Abstract
The Hong–Ou–Mandel effect is a demonstration of destructive quantum interference between pairs of indistinguishable bosons, realised so far only with massless photons. Here we propose an experiment to realize this effect in the matterwave regime using paircorrelated atoms produced via a collision of two Bose–Einstein condensates and subjected to two laserinduced Bragg pulses. We formulate a measurement protocol for the multimode matterwave field, which—unlike the typical twomode optical case—bypasses the need for repeated measurements under different displacement settings of the beam splitter, markedly reducing the number of experimental runs required to map out the interference visibility. Although the protocol can be used in related matterwave schemes, we focus on condensate collisions. By simulating the entire experiment, we predict a Hong–Ou–Mandel dip visibility of ~69%. This visibility highlights strong quantum correlations between the atoms, paving the way for a possible demonstration of a Bell inequality violation with massive particles in a related Rarity–Tapster setup.
Introduction
Since its first demonstration, the Hong–Ou–Mandel (HOM) effect^{1} has become a textbook example of quantum mechanical twoparticle interference using pairs of indistinguishable photons. When two such photons enter a 50:50 beam splitter, with one photon in each input port, they both preferentially exit from the same output port, even though each photon individually had a 50:50 chance of exiting through either output port. The HOM effect was first demonstrated using optical parametric downconversion^{1}; the same setup, but with an addition of linear polarisers, was subsequently used to demonstrate a violation of a Bell inequality^{2} that is of fundamental importance to validating some of the foundational principles of quantum mechanics such as quantum nonlocality and longdistance entanglement.
The HOM effect is a result of destructive quantum interference in a (bosonic) twinphoton state that leads to a characteristic dip in the photon coincidence counts at two photodetectors placed at the output ports of a beam splitter. The destructive interference occurs between two indistinguishable paths corresponding to the photons being both reflected from, or both transmitted through, the beam splitter. Apart from being of fundamental importance to quantum physics, the HOM effect underlies the basic entangling mechanism in linear optical quantum computing^{3}, in which a twinphoton state 1,1› is converted into a quantum superposition —the simplest example of the elusive ‘NOON’ state^{4}. Whereas the HOM effect with (massless) photons has been extensively studied in quantum optics (see refs 5, 6 and references therein), twoparticle quantum interference with massive particles remains largely unexplored. A matterwave demonstration of the HOM effect would be a major advance in experimental quantum physics, enabling an expansion of foundational tests of quantum mechanics into previously unexplored regimes.
Here we propose an experiment that can realize the HOM effect with matter waves using a collision of two atomic Bose–Einstein condensates (BECs) (as in refs 7, 8, 9, 10, 11) and a pair of laserinduced Bragg pulses. The HOM interferometer uses paircorrelated atoms from the scattering halo that is generated during the collision through the process of spontaneous fourwave mixing. The paircorrelated atoms are mixed using two separate Bragg pulses^{12,13} that realize an atomoptics mirror and beamsplitter elements—in analogy with the use of twinphotons from parametric downconversion in the optical HOM interferometer scheme. The HOM effect is quantified via the measurement of a set of atom–atom pair correlation functions between the output ports of the interferometer. Using stochastic quantum simulations of the collisional dynamics and the application of Bragg pulses, we predict a HOM dip visibility of ~69% for realistic experimental parameters. A visibility larger than 50% is indicative of stronger than classical correlations between the atoms in the scattering halo^{7,11,14,15,16}, which in turn renders our system as a suitable platform for demonstrating a Bell's inequality violation with matter waves using a closely related Rarity–Tapster scheme^{17}.
Results
Setup
The schematic diagram of the proposed experiment is shown in Fig. 1. A highly elongated (along the x axis) BEC is initially split into two equal and counterpropagating halves travelling with momenta ±k_{0} along z in the centreofmass frame. Constituent atoms undergo binary elastic collisions that produce a nearly spherical swave scattering halo of radius (ref. 8) in momentum space due to energy and momentum conservation. The elongated condensates have a diskshaped density distribution in momentum space, shown in Fig. 1b on the north and south poles of the halo. After the end of the collision (which in this geometry corresponds to complete spatial separation of the condensates in position space), we apply two counterpropagating lasers along the x axis whose intensity and frequency are tuned to act as a resonant Bragg πpulse with respect to two diametrically opposing momentum modes, k_{1} and k_{2}=−k_{1}, situated on the equatorial plane of the halo and satisfying k_{1,2}=k_{r}.
