Abstract Periodic band structures are a hallmark phenomenon of condensed matter physics. While often imposed by external potentials, periodicity can also arise through the interplay of couplings that are not necessarily spatially periodic on their own, but this option is generally less explored than the fully-periodic counterpart. Here, we investigate dynamics in a lattice structure that emerges from the simultaneous application of Raman and radio frequency coupling to a dilute-gas Bose-Einstein condensate. We elaborate on the role of Galilean invariance in this system and demonstrate a variety of techniques, including Bloch oscillations and lattice shaking with spin and momentum resolved measurements. This combined coupling scheme allows for tunability and control, enabling future investigations into unconventional band structures such as quasi-flat ground bands and those with semimetal-like band gaps.
Abstract Periodic band structures are a hallmark phenomenon of condensed matter physics. While often imposed by external potentials, periodicity can also arise through the interplay of couplings that are not necessarily spatially periodic on their own, but this option is generally less explored than the fully-periodic counterpart. Here, we investigate dynamics in a lattice structure that emerges from the simultaneous application of Raman and radio frequency coupling to a dilute-gas Bose-Einstein condensate. We elaborate on the role of Galilean invariance in this system and demonstrate a variety of techniques, including Bloch oscillations and lattice shaking with spin and momentum resolved measurements. This combined coupling scheme allows for tunability and control, enabling future investigations into unconventional band structures such as quasi-flat ground bands and those with semimetal-like band gaps.
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 report on the Bose-Einstein condensate in the Bloch bands with off-diagonal periodic potential (ODPP), which simultaneously plays the role of spin-orbit coupling and Zeeman field. This model can be realized using two independent Raman couplings in the same three-level system, in which the time-reversal symmetry ensures the energy degeneracy between the two states with opposite momenta. We find that these two Raman couplings can be used to tune the spin polarization in momentum space, thus greatly modifying the effective scatterings over the Bloch bands. We observe a transition from the Bloch plane wave phase with condensate at one wave vector to the Bloch stripe phase with condensate at the two Bloch states with opposite wave vectors. These two phases will exhibit different spin textures and density modulations in real space, which are totally different from that in free space. In momentum space, multiple peaks differing by some reciprocal lattice vectors can be observed in both phases, reflecting the periodic structure of the ODPP. A three-band effective model is proposed to understand these observations. This ODPP will never approach the tight-binding limit, thus can provide an alternative platform in the investigations of various physics, such as collective excitations, polaron, and topological superlfuids, over the Bloch bands.
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.
Self-trapping is a hallmark phenomenon of nonlinear dynamics. It has significant applications in modern physics, including band structure engineering, phase transition dynamics, quantum metrology, and more. Dilute-gas Bose-Einstein condensates (BECs), in which self-trapping can arise from interatomic interactions, are a prime testbed for probing nonlinear dynamics. In this Letter, we report the observation of self-trapping in a spin-orbit coupled BEC subjected to a stationary optical lattice. We employ Raman-induced spin-orbit coupling, complemented by a matching optical lattice that facilitates coupling between momentum eigenstates of the spin-orbit coupled system. By ramping the Raman detuning, we probe atomic current flow between these eigenstates and identify a clear distinction between a delocalized mixed state and a self-trapped regime. Following a quench of the Raman detuning, the time-averaged atomic current exhibits non-analytic behavior across the transition between these two regimes in certain parameter ranges, signaling a dynamical phase transition in the system.
We review our recent experimental realization and investigation of a spin-orbit (SO) coupled Bose-Einstein condensate (BEC) and quantum degenerate Fermi gas. By using two counter-propagating Raman lasers and controlling the different frequency of two Raman lasers to engineer the atom-light interaction, we first study the SO coupling in BEC. Then we study SO coupling in Fermi gas. We observe the spin dephasing in spin dynamics and momentum distribution asymmetry of the equilibrium state as hallmarks of SO coupling in a Fermi gas. To clearly reveal the property of SO coupling Fermi gas, we also study the momentum-resolved radio-frequency spectroscopy which characterizes the energy-momentum dispersion and spin composition of the quantum states. We observe the change of fermion surfaces in different helicity branches with different atomic density, which indicates that a Lifshitz transition of the Fermi surface topology change can be found by further cooling the system. At last, we study the momentum-resolved Raman spectroscopy of an ultracold Fermi gas.
Flat bands and dynamical localization of binary mixtures of Bose-Einstein condensates, with spin-orbit coupling subjected to a deep optical lattice which is shaking in time and to a periodic time modulation of the Zeeman field, are investigated. In contrast with usual dynamical localization in the absence of spin-orbit coupling, we find that to fully suppress the tunneling in the system the optical lattice shaking is not enough, and a proper tuning of the spin-orbit term, achievable via the Zeeman field modulation, is also required. This leads to a sequence of Zeeman parameter values where energy bands become flat, the tunneling in the system is suppressed, and the dynamical localization phenomenon occurs. Exact wave functions at the dynamical localization points show that the binary mixture localizes on a dimer with the two components occupying different sites. This type of localization occurs in exact form also for the ground state of the system at the dynamical localization points in the presence of nonlinearity and remains valid, although in approximate form, for a wide range of the Zeeman parameter around these points. The possibility of observing the above phenomena in real experiments is also briefly discussed.