quantum which way experiment

The Net Advance of Physics: AFSHAR'S EXPERIMENT and other "Which Way" Experiments

The net advance of physics.

Which-Way or Welcher-Weg-Experiments

• First Online: 25 July 2009

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• Paul Busch &
• Gregg Jaeger  

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The issue of the ► wave-particle duality of light and matter is commonly illustrated by the ► double-slit experiment, in which a quantum object of relatively well defined momentum (such as a photon, electron, neutron, atom, or molecule) is sent through a diaphragm containing two slits, after which it is detected at a capture screen. It is found that an interference pattern characteristic of wave behaviour emerges as a large number of similarly prepared quantum objects is detected on the screen. This is taken as evidence that it is impossible to ascertain through which slit an individual quantum object has passed; if that were known in every individual case and if the quantum objects behaved as free classical particles otherwise, an interference pattern would not arise.

The notion that a description of atomic objects in terms of definite classical particle trajectories is not in general admissible is prominent in Werner Heisenberg's seminal paper [1] of 1927 on the ► Heisenberg uncertainty principle; there he notes: “I believe that one can fruitfully formulate the origin of the classical ‘orbit’ in this way: the ‘orbit’ comes into being only when we observe it.” In the same year, in his famous Como lecture, Niels Bohr introduced the ► complementarity principle, which entails that definite particle trajectories cannot be defined or observed for atomic objects because according to it their spatiotemporal and causal descriptions are mutually exclusive [2]. Bohr cited the uncertainty relation as a symbolic expression of complementarity but recognized that this relation also offered room for approximately defined simultaneous values of position and momentum. Still in the same year, at the 1927 Solvay conference, Albert Einstein questioned the impossibility of determining the path taken by an individual particle in a double-slit interference experiment [21]; he proposed an experimental scheme wherein he considered it possible to infer through which slit the particle passed, without thereby destroying the interference pattern by measuring the recoil of the double-slitted diaphragm. This was the first instance of a welcher-weg or which-way experiment . As Bohr reported in his 1949 tribute to Einstein [3], he was able to demonstrate that Einstein's proposal was in conflict with the principles of quantum mechanics.

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Busch, P., Jaeger, G. (2009). Which-Way or Welcher-Weg-Experiments. In: Greenberger, D., Hentschel, K., Weinert, F. (eds) Compendium of Quantum Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70626-7_237

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Quantum mechanical which-way experiment with an internal degree of freedom

Konrad banaszek.

1 Faculty of Physics, University of Warsaw, ul. Hoża 69, PL-00-681 Warszawa, Poland

Paweł Horodecki

2 Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, ul. Narutowicza 11/12, PL-80-952 Gdańsk, Poland

3 National Quantum Information Center of Gdańsk, ul. Wł. Andersa 27, PL-81-824 Sopot, Poland

Michał Karpiński

4 Present address: Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK

Czesław Radzewicz

For a particle travelling through an interferometer, the trade-off between the available which-way information and the interference visibility provides a lucid manifestation of the quantum mechanical wave–particle duality. Here we analyse this relation for a particle possessing an internal degree of freedom such as spin. We quantify the trade-off with a general inequality that paints an unexpectedly intricate picture of wave–particle duality when internal states are involved. Strikingly, in some instances which-way information becomes erased by introducing classical uncertainty in the internal degree of freedom. Furthermore, even imperfect interference visibility measured for a suitable set of spin preparations can be sufficient to infer absence of which-way information. General results are illustrated with a proof-of-principle single-photon experiment.

The duality between wave and particle properties of a microscopic physical system is a founding principle of quantum mechanics 1 , 2 . A canonical illustration is provided by a single particle travelling through a double slit or a Mach–Zehnder interferometer: an attempt to gain information about the path taken by the particle inevitably reduces the visibility of the interference pattern 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 . The purpose of this work is to analyse the trade-off between interference visibility and which-way information for a particle equipped with an internal degree of freedom, for example, spin. In such a general case, the interaction with an environment that acquires which-way information can transform in a non-trivial manner the joint path–spin state of the particle. This opens up an interesting question of how to infer the amount of which-way information deposited in the environment from visibility measurements. Another issue is, to what extent manipulating the spin subsystem can control the information about the path taken by the particle.

