Erratum: Ground-state properties of few dipolar bosons in a quasi-one-dimensional harmonic trap [Phys. Rev. A81, 063616 (2010)]

Abstract

In the Appendix of our paper, we derive the effective onedimensional (1D) dipole-dipole interaction (DDI) potential. The derivation and the result are incomplete, since the singularity of the three-dimensional (3D) DDI at zero distance is not properly accounted for. To see this, we introduce a small > 0 in the denominator of the 3D DDI. The first steps of the calculation are analogous to those shown in the Appendix until one arrives at 1with w = ρ/ l ⊥ and u = x/ l ⊥ . This is equal towith v = √ u 2 + 2 andIn the limit → 0, one finally obtainsi.e., apart from the second term in square brackets, which is already given in the paper, there is an additional δ term present in the 1D DDI. This term is missing in Eqs. (2)-(4) of our paper. Let us briefly show that lim →0 ∞ −∞ du δ (u) = 1. One finds thatThe third step follows from the substitution t = (w 2 + 2 )/2, the fourth step from an integration by parts, and the last step from Ei(x) ≈ ln(x) and x 2 ln(x) → 0 for x → 0, where Ei(x) is the exponential integral. It can be seen from Eq. (4) that the strength of the δ interaction depends on the dipole orientation with respect to the weak trap axis (the angle θ ). This is different from the δ contribution, which stems from the point limit of a real (extended) dipole [1], where the strength depends on the relative orientation of the two interacting dipoles. In total there are three δ terms, which originate from the van der Waals interaction, the point limit of a real dipole, and the integration over the transverse directions.In our paper, we study the effect of the 1D DDI without the δ terms and perform a sweep of the interaction strength. Such a sweep can be performed experimentally, when the strength of all δ terms is tuned to zero by means of a Feshbach resonance. Then, the strength of the interaction term in square brackets with respect to the level spacing can be tuned by changing the axial angular frequency, as described at the end of Sec. III.

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

“…There are many works studying fermionization in 1D, for instance, in optical lattices [8], in few-atom mixtures [9][10][11], for attractive interactions [12] and for few dipolar bosons [13]. In other cases, the focus are quantum correlations [14,15], its effects in mixtures of distinguishable and identical particles [16] and analytic ansatz to capture the physics in all interaction regimes [17].…”

Dipolar-induced resonance for ultracold bosons in a quasi-one-dimensional optical lattice

“…Figure 2(bottom) shows the r * -dependence of the lowest eigenvalues of H 2B in two different situations: (i) the s-wave interaction term g 1D = 0 and (ii) g 1D = 0 cancels the contact term in Eq. (1) completely [17]. The r * -dependence of the energy levels in these two situations is completely different.…”

“…Trapping a dipolar system into lower dimensions stabilizes it with respect to two-body [11] and many-body [12] instabilities caused by the attractive part of the 3D DDI. This has prompted detailed studies of dipolar systems in 2D and quasi-2D [13,14], bilayer [15], and quasi-1D [16][17][18][19] geometries.…”

“…In our quasi-1D geometry, we model the DDI using an effective potential obtained by averaging the transverse degrees of freedom into the harmonic-oscillator ground state [16,17]: interaction for a 3D scattering length a 3D [35], which can be manipulated using a Feshbach resonance [36]. It competes with the DDI to determine the stability and the phase of the system [37].…”

Wigner crystal induced by dipole-dipole interaction in one-dimensional optical lattices

“…In the single-mode approximation with only the transversal ground state considered, we can integrate over transversal directions and get the effective 1D DDI given by [19,22]. The Hamiltonian for dipolar fermions in optical lattices is given by…”