Einstein-Podolsky-Rosen Correlations of Ultracold Atomic Gases
Nir Bar-Gill, Christian Gross, Gershon Kurizki, Igor Mazets, Markus Oberthaler
aa r X i v : . [ qu a n t - ph ] S e p Einstein-Podolsky-Rosen Correlations of Ultracold Atomic Gases
Nir Bar-Gill , Christian Gross , Gershon Kurizki , Igor Mazets , and Markus Oberthaler Weizmann Institute of Science, Rehovot, Israel. Kirchhoff-Institut f¨ur Physik, Universit¨at Heidelberg, 69120 Heidelberg, Germany. and Atominstitut ¨osterreichischer Universit¨aten, TU Wien, Vienna, Austria.
Einstein, Podolsky & Rosen (EPR) pointed out [1] that correlations induced between quantumobjects will persist after these objects have ceased to interact. Consequently, their joint continu-ous variables (CV), e.g., the difference of their positions and the sum of their momenta, may bespecified, regardless of their distance, with arbitrary precision. EPR correlations give rise to twofundamental notions[2–6]: nonlocal “steering” of the quantum state of one object by measuring theother, and inseparability ( entanglement ) of their quantum states. EPR entanglement is a resource ofquantum information (QI)[6–8] and CV teleportation of light[9, 10] and matter waves[11, 12]. It haslately been demonstrated for collective
CV of distant thermal-gas clouds, correlated by interactionwith a common field [13, 14]. Here we demonstrate that collective CV of two species of trappedultracold bosonic gases can be EPR-correlated (entangled) via inherent interactions between thespecies. This paves the way to further QI applications of such systems, which are atomic analogs ofcoupled superconducting Josephson Junctions (JJ)[15, 16]. A precursor of this study has been theobservation of quantum correlations (squeezing) in a single bosonic JJ [17].
EPR criteria –
In studying continuous variable en-tanglement (CVE), it is instructive to draw an analogywith the original EPR scenario [1], wherein two particles,1 and 2, are defined through their position and momen-tum variables x , , p , . EPR saw as paradox the factthat depending on whether we measure x or p of par-ticle 1, one can predict the measurement result of x or p , respectively, with arbitrary precision, unlimited bythe Heisenberg relation ∆ x ∆ p ≤ / h = 1).This nonlocal dependence of the measurement results ofparticle 2 on those of particle 1 has been dubbed “steer-ing” by Schr¨odinger[18]. Equivalently, the EPR state isdeemed entangled in the continuous translational vari-ables of the two particles. The entanglement is exhibitedby the collective operators ˆ x ± = ˆ x ± ˆ x and ˆ p ± = ˆ p ± ˆ p .In quantum optics these variables are associated with thesum and difference of field quadratures of two light modesmixed by a symmetric beam splitter [6, 19] (Fig. 1a).In order to quantify the EPR correlations, one mayadopt two distinct criteria. The first criterion imposesan upper bound on the product of the variances of EPR-correlated commuting dimensionless operators, ˆ x + andˆ p − or ˆ x − and ˆ p + [7, 11]: h ∆ˆ x ± ih ∆ˆ p ∓ i ≡ s ≤ , (1)The EPR correlation is then measured by the two-modesqueezing factor ∞ > s >
1. The second is the insepara-bility (entanglement) criterion for gaussian states [8, 14],related to the sum of the variances of the correlated ob-servables ǫ ≡ h ∆ˆ x ± i + h ∆ˆ p ∓ i − <
0. Here the max-imal entanglement corresponds to the most negative ǫ obtainable. In what follows we inquire: to what extentdo these EPR criteria apply to the system at hand, i.e.,a two-species BEC in a symmetric double-well potential? Scheme for global-mode EPR correlations in bosonicJJs –
We first consider the correlation dynamics of thetwo species (two internal states of the atom), in the pres- ence of tunnel coupling between the wells. We shall an-alyze the EPR correlations in the basis of two globalinternal-state modes that are not spatially separated be-tween the two wells.Since there is no population exchange between the in-ternal states | A i and | B i , the numbers of atoms N A and N B in these states are constants of motion. TheHamiltonian (Supplement) can be then written in thisbasis in terms of the left-right atom-number differencesin the two internal states, ˆ n A = (cid:16) ˆ a † L ˆ a L − ˆ a † R ˆ a R (cid:17) / n B = (cid:16) ˆ b † L ˆ b L − ˆ b † R ˆ b R (cid:17) /
