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arXiv:cond-mat/0412222 · Physical Review A

Properties of fragmented repulsive condensates

Alexej I. Streltsov, L. S. Cederbaum

Published Jun 22, 2005• 18 citations
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Abstract

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.

DOI
10.1103/physreva.71.063612

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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.
The fragmentation of the ground state of a repulsive condensate immersed into a double-trap potential is found to be a general and critical phenomenon. It takes place for a given number of bosons if their scattering length is larger than some critical value or for a given value of the scattering length if the number of bosons is above some critical number. We demonstrate that the geometry of the inner trap determines these critical parameters while the number of the fragments and the fraction of bosons in the various fragments can be manipulated by the outer trap. There is also a maximal number of bosons for which the ground state is fragmented. If this number is exceeded, the fragmented state becomes a very low-lying excited state of the condensate. This maximal number of bosons can be substantially manipulated by varying the inner and outer traps. To study threefold fragmentation we have chosen a potential well with two barriers as the inner trap and embedded in two types of outer ones. A manifold fragmentation is also addressed.
Linear response of simple (i.e., condensed) Bose-Einstein condensates is known to lead to the Bogoliubov--de Gennes equations. Here, we derive linear response for fragmented Bose-Einstein condensates, i.e., for the case where the many-body wave function is not a product of one, but of several single-particle states (orbitals). This gives one access to excitation spectra and response amplitudes of systems beyond the Gross-Pitaevskii description. Our approach is based on the number-conserving variational time-dependent mean-field theory [Alon, Streltsov, and Cederbaum, Phys. Lett. A 362, 453 (2007)], which describes the time evolution of best-mean-field states. Correspondingly, we call our linear-response theory for fragmented states LR-BMF. In the derivation it follows naturally that excitations are orthogonal to the ground-state orbitals. As applications excitation spectra of Bose-Einstein condensates in double-well potentials are calculated. Both symmetric and asymmetric double wells are studied for several interaction strengths and barrier heights. The cases of condensed and twofold fragmented ground states are compared. Interestingly, even in such situations where the response frequencies of the two cases are computed to be close to each other, which is the situation for the excitations well below the barrier, striking differences in the density response in momentum space are found. For excitations with an energy of the order of the barrier height, both the energies and the density response of condensed and fragmented systems are very different. In fragmented systems there is a class of ``swapped'' excitations where an atom is transferred to the neighboring well. The mechanism of its origin is discussed. In asymmetric wells, the response of a fragmented system is purely local (i.e., finite in either one or the other well) with different frequencies for the left and right fragments. This finding is in stark contrast to that for condensed systems.
Self-consistent excited states of condensates are solutions of the Gross-Pitaevskii (GP) equation and have been amply discussed in the literature and related to experiments. By applying a recently introduced more general mean field which includes the GP one as a special case, we find a new class of self-consistent excited states. In these states macroscopic numbers of bosons reside in different one-particle functions, i.e., the states are fragmented. Still, a single chemical potential is associated with the condensate. A numerical example is presented, illustrating that the energies of the new, fragmented states are much lower than those of the GP excited states, and that they are stable to variations of the particle number and shape of the trap potential.
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.
Fragmentation of a two-boson system in an external potential with contact interaction (i.e., $\ensuremath{\delta}$-function interaction) is studied using a full variational treatment. The results are compared to those obtained by the standard mean-field method (Gross-Pitaevskii equation) and the more recently developed best mean-field method. The numerically exact calculations show a transition to fragmentation as a function of the boson-boson interaction strength. The Gross-Pitaevskii solution cannot describe fragmentation. The respective energy exhibits a bifurcation which can be viewed as a hint that fragmentation may take place in reality. The best mean-field approach, on the other hand, can predict the fragmentation of the system and give the correct physical description of the fragmented state. The Gross-Pitaevskii energy diverges as the boson-boson interaction goes to infinity, while the exact and the best mean-field energies saturate.
We present a theoretical study of the ground state of a Bose-Einstein condensate with repulsive interparticle interactions in a double-well potential, using a restricted variational principle. Within such an approach, there is a transition from a single condensate to a fragmented condensate as the strength of the central barrier of the potential is increased. We determine the nature of this transition through an approximate analytic solution as well as a numerical solution of our model, in the regime where the interparticle interactions can be treated perturbatively. The degree of fragmentation of the condensate is characterized by the degrees of first- and second-order spatial coherence across the barrier.