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arXiv:1004.3954 · Physical Review A

Fragmented many-body states of definite angular momentum and stability of attractive three-dimensional condensates

Marios C. Tsatsos, Alexej I. Streltsov, Ofir E. Alon, L. S. Cederbaum

Published Sep 20, 2010• 15 citations
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Abstract

A three-dimensional attractive Bose-Einstein condensate is expected to collapse when the number of the particles $N$ in the ground state or the interaction strength ${\ensuremath{\lambda}}_{0}$ exceeds a critical value. We study systems of different particle numbers and interaction strength and find that even if the overall ground state is collapsed there is a plethora of fragmented excited states that are still in the metastable region. Utilizing the configuration interaction expansion we determine the spectrum of the ground (``yrast'') and excited many-body states with definite total angular-momentum quantum numbers $0\ensuremath{\leqslant}L\ensuremath{\leqslant}N$ and $\ensuremath{-}L\ensuremath{\leqslant}{M}_{L}\ensuremath{\leqslant}L$, and we find and examine states that survive the collapse. This opens up the possibility of realizing a metastable system with overcritical numbers of bosons in a ground state with angular momentum $L\ensuremath{\ne}0$. The multiorbital mean-field theory predictions about the existence of fragmented metastable states with overcritical numbers of bosons are verified and elucidated at the many-body level. The descriptions of the total angular momentum within the mean-field and the many-body approaches are compared.