2, and their canonically conjugatephase operators ˆ φ A,B , obeying the commutation relations h ˆ φ α , ˆ n α ′ i = iδ αα ′ ( α, α ′ = A, B ). For simplicity we as-sume from now on that N A = N B ≡ N (generalizationto N A = N B is straightforward), and consider small in-terwell number differences such that h ˆ n A,B i << N . TheHamiltonian[20] then becomes H = ( E c ) AA ˆ n A + ( E c ) BB ˆ n B + 2( E c ) AB ˆ n A ˆ n B − JN (cid:16) cos ˆ φ A + cos ˆ φ B (cid:17) + 2 JN (cid:16) ˆ n A cos ˆ φ A + ˆ n B cos ˆ φ B (cid:17) . (2)Here the nonlinearity coefficients (“charging” energies)( E c ) AA , ( E c ) BB and ( E c ) AB are determined respectivelyby the intra- and inter-species s-wave scattering lengths.The tunneling energy J is the same for the atoms in theinternal states | A i and | B i .Equation (2) displays the full dynamics used in our nu-merics (Fig. 1), that of two quantum nonlinear pendulacoupled via 2( E c ) AB ˆ n A ˆ n B . This coupling is the key toEPR correlations of modes A and B .We may, for didactic purposes, simplify (2) by expand-ing the cosine terms. In the lowest-order approxima-tion cos ˆ φ A,B ≃ − ˆ φ A,B /
2, the system is described bytwo coupled harmonic oscillators. This suggests that thesystem under study can indeed satisfy the entanglementor two-mode squeezing criteria, if the relevant collectivevariables in our system are mapped onto those of twofield modes mixed by a symmetric beam splitterˆ n ± = 1 √ n A ± ˆ n B ) ↔ ˆ x ± , ˆ φ ± = 1 √ (cid:16) ˆ φ A ± ˆ φ B (cid:17) ↔ ˆ p ± . (3)Using the collective variables defined in (3), we canrewrite Eq. (2) in the harmonic approximation, assuming( E c ) AA ≃ ( E c ) BB = E c , as:ˆ H = (cid:18) E c + ( E c ) AB + 2 JN (cid:19) ˆ n + JN φ + (cid:18) E c − ( E c ) AB + 2 JN (cid:19) ˆ n − + JN φ − . (4)Hence, the transformed Hamiltonian describes two un-coupled harmonic modes in the collective basis. The “+”-mode corresponds to Josephson oscillations of the totalatomic population (regardless of the internal state) be-tween the two wells, such that the inter-species ratio ineach well is constant (in-phase oscillations of the A, B species). The “-”-mode corresponds to oscillations of theinter-species ratio between the two wells, such that thetotal population imbalance does not change (out-of-phaseoscillations of the 1 , ω ± (see Supplement).We may then wonder: do the EPR correlation cri-teria hold in the uncoupled ( ± modes) basis? Indeed,they do: for ( E c ) AB > ± modes to satisfy the EPR criteria, yielding s = [(2 J/N + E c + ( E c ) AB )] / [(2 J/N + E c − ( E c ) AB )]. We then obtain s >> E c ≃ ( E c ) AB >> J/N and the ground statesof both modes, approaching the ideal
EPR limit s → ∞ of full CV entanglement. Thus, the fact that there is cou-pling between the original ( A and B ) modes suffices tocreate EPR correlations between the collective ± modes,although there is no coupling in the latter basis.Beyond the lowest-order approximation that has led to(4), there is parametric coupling of the collective modesthat may induce nontrivial dynamics of CV wavepackets:the slow, − , mode can be “frozen” at a low-temperaturestate, while the fast, +, mode may be kept at its groundstate, conforming to the Born-Oppenheimer couplingregime (see Supplement). The occupations of thermallyexcited + mode states must be low compared to itsground state, in order to satisfy the EPR criteria (seeSupplement).For exact calculation of the dynamics we must resortto the angular momentum operators that describe thefull system (Supplement). The entanglement criterion isthen[14] 1 |h ˆ L x i| (cid:16) h ∆ ˆ L y ± ih ∆ ˆ L z ∓ i (cid:17) ≡ s < . (5)This entanglement criterion differs from those used forthe number-phase operators only for significant nonlin-ear phase diffusion, which reduces |h ˆ L x i| compared to 1 and thus diminishes the ideal limit of s . Since such phasediffusion occurs due to the interatomic (nonlinear) inter-action, which is also responsible for the entanglement,one needs to find the optimal charging energies in (2)and state-preparation that would yield the largest EPRcorrelations (see Methods).The optimal sudden sequence for state preparationconsists of (Fig. 1a): (a) filling the original trap by aBEC in internal state | A i ; (b) sudden ramping up ofthe inter-well potential barrier, thus creating a two-wellsymmetric superposition; (c) transforming state | A i intoa symmetric superposition of | A i and | B i by a fast π/ coherent state in thetwo original modes A , B , whose EPR