# The Franson Experiment as an Example of Spontaneous Breaking of Time-Translation Symmetry

## Abstract

**:**

## 1. Introduction

## 2. The Franson Experiment

## 3. The Actual Experimental Set-Up

## 4. The Statistical Model

- If $\tilde{\mathsf{\Delta}}\in [0,\pi )$, then we have the following:$$\begin{array}{c}\hfill \phantom{\rule{-10.84006pt}{0ex}}L(\phi ;\tilde{\mathsf{\Delta}})=\left\{\begin{array}{c}\phantom{\rule{0.72229pt}{0ex}}q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(-cos\left(\tilde{\mathsf{\Delta}}\right)-cos\left(\phi \right)-1\right),\\ \phantom{\rule{63.59795pt}{0ex}}\mathrm{if}\phantom{\rule{7.22743pt}{0ex}}-\pi \phantom{\rule{11.56346pt}{0ex}}\le \phi <\tilde{\mathsf{\Delta}}-\pi ,\\ \phantom{\rule{0.72229pt}{0ex}}q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(+cos\left(\tilde{\mathsf{\Delta}}\right)+cos\left(\phi \right)-1\right),\\ \phantom{\rule{49.50476pt}{0ex}}\mathrm{if}\phantom{\rule{3.61371pt}{0ex}}\tilde{\mathsf{\Delta}}-\pi \phantom{\rule{5.78172pt}{0ex}}\le \phi <\phantom{\rule{7.58803pt}{0ex}}0,\\ \phantom{\rule{0.72229pt}{0ex}}q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(+cos\left(\tilde{\mathsf{\Delta}}\right)-cos\left(\phi \right)+1\right),\\ \phantom{\rule{49.86647pt}{0ex}}\mathrm{if}\phantom{\rule{18.06749pt}{0ex}}0\phantom{\rule{13.00806pt}{0ex}}\le \phi <\phantom{\rule{4pt}{0ex}}\tilde{\mathsf{\Delta}},\\ \phantom{\rule{0.72229pt}{0ex}}q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(-cos\left(\tilde{\mathsf{\Delta}}\right)+cos\left(\phi \right)+1\right),\\ \phantom{\rule{52.03448pt}{0ex}}\mathrm{if}\phantom{\rule{15.17719pt}{0ex}}\tilde{\mathsf{\Delta}}\phantom{\rule{13.73148pt}{0ex}}\le \phi <+\pi ,\end{array}\right.\end{array}$$
- If $\tilde{\mathsf{\Delta}}\in [-\pi ,0)$, then we have the following:$$\begin{array}{c}\hfill \phantom{\rule{-10.84006pt}{0ex}}L(\phi ;\tilde{\mathsf{\Delta}})=\left\{\begin{array}{c}q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(-cos\left(\tilde{\mathsf{\Delta}}\right)+cos\left(\phi \right)+1\right),\\ \phantom{\rule{46.97505pt}{0ex}}\mathrm{if}\phantom{\rule{7.94974pt}{0ex}}-\pi \phantom{\rule{10.84006pt}{0ex}}\le \phi <\tilde{\mathsf{\Delta}},\\ q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(+cos\left(\tilde{\mathsf{\Delta}}\right)-cos\left(\phi \right)+1\right),\\ \phantom{\rule{47.69846pt}{0ex}}\mathrm{if}\phantom{\rule{21.68121pt}{0ex}}\tilde{\mathsf{\Delta}}\phantom{\rule{7.22743pt}{0ex}}\le \phi <\phantom{\rule{3.61371pt}{0ex}}0,\\ q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(+cos\left(\tilde{\mathsf{\Delta}}\right)+cos\left(\phi \right)-1\right),\\ \phantom{\rule{66.48827pt}{0ex}}\mathrm{if}\phantom{\rule{16.62178pt}{0ex}}0\phantom{\rule{15.89948pt}{0ex}}\le \phi <\tilde{\mathsf{\Delta}}+\pi ,\\ q\left(\phi \right)\xb7\mathrm{arc}-\mathrm{cos}\left(-cos\left(\tilde{\mathsf{\Delta}}\right)-cos\left(\phi \right)-1\right),\\ \phantom{\rule{54.2025pt}{0ex}}\mathrm{if}\phantom{\rule{11.56346pt}{0ex}}\tilde{\mathsf{\Delta}}+\pi \phantom{\rule{0.0pt}{0ex}}\le \phi <+\pi ,\end{array}\right.\end{array}$$

