Quantum correlations and degeneracy of identical bosons in a two-dimensional harmonic trap

Abstract

We consider a few number of identical bosons trapped in a 2D isotropic harmonic potential and also the N -boson system when it is feasible. The atom-atom interaction is modelled by means of a finite-range Gaussian interaction. The spectral properties of the system are scrutinized, in particular, we derive analytic expressions for the degeneracies and their breaking for the lower-energy states at small but finite interactions. We demonstrate that the degeneracy of the low-energy states is independent of the number of particles in the noninteracting limit and also for sufficiently weak interactions. In the strongly interacting regime, we show how the many-body wave function develops holes whenever two particles are at the same position in space to avoid the interaction, a mechanism reminiscent of the Tonks-Girardeau gas in 1D. The evolution of the system as the interaction is increased is studied by means of the density profiles, pair correlations and fragmentation of the ground state for N = 2, 3, and 4 bosons.

Energy-level splitting for weakly interacting bosons in a harmonic trap

“…Eigenvalues outside this kernel in a given (N, E, M )-block always descend through the action of Z ± and W from the kernel eigenvalues of a lower (N, E, M )-block. With this picture in mind, we have plotted the unfolded level spacing distribution in the (Z † ± , W † )kernel of the (N, E, M ) = (7,22,16) block, given in Fig. 3.…”

“…Energy levels of quantum identical interacting bosons in harmonic traps have often been studied in the literature [1][2][3][4][5][6][7][8][9][10], motivated in particular by the physics of cold atomic gases. Contact interactions between the bosons commonly appear in such studies as the simplest possible choice for the twoparticle potential that is expected to retain realistic features.…”

Are smooth pseudopotentials a good choice for representing short-range interactions?

“…A straightforward and generalist approach to representing the many-body problem for computational treatment is to introduce a discrete and necessarily finite basis of smooth single-particle wave functions from which a finite but still potentially very large Fock-space is constructed to represent the many-body Hamiltonian as a matrix. Finding eigenstates and eigenvalues of the full matrix is known as exact diagonalization or full * jeszenszki.peter@gmail.com † A.Alavi@fkf.mpg.de ‡ J.Brand@massey.ac.nz configuration-interaction [22][23][24][25][26], but many different approximation schemes have also been followed [27]. In particular, standard approaches of ab initio quantum chemistry or nuclear physics like the coupled-cluster [28] or multi configurational self-consistent field theory [29] all can be formulated in this language as they rely on an underlying single-particle basis.…”

Path integral molecular dynamics for bosons

“…We thank P. Mujal for sharing his raw data from ref. [22] with us. BH would like to thank O. Buchman, J. Runeson and M. Nava for useful conversations and comments.…”

“…In the following, we first present the method and benchmark it against analytical results for up to 64 noninteracting Bosons and numerical diagonalization of the Hamiltonian for small interacting systems [22]. We also apply the method to 32 interacting particles in a 2D trap, for which the number of permutations to be considered is already prohibitive.…”