# Two-boson quantum interference in time

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Edited by Marlan O. Scully, Texas A&M University, College Station, TX, and approved October 19, 2020 (received for review May 27, 2020)

## Significance

We uncover an unsuspected quantum interference mechanism, which originates from the indistinguishability of identical bosons in time. Specifically, we build on the Hong–Ou–Mandel effect, namely the “bunching” of identical bosons at the output of a half-transparent beam splitter resulting from the symmetry of the wave function. We establish that this effect turns, under partial time reversal, into an interference effect in a quantum amplifier that we ascribe to time-like indistinguishability (bosons from the past and future cannot be distinguished). This hitherto unknown effect is a genuine manifestation of quantum physics and may be observed whenever two identical bosons participate in Bogoliubov transformations, which play a role in many facets of physics.

## Abstract

The celebrated Hong–Ou–Mandel effect is the paradigm of two-particle quantum interference. It has its roots in the symmetry of identical quantum particles, as dictated by the Pauli principle. Two identical bosons impinging on a beam splitter (of transmittance 1/2) cannot be detected in coincidence at both output ports, as confirmed in numerous experiments with light or even matter. Here, we establish that partial time reversal transforms the beam splitter linear coupling into amplification. We infer from this duality the existence of an unsuspected two-boson interferometric effect in a quantum amplifier (of gain 2) and identify the underlying mechanism as time-like indistinguishability. This fundamental mechanism is generic to any bosonic Bogoliubov transformation, so we anticipate wide implications in quantum physics.

The laws of quantum physics govern the behavior of identical particles via the symmetry of the wave function, as dictated by the Pauli principle (1). In particular, it has been known since Bose and Einstein (2) that the symmetry of the bosonic wave function favors the so-called bunching of identical bosons. A striking demonstration of bosonic statistics for a pair of identical bosons was achieved in 1987 in a seminal experiment by Hong, Ou, and Mandel (HOM) (3), who observed the cancellation of coincident detections behind a 50:50 beam splitter (BS) when two indistinguishable photons impinge on its two input ports (Fig. 1*A*). This HOM effect follows from the destructive two-photon interference between the probability amplitudes for double transmission and double reflection at the BS (Fig. 1*B*). Together with the Hanbury Brown and Twiss effect (4, 5) and the violation of Bell inequalities (6, 7), it is often viewed as the most prominent genuinely quantum feature: it highlights the singularity of two-particle quantum interference as it cannot be understood in terms of classical wave interference (8, 9). It has been verified in numerous experiments over the last 30 y (see, e.g., refs. 10⇓⇓–13), even in case the single photons are simultaneously emitted by two independent sources (14⇓–16) or within a silicon photonic chip (17, 18). Remarkably, it has even been experimentally observed with 4He metastable atoms, demonstrating that this two-boson mechanism encompasses both light and matter (19).

Here, we explore how two-boson quantum interference transforms under reversal of the arrow of time in one of the two bosonic modes (Fig. 2*A*). This operation, which we dub partial time reversal (PTR), is unphysical but nevertheless central as it allows us to exhibit a duality between the linear optical coupling effected by a BS and the nonlinear optical (Bogoliubov) transformation effected by a parametric amplifier. As a striking implication of these considerations, we predict a two-photon interferometric effect in a parametric amplifier of gain 2 (which is dual to a BS of transmittance 1/2). We argue that this unsuspected effect originates from the indistinguishability between photons from the past and future, which we coin “time-like” indistinguishability as it is the partial time-reversed version of the usual “space-like” indistinguishability that is at work in the HOM effect.

Since Bogoliubov transformations are ubiquitous in quantum physics, it is expected that this two-boson interference effect in time could serve as a test bed for a wide range of bosonic transformations. Furthermore, from a deeper viewpoint, it would be fascinating to demonstrate the consequence of time-like indistinguishability in a photonic or atomic platform as it would help in elucidating some heretofore overlooked fundamental property of identical quantum particles.