## 5. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wiseman, H. Quantum physics: Death by experiment for local realism. Nature
**2015**, 526, 649. [Google Scholar] [CrossRef] [PubMed] - Einstein, A.; Podolsky, B.; Rosen, N. Can quantum mechanical description of physical reality be considered complete? Phys. Rev.
**1935**, 47, 777–780. [Google Scholar] [CrossRef] [Green Version] - Bohm, D. Quantum Theory; Prentice-Hall: New York, NY, USA, 1951. [Google Scholar]
- Bell, J.S. On the Einstein-Podolsky-Rosen paradox. Physics
**1964**, 1, 195–200. [Google Scholar] [CrossRef] [Green Version] - Bell, J.S. On the problem of hidden variables in quantum mechanics. Physics
**1966**, 38, 447–452. [Google Scholar] - Clauser, J.F.; Horne, M.A.; Shimony, A.; Holt, R.A. Proposed experiment to test local hidden variables theories. Phys. Rev. Lett.
**1969**, 23, 880–884. [Google Scholar] [CrossRef] [Green Version] - Fine, A. Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett.
**1982**, 48, 291. [Google Scholar] [CrossRef] - Oaknin, D.H. The Bell theorem revisited: Geometric phases in gauge theories. Front. Phys.
**2020**, 12, 00142. [Google Scholar] [CrossRef] - Oaknin, D.H. Are models of local hidden variables for the singlet polarization state necessarily constrained by the Bell inequality? Mod. Phys. Lett. A
**2020**, 35, 2050229. [Google Scholar] [CrossRef] - Oaknin, D.H. Solving the EPR paradox: An explicit statistical local model of hidden variables for the singlet state. arXiv
**2014**, arXiv:1411.5704. [Google Scholar] - Aspect, A.; Dalibard, J.; Roger, G. Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett.
**1982**, 49, 1804. [Google Scholar] [CrossRef] [Green Version] - Weihs, G.; Jennewein, T.; Simon, C.; Weinfurter, H.; Zeilinger, A. Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett.
**1998**, 81, 5039. [Google Scholar] [CrossRef] [Green Version] - Hensen, B.; Bernien, H.; Dréau, A.E.; Reiserer, A.; Kalb, N.; Blok, M.S.; Ruitenberg, J.; Vermeulen, R.F.L.; Schouten, R.N.; Hanson, R.; et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature
**2015**, 526, 682. [Google Scholar] [CrossRef] [PubMed] - Franson, J.D. Bell inequality for position and time. Phys. Rev. Lett.
**1989**, 62, 2205. [Google Scholar] [CrossRef] [PubMed] - Franson, J.D. Violations of a Simple Inequality for Classical Fields. Phys. Rev. Lett.
**1991**, 67, 290. [Google Scholar] [CrossRef] [PubMed] - Belinskii, A.V.; Klyshko, D.M. Interference of light and Bell’s theorem. Usp. Fix. Nauk.
**1993**, 163, 1. [Google Scholar] [CrossRef] - Kwiat, P.G.; Steinberg, A.M.; Chiao, R.Y. High-visibility interference in a Bell-inequality experiment for energy and time. Phys. Rev. A
**1993**, 47, 2472. [Google Scholar] [CrossRef] [PubMed] - Kwiat, P.G.; Vareka, W.A.; Hong, C.K.; Nathel, H.; Chiao, R.Y. Correlated two-photon interference in a dual-beam Michelson interferometer. Phys. Rev. A
**1990**, 41, 2910. [Google Scholar] [CrossRef] [PubMed] - Su, C.; Wodkiewicz, K. Quantum versus stochastic or hidden-variable fluctuations in two-photon interference effects. Phys. Rev. A
**1990**, 44, 6097. [Google Scholar] [CrossRef] - Jogenfors, J.; Larsson, J.A. Energy-time entanglement, Elements of Reality, and Local Realism. J. Phys. A Math. Theor.
**2014**, 47, 424032. [Google Scholar] [CrossRef] [Green Version] - Aerts, S.; Kwiat, P.; Larsson, J.A.; Zukowski, M. Two-photon Franson-type experiments and local realism. Phys. Rev. Lett.
**1999**, 83, 2872. [Google Scholar] [CrossRef] [Green Version] - Elitzur, S. Impossibility of spontaneously breaking local symmetries. Phys. Rev. D
**1975**, 12, 3978. [Google Scholar] [CrossRef] - Ham, B.S. The origin of correlation fringe in Franson-type experiments. arXiv
**2020**, arXiv:2005.14432. [Google Scholar] - Yongram, N. Spin correlations in e
^{+}e^{-}pair creation by two-photons and entanglement in QED. Int. J. Theor. Phys.**2011**, 50, 838. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematic setup for an ideal Franson experiment. A source produces pairs of photons locked in time and energy, which are then sent through two unbalanced, perfectly calibrated Mach–Zender interferometers. At their exit, each one of the photons is registered by either one of two detectors located at their corresponding ends (${D}_{A}$, ${D}_{{A}^{\prime}}$ and ${D}_{B}$, ${D}_{{B}^{\prime}}$, respectively), which record their times of arrival. This figure has been adapted from reference [14]. Copyright 1989 by the American Physical Society. Adapted with permission.

