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week ending PRL 101, 260503 (2008) PHYSICAL REVIEW LETTERS 31 DECEMBER 2008

Bell Test for the Free of Material Particles

Clemens Gneiting and Klaus Hornberger Center for Theoretical , Ludwig-Maximilians-Universita¨tMu¨nchen, Theresienstraße 37, 80333 Munich, Germany (Received 9 July 2008; published 24 December 2008) We present a scheme to establish nonclassical correlations in the motion of two macroscopically separated massive particles without resorting to entanglement in their internal degrees of freedom. It is based on the dissociation of a diatomic molecule with two temporally separated Feshbach pulses generating a motional state of two counterpropagating atoms that is capable of violating a Bell inequality by means of correlated single-particle interferometry. We evaluate the influence of dispersion on the Bell correlation, showing it to be important but manageable in a proposed experimental setup. The latter employs Bose-Einstein condensation of fermionic lithium atoms, uses laser-guided atom interferometry, and seems to be within the reach of present-day technology.

DOI: 10.1103/PhysRevLett.101.260503 PACS numbers: 03.67.Bg, 03.65.Ud, 03.75.Gg, 37.25.+k

Introduction.—The possibility of entangling macro- Here we aim instead at producing a motional analogue scopically separate, noninteracting particles challenges of the ji and implementing a which our classical view of the world by putting into question requires a simple position measurement in the end. The the concepts of realism and locality [1]. Nowadays en- idea is to expose a diatomic molecule to a sequence of two tangled states are routinely established with [2]. temporally separated dissociation pulses. Each of the two Also the internal states of material particles have been counterpropagating dissociated atoms then has an early entangled, e.g., using nonclassical light as a carrier for and a late wave packet component corresponding to the the correlations [3–7], or their Coulomb interac- two possible dissociation times, and their correlation may tion in an ion trap [8,9]. However, the original discussion of be called ‘‘dissociation time entanglement’’ (DTE), see entanglement focused on the motional state of massive Fig. 1(a). If one regards these components as state particles, whose spatial separation is a dynamic feature of the entangled two-particle [10,11]. The latter is spatially extended, unlike with internal entangle- ment, where the positions only play the passive role of separating the parties. Since the positions and momenta are quantum with a direct classical analog, an observation of nonclassical correlations arising from mac- roscopically distinct phase space regions would therefore be a striking demonstration of the failure of classical . A convincing demonstration of nonclassical correlations between two parties requires the experimental violation of a Bell-type inequality [12]. In the simplest case, it involves detecting dichotomic properties on each side, such as the = of spin-1 2 systems, where a maximal viola- FIG. 1 (color online). (a) The two particles in a DTE state are tion is obtained if the spinsffiffiffi are in a Bell state, say ji¼ each characterized by an early and a late wave packet component i p ðj"i1j"i2 þ e j#i1j#i2Þ= 2. Such a dichotomic property is resulting from two different dissociation times. (b) Each particle not readily available in the motional state of free, structure- is passed through an unbalanced Mach-Zehnder interferometer less particles. One possibility is to consider observables with a switch deflecting the early wave component into the long which have no classical analogue, such as the displaced arm, while conducting the late component through the short arm. parity or pseudospin operators used to discuss entangle- (c) The path difference is chosen such that the early and the late wave packets overlap. Detecting the particle in an output port, at ment in the ‘‘original EPR state’’ [13,14]. While they are a given phase ’ and splitting ratio #, amounts to a dichotomic expedient for characterizing the quadratures of light fields, measurement analogous to a spin 1=2 detection in an arbitrary their experimental implementation seems exceedingly dif- orientation. The existence of DTE is established, implying a ficult in the case of free, macroscopically separated mate- macroscopically delocalized two-particle state, if the correla- rial particles, where only position measurements are easily tions at the single-particle interferometers violate a Bell inequal- realized. ity.

