein-Podolsky-Rosen paradox [1, 7–9]. These applications rely on techniques to manipulateand detect coherences of quantum fields, the quadratures. While in optics coherent homodyne de-tection [10] of quadratures is a standard technique, for massive particles a corresponding methodwas missing. Here we report on the realization of an atomic analog to homodyne detection for themeasurement of matter-wave quadratures. The application of this technique to a quantum stateproduced by spin-changing collisions in a Bose-Einstein condensate [11, 12] reveals continuous vari-able entanglement, as well as the twin-atom character of the state [13]. With that we present anew system in which continuous variable entanglement of massive particles is demonstrated [6, 14].The direct detection of atomic quadratures has applications not only in experimental quantum atomoptics but also for the measurement of fields in many-body systems of massive particles [15].
Continuous variable entangled states, whose inter-mode entanglement is reflected in quadrature as well aspopulation correlations, are routinely generated by para-metric downconversion in optical parametric amplifiers(OPA) [10]. Spin-changing collisions in ultracold bosonicquantum gases provide a similar nonlinear process formatter-waves generating atom pairs with spin up anddown – twin-atoms [16–19]. Correlated photon or atompairs in the signal | ↑(cid:105) and idler | ↓(cid:105) modes are createdfrom a pump field via nonlinear interactions describedby the Hamiltonian (cid:126) α ( a †↓ a †↑ + a ↓ a ↑ ), where a † k is thecreation operator of the respective mode k and 2 π (cid:126) isPlanck’s constant. This Hamiltonian presumes a largeamplitude coherent pump field, whose depletion or distor-tion by the signal and idler modes is negligible such thatit can be treated classically. In optics the effective non-linearity α is proportional to the amplitude of this pumpfield and the susceptibility of the medium. In the analo-gous regime for atoms the effective nonlinearity originatesfrom interatomic interactions and is proportional to thepopulation in the pump mode, that is, the Bose-Einsteincondensate (BEC). For initially empty signal and idlermodes quantum fluctuations are amplified and the out-put is the so called two-mode squeezed vacuum [10]. Thisstate is characterized by the vanishing of the mean fieldamplitude (cid:104) E k (cid:105) = 0 in each of the modes, while the meanintensity is nonzero (cid:104) I k (cid:105) ∝ (cid:104) E † k E k (cid:105) >
0. The associatedtwo-mode entanglement is revealed by the quadratures X ± ( ϕ ) = a ↓ e − iϕ ± a †↑ e iϕ + h . c . , the signature being thesqueezing of their variance for appropriately chosen ϕ .To access the two-mode quadratures the coherent ho-modyne measurement technique is employed, whereby the conjugate canonical quadratures X k = a † k + a k and Y k = i ( a † k − a k ) are measured by mixing the quantummode with a strong classical reference field, the local os-cillator a † LO ≈ √ N LO e − iϕ with large amplitude √ N LO .Experimental control of the local oscillator phase ϕ pro-vides access to the continuous distribution of single modequadratures X k ( ϕ ) = e iϕ a † k + e − iϕ a k and their two-modecounterparts X ± ( ϕ ) ( X k and Y k are used as abbrevia-tions for X k (0) and X k ( π )). For the squeezed vacuumstate the quadrature distribution of individual modes isisotropic and centered around the origin reflecting theundefined phase. The correlation of the modes leads toanisotropic and squeezed distributions of the two-modequadratures X ± ( ϕ ) as seen in figure 1a [10, 20].Here we report on the development of atomic ho-modyne detection and apply this technique to measurequadratures of twin-atom matter-wave fields generatedby controlled spin-changing collisions in a BEC. The ex-periments involve a spinor BEC of Rubidium 87 trappedin a 1D optical lattice with few hundred atoms per siteand a lattice spacing of 5 . µ m (Fig. 1b). A high latticepotential assures that tunneling between sites is negli-gible on the experimental timescale such that the con-densates in the different sites are independent [21]. Thedensity in the center of the lattice sites is in the or-der of 2 × cm − resulting in a minimal spin healinglength of ξ ≈ µ m, which is comparable to the exten-sion of the on-site wave function (approximately 1 . µ mFWHM). The spatial degrees of freedom are thereforefrozen and the dynamics happens exclusively in the hy-perfine spin [22]. The condensate is initially prepared a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec ac b photonsphotons −200−1000100200 a t o m nu m be r atoms FIG. 1.
