An Experimental Proposal to Test Dynamic Quantum Non-locality with Single-Atom Interferometry
aa r X i v : . [ qu a n t - ph ] S e p An Experimental Proposal to Test Dynamic Quantum Non-localitywith Single-Atom Interferometry
Shi-Liang Zhu , Zheng-Yuan Xue , Dan-Wei Zhang , and Lu-Ming Duan Laboratory of Quantum Information Technology, ICMP and SPTE, South China Normal University, Guangzhou, China Department of Physics and MCTP, University of Michigan, Ann Arbor, Michigan 48109, USA
Quantum non-locality based on the well-known Bell inequality is of kinematic nature. A different typeof quantum non-locality, the non-locality of the quantum equation of motion, is recently put forward withconnection to the Aharonov-Bohm effect [Nature Phys. 6, 151 (2010)]. Evolution of the displacement operatorprovides an example to manifest such dynamic quantum non-locality. We propose an experiment using single-atom interferometry to test such dynamic quantum non-locality. We show how to measure evolution of thedisplacement operator with clod atoms in a spin-dependent optical lattice potential and discuss signature toidentify dynamic quantum non-locality under a realistic experimental setting.
PACS numbers: 03.65.Ud, 03.65.Vf, 03.75.Dg, 37.10.Jk
Non-locality dramatically exemplified in the Einstein-Podolsky-Rosen (EPR) paradox [1] is a fundamental con-cept of quantum mechanics that distinguishes it from classi-cal physics. Quantum non-locality based on the EPR correla-tion was later formulated into an experimentally testable resultknown as the Bell inequality [2]. Quantum non-locality basedon the EPR correlation and the Bell inequality has been veri-fied in many experiments involving different physical systems[3].Recently, a different type of quantum non-locality, the non-locality of the quantum equation of motion [4], implied inthe famous Aharonov-Bohm effect [5], was put forward byPopescu [6]. Non-locality based on violation of the Bell in-equality comes from the Hilbert space structure of quantummechanics and thus is purely kinematic, while non-localityimplied in the Aharonov-Bohm effect is from non-locality ofquantum equations of motion in the Heisenberg picture andthus is of dynamic nature [6]. Another significant differencebetween these two kinds of non-localities is that non-localityfrom the EPR correlation is assumed to be an exclusive quan-tum property of two or more well separated but entangledparticles, while the dynamic quantum non-locality (DQNL)considered by Aharonov et al. and Popescu can be demon-strated even with evolution of a single particle in a super-position state of two distinct locations. The evolution of thedisplacement operator provides an explicit example to clearlyshow the DQNL [6], however, experiment is still lacking inthis direction due to the difficulty to measure the displacementoperator.In this paper, we propose a feasible experiment using coldatoms in a spin-dependent optical lattice potential to test theDQNL. We figure out a configuration where the DQNL in-herent in the Heisenberg equation leads to a detectable sig-nal qualitatively different from that of the corresponding local(classical) evolution equation, and propose a method to di-rectly measure evolution of the displacement operator in thereal experimental system. The required ingredients in this pro-posed experiment, such as the double well optical lattice andthe spin-dependent movement of a particle, have all been real-ized in previous experiments [7–9], and thus the proposal well fits with the status of the current technology.Before explaining the proposal, first we briefly recall theconcept of DQNL elaborated in Ref. [6]. The Schrodingerequation describing evolution of the wave function of a quan-tum system is always a local differential equation, however,the wave function by itself is not directly observable. To seethe DQNL, one needs to look at the Heisenberg equationswhich describe