## Hong–Ou–Mandel Effect

The HOM effect is a landmark in quantum optics as it is the most spectacular manifestation of boson bunching. It is a two-photon intrinsically quantum interference effect where the probability amplitude of both photons being transmitted cancels out the probability amplitude of both photons being reflected. A 50:50 BS effects the single-photon transformations (for details, see *Materials and Methods, Gaussian Unitaries for a BS and PDC*)*Materials and Methods, Two-Photon Interference in a BS and PDC*)*C*).

## Partial Time Reversal

Bogoliubov transformations on two bosonic modes comprise passive and active transformations. The BS is the fundamental passive transformation, while parametric down conversion (PDC) gives rise to the class of active transformations (also called nondegenerate parametric amplification). Although the involved physics is quite different (a simple piece of glass makes a BS, while an optically pumped nonlinear crystal is needed to effect PDC), the Hamiltonians generating these two unitaries are amazingly close, namely

The underlying concept of PTR will be formalized in Eq. **7**, but we first illustrate this duality between a BS and PDC with the simple example of Fig. 2*A*, where n photons impinge on port *Gaussian Unitaries for a BS and PDC*). The path where all photons are reflected (*A*) leads us to consider the transition probability amplitude for a PDC of gain *Gaussian Unitaries for a BS and PDC*). Strikingly, the above two amplitudes can be made equal (up to a constant *Materials and Methods, Example of PTR*. Conditioning again the output port

These examples reflect the existence of a general duality between a BS and PDC. Indeed, as demonstrated in *Materials and Methods, Proof of PTR Duality*, partial transposition in Fock basis gives rise to PTR duality**24** or **25**). The PTR duality is nicely evidenced by the conservation rules exhibited by the BS and PDC transformations: the former conserves the total photon number, while the latter conserves the difference between the photon numbers. In Eq. **7**, the only nonzero matrix elements for a BS are those satisfying

The notion of time reversal can be conveniently interpreted using the so-called “retrodictive” picture of quantum mechanics (21). Along this line, PTR must be understood here as the fact that the “retrodicted” state of mode *Materials and Methods, Retrodictive Picture of Quantum Mechanics*). As shown in Fig. 2*B*, the PTR duality can be made operational by sending half of a so-called Einstein–Podoslky–Rosen (EPR) entangled state on mode

## Two-Boson Interference in an Amplifier

Due to this duality, the HOM effect for a BS of transmittance 1/2 immediately suggests the possible existence of a related interferometric suppression effect in a PDC of gain 2, namely *Two-Photon Interference in a BS and PDC*):*A*.

The dependence of the probability of detecting a single pair (*Two-Photon Interference in a BS and PDC*)**3** and divide by g, we get exactly Eq. **9**, as implied by PTR duality. More generally, we show in *Materials and Methods, Extension to a PDC with Integer Gain* that this interferometric suppression effect actually extends to any larger integer value of the gain (e.g., *B*, the probability of detecting n photons simultaneously on each output port vanishes when the gain

## Space-Like vs. Time-Like Indistinguishability

The origin of the two-boson quantum interference effect that we predict can be traced back to boson indistinguishability, similarly as for the HOM effect albeit in a time-like version (involving bosons from the past and future). We first recall that the HOM effect originates from what can be viewed as space-like indistinguishability (Fig. 4, *Upper*). When two photons impinge on a BS of transmittance η, each photon has a probability amplitude

We now argue that it is the exchange of indistinguishable photons in time that is responsible for the interference effect in an amplifier (Fig. 4, *Lower*). When two photons impinge on a PDC with gain g, they can be both transmitted without triggering a stimulated event, which is dual to the double transmission in a BS (where η is substituted by