Previous experiments and theoretical work^{7,9,10,11,18,19,20,21,22,23} have shown the existence of strong atom–atom correlation between such diametrically opposite modes, similar to the correlation between twinphotons in parametric downconversion. Applying the Bragg πpulse to the collisional halo replicates an optical mirror and reverses the trajectories of the scattered atoms with momenta k_{1} and k_{2}, and a finite region around them. We assume that the pulse is tuned to operate in the socalled Bragg regime of the KapitzaDirac effect^{13,24} (diffraction of a matter wave from a standing light field), corresponding to conditions in which second and higherorder diffractions are suppressed. The system is then allowed to propagate freely for a duration so that the targeted atomic wavepackets regain spatial overlap in position space. We then apply a second Bragg pulse—a π/2pulse—to replicate an optical 50:50 beam splitter, which is again targeted to couple k_{1} and k_{2}, thus realising the HOM interferometer.
The timeline of the proposed experiment is illustrated in Fig. 2a, whereas the results of numerical simulations (see Methods) of the collision dynamics and the application of Bragg pulses are shown in Fig. 2b–d: panel b shows the equatorial slice of the momentumspace density distribution n(k,t) of the scattering halo at the end of collision; panels c and d show the halo density after the application of the π and π/2 pulses, respectively. The ‘banana’ shaped regions in panel c correspond to ‘kicked’ populations between the targeted momenta around k_{1} and k_{2} in the original scattering halo, whereas panel d shows the density distribution after mixing. The density modulation in panel c is simply the result of interference between the residual and transferred atomic populations after the πpulse upon their recombination on the beam splitter. The residual population is due to the fact that the pairs of offresonance modes in these parts of the halo (which are coupled by the same Bragg pulses as they share the same momentum difference 2k_{r} as the resonant modes k_{1} and k_{2}) no longer satisfy the perfect Braggresonance condition and therefore the population transfer during the πpulse is not 100% efficient. As these components have unequal absolute momenta, their amplitudes accumulate a nonzero relative phase due to phase dispersion during the free propagation. The accrued relative phase results in interference fringes upon the recombination on the beam splitter, with an approximate period of .
Owing to the indistinguishability of the paths of the Braggresonant modes k_{1} and k_{2} through the beam splitter and the resulting destructive quantum interference, a measurement of coincidence counts between the atomic populations in these modes will reveal a suppression compared with the background level. To reveal the full structure of the HOM dip, including the background level where no quantum interference occurs, we must introduce path distinguishability between the k_{1} and k_{2} modes. One way to achieve this, which would be in a direct analogy with shifting the beam splitter in the optical HOM scheme, is to change the Braggpulse resonance condition from the (k_{1}, k_{2}) pair to (k_{1}, k_{2}+ê_{x}δk), where ê_{x} is the unit vector in the xdirection. The approach to the background coincidence rate between the populations in the k_{1} and k_{2} modes would then correspond to performing the same experiment for increasingly large displacements δk. Taking into account that acquiring statistically significant results for each δk requires repeated runs of the experiment (typically thousands), this measurement protocol could potentially pose a significant practical challenge because of the very large total number of experimental runs required.
