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arXiv:0804.1672 · Physical Review A

Criterion for Bose-Einstein condensation in traps and self-bound systems

T. Yamada, Yasuro Funaki, H. Horiuchi, GERD RÖPKE, Peter Schuck, A. Tohsaki

Published Sep 18, 2008• 19 citations
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Abstract

The internal one-particle density matrix is discussed for Bose-Einstein condensates with a finite number of particles in a harmonic trap. It is found that the outcome of the diagonalization of the single-particle density matrix depends on the choice of the internal coordinates: The Pethick-Pitaevskii (PP) type of internal density matrix, whose analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented condensate, while the Jacobi type of internal density matrix leads to an ideal condensate. We give a criterion for the choice of the internal coordinates: In the macroscopic limit the internal density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame, this being a very physical condition. One choice satisfying this boundary condition is given by the internal Jacobi coordinates, while the internal coordinates with respect to the center of mass of the PP density matrix do not satisfy this condition. Based on our criterion, a general definition of the internal one-particle density matrix is presented in a self-bound system consisting of interacting bosons.

DOI
10.1103/physreva.78.035603

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The internal one-particle density matrix is discussed for Bose-Einstein condensates with a finite number of particles in a harmonic trap. It is found that the outcome of the diagonalization of the single-particle density matrix depends on the choice of the internal coordinates: The Pethick-Pitaevskii (PP) type of internal density matrix, whose analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented condensate, while the Jacobi type of internal density matrix leads to an ideal condensate. We give a criterion for the choice of the internal coordinates: In the macroscopic limit the internal density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame, this being a very physical condition. One choice satisfying this boundary condition is given by the internal Jacobi coordinates, while the internal coordinates with respect to the center of mass of the PP density matrix do not satisfy this condition. Based on our criterion, a general definition of the internal one-particle density matrix is presented in a self-bound system consisting of interacting bosons.
Investigations on the internal one-particle density matrix in the case of Bose-Einstein condensates with a finite number (NNN) of particles in a harmonic potential are performed. We solve the eigenvalue problem of the Pethick-Pitaevskii-type internal density matrix and find a fragmented condensate. On the contrary the condensate Jacobi-type internal density matrix gives complete condensation into a single state. The internal one-particle density matrix is, therefore, shown to be different in general for different choices of the internal coordinate system. We propose two physically motivated criteria for the choice of the adequate coordinate systems that give us a unique answer for the internal one-particle density matrix. One criterion is that in the infinite particle number limit ($N=\ensuremath{\infty}$) the internal one-particle density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame. The other criterion is that the coordinate of the internal one-particle density matrix should be orthogonal to the remaining $(N\ensuremath{-}2)$ internal coordinates, though the $(N\ensuremath{-}2)$ coordinates, in general, do not need to be mutually orthogonal. This second criterion is shown to imply the first criterion. It is shown that the internal Jacobi coordinate system satisfies these two criteria while the internal coordinate system adopted by Pethick and Pitaevskii for the construction of the internal one-particle density matrix does not. It is demonstrated that these two criteria uniquely determine the internal one-particle density matrix that is identical to that calculated with the Jacobi coordinates. The relevance of this work concerning $\ensuremath{\alpha}$-particle condensates in nuclei, as well as bosonic atoms in traps, is pointed out.
We consider the criterion for Bose-Einstein condensation for particles in a harmonic trap. For a fixed angular momentum, the lowest energy state for a cloud of bosons with attractive interactions is the ground state of the cloud with all the angular momentum in the center-of-mass motion, and the one-particle reduced density matrix generally does not have a single large eigenvalue, but a number of them, suggesting that the state is an example of a fragmented condensate [N. K. Wilkin, J. M. F. Gunn, and R. A. Smith, Phys. Rev. Lett. 80, 2265 (1998)]. We show that a convenient way to describe correlations in the system is by defining an internal one-particle reduced density matrix, in which the center-of-mass motion is eliminated, and that this has a single eigenvalue equal to the number of particles for the problem considered here. Our considerations indicate that care is necessary in formulating a criterion for Bose-Einstein condensation.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
Using a model many-body wave function I analyze the standard criterion for Bose-Einstein condensation and its relation to coherence properties of the system. I pay special attention to an attractive condensate under such a condition that a characteristic length scale of the spatial extension of its center of mass differs significantly from length scales of relative coordinates. I show that although no interference fringes are produced in the two-slit Young interference experiment performed on this system, fringes of a high visibility can be observed in a conditional simultaneous detection of two particles.
We rigorously revisit a textbook model used to figure out the Bose-Einstein condensation (BEC) phenomenon created by dilute cold alkali atoms gases in a magnetic-optical trap. It consists of a d-dimensional (d=1,2,3) ideal non-relativistic spin-0 Bose gas confined in a box and trapped in an isotropic harmonic potential. Throughout we review and clarify a series of methods involved in the derivation of the thermodynamics in the grand-canonical situation. To make the derivation consistent with the usual rules of the statistical mechanics, we assign through our open-trap limit approach the role of canonical parameter to a rescaled number of particles (instead of an effective density involving the pulsation of the trap). Within this approach, we formulate an Einstein-like and Penrose-Onsager-like criterion of BEC and show their equivalence. Afterwards, we focus on the spatial localization of the condensate/thermal gas. When dealing with the reduced density matrix, our method is similar to the loop path approach.