Symmetry Allows for Distinguishability in Totally Destructive Many-Particle Interference

We investigate, in a four-photon interference experiment in a laser-written waveguide structure, how symmetries control the suppression of many-body output events of a $J_x$ unitary. We show that totally destructive interference does not require mutual indistinguishability between all but only between symmetrically paired particles, in agreement with recent theoretical predictions. The outcome of the experiment is well described by a quantitative simulation that accounts for higher-order emission of the photon source, imbalances in the scattering network, partial distinguishability, and photon loss.

Popular Summary

Symmetries are key to our understanding of nature. They allow us to categorize all particles in the universe into bosons and fermions, whose quantum-mechanical wave function is symmetric or antisymmetric under the exchange of any two indistinguishable particles, respectively. As a result, there can be interference between indistinguishable many-particle paths, leading to intricate counting statistics. However, partially distinguishable particles only hold reduced symmetry properties, such that interference diminishes and the counting statistics lose their quantum-mechanical character. While totally destructive many-particle interference (aka suppression) has commonly been believed to require perfectly indistinguishable particles, here we show that only particular symmetry properties of the many-particle state matter. Hence, suppression can occur even for partially distinguishable particles, as long as the required symmetry is met. This has profound consequences for photonic quantum information processing; for example, in the validation of boson sampling.

We experimentally demonstrate the symmetry dependence of the many-body suppression effect via the interference of four photons. The photons are generated from a spontaneous parametric down-conversion source in a multimode interferometer, which is implemented with a femtosecond-laser-written waveguide structure in glass. Our results are in agreement with recent theoretical predictions and follow a quantitative simulation of the experiment, taking into account typical systematic errors.

Our findings have immediate consequences for established methods on the validation of photonic quantum protocols, such as boson sampling. The results are key for the characterization of many-body indistinguishability and, thus, of relevance for future quantum information processing with identical particles, even beyond quantum photonics.

Figure 1

The experimentally investigated input states of the $J_x$ unitary. (a) Photons are input to the odd modes of the unitary. Four different symmetry scenarios with respect to the mirror axis (central mode) are illustrated. Suppression is expected only for the upper two scenarios. The internal states of the particles are illustrated by their coloring. (b) The occupation of the permutation cycles $c_j$. (c) The photonic waveguide structure implementing the $J_x$ unitary.

Figure 2

A schematic illustration of the four-photon SPDC source, of the multiphoton interference, and of the detection setup. A pulsed $\mathrm{Ti}$:$\mathrm{Sa}$ laser is frequency doubled by second-harmonic generation in a BiBO crystal in collinear configuration. The residual pump light is filtered by dichroic mirrors, while the up-converted light is focused into a BBO crystal, where pairs of photons are generated in a type-I SPDC process. The emitted and spectrally filtered photon pairs are collected via four polarization-maintaining single-mode fibers, labeled as channels (Ch.) A–D. Three free-space optical delay stages are used to adjust the temporal delay between photons from different channels. The light is coupled to the waveguide chip via a fiber array. The output of the waveguide structure is collected with multimode fibers and measured using APDs and a coincidence counter (Coin. Counter).

Figure 3

The experimental and simulated output statistics of all fourfold coincidences between distinct output modes. (a),(b) The output statistics obtained for a mutually indistinguishable and for an intercycle distinguishable input state, respectively, for which all output events in the gray shaded area are ideally suppressed. (c),(d) The output statistics obtained for an intracycle distinguishable and for a mutually distinguishable input state, respectively. Here, no suppression effect is expected, even in the ideal case. The green bars correspond to experimental (Exp) data and the error bars indicate one standard deviation of the Poissonian counting statistics. The bars to the right of the experimental data correspond to a simulation of the experiment, with the dark blue, blue, light blue, and gray parts of the bars indicating the contribution stemming from the input-mode occupation ${\overrightarrow{R}}_1$, ${\overrightarrow{R}}_2$, ${\overrightarrow{R}}_3$ [see Eq. (5a)] and from the six-photon background of the photon source, respectively.

Figure 4

The reconstructed magnitudes of the $J_x$ unitary. (a)–(g) We launch single photons into each input mode of the waveguide structure and measure the relative output intensity in each output mode. From this, we reconstruct $|[U_{\mathrm{recon}}{]}_{k,j}{|}^2$, following the algorithm from Ref. [61]. (h) The fidelities $F_j={\left(\sum _{k=1}^7\sqrt{|[U_{\mathrm{recon}}{]}_{k,j}{|}^2\times |[U^J{]}_{k,j}{|}^2}\right)}^2$ of the reconstructed magnitudes, which benchmark how well the physical unitary agrees with the ideal $J_x$ unitary.

Figure 5

Selected twofold coincidence counts and single-photon count rates during the measurement of the mutually indistinguishable particle scenario. Counts are averaged over an interval of 60 s for a total measurement time of 88 h. (a) The time traces of two output states that have approximately equal contributions from input state $\overrightarrow{R_4}=(0,0,1,0,1,0,0)$ and $\overrightarrow{R_5}=(1,0,0,0,0,0,1)$ according to the theoretical predictions. The observed coincidence rate fluctuates heavily, suggesting that the relative phase of these two contributions fluctuates. (b) The time traces of the single-photon count rates of all seven output channels. (c) Selected time traces of output states that are dominated by contributions of input state $\overrightarrow{R_4}$. These output states are ideally suppressed according to the suppression law. We attribute the slow drift in the observed coincidence rates to a change in the indistinguishability of the photons caused by changes of the optical path length. Hardly any fast fluctuations in the coincidence rate are observed, since mostly one input state contributes. (d) Selected time traces of the output states, which are dominated by contributions of input state $\overrightarrow{R_5}$. Neither strong fast fluctuations nor significant slow drifts of the coincidence rates are observed.

Figure 6

The calculated fidelity for an output probability distribution of all twofold nonbunching coincidence events as a function of the indistinguishabilities $V_{AB}$ and $V_{CD}$.

Figure 7

The output probability distribution of all twofold nonbunching coincidence events. For the simulated result, we use optimized time-averaged indistinguishabilities of $V_{AB}=0.94$ and $V_{CD}=0.90$. The fidelity is $F=0.991$. The experimental and simulated degree of suppression violation are $D_{\exp }=0.198$ and $D_{\mathrm{sim}}=0.202$, respectively. The experimental error bars from the counting statistics are too small to be visible. The inset shows the discretized JSA of channels A and B (the JSA of channels C and D is similar).