Ground state properties of trapped boson system with finite-range Gaussian repulsion: exact diagonalization study

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 Lz 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 Lz = N) nucleates at a lower value of the rotational angular velocity Ωc1 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.

Novel phases in rotating Bose condensed gas: vortices and quantum correlation

“…The first two terms in the Hamiltonian (1) correspond to the kinetic and potential energies respectively. The third term U (r i , r j ) arises from the atom-atom interaction, is modelled by the Gaussian potential with parameter σ (scaled by a r ) being the effective range of the potential [52,[58][59][60][61][62]]…”

“…For a given value of σ, the s-wave scattering length in the parameter g 2 = 4πa s /a r is adjusted in such a way that N a s /a r becomes relevant to the experimental value [4,36]. In addition to being physically more realistic, the finite-range Gaussian potential (2) is expandable within a finite subspace of single-particle basis functions [63] and hence computationally more feasible compared to the zero-range δ-function potential [52,[58][59][60].…”

“…An interesting and useful measure of the quantum phase correlation in the many-body ground state of a confined system is provided by the von Neumann entanglement entropy [53][54][55][56][57][58], an extension to the classical Gibbs entropy, defined as…”

“…Appendix C: Degree of condensation It is to be noted that the usual definition of condensation for a macroscopic system, given by the largest eigenvalue λ 1 of the SPRDM, is not appropriate for systems with small number of particles being studied here [45,58,61]. For example, in the absence of condensation, there is no macroscopic occupation of a single quantum state and all levels are equally occupied, such a definition would imply a condensate though with small magnitude.…”

“…[48][49][50][51][52]. To examine quantum mechanical phase coherence of the Bose-condensed gas, recent studies has been elucidated that the quantum entanglement in terms of von Neumann entropy, can be used as a probe to analyze the novel phases of many-body quantum states [52][53][54][55][56][57][58].…”