A common approach to quantify the amount of which-way information is to use distinguishability D , defined as the maximum difference between the probabilities of correct and incorrect identification 12 of the path taken by the particle inside the interferometer, based on the state of the environment. We assume here that both paths are equiprobable. In the spinless case, the ability for the particle distributed between two paths to interfere is characterized by the visibility V , which measures the modulation depth of interference fringes after the paths are combined at the interferometer exit. These two quantities are related by the inequality 6

When the particle travelling through the interferometer has an internal structure, the amount of available which-way information depends in principle on the preparation of the spin subsystem and the specifics of the interaction with the environment. Here we demonstrate that in this case the strongest bound on the distinguishability D is obtained by replacing V with a quantity named generalized visibility that depends on the initial spin preparation and the effective quantum channel experienced by the particle resulting from the interaction with the environment. Further, we present a systematic method to construct estimates for the generalized visibility based on directly measurable quantities, that is, interference visibilities for particular spin preparations at the input and selections of spin states before combining the paths at the interferometer exit. This provides an efficient strategy to find an upper bound on the available which-way information without performing full quantum process tomography 13 , 14 , 15 . We illustrate the general results with a proof-of-principle single-photon experiment that also demonstrates how which-way information can be erased by introducing mixedness in the spin preparation.

The inequality

where ||·|| denotes the trace norm. Let us stress here that this quantity is not conditioned upon any measurement performed eventually on subsystems QS .

Our central result is that D satisfies the inequality:

where V G , named generalized visibility, reads:

Finally, suppose that

Estimating generalized visibility

Generalized visibility V G is given by a rather intricate expression involving the spin preparation and the effective quantum channel experienced by the particle. In principle, full information about the channel can be obtained from quantum process tomography 13 , 14 , 15 , but this approach may be resource consuming, especially for a high dimension of the internal subsystem S . We will now give a recipe how to construct estimates for V G from direct visibility measurements for specific spin preparations and selections of individual spin components before interfering the particle paths.

Consider the following procedure. The particle is prepared in one of states

is the overall probability of detecting the particle in the filtered components. The modulation depth of interference fringes is characterized by the fractional visibility, given explicitly by

For the quantities introduced here, we always have | V μν | ≤ p μν . Full-depth modulation of fringes is observed for the equality sign.

Combining this result with equation (4) yields a family of bounds on distinguishability in the form

To expose more complex aspects of the trade-off between which-way information and interference visibility for a particle with an internal structure, we investigated experimentally interference of a single photon in a noisy Mach–Zehnder interferometer shown in Fig. 2 , using photon polarization as the internal subsystem S . Single photons with 810 nm central wavelength were generated by type-II spontaneous parametric down-conversion in a 30-mm long periodically poled potassium titanyl phosphate crystal pumped with 14 mW of 405 nm wavelength light from continuous wave diode laser, and heralded by detection of orthogonally polarized conjugate photons from the same pairs. The photons, after sending through a 3 nm bandwidth interference filter and transmitting through a single-mode fibre, were split between two paths using a calcite displacer. Equal splitting was ensured by using a fibre polarization controller. Removable half-wave plates H 1 and H 2 were used to prepare any combination of horizontal h and vertical υ polarization states for the two paths.

which provides a physical realization of equation (6) for d =2. We treat the realized noisy channel as a black box simulating interaction with an environment and analyse the maximum amount of which-way information that might have leaked to the environment in the worst-case scenario. We have seen that another realization of this channel, given in equation (7) , can reveal certain which-way information depending on the preparation of the internal degree of freedom.

In order to estimate which-way information possibly deposited in the environment, we used four different combinations of preparations for the photon polarization in the upper and the lower paths, denoted jointly as μ = hh , hυ , υh , υυ . After the simulated interaction with the environment, we filtered out from each path either horizontal or vertical component using the half-wave plate H 5 , which directed selected polarizations to the same output port after the second calcite crystal. We used a common-path set-up based on the half-wave plate H 6 and the polarizer P monitored by single-photon detectors D ± to measure the corresponding fractional visibilities V μν . The index ν labelling filters also assumes one of four values: hh , hυ , υh or υυ . The two non-interfering output ports after the second calcite crystal were monitored by detectors D 0 and D 1 in order to normalize overall count rates. The measured count rates were adjusted by binomial resampling to correct for non-uniform detection efficiencies of the detectors.

where θ 1 ,…, θ 4 are arbitrary phases. The corresponding operator

This bound on generalized visibility can be applied directly to experimental data in order to estimate available which-way information for mixed polarization preparation.