Analogy to optics and measured population correlations of twin-atom states. a,
Parametric downconver-sion with light and the analogy to atomic spin changing collisions. In a nonlinear medium effective interactions result in thecreation of photon pairs in the signal (red) and idler (blue) modes from pump mode photons (green). The quadratures of theoutput modes are centered around the origin – the individual ones are isotropic, while the two-mode quadratures are squeezedreflecting their correlations (purple). Pair creation due to spin changing collisions in tight traps is an analogous process inquantum atom optics. b, Experimental system and exemplary raw absorption pictures. The illustration on the left showsschematically the 1D optical lattice which is superimposed by an optical dipole trap to tightly confine the atoms in all direc-tions. In each site selected Zeeman levels of the hyperfine spin are the only accessible degrees of freedom. The states involvedin the spin changing collision process are highlighted. Three typical experimental pictures after spin changing collisions arepresented on the right where the clear spatial separation of sites and states can be seen. c, Population correlations. Populationcorrelations are visible directly on the raw data where the atom numbers in the signal and idler modes fluctuate strongly (blue,red) while their difference (black) shows small noise. The left part of the graph is a zoom into 25 experimental realizationsclearly showing that the fluctuations are common mode. in the (
F, m F ) = (2 ,
0) hyperfine state serving as theanalog to the pump mode in an OPA. Making use ofa combination of quadratic Zeeman shift and state de-pendent microwave dressing [23, 24] the spin dynamicsis energetically restricted to a spin 1 subspace definedby the 0 , ± − ↑ ) and idler ( ↓ ) modes. In op-tics the integrated nonlinearity is restricted by short in-teraction times, a limit which is surpassed in the atomicsystem, such that continuous variable entangled stateswith large population can be realized. Since the under-lying process generates pairs of atoms in the signal andidler modes – twin-atoms – the mean and variance ofthe population difference N − = N ↓ − N ↑ should ideallyvanish although the total population N + = N ↓ + N ↑ fluc-tuates strongly. This can be experimentally confirmedby absorption imaging of the atomic cloud [21] wherestate selectivity is achieved by Stern-Gerlach separation(Fig. 1b, c).A quantitative analysis of the variance of the popula-tion difference ∆ N − and the distribution of the popula- tion sum N + gives a first indication of the quantum stateof the system (Fig. 2). Due to the pump mode populationdependent nonlinearity the mean atom number in signaland idler mode (cid:104) N + (cid:105) increases nonlinearly with the totalatom number N , resulting in a growing fraction (cid:104) N + (cid:105) /N .In the small N (cid:46)
300 limit the observed distribution ofthe total population N + matches the prediction for thesqueezed vacuum (taking detection noise into account byconvolution with a gaussian) with maximal squeezing pa-rameter r ≈ (cid:104) N + (cid:105) ≈
25 and the data shows a comparably large stan-dard deviation – in contrast, the noise in the populationdifference N − almost vanishes (∆ N − < . N the pump depletion can-not be neglected any more resulting in a breakdown ofthe analogy to the optical parametric amplification pic-ture and the observed distributions of N + differ from thesqueezed vacuum (insets in Fig. 2) [22]. In this regimethe well controllable spin dynamics offer prospects forthe deterministic generation of non-gaussian entangledtwin-atom states [25]. Density dependent two-body spinrelaxation loss deteriorates the perfect pair correlationssuch that the relative population noise grows with N .
100 300 500 7000100200300400 a t o m nu m be r ( m ean and v a r i an c e ) total atom number c oun t s c oun t s c oun t s FIG. 2.
Population of the signal and idler modes.
The red data shows the mean population (cid:104) N + (cid:105) = (cid:104) N ↑ + N ↓ (cid:105) versus total atom number N . The bars indicate the fluctuations (1-s.d.) of the data. The large uncertainty is a feature of thenonlinear process itself which leads to a non-gaussian distribution of the mode population. The gray area is the sub-poissonregime and its upper bound is obtained by a spline fit to (cid:104) N + (cid:105) . Blue data points show the variance of the population difference∆ N − , which is compatible with zero for small N (bars indicate the 1-s.d. statistical uncertainty). Its increase with N isreproduced when taking particle loss due to spin relaxation into account (black line). The insets show the distribution of themode population for indicated total atom numbers. For small N the distributions match the prediction for a non-depletedpump with the measured mean population (cid:104) N + (cid:105) (black lines) while for larger N they clearly differ. The fitted squeezed vacuumdistribution for 250 < N <
