# Interference with Non-Interacting Free Particles and a Special Type of Detector

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## Abstract

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## 1. Recorded vs. Detected Particles

## 2. The Gaussian Free-Particle Wavepacket

## 3. Two Oppositely Moving Free-Particle Gaussian Wavepackets

## 4. Free-Particle Gaussian Wavepacket Reflecting off a Wall

## 5. Discussion

- The detector will be turned on and we will wait till a single photon is detected;
- The detector will then be turned off and on, expecting that it thus loses all memory of previous photon detections, and in particular all information about their phases (this expectation must be confirmed, and we may have to wait for some longer time interval before the detector is turned on again);
- The above procedure will be automated and will be repeated until a large number of photons is recorded (e.g., 1000 photons);
- The time ${t}_{\mathrm{max}}$ it takes to complete that part of the experiment will be measured;
- The detector will then be moved to the position of the nearby minimum, and steps 1 to 3 will be repeated. The time ${t}_{\mathrm{min}}$ it takes to complete that second part of the experiment will also be measured.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Schematic of a double-slit experiment with f individual non-interacting photons emitted horizontally per unit time from each slit. $l\gg D$.

**Figure A2.**Gradual buildup of the double-slit interference pattern for 100, 1000, 10,000, and 100,000 photon detections, respectively, similarly to Figure 3. Shown are the distributions of photons along the 28,100 $\mathsf{\mu}$m-wide segments of the detector in the actual experiment of [19]. Detection pixels have width $\delta x=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}\sim \lambda $. Solid line: analytic solution.

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**Figure 1.**Free-particle Gaussian distributions with average velocity ${\mathrm{v}}_{o}=5$ at times $t=0.1,1,2,4,6,8,10$. Here, velocities ${\mathrm{v}}_{o}$ are expressed in units of ${\sigma}_{{v}_{o}}$, spatial scales x in units of $\hslash /\left(m{\sigma}_{{v}_{o}}\right)$, and times t in units of $\hslash /\left(m{\sigma}_{{v}_{o}}^{2}\right)$ (see main text for details). Black lines: solution of the Schrödinger Equation (Equation (6)). Blue lines: the particle distribution $\overline{N}(x;t)/\left(\mathcal{N}\delta x\right)$ obtained with $\mathcal{N}$ = 10,000,000 particles and detection bin sizes $\delta x=0.01$ in our numerical experiment. Red lines: simple analytic approximation (Equation (13)). The three distributions are practically indistinguishable beyond time $t>\phantom{\rule{-7.77771pt}{0ex}}$2.

**Figure 2.**Two free-particle Gaussian distributions moving against each other with average velocities $\pm {\mathrm{v}}_{o}=5$ at times $t=0.1,1,2,4,6,8,10$. Line colors as in Figure 1. Analytic approximation according to Equation (24). Once again, the three distributions are practically indistinguishable beyond time $t\gtrsim 2$. Green line: the particle distribution $N(x;t)/\left(\mathcal{N}\delta x\right)$ obtained numerically without the phase information. We see clearly two Gaussian distributions passing each other without interference.

**Figure 3.**Development of the interference pattern shown in Figure 2 at time $t=4$ for various values of the detection bin size $\delta x=0.001,0.01,0.1,0.2,0.4$ as the number $\mathcal{N}$ of particles/experiments increases. Black line: solution to the Schrödinger equation at time $t=4$, as in Figure 2. The red line corresponds to the bin size implemented to obtain Figure 2, namely $\delta x=0.01$. As $\delta x\to 0$, it takes a very long time for the interference pattern to develop clearly. For $\delta x\gtrsim 0.1$, particles are undercounted because of phase mixing in each detection bin. The interference pattern disappears altogether when δx is too large.

**Figure 4.**One free-particle Gaussian distribution with average velocity ${\mathrm{v}}_{o}=5$ reflecting off a wall at $x=0$. Left plot: shown times $t=0.1,1,2,4$. Right plot: shown times $t=6,8,10$. Black, blue, red line colors as in Figure 1. Analytic approximation according to Equation (32). Once again, the three distributions are practically indistinguishable beyond time $t\gtrsim 2$. Green line: the particle distribution $N(x;t)/\left(\mathcal{N}\delta x\right)$ without the phase information, thus also without interference.

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**MDPI and ACS Style**

Contopoulos, I.; Tzemos, A.C.; Zanias, F.; Contopoulos, G. Interference with Non-Interacting Free Particles and a Special Type of Detector. *Particles* **2023**, *6*, 121-133.
https://doi.org/10.3390/particles6010005

**AMA Style**

Contopoulos I, Tzemos AC, Zanias F, Contopoulos G. Interference with Non-Interacting Free Particles and a Special Type of Detector. *Particles*. 2023; 6(1):121-133.
https://doi.org/10.3390/particles6010005

**Chicago/Turabian Style**

Contopoulos, Ioannis, Athanasios C. Tzemos, Foivos Zanias, and George Contopoulos. 2023. "Interference with Non-Interacting Free Particles and a Special Type of Detector" *Particles* 6, no. 1: 121-133.
https://doi.org/10.3390/particles6010005