DOI
10.1103/physreva.82.033613

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A three-dimensional attractive Bose-Einstein condensate is expected to collapse when the number of the particles NNN in the ground state or the interaction strength \ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ exceeds a critical value. We study systems of different particle numbers and interaction strength and find that even if the overall ground state is collapsed there is a plethora of fragmented excited states that are still in the metastable region. Utilizing the configuration interaction expansion we determine the spectrum of the ground (``yrast'') and excited many-body states with definite total angular-momentum quantum numbers 0\ensuremath⩽L\ensuremath⩽N0\ensuremath{\leqslant}L\ensuremath{\leqslant}N0\ensuremath⩽L\ensuremath⩽N and \ensuremath−L\ensuremath⩽ML\ensuremath⩽L\ensuremath{-}L\ensuremath{\leqslant}{M}_{L}\ensuremath{\leqslant}L\ensuremath−L\ensuremath⩽ML​\ensuremath⩽L, and we find and examine states that survive the collapse. This opens up the possibility of realizing a metastable system with overcritical numbers of bosons in a ground state with angular momentum L\ensuremath≠0L\ensuremath{\ne}0L\ensuremath=0. The multiorbital mean-field theory predictions about the existence of fragmented metastable states with overcritical numbers of bosons are verified and elucidated at the many-body level. The descriptions of the total angular momentum within the mean-field and the many-body approaches are compared.
In this Dissertation the physics of trapped, attractive Bose-Einstein condensates in two and three spatial dimensions are examined. Particular emphasis is put on the collapse of the gas, when the attractive interparticle interaction is raised above a critical value. The states of the system that possess good angular momentum quantum numbers are scrutinized and connections between the angular momentum and the fragmentation of the states are investigated. In two spatial dimensions mean-field states can describe states of good angular momentum. Using a mean-field ansatz, analytical results for the energy, the occupation numbers and the stability of the ground state carrying L quanta of angular momentum are obtained. In three dimensions, however, a mean-field theory does not suffice for the proper description of angular momentum states. The eigenstates of both total angular momentum operators (L^2 and L_z) are derived and it is shown that they are generally many-body states. It is shown, moreover, that angular momentum has a general stabilizing effect on the gas and this connection is expressed quantitatively. Finally, the ground state of the gas is examined, when its container is set into external rotation. It is found that due to the attraction the symmetry of the ground state does not change, angular momentum is not transfered from the rotation to the gas and thus no significant impact in the stability is seen.
An attractive Bose-Einstein condensate in two spatial dimensions is expected to collapse for supercritical values of the interaction strength. Moreover, it is known that for nonzero quanta of angular momentum and infinitesimal attraction the gas prefers to fragment and distribute its angular momentum over different orbitals. In this work we examine the two-dimensional trapped Bose gas for finite values of attraction and describe the ground state in connection to its angular momentum by theoretical methods that go beyond the standard Gross-Pitaevskii theory. By applying the best-mean-field approach over a variational ansatz whose accuracy has been checked numerically, we derive analytical relations for the energy, the fragmentation of the ground states, and the critical (for collapse) value of the attraction strength as a function of the total angular momentum LLL.
The theoretical investigation of rotating Bose-Einstein condensates has mainly focused on the emergence of quantum vortex states and the condensed properties of such systems. In the present work, we concentrate on other facets by examining the impact of rotation on the ground state of weakly interacting bosons confined in anharmonic potentials computed both at the mean-field level and particularly at the many-body level of theory. For the many-body computations, we employ the well-established many-body method known as the multiconfigurational time-dependent Hartree method for bosons. We present how various degrees of fragmentation can be generated following the breakup of the ground state densities in anharmonic traps without ramping up a potential barrier for strong rotations. The breakup of the densities is found to be associated with the acquisition of angular momentum in the condensate due to the rotation. In addition to fragmentation, the presence of many-body correlations is examined by computing the variances of the many-particle position and momentum operators. For strong rotations, the many-body variances become smaller than their mean-field counterparts, and one even finds a scenario with opposite anisotropies of the mean-field and many-body variances. Further, it is observed that for higher discrete symmetric systems of order k, namely three-fold and four-fold symmetry, breakup to k sub-clouds and emergence of k-fold fragmentation take place. All in all, we provide a thorough many-body investigation of how and which correlations build up when a trapped Bose-Einstein condensate breaks up under rotation.
Repulsive Bose-Einstein condensates immersed into a double-well trap potential are studied within the framework of the recently introduced mean-field approach which allows for bosons to reside in several different orthonormal orbitals. In the case of a one-orbital mean-field theory (Gross-Pitaevskii) the ground state of the system reveals a bifurcation scenario at some critical values of the interparticle interaction and/or the number of particles. At about the same values of the parameters the two-orbital mean field predicts that the system becomes twofold fragmented. By applying the three-orbital mean field we verify numerically that for the double-well external potential studied here the overall best mean field is achieved with two orbitals. The variational principle minimizes the energy at a vanishing population of the third orbital. To discuss the energies needed to remove a boson from and the energies gained by adding a boson to the condensate, we introduce boson ionization potentials and boson affinities and relate them to the chemical potentials. The impact of the finite number of bosons in the condensate on these quantities is analyzed. We recall that within the framework of the multiorbital mean-field theory each fragment is characterized by its own chemical potential. Finally, the stability of fragmented states is discussed in terms of the boson transfer energy which is the energy needed to transfer a boson from one fragment to another.
We consider a system of NNN interacting bosons in a rotating harmonic trap in R2\mathbb{R}^2R2, where the two-body interaction is attractive and scaled as N2αU(Nαx)N^{2α}U(N^αx)N2αU(Nαx) with 0<α<1/120<α<1/120<α<1/12, and the three-body interaction is repulsive and scaled as N4βW(Nβx,Nβy)N^{4β}W(N^βx,N^βy)N4βW(Nβx,Nβy) with 0<β<1/240<β<1/240<β<1/24. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schrödinger functional. We analyze the collapse regime where the two-body coupling aaa approaches the critical value a∗a_*a∗​ and the three-body coupling bbb tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies ENLS=(1−ζ/4+o(1))Qpotℓn2E^{\mathrm{NLS}} = (1-ζ/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2ENLS=(1−ζ/4+o(1))Qpotℓn2​ with ζ⩾0ζ\geqslant0ζ⩾0 determined by the relative rates of an→a∗a_n\to a_*an​→a∗​ and bn↘0b_n\searrow0bn​↘0. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.