300 corresponds to a squeezing parameter of r ≈ Averaging the results for all total atom numbers we findthe noise ∆ N − suppressed by 6 . (cid:104) N + (cid:105) .The observable populations do not suffice to charac-terize the two-mode state; we must access coherences be-tween the modes, the two-mode quadratures. We em-ploy atomic homodyning by analogy to the experimentaltechnique used in optics, whereby the quadratures aremeasured relative to a reference field. Yet in the atomiccase the population of available local oscillator (LO) ref-erence fields is often limited to rather small atom num-bers. This has important consequences for the observ-able quadratures (cid:101) X k = ( a † k a LO + a k a † LO ) / (cid:113) (cid:104) a † LO a LO (cid:105) ,which in general differ from the canonical ones. A mea-sure for this difference is given by the ratio of the quan-tum mode population to the local oscillator population,by which the Heisenberg uncertainty relation is modified∆ (cid:101) X k ∆ (cid:101) Y k ≥ − (cid:104) N k (cid:105) / (cid:104) a † LO a LO (cid:105) [26].For the twin-atom state one expects peculiar coherenceproperties: The quadratures of the individual modes fluc-tuate strongly around a vanishing mean, while the two-mode quadratures X ± ( ϕ ) show reduced fluctuations forcertain ϕ reflecting the quantum correlations. To re- veal these characteristics in the coherences of the gen-erated quantum fields we employ a homodyning schemesuch that information about the two-mode quadratures (cid:101) X ± ( ϕ ) can be obtained. Measurement of these observ-ables requires comparison of each mode with a local os-cillator. We ensure relative local oscillator phase sta-bility by using the pump mode (2 ,
0) as the simultane-ous reference, whose phase ϕ can be controlled by mi-crowave dressing. An atomic three-port beam splitter isrealized by radio-frequency coupling of signal and idlermodes to the pump. Analysis of the fluctuations in atomnumber sum and difference N (2 , − ± N (2 , between the(2 , −
1) and (2 ,
1) Zeeman states after the mixing pro-vides, after proper normalization, an upper bound (ub)for the variance of the two-mode quadratures ∆ (cid:101) X ub ± ( ϕ ).The measured fluctuations include further noise contri-butions governed by population fluctuations in the signaland idler modes, which are small only for the quadraturedifference. Further details about the analysis in the threemode framework and the choice of the normalizations canbe found in the supplementary information. Figures 3aand b show raw data in the regime of small pump deple-tion of the measured normalized atom number difference b ca k l −40040 no r m . a t o m nu m be r d i ff e r en c e −2020 local oscillator phase no r m . a t o m nu m be r s u m s i t e l s i t e k −101 c o rr e l a t i on s i t e l s i t e k −101 c o rr e l a t i on local oscillator phase FIG. 3.
Atomic homodyning. a,
Homodyne measurement of the quadrature difference. After symmetric coupling to the m F = 0 state (inset in b) the normalized population difference in the m F = ± b, Homodyne measurement of the quadrature sum. An upper bound for the varianceof the two-mode quadrature sum is obtained from the fluctuations of the normalized population sum after the coupling. Thefluctuations of the raw data are strongly phase dependent while the mean (blue solid line) stays constant. The variance minimumis shifted by π as compared to a indicating noise suppression in the orthogonal two-mode quadrature. The data shown in a andb corresponds to all measurements with total numbers 150 < N <
200 and the red lines indicate the inferred standard deviationabove and below the mean after subtraction of the readout noise. c, Classical vs. quantum preparation – inter-site correlationmeasurement. Characterization of classical phase noise is crucial to obtain information about the individual quadratures of thesignal and idler modes. For initial states prepared classically, that is by linear coupling, strong inter-site correlations betweendifferent sites in the optical lattice after mixing with the local oscillator (illustrations) reveal classically washed out coherences(top panel). In contrast, when the ( F = 2 , m F = ±
1) states are populated by spin changing collisions no correlations areobserved within the statistical uncertainty showing that (cid:104) (cid:101) X ↑ ( ↓ ) (cid:105) = 0 as a direct result of the spin dynamics. and sum versus local oscillator phase ϕ . No phase depen-dence is visible in the mean, nevertheless, the fluctuationsare strongly modulated indicating phase correlations be-tween the two atomic modes.We set up a different experiment to obtain the quadra-ture fluctuations of the signal and idler modes individu-ally. As local oscillators we prepare population in the(1 , ∓
1) states before initiating the dynamics. Mixingof the local oscillators with the quantum modes is doneby a two-photon microwave and radio-frequency π cou-pling pulse. For both, the signal and idler mode, we findcoupling phase independent fluctuations of the atomicquadratures (cid:101) X ↑ ( ↓ ) = ( N (2 , ± − N (1 , ∓ ) / (cid:112) (cid:104) N LO (cid:105) and avanishing mean. The mean local oscillator population (cid:104) N LO (cid:105) of the (1 , ∓
1) state was measured by omitting themixing pulse.To assure that our observations are indeed resulting di-rectly from the spin changing collision process the tech-nical phase noise has to be characterized. The 1D op-tical lattice setup allows for this characterization by areference measurement in which the population in the(2 , ±