Proposed measurement protocol
To overcome this challenge, we propose an alternative measurement protocol that can reveal the full structure of the HOM dip from just one Braggresonance condition, requiring only one set of experimental runs. The protocol takes advantage of the broadband, multimode nature of the scattering halo and the fact that the original Braggpulse couples not only the targeted momentum modes k_{1} and k_{2}, but also many other pairs of modes that follow distinguishable paths through the beam splitter. One such pair, k_{3}=(k_{x},k_{y},k_{z})=k_{r}(cos(θ),sin(θ),0) and k_{4}=−k_{3}, located on the halo peak, is shown in Fig. 3a and corresponds to a rotation by angle θ away from k_{1} and k_{2}. The modes k_{3} and k_{4} are equivalent to the original pair in the sense of their quantum statistical properties and therefore, these modes can be used for the measurement of the background level of coincidence counts, instead of physically altering the paths of the k_{1} and k_{2} modes. The angle θ now serves the role of the ‘displacement’ parameter that scans through the shape of the HOM dip. A topologically equivalent optical scheme is shown in Fig. 3b,c, which is in turn similar to the one analysed in the study by Rarity and Tapster^{25} using a broadband source of angleseparated pairphotons and directionally asymmetric apertures.
In the proposed protocol, detection (after the final Bragg pulse) of atom coincidences at the pair of originally correlated momenta k_{3} and k_{4} corresponds to both paths being separately reflected on the beam splitter (Fig. 3c). Apart from this outcome, we need to take into account the coincidences between the respective Braggpartner momenta, k_{6} and k_{5} (separated, respectively, from k_{3} and k_{4} by the same difference 2k_{r} as k_{1} from k_{2}). Coincidences at k_{6} and k_{5} correspond to atoms of the originally correlated momenta k_{3} and k_{4} being both transmitted through the beam splitter (Fig. 3c). Finally, in order to take into account all possible channels contributing to coincidence counts between the two arms of the interferometer, we need to measure coincidences between k_{3} and k_{6}, as well as between k_{4} and k_{5}. This ensures that the total detected flux at the output ports of the beam splitter matches the total input flux. In addition to this, we normalize the bare coincidence counts to the product of singledetector count rates, that is, the product of the average number of atoms in the two output arms of the interferometer. We use the normalized correlation function as the total population in the four relevant modes varies as the angle θ is increased, implying that the raw coincidence rates are not a suitable quantity to compare at different angles.
HOM effect and visibility of the dip
With this measurement protocol in mind, we quantify the HOM effect using the normalized secondorder correlation function after the π/2pulse concludes at t=t_{4}. Here, and correspond to the number of atoms detected, respectively, on the two (right and left) output ports of the beam splitter, with the detection bins centred around the four momenta of interest k_{i} (i=3, 4, 5 and 6), for any given angle θ (Fig. 2e). More specifically, is the atom number operator in the integration volume centred around k_{i}, where is the momentumspace density operator, with and the corresponding creation and annihilation operators (the Fourier components of the field operators and , see Methods). The doublecolon notation in indicates normal ordering of the creation and annihilation operators.
The integrated form of the secondorder correlation function, which quantifies the correlations in terms of atom number coincidences in detection bins of certain size rather than in terms of local densitydensity correlations, accounts for limitations in the experimental detector resolution, in addition to improving the signaltonoise ratio that is typically low owing to the relatively low density of the scattering halo; in typical condensate collision experiments and in our simulations, the low density translates to a typical halomode occupation of ~0.1. We choose to be a rectangular box with dimensions corresponding to the r.m.s. width of the initial momentum distribution of the trapped condensate, which is a reasonable approximation to the mode (or coherence) volume in the scattering halo^{10,22}.
The secondorder correlation function, quantifying the HOM effect as a function of the pathdistinguishability angle θ, is shown in Fig. 4. For θ=0, where k_{3(4)}=k_{1(2)}, we observe maximum suppression of coincidence counts relative to the background level because of the indistinguishability of the paths. As we increase θ>0, we no longer mix k_{3} and k_{4} as a pair and their paths through the beam splitter become distinguishable; the path interference is lost, and we observe an increase in the magnitude of the correlation function to the background level. We quantify the visibility of the HOM dip via , where occurs for θ=0 and for sufficiently large θ such that momenta k_{5,6} lie outside the scattering halo. Owing to the oscillatory nature of the wings (see below) we take to correspond to the mean of for . Using this definition, we measure a visibility of , where the uncertainty of ±0.08 corresponds to taking into account the full fluctuations of about the mean in the wings rather than fitting the oscillations (Supplementary Fig. 1 and Supplementary Notes 1 and 2). The visibility >0.5 is consistent with the nonclassical effect of violation of Cauchy–Schwarz inequality with matter waves^{22}, observed recently in condensate collision experiments^{11}. The exact relationship between the visibility and the Cauchy–Schwarz inequality is discussed further in Supplementary Note 3, as are simple (approximate) analytic estimates of the magnitude of the HOM dip visibility (Supplementary Notes 1–5 and Supplementary Fig. 2).
