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arXiv:1111.1035

Role of particle-number statistics in interference of independent Bose fields

Toru Kawakubo, Katsuji Yamamoto

Published Nov 4, 2011• 0 citations
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Abstract

We elucidate generally the interference of independent Bose fields in view of the conditional probability for the particle number measurements, and clarify its relation to the source number statistics. Despite lack of intrinsic phases, the interference phase can be inferred from the particle number registered at one detector by using the classical mean fields. If the conditional number distributions for the other detectors, given the outcome of the first detector, exhibit sufficiently narrow peaks around the values specified by the estimated phases, the mean field description is valid in a single run of interference. The widths in the conditional distribution are determined by the number statistics of the sources, among which notable scaling behavior is found depending on the detector configurations with the boundary at the Poissonian. The mean field description is found to be applicable to Poissonian and sub-Poissonian sources, whereas for super-Poissonian sources it is likely invalidated with the rather broad conditional distribution.

DOI
10.48550/arxiv.1111.1035

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We elucidate generally the interference of independent Bose fields in view of the conditional probability for the particle number measurements, and clarify its relation to the source number statistics. Despite lack of intrinsic phases, the interference phase can be inferred from the particle number registered at one detector by using the classical mean fields. If the conditional number distributions for the other detectors, given the outcome of the first detector, exhibit sufficiently narrow peaks around the values specified by the estimated phases, the mean field description is valid in a single run of interference. The widths in the conditional distribution are determined by the number statistics of the sources, among which notable scaling behavior is found depending on the detector configurations with the boundary at the Poissonian. The mean field description is found to be applicable to Poissonian and sub-Poissonian sources, whereas for super-Poissonian sources it is likely invalidated with the rather broad conditional distribution.
We investigate interference of optical fields by examining the probability distribution of photon detection. The usual description of interference patterns in terms of superposition of classical mean fields with definite phases is elucidated in quantum fashion. Especially, for interference of two independent mixtures of number states with Poissonian or sub-Poissonian statistics, despite lack of intrinsic phases, it is found that the joint probability has a distinct peak manifold in the multi-dimensional space of the detector outcomes, which is along the trajectory of the mean-field values as the relative phase varies on the unit circle. Then, an interference pattern should mostly appear in each shot of measurement as a point in the peak manifold with a randomly chosen relative phase. On the other hand, for super-Poissonian sources the mean-field description is likely invalidated with rather broad probability distributions.
A method is presented that is able to predict the probability of outcomes of snapshot measurements, such as the images of the instantaneous particle density distribution in a quantum many-body system. It is shown that a gauge-like transformation of the phase of the many-body wave function allows one to construct a probability generating functional, the Fourier transform of which with respect to the “gauge” field returns the joint probability distribution to detect any given number of particles at various locations. The method is applied to the problem of interference of two independent clouds of Bose-Einstein condensates, where the initially separated clouds with fixed boson numbers expand and the density profile image of the overlapping clouds is registered. In the limit of large particle numbers, the probability to observe a particular image of the density profile is shown to be given by a sum of partial probability distributions, each of which corresponds to a noisy image of interference of two matter waves with definite phase difference. In agreement with earlier theoretical arguments, interference fringes are, therefore, expected in any single shot measurement, the fringe pattern randomly varying from run to run. These results conform to the physical picture where the Bose-Einstein clouds are in spontaneously symmetry broken states, the hidden phases of which are revealed by the density profile measurement via the position of the interference fringes.
The interference of two Bose-Einstein condensates, initially in Fock states, can be described in terms of their relative phase, treated as a random unknown variable. This phase can be understood either as emerging from the measurements or pre-existing to them; in the latter case, the originating states could be phase states with unknown phases, so an average over all their possible values is taken. Both points of view lead to a description of probabilities of results of experiments in terms of a phase angle, which plays the role of a classical variable. Nevertheless, in some situations, this description is not sufficient: another variable, which we call the ``quantum angle,'' emerges from the theory. This article studies various manifestations of the quantum angle. We first introduce the quantum angle by expressing two Fock states crossing a beam splitter in terms of phase states and relate the quantum angle to off-diagonal matrix elements in the phase representation. Then we consider an experiment with two beam splitters, where two experimenters make dichotomic measurements with two interferometers and detectors that are far apart; the results lead to violations of the Bell-Clauser-Horne-Shimony-Holt inequality (valid for local-realistic theories, including classical descriptions of the phase). Finally, we discuss an experiment where particles from each of two sources are either deviated via a beam splitter to a side collector or proceed to the point of interference. For a given interference result, we find ``population oscillations'' in the distributions of the deviated particles, which are entirely controlled by the quantum angle. Various versions of population oscillation experiments are discussed, with two or three independent condensates.
Photon anti-bunching, measured via the Hanbury-Brown-Twiss experiment, is one of the key signatures of quantum light and is tied to sub-Poissonian photon number statistics. Recently, it has been reported that photon anti-bunching or conditional sub-Poissonian photon number statistics can be obtained via second-order interference of mutually incoherent weak lasers and heralding based on photon counting [Phys. Rev. A92, 033855 (2015)10.1103/PhysRevA.92.033855; Opt. Express24, 19574 (2016)10.1364/OE.24.019574; https://arxiv.org/abs/1601.08161]. Here, we report theoretical analysis on the limits of manipulating conditional photon statistics via interference of weak lasers. It is shown that conditional photon number statistics can become super-Poissonian in such a scheme. We, however, demonstrate explicitly that it cannot become sub-Poissonian, i.e., photon anti-bunching cannot be obtained in such a scheme. We point out that incorrect results can be obtained if one does not properly account for seemingly negligible higher-order photon number expansions of the coherent state.
We investigate the role of quantum interference phenomena on the characteristics of the fields radiated by an array of quantum emitters. In analogy to, but distinct from, classical outcomes, we demonstrate that the array geometry empowers control over direction-dependent photon statistics of arbitrary order. Our formulation enables the recognition of configurations providing spatial correlations with no classical counterpart. For example, we identify a system in which the angular distribution of the average number of photons is independent of the number and position of the emitters, while its higher-order photon statistics exhibit a directional behavior. These results extend our understanding of the fields generated by ensembles of quantum emitters, with potential applications to nonclassical light sources.
The optical interference constitutes a paramount resource in modern physics. At the scale of individual atoms and photons, it is a diverse concept that causes different coherent phenomena. We present the experimental characterization of both coherent and statistical properties of light emitted from ensembles of trapped ions increasing with a number of contributing phase-incoherent independent atomic particles ranging from a single to up to several hundreds. It conclusively demonstrates how super-Poissonian quantum statistics non-trivially arises purely from the finite number of indistinguishable single-photon emitters in the limit of a single detection mode. The achieved new optical emission regime in which these independent atoms contribute coherently to the super-Poissonian statistics provides a new perspective on the emergence of optical coherence at the atomic scale and constitutes a unique toolbox for its generation and control at the most microscopic level.