1) states is also prepared by electromagnetic cou-pling and spin changing collisions are tuned off-resonanceduring the evolution time. The inter-site correlations ob-served after the mixing are shown in figure 3c for thisreference experiment (top) and for the quantum state prepared by spin changing collisions (bottom). In bothcases we observe similarly large on-site fluctuations in theatom number difference. However, in the reference ex-periment we can use the inter-site correlations to reducethese fluctuations by a factor of 20 such that the remain-ing fluctuations are close to the expected noise limit fortwo coherent modes. This shows that inter-site correla-tions detect finite coherences that have been washed outby classical phase noise in our measurements. In caseof spin changing collisions no correlations are observedwithin the statistical uncertainty, meaning that the van-ishing coherences are indeed caused by the process it-self. Given this result, technical phase noise does notinfluence the sum of the orthogonal quadrature variances∆ (cid:101) X ↑ ( ↓ ) + ∆ (cid:101) Y ↑ ( ↓ ) = 2∆ (cid:101) X ↑ ( ↓ ) (the measured quadra-ture variance is LO phase independent), such that it isa useful observable to characterize the quadrature fluc-tuations. For further details on the inter-site correlationmeasurement we refer to the supplementary information.Inter-mode entanglement of the twin-atom state is re-flected in suppressed noise in the two-mode quadraturescompared to the quadrature fluctuations of the individualmodes ∆ (cid:101) X − +∆ (cid:101) Y + < ∆ (cid:101) X ↑ +∆ (cid:101) X ↓ +∆ (cid:101) Y ↑ +∆ (cid:101) Y ↓ [13].When investigating this inequality in our experiment,care has to be taken to assure comparability of the single-and two-mode measurements [26]. In both measurements a bc local oscillator phase v a r i an c e v a r i an c e s and
01 200 400300 500 r a t i o -2614150 250 35021018 FIG. 4.
Two-mode quadrature fluctuations and mode inseparability. a,
Two-mode sum and difference quadratures.The measured variances ∆ (cid:101) X ub+ ( ϕ ) (blue) and ∆ (cid:101) X ub − ( ϕ ) (red) – upper bounds for the sum and difference two-mode quadraturevariances – are plotted versus local oscillator phase for total atom numbers 150 < N < b, Continuous variable quadrature entanglement. Inseparability is detected in the gray shaded region whereΣ∆ , the upper bound for the sum of the two-mode quadrature variance minima (green), is smaller than Σ∆ , the sum of thevariance in the individual quadratures (black). The inset shows the ratio Σ∆ / Σ∆ . c, Minimum of the two-mode differencequadrature fluctuations. For small total atom numbers N – in the non-depleted pump regime and where spin relaxation lossis small – the minimum two-mode quadrature variance ∆ (cid:101) X ub − ( ϕ ) inferred from a local quadratic fit is at the noise limit fortwo coherent fields (dashed black line). The inset shows a zoom into the small N region. All error bars correspond to 1-s.d.uncertainties. the ratio of the sum of the population in the signal andidler modes to the reference mode population is below10% for small N <
200 and grows to approximately 55%for the largest N . In figure 4a we plot the measured up-per bound for the two-mode quadrature variances versuslocal oscillator phase ∆ (cid:101) X ub ± ( ϕ ) for an exemplary totalatom number. The sum of the minima at ϕ and ϕ − π provides an upper bound for the sum of the orthogonaltwo-mode quadrature variances Σ∆ = ∆ (cid:101) X ub − + ∆ (cid:101) Y ub+ ,while the measurement of the single-mode quadraturevariances Σ∆ = 2(∆ (cid:101) X ↑ + ∆ (cid:101) X ↓ ) has been discussedabove. The generated quantum states fulfill the inequal-ity Σ∆ < Σ∆ for a wide range of total atom numbers(Fig. 4b) showing that the produced twin-atom state isinseparable – the observed minimal ratio is Σ∆ / Σ∆ ≈ . (cid:101) X ub − ( ϕ ) = 2. For the moreprecisely detectable variance of the quadrature difference we find the noise minimum for small total N compara-ble to this level, limited by the experimental signal tonoise ratio. In figure 4c we show the dependence of thisnoise minimum on the total atom number N – detectionnoise subtraction is crucial here and without it we find∆ (cid:101) X ub − ( ϕ ) ≈
17 as the minimal value.In conclusion, we have developed an atomic homo-dyne detection method, which allows for the measure-ment of quadrature correlations of twin-atom quan-tum fields. The continuous variable entangled stateshave been produced in a deterministic manner utiliz-ing controlled atomic spin interactions – for small to-tal atom numbers the observed populations distributionsare compatible with the atomic two-mode squeezed vac-uum. Einstein-Podolsky-Rosen (EPR) entanglement isrevealed by ∆ ( X ↑ − X ↓ )∆ ( Y ↑ + Y ↓ ) < ±
17 for this value, showing that EPR-entanglement foratomic quadratures is within reach. In a future extensionof our scheme the local oscillator might be split before themixing, enabling precise detection also of the two-modequadrature sum [26]. Spatial separation of the modes canbe implemented by employing the Stern-Gerlach tech-nique, being an alternative approach to recently reportedexperiments [27, 28]. The regime of large average pairpopulation is accessible such that highly non-classicalquantum states might be generated by controlled quan-tum spin evolution [25].We note that in parallel to this work, sub-poissoniannumber fluctuations after spin changing collisions havebeen observed by two other groups [29, 30] and it hasbeen shown that the generated quantum correlated statesare useful for noise interferometry beyond the standardquantum limit [30].