Width of the HOM dip
The broadband, multimode nature of the scattering halo implies that the range of the pathlength difference over which the HOM effect can be observed is determined by the spectral width of the density profile of the scattering halo. Therefore, the width of the HOM dip is related to the width of the halo density. This is similar to the situation analysed in Rarity and Tapster^{25} using pairphotons from a broadband parametric downconverter. The angular width of the HOM dip extracted from Fig. 4 is approximately radians, which is indeed close to the width (full width at half maximum) of the scattering halo in the relevant direction, radians (see also Supplementary Note 2 for simple analytic estimates). The same multimode nature of the scattering halo contributes to the oscillatory behaviour in the wings of the HOM dip profile: here we mix halo modes with unequal absolute momenta and the resulting phase dispersion from free propagation leads to oscillations similar to those observed with twocolour photons^{25}.
Comparison with optical parametric downconversion
We emphasize that the input state in our matterwave HOM interferometer is subtly different from the idealized twinFock state 1,1› used in the simplest analytic descriptions of the optical HOM effect. This idealized state stems from treating the process of spontaneous optical parametric downconversion (SPDC) in the weakgain regime. We illustrate this approximation by considering a twomode toy model of the process, which in the undepleted pump approximation is described by the Hamiltonian that produces perfectly correlated photons in the â_{1} and â_{2} modes, where is a gain coefficient related to the quadratic nonlinearity of the medium and the amplitude of the coherent pump beam. (In the context of condensate collisions, the coupling corresponds to at the same level of ‘undepleted pump’ approximation^{10,22}; see Methods for the definitions of U and ρ_{0}.) The full output state of the SPDC process in the Schrödinger picture is given by , where α=tanh(gt) and t is the interaction time^{26}. In the weakgain regime, corresponding to , this state is well approximated by , that is, by truncating the expansion of and neglecting the contribution of the 2,2› and higher n components. This regime corresponds to mode populations being much >1, . The truncated state itself is qualitatively identical to the idealized state 1,1› as an input to the HOM interferometer: both result in a HOM dip minimum of and in the wings (where for the 1,1› case), with the resulting maximum visibility of V=1. If, on the other hand, the contribution of the 2,2› and higher n components is not negligible (which is the case, for example, of then the raw coincidence counts at the HOM dip and the respective normalized correlation function no longer equal to zero; in fact, the full SPDC state for arbitrary α<1 leads to a HOM dip minimum of and in the wings, which in turn results in a reduced visibility of . Despite this reduction, the visibility is still very close to V=1 in the weakgain regime (), where it scales as .
The process of fourwave mixing of matter waves gives rise to an output state analogous to the above SPDC state for each pair of correlated modes (see, for example, refs 10, 22 and Supplementary Note 1). Indeed, the fraction of atoms converted from the source BEC to all scattering modes is typically <5%, which justifies the use of the undepleted pump approximation. The typical occupation numbers of the scattered modes are, however, beyond the extreme of a very weak gain. In our simulations, the mode occupation on the scattering halo is on the order of 0.1 and therefore, even in the simplified analytical treatment of the process, the output state of any given pair of correlated modes cannot be approximated by the truncated state 0,0›+α1,1› or indeed the idealized twinFock state 1,1›.
