Spin-orbit-coupled bosons interacting in a two-dimensional harmonic trap

Abstract

A system of bosons in a two-dimensional harmonic trap with the presence of Rashba-type spinorbit coupling is investigated. An analytic treatment of the ground state of a single atom in the weak-coupling regime is presented and used as a basis for a perturbation theory in the interacting two-boson system. The numerical diagonalization of both the single-particle and the two-boson Hamiltonian matrices allows us to go beyond those approximations and obtain not only the ground state, but also the low-energy spectra and the different energy contributions separately. We show that the expectation value of the spin-orbit term is related to the expectation value ofσzLz for the eigenstates of the system, regardless of the trapping potential. The ground state of the repulsively interacting two-boson system is characterized. With the presence of a sufficiently strong interaction and spin-orbit coupling strength, there is an energy-level crossing in the ground state of the system that changes its structure. This is reflected in a discontinuity in the different energy terms and it is signaled in the spatial density of the system.

Direct diagonalization method for a few particles trapped in harmonic potentials

“…The solid lines correspond to profiles obtained with corrected direct diagonalization calculations with 20 single-particle modes, whereas the black dash-dotted lines are exact results given by Eq. (19). The inset shows a correlation parameter Sρ between the exact profiles (19) and the profiles obtained with the truncated diagonalizations (squares with dotted lines) and the corrected ones (circles with solid lines) as a function of the number of single-particle modes M .…”

“…(19). The inset shows a correlation parameter Sρ between the exact profiles (19) and the profiles obtained with the truncated diagonalizations (squares with dotted lines) and the corrected ones (circles with solid lines) as a function of the number of single-particle modes M .…”

“…Then, 2) truncate the single-particle basis to M modes, and 3) build the corresponding many-body basis performing a truncation on the total energy of the non-interacting many-body states [15]. This technique has been recently used to study small 1D bosonic mixtures [16,17], fermionic systems [14], 2D bosonic systems with and without spin-orbit coupling [18,19]. Even though direct diagonalization calculations are limited to a small number of particles, they offer several advantages compared to other approaches.…”

On the direct diagonalization method for a few particles trapped in harmonic potentials

“…( 19). The inset shows a correlation parameter Sρ between the exact profiles (19) and the profiles obtained with the truncated diagonalizations (crosses with dashed lines) and the corrected ones (circles with solid lines) as a function of the number of single-particle modes M . ization calculation for g → +∞, the resulting energy is greater than the Tonks solution for infinite repulsion [14].…”

Spin–orbit coupling in the presence of strong atomic correlations

“…While solving large many-body systems with approaches beyond mean-field is a very difficult task and only possible in special cases [12][13][14], few-particle systems can actually be amenable to exact treatments across the whole range of interactions and correlation strengths [15][16][17][18][19]. Several treatments of SOC in such systems have already been carried out [20][21][22][23][24][25][26] and, for example, a mapping to an effective spin model was recently suggested by a perturbative approach to account for weak Raman coupling [21]. It was also shown that, while there is no entanglement in the mean-field regime, in two-particle systems the ground state can be maximally entangled in the pseudo-spin space [23].…”