Supplementary InformationExperimental sequence
We routinely produce Bose-Einstein condensates(BEC) of Rubidium 87 in the low field seeking (
F, m F ) =(1 , −
1) hyperfine state with an experimental cycle timeof approximately 40 s. Before the final evaporation rampin an optical trap we turn up a 1-dimensional optical lat-tice slicing the atomic cloud into eight pieces. Since thelattice potential does not allow for tunneling between dif-ferent sites on the experimental timescale, we start witheight independent samples with different mean total atomnumber N per site on which the spin dynamics experi-ment is done in parallel.First we transfer the atoms to the (1 ,
0) state by a radiofrequency (rf) π pulse. A homogeneous magnetic off-set field of approximately 9 G is applied such that thesecond-order Zeeman shift is sufficient to resolve the dif-ferent rf transitions in the F = 1 hyperfine multiplet.Afterwards we remove spurious population in the (1 , ± B ≈ . , −
1) and the (2 ,
1) states, as well as powerbroadened symmetric rf coupling between the (2 ,
0) andthe (2 , ±
1) states are possible. For the homodyning ofsingle-mode quadratures local oscillators in (1 , ±
1) arenecessary, which we prepare by transferring a part of thepopulation via symmetric rf coupling. The next step is amicrowave π pulse from (1 ,
0) to (2 , δ = 162 Hz)with microwave dressing of the (2 ,
0) level [23, 24]. Thedressing field is 98 kHz blue detuned to the (1 , ↔ (2 , m F = 0 , ± m F = ± π/ , ∓ ↔ (2 , ±
1) transitions, where changing the phaseof these pulses is equivalent to a change of the local os-cillator phase.In order to realize the three-port beamsplitter scheme,i.e. to measure the two-mode quadratures after the spinevolution, we apply a single rf field for 60 µ s, which cou-ples the (2 ,
0) and (2 , ±
1) states. This corresponds to a π/ ϕ (of the m F = 0 state) a b -2-1012-101 100 μm-2-1012-101 100 μm FIG. 5.
Experimental raw data.
Absorption pictureof atoms in the F = 1 ( a ) and F = 2 ( b ) multiplet. Thecounting boxes for regions with atomic signal are shown inred, the boxes for technical noise measurement in green. Thenumbers indicate the Zeeman sub state, which is the same ineach row. Due to the opposite sign of the magnetic momentsthese numbers are inverted between F = 1 and F = 2. Thesize of the image in real space is given. Note that the cloudsfor F = 1 atoms are more extended in vertical direction dueto the imaging sequence [21]. we switch off the dressing field for a variable time t (be-tween 0 ms and 4 . ϕ = 2 πδt . In the case of the measurement of popu-lation difference N − no coupling is applied after the spinchanging collisions.After the experimental sequences described above thepopulation in the different Zeeman states is measured.State and site selective imaging is achieved by a combi-nation of the Stern-Gerlach technique and a high spatialresolution imaging system [21, 31]. The Stern-Gerlachpulse, which separates the m F states, is aligned carefullywith the magnetic offset field such that the following ab-sorption imaging is a projective measurement of the pop-ulation in the Zeeman states m F defined with the quan-tization axis along the direction of the offset field. Data analysis
Imaging is done in the high intensity regime and thecolumn density (atom number per pixel) is calculatedfrom the raw absorption data following the recipe in [32].Care is taken to assure a linear and well calibrated imag-ing system [21, 31]. To obtain the atom number per stateand lattice site we define counting boxes in which thecolumn density is integrated. Figure 5 shows a typicalsingle run picture with counting boxes indicated. Twotypes of technical noise add to the atomic signal in theimaging process. The first is unavoidable photon shotnoise ( √ ∆ N psn ≈ . √ ∆ N fr ≈ . N tech − ≈ N (cid:38)
450 where (cid:104) N + (cid:105) > ∆ N tech − (seefigure 2 of the main text). Here it is important to notethat the uncertainties on the experimental data given inthis manuscript include the statistical uncertainty due tothe noise subtraction. As a test for our noise calibra-tion we analyzed pictures without atoms in the countingboxes and found possible systematic errors smaller thanthe statistical uncertainties.The dependence of all presented observables on the to-tal atom number N is obtained by binning the data inintervals of size δN = 50. Inter-site correlation analysis
The use of a single site resolved optical lattice in ourexperiments has various advantages. Two of them arerather obvious, the boost in statistics due to the parallelrealization of eight experiments and the increased localconfinement which is important for the validity of the sin-gle mode approximation. Here we point out that inter-site correlations between the observables can be used toextract phase noise contributions stemming from envi-ronmental fluctuations.At a finite magnetic field B the energy of the differentZeeman states shifts differentially with B which leads tophase noise in Ramsey type experiments [21], in whichthe integrated phase difference is mapped onto an observ-able population difference. The homodyning experimentsreported in this manuscript are of similar type and theobserved atom number differences are in principle sensi-tive to environmentally induced phase noise. Magneticfield or microwave phase fluctuations in our experimentare mainly low frequency and homogeneous over a spatialregion much bigger than the size of the entire lattice sys-tem. Therefore they result in shot-to-shot fluctuationswhich are correlated between different lattice sites k . a b s i t e l s i t e k s i t e l s i t e k c o rr e l a t i on FIG. 6.
