Harmonically trapped atoms with spin–orbit coupling

Abstract

We study harmonically trapped one-dimensional atoms subjected to an equal combination of Rashba and Dresselhaus spin-orbit coupling induced by Raman transition. We first examine the wave function and the degeneracy of the single-particle ground state, followed by a study of two weakly interacting bosons or fermions. For the two-particle ground state, we focus on the effects of the interaction on the degeneracy, the spin density profiles, and the density-density correlation functions. Finally we show how these studies help us to understand the many-body properties of the system.

Collective excitation of a trapped Bose-Einstein condensate with spin-orbit coupling

“…For strong inter-spin interaction with g ↑↓ ≥ g, the system is in the PW (ZM) phase if Ω < Ω P −Z C (Ω > Ω P −Z C ) and the ST phase is absent. The critical value Ω P −Z C = 4E L for a homogeneous system, and is slightly down shifted by the trap [19]. For weaker inter-spin interaction with g ↑↓ < g, the ST phase is also present at small Raman coupling strength Ω < Ω S−P C , the PW phase exists when Ω S−P C < Ω < Ω P −Z C , and the ZM phase remains at large Raman coupling strength when Ω > Ω P −Z C .…”

“…This leads to three mean-field BEC phases: For Ω > 4E L , all atoms condense to the zero momentum state and hence this phase is termed as zero-momentum phase (ZM); for Ω < 4E L , depending on the interaction strength, we may have all the atoms condense to one of the degenerate single-particle ground states and we have the plane-wave phase (PW); or the atoms can condense to an equal-weight superposition of the two degenerate single-particle ground states and we have the stripe phase (ST), as both spin components exhibit density stripes. In the presence of a weak harmonic trap, even though momentum is no longer a good quantum number, the analog of all the three phases can still be easily identified, and the main effects of the trap is to provide an overall envelop for the atomic density and the boundaries between different phases are shifted [19].…”

Spin–orbit coupling in the presence of strong atomic correlations

“…and it is known numerically that for harmonically trapped systems the value of the critical point is lower [23].…”

“…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].…”

“…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].…”