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New Journal of Physics

Helicoidal spin–orbit coupling induced novel nonlinear dispersion relation and Landau–Zener tunneling dynamics

Geng Zhao, Haibin Tang, Junyan Yao, Juncheng Liang, Zi-Fa Yu, Ai Xia Zhang, Ju-Kui Xue

Published Jul 1, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1367-2630/adee3e

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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.
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.
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.
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.
Abstract The effects of modulating spin-orbit coupled Bose–Einstein condensates are analytically studied. A sinusoidal driving of the coupling amplitude is shown to induce significant changes in the energy bands and in the associated spin-momentum locking. Moreover, in agreement with recent experimental results, gravitational acceleration of the modulated system is found to generate transitions between the modified energy bands. The applicability of the Landau–Zener (LZ) model to the understanding of the experimental findings is rigorously traced. Through a sequence of unitary transformations and the reduction to the spin space, the modulated Hamiltonian, with the gravitational potential incorporated, is shown to correspond to an extended version of the LZ scenario. The generalization of the basic LZ model takes place along two lines. First, the dimensionality is enlarged to combine the description of the external dynamics with the internal-state characterization. Second, the model is extended to incorporate two avoided crossings emerging from the changes induced in the energy bands by the modulation. Our approach allows a first-principle derivation of the effective model-system parameters. The obtained analytical results provide elements to control the transitions.
We theoretically study the band structure, tunneling dynamics, and tunneling probability of tunable spin-orbit-coupled Bose-Einstein condensates under the periodic driving of Raman coupling. The time-independent Floquet Hamiltonian is obtained in the high-frequency approximation. It is found that the periodic driving can effectively tune spin-orbit coupling and nonlinear interaction. The system is mapped to a standard nonlinear two-level model, and the critical condition for the appearance of the loop in energy band structure and the width of the loop are obtained analytically. When the interspecies atomic interaction is equal to the intraspecies atomic interaction, there is no loop. However, when the intraspecies atomic interaction is smaller (larger) than the interspecies atomic interaction, the loop appears in the lower (upper) energy band. In this case, both spin-orbit coupling and Raman coupling will suppress the appearance of loop. In particular, the critical condition for the appearance of loop structure can be controlled by adjusting external driving. We also study the tunneling dynamics of Bose-Einstein condensate with tunable spin-orbit coupling. More importantly, by tuning the periodic driving, the tunneling dynamics of the system and the location of nonlinear Landau-Zener tunneling can be controlled. We also find that the spin components of the system can be reversed. Finally, the Landau-Zener tunneling probability of the system is calculated. The research shows that the periodic driving can effectively change the tunneling probability of the system.
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.