Scaling with mode population and experimental considerations
At the basic level, our proposal only relies on the existence of the aforementioned paircorrelations between scattered atoms, with the strength of the correlations affecting the visibility of the HOM dip. For a sufficiently homogeneous source BEC^{22,27}, the correlations and thus the visibility V effectively depend only on the average mode population in the scattering halo, with a scaling of V on given by by our analytic model. Dependence of on system parameters such as the total number of atoms in the initial BEC, trap frequencies and collision duration is well understood both theoretically and experimentally^{7,8,10,11}, and each can be sufficiently controlled such that a suitable mode population of can, in principle, be targeted. There lies, however, a need for optimization: very small populations are preferred for higher visibility, but they inevitably lead to a low signal to noise, hence requiring a potentially very large number of experimental runs for acquiring statistically significant data. Large occupations, on the other hand, lead to higher signal to noise, but also to a degradation of the visibility towards the nonclassical threshold of V=0.5. The mode population of ~0.1 resulting from our numerical simulations appears to be a reasonable compromise; following the scaling of the visibility with predicted by the simple analytic model, it appears that one could safely increase the population to ~0.2 before a nonclassical threshold is reached to within a typical uncertainty of ~12% (as per quoted value of ) obtained through our simulations.
The proposal is also robust to other experimental considerations such as the implementation of the Bragg pulses; for example, one may use square Bragg pulses rather than Gaussians. Furthermore, experimental control of the Bragg pulses is sufficiently accurate to avoid any degradation of the dip visibility. Modifying the relative timing of the π and π/2 pulses by few percent in our simulations does not explicitly affect the dip visibility, rather only the period of the oscillations in the wings of . This may lead to a systematic change in the calculated dip visibility, however, this is overwhelmed by the uncertainty of 12% that accounts for the fluctuations of about the mean.
Importantly, we expect that the fundamentally new aspects of the matterwave setup, namely the multimode nature of the scattering halo and the differences from the archetypical HOM input state of 1,1›, as well as the specific measurement protocol we have proposed for dealing with these new aspects, are broadly applicable to other related matterwave setups that generate paircorrelated atoms. These include molecular dissociation^{19}, an elongated BEC in a parametrically shaken trap^{14}, or degenerate fourwave mixing in an optical lattice^{28,29}. In the present work, we focus on condensate collisions only because of the accurate characterization, both experimental and theoretical, of the atom–atom correlations, including in a variety of collision geometries^{7,8,9,10,11}.
Discussion
In summary, we have shown that an atomoptics analogue of the HOM effect can be realised using colliding condensates and laserinduced Bragg pulses. The HOM dip visibility >50% implies that the atom–atom correlations in this process cannot be described by classical stochastic random variables. Generation and detection of such quantum correlations in matter waves can serve as precursors to stronger tests of quantum mechanics such as those implied by a Bell inequality violation and the Einstein–Podolsky–Rosen paradox^{30}. In particular, the experimental demonstration of the atomoptics HOM effect would serve as a suitable starting point to experimentally demonstrate a violation of a Bell inequality using an atomoptics adaptation of the Rarity–Tapster setup^{17}. In this setup, one would tune the Bragg pulses as to realize two separate HOM interferometer arms, enabling to mix two angleresolved pairs of momentum modes from the collisional halo, such as (k,q) and (−k,−q), which would then form the basis of a Bell state (ref. 31).
Methods
Stochastic Bogoliubov approach for simulations
To simulate the collision dynamics, we use the timedependent stochastic Bogoliubov approach^{8,23} used previously to accurately model a number of condensate collision experiments^{8,7,11}. In this approach, the atomic field operator is split into , where is the meanfield component describing the source condensates and is the fluctuating component (treated to lowest order in perturbation theory) describing the scattered atoms. The meanfield component evolves according to the standard timedependent Gross–Pitaevskii equation, where the initial state is taken in the form of . This models an instantaneous splitting at t=0 of a zerotemperature condensate in a coherent state into two halves that subsequently evolve in free space, where ρ_{0}(r) is the particle number density of the initial (trapped) sample before splitting.
