Asymptotic Gaussian law for noninteracting indistinguishable particles in random networks

Valery S Shchesnovich

doi:10.1038/s41598-017-00044-8

Abstract

For N indistinguishable bosons or fermions impinged on a M-port Haar-random unitary network the average probability to count n1, … nr particles in a small number r ≪ N of binned-together output ports takes a Gaussian form as N ≫ 1. The discovered Gaussian asymptotic law is the well-known asymptotic law for distinguishable particles, governed by a multinomial distribution, modified by the quantum statistics with stronger effect for greater particle density N/M. Furthermore, it is shown that the same Gaussian law is the asymptotic form of the probability to count particles at the output bins of a fixed multiport with the averaging performed over all possible configurations of the particles in the input ports. In the limit N → ∞, the average counting probability for indistinguishable bosons, fermions, and distinguishable particles differs only at a non-vanishing particle density N/M and only for a singular binning K/M → 1, where K output ports belong to a single bin.

Highlights

Indistinguishable identical particles show correlated behavior due to their quantum statistics even in the absence of interactions: indistinguishable bosons show bunching, e.g., leave the balanced beam splitter in the same port, as demonstrated in the famous experiment with single photons[1] and recently with massive bosons[2], while indistinguishable fermions show anti-bunching[3]
A natural question arises: Is there a universal statistics-dependent law in behavior of noninteracting indistinguishable identical bosons in multi-port networks? It is shown below that the probability of counting indistinguishable particles in binned-together output ports of a unitary M-port, averaged over the Haar-random unitary matrix representing the multiport, has a statistics-dependent asymptotic Gaussian form as N ≫ 1, where the quantum statistics enters through the particle density N/M. (Our main interest is the bosonic case, since indistinguishable bosons and distinguishable particles can share the same port in a multiport, whereas we consider fermions to identify the contribution of the quantum statistics)
M-port network U Fig. 1, whose output ports are partitioned into r bins having K ≡ (K1, ..., Kr) ports, with N noninteracting identical particles at the input

Summary

Introduction

Indistinguishable identical particles show correlated behavior due to their quantum statistics even in the absence of interactions: indistinguishable bosons show bunching, e.g., leave the balanced beam splitter in the same port, as demonstrated in the famous experiment with single photons[1] and recently with massive bosons[2], while indistinguishable fermions show anti-bunching[3]. Indistinguishable particles in multi-port networks show correlated behavior beyond the quantum statistics, e.g., in the symmetric (Bell type) multiports complex multi-particle interference results in common forbidden output configurations both for bosons and fermions[10], confirmed recently with photons[16]. (Our main interest is the bosonic case, since indistinguishable bosons and distinguishable particles can share the same port in a multiport, whereas we consider fermions to identify the contribution of the quantum statistics).

Methods

Results

Conclusion