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Physical Review A

Bose-Einstein atoms in atomic traps with predominantly attractive two-body interactions

Mahir Saleh Hussein, Oleg Vorov

Published Apr 26, 2002• 6 citations
Publisher

Abstract

Using the Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn, and Smith [Phys. Rev. Lett. 80, 2265, (1998)] for the ground states at angular momentum $L$ of $N$ harmonically trapped Bose atoms, interacting via weak attractive ${\ensuremath{\delta}}^{2}(r)$ forces, are valid for a broad class of predominantly attractive interactions $V(r),$ not necessarily attractive for any $r.$ This class is described by sufficient conditions on the two-body matrix elements of the potential $V(r).$ It includes, in particular, the Gaussian attraction of arbitrary radius, $\ensuremath{-}1/r$-Coulomb and $\mathrm{log}(r)$-Coulomb forces, as well as all the short-range interactions satisfying inequality $\ensuremath{\int}{d}^{2}\stackrel{\ensuremath{\rightarrow}}{r}V(r)<0.$ In the precollapse regime, the angular momentum $L$ is concentrated in the collective ``center-of-mass'' mode, and there is no condensation at high $L.$

DOI
10.1103/physreva.65.053608

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Using the Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn, and Smith [Phys. Rev. Lett. 80, 2265, (1998)] for the ground states at angular momentum LLL of NNN harmonically trapped Bose atoms, interacting via weak attractive \ensuremathδ2(r){\ensuremath{\delta}}^{2}(r)\ensuremathδ2(r) forces, are valid for a broad class of predominantly attractive interactions V(r),V(r),V(r), not necessarily attractive for any r.r.r. This class is described by sufficient conditions on the two-body matrix elements of the potential V(r).V(r).V(r). It includes, in particular, the Gaussian attraction of arbitrary radius, \ensuremath−1/r\ensuremath{-}1/r\ensuremath−1/r-Coulomb and log(r)\mathrm{log}(r)log(r)-Coulomb forces, as well as all the short-range interactions satisfying inequality \ensuremath∫d2r\ensuremath→V(r)<0.\ensuremath{\int}{d}^{2}\stackrel{\ensuremath{\rightarrow}}{r}V(r)<0.\ensuremath∫d2r\ensuremath→V(r)<0. In the precollapse regime, the angular momentum LLL is concentrated in the collective ``center-of-mass'' mode, and there is no condensation at high L.L.L.
Using Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn and Smith [1] for the ground states of N Bose atoms rotating at the angular momentum L in a harmonic atomic trap with frequency omega interacting via attractive delta^2(r) forces, are valid for a broad class of predominantly attractive interactions V(r), not necessarily attractive for any r. The sufficient condition for the interaction is that all the two-body matrix elements allowed by the conservation of angular momentum k+l = m+n, are negative. This class includes, in particular, the Gaussian attraction of arbitrary radius, -1/r - Coulomb and log(r)-Coulomb forces, as well as all the short-range R << omega^{-1/2} interactions satisfying inequality int d^2r V(r) < 0. There is no condensation at L>> 1, and the angular momentum is concentrated in the collective ``center-of-mass'' mode.
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.
In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.
In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.
We study a class of interacting, harmonically trapped boson systems at angular momentum L. The Hamiltonian leaves a L-dimensional subspace invariant, and this permits an explicit solution of several eigenstates and energies for a wide class of two-body interactions. PACS numbers: 05.30.Jp, 03.75.Fi, 03.65.Fd, 67.40.Db Typeset using REVTEX 1 The study of vortices in Bose-Einstein condensates of trapped atomic vapors is of considerable experimental [1,2] and theoretical [3–13] interest. Present experiments and much theoretical work [3–7] focus on the Thomas-Fermi regime of short coherence length [14]. The opposite limit of perturbatively weak contact interactions is not yet realized experimentally but is of theoretical interest as well [8–13]. The case of attractive interactions was studied by Wilkin et al. [9]. Mottelson developed a theory for repulsive interactions [10], and Kavoulakis et al. [12] and Jackson et al. [13] compared mean field and exact numerical results. Exact diagonalization techniques by us showed that the ground state energy depends linearly on angular momentum and led to an analytical expression for the ground state wave function [11]. Recently, the form of this wave function and its eigenenergy were confirmed in analytic calculations [15–17]. However, we still lack insight about why certain eigenstates turn out to be simple analytical functions. We show in this paper that there is a subspace structure that explains these findings. Interestingly, this structure is not limited to the two-body contact interaction considered in previous investigations but present in a rather large class of Hamiltonians. Let us consider a system of N harmonically trapped bosons in two spatial dimensions 1 at angular momentum
We calculated, within the Gross-Pitaevskii formalism, the critical number of atoms for Bose-Einstein condensates with two-body attractive interactions in cylindrical traps with different frequency ratios. In particular, by using the trap geometries considered by Roberts et al. [Phys. Rev. Lett. 86, 4211 (2001)], we show that the theoretical maximum critical numbers are given approximately by Nc=0.55(l0/∣a∣).{N}_{c}{=0.55(l}_{0}/|a|).Nc​=0.55(l0​/∣a∣). Our results also show that, by exchanging the frequencies ${\ensuremath{\omega}}_{z}$ and ${\ensuremath{\omega}}_{\ensuremath{\rho}},$ the geometry with ${\ensuremath{\omega}}_{\ensuremath{\rho}}<{\ensuremath{\omega}}_{z}$ favors the condensation of larger number of particles. We also simulate the time evolution of the condensate when changing the ground state from a=0a=0a=0 to a<0a<0a<0 using a 200 ms ramp. A conjecture on higher-order nonlinear effects is also added in our analysis with an experimental proposal to determine its signal and strength.