Condensates and Bose-Einstein correlations

We discuss the conditions under which Bose-Einstein correlation can be used to determine the presence of condensates. One starts from the fact that any condensate is a coherent state and uses then the formalism of quantum optics. The case of final state interactions is considered explicitly in a Landau-Ginzburg type approach as used in laser physics. By applying a mean field Hartree approximation for the many mode problem we find that the second order correlation function for identical bosons $\text{C}{}_2\text{(}\text{p}{}_1\text{?}\text{p}{}_2\text{=0)}$ does effectively not depend on the phenomenological coupling constant g and thus C₂(0) can be used to determine the amount of coherence and hence the existence of condensates. On the other hand $\text{C}{}_2\text{(}\text{p}{}_1\text{?}\text{p}{}_2\text{≠0)}$ does depend on g and therefore its measurement provides information about g. The implications of these findings for the determination of the size and lifetime of the source á la Hanbury-Brown-Twiss are discussed. Our conclusions apply both to condensates in nuclear (pion condensates, alpha and deuteron condensates) and hadronic matter.     

Bose-Einstein correlations and isospin

Bose-Einstein correlations between pions are analysed in terms of isospin. The lack of Bose-Einstein correlation between π⁺ and π⁰ is explicitly demonstrated.     

Coulomb and strong interactions for Bose-Einstein correlations

We present an analytical formula for the Bose-Einstein correlations (BEC) which includes effects of both Coulomb and strong final state interactions (FSI). It was obtained by using Coulomb wave function together with the scattering partial wave amplitude of the strong interactions describing data on thes-wave phase shift. We have proved numerically that this method is equivalent to solving Schrödinger equation with Coulomb and thes-wave strong interaction potentials. As an application we have analysed, using our formula which includes the degree of coherence and the long range correlation, the data fore ⁺ e ^? annihilations. We have found that the degree of coherence present in our formula approaches approximately unity whereas the long range correlation parameter becomes approximately zero. These results suggest that the physical meanings of the fractional degree of coherence and the long range correlation observed in various other analyses can most probably be attributed to FSI.     

Bose-Einstein correlations and color strings

It is shown that correlations between like-sign bosons (the “Bose-Einstein effect”) arise naturally in string fragmentation models for e⁺e⁻ annihilation. Via the area dependence of Wilson loop integrals, the correlation length in momentum space can be related to the string tension.     

Bose-Einstein correlations of pion wavepackets

A wavepacket model for a system of free pions, which takes into account the full permutation symmetry of the wavefunction and which is suitable for any phase space parametrization is developed. The properties of the resulting mixed ensembles and the two-particle correlation function are discussed. A physical interpretation of the chaoticity λ as localization of the pions in the source is presented. Two techniques to generate test-particles, which satisfy the probability densities of the wavepacket state, are studied:

1. A Monte Carlo procedure in momentum space based on the standard Metropolis technique.
2. A molecular dynamic procedure using Bohm’s quantum theory of motion. In order to reduce the numerical complexity, the separation of the wavefunction into momentum space clusters is discussed. In this context the influence of an unauthorized factorization of the state, i. e. the omission of interference terms, is investigated. It is shown that the correlation radius remains almost uneffected, but the chaoticity parameter decreases substantially. A similar effect is observed in systems with high multiplicities, where the omission of higher order corrections in the analysis of two-particle correlations causes a reduction of the chaoticity and the radius. The approximative treatment of the Coulomb interaction between pions and the source is investigated. The results suggest that Coulomb effects on the correlation radii are not symmetric for pion pairs of different charges. For (π^?,π^?) pairs the radius, integrated over the whole momentum spectrum, increases substantially, while for (π⁺,π⁺) pairs the radius remains almost unchanged.

     

Bose-Einstein correlations in neutrino and antineutrino interactions with nucleons

Four data sets of charged current neutrino and antineutrino interactions with neon, deuterium and hydrogen collected from BEBC and the 15 Foot Bubble Chamber are used to study the Bose-Einstein correlations between like-sign charged pions. Two forms of the parametrization for the effect are used. No substantial differences are found between the data sets obtained with neon, deuterium and hydrogen targets. The Lorentz invariant parametrization of Goldhaber gives for the radius of the pion emission region the valuer _G =0.80 ±0.04±0.16 fm and for the chaoticity parameter the value =0.61±0.04±0.15. Using the Kopylov-Podgoretskii parametrization yieldsr _K =1.27±0.06±0.12 fm, =0.58±0.03±0.12 and for the pion source lifetimec =0.52±0.05±0.12 fm. The Goldhaber parametrization was used to study the effect in further detail. The same emission radius and the same strength of the effect were found for particles produced in interactions on neutrons and protons. The data are compatible with a spherical shape of the pion emission region. No multiplicity or forward-backward dependences are found. No dependence of the effect on the event kinematical variables is seen.     

