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Mahler, D. H., Rozema, L., Fisher, K., Vermeyden, L., Resch, K. J., Wiseman, H. M., & Steinberg, A. (2016). Experimental nonlocal and surreal Bohmian trajectories. Science Advances, 2(2), [e1501466]. https://doi.org/10.1126/sciadv.1501466 Publisher's PDF, also known as Version of record License (if available): CC BY-NC Link to published version (if available): 10.1126/sciadv.1501466 Link to publication record in Explore Bristol Research PDF-document This is the final published version of the article (version of record). It first appeared online via Science at http://advances.sciencemag.org/content/2/2/e1501466. Please refer to any applicable terms of use of the publisher. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ RESEARCH ARTICLE QUANTUM MECHANICS 2016 © The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. Distributed Experimental nonlocal and surreal under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC). Bohmian trajectories 10.1126/sciadv.1501466 Dylan H. Mahler,1,2* Lee Rozema,1,2 Kent Fisher,3 Lydia Vermeyden,3 Kevin J. Resch,3 Howard M. Wiseman,4* Aephraim Steinberg1,2 Weak measurement allows one to empirically determine a set of average trajectories for an ensemble of quantum particles. However, when two particles are entangled, the trajectories of the first particle can depend nonlocally on the position of the second particle. Moreover, the theory describing these trajectories, called Bohmian mechanics, predicts trajectories that were at first deemed “surreal” when the second particle is used to probe the position of thefirstparticle.Weentangletwophotonsanddeterminea set of Bohmian trajectories for one of them using weak measurements and postselection. We show that the trajectories seem surreal only if one ignores their manifest Downloaded from nonlocality. INTRODUCTION to the theory (as in other interpretations of quantum mechanics) but is The concept of a trajectory, the path followed by a particle, is ubiquitous rather due to the evolution of the global wave function, describing the in classical mechanics. In orthodox quantum mechanics, however, a interaction of the particle of interest with a measurement device that has particle does not follow a trajectory, because it does not have a simul- its own Bohmian degrees of freedom. Although a single particle’stra- http://advances.sciencemag.org/ taneous position and momentum. Nonetheless, it is possible to re- jectory cannot be directly observed, a set of trajectories of an ensemble of interpret the quantum formalism as describing particles following particles can be mapped out. This can be done by making a so-called definite trajectories, each with a precisely defined position at each instant weak measurement of the momentum of a particle at a given instant in in time. However, in this interpretation, called Bohmian mechanics (1–4), time. The weakness ensures that the system is not disturbed appreciably, orthedeBroglie–Bohm interpretation (5, 6), the trajectories of the par- so that it is sensible to make a subsequent measurement of position. ticles are quite different from those of classical particles, because they Repeating the experiment many times, one can calculate the average mo- are guided by the wave function. This allows for phenomena such as mentum as a function of position. This entire process can then be re- double-slit interference, as has been investigated experimentally for peated at many instants in time, allowing a set of average trajectories single photons (7). Note that this is very different from the Feynman to be reconstructed. It was shown by Wiseman (9) how these trajectories, path formalism of quantum mechanics (8), where the transition prob- in the limit of very weak measurements, correspond exactly to the tra- ability between two points in phase space is calculated using all pos- jectories obtained from the Bohmian interpretation. sible paths between those two points. In contrast to the Feynman To explain nonlocal phenomena such as Bell nonlocality (10), any formalism, Bohmian mechanics says that each quantum particle in a realistic interpretation of quantum mechanics must also be nonlocal, on October 17, 2016 given experiment follows a trajectory in a deterministic manner. Thus, and Bohmian mechanicsisnoexception(2).Thiscanbeseeninthe much of the intuition of classical mechanics is regained. Bohmian velocity law (shown later), where the velocity of a particle can As with any interpretation of quantum mechanics, the experimental depend explicitly on the position of a second particle, even when the predictions of Bohmian mechanics are