Exchange Symmetry in Multiphoton Quantum Interference

TL;DR

This work investigates how exchange symmetry can be distributed across internal and external degrees of freedom to realize mixed-symmetry configurations among three photons. The authors develop a theoretical framework using a phase parameter $\phi$ to interpolate between symmetric and mixed-symmetry components in $\ket{\Psi^{\phi}}_{a,a,b}$, and implement an experiment with a generalized Bell state and a heralded photon to observe interference via a variable beam splitter. They report that three-photon interference is strong when the symmetric component dominates ($\phi \approx 0$) and vanishes for the pure mixed-symmetry case ($\phi = \pi$), with intermediate behavior at $\phi = \pi/2$. The work demonstrates a scalable photonic platform for emulating generalized quantum statistics beyond bosons, including immanonic regimes, and suggests extensions to larger multiphoton states or qutrit encodings to explore richer exchange-statistics scenarios.

Abstract

Photons are bosons, and yet, when prepared in specific entangled states, they can exhibit non-bosonic behaviour. While this phenomenon has so far been studied in two-photon systems, exchange symmetries and interference effects in multi-photon scenarios remain largely unexplored. In this work, we show that multi-photon states uncover a rich landscape of exchange symmetries. With three photons already, multiple pairwise combinations are possible, where each pair of photons can exhibit either bosonic, fermionic, or anyonic exchange symmetry. This gives rise to mixed symmetry systems that are not possible to achieve with two photon alone. We experimentally investigate how these symmetry configurations manifest themselves in the observed interference of three photons. We show that multi-photon interference can be effectively turned on and off by tuning the symmetry of the constituent pairs. The possibility of accessing and tuning new quantum statistics in a scalable photonic platform not only deepens our understanding of quantum systems, but is also highly relevant for quantum technologies that rely on quantum interference.

Figures (5)

• Figure 1: Probing exchange symmetry in photon interference: (a) The bosonic symmetry of the wavefunction, $\ket*{\Psi^{(n)}(\vec{\phi}}$, for $N$ photons can be distributed over different degrees of freedom using a set of parameters $\vec{\phi}$. This allows access to non-bosonic pairwise exchange statistics when the state is evolved under some unitary $U$ and then measured. (b) A state of three photons, $\ket*{\Psi^{(3)}(\vec{\phi}}$, with a relative phase, $\phi$, to directly adjust the exchange symmetry and probe different particle statistics using a variable beam splitter described by the unitary $U_{VBS}$ (see Eq.\ref{['eq:VBS']}).
• Figure 2: Overview of experimental setup: A pico-second pulsed laser at $\lambda_{p}=775~\textrm{nm}$ is used to pump two spontaneous parametric down-conversion sources consisting of periodically poled potassium titanyl phosphate (ppKTP 1, ppKTP 2) crystals. The resulting down-converted signal and idler photons are at $\lambda_{s/i}=1550~\textrm{nm}$. The first source (ppKTP 1) is a linear source, gives us the heralded single photon, while the other source (ppKTP 2) is a Sagnac-type source, which results in the generation of a Bell state. The heralded single photon and one half of the Bell state are sent to the two input ports of a fibre-based 50:50 beam splitter. One of the outputs of the 50:50 beam splitter is directed to one input to the VBS. Only the cases where both photons are directed to the VBS and no photons are detected at the other output are considered to be successful measurement runs. The other half of the Bell state is directed to the other input of the VBS. Each output of the VBS is demultiplexed to four superconducting nano-wire single photon detector (SNSPD) channels to have pseudo photon number resolution.
• Figure 3: Scattering probabilities at the variable beam splitter (VBS):$P(x,y)$ denotes the probability of detecting $x$ photons in the first output mode and $y$ photons in the second. The VBS can be tuned from full transmission ($\theta=0 \text{ and } \theta=\pi$) over a balanced beam splitter ($\theta=\pi/4$ and $\theta=3\pi/4$) to full reflection ($\theta=\pi/2$). (a) Scattering statistics for a heralded single photon. (b) Scattering statistics for a generalized $\ket{\Psi}$-type Bell state, $\ket{\Psi}_{a,b}^{\phi} = \frac{1}{\sqrt{2}}(a^{\dagger}_{0}b^{\dagger}_{1} + e^{i\phi}a^{\dagger}_{1}b^{\dagger}_{0})\ket{\text{vac}}$, for phases $\phi = \pi$, $\pi/2$, and $0$. (c) The product of the independent scattering statistics from panels (a) and (b), denoted $\xi(x,y)$, shown for each value of $\phi$. (d) Joint scattering statistics for the three-photon entangled state $\ket{\Psi^{\phi}}_{a,a,b} = \frac{1}{\sqrt{3}}(a^{\dagger}_{0}a^{\dagger}_{0}b^{\dagger}_{1} + e^{i\phi}a^{\dagger}_{0}a^{\dagger}_{1}b^{\dagger}_{0})\ket{\text{vac}}$, also for $\phi = \pi$, $\pi/2$, and $0$. (e) The difference $\Delta P = \xi(x,y) - P(x,y)$, highlighting minimal deviation for $\phi = \pi$ and maximal deviation for $\phi = 0$. Error bars indicate Poissonian uncertainties arising from photon-counting statistics.
• Figure 4: Schematic of the experimental setup: A single photon is injected into the first mode of an entangled state via a 50:50 beam splitter. The desired three-photon state is post-selected by selecting events in which the injected photon and the first photon of the entangled pair occupy the same spatial mode, forming the variable beam splitter (VBS). The splitting ratio is controlled at the VBS, and the resulting scattering statistics are measured using pseudo–photon-number resolution, achieved by demultiplexing each VBS output into four detector channels.
• Figure 5: Hong–Ou–Mandel (HOM) interference dips measured for photons from a single source (here, the Sagnac source) and from two independent sources.