## Abstract

Principle of quantum superposition permits a photon to interfere with itself. As per the principle of causality, a photon must pass through the double-slit prior to its detection on the screen to exhibit interference. In this paper, a double-slit quantum interference experiment with reversible detection of Einstein–Podolsky–Rosen quantum entangled photons is presented. Where a photon is first detected on a screen without passing through a double-slit, while the second photon is propagating towards the double-slit. A detection event on the screen cannot affect the second photon with any signal propagating at the speed of light, even after its passage through the double-slit. After the detection of the first photon on the screen, the second photon is either passed through the double-slit or diverted towards a stationary photon detector. Therefore, the question of whether the first photon carries the which-path information of the second photon in the double-slit is eliminated. No single photon interference is exhibited by the second photon, even if another screen is placed after the double-slit.

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## Introduction

In 1801, Young’s double-slit interference experiment proved that light behaves as a wave^{1,2}. A first experimental observation of interference with very low intensity of light was reported by Taylor^{3}. However, quantum mechanically, light consists of discrete energy packets known as photons, which can exhibit particle- or wave-like behaviour. According to the principle of quantum superposition, a single particle can exist at different locations simultaneously^{4,5}. This counter-intuitive law of nature gives wave nature to a particle, and by its consequence, a particle can interfere with itself, i.e. all quantum superimposed states of a particle interfere with each other. In a single particle quantum double-slit experiment, a single particle is passed through a double-slit, and an interference pattern gradually emerges on the screen by accumulating particle detections by repeating the experiment. Each detection of a particle on the screen determines its position on the screen, whereas this measurement cannot determine the path of a particle in the double-slit. However, the interference pattern cannot be formed, when the which-path information of a particle is measured or stored by modifying the experiment. The casual temporal order of this experiment signifies a detection on the screen after the passage of a particle through the double-slit. This casual self interference was demonstrated experimentally with electrons^{6,7,8,9}, neutrons^{10,11}, photons^{12,13} and positrons^{14}. Quantum mechanics exerts no restriction on the self interference of macromolecules, which is experimentally demonstrated^{15,16,17,18,19,20}. An experiment demonstrating self interference of two-photon amplitudes in a double-double-slit is performed with momentum entangled photons^{21,22}. Another version of a double-slit experiment is realised by placing a double-slit in the path of one photon, where the interference pattern is formed only when both photons are measured^{23,24,25,26,27,28,29}. However, in these experiments, both photons are measured after one of them is passed through the double-slit.

In this paper, a quantum double-slit experiment with reversible detection of photons is presented, which is carried out with continuous variable Einstein–Podolsy–Rosen (EPR)^{30} quantum entangled photon pairs. A reversible detection implies that a photon of a quantum entangled pair is first detected on a screen while the other photon is propagating towards a double-slit, and later it can pass either through the double-slit or it can be diverted towards a stationary single photon detector. The experiment is configured such that the detection of a first photon on the screen cannot affect the second photon through any local communication, even after its interaction with the double-slit. This is because the second photon is separated from the detection event by a lightlike interval. Since the second photon passes through the double-slit after the detection of the first photon therefore, the first photon carried no which-path information of the second photon in the double-slit. The second photon is detected by stationary single photon detectors, which are placed at fixed locations throughout the interference experiment. The interference pattern is produced on the screen by repeating the experiment each time with a new EPR entangled pair, provided those photon detections on the screen are considered when the second photon is detected by a stationary detector positioned after the double-slit. However, position measurements of individual photons do not produce any interference pattern, even if the detector placed after the double-slit is displaced gradually to count photons at different locations. The experiment is performed with continuous variable EPR entangled photon pairs produced simultaneously in a Beta Barium Borate (BBO) nonlinear crystal by Type-I spontaneous parametric down conversion (SPDC) in a noncollinear configuration^{31,32,33,34,35,36,37}. A real double-slit is used in this experiment, whereas the quantum entangled pair production rate is intentionally reduced to keep one entangled photon pair in the experiment until its detection. The second EPR entangled pair of photons is produced considerably later than the detection of the first entangled pair of photons. In addition, this paper presents a theoretical analysis of the experiment.

