The Franson experiment as an example of spontaneous breaking of time-translation symmetry

We describe an explicit statistical model of local hidden variables that reproduces the predictions of quantum mechanics for the ideal Franson experiment and sheds light on the physical mechanisms that might be involved in the actual experiment. The crux of our model is the spontaneous breaking of the time-translation gauge symmetry by the hidden configurations of the pairs of photons locked in time and energy involved in the experiment, which acquire a non-zero geometric phase through certain cyclic transformations.


I. INTRODUCTION
It is a widely accepted wisdom that quantum phenomena cannot be fully described within the framework of any physical theory that shares the same notions of reality and relativistic causality that we acknowledge as a given in our classical descriptions of the macroscopic world [1]. This wisdom is precisely formulated through the Bell theorem on the attainable correlations between the outcomes of polarization measurements performed on pairs of photons prepared in a singlet polarization state [2,3]. The theorem draws a solid line (the Bell inequality) that allows to experimentally discriminate between the predictions of quantum mechanics for these correlations and those of models of local hidden variables that fulfill certain physically intuitive requirements [4][5][6][7].
In a series of recent papers we have shown, however, that the proof of the Bell theorem relies crucially on a subtle implicit assumption that is not required by fundamental physical principles and, therefore, the Bell inequality does not necessarily hold for models of local hidden variables that do not comply with the said unjustified assumption [8][9][10]. In conse-quence, such models cannot be ruled out by the experimental evidence for the violation of the inequality [11][12][13].
The Franson experiment is often regarded as an alternative demonstration of the impossibility of describing quantum phenomena within the framework of any local model of hidden variables [14][15][16][17]. As in the case of the Bell experiment, it has been shown that certain features of the predictions of quantum mechanics for the Franson experiment cannot be reproduced within the framework of any model of local hidden variables that shares certain intuitive requirements [18][19][20][21].
In this paper we argue, however, that as in the case of the Bell experiment the models of local hidden variables whose predictions for the Franson experiment can be distinguished from those of quantum mechanics all share an assumption that is not required by any fundamental principle. Indeed, we explicitly describe a model of local hidden variables that does not comply with the disputed assumption and, hence, it successfully reproduces the predictions of quantum mechanics for the experiment. The crux of our model is the spontaneous breaking of the time-translation gauge symmetry by the hidden configurations of the pairs of photons locked in time and energy involved in the experiment, which acquire due to a holonomy a non-zero geometric phase through certain cyclic coordinate transformations.
Let us stress that the gauge symmetry is spontaneously broken when each one of the possible hidden configurations of the pair of photons is considered separately (that is, for every single realization of the experiment), but it is statistically restored over the whole population of all possible hidden configurations (that is, over a long sequence of repetitions). Thus, the expected average correlations do only depend on the gauge-independent physical parameters that describe the settings of the experiment, in agreement with Elitzur's theorem [22].
The model discussed here for the Franson experiment closely resembles the model of hidden variables introduced in [8][9][10] for the Bell experiment.
The paper is organized as follows. In section II we present a (somewhat simplistic) description of the ideal Franson experiment within the framework of quantum mechanics and summarize its predictions (see [16] for a general review of optical tests of the Bell inequality). In section III we present a detailed description of the setup of an actual experiment.
In section IV we describe a explicit model of local hidden variables that reproduces the predictions and the collected experimental data described in the two previous sections. Finally, in section V we discuss and summarize our findings.

II. THE FRANSON EXPERIMENT
In a Franson experiment a source produces pairs of photons, A and B, whose state is described by a wavefunction of the form where {|ξ 1 , |ξ 2 } (A,B) are orthonormal bases in their respective single-particle Hilbert spaces, and φ A , φ B are phases that, in principle, can be controlled and set at will.
A projective measurement is then performed on the pair of photons along the orthonormal basis in their joint Hilbert space {|c 1 , |c 2 , |c 3 , |c 4 } defined by the vectors so that the probabilities p i = Ψ|c i c i |Ψ for each one of the four possible outcomes are given, respectively, by: Outcomes #1 and #2, to which we shall refer as simultaneous events for reasons that will be clear later on, account for half of all the events, while outcomes #3 and #4, to which we shall refer as non-simultaneous events, account for the other half. The experiment is schematically described in Fig. 1. This figure is a reproduction of Fig. 1 of reference [14].
We are interested here in the pattern of interference fringes shown by the probabilities of the simultaneous events, p 1 and p 2 , as a function of the total phase ∆ = φ A + φ B .
In particular, the probability p 1 is equal (up to a normalization factor) to the probability of 'equal' outcomes -either (−1, −1) or (+1, +1) -at the two polarization measurement devices in a Bell experiment with photons prepared in a singlet polarization state, while the probability p 2 is equal (up to the same normalization factor as above) to the probability for 'non-equal' outcomes -either (+1, −1) or (−1, +1) -in the two devices. Hence, following the Bell theorem, it is claimed that these probabilities cannot be reproduced within the framework of any model of local hidden variables [18][19][20].
Notwithstanding, some authors have raised questions regarding the origin of the claimed 'non-classical' features of the Franson experiment [23], since the pairs of photons are initially prepared in a separable state and they become entangled only when they both are measured, well after they have left their source and do not further interact with each other [17]. In this paper we explore these and other questions with the help of an explicit model of local hidden variables that reproduces the predictions of quantum mechanics summarized in (4).
Before we proceed we make the following important observation. The orthonormal singleparticle eigenstates |ξ 1 (A,B) , |ξ 2 (A,B) are defined each up to a global phase (as any normalized eigenvector of any linear operator) and, therefore, the phases φ A and φ B in the wavefunction (1) have not been properly defined yet. In order to do so it is necessary to set an arbitrary setting of the actual experiment as a reference and measure the resulting probabilities. This reference setting, thus, fixes a reference value for the phase φ A + φ B , with respect to which we can properly define a subsequent change. On the other hand, in the above description the phase difference φ A − φ B cannot be properly defined and it is, therefore, a spurious degree of freedom, which we can set to φ A − φ B = 0. In other words, in the quantum description the setting of the experiment is actually described by a single physical parameter, φ A + φ B , rather than two independent parameters, φ A and φ B .