entanglement thenbuilds up with time according to their coupled-penduladynamics (Eq. (2)). By contrast, slower ramping up ofthe barrier causes them to be exposed to both nonlinearphase diffusion and environment-induced dephasing (seebelow) much longer, thus spoiling the entanglement crite-rion (5) (Fig. 1(b),(c)). We find an optimal value for thecharging energy which results in the largest amount ofEPR correlations, closest to the ideal inseparability (ob-tainable in the absence of nonlinear phase diffusion andfor the ground-state of the coupled two-mode system).We note that it is not advantageous in this scheme tocreate a single-mode squeezed state in each well as an ini-tial condition. Intuitively, this is due to the fact that suchsqueezing does not translate into correlations between thewells, and thus does not induce reduced variances of thetwo-mode coordinates. In more detail, an initial coher-ent state provides minimal non-correlated variances inthe combined variables h ∆ ˆ L y ± ih ∆ ˆ L z ± i / (cid:12)(cid:12)(cid:12) h ˆ L x i (cid:12)(cid:12)(cid:12) = 1 / Scheme for local-mode correlations and “steering” inbosonic JJs –
We now present an approach based oncorrelations of two squeezed local (left- and right- well)modes (Fig. 2(a). The system is initialized in the left well(L) of a double-well potential, in a single internal state(1). Then, the barrier is suddenly dropped in order tocreate a coherent superposition of the vibrational ground-state | g i and first excited state | e i of the new potential.Next, a π/ A, B of the atoms.We assume that there is no exchange term, since tolowest order the cross-coupling terms cancel, and there-fore the number of particles in each external state is con-served. This conservation allows us to rewrite the Hamil-tonian in terms of the internal-state number differenceoperator in each vibrational state, ˆ n g = (ˆ n g ) A − (ˆ n g ) B and ˆ n e = (ˆ n e ) A − (ˆ n e ) B . The Hamiltonian then becomes time [ms] s ( s qu ee z i ng f ac t o r ) (E c ) AB =E c (E c ) AB =0(E c ) AB =E c (adiabatic: t ramp =20ms)0 50 100 150 20000.511.52 time [ms] ε ( I n se p a r a b ili t y ) (E c ) AB =E c (E c ) AB =0(E c ) AB =E c (adiabtic: t ramp =20ms) 0 0.5 1 1.5 2x 10 −3 −0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10 Ec [Hz] I n se p a r a b ili t y c r i t e r i on Maximal EntanglementSeparability Limit
FIG. 1: (a) State preparation scheme: the condensate is splitboth in real-space and in the internal-state basis, to createtwo-coupled modes. Then entanglement dynamics take placeas a function of time, and the measurement is done in thecollective beam-splitter basis. (b)-(c) Dynamics of the entan-glement defined by EPR criteria for both sudden and slowintermode coupling barrier ramping-up, found through ex-act simulation of Eq. (1) in the Supplement. In each fig-ure we plot the case of a coupled system ( E c = E c , solidblue), an uncoupled system ( E c = 0, solid green), slowramp-up (solid black) and the classical limit (solid red). Thetradeoff between entanglement and nonlinear phase diffusionis better for the sudden coupling. (d) Maximal amount ofentanglement (blue dash-dotted line) reached through thedynamics, measured by the inseparability criterion in Eq.(5). We plot the maximal inseparability as a function ofthe charging energy, assuming that all coefficients are equal( E c ) AA = ( E c ) BB = ( E c ) AB ≡ E c [21, 22]. It can be seenthat for small charging energy E c the entanglement grows,but stronger interactions cause significant nonlinear phasediffusion, and therefore reduce the EPR correlations. Fromthe competition between charging-induced entanglement andcharging-induced nonlinear phase diffusion, we find the charg-ing energy which gives maximal entanglement (see Methods).Red-dashed line indicates the separability limit, green-solidline indicates maximal inseparability for the ideal case (seetext). (see Supplement):ˆ H = (( E c ) AA + ( E c ) BB − E c ) AB ) (cid:0) ˆ n g + ˆ n e (cid:1) . (6)Thus, the system evolves separately in the two vibrationalmodes, each undergoing dynamical single-mode squeez-ing in the internal-state basis[23, 24]. Such internal-statesqueezing in each mode was demonstrated recently [17].Following an evolution during which each vibrationalmode separately experiences internal-state squeezing, weraise the barrier quickly to create two separate symmet-ric wells, denoted L (left) and R (right). This suddenprojection creates a BS-like transformation: | L i = 1 √ | g i + | e i ) , | R i = 1 √ | g i − | e i ) . (7)Therefore, we now have two-mode squeezing, or EPR-like entanglement, between the left and right wells. The (a) time [ms] l og ( s ) − s i ng l e − m od e s qu ee z i ng (b)
