General Rules for Bosonic Bunching in Multimode Interferometers

We perform a comprehensive set of experiments that characterize bosonic bunching of up to three photons in interferometers of up to 16 modes. Our experiments verify two rules that govern bosonic bunching. The first rule, obtained recently, predicts the average behavior of the bunching probability and is known as the bosonic birthday paradox. The second rule is new and establishes a $n!$-factor quantum enhancement for the probability that all $n$ bosons bunch in a single output mode, with respect to the case of distinguishable bosons. In addition to its fundamental importance in phenomena such as Bose-Einstein condensation, bosonic bunching can be exploited in applications such as linear optical quantum computing and quantum-enhanced metrology.

Figure 1

Experimental platform for photonic bunching experiments. (a) Input bosons evolve via a linear interferometer described by an $m\times m$ unitary $U$. A bunching event results when two (or more) bosons emerge from the same output port. (b) Architectures of the integrated linear optical interferometers exploited in the experimental verification. Red spots represent phase shifters, while the directional couplers perform the beam splitter transformation, whose reflectivity can be varied by modulating the coupling coefficient in the interaction region. (c) Experimental layout evidencing the generation and detection of single photons. (Legend—SHG, second harmonic generation; PDC, parametric down-conversion; C, walk-off compensation; IF, interference filter; PBS, polarizing beam splitter; FPC, fiber polarization compensator; DL, delay line; SMF, single-mode fiber; MMF, multimode fiber; MMFBS, multimode fiber beam splitter; APD, avalanche photodiode; and black box, electronic coincidence apparatus.) See Ref. 27 for more experimental details.

Figure 2

Two-photon photonic bunching data. (a) Bunching probability $p_b$ as a function of the number of modes $m$ for two indistinguishable photons ($p_b^{(q)}$, red points) and two distinguishable photons ($p_b^{(c)}$, green points). We performed experiments with different unitaries ($m=3$, $m=8$, $m=12$) or different input states ($m=3$, $m=5$). Shaded areas correspond to the interval $[\overline{p_b}-1.5\sigma ;\overline{p_b}+1.5\sigma ]$ obtained with a numerical sampling over 10 000 uniformly random unitaries, $\overline{p_b}$ being the average bunching probability and $\sigma$ its standard deviation. Red area: indistinguishable photons. Green area: distinguishable photons. (b)–(g) Experimental results (points) together with histograms showing the distribution of the bunching probabilities obtained with the numerical simulation. (h) Results for the bunching probability $p_f$ as a function of the number of modes $m$ for two indistinguishable fermions ($p_f^{(q)}$, black points) and two distinguishable particles ($p_f^{(c)}$, green points). Nonzero bunching probabilities have to be attributed to imperfections in the state preparation. Error bars in the experimental data are due to the Poissonian statistics of the measured events and where not visible are smaller than the symbol.

Figure 3

Three-photon photonic bunching data. Experimental results for the three-photon photonic bunching experiments as a function of the number of modes $m$ for three indistinguishable photons ($p_b^{(q)}$, red points) and three distinguishable photons ($p_b^{(c)}$, green points). Shaded areas correspond to the interval $[\overline{p_b}-1.5\sigma ;\overline{p_b}+1.5\sigma ]$ with a numerical sampling over 10 000 uniformly random unitaries, $\overline{p_b}$ being the average bunching probability and $\sigma$ its standard deviation. Red area: simulation taking into account the partial indistinguishability parameter $\alpha$ of our source. Green area: three distinguishable photons. Error bars in the experimental data are due to the Poissonian statistics of the measured events and where not visible are smaller than the symbol. Inset: numerical simulation of the effect of photon distinguishability on the bunching probability $p_b$. Grey area: perfectly indistinguishable photons ($\alpha =1$).

Figure 4

Ratio $r_{fb}$ between quantum and classical full-bunching probabilities. Here, we report $r_{fb}$ for two- and three-photon experiments on a number of photonic chips. Blue data: $r_{fb}$ for two-photon experiments in chips with different numbers of modes $m$. Blue area: expected value for two-photon full-bunching ratio. Two-photon measurements have been carried out in two different regimes, where the measured photon indistinguishability was $\beta \simeq 0.95$ (for the interferometers with $m=3$, 5, 7) and $\beta \simeq 0.99$ (for the interferometers with $m=2$, 8, 12, 16). The corresponding values of the expected two-photon bunching ratio are given by $r_{fb}=\beta 2!+(1-\beta )(2-1)!$ and range between $r_{fb}\simeq 1.95$ and $r_{fb}\simeq 1.99$. Red data: $r_{fb}$ for three-photon experiments in various random chips. Red area: expected value for three-photon full-bunching ratio taking into account the partial indistinguishability parameter $\alpha$. Error bars in the experimental data are due to the Poissonian statistics of the measured events and where not visible are smaller than the symbol.