III. THE ACTUAL EXPERIMENTAL SET-UP
A general review of the actual setup of the Franson experiment and other related optical tests can be found in reference [16]. Fig. 2 shows the setup of the Franson experiment described in reference [17]. In this experiment pairs of photons locked in time and energy are produced via parametric down-conversion by splitting photons from a single-mode laser pulse with wavelength λ p = 351.1 nm inciding in a crystal possessing a χ (2) non-linearity.
The laser pulse has a typical time width T ∼ 20 ns.
The produced photons, A and B, have a typical coherence time τ ∼ 36 µm/c ∼ 10 −4 ns and a precisely defined total energy, equal to that of the splitted incident photon: with The two photons are then sent in opposite directions into two unbalanced Mach-Zender-type interferometers with a longer arm and a shorter arm. The length differences between the two arms of each interferometer are set to ∆L A,B ∼ 63 cm, which is much longer than the typical coherence length of the propagating photons, Hence, there is no single-particle interference in the unbalanced interferometers. Yet, the time delay that these length differences introduce is much shorter than the time width of the incident laser pulse These length differences ∆L A,B can be precisely controlled and modified at each one of the unbalanced interferometers for different settings of the experiment, but they are always set at equal values: with high precision, This length difference ∆L, which is the only parameter in the described experimental setting, introduces a relative total phase or by detectors D A and D B they are counted as event #2. As shown in Fig. (3), which has been taken and reproduced here from reference [17], these events occur with probabilities that show a characteristic pattern of interference fringes as a function of the length difference ∆L introduced in the experimental setting (9), even though the total number of photons As it can be seen from Fig. (3) and eq. (11), the period of the interference pattern in these probabilities is fixed by the wavelength of the incident photon splitted via parametric downconversion at the non-linear crystal, The described interference pattern in the probabilities of simultaneous events is at-