## Concept and analysis

The EPR state is a continuous variable entangled quantum state of two particles, where both particles are equally likely to exist at all position and momentum locations. A one-dimensional EPR state in position basis is written as \(|\alpha \rangle =\int ^{\infty }_{-\infty }|x\rangle _{1}|x+{\textbf {x}}_{o}\rangle _{2} \textrm{d}x\), where subscripts 1 and 2 represent particle-1 and particle-2, respectively. A constant \({\textbf {x}}_{o}\) corresponds to the position difference of particles. The same EPR state is expressed in momentum basis as \(|\alpha \rangle =\int ^{\infty }_{-\infty }e^{i \frac{p x_{o}}{\hslash }}|p\rangle _{1}|-p\rangle _{2}\textrm{d}p\), where particles have opposite momenta, and \(\hslash =h/2\pi\) is the reduced Planck’s constant. Therefore, both position and momentum of each particle are completely unknown. If the position of any one particle is measured, then the EPR state is randomly collapsed onto \(|x'\rangle _{1}|x'+{\textbf {x}}_{o}\rangle _{2}\) where a prime on *x* indicates a single measured position value from the integral range. Therefore, the measured positions of particles are correlated, *i.e.* they are separated by \({\textbf {x}}_{o}\) irrespective of \(x'\). Instead of position, if momentum, which is a complementary observable to the position, of a particle is measured, then the EPR state is randomly collapsed onto \(| p'\rangle _{1}|-p'\rangle _{2}\), where both the particles exhibit opposite momenta irrespective of \(p'\) thus, their measured momenta are correlated.

In this paper, the EPR state of two photons in three-dimensions, propagated away from a finite size of source, is evaluated as follows: Consider a schematic of the experiment shown in Fig. 1, where EPR entangled photons are produced by a finite source size. Photon-1 is detected on a screen, which can record the position of a detected photon as a point on the screen. This measurement corresponds to a position measurement of photon-1 while photon-2 is propagating towards a double-slit, and later it passes through a 50:50 beam splitter. The reflected probability amplitude of photon-2 is incident on a single photon detector-3, which is placed at the focal point of a convex lens. Whereas the transmitted probability amplitude is passed through a double-slit and incident on a single photon detector-2. The detector-2 is stationary, and it measures the position of photon-2 behind the double-slit. Each single photon detector is equipped with a very narrow aperture in order to measure the position of a photon around a location.

To evaluate the EPR state of two photons emanating from a three-dimensional source of finite extension, consider a source placed around the origin of a right-handed Cartesian coordinate system, as shown in Fig. 1. Two photons are produced simultaneously from each point in the source as a consequence of the EPR constraint. Corresponding to an arbitrary point \(\mathbf {r'}\) within the source, a two-photon probability amplitude to find photon-1 at a point \(o_{a}\) in region-*a* left to the source and photon-2 at a point \(o_{b}\) in region-*b* right to the source is written as \(\frac{e^{ip_{1}|\textbf{r}_{a}-\textbf{r}'|/\hslash }}{|\textbf{r}_{a}-\textbf{r}'|}\frac{e^{ip_{2}|\textbf{r}_{b}-\textbf{r}'|/\hslash }}{|\textbf{r}_{b}-\textbf{r}'|}\)^{38,39}, where \(p_{1}\) and \(p_{2}\) are magnitudes of momentum of photon-1 and of photon-2, respectively, and \(\textbf{r}_{a}\) and \(\textbf{r}_{b}\) are the position vectors of points \(o_{a}\) and \(o_{b}\) from the origin. Since the source size is finite, the total finite amplitude to find a photon at \(o_{a}\) and a photon at \(o_{b}\) is a linear quantum superposition of amplitudes originating from all points located in the source, which is written as