The fluctuating component is simulated using the stochastic counterpart of the Heisenberg operator equations of motion^{8,22}, , in the positive Prepresentation with the vacuum initial state. Here represents the kinetic energy term, an effective meanfield potential, plus the lattice potential V_{BP}(r,t) imposed by the Bragg pulses, whereas is an effective coupling responsible for the spontaneous pairproduction of scattered atoms. The interaction constant U is given by U=4πℏ^{2}a/m, where m is the atomic mass and a is the swave scattering length.
Details of Bragg pulses
The Bragg pulses are realised by two interfering laser beams (assumed for simplicity to have a uniform intensity across the atomic cloud and zero relative phase) that create a periodic lattice potential , where V_{L}(t) is the lattice depth and is the lattice vector determined by the wave vectors k_{L,i} (i=1, 2) of the two lasers, and tuned to k_{L}=k_{r}. The Bragg pulses couple momentum modes k_{i} and k_{j}=k_{i}−2k_{L}, satisfying momentum and energy conservation (up to a finite width owing to energytime uncertainty^{24}). The lattice depth is ramped up (down) according to , where t_{2(3)} is the pulse centre, whereas τ_{π(π/2)} is the pulse duration that governs the transfer of atomic population between the targeted momentum modes: a πpulse is defined by and converts the entire population from one momentum mode to the other, whereas a π/2pulse is defined by and converts only half of the population.
Aspects of measurement after expansion
In practice, the atom–atom correlations quantifying the HOM interference are measured in position space after the lowdensity scattering halo expands ballistically in free space and falls under gravity onto an atom detector. The detector records the arrival times and positions of individual atoms, which is literally the case for metastable helium atoms considered here^{7,8,9,11,32}. The arrival times and positions are used to reconstruct the threedimensional velocity (momentum) distribution before expansion, as well as the atom–atom coincidences for any desired pair of momentum vectors. In our simulations and the proposed geometry of the experiment, the entire system (including the Bragg pulses) maintains reflectional symmetry about the yzplane, with z being the vertical direction. Therefore, the effect of gravity can be completely ignored as it does not introduce any asymmetry to the momentum distribution of the atoms and their correlations on the equatorial plane of the halo or indeed any other plane parallel to it.
Additional information
How to cite this article: LewisSwan, R. J. & Kheruntsyan, K. V. Proposal for demonstrating the Hong–Ou–Mandel effect with matter waves. Nat. Commun. 5:3752 doi: 10.1038/ncomms4752 (2014).
References
Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044–2046 (1987).
Ou, Z. Y. & Mandel, L. Violation of Bell's inequality and classical probability in a twophoton correlation experiment. Phys. Rev. Lett. 61, 50–53 (1988).
Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001).
Kok, P., Lee, H. & Dowling, J. P. Creation of largephotonnumber path entanglement conditioned on photodetection. Phys. Rev. A 65, 052104 (2002).
Duan, L.M. & Monroe, C. Colloquium: Quantum networks with trapped ions. Rev. Mod. Phys. 82, 1209–1224 (2010).
Lang, C. et al. Correlations, indistinguishability and entanglement in HongOuMandel experiments at microwave frequencies. Nat. Phys. 9, 345–348 (2013).
Jaskula, J. et al. SubPoissonian number differences in fourwave mixing of matter waves. Phys. Rev. Lett. 105, 190402 (2010).
Krachmalnicoff, V. et al. Spontaneous fourwave mixing of de Broglie waves: Beyond optics. Phys. Rev. Lett. 104, 150402 (2010).
Perrin, A. et al. Observation of atom pairs in spontaneous fourwave mixing of two colliding BoseEinstein condensates. Phys. Rev. Lett. 99, 150405 (2007).
Perrin, A. et al. Atomic fourwave mixing via condensate collisions. New J. Phys. 10, 045021 (2008).
Kheruntsyan, K. V. et al. Violation of the CauchySchwarz inequality with matter waves. Phys. Rev. Lett. 108, 260401 (2012).
Kozuma, M. et al. Coherent splitting of BoseEinstein condensed atoms with optically induced Bragg diffraction. Phys. Rev. Lett. 82, 871–875 (1999).
Meystre, P. Atom Optics SpringerVerlag (2001).
Bücker, R. et al. Twinatom beams. Nat. Phys. 7, 608–611 (2011).
