Exact eigenstates for trapped weakly interacting bosons in two dimensions

Abstract

A system of N two-dimensional weakly interacting bosons in a harmonic trap is considered. When the two-particle potential is a delta function Smith and Wilkin have analytically proved that the elementary symmetric polynomials of particle coordinates measured from the center of mass are exact eigenstates. In this study, we point out that their proof works equally well for an arbitrary two-particle potential which possesses the translational and rotational symmetries. We find that the interaction energy associated with the eigenstate with angular momentum L is equal to aN (N −1)/2+(b − a) N L/2, where a and b are the interaction energies of two bosons in the lowest-energy one-particle state with zero and one unit of angular momentum, respectively. Additionally, we study briefly the case of attractive quartic interactions. We prove rigorously that the lowest-energy state is the one in which all angular momentum is carried by the center of mass motion. 03.75.Fi,05.30.Jp,67.40.Db The study of Bose-Einstein condensation in trapped atomic gases has attracted a great deal of attention in the past few years [1][2][3][4]. One of the central issues has been the possibility of creating quantized vortices in these dilute atomic gases. Recently, vortex states in various systems have been experimentally observed [5,6]. Some theoretical investigations on quantized vortices and on rotating Bose condensates have also been carried out both in the Thomas-Fermi limit of strong interactions [7][8][9][10] and in the limit of weak interactions [11][12][13][14][15][16][17][18] between the atoms. For the weakly interacting bosons one is naturally led to study the model of N two-dimensional bosons in a harmonic trap with weak repulsive delta-function interactions [11]. One important theoretical problem here is to understand the properties of the system with a given total angular momentum L. Numerical studies have shown that when L > N the lowest-energy state is the one where an array of singly quantized vortices is formed [15,8]. However, due to the complexity of these states, very few analytic results are known. In the range 0 ≤ L ≤ N the structure of lowest-energy states turns out to be simpler. Numerical computations by Bertsch and Papenbrock [13] showed that the interaction energy of the lowest-energy states has a very simple form and decreases linearly with L (the only exception being that for L = 1). They also noted that the wave functions for the lowest-energy states are simply the elementary symmetric polynomials of complex coordinates relative to the center of mass. Very recently, this remarkable formula for interaction energy was derived analytically by Jackson and Kavoulakis [17]. Moreover, Smith and Wilkin [18] proved that these symmetric polynomials are indeed exact eigenstates (see Ref. [19] for an alternative proof). In this note, we point out that the analytic proof by Smith and Wilkin works not only for a delta-function potential but also for an arbitrary potential which possesses translational and rotational symmetri...

Rotational properties of two-component Bose gases in the lowest Landau level

“…In addition, the large number of compact candidate determinants quickly becomes difficult to handle. As an example, the dimension of the TI-HW eigenspace at (N,M) = (3,9) and L = 8 is only 14, while the number of compact candidates is 1799 (see Table II). We therefore seek to identify a subset of the CF states, to reduce their number without losing the high overlaps with the lower energy exact states.…”

“…The simple states have been found to accomplish just this. 043625-5 In Table II we compare, for different L, the number of candidate determinants for the compact states to the corresponding number of simple state candidates at (N,M) = (3,9). As we see, the latter is significantly lower [29].…”

“…We clearly see that the simple states give very good TABLE II. Comparison of the number of compact candidates to the number of simple candidates for the case of (N,M) = (3,9). approximations to the lowest-lying states, while the higher excitations are excluded.…”

“…Consequently, a large body of work has focused on the rotational properties of atomic Bose gases. In this context, several groups have studied the rotational properties of bosons in the lowest Landau level also at the lowest angular momenta [6][7][8][9][10][11][12][13] (L N where N is the number of particles). Notably, analytically exact ground state wave functions were found in this regime [7][8][9][10].…”

Fragmentation of spin-orbit-coupled spinor Bose-Einstein condensates

“…To explore the formation and properties of vortices in an atomic boson system, the non-vanishing angular momentum states of weakly interacting Bose gase in harmonic trap have attracted considerable attentions [22][23][24][25][26]. In Ref [26], Xia-Ji Liu et al studied the ground state for a weakly interacting, harmonically trapped Nboson system and found that the ground state is generally a fragmented condensate because of the orbital an-gular momentum conservation.…”