Bose-Einstein correlations in thermal field theory

Two-particle correlation functions are calculated for bosons emitted from a localized thermal source (the “glow” of a “hot spot”). In contrast to existing work, non-equilibrium effects up to first order in gradients of the particle distribution function are taken into account. The spectral width of the bosons is shown to be an important quantity: if it is too small, they do not equilibrate locally and therefore strongly increase the measured correlation radius.     

Coherence influence on the Bose-Einstein correlations

Two-particle correlations for both like and unlike pion pars are considered using the method of ensemble of sources with partially random phases. The dependence of both the interferometric source size and the coherence parameter on the source size and on the meansquared momentum of the emitted pions is found.     

Bose-Einstein correlations in the Lund model

By providing the Lund model fragmentation process with a quantum mechanical framework we extend the results of Andersson and Hofmann [Phys. Lett. B 169 (1986) 364] to situations where there are very many identical bosons. We investigate the features of the weight distributions in some detail and in particular exhibit three-particle BE correlations, the influence on the ϱ-spectrum and the difference between charged and neutral pion correlations.     

Analyses of multi-pion Bose-Einstein correlations for granular sources with coherent pion-emission droplets

The ALICE Collaboration measured three-and four-pion Bose-Einstein correlations(BECs)for Pb-Pb collisions at the Large Hadron Collider(LHC).It is speculated that the observed significant suppression of multi-pion BECs is owing to a considerable degree of coherent pion emission in these collisions.Here,we study multi-pion BEC functions for granular sources with coherent pion-emission droplets.We find that the intercepts of the multi-pion correlation functions at the relative momenta near zero are sensitive to the number of droplets in the granular source.They decrease with the droplet number.The three-pion correlation functions for evolving granular sources with momentum-dependent partially coherent pion-emission droplets basically agree with the experimental data for Pb-Pb collisions at √SNN=2.76 TeV at the LHC.However,the model results for the four-pion correlation function are inconsistent with the experimental data.Investigations into normalized multi-pion correlation functions of granular sources suggest an interesting enhancement of the normalized four-pion correlation function in the moderate relative-momentum region.     

Bose-Einstein condensates with large number of vortices

We show that as the number of vortices in a three dimensional Bose-Einstein condensate increases, the system reaches a "quantum Hall" regime where the density profile is a Gaussian in the xy plane and an inverted parabolic profile along z. The angular momentum of the system increases as the vortex lattice shrinks. However, Coriolis force prevents the unit cell of the vortex lattice from shrinking beyond a minimum size. Although the recent MIT experiment is not exactly in the quantum Hall regime, it is close enough for the present results to be used as a guide. The quantum Hall regime can be easily reached by moderate changes of the current experimental parameters.     

Quantum correlations among superradiant Bose-Einstein condensate atoms

Quantum correlations among atoms in superradiant Bose-Einstein condensates are discussed. It is shown that atoms in the superradiant atomic condensate can exhibit continuous variable quantum entanglement analogous to Einstein-Podolsky-Rosen (EPR)-type quantum correlations. Comparison to quantum entanglement in the Dicke model in thermal equilibrium is provided.     

Some forgotten features of the Bose-Einstein correlations

Notwithstanding the visible maturity of the subject of Bose-Einstein correlations (BEC), as witnessed nowadays, we would like to bring to one’s attention two points which apparently did not receive the attention they deserve: the problem of the choice of the form of C ₂(Q) correlation function when effects of partial coherence of the hadronizing source are to be included and the feasibility to model effects of Bose-Einstein statistics, in particular, the BEC, by direct numerical simulations. The text was submitted by the authors in English.     

Quantum clan model description of Bose-Einstein correlations

We propose a novel numerical method of <i≯modeling</i≯ Bose–Einstein correlations (BEC) observed among identical (bosonic) particles produced in multiparticle production reactions. We argue that the most natural approach is to work directly in the momentum space in which the Bose statistics of secondaries reveals itself in their tendency to bunch in a specific way in the available phase space. Because such procedure is essentially identical to the clan model of multiparticle distributions proposed some time ago, therefore we call it the <i≯Quantum Clan Model</i≯.     

On the directional dependence of Bose-Einstein correlations

A simple method to measure the shape of the particle emission region in high energy particle collisions by interferometry is discussed. Results from an application to data on inelasticpp scattering are compared with results based on determinations of the radial extension of the source. Both show that in the c.m.s., the source is elongated in the beam direction.