Experimentally observed inter-site correlations. a,
Initially coherent state. Strong inter-site correlations C kl are detected for an initially coherent state, which are caused by spatially large scale technical fluctuations resulting insite correlated phase-noise. b, Spin-changing collisions experiment. After spin dynamics a random phase is observed and theinter-site correlations vanish within the statistical uncertainty C kl ≈
0. The striking difference in the correlation signal betweenthe two measurements shows that environmental noise can neither explain the random phase of the individual modes nor thecorrelations between signal and idler modes observed after spin-changing collisions.
Figure 6a shows the measured inter-site correlations for areference homodyning experiment with the local oscilla-tor in the (1 , −
1) Zeeman state and a coherently preparedstate in the (2 ,
1) Zeeman state. The inter-site correla-tions C kl are calculated for the observed atom numberdifferences n k = N ,k − N ,k (1 and 2 label the two Zee-man states involved) C kl = (cid:104) (cid:101) n k (cid:101) n l (cid:105) ∆ n k ∆ n l (1)where (cid:101) n k = n k − (cid:104) n k (cid:105) is the atom number differencein site k corrected for its mean and ∆ n k the standarddeviation of n k . The experimental sequence is similar tothe spin-changing collision evolution described above butspin dynamics was suppressed by tuning the process offresonance. Since the relative phases are in this case ini-tially defined by the coherent preparation (correspondingto a non-vanishing mean of the single-mode quadrature),finite inter-site correlations | C kl | > N k − N l which is due to the N k depen-dent mean field shift. In our experiment N k fluctuateslittle from shot to shot. Thus, this population dependentenergy offset results in a fixed difference in the integratedrelative phases in different sites which causes correlationsor anti-correlations depending on its magnitude. Usingthe inter-site correlations to remove these technically in-duced phase fluctuations we reduce the observed fluctua-tions ∆ n k (cid:104) N ,k + N ,k (cid:105) from approximately 50 to 2 . C kl ≈
0. In figure 6b we plot theinter-site correlations for the signal mode homodyningexperiment. Here the (2 ,
1) Zeeman state was populated by spin evolution. No statistically significant correlationsare observed, excluding that the phase is randomized byenvironmental noise. We also tested for inter-site corre-lations in the signal-idler mode entanglement measure-ments (Fig. 3a, b and 4 of the main text) and found avanishing signal C kl ≈ Two-mode quadratures
The two-mode quadratures of signal ( a ↑ ) and idler ( a ↓ )modes are measured by simultaneous symmetric radio-frequency coupling to the pump mode a which serves asthe local oscillator. This process is described by a uni-tary rotation U cpl = e − iH cpl t/ (cid:126) in the three mode systemgenerated by the Hamiltonian H cpl = (cid:126) Ω2 √ a a †↓ + a † a ↑ + a † a ↓ + a a †↑ ) (2)where 2 π (cid:126) is Planck’s constant.Experimentally the coupling is switched on and offnon-adiabatically with a total pulse duration τ cpl . Therotation (mixing) angle of the pulse is given by γ = Ω τ cpl ,which is defined such that a γ = π pulse transfers all pop-ulation from the m F = 0 state to the m F = ± γ ≈ π/
