Single-particle dynamics of a spin-orbit-coupled Bose-Einstein condensate has recently been investigated in experiments that explore the physics of Landau-Zener tunneling and of the Zitterbewegung. In this paper, we study the influence of a nonlinearity due to interactions on these dynamics and show that the dispersion relation develops interesting loop structures. The atoms can move along the nonlinear dispersion curve in the presence of a weak acceleration force, and we show that the loops lead to straightforward nonlinear Landau-Zener tunneling. However, we find that for the Zitterbewegung, induced by a sudden quench in the spin-orbit-coupling parameters, the nonlinear dispersion is irrelevant.
Single-particle dynamics of a spin-orbit-coupled Bose-Einstein condensate has recently been investigated in experiments that explore the physics of Landau-Zener tunneling and of the Zitterbewegung. In this paper, we study the influence of a nonlinearity due to interactions on these dynamics and show that the dispersion relation develops interesting loop structures. The atoms can move along the nonlinear dispersion curve in the presence of a weak acceleration force, and we show that the loops lead to straightforward nonlinear Landau-Zener tunneling. However, we find that for the Zitterbewegung, induced by a sudden quench in the spin-orbit-coupling parameters, the nonlinear dispersion is irrelevant.
Nonlinear Landau-Zener tunneling is an important nonlinear phenomenon. We propose to stimulate the nonlinear tunneling in a spin-orbit-coupled spinor Bose-Einstein condensate. The system provides an experimentally tunable nonlinearity as well as multiple avoided crossings with tunable gap size in nonlinear dispersion relations. The nonlinearity generates tilted cusp and loop structures around the avoided crossings, and the physical consequence of these nonlinear structures is the nonlinear Landau-Zener tunneling. The spin-momentum locking induced by the spin-orbit coupling leads to a time-resolved observation of the nonlinear tunneling by measuring atom populations.
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 nonadiabatic dynamics of a driven spin-orbit-coupled Bose-Einstein condensate in both weak and strong driven force. It is shown that the standard Landau-Zener (LZ) tunneling fails in the regime of weak driven force and/or strong spin-orbital coupling, where the full nonadiabatic dynamics requires a new mechanism through multichannel effects. Beyond the semiclassical approach, our numerical and analytical results show an oscillating power-law decay in the quantum limit, different from the exponential decay in the semiclassical limit of the LZ effect. Furthermore, the condensate density profile is found to be dynamically fragmented by the multichannel effects and enhanced by interaction effects. Our work therefore provides a complete picture to understand the nonadiabatic dynamics of a spin-orbit coupled condensate, including various ranges of driven force and interaction effects through multichannel interference. The experimental indication of these nonadiabatic dynamics is also discussed.
Abstract Spin–orbit (SO) coupling enriches the single-particle dispersion relation, inducing dramatic change of the ground state phase and non-equilibrium quantum dynamics. Here, the nonlinear energy spectrum and Landau–Zener (LZ) tunneling dynamics in helicoidal SO coupled Bose–Einstein condensates are studied. The interplay of atomic interactions and helicoidal SO coupling results in novel nonlinear energy spectrum and LZ tunneling dynamics. Two distinct helicoidal SO avoided crossing points in the energy spectrum are predicted, i.e. a normal avoided crossing point characterized with degenerate eigenenergies and non-degenerate eigenstates, and an abnormal avoided crossing point analogous to the exceptional point in non-Hermitian system characterized with both degenerate eigenenergies and eigenstates. When the effective atomic interaction beyond a threshold, normal and abnormal loops emerge at the corresponding crossing points successively. Particularly, by adjusting the system parameters (e.g. atomic interactions), the two avoided crossing points and loops can co-exist in different bands, and the two loops even collide and merge. When the atoms are accelerated and move adiabatically along the energy band, atoms can steadily cross the normal avoided crossing point and normal LZ tunneling dynamics takes place at the edge of the normal loop, while an abnormal LZ tunneling dynamics is excited successively at the abnormal avoided crossing point and the edge of the abnormal loop. The two avoided crossing points and loops result in distinct LZ tunneling dynamics. Furthermore, the helicoidal SO coupling induces spatial symmetry broken of energy band, resulting in nonreciprocal LZ tunneling, i.e. the tunneling probabilities for right-ward and left-ward sweeping are different. The novel energy spectrum and tunneling dynamics are further understood with the help of the evolution of fixed points in classical Hamiltonian phase space. We provide a theoretical evidence for experimental probing and manipulation of nonlinear dispersion relation and quantum tunneling dynamics in ultracold atomic gases.
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.
Spin-orbit-coupled ultracold atoms provide an intriguing new avenue for the study of rich spin dynamics in superfluids. In this Rapid Communication, we observe Zitterbewegung, the simultaneous velocity (thus position) and spin oscillations, of neutral atoms between two spin-orbit-coupled bands in a Bose-Einstein condensate (BEC) through sudden quantum quenches of the Hamiltonian. The observed Zitterbewegung oscillations are perfect on a short time scale but gradually damp out on a long time scale, followed by sudden and strong heating of the BEC. As an application, we also demonstrate how Zitterbewegung oscillations can be exploited to populate the upper spin-orbit band and observe a subsequent dipole motion. Our experimental results are corroborated by a theoretical and numerical analysis and showcase the great flexibility that ultracold atoms provide for investigating rich spin dynamics in superfluids.