Researchers at Université Paris Cité and CNRS have revealed a connection between spatial symmetry and the performance of quantum devices using few photons. The work reinterprets a two-photon Mach-Zehnder interferometer as a generalized Hong-Ou-Mandel experiment, demonstrating that its efficiency stems from the spatial symmetry of an “effective probe state.” The team further establishes a direct link between spatial symmetry and source indistinguishability when analyzing the output of two coherently pumped parametric photon-pair sources, extending the symmetry principles to describe three-photon interference in tritter configurations.

Symmetry-Based Framework for Generalized Hong-Ou-Mandel Interference

The team’s framework demonstrates that the photon-counting statistics and metrological performance of this common interferometer are a result of underlying spatial symmetries. The research extends beyond simply reinterpreting existing setups. By deriving the complete output distribution of two coherently pumped sources, the scientists show that the degree to which photons from these sources can be distinguished directly correlates with the spatial symmetry present in the system.

This finding offers a new way to assess and optimize the performance of these sources, which are fundamental building blocks for many quantum applications. The team’s analysis bypasses complex propagation calculations, instead relying on a symmetry-based derivation of the spectral response and Fisher information associated with photon counting.

The applicability of this symmetry-based framework extends beyond two-photon interference, encompassing three-photon interference in tritter configurations, three-mode interferometers. This demonstrates the broad potential of the approach to provide a unified description of quantum interference at different scales.

The team’s work shows that the mapping of input spatial symmetry onto constraints on the output photon-number distribution by discrete Fourier-transform interferometers provides a common interpretation of seemingly disparate interference phenomena. In their publication in Physical Review Letters, the researchers state that they have “applied the spatial-symmetry framework developed in our previous studies to several few-photon interferometric configurations.” This framework, they claim, offers a powerful tool for understanding and optimizing quantum interference experiments.

Two-Photon Mach-Zehnder Interferometer as Generalized HOM Experiment

A balanced Mach-Zehnder interferometer, commonly used in optical metrology, can be understood as a generalized Hong-Ou-Mandel (HOM) interferometer, according to analysis of its underlying symmetries. This reinterpretation reveals that the interferometer’s performance, specifically its photon-counting statistics and ability to precisely measure physical quantities, is directly determined by the exchange symmetry of what researchers term an “effective probe state”. The analysis bypasses complex calculations of photon propagation within the interferometer by focusing instead on these fundamental symmetry properties.

Researchers found that by absorbing the initial beam splitter into the initial state preparation, the remaining interferometer functions as a generalized HOM setup, probing an “effective two-photon input state”. This simplification allows for a more intuitive understanding of the interference phenomena at play and offers a pathway to predict and control the interferometer’s behavior.

The team’s framework demonstrates that the spectral response of this two-photon configuration is determined by the exchange symmetry of the joint spectral amplitude. Beyond simply reclassifying the setup, the symmetry framework yields insights into metrological applications. The work identifies fixed-total-frequency states as optimal probes for estimating local time delays, a critical capability in precision measurement.

This connection between symmetry and metrology suggests that careful control of the input state’s symmetry can enhance the sensitivity of the interferometer. The researchers also extended their analysis to two coherently pumped parametric photon-pair sources, deriving a complete description of their output distribution and linking the sources’ indistinguishability to spatial symmetry.

Spatial Symmetry and Photon-Counting Statistics in Two-Photon Sources

The analysis bypasses complex propagation calculations, instead focusing on the inherent symmetries within the system to determine outcomes. Within this approach, the beam splitter maps exchange symmetry onto photon-number parity, so that a simple photon-counting measurement directly probes a symmetry property of the incoming light. This approach provided a natural way of assessing various interferometric situations and metrology protocols without first expanding every output amplitude.

The coincidence probability, a key metric in these experiments, is controlled by the spatial symmetry of the input state, as expressed in the equation $P_{c}$ =. This means that the likelihood of detecting coincident photons, those arriving simultaneously at the detectors, is directly determined by how symmetrical the initial state of the photons is. The Fisher information, a measure of the precision of a measurement, is sensitive only to aspects of the evolution that break the cyclic symmetry between the three spatial modes.

