A test of Wheeler. Local-realistic explanation of interferometry

(Cover image from: A.G. Manning, R.I. Khakimov, R.G. Dall, and A.G. Truscott, "Wheeler’s delayed-choice gedanken experiment with a single atom", Nature Physics, vol. 11, July 2015, DOI: 10.1038/NPHYS3343)

In previous posts, we have discussed several quantum mechanics scenarios (for example here, here, here, etc.) and seen how they are perfectly reproduced by our local-realistic model, both theoretically and numerically.

In this post we shall describe a further test for our model, that is reproducing the non-classical behavior of interferometers. We shall consider in particular atom interferometers, and leave those operating with photons to when we shall treat quantum electrodynamics.

A rather general interferometer scheme, shown in the figure below, consists of: (i) a source S where a beam in a particular state is prepared, (b) a first splitting of the beam into two paths, BS1, with different momentum states and phases (iii) a recombination M of the two paths (iv), a second beam splitter BS2 that creates interference between pairs of overlapping beams coming from different paths, and (v) a detector (or a pair of detectors D1 and D2) counting the arrivals of single atoms. We shall consider further an additional phase that is inserted (often, for the purpose of measuring it) in the upper path.

The reader can refer to the experiment of Manning et al. for an actual interferometer aimed at testing the famous Wheeler's delayed choice gedanken experiment. In this arrangement, the beams' split and recombination are realized with Bragg pulses. Each of the three stages shifts the momentum of the incoming atoms of a certain quantity along direction x transversal to the beam propagation. We shall consider that momentum along the prevalent beam direction is unaffected (e.g., determined by gravity).

The QM description of such appratuses is well known and is based on plane wave amplitudes and matrix transformations. At the detector, if one spans the direction x, one observes fringes with a spatial period d = G/2π, where G is the resultant momentum difference between the two arms, and a phase that corresponds to the inserted phase φ. Alternatively, one can count arrivals at a given position (detector): the result would be proportional to 1+cos(φ) for one arm, and 1-cos(φ) for the other, thus with a perfect anticorrelation. If, however, the BS2 is not active, no interference is generated and both detectors count the same number of arrivals that is independent of φ.

Now, let us see how our model represents this situation. We assume that the incoming beam is the resut of a Gaussian wave preparation (see previous post) with a position-dependent source phase v_mx. After a sufficiently long time of flight, the momentum propensity will be distributed as a Gaussian too, around the value v_m.

At the first BS, external forces act on particles with probability P = 0.5. The result of such interaction is a momentum shift f₁ and a phase shift of 1/2. Therefore, particles affected will move along path 2 (orange in the figure) with a Gaussian-distributed momentum propensity around v₂ and a phase 1/2 plus the "sample" value φ.

At mirrors, momenta in path 1 are shifted by +f₂, while momenta of path 2 are shifted by -f₂. Phases are shifted by one unity in both paths.

The second BS acts like the first, but now it creates two pair of beams, one "transmitted" and the other "reflected" for each incident beam. The transmitted particles keep their momentum, while the reflected ones have their momentum flipped by 土f₃ and their phase shifted by 1/2 again.

Then the interference mechanism discussed in many previous posts intervene. Atoms feel the traces left by previous atoms that have followed the same path and, through "boson" creation, a momentum transfer takes place. The span difference seen by atoms in both resulting paths is equal to Gt, where t is the time of flight and

in perfect analogy with QM. The relative phase is equal to π in path 1 (red and light blue in the figure) and 0 in path 2 (orange and dark blue in the figure).

At the end, momentum propensity in the two beams will be shifted with respect to the source momentum by a quantity due to such bosons, see this post. Since the source momentum v₀ is Gaussian distributed as said above, the pdf of momentum propensity and position can be calculated using the method discussed in this post, and will be eventually

As a consequence, fringes in the direction transverse to the beam propagation appear, with the period predicted by standard QM.

If now we place a detector at x = 0 downstream of each beam (D1 and D2 in the figure), then

 

which is again the exact result of standard QM calculations made with wavefunctions.

Some of these models will be useful to describe more complex apparatuses, for example interferometers with entangled paths, which will be the subject of a future post.