[Paper Review] Direct measurement of particle statistical phase

Yan Wang, Matteo Piccolini+9 more|arXiv (Cornell University)|Feb 1, 2022

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper proposes a general, all-optical scheme for the direct measurement of the particle statistical phase (exchange phase) in identical particles—bosons, fermions, and anyons—using spatially localized operations and classical communication. The method experimentally confirms the symmetrization postulate by directly measuring the phase shift upon particle exchange, achieving high-precision detection of bosonic (ϕ=0) and anyonic (ϕ=π/2) exchange phases, and demonstrating robustness under particle mixture conditions.

ABSTRACT

The symmetrization postulate in quantum mechanics is formally reflected in the appearance of an exchange phase ruling the symmetry of identical particle global states under particle swapping. Many indirect measurements of this fundamental phase have been reported so far, while a direct observation has been only recently achieved for photons. Here we propose a general scheme capable to directly measure the exchange phase of any type of particles (bosons, fermions, anyons), exploiting the operational framework of spatially localized operations and classical communication. We experimentally implement it in an all-optical platform providing proof-of-principle for different simulated exchange phases. As a byproduct, we supply a direct measurement of the real bosonic exchange phase of photons. Also, we analyze the performance of the proposed scheme when mixtures of particles of different nature are injected. Our results confirm the symmetrization tenet and provide a tool to explore it in various scenarios.

Motivation & Objective

• To develop a general, direct method for measuring the exchange phase of identical particles, including bosons, fermions, and anyons, overcoming limitations of indirect measurements.
• To experimentally validate the symmetrization postulate by directly observing the exchange phase in photons and simulated anyonic statistics.
• To investigate the role of spatial indistinguishability in the measurement of exchange phases using a no-label formalism.
• To demonstrate the feasibility of measuring exchange phases in mixed ensembles of particles with different statistical natures.
• To provide a robust, scalable framework for future exploration of quantum statistics in diverse quantum systems.

Proposed method

• The scheme employs spatially localized operations and classical communication (sLOCC) to perform a direct measurement of the exchange phase via interference of two identical particles distributed over two spatial regions (L and R).
• Particles are prepared in superposition states with tunable spatial overlap via a half-wave plate (HWP) setting β, controlling the degree of spatial indistinguishability 𝒫.
• A coincidence measurement is performed at the output ports to measure the observable ⟨O⟩, which directly encodes the exchange phase ϕ through a sinusoidal dependence on the relative phase introduced by a movable plate.
• Theoretical predictions for ⟨O⟩ are derived using the no-label formalism, where the particle statistics manifest in the probability amplitudes rather than state symmetrization.
• Experimental noise is modeled by a fidelity factor F ≈ 1 − 10⁻³, which is used to correct theoretical predictions and improve measurement accuracy.
• The method is extended to mixed states by simulating classical mixtures of particles with different exchange phases (e.g., bosons and anyons with ϕ=π/2, or bosons and fermions with ϕ=π), using variable input HWP angles to tune the mixture probability p.

Experimental results

Research questions

• RQ1 Can the exchange phase of identical particles be directly measured without relying on indirect inference or state tomography?
• RQ2 How does spatial indistinguishability affect the sensitivity and accuracy of exchange phase measurements?
• RQ3 Can the proposed scheme be generalized to measure anyonic exchange phases, including fractional phases?
• RQ4 How does the presence of mixed particle statistics (e.g., bosons and fermions) affect the measurement of the exchange phase?
• RQ5 To what extent does experimental noise impact the fidelity and reliability of the direct measurement protocol?

Key findings

• The experiment successfully demonstrated a direct measurement of the bosonic exchange phase (ϕ=0) in photons with high precision, confirming the symmetrization postulate.
• The scheme achieved direct observation of a simulated anyonic exchange phase of ϕ=π/2, marking the first direct measurement of a non-bosonic, non-fermionic exchange phase.
• The measured values of ⟨O⟩ showed excellent agreement with theoretical predictions, both in ideal and noisy scenarios, with experimental error fluctuations on the order of 10⁻³.
• As spatial indistinguishability decreased (from β=45° to β=10°), the dynamic range of ⟨O⟩ narrowed, confirming reduced measurement sensitivity at lower overlap.
• For classical mixtures of particles, the method accurately measured the expectation value ⟨O⟩ as a function of the mixture probability p, with results matching theoretical predictions within experimental error.
• The fidelity factor F was estimated to be approximately 1 − 10⁻³, indicating high robustness against experimental noise and validating the correction model for enhanced accuracy.