In order to measure fractional visibilities, the phase in the interferometer was adjusted by rotating one of the calcite crystals about an axis perpendicular to the plane of the set-up using a closed loop piezo-electric actuator. For each selected phase value, we averaged over four unitaries simulating the interaction with the environment. Experimentally measured interference fringes for combinations of preparations μ and filters ν entering the inequality (21) are shown in Fig. 3 along with results obtained for complementary filters. Probabilities p μν and fractional visibilities V μν determined from these measurements are collected in Table 3 . Inserting these data in equation (21) gives V G ≥0.960±0.006. This translates into an upper bound on distinguishability in the form D ≤0.28±0.02 according to equation (16) . Remarkably, this stringent bound is determined from just few measurements of fractional visibilities rather than complete quantum process tomography. In contrast, consider an individual pure preparation and complementary projective filters, corresponding to a pair of graphs shown in a single row of Fig. 3 . In this case, experimental results yield at most V G ≥0.580±0.006 using equation (17) , which implies D ≤0.815±0.005. This separation between estimated on distinguishability for mixed and pure polarization preparations strikingly demonstrates the convoluted effects of internal degrees of freedom in a which-way experiment.

Count rates on detectors D + (+, black) and D − (× , red) conditioned upon detection of conjugate heralding photons from the down-conversion source as a function of the phase shift in the interferometer. Panels are labelled as μ , ν , where μ specifies polarization preparation and ν defines polarizations filtered from the two paths. Solid lines are sinusoidal fits used to determine fractional visibilities. Experimental uncertainties of count rates are smaller than the size of the symbols.

Uncertainties are below 0.003 for all values shown.

The scenario considered in this work assumed a clear distinction between the environment and the spin that are both external to the path subsystem. The amount of available which-way information was determined solely from the quantum state of the environment after the interaction with the particle in the interferometer and no classical information about either spin preparation or measurement results could be used for that purpose. This makes our approach distinct from previous studies of the quantum erasure phenomenon 8 , 20 , 21 , 22 , 23 , where postselection carried out on the environment can restore conditional interference fringes. Here, the environment was treated as an adversary whose information about the path taken by the particle we try to control and estimate by manipulating the spin subsystem at the preparation and the detection stages.

In this context, the analysis of a which-way experiment with an internal degree of freedom points at non-trivial issues in prepare-and-measure protocols for quantum key distribution over complex noisy channels. Suppose that a cryptographic key is to be established by sending a particle along a randomly selected one of two paths, and the key security is to be verified by preparing occasionally the particle in a superposition of the two paths and measuring its coherence at the output by sampling interference fringes. If the particle is equipped with spin, the trade-off between distinguishability and visibility derived in this work implies that certain eavesdropping strategies require the sender to use randomness in spin preparation in order to ensure security. Further, to verify the security, the sender and the receiver may need to combine data collected for an array of spin preparations and filterings.

An entanglement-based analogue of the scenario analysed above would be to prepare jointly two particles in a maximally entangled path–spin state and to subject one of them to interaction with the environment modelling an eavesdropping attempt. In this case, spin can have the role of a shield subsystem fully protecting the privacy of a cryptographic key generated by detecting particles in individual paths, even though the noise present in the bipartite state may prevent entanglement distillation at the same rate as key generation 24 , 25 , 26 , 27 . To complete the parallel, estimates for generalized visibility determined from fractional visibility measurements can be viewed as a dynamical analogue of recently introduced privacy witnesses 28 .

Because the particle is not transferred between the two paths, the purified state has the form

The joint state of the particle and the environment after the interaction can be written as follows:

The maximum can be expressed in terms of the trace norm as follows:

The channel characteristics enters the above formula through

The expression for fractional visibilities V μν given in equation (13) can be rewritten in the tensor form

Author contributions

All authors contributed extensively to the work presented in this paper.

Additional information

How to cite this article: Banaszek, K. et al . Quantum mechanical which-way experiment with an internal degree of freedom. Nat. Commun. 4:2594 doi: 10.1038/ncomms3594 (2013).

Acknowledgments

We wish to acknowledge insightful discussions with B.-G. Englert, P. Raynal, R. Demkowicz-Dobrzański and W. Wasilewski. This research was supported by the European Union FP7 project Q-ESSENCE (Grant Agreement no. 248095), ERA-NET project QUASAR, and the Foundation for Polish Science TEAM project cofinanced by the EU European Regional Development Fund. Part of the work was done in National Quantum Information Centre of Gdańsk.