where both photons have the same linear polarisation state. A case for different polarisation states of photons leads to a hyper-entangled state, which is reported in Refs.^{40,41}. However, for this experiment, an EPR entanglement is sufficient. Therefore, both photons are assumed to have the same linear polarisation state along the *y*-axis, which is omitted in this analysis. In Eq. (1), \(A_{o}\) is a constant, and \(\psi (x',y',z')\) is the probability amplitude of a pair production at a position \(r'(x',y',z')\) in the source. This amplitude is constant for an infinitely extended EPR state at any arbitrary position vector \(\textbf{r}'\) in the source. This integral represents the amplitude of two photons emanating from a three-dimensional photon pair source of finite size. It leads to a two-photon amplitude, which corresponds to the probability amplitude to find two photons together at different locations. Further, the magnitudes of momenta of photons are considered to be equal \(p_{1}=p_{2}=p\), for the degenerate photon pair production. The amplitude of pair production \(\psi (x',y',z')\) is considered to be a three-dimensional Gaussian function such that, \(\psi (x',y',z')= a e^{-(x'^2+y'^2)/\sigma ^2}e^{-z'^2/w^2}\), where *a* is a constant, \(\sigma\) and *w* are the widths of the Gaussian.

To evaluate the integral, consider two planes oriented perpendicular to the *z*-axis such that a plane-1 is located at a distance \(s_{1}\) and a plane-2 is located at a distance \(s_{2}\) from the origin. These planes are not shown in Fig. 1 however, a screen can be placed in a plane-1 and a double-slit can be placed in a plane-2. The amplitude to find photon-1 on plane-1 and photon-2 on plane-2 is evaluated as follows: Consider the distances of planes from the origin are such that, \(\sigma ^{2}p/h s_{1}\ge 1\) and \(\sigma ^{2}p/h s_{2} \ge 1\), where the magnitudes of \(s_{1}\) and \(s_{2}\) are considerably larger than \(\sigma\) and *w*. This approximation is valid for the experimental considerations of this paper. Since the double-slit and the detectors are placed close to the *z*-axis therefore, Eq. (1) can be written as

where \(\Phi (x_{1},y_{1}; x_{2}, y_{2})\) is a two-photon position amplitude with variables of its argument separated by a semicolon denoting a position of photon-1 on plane-1 and of photon-2 on plane-2. After solving the integral, \(\Phi (x_{1},y_{1}; x_{2}, y_{2})\) is written as

where \(c_{n}\) is a constant and tan(\(\phi\))=–p(s_{1}+s_{2})σ^{2}/2*ℏ*s_{1}s_{2}. It is evident from Eq. (3) that both photons can be found at arbitrary positions. Once a photon is detected at a well-defined location (\(x'_{i}, y'_{i}\)), where a label \(i\in \{1,2\}\) corresponds to any single measured photon, then its position is determined. This measurement collapses the total wavefunction of both photons. Note that when a photon is detected at a well-defined position, even then the amplitude to find the other photon in the position space is delocalised, *i.e.* the projected position wavefunction has a nonzero spread. This wavefunction projection happens immediately once a photon is detected.

The second order quantum interference is exhibited if photon-1 detections are retained on the screen with the condition that photon-2 is detected after the double-slit by a stationary detector-2 as shown in Fig. 1. However, this stationary detector will not always detect photon-2 since, photon has a nonzero amplitude to exist at different positions even after passing through the double-slit. The conditional detection corresponds to a joint measurement of photons. If all photon-1 detections on the screen are considered, then the interference pattern does not appear. Single photon interference is suppressed on the screen as well as after the double-slit, since photons are EPR entangled. To evaluate the second order interference pattern, consider a screen placed at \(z=-s_{1}\) and a double-slit placed at \(z=s_{2}\) with their planes oriented perpendicular to the *z*-axis. If the transmission function of the double-slit is \(A_{T}(x_{2},y_{2})\) then the joint amplitude to detect photon-1 on the screen at a position \((x_{1}, y_{1})\) and photon-2 by a stationary detector-2 is written as