50 100 150 200−0.0500.050.10.150.20.250.3 time [ms] ∆ n + ( T ) − ∆ n + ( ) T=20nKT=30nK (c) time [ms] ∆ φ − ( T ) − ∆ φ − ( ) T=30nKT=20nK (d)
FIG. 2: (a) Schematic sequence for the creation of “non-local”two-mode entanglement in analogy with the BS approach. (b)Single-mode squeezing dynamics as a function of time, for N = 100. Decoherence effects: the variance of n + (c) andof φ − (d) as a function of time, in the presence of properdephasing. We subtract the variance of the Hermitian dy-namics (without dephasing) in order to single-out the effectof dephasing on the dynamics of the variances. The coherencetime is here estimated to be ∼
100 ms at 20 nK. mode in each well is defined by the number and phase dif-ferences of the internal states. Local measurements maybe done in the internal-state basis in each well separately ,exhibiting non-classical correlations between the | L i and | R i spatially separated modes, in the spirit of “steering”.The scheme presented above is analogous to the quan-tum optics approach[6], in which two independent single-mode squeezed states are injected into the input ports ofa beam splitter (BS), thereby creating entangled modesat the output ports of the BS. However, the intrinsic nonlinearity of each BEC mode causes their unwarrantedmixing even before the BS-like transformation, causing fi-delity loss (see Methods).In this sequence we wait for the maximal single-modesqueezing to develop separately, before raising the barrierto project the | g i and | e i states onto the | L i and | R i basis. Therefore, we can immediately use the maximalsqueezing factor s calculated for each single-mode[6], toextract the two-mode squeezing parameter. Then thecollective two-mode squeezing is given by h ∆ˆ n i = h ∆ (ˆ n L + ˆ n R ) i = (cid:28) ∆ (cid:16) n (0)+ (cid:17) (cid:29) s , h ∆ ˆ φ − i = h ∆ (cid:16) ˆ φ L − ˆ φ R (cid:17) i = (cid:28) ∆ (cid:16) φ (0) − (cid:17) (cid:29) s , (8)namely, the two-mode squeezing parameter is equal tothat of single-mode squeezing. This squeezing parame-ter now characterizes the knowledge obtained about vari-ables in one well having measured their counterparts inthe other well. Decoherence effects –
We now turn to the effect ofenvironment-induced decoherence on the robustness ofEPR entanglement in this system. We assume properdephasing created by independently fluctuating (stochas-tic) energy shifts of atoms in each internal state and well1(2) l ( R ), caused by the thermal atomic or electromag-netic environment. Due to the spectroscopic similarity ofthe two BEC species, we reduce the number of indepen-dent stochastic processes by setting ǫ L /ǫ L = ǫ R /ǫ R =(1 − ξ ) / (1 + ξ ), and assuming a ”symmetrized environ-ment” , i.e. ξ <<
1. Due to the small value of ξ , the vari-ance of ˆ φ − almost does not change (in either the global orlocal scheme), while the variance of ˆ n + increases linearly,and is responsible for the growing loss of entanglement.Hence, we may manipulate the system as we see fit within the coherence time. Discussion –
We have addressed EPR effects in anultracold-atom analog of two coupled
Josephson junctions(JJs): a two-species
Bose-Einstein condensate (BEC),each species corresponding to a different sublevel of theatomic internal ground state [25], trapped in a tunnel-coupled double-well potential (Fig. 1a). We have shownthat such bosonic coupled JJs can induce EPR entan-glement of appropriate combinations of collective contin-uous (phase and atom-number) variables. This entan-glement has been shown to be resilient to environmen-tal noise (decoherence). It exhibits intriguing dynam-ics under conditions analogous to the molecular
Born-Oppenheimer regime for coupled slow and fast variables[26]. Alternatively, it can dynamically realize beam-splitter mixing of two squeezed modes.We acknowledge the support of GIF, DIP and EC (MI-DAS STREP, FET Open), and the Humboldt Founda-tion (G.K.). [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] A. Mann and M. Revzen, eds., The dilemma of Einstein,Podolsky and Rosen - 60 years after (IOP Publishing,1996).[3] J. S. Bell,