Two-boson quantum interference in time

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. 1A). This HOM effect follows from the destructive two-photon interference between the probability amplitudes for double transmission and double reflection at the BS (Fig. 1B). 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).Open in a separate windowFig. 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 coincident 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 correlation function exhibits an HOM dip when the time difference Δt between the two detected photons is close to zero (i.e., when they tend to be indistinguishable).Here, we explore how two-boson quantum interference transforms under reversal of the arrow of time in one of the two bosonic modes (Fig. 2A). 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.Open in a separate windowFig. 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 on port â), and we condition on all photons being reflected. The retrodicted state of mode b^′ (initially the vacuum state 0) back propagates 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 of gain g=1/η with input state 0,0 and output state n,n. PDC is an active Bogoliubov transformation, requiring a pump beam (represented in blue). Note that the PTR duality is rigorously valid when this pump beam is of high intensity (i.e., treated as a classical light beam) since the Hamiltonian HPDC of Eq. 4 holds in this limit only. (B) Operational view of the PTR duality. As noted in ref. 20, if we prepare the entangled (EPR) state Ψb,b′′∝∑n=0∞n,n and send mode b^ in the BS, we get the output state Ψa′,b′′∝∑n=0∞⁡sinn θ n,n, which is precisely the two-mode squeezed vacuum state produced by PDC when the signal and idler modes are initially in the vacuum state.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.