where an integration represents a projection onto a quantum superposition of position states of the photon-2 in the plane of the double-slit. A phase multiplier \(B(x_{2}, y_{2})=e^{ipr_{d}/\hslash }\) represents a phase acquired by a photon to reach detector-2 from the double-slit plane. The distance between detector-2 location (\(x_{o}, y_{o}, z_{o}\)) and an arbitrary point location (\(x_{2}, y_{2}, s_{2}\)) in the double-slit plane is \(r_{d}=(D^{2}+(x_{2}-x_{o})^{2}+ (y_{2}-y_{o})^{2})^{1/2}\), where \(D=z_{o}-s_{2}\) is the distance of detector-2 from the double-slit. Note that \(x_{o}\) is different than the symbol \({\textbf {x}}_{o}\) which is denoting separation of particles in the one-dimensional EPR state. Thus, the second order interference pattern depends on the position of detector-2. For a double-slit with slit separation *d* along the *x*-axis and infinite extension along the *y*-axis, the transmission function is given by \(A_{T}(x_{2},y_{2})= [\delta (x_{2}-d/2)+\delta (x_{2}+d/2)]/\sqrt{2}\). In the following experiment, each slit of the double-slit is largely extended along the *y*-axis as compared to its width. The effect of slit width and position resolution of single photon detectors is considered in the analysis of the following experiment. It is also evident that the two-photon interference pattern exhibits a shift when the position of the stationary detector-2 is shifted.

## Experimental results

An experiment is performed with continuous variable EPR entangled photons of equal wavelength 810 nm, which are produced by the Type-I SPDC in a negative-uniaxial BBO nonlinear crystal. An experimental diagram of the setup is shown in Fig. 2, where the *x*-axis is perpendicular to the optical table passing through the crystal. This experimental setup is a folded version of a diagram shown in Fig. 1, where folding is along the *x*-axis such that photons propagate close to the angle of the conical emission pattern in a horizontal plane parallel to the optical table. Furthermore, photons propagating at a small inclination *w.r.t.* a horizontal plane pass through the double-slit. A vertical linearly polarised laser beam, along the *x*-axis, of wavelength 405 nm is expanded ten times to obtain a beam diameter of 8 mm at the full-width-half-maximum. The expanded laser beam is passed through the BBO crystal, whose optic-axis can be precisely tilted in a vertical plane passing through the crystal. This configuration results in noncollinear spontaneous down-converted photon pair emission in a broad conical pattern, where both the photons of each pair have the same linear polarisation state perpendicular to the polarisation state of the pump photons. The down-converted photons are EPR entangled in a plane perpendicular to the symmetry axis of the cone. The pump laser beam, after passing through the nonlinear crystal, is absorbed by a beam dumper to minimise unwanted background light.

A screen is represented by a movable single photon detector-1 (\(D_{1}\)), which is placed close to the crystal at a distance of 26.4 cm to detect photon-1 at about 5.68 ns prior to the detection of photon-2. The aperture of a single photon detector \(D_{1}\) is an elongated single-slit of width 0.1 mm along the *x*-axis, which represents an effective detector width. It also corresponds to the resolution of the position measurements along the *x*-axis. This detector can be displaced parallel to the *x*-axis in steps of 0.1 mm to detect photons at different positions. Photon-1 is passed through a band-pass filter of band-width 10 nm at the centre wavelength 810 nm prior to its detection. Photon-2 is incident on a 50:50 polarisation independent beam splitter, which is placed at a distance of 93.8 cm from the crystal. A double-slit with an orientation of single slits perpendicular to the *x*-axis is placed after the beam splitter at a distance of 3 cm. Another elongated single-slit aperture of width 0.1 mm along the *x*-axis is placed after the double-slit at a distance of 23 cm from the double-slit in front of an optical fibre coupler. After passing through the double-slit, photon-2 is filtered by a band-pass filter of band-width 10 nm at the centre wavelength 810 nm. It is then passed through the aperture and directed towards a single photon detector-2 (\(D_{2}\)) with a multimode optical fibre of length 0.5 m. This single-slit aperture can be displaced along the *x*-axis with a resolution of 0.1 mm. However, it is positioned at a predetermined location during one complete interference pattern data collection. In this experimental configuration, photon-1 is detected much earlier while photon-2 is propagating towards the beam splitter. Detection of photon-1 cannot affect photon-2 through any signalling limited by the speed of light until it reaches at an optical fibre coupler placed after the double-slit. Photon-2 arrives at the beam splitter 2.26 ns after the detection of photon-1, and from the beam splitter, its transmitted amplitude takes about 0.1 ns to arrive at the double-slit. The reflected amplitude of photon-2 is detected after passing through a band-pass filter by another optical fibre coupled single photon detector-3 (\(D_{3}\)) without any aperture. Photons are focused on an optical fibre input with a convex lens, which projects the incident quantum state of photon-2 onto an eigen-state of the transverse momentum, provided photon-2 is detected by a single photon detector \(D_{3}\). Distance of the lens from the beam splitter is 25 cm, where this lens and the single photon detector \(D_{3}\) are positioned at predetermined fixed locations throughout the experiment.

