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Two-Boson Quantum Interference in Time Two-boson quantum interference in time Nicolas J. Cerfa,1 and Michael G. Jabbourb aCentre for Quantum Information and Communication, Ecole polytechnique de Bruxelles, Universite´ libre de Bruxelles, 1050 Bruxelles, Belgium; and bDepartment of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom Edited by Marlan O. Scully, Texas A&M University, College Station, TX, and approved October 19, 2020 (received for review May 27, 2020) The celebrated Hong–Ou–Mandel effect is the paradigm of two- time could serve as a test bed for a wide range of bosonic trans- particle quantum interference. It has its roots in the symmetry of formations. Furthermore, from a deeper viewpoint, it would be identical quantum particles, as dictated by the Pauli principle. Two fascinating to demonstrate the consequence of time-like indistin- identical bosons impinging on a beam splitter (of transmittance guishability in a photonic or atomic platform as it would help in 1/2) cannot be detected in coincidence at both output ports, as elucidating some heretofore overlooked fundamental property confirmed in numerous experiments with light or even matter. of identical quantum particles. Here, we establish that partial time reversal transforms the beam splitter linear coupling into amplification. We infer from this dual- Hong–Ou–Mandel Effect ity the existence of an unsuspected two-boson interferometric The HOM effect is a landmark in quantum optics as it is effect in a quantum amplifier (of gain 2) and identify the underly- the most spectacular manifestation of boson bunching. It is a ing mechanism as time-like indistinguishability. This fundamental two-photon intrinsically quantum interference effect where the mechanism is generic to any bosonic Bogoliubov transformation, probability amplitude of both photons being transmitted cancels so we anticipate wide implications in quantum physics. out the probability amplitude of both photons being reflected. A 50:50 BS effects the single-photon transformations (for quantum interference j boson bunching j time reversal details, see Materials and Methods, Gaussian Unitaries for a BS and PDC) he laws of quantum physics govern the behavior of identical 1 1 particles via the symmetry of the wave function, as dictated T j1ia ! p (j1ia − j1ib ), j1ib ! p (j1ia + j1ib ), [1] by the Pauli principle (1). In particular, it has been known since 2 |{z} |{z} 2 |{z} |{z} PHYSICS Bose and Einstein (2) that the symmetry of the bosonic wave trans ref ref trans function favors the so-called bunching of identical bosons. A a b j1i striking demonstration of bosonic statistics for a pair of iden- where and label the bosonic modes and a=b stands for tical bosons was achieved in 1987 in a seminal experiment by a single-photon state in mode a=b (here, “trans” stands for Hong, Ou, and Mandel (HOM) (3), who observed the cancel- transmitted and “ref” for reflected). Hence, the state of two lation of coincident detections behind a 50:50 beam splitter (BS) indistinguishable photons (one in each mode) transforms as when two indistinguishable photons impinge on its two input 1 ports (Fig. 1A). This HOM effect follows from the destructive j1i j1i ! p (j1i j1i − j1i j1i ) [2] two-photon interference between the probability amplitudes for a b 2 a a b b double transmission and double reflection at the BS (Fig. 1B). Together with the Hanbury Brown and Twiss effect (4, 5) and since the double-transmission term j1ia j1ib cancels out the the violation of Bell inequalities (6, 7), it is often viewed as the double-reflection term j1ib j1ia . More generally, the probability most prominent genuinely quantum feature: it highlights the sin- gularity of two-particle quantum interference as it cannot be understood in terms of classical wave interference (8, 9). It has Significance been verified in numerous experiments over the last 30 y (see, e.g., refs. 10–13), even in case the single photons are simulta- We uncover an unsuspected quantum interference mecha- neously emitted by two independent sources (14–16) or within nism, which originates from the indistinguishability of iden- a silicon photonic chip (17, 18). Remarkably, it has even been tical bosons in time. Specifically, we build on the Hong– experimentally observed with 4He metastable atoms, demon- Ou–Mandel effect, namely the “bunching” of identical bosons strating that this two-boson mechanism encompasses both light at the output of a half-transparent beam splitter result- and matter (19). ing from the symmetry of the wave function. We establish Here, we explore how two-boson quantum interference trans- that this effect turns, under partial time reversal, into an forms under