0031-9007=08=101(26)=260503(4) 260503-1 Ó 2008 The American Physical Society week ending PRL 101, 260503 (2008) PHYSICAL REVIEW LETTERS 31 DECEMBER 2008 analogues in the motion (say, early corresponds to spin up, come is characterized by the two-time probability density ; t ;t late to spin down), the DTE state shows the same structure prð 1 2; 1 2Þ for detecting particles behind the respec- as the Bell state ji. Switched, unbalanced Mach-Zehnder tive ports at the times t1, t2. It depends on the phase settings interferometers on each side then serve to mimic arbitrary ’j and mirror transmission angles #j of the interferome- spin rotations, and the detection of each particle in one of ters, and the dispersive evolution of the DTE state leads to t t the interferometer output ports completes the Bell test, see a complicated fringe pattern as a function of 1, 2, and . Figs. 1(b) and 1(c). However, as suggested by the above analogy with a dis- The DTE state is a variant of the ‘‘energy-time en- crete Bell test, a robust quantity characterizing entangle- tangled’’ state introduced in [15], and it is closely related ment is obtained by integratingR the port-specific proba- P ¼ ð ; t ;t Þdt dt to the ‘‘time-bin entanglement’’ of photons [2,16,17], bilities over all times, 1;2 pr 1 2; 1 2 1 2. which has been used, e.g., for establishing nonlocal corre- For any reasonable model of the time-of-arrival detection, lations over fiber distances of more than 50 km [18]ina the correlation function can be equally evaluated by means similar setup as in Fig. 1. As a main difference, the DTE ^ ^ ^ 1 of the projections 1 2 2 onto the (un- state is not composed of two single-particle product states, bounded) spatial regions behind the respective ports, but it superposes their relative coordinate, while the center- P ¼ k ^ 1 U^ j ik2 1;2 limt!1 2 z;t DTE . of-mass state remains separable during the two-pulse dis- Since we are only interested in the interferometer output, sociation process. the effect of dispersion is best incorporated by time- Our use of DTE reflects the necessity to come up with a dependent , which separates the ‘‘raw state generation scheme appropriate to material particles. action’’ of the interferometers, described by the S matrices Their finite mass and internal structure entail substantial ð0Þ S^j, from the free dispersive time evolution U^ z;t . The pro- complications which require a careful investigation of jection of the DTE state component jc c:m:ijc reli onto a whether a Bell violation can be expected at all. Most 0 0 particular output-port combination , then takes the prominently, one must account for the unavoidable wave 1 2 form packet dispersion, and only the recent progress in manipu- lating ultracold molecules (such as their condensation [19] 1 ðonÞ c:m: rel ð0Þ ðonÞ ðonÞ c:m: rel ^ U^ z;t jc ijc i¼U^ z;t ½S^ S^ jc ijc i: and controlled dissociation [20]) suggests the possibility to 2 0 0 1 2 0 0 generate motional states that allow one to keep the detri- (1) mental effect of dispersion under control. We note that Here we assume t to be sufficiently large, so that the wave other ways to demonstrate nonlocal correlations of mo- packets have passed the interferometers, and we take lecular dissociation products have been proposed in the entrance switches to be in the ‘‘on’’ configuration, [21,22]. i.e., routing towards the long arms. Implementing the We will show below that, by appropriately choosing a phase shifts by the arm-length variations ‘j, the projected sequence of magnetic field pulses, the Feshbach-induced ðonÞ ðonÞ ðonÞ S matrices S^ ¼ ^ S^j are given by S^ ¼ dissociation of an ultracold 6Li molecule within a guid- j j j¼þ1 2 ðonÞ ðipj‘j=@Þ #j S^ ¼ ðipj‘j=@Þ #j ing laser beam can generate a motional state of macro- exp ^ cos and j¼1 exp ^ sin . scopically entangled atoms that is capable of violating a For the ‘‘off’’ configuration we have correspondingly Bell inequality. But first, to clarify the implications of S^ðoffÞ # S^ðoffÞ # j¼þ1 ¼ sin j and j¼1 ¼cos j. Analogous rela- dispersion, we determine the correlation function for a : : tions hold for jc c m iP^jc reli. generic DTE state subject to correlated single-particle 0 0 The setup requires the dispersion-induced broadening of interferometry. the wave packets to remain much smaller than the separa- The general DTE Bell test.—It is natural to take the tion between the early and the late components, so that the bound molecular two-particle state to be separable in the switches can be changed in between. In this case the center-of-mass (c.m.) and the relative (rel) coordinate. correlation function P ; can be evaluated by using Eq. Denoting by the period between the two dissociation 1 2 (1) and its variants even for nonpure and nonseparable pulses and assuming the transverse motion to be frozen initial two-particle states, j ih j!% . One obtains in the of the guiding laser beam, the longitu- 0 0 0   Z Z dinal part of a pure DTE stateffiffiffi then takes the form 1 i i p P ; ð‘ ;‘ Þ¼ 1 þ Re e dp dp U^ ð0Þ e = 1 2 1 2 1 2 1 2 jDTEi¼ð z;j0iþ pffiffiffij0iÞ 2, where j0i¼ 4 c c:m: c rel P^ c rel =  ~   j 0 iðj 0 iþ j 0 iÞ 2. It involves the free time- p~ ‘ p~ 2 U^ ð0Þ P^ exp i i prðp ;p Þ ; (2) evolution z;t , the parity operator , and a rela- @ 2m@ 1 2 tive phase determined by the details of the two-pulse p~ p ;p T ‘~ ‘ ;‘ T p ;p dissociation process. The center-of-mass state of the origi- with ¼ð 1 2Þ , ¼ð 1 2Þ , and prð 1 2Þ¼ c c:m: p ;p % p ;p nal molecule j 0 i is taken to rest in a waveguide, while h 1 2j 0j 1 2i the momentum distribution function. c rel j 0 i propagates into positive direction. For simplicity we take here the beam splitters to be sym- Indicating the output ports of the two interferometers metric (#j ¼ =4) and the particles to have equal mass m. j ¼ 1; 2,byj ¼1, the immediate experimental out- Note that (2) is independent of the total time of flight and 260503-2 week ending PRL 101, 260503 (2008) PHYSICAL REVIEW LETTERS 31 DECEMBER 2008

