The Hong-Ou-Mandel experiment and Bohmian mechanics

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Dated: 25 July 2017

This essentially arose from a discussion between Sofia Wechsler and myself. She had written a paper, in which she tried to use a version of the Hong-Ou-Mandel experiment, employing atoms rather than the standard photons, to refute Bohmian trajectories.

As far as I can tell, this was motivated by an earlier article by P. Ghose [1], claiming that Bohmian mechanics does not correctly treat correlations between identical particles as the Bohmian trajectories would allow to distinguish the particles, at least in principle, via the initial condition and their trajectories. According to standard quantum mechanics, that claim must be wrong, because the (full N-particle) wave function is assumed to encode the maximum information possible about a quantum system. Therefore, all purely spatial (i.e. one-time) correlations appearing in the particle system must be correctly representable from the wave function. Moreover, indistuingishability of the constituent particles is a symmetry property of the many-body wave function of a many-particle system, so it is also encoded in the wave function. Bohmian mechanics is so constructed that all its experimentally verifiable predictions are the same as that of Schrödinger theory.

There are additional predictions or "postdictions" of Bohmian mechanics, referring to the trajectories, that cannot be tested in any strict sense. (Therefore, the trajectories cannot be used to distinguish particles, even in principle, as long as the initial wave function of a Bohmian ensemble corresponds to the probability distribution of its particles.)

In order to find predictions of Bohmian mechanics that might not agree with standard quantum mechanics, one might consider multiple-time correlation functions that are defined in the Heisenberg picture. If this lead to a contradiction with Bohmian mechanics, it would presumably also have to contradict Schrödinger picture predictions. Since the prospect is dim to find any such differences between the Schrödinger and Heisenberg pictures, it is not very likely that a route towards demonstrating errors in Bohmian mechanics can be found this way.

It is, however, possible to render the Bohmian construction of particle trajectories (i.e., the domain of unobservable predictions) less plausible than they may appear at first sight. Precisely this was done in a famous paper by Englert, Scully, Süssmann, and Walther (ESSW) [2], where they suggested a thought experiment with an incomplete Stern-Gerlach interferometer and the storage of which-way information in a microcavity by a single photon, emitted when an atom traverses the cavity. The arrangement of the interferometer has a symmetry plane, and Bohmian mechanics states that a particle trajectory cannot traverse this plane. Standard quantum mechanics predicts that an atom that has triggered the which-way detector above the symmetry plane will arrive on the screen below it and vice versa. The standard quantum mechanical interpretation thus suggests that particles cross the symmetry plane whereas the Bohmian trajectories don't do so. Therefore, if we measure the position of a particle twice with a sufficient time interval between them, the two positions will not lie on the same Bohmian trajectory, which hence is a metaphysical rather than a physical concept, or surrealistic rather than realistic.

Bohmianists have, of course, attacked the ESSW setup and argued that the measurement via a which-way detector, which corresponds to a rather microscopic process (a single photon is deposited in the cavity), did not constitute a measurement. ESSW explicitly state that they do not claim Bohmian mechanics to make predictions that differ from those of standard quantum mechanics [3]. Their argument rather attacks the plausibility of the (unobservable) Bohmian trajectories. They also do not deny the mathematical existence of these trajectories [4], they just consider their interpretation as particle trajectories doubtful. This kind of objection is not really removed by newer work based on weak measurements [5].

Considering the way Bohmian trajectories are constructed, it is obvious that in a stationary situation they just correspond to the streamlines of the probability current density. These streamlines can be measured via weak measurement techniques. If they are considered "average particle" trajectories, we then have a measurement method for Bohmian trajectories [5]! Nevertheless, it is not clear at all to what extent this gives us information on true particle trajectories. In hydrodynamics, streamlines indicate the local direction of motion of fluid volume elements. In the stationary case, they can be used to construct trajectories of volume elements. But of particles of the liquid? No. The particles diffuse in addition to the motion indicated by the streamlines and move arbitrarily far away from them. The average has nothing to do anymore with the true trajectory, after a sufficient amount of time. The trajectories are nondifferentiable, the streamlines are smooth. There is no proof that in quantum mechanics particle trajectories coincide with the streamlines of the probability current density. Hence, the conclusion that the measurement of these streamlines makes the Bohmian trajectories less surreal may be considered doubtful.