evolution of observable physical quantities.The Heisenberg equation for the displacement operator ˆ D Q provides an example to explicitly show this kind of DQNLfor a single particle [6]. The displacement operator ˆ D Q is de-fined as ˆ D Q ≡ exp[ i ˆ pL/ ~ ] with ˆ p being the momentum oper-ator of the particle. This operator shifts the particle by a finitedistance L . For simplicity, we consider a one-dimensionalsituation where the Hamiltonian of the particle is given by H = ˆ p m + V ( x ) with m being the mass of the particle and V ( x ) being the potential. For this system, the classical andquantum equations of motion of the displacement operator arequite different [6]. In classical mechanics, we can apply thechain rule for differentiation of a function, and evolution ofthe quantity D C ≡ exp[ ipL/ ~ ] is given by dD C dt = de ipL/ ~ dp dpdt = Li ~ e ipL/ ~ dV ( x ) dx , (1)which is a local differential equation. However, quantum me-chanically, the displacement operator ˆ D Q is governed by theHeisenberg equation, which leads to d ˆ D Q dt = 1 i ~ h ˆ D Q , H i = 1 i ~ [ V ( x + L ) − V ( x )] ˆ D Q , (2)where we have used e i ˆ pL/ ~ V ( x ) = V ( x + L ) e i ˆ pL/ ~ . Thisevolution equation is clearly nonlocal as the time derivativeof the quantity depends on the potential at two distinct (andpossibly remote) locations x and x + L .To demonstrate this kind of DQNL, we need to figure outa configuration where the classical and the quantum evolu-tion equations (1) and (2) for the displacement operator showclear qualitative difference. We also need to find a method tomeasure the displacement operator in real experimental sys-tems. The evolution operator ˆ D Q is non-Hermitian, so it isnot directly observable. However, we can look at the realand imaginary parts of ˆ D Q , and they correspond to observ-able quantities and still satisfy nonlocal evolution equationsin quantum mechanics. To have a configuration that manifeststhe DQNL represented by Eq. (2), we consider a particle con-fined in one dimension with a double-well potential, as shownin Fig. 1(a-c). The two potential wells are identical, exceptthat the bottom of one of the wells may be shifted with that ofthe other well by a constant energy ∆ . For classical particlesin either of these wells, they see identical force and cannot tellthe difference of the wells. The local dynamic equation shouldbe independent of the energy shift ∆ . However, for quantumparticles in a superposition state, the evolution of the displace-ment operator can sense this nonlocal constant energy shift ∆ .To be explicit, let us assume that the potential V ( x ) aroundthe two minima ± L/ can be described by the identical har-monic trap, with V ( x ) = (cid:0) mω / (cid:1) ( x + L/ , V ( x ) = (cid:0) mω / (cid:1) ( x − L/ + ∆ , where ω is the characteristic trapfrequency. Let | Φ( x ) i denote an eigenstate of the harmonictrap (for convenience, it can be taken as the ground state). Wetake the initial state of the particle at time t = 0 as the follow-ing superposition of two localized wave packets, | Ψ( x, i = (cid:2) | Φ( x + L/ i + | Φ( x − L/ i e iθ (cid:3) / √ , (3)where θ is an arbitrary initial phase difference. The size of thewave packet | Φ( x ) i at each well, estimated by p ~ /mω , isassumed to be significantly smaller than L so that the overlap R Φ ∗ ( x + L/ x − L/ dx ≈ . For this state, the quantumHeisenberg equation (2) for the displacement operator directlygives * d ˆ D Q dt + = − iω d D ˆ D Q E , (4)where ω d = ∆ / ~ . It has the straightforward solution h ˆ D Q ( t ) i = D ˆ D Q (0) E e − iω D t = (1 / e iθ − iω d t . (5)So the evolution of the quantum displacement operator is sen-sitive to the nonlocal constant energy shift ∆ . In contrast, forclassical particles with the dynamic equation (1), even if theyare distributed over the two wells, as long as the distributionfunction in each well is symmetric with respect to the