Ultracold quantum gases dressed by laser light are a powerful tool to investigate the dynamics of quantum mechanical Hamiltonians. When a Bose-Einstein condensate is placed in appropriately tailored laser fields, spin-orbit coupling as well as periodic band structures can be generated. We have developed an effective band structure picture to describe such Bose-Einstein condensates in a spin-orbit coupled lattice. Our model predicts the location of the resulting band edges. By performing atom loss measurements, we experimentally probe the predicted band structures and find excellent agreement between our experiment and our theoretical predictions.
Ultracold quantum gases dressed by laser light are a powerful tool to investigate the dynamics of quantum mechanical Hamiltonians. When a Bose-Einstein condensate is placed in appropriately tailored laser fields, spin-orbit coupling as well as periodic band structures can be generated. We have developed an effective band structure picture to describe such Bose-Einstein condensates in a spin-orbit coupled lattice. Our model predicts the location of the resulting band edges. By performing atom loss measurements, we experimentally probe the predicted band structures and find excellent agreement between our experiment and our theoretical predictions.
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.
In a recent experiment (Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, arXiv:1511.08170 [cond-mat.quant-gas]), a Raman-assisted two-dimensional spin-orbit coupling has been realized for a Bose-Einstein condensate in an optical lattice potential. In light of this exciting progress, we study in detail key properties of the system. As the Raman lasers inevitably couple atoms to high-lying bands, the behaviors of the system in both the single- and many-particle sectors are significantly affected. In particular, the high-band effects enhance the plane-wave phase and lead to the emergence of ``roton'' gaps at low Zeeman fields. Furthermore, we identify high-band-induced topological phase boundaries in both the single-particle and the quasiparticle spectra. We then derive an effective two-band model, which captures the high-band physics in the experimentally relevant regime. Our results not only offer valuable insights into the two-dimensional lattice spin-orbit coupling, but also provide a systematic formalism to model high-band effects in lattice systems with Raman-assisted spin-orbit couplings.
The experimental and theoretical research of spin–orbit-coupled ultracold atomic gases has advanced and expanded rapidly in recent years. Here, we review some of the progress that either was pioneered by our own work, has helped to lay the foundation, or has developed new and relevant techniques. After examining the experimental accessibility of all relevant spin–orbit coupling parameters, we discuss the fundamental properties and general applications of spin–orbit-coupled Bose–Einstein condensates (BECs) over a wide range of physical situations. For the harmonically trapped case, we show that the ground state phase transition is a Dicke-type process and that spin–orbit-coupled BECs provide a unique platform to simulate and study the Dicke model and Dicke phase transitions. For a homogeneous BEC, we discuss the collective excitations, which have been observed experimentally using Bragg spectroscopy. They feature a roton-like minimum, the softening of which provides a potential mechanism to understand the ground state phase transition. On the other hand, if the collective dynamics are excited by a sudden quenching of the spin–orbit coupling parameters, we show that the resulting collective dynamics can be related to the famous Zitterbewegung in the relativistic realm. Finally, we discuss the case of a BEC loaded into a periodic optical potential. Here, the spin–orbit coupling generates isolated flat bands within the lowest Bloch bands whereas the nonlinearity of the system leads to dynamical instabilities of these Bloch waves. The experimental verification of this instability illustrates the lack of Galilean invariance in the system.
Since the first experimental realization of Bose-Einstein condensates in a spin-orbit-coupled Zeeman lattice, a wide range of applications have been found in these systems. Here, we systematically study the ground-state phase diagram of the systems. The spin-orbit-coupled Zeeman lattice generates multiple energy minima in the lowest Bloch band. These minima may locate at Brillouin-zone edges and center, or inside the Brillouin zone. Interacting atoms condensing into these minima constitute ground-state quantum phases. Among these phases, the band-edge condensation states have no analog in either spin-orbit-coupled homogeneous systems or lattice systems without the spin-orbit coupling. They exist in a very broad parameter regime of the spin-orbit-coupled Zeeman lattice system. We address superfluidity of the band-edge states by analyzing their elementary excitations and superfluid fraction.
The quantum emulation of spin-momentum coupling, a crucial ingredient for the emergence of topological phases, is currently drawing considerable interest. In previous quantum gas experiments, typically two atomic hyperfine states were chosen as pseudospins. Here, we report the observation of a spin-momentum coupling achieved by loading a Bose-Einstein condensate into periodically driven optical lattices. The s and p bands of a static lattice, which act as pseudospins, are coupled through an additional moving lattice that induces a momentum-dependent coupling between the two pseudospins, resulting in s-p hybrid Floquet-Bloch bands. We investigate the band structures by measuring the quasimomentum of the Bose-Einstein condensate for different velocities and strengths of the moving lattice, and compare our measurements to theoretical predictions. The realization of spin-momentum coupling with lattice bands as pseudospins paves the way for engineering novel quantum matter using hybrid orbital bands.