Anyway, let me return to Ghose's paper. He considers an interferometer with a symmetry plane. He then infers correctly that two Bohmian particles starting from symmetrical initial positions about the symmetry plane will always hit the screen of the interferometer on different sides of the plane. Considering the initial wave function, creating the distribution from which the particle pairs were taken, he then concludes that in standard quantum mechanics the two particles can reach the screen on the same side of the symmetry plane and hence there is a contradiction between Bohmian mechanics and standard quantum mechanics. There is a small problem however: symmetry of the initial wave function does not guarantee that two Bohmian particles drawn from the corresponding distribution are symmetrically arranged with respect to the symmetry plane. The distribution contains many more possible initial particle combinations, corresponding to asymmetric initial arrangements. In order to compare predictions of Bohmian mechanics and standard quantum mechanics, however, the probability distribution of the particles must be identical to the absolute square of the wave function. In restricting himself to a subensemble of particle pairs that are symmetrically arranged, Ghose does not consider the distribution corresponding to the initial wave function but a distribution obtained from it via multiplication by a delta function of the form δ(x1+x2).

Clearly, the predictions of Bohmian mechanics gathered from this distribution need not agree with those of standard quantum mechanics from a distribution without the delta function factor. In order to have a valid comparison, either the ensemble of Bohmian particles must correspond to the original Schrödinger wave function, meaning that all possible pairs of non-symmetrically arranged Bohmian particles compatible with that wave function must be factored into the prediction with their proper weights. Or else, a Schrödinger wave function must be considered that actually corresponds to the subensemble of Bohmian particles considered. (One might replace the delta function by a strong but finite peak to have something regular.) Ehrenfest's theorem strongly suggests that in this case the standard quantum mechanical prediction is also that the two particles will arrive on different sides of the symmetry plane. (For the regularized wave function, a few pairs may arrive on one side of the plane, but the probability for this to happen should go to zero, as the peak is made narrower and larger.) But then Ghose's claim is simply wrong.

In her paper, Sofia discussed the Hong-Ou-Mandel experiment with two bosonic and identical atoms. According to standard quantum mechanics, these should always come out of the apparatus together, whereas Sofia believed she could show that two Bohmian atoms would always have different velocities and therefore be separated in the apparatus. Her first attempt suffered from the wave functions being assumed to be plane waves, which is not very helpful for predictions of positions. Later that was changed to Gaussian wave packets but the then necessary step of considering the whole distribution of possible Bohmian pairs making up that initial wave function was never taken, as far as I can tell. This is an error similar to that in Ghose's approach, although the systems considered are pretty different.

Anyway, my essay gives a qualitative discussion of how the two-particle wave function may be used to demonstrate that Bohmian mechanics reproduces the predictions of standard quantum mechanics. I also discuss that certain pairs of Bohmian particles show weird behavior constituting a deviation from what is expected in standard quantum mechanics. But the probability measure of these events is zero, so they will not lead to observable discrepancies.

[1] P. Ghose, Incompatibility of the de Broglie-Bohm Theory with Quantum Mechanics, arXiv: quant-ph/0001024v3, 23 April 2003.
[2] B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, Surrealistic Bohm Trajectories, Z. Naturforschung 47a, 1175-1186 (1992).
[3] B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, Reply to Comment on "Surrealistic Bohm Trajectories", Z. Naturforschung 48a, 1263-1264 (1993).
[4] M. O. Scully, Do Bohm Trajectories Always Provide a Trustworthy Physical Picture of Particle Motion?, Physica Scripta T76, 41-46 (1998).
[5] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, Experimental nonlocal and surreal Bohmian trajectories, Sci. Adv. 2: e1501466 (2016).

The Hong-Ou-Mandel experiment and Bohmian mechanics, 25.07.2017

Next: Quantum mechanical pictures, time dependent   Up: Introduction science education project    Previous: Superluminal communication scheme wrong

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