Specifically, if the evolution preserves cyclic permutations, meaning it affects all three modes identically, the Fisher information vanishes, indicating an undetectable change in the output photon-number distribution. “If H is invariant under cyclic permutations, such that $P H P^{- 1}$ = H, the Fisher information vanishes,” the paper states, highlighting the importance of symmetry breaking for detection.

Recovering Extended Hong-Ou-Mandel Interference via Symmetry-Parity

This reframing, detailed in work published in Physical Review Letters, demonstrates that the nodal line characteristic of extended Hong-Ou-Mandel interference can be recovered without complex calculations of the full photon-number distribution. Researchers found the symmetry-parity correspondence provides a physically transparent explanation for results previously obtained through more involved methods, simplifying the analysis of interference patterns. The team showed that the coincidence probability, a key metric for these sources, directly measures the overlap between the complete two-photon states they generate.

This applies to both spontaneous parametric down-conversion and spontaneous four-wave mixing, indicating a broad applicability of the symmetry-based framework. The framework also explains the appearance of a nodal line in generalized Hong-Ou-Mandel interference when using inputs with an odd number of photons, offering a deeper understanding of the underlying physics.

Three-Photon Interference Analysis in Balanced Tritter Configurations

Extending established principles of quantum interference, researchers have demonstrated a symmetry-based description of three-photon interactions within balanced tritters, a type of three-mode interferometer. This work builds upon a previously developed framework connecting spatial symmetry to generalized Hong-Ou-Mandel interference, revealing how the same principles govern interference with increasing numbers of photons. The analysis focuses on how a modular quantity, a measurement observable, resolves the eigenspaces of the cyclic permutation operator governing the tritter’s behavior.

The tritter, composed of beam splitters and phase shifters or a three-dimensional waveguide coupling, allows for the study of multiparticle interference and the permutation structure of interfering paths. Researchers show that the average cyclic symmetry of the input photons is directly measurable through this modular quantity, with different phase quadratures selected by the tritter’s photon-number sectors. This means the measurement isn’t simply detecting photons, but also revealing information about the underlying symmetry of the quantum state.

The team’s approach doesn’t rederive existing tritter interference theory, but instead provides a new perspective rooted in spatial symmetry, explaining previously established results like the suppression of specific output photon-number configurations. The framework accounts for both independent photons and correlated states, connecting source indistinguishability with spatial symmetry. For independent photons, the symmetry expectation value is the cyclic product of their pairwise overlaps, revealing a specific “triad phase”.

When dealing with correlated states, the same measurement directly probes the cyclic symmetry of the complete three-photon spectrum. This connection is significant because it suggests a unified way to understand interference phenomena across different quantum states and complexities. The product formulas used in the analysis can accommodate scenarios where the internal states of the three photons are not independent, allowing for a more general description of three-photon interference.

Triad Phase and Cyclic Symmetry of Three-Photon Spectra

A balanced tritter, a three-mode interferometer, reveals the connection between cyclic symmetry and the measurement of three-photon interference, according to recent analysis. Researchers recovered this known three-photon invariant within their framework, identifying it with the expectation value of the cyclic spatial permutation measured by a Fourier interferometer. The analysis extends beyond independent photons to encompass correlated states, revealing how the modular output probabilities relate to the cyclic symmetry properties of the three-photon spectral amplitude.

If the spectral amplitude belongs to a specific symmetry sector, the corresponding modular outcome occurs with unit probability, effectively transforming the tritter and photon counting setup into a measurement of the complete three-photon spectrum’s cyclic symmetry. Optimal probe states, those with a perfectly defined total frequency, satisfy a specific condition, equality between two quantities and possess a spectral amplitude proportional to a function with required cyclic symmetry.

These states represent a three-photon generalization of perfectly frequency anticorrelated states used in optimal Hong-Ou-Mandel time-delay measurements. For local time-delay estimation, the tritter measurement achieves optimality when the total frequency of the three photons is perfectly defined, further solidifying the framework’s predictive power and experimental relevance.

👉 More information
🗞 Spatial symmetry in few-photon interference and quantum metrology
✍️ Éloi Descamps and Pérola Milman
🧠 DOI: http://link.aps.org/doi/10.1103/nrf9-rx7w

More like this