## Discussion and Conclusion

The role of time reversal in quantum physics has long been a fascinating subject of questioning (see, e.g., ref. 22 and references therein), but the key idea of the present work is to consider a bipartite quantum system (two bosonic modes) with counterpropagating times. Incidentally, we note that the notion of time reversal has been exploited in the context of defining separability criteria (23, 24), but this seems to be unrelated to PTR duality. Further, the link between time reversal and optical-phase conjugation has been mentioned in the quantum optics literature (see, e.g., ref. 25), but it exploits the fact that the complex conjugate of an electromagnetic wave is the time-reversed solution of the wave equation (the phase conjugation time-reversal mirror concerns one mode only). The PTR duality introduced here bears some resemblance with an early model of lasers (26) based on the coupling of an “inverted” harmonic oscillator (having a negative frequency ω) with a heat bath. The inverted harmonic oscillator (**25**, namely

In this work, we have promoted PTR as the proper way to approach the duality between passive and active bosonic transformations. As a compelling application of PTR duality, we have unveiled a hitherto unknown quantum interference effect, which is a manifestation of quantum indistinguishability for identical bosons in active transformations (space-like indistinguishability, which is at the root of the HOM effect, transforms under PTR into time-like indistinguishability). The interferometric suppression of the coincident 1,1 term is induced by the indistinguishability between a photon pair originating from the past and a photon pair going to the future. Stated more dramatically, while the two photons may cross the amplification medium and be detected, the sole fact that they could instead be annihilated and replaced by two other photons makes the detection probability drop to zero when

The experimental verification of this effect can be envisioned with present technologies (see *Experimental Scheme*). A coincidence probability lower than 25% would be sufficient to rule out a classical interpretation, which could in principle be reached with a moderate gain of 1.28 (see *Classical Baseline*). Observing time-like two-photon interference in experiments involving active optical components would then be a highly valuable metrology tool given that the HOM dip is commonly used today as a method to benchmark the reliability of single-particle sources and mode matching. More generally, the interference of many photons scattered over many modes in a linear optical network has generated a tremendous interest in the recent years, given the connection with the “boson sampling” problem [i.e., the hardness of computing the permanent of a random matrix (29)], and technological progress in integrated optics now makes it possible to access large optical circuits (see, e.g., ref. 30). In this context, it would be exciting to uncover new consequences of PTR duality and time-like interference.

Finally, we emphasize that our analysis encompasses all bosonic Bogoliubov transformations, which are widespread in physics, appearing in quantum optics, quantum field theory, or solid-state physics, but also in black hole physics or even in the Unruh effect (describing an accelerating reference frame). This suggests that time-like quantum interference may occur in various physical situations where identical bosons participate in such a transformation. Beyond bosons, let us point out an intriguing connection with the notion of “crossing” in quantum electrodynamics (31, 32). Crossing symmetry refers to a substitution rule connecting two scattering matrix elements that are related by a Wick rotation (antiparticles being turned into particles going backward in time). For example, the scattering of a photon by an electron (Compton scattering) and the creation of an electron–positron pair by two photons are processes that are related to each other by such a substitution rule (see, e.g., ref. 33). This is in many senses analogous to the PTR duality described here: since a photon (or truly neutral boson) is its own antiparticle, we may view PTR duality as a substitution rule connecting the BS diagram to the PDC diagram. We hope that this connection with quantum electrodynamics may open up even broader perspectives.

## Materials and Methods

### Gaussian Unitaries for a BS and PDC.

Passive and active Gaussian unitaries are effected by linear optical interferometry or parametric amplification, respectively (34). The fundamental passive two-mode Gaussian unitary, namely the BS unitary

The action of **11**. For example, when n photons impinge on one of the input ports, each photon may be transmitted or reflected, so we get the binomial state

### Example of PTR.

We illustrate the PTR duality between a BS and PDC by considering the additional example of a BS with m photons impinging on input port â and n photons impinging on input port *A*). If we condition on the vacuum on mode **21** with the probability amplitude**19**). Now, if we make the substitution **21** and **22** are dual under PTR, namely*B*, if the input mode **19**).