The experiment is performed with 19 mW power of the pump laser beam, which is incident on the crystal. Each single photon detector output is connected to an electronic time correlated single photon counter (TCSPC), which measures the single and coincidence photon counts with 81 ps temporal resolution. A selected width of time window for the coincidence detection of photons is 81 ns. Single photon counts of each detector and coincidence photon counts of \(D_{1}\) and \(D_{2}\), \(D_{1}\) and \(D_{3}\) are measured for 60 s. These measurements are repeated ten times to obtain an average of photon counts. A two-photon quantum interference pattern with a reversible detection of photons is shown in Fig. 3a, where open circles represent the measured coincidence photon counts of single photon detectors \(D_{1}\) and \(D_{2}\) and the single photon counts of \(D_{1}\). Whereas, a solid line corresponds to the theoretical calculation of two-photon quantum interference using Eq. (4) by considering the finite width of each slit of the double-slit and position resolution of \(D_{2}\). A position of \(D_{2}\) relative to the double slit is \((x_{o}, D)\) in a vertical plane with \(D=23\) cm. The two-photon quantum interference pattern exhibits a shift as the position \(x_{o}\) of \(D_{2}\) is displaced, which is due to the phase-shift multiplier term \(B(x_{2}, y_{2})=e^{ipr_{d}/\hslash }\) in Eq. (4). The slit separation of the double-slit is 0.75 mm and the width of each slit is 0.15 mm. A fixed position of a single photon detector \(D_{2}\) is taken to be the reference point with \(x_{o}=0\). There is no single photon interference pattern produced in this experiment by scanning the detector \(D_{1}\) or \(D_{2}\). When photon-1 is detected, photon-2 is still propagating towards the beam splitter, and later its transmitted amplitude is detected by a fixed position single photon detector \(D_{2}\) at a time lapse of 5.68 ns after the detection of a photon-1. On the other hand, if the reflected amplitude of photon-2 is detected by a single photon detector \(D_{3}\) then \(D_{2}\) will not measure any photon. In this case, photon-2 is not passed through the double-slit, and therefore, no two-photon quantum interference results as shown in Fig. 3b, which shows coincidence counts of single photon detectors \(D_{1}\) and \(D_{3}\) and the single counts of \(D_{1}\). A choice of whether to detect a photon after the double-slit or not is naturally and randomly occurring due to the presence of a beam splitter in the path of photon-2 after its detection. A path superposition quantum state of photon-2 after the beam splitter is projected either onto the transmitted or the reflected path due to a single photon detection by a detector \(D_{2}\) or \(D_{3}\), respectively. The main characteristic of two-photon quantum interference is that it exhibits a shift of the entire pattern as the single photon detector \(D_{2}\) is displaced to another fixed position. This shift in the pattern is shown in Fig. 4 when, (a) \(D_{2}\) is placed at a position \(x_{o}= 0.11\) mm, (b) \(D_{2}\) is placed at a position \(x_{o}=-0.11\) mm with same *D*. This shift is also observed experimentally in the quantum ghost interference experiment by Strekalov et al.^{23}. As a consequence of the EPR entanglement, there is no single photon interference. Therefore, the experiment in this paper presents a quantum two-photon interference with a reversible detection of photons, which has no classical counterpart.