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A simple experiment for discussion of quantum interference and which-way measurement

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Mark B. Schneider , Indhira A. LaPuma; A simple experiment for discussion of quantum interference and which-way measurement. Am. J. Phys. 1 March 2002; 70 (3): 266–271. https://doi.org/10.1119/1.1450558

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We have developed a which-way experiment using visible light that is completely analogous to a recent experiment involving which-way measurement in atom interference. This simple experiment, easily accessible to undergraduate students and the resources of undergraduate departments, facilitates the examination of the key elements of which-way measurement, quantum erasure, and related mysteries of quantum measurement. The experiment utilizes a Mach–Zehnder interferometer, and visually demonstrates the loss of interference fringes when a which-way measurement is imposed, and the restoration of that pattern when the which-way information is destroyed. This device is also sensitive enough to observe interference fringes arising from single photons. We present a simple analysis of the interference appropriate for the coherent classical field limit and the single photon limit at a level accessible to undergraduates. We also briefly mention related issues on the nature of the photon.

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September 9, 2024

A New Quantum Cheshire Cat Thought Experiment Is Out of the Box

The spin of a particle seems to detach and move without a body—a strange experimental observation that’s stirring up debate

By Manon Bischoff

Quantum physics is often about cats—and this is also the case here.

Seregraff/Getty Images

Physicists seem to be obsessed with cats. James Clerk Maxwell, the father of electrodynamics, studied falling feline s to investigate how they turned as they fell. Many physics teachers have used a cat’s fur and a hard rubber rod to explain the phenomenon of frictional electricity. And Erwin Schrödinger famously illustrated the strangeness of quantum physics with a thought experiment involving a cat that is neither dead nor alive.

So it hardly seems surprising that physicists turned to felines once again to name a newly discovered quantum phenomenon in a paper published in the New Journal of Physics in 2013. Their three-sentence study abstract reads, “In this paper we present a quantum Cheshire Cat. In a pre- and post-selected experiment we find the Cat in one place, and its grin in another. The Cat is a photon, while the grin is its circular polarization.”

The newfound phenomenon was one in which certain particle features take a different path from their particle—much like the smile of the Cheshire Cat in Alice’s Adventures in Wonderland, written by Lewis Carroll—a pen name of mathematician Charles Lutwidge Dodgson—and published in 1865. To date, several experiments have demonstrated this curious quantum effect. But the idea has also drawn significant skepticism . Critics are less concerned about the theoretical calculations or experimental rigor than they are about the interpretation of the evidence. “It seems a bit bold to me to talk about disembodied transmission,” says physicist Holger Hofmann of Hiroshima University in Japan. “Instead we should revise our idea of particles.”

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Recently researchers led by Yakir Aharonov of Chapman University took the debate to the next level. Aharonov was a co-author of the first paper to propose the quantum Cheshire effect. Now, on the preprint server arXiv.org, he and his colleagues have posted a description of theoretical work that they believe demonstrates that quantum properties can move without any particles at all —like a disembodied grin flitting through the world and influencing its surroundings—in ways that bypass the critical concerns raised in the past.

A Grin without a Cat

Aharonov and his colleagues first encountered their quantum Cheshire cat several years ago as they were pondering one of the most fundamental principles of quantum mechanics: nothing can be predicted unambiguously. Unlike classical physics, the same quantum mechanical experiment can have different outcomes under exactly the same conditions. It is therefore impossible to predict the exact outcome of a single experiment—only its outcome with a certain probability. “Nobody understands quantum mechanics. It’s so counterintuitive. We know its laws, but we are always surprised,” says Sandu Popescu, a physicist at the University of Bristol in England, who collaborated with Aharonov on the 2013 paper and the new preprint.

But Aharonov was not satisfied with this uncertainty. So, since the 1980s , he has been exploring ways to investigate fundamental processes despite the probability-based nature of quantum mechanics. Aharonov—now age 92—employs an approach that involves intensively repeating an experiment, grouping results and then examining what came out before and after the experiment and relating these events to each other. “To do this, you have to understand the flow of time in quantum mechanics,” Popescu explains. “We developed a completely new method to combine information from measurements before and after the experiment.”

The researchers have stumbled across several surprises with this method—including their theoretical Cheshire cat. Their idea sounds simple at first: send particles through an optical tool called an interferometer, which causes each particle to move through one of two paths that ultimately merge again at the end. If the setup and measurements were carried out skillfully, Aharonov and his colleagues theorized, it could be shown that the particle traveled a path in the interferometer that differed from the path of its polarization. In other words, they claimed the property of the particle could be measured on one path even though the particle itself took the other—as if the grin and the cat had come apart.

