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arXiv:1009.5241 · Physical Review A

Counting statistics of many-particle quantum walks

K. Ulrich Mayer, Malte C. Tichy, Florian Mintert, Thomas Konrad, Andreas Buchleitner

Published Jun 7, 2011• 74 citations
arXivHTMLPDF

Abstract

We study quantum walks of many noninteracting particles on a beam splitter array as a paradigmatic testing ground for the competition of single- and many-particle interference in a multimode system. We derive a general expression for multimode particle-number correlation functions, valid for bosons and fermions, and infer pronounced signatures of many-particle interferences in the counting statistics.

DOI
10.1103/physreva.83.062307

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We study quantum walks of many noninteracting particles on a beam splitter array as a paradigmatic testing ground for the competition of single- and many-particle interference in a multimode system. We derive a general expression for multimode particle-number correlation functions, valid for bosons and fermions, and infer pronounced signatures of many-particle interferences in the counting statistics.
Identical particles exhibit correlations even in the absence of inter-particle interaction, due to the exchange (anti)symmetry of the many-particle wavefunction. Two fermions obey the Pauli principle and anti-bunch, whereas two bosons favor bunched, doubly occupied states. Here, we show that the collective interference of three or more particles leads to much more diverse behavior than expected from the boson–fermion dichotomy known from quantum statistical mechanics. The emerging complexity of many-particle interference is tamed by a simple law for the strict suppression of events in the Bell multiport beam splitter. The law shows that counting events are governed by widely species-independent interference, such that bosons and fermions can even exhibit identical interference signatures, while their statistical character remains subordinate. Recent progress in the preparation of tailored many-particle states of bosonic and fermionic atoms promises experimental verification and applications in novel many-particle interferometers.
The present thesis deals with the interrelation between entanglement of identical particles and many-particle interference in multimode scattering scenarios. The processes that govern the scattering are shown to characterize the quantum correlations that result from coherent deletion of many-particle which-way information. Part I treats the constitutive situation of two identical particles which are measured by two detectors to provide a consistent definition of identical-particle entanglement. The key quantities that determine the strength and nature of correlations are diagnosed, which allows a formulation of the problem as scattering scenario. Quite counter-intuitively, the effective indistinguishability of the particles is not necessarily reflected monotonically by the strength of quantum correlations, which is explained by an interplay of the entanglement encoded in the initial state with measurement-induced correlations. The subject of Part II is the scattering of many non-interacting particles that are initially injected in an arbitrary number of modes. A comparison of bosons, fermions and distinguishable particles reveals that the dichotomy of bosons and fermions, known from the two-particle case, does, surprisingly, not prevail for many particles: A given output event can be suppressed or enhanced for bosons and fermions simultaneously. Consequently, the bunching behavior of bosons is not apparent at the level of single counting events, where dynamical many-particle interference dominates, and it is only reflected in more coarse-grained quantities that are related to the occupation statistics, for which efficient approximate treatments are derived. The general complexity of the many-particle problem is overcome for the special case of a Fourier multiport beam splitter. The exploitation of its scattering matrix’ symmetries yields a systematic event suppression law for, both, bosons and fermions, which provides a generalization of the two-photon Hong-Ou-Mandel effect to an arbitrary number of particles and modes. As a further characteristic trait of many-particle interference, the transition between indistinguishable and distinguishable particles is reflected non-monotonically by the event probabilities, which is also confirmed by an experiment that was suggested by the present work. Part III combines the formalism for identical-particle entanglement and the tools and notions developed in the context of many-particle scattering for a description of those multipartite entangled states that can be created via the scattering of many identical particles. The formalism is general enough to include virtually any possible scheme for the multipartite entanglement creation with single photons. An analytical treatment reveals fundamental constraints for the set of accessible states. In particular, an intuitive combinatorial bound for the Schmidt number of the target state follows from the finite number of combinations to distribute the particles among the output modes. Furthermore, bosons and fermions have very distinct potential for entanglement creation: Fewer states are accessible for fermions, whereas the success probability of entanglement production is strongly enhanced with respect to bosons.
Quantum interference between identical single particles reveals the intrinsic quantum statistic nature of particles, which could not be interpreted through classical physics. Here, we demonstrate quantum interference between nonidentical bosons using a generalized beam splitter based on a quantum memory. The Hong-Ou-Mandel type interference between single photons and single magnons with high visibility is demonstrated, and the crossover from the bosonic to fermionic quantum statistics is observed by tuning the beam splitter to be non-Hermitian. Moreover, multiparticle interference that simulates the behavior of three fermions by three input photons is realized. Our work extends the understanding of the quantum interference effects and demonstrates a versatile experimental platform for studying and engineering quantum statistics of particles.
We show that kkk-point correlation measurements on output of a non-interacting, multimode, random unitary allow to quantify the kkk-particle coherence of N≥kN\geq kN≥k identical (bosonic or fermionic) particles. For separable many-particle input states, we establish an on average strictly monotonic relationship between such kkk-particle coherences, the interference contrast in the experimentally accessible counting statistics and the degree of the particles' mutual distinguishability, as controlled by their internal degrees of freedom. Non-separability on input can be unveiled by comparison of correlation measurements of different orders.
The counting statistics (CS) for charges passing through a coherent conductor comprise the most general quantity that characterizes electronic transport. The CS depend not only on the transport properties of the conductor, but also on the correlations among particles which compose the incident beam. In this paper we present general results for the CS of entangled electron pairs traversing a beam splitter, and we show that the probability that Q charges have passed is not binomial, as in the uncorrelated case, but is symmetric with respect to the average transferred charge. We furthermore consider the joint probability for transmitted charges of a given spin, and we show that the signature of entanglement distinctly appears in a correlation which is not present for a nonentangled case.