**Figure 2.**Illustration of the setup of the Franson experiment described in reference [17]. In this experiment, only events recorded at one of the two detectors at each end of the optical device are registered, since, by symmetry considerations $p({D}_{A}\bigcap {D}_{B})=p({D}_{A}^{\prime}\bigcap {D}_{B}^{\prime})$, so these two detectors are sufficient to test the expected fringes of interference. The figure is reprinted from reference [17]. Copyright 1993 by the American Physical Society. Reprinted with permission.

**Figure 3.**Experimental data collected in the Franson experiment described in reference [17]. The relative number of simultaneous events collected in a given pair of detectors shows a characteristic pattern of interference fringes with characteristic period of ${\mathit{l}}_{p}\sim {\omega}_{p}/c\sim 0.35\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$, while the total number of events in each detector shows no such pattern. The figure is reprinted from reference [17]. Copyright 1993 by the American Physical Society. Reprinted with permission.

**Figure 4.**Plot of the transformation law ${\phi}_{A}\to {\phi}_{B}=L({\phi}_{A};\tilde{\mathsf{\Delta}})\phantom{\rule{4pt}{0ex}}for\phantom{\rule{4pt}{0ex}}\tilde{\mathsf{\Delta}}=\pi /3$ (solid line), compared to the corresponding linear transformation (dotted line).

**Table 1.**Characteristic time and length scales in the Franson experiment reported in reference [17].

Time Scale | Length Scale | |
---|---|---|

Laser pulse | ∼20 ns | ∼6 m |

Arms imbalance | ∼2 ns | ∼60 cm |

Photons coherence | ∼${10}^{-4}$ ns | ∼36 $\mathsf{\mu}$m |

Interference fringes | ∼${10}^{-6}$ ns | ∼0.3 $\mathsf{\mu}$m |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Oaknin, D.H.
The Franson Experiment as an Example of Spontaneous Breaking of Time-Translation Symmetry. *Symmetry* **2022**, *14*, 380.
https://doi.org/10.3390/sym14020380

**AMA Style**

Oaknin DH.
The Franson Experiment as an Example of Spontaneous Breaking of Time-Translation Symmetry. *Symmetry*. 2022; 14(2):380.
https://doi.org/10.3390/sym14020380

**Chicago/Turabian Style**

Oaknin, David H.
2022. "The Franson Experiment as an Example of Spontaneous Breaking of Time-Translation Symmetry" *Symmetry* 14, no. 2: 380.
https://doi.org/10.3390/sym14020380