Lücke, B. et al. Twin matter waves for interferometry beyond the classical limit. Science 334, 773–776 (2011).
Gross, C. et al. Atomic homodyne detection of continuousvariable entangled twinatom states. Nature 480, 219–223 (2011).
Rarity, J. G. & Tapster, P. R. Experimental violation of Bell's inequality based on phase and momentum. Phys. Rev. Lett. 64, 2495–2498 (1990).
Norrie, A. A., Ballagh, R. J. & Gardiner, C. W. Quantum turbulence and correlations in BoseEinstein condensate collisions. Phys. Rev. A 73, 043617 (2006).
Savage, C. M., Schwenn, P. E. & Kheruntsyan, K. V. Firstprinciples quantum simulations of dissociation of molecular condensates: Atom correlations in momentum space. Phys. Rev. A 74, 033620 (2006).
Deuar, P. & Drummond, P. D. Correlations in a BEC collision: firstprinciples quantum dynamics with 150,000 atoms. Phys. Rev. Lett. 98, 120402 (2007).
Mølmer, K. et al. Hanbury Brown and Twiss correlations in atoms scattered from colliding condensates. Phys. Rev. A 77, 033601 (2008).
Ogren, M. & Kheruntsyan, K. V. Atomatom correlations in colliding BoseEinstein condensates. Phys. Rev. A. 79, 021606 (2009).
Deuar, P., Chwedeńczuk, J., Trippenbach, M. & Ziń, P. Bogoliubov dynamics of condensate collisions using the positiveP representation. Phys. Rev. A 83, 063625 (2011).
Batelaan, H. The KapitzaDirac effect. Contemp. Phys. 41, 369–381 (2000).
Rarity, J. G. & Tapster, P. R. Twocolor photons and nonlocality in fourthorder interference. Phys. Rev. A 41, 5139–5146 (1990).
Braunstein, S. L. & van Loock, P. Quantum information with continuous variables. Rev. Mod. Phys. 77, 513–577 (2005).
Ögren, M. & Kheruntsyan, K. V. Role of spatial inhomogeneity in dissociation of trapped molecular condensates. Phys. Rev. A 82, 013641 (2010).
Hilligsøe, K. M. & Mølmer, K. Phasematched four wave mixing and quantum beam splitting of matter waves in a periodic potential. Phys. Rev. A 71, 041602 (2005).
Bonneau, M. et al. Tunable source of correlated atom beams. Phys. Rev. A 87, 061603 (2013).
Kofler, J. et al. EinsteinPodolskyRosen correlations from colliding BoseEinstein condensates. Phys. Rev. A 86, 032115 (2012).
LewisSwan, R. J. & Kheruntsyan, K. V. Book of Abstracts of ICAP 2012—The 23rd International Conference on Atomic Physics 291 (Palaiseau, France (2012).
Vassen, W. et al. Cold and trapped metastable noble gases. Rev. Mod. Phys. 84, 175–210 (2012).
Acknowledgements
We acknowledge discussions with C.I. Westbrook and D. Boiron. K.V.K. acknowledges support by the Australian Research Council Future Fellowship award FT100100285.
Author information
Authors and Affiliations
Contributions
R.J.L.S. conducted the numerical simulations and derived the analytic treatment. Both authors contributed extensively to the conceptual formulation of the physics, the interpretation of the data and writing the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 12, Supplementary Notes 15 and Supplementary References (PDF 239 kb)
Rights and permissions
About this article
Cite this article
LewisSwan, R., Kheruntsyan, K. Proposal for demonstrating the Hong–Ou–Mandel effect with matter waves. Nat Commun 5, 3752 (2014). https://doi.org/10.1038/ncomms4752
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms4752
This article is cited by

On the survival of the quantum depletion of a condensate after release from a magnetic trap
Scientific Reports (2022)

Protocol designs for NOON states
Communications Physics (2022)

Ghost imaging with atoms
Nature (2016)

Atomic Hong–Ou–Mandel experiment
Nature (2015)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.