IV. THE STATISTICAL MODEL
In this section we describe an explicit model of local hidden variables that reproduces the predictions of quantum mechanics for the ideal Franson experiment, as summarized in eq.
(4). The model closely resembles the model introduced in [8-10] to reproduce the predictions of quantum mechanics for the Bell experiment. It also bears some similarities as well as many crucial differences, which we highlight below, with the model of hidden variables introduced by Aerts et al. in [21]. The comparison between the two models will help us to make clear the novel features of our model.
The crux of our model is the spontaneous breaking of the time-translation gauge symmetry by the hidden configurations of the pair of photons produced in the non-linear crystal. The breaking of the time-translation symmetry in this model is tantamount, as we show below, to the impossibility to set the time of emission of the photons with precision better than roughly 10% of the period of the observed interference fringes (12), that is, ∼ 0.03 µm/c ∼ 10 −16 s. At the origin of this uncertainty is a geometric phase associated with a holonomy, as intuitively illustrated in Fig.3 in [8]. In the example shown in that figure, three parties located on the surface of a sphere cannot agree on the orientation of a tangent vector shared between them to a precision better than the geometric phase that the vector acquires when transported over the closed loop that connects the parties. In the model discussed in this paper, the three parties are the source of the pairs of photons and the detectors at both ends of the optical device, who cannot agree on the phase of the photons shared between them due to a similar holonomy.
Since the time uncertainty associated to this holonomy is shorter by three orders of magnitude than the coherence time of the two propagating photons, see Table I We consider a statistical model in which the space of possible hidden configurations of the pairs of photons consists of two separated sub-populations, each one of them occurring with a probability of one half. The first sub-population accounts for events in which the two photons of the pair are detected simultaneously, that is, events #1 and #2 in (4), whose probabilities depend on the total phase defined by the setting of the experiment, while the second sub-population accounts for events in which the two photons of the pair are detected at distinct times, that is, events #3 and #4, whose probabilities do not depend on the setting of the experiment.
The statistical space consists of an infinitely large number of possible hidden configurations distributed over the unit circle, with a density of probability given by where ϕ A ∈ [−π, π) is an angular coordinate over the circle, i.e. a phase, which determines if photon A will be detected either at detector D A or at detector D A , according to: Since g(ϕ A ) = g(−ϕ A ) each one of the two possibilities occurs with a probability of one half.
Let us notice that the density of probability (14) is normalized to This coordinate is somewhat similar to the angular coordinate θ used in the model of hidden variables built by Aerts et al. in [21]. In their model the random variable θ is distributed uniformly over its range [−π, π), but they introduce a non-uniform boundary to produce the correct distribution.
Each one of these possible hidden configurations may appear in four possible shapes, labelled as (η A = ±1, η B = ±1), each one with a probability of 1/4. These shapes determine if the photons are detected either at the earlier or the later time slot, according to: where N = A, B. For example, photon A of a pair with shape defined by η A = +1, η B = −1 will be detected in the later time slot, while photon B of the same pair will be detected at the earlier time slot. These binary variables come instead the continuous coordinate r ∈ [0, 1] considered in the model discussed by Aerts et al. in [21].
In simultaneous events the two photons of the pair acquire equal phases φ A = φ B = π ω p ∆L/c as they go through their respective interferometers, which add up to a total phase that depends on the setting of the experiment (13), while in not-simultaneous events the two photons acquire opposite phases φ A = −φ B , so that the total phase between the two does not depend on the experimental setting, A hidden configuration characterized by a phase ϕ A at the exit of interferometer A is described at the exit of interferometer B by a phase ϕ B related to the former by the coordinate transformation, where, • If ∆ ∈ [0, π), • If ∆ ∈ [−π, 0), and q(ϕ) = sign((ϕ − ∆)mod([−π, π))), with the function y = arc-cos(x) defined in its main branch, such that y ∈ [0, π] while x ∈ [−1, +1].
We have shown in [8][9][10] that this coordinates transformation fulfills the constraint so that the phases ϕ B are distributed over the circle with a density of probability that is functionally identical to the density of probability for the phases ϕ A , as it should be expected from symmetry considerations. Furthermore, the constraint (23) states that the probability to occur of each possible configuration is independent -as it must be -from the set of coordinates used to describe them.
In order to keep the symmetry between the two involved parties we stipulate that photon B is detected either at detector D B or at detector D B according to the same response function defined above for photon A, that is, Therefore, for simultaneous events which exactly reproduces the probabilities (4) for events #1 and #2. For non-simultaneous events, on the other hand, we get which also corresponds to the probabilities (4) for events #3 and #4.
In order to obtain these results we notice that L(ϕ A ; ∆) changes sign at ϕ A = ∆ and at Fig.4) and, therefore, The coordinate transformation (20) as defined in (21)(22) introduces the holonomy responsible for the time uncertainty mentioned above. This non-linear transformation generalizes the linear transformation ϕ B = ϕ A + ∆ assumed as an unavoidable must in the model of hidden variables discussed by Aerts et al. in [21]. In Fig. 4 the transformation (20) is plotted against the linear transformation for the particular value ∆ = π/3 for the sake of illustration. The maximum difference between the actual transformation L(ϕ; ∆) and the linear transformation, which bounds the geometric phase that can be accumulated in a cycle, is roughly a 10% of the period of the transformation, that is, ∼ 0.1ω −1 p ∼ 10 −16 s.
As already noticed, this model closely resembles the model of local hidden variables introduced in [8][9][10]. The crux of both models is the spontaneous breaking of a gauge symmetry by the hidden configuration of the described pairs of photons, which acquires a non-zero geometric phase through certain cyclic transformations. In the model discussed here the spontaneously broken gauge symmetry is the time-translation symmetry, or equivalently the rotational symmetry of the phases ϕ A , ϕ B of the hidden configurations, which cannot be described at once with respect to the two splitters and the source of the photons due to the holonomy of the model.
In both cases, however, the gauge symmetries are statistically restored when considered over the whole population of possible hidden configurations, in agreement with Elitzur's theorem that forbids any gauge-dependent magnitude to get a non-invariant expected value [22]. Thus, in the model discussed in this paper the expected probabilities (4) predicted by the model depend only on the well-defined physical parameter φ A + φ B that describes the experimental setting. The crux of both models is the spontaneous breaking of a gauge symmetry by the hidden certain cyclic coordinates transformations. In the model presented in this paper the broken gauge symmetry is the time-translation symmetry. The symmetry is broken at the scale of ∼ 10 −16 s, which is roughly the inverse of the frequency of the photon splitted via parametric down-conversion and much shorter than the coherence time of each one of the photons of the resulting pair.
The insight described here for pairs of photons produced via parametric down-conversion at a non-linear crystal might also be applied to study pairs of photons (or Z-bosons) produced in the decay of scalar massive particles like, for example, positronium or neutral pions [24].