Distinguishability and Many-Particle Interference

Quantum interference of two independent particles in pure quantum states is fully described by the particles’ distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability, taking on a surprisingly rich character. Here, we study many-particle interference using three photons. We show that the distinguishability between pairs of photons is not sufficient to fully describe the photons’ behavior in a scattering process, but that a collective phase, the triad phase, plays a role. We are able to explore the full parameter space of three-photon interference by generating heralded single photons and interfering them in a fiber tritter. Using multiple degrees of freedom—temporal delays and polarization—we isolate three-photon interference from two-photon interference. Our experiment disproves the view that pairwise two-photon distinguishability uniquely determines the degree of nonclassical many-particle interference.

Figure 1

Interference of photons in balanced beam splitters and tritters. (a),(b) The output statistics of two photons interfering in a beam splitter can be described via the pairwise distinguishability of the photons. (c) In the case of a tritter, the output statistics depend on an additional phase $\phi$. (d) This triad phase $\phi$ is defined by the arguments of the pairwise complex scalar products. (e) $\phi$ only occurs in the interference of more than two photons.

Figure 2

Scheme of the experimental setup. The relative temporal delays of the three photons are adjusted using delay stages. We use sets of quarter-wave plates (QWPs) and half-wave plates (HWPs) to prepare the polarization state of each photon and to compensate for polarization rotations in the fibers. The outputs of the fiber tritter are monitored using multiplexed commercial avalanche photodiodes.

Figure 3

Experimental heralded three-photon coincidences at the output of a fiber tritter for two values of the triad phase $\phi$. (a),(b) We choose two polarization configurations so that $\phi =0$ (a) and $\phi =\pi$ (b), see Eqs. (7) and (8). (c),(d) We measure heralded threefold coincidences ($\propto P_{111}$) between the different output ports of the tritter while varying the temporal delays of the photons. As shown pictorially beneath the plots, we start in a configuration where the photons are completely distinguishable in time; two of the photons are then scanned symmetrically across the third photon ($t_1=t_2-\tau /2$, $t_3=t_2+\tau /2$). The gray boxes show the region of temporal overlap of the photons. The nonmonotonic behavior in (c) arises because $\phi =0$ causes the three-photon interference term in Eq. (4) to have a contribution of opposite sign to those of the two-photon terms described by $r_{ij}^2$. In (d) $\phi =\pi$ and so the contribution is of the same sign, resulting in monotonic behavior of the statistic. The dashed gray curve shows the theoretical prediction and the black curve is calculated using a model which includes experimental imperfections (see main text for details). The absolute number of counts per point were between 200 and 350 (250 and 450) for (a) [(b)]. Error bars are calculated from repeated measurements.

Figure 4

Isolating two-photon from three-photon interference. (a) We vary the triad phase by rotating the polarization of photon $|{\varphi }_1^{\prime \prime }⟩$ as given in Eq. (9), leaving the polarization states of the two other photons fixed. To keep the moduli $r_{12}$ and $r_{31}$ constant, we adapt the temporal overlaps of the photons by tuning $|t_1⟩$. (b) The three-photon signal $P_{111}$ varies with the triad phase (absolute number of counts per data point is between 330 and 515). The plotted curve is a theory curve calculated based on our model of the experiment. (c) We plot a subset of two-photon distinguishability terms to demonstrate that these are kept constant.