5, since we have tomake a tradeoff between signal-to-noise ratio (optimal for γ = π ) and an unwanted spurious coupling between the m F = ± m F = ± γ .The unitary transformation generated by the couplingHamiltonian (2) is described by the matrix U = c γ +12 − i s γ √ γ − − i s γ √ c γ − i s γ √ γ − − i s γ √ γ +12 (3)with s γ = sin( γ ), c γ = cos( γ ). The mode oper-ators ( a (cid:48)↓ , a (cid:48) , a (cid:48)↑ ) T in the Heisenberg-picture after thecoupling pulse are obtained by applying the matrix U to ( a ↓ , a , a ↑ ) T . The phase shift ϕ of the local oscil-lator before the pulse is described by the substitution a → ˜ a ( ϕ ) = a e iϕ . For the sake of clarity this abbrevi-ation is used in the following without the explicit phasedependence. Two-mode quadrature difference
For the experimentally measured population difference N (cid:48)− = a (cid:48)†↓ a (cid:48)↓ − a (cid:48)†↑ a (cid:48)↑ in the m F = ± N (cid:48)− ( ϕ ) = c γ N − + i s γ √ a ↓ ˜ a † + a †↑ ˜ a − a †↓ ˜ a − a ↑ ˜ a † ) (4)The second term is, up to a π phase shift of the a mode,proportional to the generalized two-mode quadratures (cid:101) X − ( ϕ ) = ( a ↓ ˜ a † − a †↑ ˜ a + h . c . ) / (cid:113) (cid:104) ˜ a † ˜ a (cid:105) = (cid:101) X ↓ ( ϕ ) − (cid:101) X ↑ ( ϕ ) (5)In the limit of a large local oscillator (˜ a ≈ (cid:112) (cid:104) N (cid:105) e iϕ )the generalized two-mode quadratures correspond to thecanonical ones [26]. With the normalization factor (cid:104) N (cid:105) s γ / (cid:104) N − (cid:105) = 0 and t γ = tan( γ ) weobtain the measured upper bound∆ (cid:101) X ub − ( ϕ ) = ∆ N (cid:48)− ( ϕ ) / ( (cid:104) N (cid:105) s γ /
2) (6)= ∆ (cid:101) X − ( ϕ )+ 2t γ (cid:104) N (cid:105) ∆ N − + √ γ (cid:112) (cid:104) N (cid:105) (cid:104) N − (cid:101) X − ( ϕ ) + (cid:101) X − ( ϕ ) N − (cid:105) The symbol ∆ indicates the variance of the subse-quent variable. The local oscillator phase independentvariance offset (second term) which is proportionalto the variance of the population difference ∆ N − issmall (approximately 0 .
1) for small total atom numbers(150 < N < . < N < . (cid:104) N − (cid:101) X − (cid:105) ≈ Two-mode quadrature sum
The second important two-mode quadrature (cid:101) X + ( ϕ ) = ( a ↓ ˜ a † + a †↑ ˜ a + h . c . ) / (cid:113) (cid:104) ˜ a † ˜ a (cid:105) = (cid:101) X ↓ ( ϕ ) + (cid:101) X ↑ ( ϕ ) , (7)i.e., the sum of the two single mode quadratures, can bemeasured less precise in the current scheme, however, anupper bound for its variance can be obtained. For thesum of the population in the m F = ± N (cid:48) + = a (cid:48)†↓ a (cid:48)↓ + a (cid:48)†↑ a (cid:48)↑ we obtain N (cid:48) + = c γ +12 N + + s γ N + i c γ s γ √ ( a ↓ ˜ a † − a †↑ ˜ a − a †↓ ˜ a + a ↑ ˜ a † )+ c γ − ( a †↓ a ↑ + a †↑ a ↓ ) (8)The third term i c γ s γ √ ( a ↓ ˜ a † − a †↑ ˜ a − a †↓ ˜ a + a ↑ ˜ a † ) is pro-portional to the two-mode quadrature (cid:101) X + ( ϕ + π ), whichis the only local oscillator phase dependent term. Covari-ance terms between this one and the three other termsin equation (8) vanish due to their strong sensitivity tomagnetic field fluctuations (see above). Hence, one ob-tains the upper bound for the two-mode quadrature sum∆ (cid:101) X ub+ ( ϕ ) = ∆ N (cid:48) + ( ϕ ) / ( (cid:104) N (cid:105) c γ s γ / (cid:101) X + ( ϕ ) + const . (9)where the constant offset due to the variance of the phaseindependent terms in equation (8) limits the obtainableprecision when deducing ∆ (cid:101) X + ( ϕ ) ≤ ∆ N (cid:48) + ( ϕ ) from ourmeasurements. Note that the normalisations for (cid:101) X + and (cid:101) X − are different. A future extension of this scheme tomeasure also the sum of the single mode quadratures withhigh precision and therefore both quadrature Einstein-Podolsky-Rosen variables [7] would require the splittingof the local oscillator prior to the coupling, such that thephase of the two coupling fields can be tuned indepen-dently [33].We acknowledge enlightening and clarifying discus-sions with P. Grangier, A. Aspect, A.J. Ferris, M.J. Davisand B.C. Sanders. This work was supported by theForschergruppe FOR760, Deutsche