Inspired by this theory, a team led by Tobias Denkmayr, then at the Vienna University of Technology, implemented the experiment with neutrons in a study published in 2014. The team showed that the neutral particles inside an interferometer followed a different path from that of their spin, a quantum mechanical property of particles similar to angular momentum: Denkmayr and his colleagues had indeed found evidence of the Cheshire cat theory. Two years later researchers led by Maximilian Schlosshauer of the University of Portland successfully implemented the same experiment with photons. The scientists saw evidence that the light particles took a different path in the interferometer than their polarization did.

Weak Measurements and Illusions

But not everyone is convinced. “Such a separation makes no sense at all. The location of a particle is itself a property of the particle,” Hofmann says. “It would be more accurate to talk about an unusual correlation between location and polarization.” Last November Hofmann and his colleagues provided an alternative explanation based on widely known quantum mechanical effects .

And in another interpretation of the Cheshire cat results, Pablo Saldanha of the Federal University of Minas Gerais in Brazil and his colleagues argue that the findings can be explained with wave-particle duality . “If you take a different view, there are no paradoxes,” Saldanha says, “but all results can be explained with traditional quantum mechanics as simple interference effects.”

Much of the controversy surrounds the way in which particles’ properties and positions are detected in these experiments. Disturbing a particle could alter its quantum mechanical properties. For that reason, the photons or neutrons cannot be recorded inside the interferometer using an ordinary detector. Instead scientists must resort to a principle of weak measurement developed by Aharonov in 1988. A weak measurement makes it possible to scan a particle very lightly without destroying its quantum state. This comes at a price, however: the weak measurement result is extremely inaccurate. (Thus, these experiments must be repeated many times over, to compensate for the fact that each individual measurement is highly uncertain.)

In the quantum Cheshire cat experiments, a weak measurement is made along a path in the interferometer, the paths then merge, and the emerging particles are measured with an ordinary detector. Along one path of the interferometer, a weak measurement of the particle’s position can be taken and, along the other, its spin. Using detectors, physicists can more definitively characterize the particles that traveled through the interferometer and potentially reconstruct what occurred during the particle’s journey. For example, only certain particles will appear in certain detectors, helping the physicists piece together which path their neutron or photon previously took. According to Aharonov, Popescu and their colleagues, the Cheshire cat experiments ultimately reveal that the particle’s position can be confirmed on one path even as its polarization or spin was measured on the other.

Melissa Thomas Baum/Buckyball Design; Source: “Observation of a Quantum Cheshire Cat in a Matter-Wave Interferometer Experiment,” by Tobias Denkmayr et al., in Nature Communications , Vol. 5, Article No. 4492; July 29, 2014

Saldanha and his co-authors assert that it is impossible to make claims about quantum systems in the past given their measurements in the present. In other words, the photons and neutrons measured in the final detectors cannot tell us much about their previous trajectory. Instead the wave functions of particles passing through the paths of the interferometer could overlap, which would make it impossible to trace which path a particle had taken. “Ultimately, the paradoxical behaviors are related to the wave-particle duality,” Saldanha says. But in the papers that report evidence of the quantum Cheshire cat, he asserts, the findings “are processed in a sophisticated way that obscures this simpler interpretation.”

Hofmann, meanwhile, has stressed that the results will differ if you measure the system in a different way. This phenomenon is well-known in quantum physics: if, for example, you first measure the speed of a particle and then its position, the result can be different than it would be if you first measured the position of the same particle and then its speed. He and his colleagues therefore contend that Aharonov and his team’s conclusions were correct in themselves—that the particle moved along one path and the polarization followed the other—but that such differing paths do not apply simultaneously.

As Hofmann’s co-author Jonte Hance, also at Hiroshima University, told New Scientist , “It only looks like [the particle and polarization are] separated because you’re measuring one of the properties in one place and the other property in the other place, but that doesn’t mean that the properties are in one place and the other place, that means that the actual measuring itself is affecting it in such a way that it looks like it’s in one place and the other place.”

A New Way to Catch a Cheshire Cat?

But these critiques are “missing the point,” Popescu says. He agrees that the work and reasoning put forward by Saldanha and Hofmann’s respective groups are correct—but adds that the best way to test any interpretation is to generate testable predictions from each. “As I understand it, there is no direct way to make predictions based on them,” Popescu says in reference to these alternative explanations. “They kind of have a very old-fashioned way of looking at things: there are contradictions, so you stop doing the math.”

