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arXiv:1112.3419 · Physical Review A

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

Zhihao Xu, Shu Chen

Published Mar 5, 2012• 4 citations
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Abstract

We demonstrate that the static structure factor, momentum distribution, and density distribution provide clear signatures of the emergence of Wigner crystal for the fermionic dipolar gas with strongly repulsive dipole-dipole interactions trapped in one-dimensional optical lattices. Our numerical evidences are based on the exact diagonalization of the microscopic effective lattice Hamiltonian of few particles interacting with long-range interactions. As a comparison, we also study the system with only nearest-neighbor interactions, which displays quite different behavior from the dipolar system in the regime of strong repulsion.

DOI
10.1103/physreva.85.033606

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We demonstrate that the static structure factor, momentum distribution, and density distribution provide clear signatures of the emergence of Wigner crystal for the fermionic dipolar gas with strongly repulsive dipole-dipole interactions trapped in one-dimensional optical lattices. Our numerical evidences are based on the exact diagonalization of the microscopic effective lattice Hamiltonian of few particles interacting with long-range interactions. As a comparison, we also study the system with only nearest-neighbor interactions, which displays quite different behavior from the dipolar system in the regime of strong repulsion.
The ground state properties of the Hubbard model with or without long-range interactions in the regime with strongly repulsive on-site interactions are investigated by means of the exact diagonalization method. We show that the appearance of N-crests in the density profile of a trapped N-fermion system is a natural result of 'fermionization' between antiparallel-spin fermions in the strongly repulsive limit and cannot be taken as the only signature of the Wigner crystal phase, as the static structure factor does not show any signature of crystallization. In contrast, both the density distribution and the static structure factor of the Hubbard model with strong long-range interactions display clear signatures of the Wigner crystal phase. Our results indicate the important role of long-range interaction in the formation of the Wigner crystal phase.
We study, by means of exact-diagonalization techniques, the ground state of a few-fermion system with strong short-range repulsive interactions trapped by a harmonic potential in one spatial dimension. Even when the ground-state density profile displays, at strong coupling, very well pronounced Wigner oscillations with a 4kF4{k}_{\mathrm{F}}4kF​ periodicity, the pair-correlation function does not show any signature of Wigner-molecule-type correlations. For the sake of comparison, we present also numerical results for few-electron systems with Coulomb interactions, demonstrating that their ground state at strong coupling is, on the contrary, a Wigner molecule.
We study the ground state of few bosons with repulsive dipole-dipole interaction in a quasi-one-dimensional harmonic trap by means of the exact diagonalization method. Up to three interaction regimes are found, depending on the strength of the dipolar interaction and the ratio of transverse to axial oscillator lengths: a regime where the dipolar Bose gas resembles a system of weakly $\ensuremath{\delta}$-interacting bosons, a second regime where the bosons are fermionized, and a third regime where the bosons form a Wigner crystal. In the first two regimes, the dipole-dipole potential can be replaced by a $\ensuremath{\delta}$ potential. In the crystalline state, the overlap between the localized wave packets is strongly reduced and all the properties of the boson system equal those of its fermionic counterpart. The transition from the Tonks-Girardeau gas to the solidlike state is accompanied by a rapid increase of the interaction energy and a considerable change of the momentum distribution, which we trace back to the different short-range correlations in the two interaction regimes.
We study the role of the dipolar-induced resonance (DIR) in a quasi-one-dimensional system of ultracold bosons. We first describe the effect of the DIR on two particles in a harmonic trap. Then, we consider a deep optical lattice loaded with ultracold dipolar bosons. In order to describe this system, we introduce a novel atom-dimer extended Bose-Hubbard model, which is the minimal model correctly accounting for the DIR. We analyze the impact of the DIR on the phase diagram at T=0T=0T=0 by exact diagonalization of a small-sized system. We show that the DIR strongly affects this phase diagram. In particular, we predict the mass density wave to occur in a narrow domain corresponding to weak nearest-neighbor interactions, and we predict the occurrence of a collapse phase for stronger dipolar interactions.
We explore the ground-state properties of a few bosons with dipole-dipole interactions in a one-dimensional optical lattice. For comparatively strong interactions, a transition from a Mott-insulator state to a crystal state occurs. Herein, we provide a detailed characterization and a detection protocol for the resulting crystal state. Using the eigenvalues of the reduced one-body density matrix, we define an order parameter that characterizes all the emergent states occurring in the finite-sized ensemble and agrees with the signatures obtained from the analysis of the one- and two-body densities. We further demonstrate that these few-body emergent states can be detected experimentally using the variance of single-shot measurements.
Strongly interacting finite ensembles of dipolar bosons in commensurately filled one-dimensional optical lattices exhibit diverse quantum phases that are rich in physics. As the strength of the long-range boson–boson interaction increases, the system transitions across different phases: from a superfluid, through a Mott-insulator and a Tonks–Girardeau gas to a crystal state. The signature of these phases and their transitions can be unequivocally identified by an experimentally detectable order parameter, recently described in Phys. Rev. A 98 235301 (2018 [ 33 ]). Herein, we calculate the momentum distributions and the normalized Glauber correlation functions of dipolar bosons in a one-dimensional optical lattice in order to characterize all their phases. To understand the behavior of the correlations across the phase transitions, we first investigate the eigenfunctions and eigenvalues of the one-body reduced density matrix as a function of the dipolar interaction strength. We then analyze the real- and momentum-space Glauber correlation functions, thereby gaining a spatially and momentum-resolved insight into the coherence properties of these quantum phases. We find an intriguing structure of non-local correlations that, independently of other observables, reveal the phase transitions of the system. In particular, spatial localization and momentum delocalization accompany the formation of correlated islands in the density as interactions become stronger. Our study showcases that precise control of intersite correlations is possible through the manipulation of the depth of the lattice, while intrasite correlations can be influenced by changing the dipolar interaction strength.