T m@=2 T invariant under momentum phase transformations istic dispersion times, c:m: ¼ 2 p;c:m: and rel ¼ hp ;p j i!exp½iðp ;p Þhp ;p j i, which includes m@= 2 1 2 0 1 2 1 2 0 2 p;rel, indicating the transition to a dispersion- spatial translations, thus rendering the correlation dominated spatial extension of the wave packets. The P ; ð‘ ;‘ Þ a robust signal. p 1 2 1 2 expectation value of the relative momentum 0;rel ¼ For generic Gaussian states in the center-of-mass and mv = @=p rel 2 defines the reduced wavelength rel ¼ 0;rel, relative motion Eq. (2) can be evaluated in closed form. which sets the scale for the nonlocal interference fringes 2 2 The variances p;c:m: and p;rel then determine character- in the explicit correlation function,

   2=T2 1=4 T ‘ ‘ v 2 T ‘ ‘ 2 1 ð1 þ c:m:Þ rel ð 1 2 relÞ c:m: ð 1 þ 2Þ P ; ð‘ ;‘ Þ¼ 1 þ exp 1 2 1 2 1 2 2 2 1=4 T2 2 v T2 2 v 4 ð1 þ =T Þ þ 2 rel rel c:m: þ 2 rel rel  rel rel  ‘ ‘ ð‘ ‘ v Þ2 ð‘ þ ‘ Þ2 ’ cos 1 2 þ 1 2 rel þ 1 2 0 ; (3) T2 2 v T2 2 v rel rel þ 2 rel rel c:m: þ 2 rel rel 2