### Proof of PTR Duality.

The PTR duality is illustrated in Table 1 for few photons. As expressed by Eq. **7**, it can be viewed as a consequence of partial transposition of the state of mode *A* and can also be interpreted by comparing the unitaries **10** and **12** or their corresponding decompositions, Eqs. **14** and **17**. In general terms, we may say that the (passive) linear coupling of two bosonic modes is dual, under PTR, to an (active) Bogoliubov transformation, which is expressed as*B*, the corresponding operational scheme is depicted, relying on the preparation of an entangled (EPR) state at the input of mode

We now prove PTR duality by reexpressing Eq. **24** in the Heisenberg picture, namely**27** with Eq. **28**, we see that**29** and **30** for

Similar equations can be derived starting from **31**, we get Eq. **26**, which concludes the proof of Eq. **24**.

Note that PTR duality can be reexpressed by using the identity**33** and using Eq. **24** implies the general relation

### Retrodictive Picture of Quantum Mechanics.

In the usual, predictive approach of quantum mechanics, one deals with the preparation of a quantum system followed by its time evolution and ultimately, its measurement. Specifically, one uses the prior knowledge on the state *A*). Specifically, one associates a retrodicted state

The retrodictive picture can be successfully exploited in different situations [for example, to characterize the quantum properties of an optical measurement device (35)], but it is always used in lieu of the predictive picture. Here, we instead combine it with the predictive picture in order to properly define PTR duality and describe a composite system that is propagated partly forward and partly backward in time, as represented in Fig. 7 *B*, *Right*. Specifically, we consider a composite system prepared in a product state *B*, *Left*), the conditional probabilities are given by**38** as**40**, the predictive picture is used for subsystem a, while the retrodictive picture is used for subsystem b.

In our analysis of a BS under PTR, we have **33** reduces to Eq. **34**. Hence, **40**, the joint state is then shown to evolve according to a PDC. Note that it is not always possible to construct an operator

### Two-Photon Interference in a BS and PDC.

The HOM effect can be simply understood by calculating the probability amplitude for coincident detection**11**, it is simple to rewrite it as

Now, we examine the corresponding quantum interferometric suppression in a PDC and its dependence in the parametric gain g. Let us calculate the probability amplitude for coincident detection**18** up to changing the sign of r, and the ket **13**. This gives **8**.

### Extension to a PDC with Integer Gain.

We may also consider the case where the gain takes a larger integer value (e.g., **46** reveals that the output term with *B* for *Left*). The latter effect is easy to understand as the interference between the amplitude with all *Right*. More generally, the transition probability **47** under PTR, namely

### Experimental Scheme.

The HOM effect is considered a most spectacular evidence of genuinely quantum two-boson interference, and we expect the same for its PTR counterpart as it admits no classical interpretation. The experimental verification of our effect can be envisioned with present technologies, as sketched in Fig. 9. We would need two single-photon sources, which could be heralded by the detection of a trigger photon at the output of a PDC with low gain (the single photon being prepared conditionally on the detection of the trigger photon in the twin beam). The two single photons would impinge on a PDC of gain 2, whose output modes should be monitored: the probability of detecting exactly one photon on each mode should be suppressed as a consequence of time-like indistinguishability. In principle, photon number resolution would be needed in order to discriminate the output term with one photon pair (n = 1) from the terms with more pairs (*Classical Baseline*), which could in principle be reached with a gain of 1.28 (i.e., a squeezing of 4.39 dB). Of course, the effect of losses should also be carefully analyzed in order to assess the feasibility of the scheme depicted in Fig. 9.