reversal of the arrow of time in one of the two interference effect in a quantum amplifier that we ascribe bosonic modes (Fig. 2A). This operation, which we dub par- to time-like indistinguishability (bosons from the past and tial time reversal (PTR), is unphysical but nevertheless cen- future cannot be distinguished). This hitherto unknown effect tral as it allows us to exhibit a duality between the linear is a genuine manifestation of quantum physics and may optical coupling effected by a BS and the nonlinear optical be observed whenever two identical bosons participate in (Bogoliubov) transformation effected by a parametric amplifier. Bogoliubov transformations, which play a role in many facets As a striking implication of these considerations, we predict a of physics. two-photon interferometric effect in a parametric amplifier of gain 2 (which is dual to a BS of transmittance 1/2). We argue Author contributions: N.J.C. designed research; and N.J.C. and M.G.J. performed that this unsuspected effect originates from the indistinguisha- research, derived the formulas, discussed the results, and wrote the paper.y bility between photons from the past and future, which we coin The authors declare no competing financial interest.y “time-like” indistinguishability as it is the partial time-reversed This article is a PNAS Direct Submission.y version of the usual “space-like” indistinguishability that is at This open access article is distributed under Creative Commons Attribution-NonCommercial- work in the HOM effect. NoDerivatives License 4.0 (CC BY-NC-ND).y Since Bogoliubov transformations are ubiquitous in quantum 1 To whom correspondence may be addressed. Email: [email protected] physics, it is expected that this two-boson interference effect in www.pnas.org/cgi/doi/10.1073/pnas.2010827117 PNAS Latest Articles j 1 of 10 Downloaded by guest on October 2, 2021 where a^ and b^ are mode operators. It is striking that a simple A y swap b^ $ b^ transforms HBS into HPDC, suggesting a deep dual- ity between a BS and a PDC by reversing the arrow of time in mode b^ (keeping mode a^ untouched). 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. 2A, where n photons impinge on port B b^ of a BS (with vacuum on port a^), resulting in the binomial output state n !1=2 BS X n k n−k U j0, ni = (sin θ) (cos θ) jk, n − ki , [5] η k k=0 with η = cos2 θ (see Gaussian Unitaries for a BS and PDC). The path where all photons are reflected (k = n) is associated with BS n the transition probability amplitude hn, 0j Uη j0, ni = sin θ. C Reversing the arrow of time on mode b^ (Fig. 2A) leads us to consider the transition probability amplitude for a PDC of gain g = cosh2 r with vacuum state on its two inputs and n photon PDC n pairs on its outputs, that is, hn, nj Ug j0, 0i = tanh r= cosh r A Fig. 1. (A) If two indistinguishable photons (represented in red and green for the sake of argument) simultaneously enter the two input ports of a 50:50 BS, they always exit the same output port together (no coinci- dent detection can be observed). (B) The probability amplitudes for double transmission (Left) and double reflection (Right) precisely cancel each other when the transmittance is equal to 1/2. This is a genuinely quantum effect, which cannot be described as a classical wave interference. (C) The correla- tion function exhibits an HOM dip when the time difference ∆t between the two detected photons is close to zero (i.e., when they tend to be indistinguishable). for coincident detections is given by (for details, see Materials and Methods, Two-Photon Interference in a BS and PDC) BS 2 2 B Pcoinc(η) = j h1, 1j Uη j1, 1i j = (2η − 1) , [3] BS where Uη denotes the unitary corresponding to a BS of transmittance η. In a nutshell, the HOM effect boils down to BS h1, 1j U1=2 j1, 1i = 0. Its experimental manifestation is the pres- ence of a dip in the correlation function, witnessing that two photons cannot be coincidently detected at the two output ports when η = 1=2 (Fig. 1C). Fig. 2. (A) BS under PTR, flipping the arrow of time in mode b^. The PTR duality is illustrated when n photons impinge on port b^ (with vacuum Partial Time Reversal on port ^a), and we condition on all photons being reflected. The retro- ^0 Bogoliubov transformations on two bosonic modes comprise dicted state of mode b (initially the vacuum state j0i) back propagates passive and active transformations. The BS is the fundamen- from the detector to the source (suggested by a wavy arrow). This yields the same transition probability amplitude (up to a constant) as for a PDC tal passive transformation, while parametric down conversion of gain g = 1/η with input state j0, 0i and output state jn, ni.
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