We present an exact diagonalization study of the breathing mode collective excitations for a rotating Bose-Einstein condensate of
N=10 spinless bosons interacting via repulsive finiterange Gaussian potential and harmonically confined in quasi-two-dimension. The yrast state and the low-lying excited states are variationally obtained in given subspaces of the quantized total angular momentum
L employing the beyond lowest Landau level approximation in slowly rotating regime with
0≤L<2N. For a given
L, the low-energy eigenspectra (bands) are obtained in weakly to moderately interacting regime. Further, for a given interaction, the split in low-lying eigenenergies with increasing
L is the precursor to spontaneous symmetry breaking of the axisymmetry associated with the entry of the first vortex. With increase in repulsive interaction, the value of the first breathing mode increases for stable total angular momentum states
L=0 and
N, but decreases for intermediate
0<L<N metastable states. The position of the observed first breathing modes in the eigenspectrum remains unchanged as the interaction is varied over several orders of magnitude.
Abstract We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum L z is carried out to obtain the N -body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ( N = 2) to many-body ( N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson ( N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N . For a given N , there is found an optimal value of the range of Gaussian potential for which the first vortex (with L z = N ) nucleates at a lower value of the rotational angular velocity Ω c1 compared to the zero-range ( δ -function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.
Abstract Using exact diagonalization, we investigate the many-body ground state for regular vortex patterns in a rotating Bose-condensed gas of N spinless particles, confined in a quasi-two-dimensional harmonic trap and interacting repulsively via finite-range Gaussian potential. The N -body Hamiltonian matrix is diagonalized in given subspaces of quantized total angular momentum L z , to obtain the lowest-energy eigenstate. Further, the internal structure of these eigenstates is analyzed by calculating the corresponding conditional probability distribution. Specifically, the quantum mechanically stable as well as unstable states in a co-rotating frame are examined in the moderately rotating regime corresponding to angular momenta 4 N ≤ L z < 5 N for N = 16 bosons. In response to externally impressed rotation, the patterns of singly quantized vortices are formed, shaping into canonical polygons with a central vortex at the trap center. The internal structure of unstable states reveals the mechanism of entry, nucleation and pattern formation of vortices with structural phase transition, as the condensate goes from one stable vortical state to the other. The stable polygonal vortex patterns having discrete p -fold rotational symmetry with p = 5 and p = 6 are observed. The hexagonal vortex pattern with p = 6 symmetry is a precursor to the triangular vortex lattice of singly quantized vortices in the thermodynamic limit. For unstable states, quantum melting of vortex patterns due to uncertainty in positions of individual vortices, is also briefly discussed.
We investigate a rotating system of
N spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large
s-wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta
Lz is carried out to obtain the
N-body ground states in slow rotating regime of
0≤Lz≤2N. We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized
Lz. The study ranges from the few-body (
N=2) system to the many-body (
N=16), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different
Lz states as well on the critical angular velocity
Ωc1 of first vortex (
Lz=N) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems (
N=2 to
8) where an understanding of few-body effects provide valuable insight into large systems (
N=8 to
16) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.
We present exact diagonalization study on a system of
10≤N≤24 spinless bosons interacting via repulsive Gaussian potential, harmonically confined in
xy-plane with an externally impressed rotation about the
z-axis. The two-body interaction strength in the Gaussian potential is taken in the strongly interacting regime with values of interaction range in the regime
0≤σ≤1. The diagonalization of the
N-body Hamiltonian matrix, in subspaces of total angular momentum in the regime
0≤Lz≤4N corresponds to the filling fraction
ν≲3.2 is carried out to obtain the variationally exact ground-state wavefunction and the corresponding eigenvalue. It is found that an increase in interaction range
σ leads to (a) a systematic decrease in energy, (b) an increase in the critical angular velocity
Ωci of the
i{th} vortex state and (c) an increase in the largest eigenvalue
λ1, (condensate fraction), of one-particle reduced density matrix (OPRDM). The von Neumann entropy
S1(Lz,σ), quantifying the quantum entanglement between the particles in the many-body ground state, is largely found to decrease with increase in
σ. Crossings in von Neumann entropy for several of the angular momentum states are observed with variation in
σ. The response of the Bose-Einstein condensate to rotation is examined through
Lz(σ)−Ω(σ) stability graph for several values of
σ. A vortex state with larger plateau length on the
Lz−Ω graph is considered to be more stable. The internal structure of the condensate, as depicted by the conditional probability distribution (CPD) in the body-fixed frame, exhibits characteristic features with interaction range
σ. One such feature is the merging of the cores of the two-vortex state.
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