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arXiv:2011.08777 · Nature Photonics

Direct observation of the particle exchange phase of photons

Konrad Tschernig, Chris Müller, Malte Smoor, Tim Kroh, Janik Wolters, Oliver J. Benson, Kurt Busch, Armando Perez-Leija

Published Jun 3, 2021• 15 citations
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Abstract

Abstract Quantum theory stipulates that if two particles are identical in all physical aspects, the allowed states of the system are either symmetric or antisymmetric with respect to permutations of the particle labels. Experimentally, the symmetry of the states can be inferred indirectly from the fact that neglecting the correct exchange symmetry in the theoretical analysis leads to dramatic discrepancies with the observations. The only way to directly unveil the symmetry of the states for, say, two identical particles is through the interference of the two-particle state and the physically permuted one, and measuring the phase associated with the permutation process, the so-called particle exchange phase. Following this idea, we have measured the exchange phase of indistinguishable photons, providing direct evidence of the bosonic character of photons.

DOI
10.1038/s41566-021-00818-7

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Abstract Quantum theory stipulates that if two particles are identical in all physical aspects, the allowed states of the system are either symmetric or antisymmetric with respect to permutations of the particle labels. Experimentally, the symmetry of the states can be inferred indirectly from the fact that neglecting the correct exchange symmetry in the theoretical analysis leads to dramatic discrepancies with the observations. The only way to directly unveil the symmetry of the states for, say, two identical particles is through the interference of the two-particle state and the physically permuted one, and measuring the phase associated with the permutation process, the so-called particle exchange phase. Following this idea, we have measured the exchange phase of indistinguishable photons, providing direct evidence of the bosonic character of photons.
The symmetrization postulate in quantum mechanics is formally reflected in the appearance of an exchange phase governing the symmetry of identical-particle global states under particle swapping. Many indirect measurements of this fundamental phase have been reported thus far, but a direct observation has been achieved only recently for photons. Here, we propose a general scheme capable of directly measuring the exchange phase of any type of particle (bosons, fermions, or anyons), exploiting the operational framework of spatially localized operations and classical communication. We experimentally implement it on an all-optical platform, providing a proof of principle for different simulated exchange phases. As a by-product, we supply a direct measurement of the real bosonic exchange phase of photons. Additionally, 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. Finally, we show that the proposed setup is suited to generating indistinguishability-driven NOON\mathrm{NOON}NOON states useful for quantum-enhanced phase estimation.
The wavefunction of two identical bosons remains invariant under particle exchange - a fundamental quantum symmetry that underlies Bose-Einstein statistics. We report the direct experimental observation of bosonic exchange symmetry in the temporal wavefunction of photon pairs generated via spontaneous four-wave mixing in a three-level cold atomic ensemble. The measured two-photon temporal correlations show excellent agreement with theoretical predictions based on symmetrized bosonic wavefunctions. In addition, we perform time-resolved two-photon interference to reconstruct the complex temporal wavefunction. Both the amplitude and phase profiles exhibit clear symmetry under photon exchange, providing a direct confirmation of bosonic exchange symmetry in the time domain.
If a system in atomic physics contains a number of particles of the same kind, e.g. a number of electrons, the particles are absolutely indistinguishable one from another. No observable change is made when two of them are interchanged … A satisfactory theory ought, of course, to count any two observationally indistinguishable states as the same state and to deny that any transition does occur when two similar particles exchange places. ( Dirac, 1958, p. 207 ) Introduction In our contribution to this volume we deal with discrete symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In physics we find that discrete symmetries frequently arise as ‘internal’, non-spacetime symmetries. Permutation symmetry is such a discrete symmetry, arising as the mathematical basis underlying the statistical behaviour of ensembles of certain types of indistinguishable quantum particle (e.g. fermions and bosons). Roughly speaking, if such an ensemble is invariant under a permutation of its constituent particles (i.e. permutation symmetric) then one doesn't ‘count’ those permutations which merely ‘exchange’ indistinguishable particles; rather, the exchanged state is identified with the original state. This principle of invariance is generally called the ‘indistinguishability postulate’ (IP), but we prefer to use the term ‘permutation invariance’ (PI).
According to the identity principle in quantum theory, states of a system consisted of identical particles should maintain unchanged under interchanging between two of the particles. The whole wavefunction should be symmetrized or antisymmetrized. This leads to statistical correlations between particles, which exhibit observable effects. We design an experiment to directly observe such effects for bosons. The experiment is performed with two photons. The effect of statistical correlations is clearly observed when the wavepackets of two photons are completely overlapped, and this effect varies with the degree of overlapping. The results of our experiment substantiate the statistical correlation in a simple way. Experiment reported here can also be regarded as another kind of two-photon Hong-Ou-Mandel interference, occurs in the polarization degree of freedom of photon.
Identical particles have dynamically invariant labels in all setups where their degrees of freedom can be partitioned into two parts, which we call internal and visible, with the visible part subject to unitary evolution, while the dynamically invariant internal part accounts for the partial distinguishability of particles. We give the explicit form of the visible state in terms of the indistinguishability function, generalizing the standard symmetrization/antisymmetrization postulate for bosons/fermions with internal modes. The indistinguishability function on the symmetric group accounts for the permutation symmetry of the dynamically invariant label state and gives a self-contained description of the symmetry spectrum of the visible state. It is shown that the indistinguishability of single particles emitted independently from a stable source is completely characterized by the projective measures on the symmetry spectrum. We reveal a hierarchy of successive upper bounds on the projective measure of indistinguishability in terms of marginal projective measures. We point out an experimentally feasible way to get the upper bounds on the indistinguishability of an arbitrarily large number of single photons by direct experimental readout of their marginal projective measures. We also analyze the indistinguishability of coherent superpositions and convex mixtures and resolve a recently posed problem.