trapbottom (which is the case for the state shown in Eq. (3)), theaverage force D dV ( x ) dx E is always zero and D dD C dt E = 0 . So,as expected, the classical dynamic equation cannot sense thenonlocal constant energy shift and there is a qualitative differ-ence in the measurement outcomes for the classical and thequantum evolution equations for the displacement operator.The phenomenon discussed above is closely related to thescalar Aharonov-Bohm effect [10, 11]: in classical physics, the constant energy shift does not lead to any physical differ-ence as long as the force is identical in space. However, fora quantum particle in a superposition state, it can sense theconstant energy shift at two remote locations even if the forceis strictly zero at any point of the particle’s trajectory. This issimilar again to the conventional Aharonov-Bohm effect [5],where a quantum charged particle senses a nonzero constantvector potential while the electromagnetic force is zero at anypoint of the particle’s trajectory.To prepare the particle in a superposition state of a dou-ble well potential and to directly measure the displacementoperator, we propose to use an optical lattice potential tocontrol cold atoms to fulfill all the requirements. We con-sider dilute atomic gas in an optical lattice with the aver-age filling number per lattice site much less than , so theatomic interaction is negligible and we just have many in-dependent copies of single-particle dynamics. To generate asuperposition state over different lattice sites, one may use adouble well lattice along the x direction, with the potential V ( x ) = − V sin ( kx/ ϕ ) − V sin ( kx ) ( V , V > )from two standing wave laser beams with the wave vector k = 2 π/λ ≡ π/L [8, 9]. Initially, we set the phase ϕ = 0 and V /V = 0 so that we have only a single lattice V andthe atom is in the ground state of this lattice well. By adiabat-ically tuning up the ratio V /V to the region with V ≫ V ,each lattice site splits into two as shown in Fig. 1 (a-b), andthe atomic state adiabatically follows the ground state config-uration and evolves into an equal superposition state of thetwo wells in the form of Eq. (3) with θ = 0 . After we haveprepared this initial state, we quickly (within a time scale t δ )tune ϕ and V so that ϕ = π/ and V = ∆ / ≪ V , andlook at evolution of the displacement operator under this dou-ble well potential with a constant energy shift ∆ (Fig. 1c).Under the lattice V , the potential well is approximated by aharmonic trap with the trapping frequency ω = 2 √ V E r / ℏ ,where E r = ~ k / m is the atomic recoil energy. We requirethe time scale t δ to satisfy the condition ω − ≪ t δ ≪ ℏ / ∆ so that on the one hand, we do not generate motional exci-tations in each well, and on the other hand, t δ is negligiblecompared with the evolution time scale (of the order of ℏ / ∆ )of the displacement operator.To demonstrate the nonlocal dynamics of the displacementoperator ˆ D Q shown in Eq. (5) under this double well lat-tice, we need to measure ˆ D Q after a controllable time delay t . The displacement operator does not correspond to a simplephysical quantity, and it is not easy to measure it directly inexperiments. To overcome this problem, we make use of theinternal (spin) states of the atoms. We show in the followingthat a Ramsey type of experiment in the internal state, togetherwith a spin-dependent movement of the lattice potential, givesa direct measurement of the expectation value of the displace-ment operator. The atoms have different hyperfine states, andwe use two of them, denoted by effective spin | ↑i and | ↓i ,respectively. The atoms are initially assumed to be in the state | ↑i . To measure the displacement operator after an evolutiontime t ( t ∼ ℏ / ∆ ), we take the following four steps as illus-trated in Fig. 1(d-f): (i) first, we apply a π/ -pulse withina time much shorter than ℏ / ∆ to the atomic internal