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Physical Review A

Criterion for Bose-Einstein condensation in a harmonic trap in the case with attractive interactions

Mariusz Gajda

Published Feb 6, 2006• 43 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.73.023603

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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.
The visibility of the density interference pattern of two Bose-Einstein condensates, which are produced in traps and overlap after the trapping potential is switched off, is investigated. Coherent wave packets are used to describe the order parameter in a second-quantization formalism. This results in a decrease of the visibility of the interference fringes with increasing time delay between the formation of the condensates and the observation of interference. In the two limiting cases of ideal and very dense gases the correlation time increases, τ →∞, and the result is identical to that obtained using an approach based on the Gross-Pitaevskii equation. Under the conditions of the experiment performed by M. R. Andrews, C. G. Townsend, H. J. Miesner et al. , Science 275 , 6367 (1997), the computed correlation time τ ≈0.2 s is much longer than the confinement time of the condensate, and it is possible to observe the predicted decrease of visibility of the interference fringes of the density of the atoms.
When Wolfgang Ketterle and his colleagues at MIT demonstrated an “atom laser” recently, the announcement excited the physics community and naturally grabbed the attention of the popular press. One essential property of a laser is, of course, its coherence, and the evidence provided by the MIT group is striking: Sharp interference fringes formed when two Bose–Einstein condensates from the same trap expanded and overlapped after being released from confinement. The fringes are reminiscent of the interference between coherent light waves emanating from a double slit. (See the figure below.)
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.
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 investigate the influence of a Bose-Einstein condensate on the coherence properties of a traversing atomic beam. We derive a master equation describing the interaction between beam atoms and the condensate in sss-wave approximation. Scattering rates are calculated and compared with the case of an ordinary Boltzmann gas. Special attention is paid to the decoherence of the beam, i.e., the decay of spatial coherence in the presence of the condensate. As an application, we explicitly calculate the visibility of the interference pattern in a double-slit experiment and indicate signs for the onset of the condensation.
We study coherence of a trapped bosonic cloud with attractive finite-range interaction in a tight harmonic trap. One-body density and pair-distribution function in the ground state for different trap sizes are calculated. We also calculate healing length and the correlation length which signify the presence of high spatial coherence in a very tight trap leading to the destruction of the condensate for a fixed particle number. This is in marked variance with the usual collapse of the attractive metastable condensate when N>NcrN>{N}_{cr}N>Ncr​. Thus we investigate the critical frequency and critical size of the trap for the existence of attractive Bose-Einstein condensation. The finite-range interaction gives a nonlocal effect in the effective many-body potential, and we observe a high-density stable branch besides the known metastable branch. Moreover, the new branch shows universal behavior even in the very tight trap.