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arXiv:2202.00575 · Physical Review Applied

Proof-of-Principle Direct Measurement of Particle Statistical Phase

Yan Wang, Matteo Piccolini, Ze-Yan Hao, Zhenghao Liu, Kai Sun, Jin‐Shi Xu, Chuan‐Feng Li, Guang‐Can Guo, Roberto Morandotti, Giuseppe Compagno, Rosario Lo Franco

Published Dec 8, 2022• 11 citations
arXivHTMLPDF

Abstract

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 $\mathrm{NOON}$ states useful for quantum-enhanced phase estimation.

DOI
10.1103/physrevapplied.18.064024

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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.
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.
Quantum coherence, an essential feature of quantum mechanics allowing quantum superposition of states, is a resource for quantum information processing. Coherence emerges in a fundamentally different way for nonidentical and identical particles. For the latter, a unique contribution exists linked to indistinguishability that cannot occur for nonidentical particles. Here, we experimentally demonstrate this additional contribution to quantum coherence with an optical setup, showing that its amount directly depends on the degree of indistinguishability, and exploiting it in a quantum phase discrimination protocol. Furthermore, the designed setup allows for simulating fermionic particles with photons, thus assessing the role of exchange statistics in coherence generation and utilization. Our experiment proves that independent indistinguishable particles can offer a controllable resource of coherence and entanglement for quantum-enhanced metrology.
In quantum theory, particles in three spatial dimensions come in two different types: bosons or fermions, which exhibit sharply contrasting behaviours due to their different exchange statistics. Could more general forms of probabilistic theories admit more exotic types of particles? Here, we propose a thought experiment to identify more exotic particles in general post-quantum theories. We consider how in quantum theory the phase introduced by swapping indistinguishable particles can be measured. We generalise this to post-quantum scenarios whilst imposing indistinguishability and locality principles. We show that our ability to witness exotic particle exchange statistics depends on which symmetries are admitted within a theory. These exotic particles can manifest unusual behaviour, such as non-abelianicity even in topologically simple three-dimensional space.
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.
The symmetrization postulate (SP) that states of more than one identical particle are either symmetric or antisymmetric under permulations is studied from the theoretical and experimental points of view. The theoretical analysis is carried out within the framework of particle quantum mechanics; the field-theory approach to identical particles using Bose, Fermi, para-Bose and para-Fermi quantization is not considered in this article. Particles not obeying SP can be accommodated in quantum mechanics, provided some modifications are made in the usual quantum-mechanical formalism. The main modification is to replace the usual ray by a many-dimensional "generalized ray" as the representative of a physical state. The properties of one-body measurements of systems having several identical particles are discussed, and the unobservability in such measurements of interferences between states having different irreducible permutation symmetries is pointed out. The condition of indistinguishability of identical particles is formulated precisely, and is analyzed both for interactions which conserve particle number and for general interactions which do not. For such general interactions, with the additional assumptions of time-reversal invariance and of coherence of the states having given values of charge, baryon number, and lepton number, it is shown that there is an absolute selection rule forbidding transitions between states ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ which contain any number of particles of species which obey SP but at most one particle of a species not obeying SP, and states which violate SP. Since only states in ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ are now avaiable as initial states of experiments, this selection rule forbids production of SP-violating states in any experiment which is feasible at present. Because of this, presently proposed experimental tests of SP are in fact tests of the quantum-mechanical description of identical particles together with time-reversal invariance and coherence of states in a given superselecting sector. The inclusion of internal variables, such as isospin, for particles violating SP is discussed. A comprehensive discussion is given of direct experimental tests of the SP selection rules. Such tests are more difficult to perform than appears at first sight, because in many cases the indistinguishability of identical particles or the conservation laws already imply the consequences of SP. Criteria for valid tests of SP are given. Several different types of tests are described, with illustrative examples of each. A survey is given of the direct experimental evidence for SP for the various particles. The Fermi character of electrons and positrons and of nucleons is accepted, as is the Bose character of photons. There is good evidence for the Bose nature of pions, especially from the absence of $2\ensuremath{\pi}$ decay of K20{{K}_{2}}^{0}K2​0. There is no direct evidence for the statistics of KKK, $\ensuremath{\Lambda}$, $\ensuremath{\Sigma}$, $\ensuremath{\Xi}$, or $\ensuremath{\mu}$. Feasible tests are proposed for the statistics of KKK and of those hyperons which have an asymmetric decay; but no such tests were found for the other hyperons or for $\ensuremath{\mu}$.
Anyons are quasiparticles with quantum statistics interpolating between those of bosons and fermions. Two distinct manifestations of anyonic behaviour have been theoretically established: fractional exchange statistics where particle exchange can produce any phase, and generalized exclusion statistics which extends the Pauli exclusion principle. While anyons exhibiting fractional exchange statistics have been observed in diverse platforms, experimental realizations of generalized exclusion statistics and direct measurements of its thermodynamic signatures have remained elusive. Here, we realize an anyonic thermodynamic ensemble obeying generalized exclusion statistics and detect its anyonic thermodynamics in a one-dimensional strongly interacting quantum gas. To achieve this, we exploit the bijective mapping between dynamical and statistical interactions in one dimension. By tuning interaction strength and temperature over a wide range, we measure the equation of state and identify clear departures from Bose-Einstein and Fermi-Dirac statistics. These deviations are quantitatively captured by generalized exclusion statistics, providing direct evidence for the generalized Pauli principle. Independent probes of other thermodynamic quantities including pressure and the Tan contact further validate this framework. Our results establish a versatile platform for engineering anyonic exclusion statistics and open the door to thermodynamic applications of anyons in quantum technologies.