’ v = =T =T with 0 ¼ rel rel þ arctanð c:m:Þþarctanð relÞþ for the BEC within the laser guide [24]. At the end of the 2. preparation steps it is very shallow (depth UT=kB ¼ These DTE correlations can violate a Bell inequality. 100 nK, !T=2 ¼ 0:5Hz) and only a small number of This is seen from the structural similarity of (3) to the molecules (on the order of 102) remains in the BEC. These correlation function of the standard spin-1=2-based setup, can be taken to be noninteracting, so that the initial longi- spin P ’ ;’ ’ ’ = tudinal center-of-mass state jc Ti of the molecules is 1;2 ð 1 2Þ¼f1 þ 1 2 cosð 1 2Þg 4, where the j ¼1 denote the spin measurement outcomes for ana- straightforwardly defined by the trap parameters. lyzers tilted by the angles ’j with respect to the Bell state Each interferometer (with path length difference v = quantization axis. It follows from this analogy that an rel 2 ’ 5mm) is implemented by two more red-detuned laser beams crossing the guide in a triangular arrangement unambiguous demonstration of entanglement requires thepffiffiffi fringe visibility of the correlation signal to exceed 1= 2 at small angles. While the crossings act as beam splitters, over at least a few periods. the required atom mirror may be realized using an evanes- The dispersive suppression of this fringe visibility is de- cent light field or a blue-detuned laser beam perpendicular scribed by those terms in (3) which depend on the charac- to the interferometer plane [25,26]. The time controlled T T appliance of such perpendicular blocking beams could also teristic times c:m: and rel. Specifically, the dispersion- induced distortion between the early and the late wave implement the switch. However, a simplified setup could packet components is reflected in the Lorentzian reduction replace the switch by an ordinary , at the cost factor and in the quadratic compression of the fringe pat- of 50% post-selection. tern, while their envelope mismatch causes the Gaussian The fluorescence detection of the slow, strongly con- suppression. Based on this result one finds that nonlocal fined atoms at the two output guides can be done with near = v unit efficiency and with single-particle resolution, since no correlations can be observed, for rel ð relÞ1, pro- 2=T2 2=T2 < particular spatial or temporal accuracy is needed. This vided ð1 þ c:m:Þð1 þ relÞ 4. In the following, we present a conceivable scenario for the generation of a single-particle resolution is crucial since events where DTE state, which meets these conditions. more than one molecule gets dissociated are disregarded Experimental scenario for a DTE Bell test.—We suggest in the present scenario, relying on post-selection. In a more to use a dilute molecular Bose-Einstein condensate (BEC) refined setup, it is conceivable to use a specially prepared produced from a 50:50 spin mixture of fermionic 6Li as a optical lattice where each site is occupied by at most one starting point. It can be prepared efficiently and with near- molecule [27]. perfect purity, displaying a huge lifetime of more than 10 s All this implies that the molecular dissociation in pres- due to the Pauli blocking of detrimental 3-body collisions ence of the waveguide must meet a number of rather [23]. A truly macroscopic time separation between the two restrictive criteria to render a demonstration of macro- pulses, say ¼ 1s, is thus conceivable, and for a realistic scopic entanglement possible. Only the transverse ground v = dissociation velocity of rel ¼ 1cm s the de Broglie state of the guide may be populated to admit the above : wavelength rel ¼ 13 3 m is large enough to pose viable quasi-one-dimensional description of the interferometers, stability requirements for the interferometers. while both the momentum spread of the wave packets and The BEC is prepared in a red-detuned, far off-resonant the dissociation probability must be sufficiently small. In laser beam (transverse trap frequency !G=2 ¼ 300 Hz), order to judge the feasibility of our experimental scenario, strong enough to guide the dissociated atoms towards the we now discuss how the dissociated part of the state, t single-particle interferometers. The intersection with a jbgð Þi, depends on the magnetic field pulse sequence second, weak laser beam creates an elongated dipole trap and the resonance parameters. 260503-3 week ending PRL 101, 260503 (2008) PHYSICAL REVIEW LETTERS 31 DECEMBER 2008