state sothat the atomic state transfers to (cid:2) ( | ↑i + | ↓i ) / √ (cid:3) | ψ ( t ) i ,where | ψ ( t ) i denotes the atomic motional state at time t . (ii)Second, we apply a spin dependent shift to the lattice po-tential with the corresponding unitary operation described by U = | ↑i h↑| ˆ D QL/ + | ↓i h↑| ˆ D Q − L/ . This kind of opera-tion has been realized before in experiments to demonstratecontrolled atomic collisions [7]. This operation needs to bedone in a time scale t δ which satisfies ω − ≪ t δ ≪ ℏ / ∆ so that the shift of the lattice does not generate motionalexcitations in each well. After this step, the atomic statebecomes (cid:16) | ↑i ˆ D QL/ + | ↓i ˆ D Q − L/ (cid:17) | ψ ( t ) i / √ . (iii) Afterthe spin-dependent lattice shift, we apply another π/ -pulse(within a time negligible compared with ℏ / ∆ ) to the atomicinternal state which transfers the atomic state to | ψ f i = h ( | ↑i + | ↓i ) ˆ D QL/ + ( | ↓i − | ↑i ) ˆ D Q − L/ i | ψ ( t ) i / . (iv) Fi-nally, we measure the total atom number N ↓ in the spin downstate minus the total number N ↑ in the spin up state. Thisnumber difference is proportional to the probability difference P ↓ − P ↑ for the state | ψ f i , which is given by P ↓ − P ↑ = Re h h Ψ ( t ) | ˆ D QL | Ψ ( t ) i i . (6)The imaginary part of the expectation value h Ψ ( t ) | ˆ D QL | Ψ ( t ) i can be measured in a similar way. The only differ-ence is that in the step (i) we add a relative phase i to the π/ -pulse which transfers spin | ↑i to the state ( | ↑i + i | ↓i ) / √ . The final state is then modified to | ψ f i = h ( | ↑i + | ↓i ) ˆ D QL/ + i ( | ↓i − | ↑i ) ˆ D Q − L/ i | ψ ( t ) i / withthe probability difference P ↓ − P ↑ = Im h h Ψ ( t ) | ˆ D QL | Ψ ( t ) i i .In the above, we have shown how to measure the expecta-tion value of the displacement operator for cold atoms in anoptical lattice. The DQNL indicates that the real (imaginary)part of this expectation value oscillates with the evolution time t as cos( ω d t ) (- sin( ω d t ) ) as predicted by Eq. (5), which issensitive to the nonlocal constant energy shift ∆ = ℏ ω d . Thissignal distinguishes it from the corresponding classical casewhere (cid:10) D C (cid:11) is independent of ∆ and shows no oscillationwith time t . For real experiments in an optical lattice, how-ever, there is inevitably a global harmonic trap potential (tak-ing the form of V t = mω t x / in the x direction) which couldcomplicate the situation [8, 9]. The measured atom numberdifference N ↓ − N ↑ involves average of the probability differ-ence P ↓ − P ↑ over all the independent double-well potentials.Due to the global trap V t , the probability difference P ↓ − P ↑ in each double well potential oscillates with slightly differ-ent frequencies, and one needs to check whether an averageover the whole lattice will wash out the oscillation signal. Forthis purpose, we simulate in Fig. (2) the averaged signal for atypical experimental configuration. The averaged probability (a) (b) (c)(d) (e) (f) D FIG. 1: (Color online) Illustration of the experimental steps for ini-tial state preparation and for detection of the displacement operator.Figs. (a)-(c) show the steps to prepare a non-local superposition state(given by Eq. (3)) in a double-well lattice. After adiabatic prepara-tion of this superposition state, a bias potential ∆ is tuned on withina time scale t δ (specified in the text) to start evolution of the dis-placement operator. Figs. (d-f) show the steps to measure the dis-placement operator after a certain evolution time t . Right beforethe measurement, the bias potential ∆ is turned off, so we have aregular optical lattice (d). The atom is then transfer to an equal su-perposition of two spin components with a π/ -pulse (e). After aspin-dependent shift of the optical lattice (f), followed by another π/ -pulse, we measure the population difference in these two spincomponents, and this difference