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Chinese Physics B

Energy band structure of spin-1 condensates in optical lattices

Zhi Li, Ai‐Xia Zhang, Juan Ma, Xue Ju-Kui

Published Oct 1, 2010• 0 citations
Publisher

Abstract

The energy band structure of spin-1 condensates with repulsive spin-independent and either ferromagnetic or antiferromagnetic spin-dependent interactions in one-dimensional (1D) periodic optical lattices is discussed. Within the two-mode approximation, Bloch bands of spin-1 condensates are presented. The results show that the Bloch bands exhibit a complex structure as the atom density of mF = 0 hyperfine state increases: bands splitting, reversion, intersection and loop structure are excited subsequently. The complex band structure should be related to the tunneling and spin-mixing dynamics.

DOI
10.1088/1674-1056/19/10/100306

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The energy band structure of spin-1 condensates with repulsive spin-independent and either ferromagnetic or antiferromagnetic spin-dependent interactions in one-dimensional (1D) periodic optical lattices is discussed. Within the two-mode approximation, Bloch bands of spin-1 condensates are presented. The results show that the Bloch bands exhibit a complex structure as the atom density of mF = 0 hyperfine state increases: bands splitting, reversion, intersection and loop structure are excited subsequently. The complex band structure should be related to the tunneling and spin-mixing dynamics.
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Abstract Employing the variational principle and the two-mode approximation, we have carried out a study on the nonlinear energy spectrum and current density of spin-1 spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) in a one-dimensional shallow optical lattice. The energy spectrum structure diagrams in the momentum space are given. When certain conditions for the system parameters are satisfied, we find that the loop structure appears at the boundary of the Brillouin zone. Furthermore, we analyze the influence of the parameters in the system on the critical condition and width of the loop structure. It is revealed that the optical lattice and Raman coupling can suppress the emergence of the loop. On the contrary, the quadratic Zeeman strength, spin-orbit-coupling and atomic interactions are favorable for the generation of the loop, which leads to the complication of the energy band structure. Intriguingly, the change of the band structure is closely related to the current density distribution of the condensates. The spin-orbit-coupling leads to a strong asymmetry in the current density distribution of different spin states in the momentum space, so as to cause a separation phenomenon. However, the optical lattice and Raman coupling weaken this asymmetry and promote the current density distribution of different spin states to re-coincidence. Our results provide a theoretical basis for further understanding the nonlinear energy spectrum characteristics of this system.
Energy band structure and stability of Bose-Einstein condensate with both two- and three-body interactions in one-dimensional periodic optical lattices are discussed. It is demonstrated that the lowest energy band can show a loop structure at the boundary of the first Brillouin zone only when the number of atoms is larger than a critical value. When the loop structure is excited, an analytical expression of the loop width is obtained. The energetical and dynamical stabilities of the system are determined by considering the behavior of the Gross-Pitaevskii energy functional when the condensate wave function diverge away from a stationary state upon small perturbation, and its dependence on the number of atoms, the magnitude of the periodic potential, especially, the strength of the three-body force are studied. In addition, a criteria of the stability for excitations with long wavelengths is given by applying the hydrodynamic approach. The influence of the three-body interaction on the band properties and the stabilities are discussed in detail. It is shown that the critical density for developing the loop structure, the loop width, and the instability criteria are modified dramatically by the three-body interaction.
We discuss the Bloch-state solutions of the stationary Gross-Pitaevskii equation and of the Bogoliubov equations for a Bose-Einstein condensate in the presence of a one-dimensional optical lattice. The results for the compressibility, effective mass and velocity of sound are analysed as a function of the lattice depth and of the strength of the two-body interaction. The band structure of the spectrum of elementary excitations is compared with the one exhibited by the stationary solutions (Bloch bands). Moreover, the numerical calculations are compared with the analytic predictions of the tight binding approximation. We also discuss the role of quantum fluctuations and show that the condensate exhibits 3D, 2D or 1D features depending on the lattice depth and on the number of particles occupying each potential well. We finally show how, using a local density approximation, our results can be applied to study the behaviour of the gas in the presence of harmonic trapping.