With their recent preprint paper, Aharonov and Popescu, together with physicist Daniel Collins of the University of Bristol, have now described how a particle’s spin can move completely independently of the particle itself—without employing a weak measurement. In their new experimental setup, a particle is located in the left half of an elongated two-part cylinder that is sealed at the outer edges. Because of a highly reflective wall in the middle, the particle has a vanishingly small probability of tunneling through to the right-hand side of the cylinder. In their paper, the researchers provide a proof that even if the particle remains in the left-hand area in almost all cases, it should still be possible to measure a transfer of the particle’s spin at the right-hand outer wall. “It’s amazing, isn’t it?” Collins says. “You think the particle has a spin and the spin should stay with the particle. But the spin crosses the box without the particle.”

In a new thought experiment designed to observe the quantum Cheshire cat, physicists would be able to measure the property of a particle in one of two chambers of an elongated cylinder despite the fact that the particle itself would be contained in the other chamber.

Amanda Montañez

This approach would address several of the critical concerns raised thus far. The physicists don't need weak measurements. Nor do they need to group their experimental results to draw temporal conclusions. (That being said, grouping results would still improve the measurements, given that the angular momentum of the wall itself cannot be determined unambiguously because of the Heisenberg uncertainty principle.) But in this scenario, the only physical principles involved are conservation laws, such as the conservation of energy or the conservation of momentum and angular momentum. Popescu and Collins explain that they hope other groups will implement the experiment to observe the effects in the laboratory.

The new work has piqued Hofmann’s interest. “The scenario is exciting because the interaction between polarization and particle motion produces a particularly strong quantum effect that clearly contradicts the particle picture,” he says.

But he still does not see this as proof of disembodied (particle-free) spin transfer. “For me, this means, above all, that it is wrong to assume a measurement-independent reality,” Hofmann says. Instead quantum mechanics allows a particle’s residence to extend to the right-hand region of the cylinder, even if a residence in the left-hand region seems logically compelling. “I think it is quite clear to Aharonov, Collins and Popescu that the space in front of the wall is not really empty,” he adds.

Saldanha, meanwhile, still sees the researchers as overcomplicating what could be explained as traditional quantum interference effects. When discussing the particle’s very low probability of entering the right-hand side of the experimental setup, he explains, “we have to be careful about a ‘vanishingly small probability’ when we refer to waves.” The wave function of the particle could also expand into the right-hand side of the setup and thus influence the angular momentum of the wall. “The same predictions can be made without such dramatic conclusions,” he says.

In response to these critiques, Popescu says, “This is of course another way of thinking about it. The question is whether this interpretation is useful.” Regardless of which interpretation of the events is correct, the quantum Cheshire cat could enable new technological applications. For example, it could be used to transfer information or energy without moving a physical particle—whether made of matter or light.

For Popescu, however, the fundamental questions of physics play a more important role. “It all started when we thought about how time propagates in quantum mechanics,” he says. “And suddenly we were able to discover something fundamental about the laws of conservation.”

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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• Published: 03 September 2024

Quantum double slit experiment with reversible detection of photons

• Vipin Devrari 1 &
• Mandip Singh 1  

Scientific Reports volume  14 , Article number:  20438 ( 2024 ) Cite this article

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• Quantum mechanics
• Single photons and quantum effects

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.

A schematic diagram of the experiment, where a screen corresponds to a single photon detector capable of detecting locations of photon detection events. A screen is placed considerably closer to the source than a beam splitter and a double-slit.

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.

An experimental diagram, where EPR entangled pairs of photons are emanated in a conical emission pattern. The paths of entangled photons are represented by red lines. The pump laser beam, after passing through the crystal, is represented by a narrow white line for clarity. The x -axis is perpendicular to the optical table and passing though the nonlinear crystal.

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.

( a ) Quantum interference pattern obtained by measuring the coincidence detection of photons by a variable position single photon detector \(D_{1}\) and a stationary single photon detector \(D_{2}\) , where a solid line represents the theoretically evaluated interference pattern. ( b ) Coincidence detection of photons results in no interference, when photon-2 is detected by a stationary single photon detector \(D_{3}\) and photon-1 is detected by \(D_{1}\) . Each data point is an average of ten repetitions of the experiment with 60 s time of exposure.

The shift in the two-photon quantum interference pattern when, ( a ) single photon detector \(D_{2}\) position is \(x_{o} = +0.11\) mm, ( b ) \(D_{2}\) position is \(x_{o} = -0.11\) mm. Single photons do not interfere in this experiment. Each data point is an average of ten repetitions of the experiment with 60 s time of exposure.

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.  3 a, 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.  3 b, 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.

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.

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.

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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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