the same as those in the oper- particles are far apart and not interacting by any conventional mecha- ationaltheory.Thestochasticnatureofmeasurementoutcomesforwhich nism and even though the second particle may be influenced, indepen- quantummechanicsisfamousisascribedtoignoranceabouttheinitial dently of the first particle, by the apparatus chosen by the experimenter. configuration of the particle(s) in the experiment, an uncertainty that is Here, as proposed by Braverman and Simon (11), we experiment described precisely by the wave function prepared by the experimenter on two entangled particles (photons) and map out the trajectories of one (1–4, 6, 9). This tidily allows features of operational quantum theory, of them as it traverses a double-slit apparatus. We show that the trajec- such as the Heisenberg uncertainty principle, to be explained within tories of this first particle (and therefore both its position and its velocity) the Bohmian formalism. A consequence of this is that the trajectory of are indeed affected by an externally controlled influence on the distant a single Bohmian particle cannot be observed in an experiment on that second particle. For some choices of that control, the second particle in particle; any measurement of a particle’spositionchangesthewave our experiment can be used to determine through which slit the first function and thus the guiding potential that the particle experiences. particle has gone. Englert, Scully, Süssmann, and Walther (ESSW) (12) Note that this change does not occur through a “collapse” that is added asserted that in the presence of such a Welcher Weg measurement (WWM) device, the particle’s Bohmian trajectories can display seem- 1Centre for Quantum Information and Quantum Control, Department of Physics, ingly contradictory behavior: There are instances when the particle’s University of Toronto, 60 Saint George Street, Toronto, Ontario M5S 1A7, Canada. 2Canadian Institute for Advanced Research, 180 Dundas Street West, Suite 1400, Bohmian trajectory goes through one slit, and yet the WWM result Toronto, Ontario M5G 1Z8, Canada. 3Institute for Quantum Computing and Department indicatesthatithadgonethroughthe other slit. ESSW concluded that of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, these trajectories predicted by Bohmian mechanics could not correspond 4 Ontario N2L 3G1, Canada. Centre for Quantum Dynamics, Griffith University, Brisbane, to reality and they dubbed them “surreal trajectories.” This serious as- Queensland 4111, Australia. 13–17 *Corresponding author. E-mail: [email protected] (D.M.); h.wiseman@griffith. sertion was discussed at length in the literature ( ), after which a edu.au (H.W.) resolution of this seeming inconsistency was proposed by Hiley et al. Mahler et al. Sci. Adv. 2016; 2 : e1501466 19 February 2016 1of7 RESEARCH ARTICLE (18). Here, we present an experimental validation of this resolution, in slit particle 1 had gone (even though, in orthodox quantum mechanics, the which the nonlocality of Bohmian mechanics comes to the fore. particle has no trajectory and hence did not “go through” either slit). In some situations, Bohmian mechanics accords with that intuition. If the WWM is read out as described above, at time tr >0,then THEORY for times t < tr, fH(x2; t) and fV (x2; t) are identical, whereas for ThedeBroglie–Bohm dynamics can be formulated in a number of dif- times t > tr, fH(x2; t)andfV(x2; t) have disjoint support. If, at the time ferent ways (1, 3–6, 9). Here, we present the formulation that is simplest tr, particle 1 is still in the near field of the double-slit apparatus, as in and most appropriate to our method of empirical determination via Fig. 1A, then the measurement outcome (the position of particle 2) is weak values (4, 9). Being a complete interpretation of quantum mechan- perfectly correlated with the origin (upper slit or lower slit) of each ics, Bohmian mechanics applies to arbitrarily many particles and allows Bohmian trajectory. The velocity formula (Eq. 1) for particle 1 and for internal degrees of freedom (such as spin). Here, we are concerned t >tr, as it traverses the apparatus toward the far field, gives with a two-particle entangled state |Y〉. Denoting the positions (in one −iℏyu′ðx1; tÞ −iℏy′lðx1; tÞ Re or Re , respectively, as expected from dimension for simplicity) of the two particles by x1 and x2, the Bohmian myuðx1; tÞ mylðx1; tÞ 4 9 velocity of particle 1 is ( , ) single-particle Bohmian mechanics. Downloaded from ˇ ESSW, however, consider a situation where the readout via the po- 〈Yjv 1jx1〉jx2〉〈x2j〈x1jY〉 v1ðx1;

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