Forschungsgemein-schaft, the German-Israeli Foundation, the HeidelbergCenter for Quantum Dynamics, Landesstiftung Baden-W¨urttemberg, the ExtreMe Matter Institute and the Eu-ropean Commission Future and Emerging TechnologiesOpen Scheme project MIDAS (Macroscopic InterferenceDevices for Atomic and Solid-State Systems). G.K. ac-knowledges support from the Humboldt-Meitner Awardand the Deutsche-Israelische Projektgruppe (DIP).Correspondence should be addressed to M.K.O. ([email protected]).0 ∗ present address: Harvard-Smithsonian CfA, HarvardUniversity Dept. of Physics, Cambridge, MA 02138, USA[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] V. Giovannetti, S. Lloyd, and L. Maccone, Science ,1330 (2004).[3] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. ,513 (2005).[4] S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. ,869 (1998).[5] T. Opatrn´y and G. Kurizki, Phys. Rev. Lett. , 3180(2001).[6] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev.Mod. Phys. , 1041 (2010).[7] M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cav-alcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, andG. Leuchs, Rev. Mod. Phys. , 1727 (2009).[8] Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng,Phys. Rev. Lett. , 3663 (1992).[9] M. D. Reid, Phys. Rev. A , 913 (1989).[10] D. Walls and G. Milburn, Quantum optics (Springer Ver-lag, 2008).[11] L.-M. Duan, A. Sørensen, J. I. Cirac, and P. Zoller, Phys.Rev. Lett. , 3991 (2000).[12] H. Pu and P. Meystre, Phys. Rev. Lett. , 3987 (2000).[13] M. G. Raymer, A. C. Funk, B. C. Sanders, andH. de Guise, Phys. Rev. A , 052104 (2003).[14] B. Julsgaard, A. Kozhekin, and E. S. Polzik, Nature ,400 (2001).[15] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[16] S. R. Leslie, J. Guzman, M. Vengalattore, J. D. Sau,M. L. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A , 043631 (2009).[17] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Hen-ninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt,Phys. Rev. Lett. , 195303 (2010). [18] M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer,K. M. Fortier, W. Zhang, L. You, and M. S. Chapman,Phys. Rev. Lett. , 140403 (2004).[19] H. Schmaljohann, M. Erhard, J. Kronj¨ager, M. Kottke,S. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, andK. Sengstock, Phys. Rev. Lett. , 040402 (2004).[20] C. M. Caves and B. L. Schumaker, Phys. Rev. A , 3068(1985).[21] C. Gross, T. Zibold, E. Nicklas, J. Est`eve, and M. K.Oberthaler, Nature , 1165 (2010).[22] C. Law, H. Pu, and N. Bigelow, Phys. Rev. Lett. ,5257 (1998).[23] J. Kronjaeger, C. Becker, P. Navez, K. Bongs, andK. Sengstock, Phys. Rev. Lett. , 110404 (2006).[24] F. Gerbier, A. Widera, S. F¨olling, O. Mandel, andI. Bloch, Phys. Rev. A , 041602 (2006).[25] R. B. Diener and T. Ho, ArXiv e-prints (2006),arXiv:cond-mat/0608732v1.[26] A. J. Ferris, M. K. Olsen, E. G. Cavalcanti, and M. J.Davis, Phys. Rev. A , 060104 (2008).[27] J.-C. Jaskula, M. Bonneau, G. B. Partridge, V. Krach-malnicoff, P. Deuar, K. V. Kheruntsyan, A. Aspect,D. Boiron, and C. I. Westbrook, Phys. Rev. Lett. ,190402 (2010).[28] R. Bucker, J. Grond, S. Manz, T. Berrada, T. Betz,C. Koller, U. Hohenester, T. Schumm, A. Perrin, andJ. Schmiedmayer, Nat. Phys. , 608 (2011).[29] E. M. Bookjans, C. D. Hamley, and M. S. Chapman,Phys. Rev. Lett. , 210406 (2011).[30] B. L¨ucke, M. Scherer, J. Kruse, L. Pezz´e, F. Deuret-zbacher, P. Hyllus, O. Topic, J. Peise, W. Ertmer, J. Arlt,L. Santos, A. Smerzi, and C. Klempt, Science , 773(2011).[31] J. Est`eve, C. Gross, A. Weller, S. Giovanazzi, and M. K.Oberthaler, Nature , 1216 (2008).[32] G. Reinaudi, T. Lahaye, Z. Wang, and D. Gu´ery-Odelin,Opt. Lett. , 3143 (2007).[33] A. J. Ferris, M. K. Olsen, and M. J. Davis, Phys. Rev.A79