## Conclusion

This paper presents a two-photon double-slit experiment with the reversible detection of photons. Continuous variable EPR entangled photons are produced by the Type-I SPDC process, where photon-1 is detected on a screen while photon-2 is propagating towards a beam splitter. At a later time, photon-2 is produced in a quantum superposition of reflected and transmitted path amplitudes at the beam splitter. The transmitted amplitude is passed through the double-slit, and if this amplitude is detected by a detector-2 then the path quantum superposition state of photon-2 is collapsed onto the transmitted path. Then detector-3 does not detect this photon. Since photon-2 interacted with the double-slit considerably later than the detection of photon-1 therefore, it is ruled out that photon-1 has carried the path information of photon-2 in the double-slit to suppress the single photon interference. In addition, a position measurement of photon-1 cannot affect photon-2 through any signal propagating with speed, which is limited by the speed of light. If photon-2 is detected by a detector-3 then the quantum superposition state is collapsed onto the reflected path. Therefore, detector-2 does not detect photon-2, which results in no interference in single photon and two-photon measurements.

## Methods

It is very important to expand the beam diameter of the pump laser beam to produce a continuous variable EPR quantum entangled state. It also leads to a broader envelope of the interference pattern. To achieve low background counts limited by the dark counts of the single photon detectors, the pump laser beam should have minimal scattering from optical components, and it should be properly dumped after passing through the crystal. A source of EPR entangled photons consists of a thin crystal in Type-1 SPDC configuration, where down-converted photons have the same linear polarisation. The nonlinear crystal is anti-reflection coated for wavelengths of pump and down-converted photons to reduce scattering and back reflection. The nonlinear crystal is kept at room temperature without any temperature control. Its optical-axis is precisely aligned *w.r.t.* the polarisation vector of the pump laser beam to obtain a broad conical emission pattern of down-converted photons with a full cone angle of 9.5°. The optical power of the pump laser beam is 19 mW, which is *x*-polarised. Single-slit apertures, which are placed in front of \(D_{1}\) and an input coupler of an optical fibre of \(D_{2}\), are attached to translational stages to displace them precisely to collect photons corresponding to different positions of apertures. To increase the number of photons passing through the double slit, a double slit consists of two elongated single slits separated by a distance of 0.75 mm along the *x*-axis, where the width of each slit is 0.15 mm. In the experimental configuration, photons are EPR entangled in a plane perpendicular to the direction of propagation of the pump laser beam. The efficiency of each single photon detector is about 65 %. The single photon detector \(D_{1}\) equipped with a convex lens is directly collecting photons and it is placed close to the crystal. Whereas, the single photon detectors \(D_{2}\) and \(D_{3}\) are coupled to multimode optical fibres of length 0.5 m. The input of each optical fibre is attached to respective optical fibre couplers, each consisting of a convex lens of diameter 1 cm. Photons are collected by the lenses after passing through the single-slit apertures to measure the position of photons by \(D_{1}\) and \(D_{2}\). However, a coupler of a single photon detector \(D_{3}\) is not equipped with any aperture. A band-pass optical filter of band-width 10 nm at centre wavelength 810 nm is placed at the input of each single photon detector to filter the unwanted scattered photons of the pump laser and background light.

## Data availability

All data generated or analysed during this study are included in this article.

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## Acknowledgements

Mandip Singh acknowledges research funding by the Department of Science and Technology, Quantum Enabled Science and Technology grant for project No. Q.101 of theme title “Quantum Information Technologies with Photonic Devices”, DST/ICPS/QuST/Theme-1/2019 (General).

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MS setup the experiment and made its theoretical model, VD took data and analysed data, MS wrote this manuscript. Both authors discussed the experiment.

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Devrari, V., Singh, M. Quantum double slit experiment with reversible detection of photons.
*Sci Rep* **14**, 20438 (2024). https://doi.org/10.1038/s41598-024-71091-1

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DOI: https://doi.org/10.1038/s41598-024-71091-1

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