gives directly the expectation valueof the displacement operator. difference is given by h P ↓ − P ↑ i = 12 N l X j cos(∆ j t/ ~ ) e − γt , (7)where ∆ j = ∆ + δ j with δ j ≈ mω t Lx j ( L = λ/ , and x j denotes the coordinate of the center of the double wells). Thesummation of j is over all the occupied double wells (withnumber N l ) in the global harmonic trap. To better modelthe experimental situation, we also add a phenomenologicaldecay e − γt to each oscillation term which corresponds to anonzero dephasing rate γ inevitable in reality. Under typi-cal experimental parameters we have damped oscillations asshown in Fig. 2. The signal is still clearly observable in thiscase. In the frequency domain, the spectrum centers at theenergy shift ∆ / ℏ that is independent of the experimental im-perfection discussed above.Before ending the paper, we briefly discuss the require-ments for the relevant experimental parameters. To assure lo-cality, we assume the wave packet overlap between differentwells is negligible. This overlap is estimated by e − L/l , where l = p ~ /mω is the size of the wave packet in each well and L = λ/ is the distance between the wells. For Rb atoms inan optical lattice with λ = 800 nm , ω = 2 √ V E r ∼ π × kHz and l ∼ nm for a typical lattice barrier V = 35 E r , the condition e − L/l ∼ e − . ≪ is well satisfied. Dur-ing the state preparation and the detection of the displace-ment operator, we require the operation time t δ to satisfy ω − ≪ t δ ≪ ℏ / ∆ . If we take ∆ ∼ . E r ∼ . ω and S pe c t r u m (b) Frequency (E r / h )(a) time t/t < P - P > FIG. 2: Simulated experiential signal in an inhomogeneous opticallattice with a global harmonic trap. (a): Averaged population dif-ference of the two spin components as a function of evolution time(in the unit of t = ~ /E r ) of the displacement operator. The av-erage is taken over occupied double wells in an global harmonictrap with the trap frequency ω t = 2 π × Hz. Other parametersinclude L = λ/ nm, the atomic mass m = 1 . × − kg for Rb atoms, and the bias potential ∆ = 0 . E r . A dephasingrate γ = 0 . E r / ~ is assumed (see Eq. (7). (b) Fourier transformof the signal in Fig. (a). Instead of a sharp line at the bias potential( ∆ / ~ ,) the curve has a broad peak due to the broadening from av-erage in the inhomogeneous global trap and the nonzero dephasingrate. However, the peak is still centered at the bias potential ∆ / ~ . t δ ∼ ω − ∼ µs , the motional excitations estimated bythe Landau-Zener formula is small and all the requirementsseem to be reasonable with the current experimental technol-ogy.In summary, we have proposed a feasible experiment us-ing cold atoms in an optical lattice to test the DQNL associ- ated with evolution of the displacement operator. The DQNLis different from and complementary to the kinetic quantumnon-locality represented by the Bell inequalities. Similarto tests of the Bell inequalities, an experimental test of theDQNL could shed new light on our understanding of funda-mentals of quantum mechanics.The work was supported by the NSF of China (No10974059), the State Key Program for Basic Research ofChina (Nos.2006CB921801 and 2007CB925204), the MURI,and the DARPA program. [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. , 777(1935).[2] J. S. Bell, Physics, , 195 (1964).[3] For a review, see M. Genovese, Phys. Rep., , 319 (2005).[4] Y. Aharonov, H. Pendleton, and A. Petersen, Int. J. Theor.Phys., , 213 (1969); Y. Aharonov in Proc. Int. Symp. Founda-tions of Quantum Mechanics and their Technical Implications(eds S. Kamefuchi et al.,) 10-19 (1984).[5] Y. Aharonov and D. Bohm, Phys. Rev. , 485 (1959).[6] S. Popescu, Nature Phys. , 151 (2010).[7] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, andI. Bloch, Phys. Rev. Lett. , 010407 (2003); Nature (London), , 937 (2003).[8] S. Foelling, S. Trotzky, P. Cheinet, M. Feld, R. 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