Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Braverman, Boris, and Christoph Simon. “Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons.” Physical Review Letters 110.6 (2013). © 2013 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevLett.110.060406 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/78322 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 110, 060406 (2013) PHYSICAL REVIEW LETTERS 8 FEBRUARY 2013 Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons Boris Braverman1,2 and Christoph Simon2 1Department of Physics, MIT-Harvard Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Institute for Quantum Information Science and Department of Physics and Astronomy, University of Calgary, Calgary T2N 1N4, Alberta, Canada (Received 11 July 2012; published 7 February 2013) Bohmian mechanics reproduces all statistical predictions of quantum mechanics, which ensures that entanglement cannot be used for superluminal signaling. However, individual Bohmian particles can experience superluminal influences. We propose to illustrate this point using a double double-slit setup with path-entangled photons. The Bohmian velocity field for one of the photons can be measured using a recently demonstrated weak-measurement technique. The found velocities strongly depend on the value of a phase shift that is applied to the other photon, potentially at spacelike separation. DOI: 10.1103/PhysRevLett.110.060406 PACS numbers: 03.65.Ud, 03.65.Ta, 42.50.Xa Bohmian mechanics [1] (BM) is the most famous and positions are distributed according to the modulus squared best developed hidden-variable theory for quantum phys- of the wave function. From the point of view of BM, the ics. It postulates the existence of both a quantum wave, theory of relativity therefore remains valid, but only in a which corresponds to the usual quantum wave function, statistical sense [7,16]. and of particles whose motion is guided by the wave, In this Letter we propose to show the nonlocal character following de Broglie [2]. The exact positions of these of BM in an experiment using entangled photon pairs. particles are the additional ‘‘hidden’’ variables compared Building on the ideas of Refs. [4,6], we propose to use to the usual quantum physical description. path-entangled photons and a double double-slit setup [17], Under the assumption that the distribution of particle with variable phase shifts between the two slits on one side. positions is given by the modulus squared of the wave We show that the Bohmian velocity field (and hence the function, which is the equilibrium state in BM [3], all trajectory) for the particle on the other side depends on statistical predictions of BM agree exactly with those of the phase shift applied to the first particle, and we discuss standard quantum mechanics. This means in particular that how this can be observed experimentally, thus allowing a the uncertainty principle applies, such that it is impossible striking demonstration of BM’s nonlocality. Note that our to precisely observe the trajectory of an individual proposal is not designed to disprove local hidden variables, Bohmian particle. which was done in Refs. [9–14]. Our goal here is to show However, in Ref. [4] it was pointed out that the velocity concretely how the nonlocality of quantum mechanics field for an ensemble of Bohmian particles, which is manifests itself in the Bohmian framework. related to the (multi-dimensional) gradient of the wave In BM, for a two-particle system the velocity field for function, can be experimentally observed in a direct and particle A is given by intuitive way using the concept of weak-value measure- jAðxA; xBÞ v ðx ; x Þ¼ ; ments [5]. This proposal was recently implemented in A A B c x ; x 2 (1) Ref. [6] for a double-slit experiment with single photons. j ð A BÞj The nonlocal character of BM was recognized by Bohm where as early as 1952 [7]. For entangled quantum states, actions i performed on one particle can have an instantaneous effect j ðx ; x Þ¼ c Ãðx ; x Þr c ðx ; x Þþc:c:; A A B 2m A B A A B (2) on the motion of another particle far away. This feature motivated Bell to study the question whether all hidden- and c ðxA; xBÞ is the two-particle wave function and m is variable theories that reproduce the statistical predictions of the mass of the particles. To obtain the velocity field for quantum mechanics have to be nonlocal. This question was particle B, the gradient is taken with respect to the position of course answered in the affirmative by Bell’s theorem [8]. of that particle. The velocity field vAðxA; xBÞ is interpreted Local hidden-variable models have since been ruled out as giving the velocity for a Bohmian particle A at position (and the predictions of quantum mechanics confirmed) by xA, provided that particle B is at position xB. It is easy to many experiments of increasing sophistication [9–15]. see that the dependence on particle B’s position disappears It should be emphasized that the superluminal influences for unentangled (product) quantum states. experienced by individual Bohmian particles cannot be It is tempting to interpret the fact that for entangled used for superluminal signaling, as long as the particle quantum systems the velocity for particle A depends on 0031-9007=13=110(6)=060406(5) 060406-1 Ó 2013 American Physical Society week ending PRL 110, 060406 (2013) PHYSICAL REVIEW LETTERS 8 FEBRUARY 2013 B the position of particle as an immediate demonstration of ðkÞ hkjp^jc i pw ¼ : (4) the nonlocality of BM. However, this is in fact not con- hkjc i clusive. BM is deterministic. This means that without external intervention the positions of the particles at all Comparing Eqs. (3) and (4) one sees that by identifying the times are uniquely determined by their initial positions system observable p^ with the momentum operator p^ A and plus the initial wave function. The apparent nonlocality the final measurement basis fjkig with the two-particle could therefore be seen as simply an unusual form of position basis fjxA; xBig, the velocity field is given by the expressing the dependence on the initial conditions. This real part of the weak values of the momentum operator. is particularly relevant in the typical case where the parti- The position-dependent (Bohmian) velocity information cles originate from the same source and were thus not far obtained in this way makes it possible to reconstruct the apart at all times. It is conceptually clearer to introduce an Bohmian trajectories. In the related single-particle experi- external local influence on one particle and study its effect ment of Ref. [6] the pointer was implemented by the on the other particle. This is the approach that we propose polarization degree of freedom of the individual photons, to pursue below. and the weak value of the momentum was inferred from the We now explain how the velocity field can be measured. rotation of the polarization; see below. In Ref. [4] it was pointed out that We will first describe the proposed experiment in con- ceptual terms, then we will discuss its implementation in 1 x ; x p^ c h A Bj Aj i more detail. We consider a source of pairs of entangled v AðxA; xBÞ¼ Re ; (3) m hxA; xBjc i particles (see Fig. 1). Particle A is emitted toward the left, and particle B toward the right. Each particle encounters a where p^ A is the momentum operator for particle A, and jc i double slit. The source is constructed in such a way that at is the two-particle quantum state, such that hxA; xBjc i¼ the time when each particle is in the plane of its respective c ðxA; xBÞ. The analogous relation holds for vBðxA; xBÞ double slit the wave function of the total system is and p^ B. Equation (3) allows one to make a link to the weak-value 1 pffiffiffi ðfuðxAÞfuðxBÞþfdðxAÞfdðxBÞÞ: (5) formalism [5]. In this approach the system under consid- 2 eration, which is initially in a given quantum state jc i,is first made to interact weakly with a pointer with an inter- Here we are only considering a single coordinate for each action Hamiltonian of the general form H ¼ p^ ^ , where particle (xA and xB, respectively), along a line connecting is the coupling strength, p^ is the observable of the system f x ^ the two slits (transverse to the direction of motion); uð AÞ that is to be measured weakly, and is an operator of the is the wave function corresponding to the upper slit for pointer. Then one performs a projective measurement of particle A. It has zero overlap with fdðxAÞ, which corre- fj ig the system in some basis k . One can show that for sponds to the lower slit, and analogously for particle B.We sufficiently weak interactions the operation performed on will also immediately introduce a phase shifter that is j i the pointer conditional on finding a final state k of the placed just behind the lower slit for particle A. It causes itpðkÞ^ system is then of the form e w , where t is the interac- a variable phase shift , leading to a modified wave ðkÞ tion time, and the weak value pw is given by function FIG.