# 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

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**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 |

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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