Demonstrating this effect would be invaluable in view of the fact that the HOM dip is widely used to test the indistinguishability of single photons and to benchmark mode matching: it witnesses the fact that the photons are truly indistinguishable (they admit the same polarization and couple to the same spatiotemporal mode). For example, HOM experiments have been used to test the indistinguishability of single photons emitted by a semiconductor quantum dot in a microcavity (10), while the interference of two single photons emitted by two independently trapped rubidium-87 atoms has been used as an evidence of their indistinguishability (15). The HOM effect has also been generalized to three-photon interference in a three-mode optical mixer (38), while the case of many photons in two modes has been analyzed in ref. 39, implying a possible application of the quantum Kravchuk–Fourier transform (40). We anticipate that most of these ideas could extend to interferences in an active optical medium.

### Classical Baseline.

The two-photon quantum interference effect in amplification cannot be interpreted within a classical model of PDC, where a pair can be annihilated or created with some probability. We have two possible indistinguishable paths (the photon pair either going through the crystal or being replaced by another one) with equal individual probabilities but opposite probability amplitudes; hence, the resulting probability vanishes (whereas the two probabilities would add for classical particles). In order to assess an experimental verification of this effect, it is necessary to establish a classical baseline, namely to determine the depletion of the probability of coincident detections that can be interpreted classically. As a guide, consider first a classical model of the HOM effect where the two input photons are distinguishable. We have to add the double-transmission probability **3**. For a 50:50 BS, **9**. For a gain 2 PDC,

## Data Availability.

There are no data underlying this work.

## Acknowledgments

We thank Ulrik L. Andersen, Maria V. Chekhova, Claude Fabre, Virginia D’Auria, Linran Fan, Radim Filip, Saikat Guha, Dmitri Horoshko, Mikhail I. Kolobov, Julien Laurat, Klaus Mölmer, Romain Mueller, Ognyan Oreshkov, Olivier Pfister, Wolfgang P. Schleich, and Sébastien Tanzilli as well as an anonymous referee for useful comments. M.G.J. acknowledges support from the Wiener-Anspach Foundation. This work was supported by the Fonds de la Recherche Scientifique - FNRS under grant PDR T.0224.18.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: nicolas.cerf{at}ulb.ac.be.

Author contributions: N.J.C. designed research; and N.J.C. and M.G.J. performed research, derived the formulas, discussed the results, and wrote the paper.

The authors declare no competing financial interest.

This article is a PNAS Direct Submission.

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

## References

- ↵
- ↵
- A. Einstein

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- M. Halder et al.

- ↵
- P. Aboussouan,
- O. Alibart,
- D. B. Ostrowsky,
- P. Baldi,
- S. Tanzilli

- ↵
- Y. Tsujimoto et al.

- ↵
- ↵
- ↵
- ↵
- J. W. Silverstone et al.

- ↵
- C. Agnesi et al.

- ↵
- ↵
- L. Chakhmakhchyan,
- N. J. Cerf

- ↵
- ↵
- O. Oreshkov,
- N. J. Cerf

- ↵
- A. Sanpera,
- R. Tarrach,
- G. Vidal

- ↵
- ↵
- J. Park,
- C. Park,
- K. Lee,
- Y. H. Cho,
- Y. Park

- ↵
- D. M. Greenberger

- R. J. Glauber

- ↵
- D. N. Klyshko

- ↵
- M. F. Z. Arruda et al.

- ↵
- S. Aaronson,
- A. Arkhipov

- ↵
- B. A. Bell,
- G. S. Thekkadath,
- R. Ge,
- X. Cai,
- I. A. Walmsley

- ↵
- ↵
- J. M. Jauch,
- F. Rohrlich

- ↵
- M. E. Peskin,
- D. V. Schroeder

- ↵
- ↵
- ↵
- A. Heuer,
- R. Menzel,
- P. W. Milonni

- ↵
- A. Heuer,
- R. Menzel,
- P. W. Milonni

- ↵
- R. A. Campos

- ↵
- H. Nakazato,
- S. Pascazio,
- M. Stobińska,
- K. Yuasa

- ↵
- M. Stobińska et al.

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