A Green function analysis within the two-channel vacuum pressure (108 mbar) suffice to suppress decoher- single-resonance approach shows that after an arbitrary ence due to scattering of off-resonant photons or back- magnetic field pulse sequence (close to an isolated reso- ground gas particles. t nance) jbgð Þi is described, for low energies, at positions A great advantage of this setup is that no interferometric far from the dissociation center, and at large times, by stability is required between the two interferometers, so t C ’c:m: ’rel U^ ð0Þ that truly macroscopic separations are feasible. Moreover, the asymptotic form jbgð Þi bgj 0;0 ij 0;0i z;t jzi, ð0Þ the DTE state reveals its entanglement robustly since where U^ z;t is the free propagator in the longitudinal direc- neither a spatial nor temporal resolution is required in the tion and where the transverse motion is frozen in the detection. harmonic ground state, j’c:m:i and j’rel i, respectively, of 0;0 0;0 This work was supported by the DFG Emmy Noether the guiding laser beam. The longitudinal state is deter- program. p ;p C~ p2 = m@ p2 =m@ mined by h c:m: reljzi¼ ð c:m: 4 þ rel þ ! p c = C~ C~ ! 2 GÞh c:m:j Ti jj jj, where ð Þ is the Fourier trans- form of the closedR channel CðtÞ C~ 2 dp dp C~ p2 = m@ p2 =m@ and k k ¼ c:m: relj ð c:m: 4 þ rel þ ! 2 p c 2 C t [1] J. S. Bell, Speakable and Unspeakable in Quantum 2 GÞj jh c:m:j Tij . ð Þ, in turn, is determined by the coupled channel dynamics as induced by the externally Mechanics (Cambridge University Press, Cambridge, controlled magnetic field BðtÞ. The dissociation probability England, 1987). [2] J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. C 2 ! a B C~ 2=@2 is given by j bgj ¼ G bg res resk k . It in- Lett. 82, 2594 (1999). volves the background scattering length abg, the resonance [3] B. Julsgaard, A. Kozhekin, and E. S. Polzik, B 413 width res, and res, the difference between the magnetic (London) , 400 (2001). moments of the resonance state and the open channel. [4] J. Sherson et al., Nature (London) 443, 557 (2006). While the association of the molecules is best done at a [5] D. N. Matsukevich et al., Phys. Rev. Lett. 96, 030405 broad resonance [23], shifting to a narrow resonance (e.g., (2006). [6] D. Moehring et al., Nature (London) 449, 68 (2007). B ¼ 1mG, ¼ 0:01B, a ¼ 100a ) one can en- res res bg 0 [7] W. Rosenfeld et al., Phys. Rev. Lett. 98, 050504 (2007). sure by choosing field pulses BðtÞ with short duration (e.g., 438 T [8] D. Leibfried et al., Nature (London) , 639 (2005). ¼ 60 ms) that about a single molecule dissociates on [9] H. Haffner et al., Nature (London) 438, 643 (2005). average. A sequence of two square pulses with base value [10] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 B B 0 and height sweeping over the resonance position (1935). B 31 res then generates a DTE longitudinal wavep packetffiffiffi of the [11] E. Schro¨dinger, Proc. Cambridge Philos. Soc. , 555 Uð0Þ ei = required form, jzi¼½^ z; þ j0i 2. The mo- (1935). 2 mentum distribution is given by jhp : :;p j ij ¼ [12] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, c m rel 0 23 p p4 2½ðp2 = þ p2 p2Þ= p2jhp jc ij2= Phys. Rev. Lett. , 880 (1969). 0 sinc c:m: 4 rel 0 c:m: T 58 p2 = p2 p2 p2 p 2 [13] K. Banaszek and K. Wodkiewicz, Phys. Rev. A , 4345 ½ð c:m: 4 þ rel 0 þ Þ , where we de- p2=m B B B U @! (1998). fine 0 ¼ resð 0 þ resÞ2 T G,as [14] Z.-B. Chen et al., Phys. Rev. Lett. 88, 040406 (2002). p2=m B p2 m@=T well as ¼ res and ¼ 2 [28]. Instead [15] J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989). of evaluating Eq. (2) directly with this momentum distri- [16] W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Phys. Rev. bution, it is more transparent to approximate the latter by Lett. 84, 4737 (2000). Gaussians, which allows one to apply the analysis follow- [17] C. Simon and J. P. Poizat, Phys. Rev. Lett. 94, 030502 B B B ing Eq. (3). For reasonable pulses 0 þ res ¼ (2005). p et al. 93 350 mG these Gaussians are centered at 0;rel ¼ [18] I. Marcikic , Phys. Rev. Lett. , 180502 (2004). mv = p [19] T. Ko¨hler, K. Go´ral, and P. Julienne, Rev. Mod. Phys. 78, rel 2 and 0;c:m: ¼ 0, withpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spreads of p;rel ¼ : m@= p T @! m= 1311 (2006). 1 196 ð 0 Þ and p;c:m: ¼ T 2, respectively. [20] T. Mukaiyama et al., Phys. Rev. Lett. 92, 180402 (2004); T : T : This yields rel ¼ 3 4s and c:m: ¼ 0 64 s, implying a S. Du¨rr, T. Volz, and G. Rempe, Phys. Rev. A 70, 031601 visibility of about 72% in the correlationpffiffiffi signal, which (2004); Greiner et al., Phys. Rev. Lett. 94, 110401 (2005). exceeds the threshold value of 1= 2. [21] K. V. Kheruntsyan, M. K. Olsen, and P. D. Drummond, Our analysis thus shows that the observation of a macro- Phys. Rev. Lett. 95, 150405 (2005). scopic DTE state is feasible with material particles, even [22] T. Opatrny´ and G. Kurizki, Phys. Rev. Lett. 86, 3180 (2001). though dispersion poses tight constraints. The technologi- 302 cal challenge is substantial, but not insurmountable. Stable [23] S. Jochim et al., Science , 2101 (2003). [24] J. Fuchs et al., J. Phys. B 40, 4109 (2007). lasers are required and the magnetic pulse sequence must 240 5 [25] C. S. Adams, M. Siegel, and J. Mlynek, Phys. Rep. , be reproducible with a relative accuracy of 10 from shot 143 (1994). to shot, so that the relative phase between the early and [26] H. Kreutzmann et al., Phys. Rev. Lett. 92, 163201 (2004). U BT late components, given by ’½2 T res þ [27] T. Volz et al., Nature Phys. 2, 692 (2006). B B =@ ! resð res 0Þ þ G , varies less than 50 mrad. [28] The normalization of the momentum distribution assumes Realistic choices of the laser wavelength (1 m) and the p, p;c:m: p0, which holds for our parameters. 260503-4