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Journal of Physics B Atomic Molecular and Optical Physics

Band structure and tunnelling dynamics of Bose–Einstein condensates in Fourier-synthesized optical lattices

Yan Chen, Heng-Tong Wang, Yong Chen

Published Oct 29, 2010• 2 citations
Publisher

Abstract

By employing a nonlinear three-mode model, we study the band structure of Bose-Einstein condensates in Fourier-synthesized optical lattices, where the nonlinearity comes from the mean field treatment of interaction between atoms. In the linear case, we present the band structure of the system. It is observed that the energy band structure is strongly dependent on the relative phase between the two lattice harmonics. In the nonlinear case, we show that the eigenenergy as a function of energy bias has a striking bowl structure in the middle energy level. It is found that there exist four critical values of interaction strength where the band structure undergoes interesting changes. Because of these topological distortions on the energy levels, we expect that the tunnelling dynamics make a dramatic change. The tunnelling probability is nonzero when the nonlinear term is very small and the energy levels keep the same topological structure as that in the linear case. Especially, the tunnelling probability as a function of the sweeping rate shows an irregular oscillation for the state on the middle level.

DOI
10.1088/0953-4075/43/22/225303

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By employing a nonlinear three-mode model, we study the band structure of Bose-Einstein condensates in Fourier-synthesized optical lattices, where the nonlinearity comes from the mean field treatment of interaction between atoms. In the linear case, we present the band structure of the system. It is observed that the energy band structure is strongly dependent on the relative phase between the two lattice harmonics. In the nonlinear case, we show that the eigenenergy as a function of energy bias has a striking bowl structure in the middle energy level. It is found that there exist four critical values of interaction strength where the band structure undergoes interesting changes. Because of these topological distortions on the energy levels, we expect that the tunnelling dynamics make a dramatic change. The tunnelling probability is nonzero when the nonlinear term is very small and the energy levels keep the same topological structure as that in the linear case. Especially, the tunnelling probability as a function of the sweeping rate shows an irregular oscillation for the state on the middle level.
The nonlinear Landau—Zener tunneling and nonlinear Rabi oscillations of Bose—Einstein condensate (BEC) with higher-order atomic interaction between the Bloch bands in an accelerating optical lattice are discussed. Within the two-level model, the tunneling probability of BEC with higher-order atomic interaction between Bloch bands is obtained. We finds that the tunneling rate is closely related to the higher-order atomic interaction. Furthermore, the nonlinear Rabi oscillations of BEC with higher-order atomic interaction between the bands are discussed by imposing a periodic modulation on the level bias. Analytical expressions of the critical higher-order atomic interaction for suppressing/enhancing the Rabi oscillations are obtained. It is shown that the critical value strongly depends on the modulation parameters (i.e., the modulation amplitude and frequency) and the strength of periodic potential.
We study the tunneling dynamics and energy bands of three Bose-Einstein condensates which are coupled weakly with each other. The study is carried out with both the mean-filed model and the second-quantized model. The results from these two models are compared and found to agree with each other when the particle number is large. Without interaction, this system possesses a Dirac point in its energy band. This Dirac point is immediately destroyed and develops into a loop structure with arbitrary small interaction. This loop structure has a strong effect on the tunneling dynamics. We find that the tunneling dynamics in this system is very sensitive to the system parameter, e.g., the interaction strength. This sensitivity is found to be caused by the chaos in the mean-field model and the avoided energy crossings with tiny gaps in the second-quantized model. This result gives a certain indication on how the classical dynamics and quantum dynamics are connected in the semi-classical limit. Our mean-field results are also valid for three mutually coupled optical nonlinear waveguides.
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 theoretically investigate the energy band structure, nonlinear Landau–Zener tunneling dynamics and tunneling probability of spin-orbit coupled Bose–Einstein condensates in a one-dimensional accelerating optical lattice by using mean-field and two-mode approximation. The critical condition for the appearance of the loop structure is obtained numerically in parameter planes. When the intraspecies atomic interaction is less than the interspecies atomic interaction, the loop only appears in the lower band. In this case, Raman coupling inhibits the appearance of loop, while spin-orbit coupling (SOC) promotes the appearance of loop. If the intraspecies atomic interaction is larger than interspecies atomic interaction, the loop can appear in either the upper band or the lower band. In this case, Raman coupling promotes the loop appearing in the lower band, while SOC suppresses the loop appearing in upper band. Interestingly, when the interspecies atomic interaction is equal to the intraspecies atomic interaction, there is a critical atomic intercation value determined by the optical lattice depth, only when the intraspecies atomic interaction is greater than the critical value, the loop will occur only in lower band. Especially, the emergence of the loop structure destroys the Bloch oscillation of the system and results in the nonlinear Landau–Zener tunneling of the system. Furthermore, the Landau–Zener tunneling probability of the system is calculated, and it is found that the nonlinear Landau–Zener dynamics and the tunneling probability can be manipulated by SOC and Raman coupling.
In a recent experiment [Hamner C, et al. 2015 Phys. Rev. Lett. 114 070401 ], spin-orbit coupled Bose-Einstein condensates in a translating optical lattice have been successfully prepared into any Bloch band, and directly proved to be the lack of Galilean invariance in the presence of the spin-orbit coupling. The energy band structure of the system becomes complicated because of the lack of Galilean invariance. At present, the energy band structure of the spin-orbit coupled Bose-Einstein condensates in optical lattice is still an open issue, especially the theoretical evidence for the in-depth understanding of the competition mechanism among the spin-orbit coupling, the Raman coupling, the optical lattice and the atomic interactions of the nonlinear energy band structure has not been clear yet. In this paper, based on the two-mode approximation and variational analysis, the nonlinear energy band structure and current density of the spin-orbit coupled Bose-Einstein condensates in the one-dimensional optical lattice are investigated. We find that when the spin-orbit coupling, the Raman coupling, the optical lattice, and the atomic interactions satisfy certain conditions, a loop structure in the Brillouin zone edge will emerge. The critical condition for the loop structure emerging in the Brillouin zone edge is obtained in a parameter space. The Raman coupling and the optical lattice suppress the emergence of the loop structure, while the spin-orbit coupling and the atomic interactions promote the emerging of the loop structure and make the energy band structure more complex. Interestingly, the atomic interactions can make the loop structure occur at both the higher-lying bands and the lowest energy band. The energy band structure is closely related to the current density of the system. The spin-orbit coupling causes the current density to be strongly asymmetric and leads the current density distributions of different spin states to be separated from each other in the momentum space near the boundary of the Brillouin zone. The optical lattice strength and the Raman coupling can weaken the asymmetry. The appearance of loop structure breaks the Bloch oscillation and gives rise to the Landau-Zener tunneling. The separation of the current density distributions of different spin states in the momentum space means the emergence of the spin exchange dynamics. Our results are beneficial to the in-depth understanding of the nonlinear dynamics of the spin-orbit coupled Bose-Einstein condensates in optical lattice.
We investigate the Landau–Zener tunnelling of two-component Bose-Einstein condensates (BECs) in optical lattices. In the neighborhood of the Brillouin zone edge, the system can be reduced to two coupled nonlinear two-level models. From the models, we calculate the change of the tunnelling probability for each component with the linear sweeping rate. It is found that the probability for each component exhibits regular oscillating behavior for the larger sweeping rate, but for smaller rate the oscillation is irregular. Moreover, the asymmetry of the tunnelling between the two components can be induced by the unbalanced initial populations or the inequality of the two self-interactions when the cross-interaction between the components exists. The result can not be found in the single component BECs.