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Chinese Physics B

Nonlinear modes coupling of trapped spin–orbit coupled spin-1 Bose–Einstein condensates

Jie Wang, Jun-Cheng Liang, Zi-Fa Yu, An-Qing Zhang, Ai‐Xia Zhang, Ju-Kui Xue

Published Apr 17, 2023• 3 citations
Publisher

Abstract

We study analytically and numerically the nonlinear collective dynamics of quasi-one-dimensional spin–orbit coupled spin-1 Bose–Einstein condensates trapped in harmonic potential. The ground state of the system is determined by minimizing the Lagrange density, and the coupled equations of motions for the center-of-mass coordinate of the condensate and its width are derived. Then, two low energy excitation modes in breathing dynamics and dipole dynamics are obtained analytically, and the mechanism of exciting the anharmonic collective dynamics is revealed explicitly. The coupling among spin–orbit coupling, Raman coupling and spin-dependent interaction results in multiple external collective modes, which leads to the anharmonic collective dynamics. The cooperative effect of spin momentum locking and spin-dependent interaction results in coupling of dipolar and breathing dynamics, which strongly depends on spin-dependent interaction and behaves distinct characters in different phases. Interestingly, in the absence of spin-dependent interaction, the breathing dynamics is decoupled from spin dynamics and the breathing dynamics is harmonic. Our results provide theoretical evidence for deep understanding of the ground sate phase transition and the nonlinear collective dynamics of the system.

DOI
10.1088/1674-1056/accd53

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We study analytically and numerically the nonlinear collective dynamics of quasi-one-dimensional spin–orbit coupled spin-1 Bose–Einstein condensates trapped in harmonic potential. The ground state of the system is determined by minimizing the Lagrange density, and the coupled equations of motions for the center-of-mass coordinate of the condensate and its width are derived. Then, two low energy excitation modes in breathing dynamics and dipole dynamics are obtained analytically, and the mechanism of exciting the anharmonic collective dynamics is revealed explicitly. The coupling among spin–orbit coupling, Raman coupling and spin-dependent interaction results in multiple external collective modes, which leads to the anharmonic collective dynamics. The cooperative effect of spin momentum locking and spin-dependent interaction results in coupling of dipolar and breathing dynamics, which strongly depends on spin-dependent interaction and behaves distinct characters in different phases. Interestingly, in the absence of spin-dependent interaction, the breathing dynamics is decoupled from spin dynamics and the breathing dynamics is harmonic. Our results provide theoretical evidence for deep understanding of the ground sate phase transition and the nonlinear collective dynamics of the system.
The collective excitations of a spin-orbit–coupled Bose-Einstein condensate are investigated by using a variational method and numerical simulation. The collective modes are depicted in zero-momentum and plane-wave phases, while the mechanism for inducing the anharmonic collective oscillations is revealed analytically. The coupling among spin-orbit coupling, Raman coupling, and atomic interaction results in multiple external collective modes, which can lead to the collective dynamics being the superposition of the multiple harmonic oscillations. When one of the external frequencies is close to the inherent frequency of the conventional condensate and the amplitudes of other external frequencies vanish, the collective oscillations are harmonic, otherwise the collective oscillations are anharmonic. Thus the collective dynamics can undergo a harmonicity-anharmonicity transition. This can provide a possible way for elaborating distinct collective oscillation modes. Moreover, the collective excitations exhibit distinct properties in different phases, while the softening of collective modes is detected at the phase transition point, which can be used to map out the phase boundaries in experiment.
Supersolidity, characterized by superfluidity and crystallization, is an exotic state of matter. The spin-orbit- (SO-) coupled Bose-Einstein condensates (BECs) provide an ideal paradigm for exploring this quantum state. Here, we present a theoretical analysis of the ground-state phases and collective modes of supersolid SO-coupled BECs in a harmonic trap. The phase diagram of three ground-state phases (i.e., the supersolid phase, plane-wave phase, and zero-momentum phase) is obtained analytically under the condition $\ensuremath{\omega}\ensuremath{\ll}{k}_{0}$ (here, $\ensuremath{\omega}$ is harmonic trap frequency, k0{k}_{0}k0​ is SO-coupling strength), showing the spin polarization, the phase diagram, and the contrast of the supersolid stripes can be effectively manipulated by trapping potential. An efficient scheme to manipulate the supersolid phase is provided. Furthermore, with particular discussion of the supersolid phase, the collective modes in all three phases are obtained with a unified approach, and softening of both dipole and breathing modes with divergence of effective mass at phase transition point is explained analytically and well confirmed by direct numerical simulations. The impact of SO coupling on local currents during the breathing mode is discussed. The anharmonic collective modes are also revealed with the appearance of the flat band in the single-particle dispersion, especially when SO-coupling strength is comparable with trapping potential. Our theoretical method provides a clear physical picture that collective modes of SO-coupled BECs can be regarded as classical harmonic oscillators with an effective mass containing the SO coupling.
We investigate the collective excitations of a Raman-induced spin-orbit coupled Bose-Einstein condensate confined in a quasi-one-dimensional harmonic trap using the Bogoliubov method. By tuning the Raman coupling strength, three phases of the system can be identified. By calculating the transition strength, we are able to classify various excitation modes that are experimentally relevant. We show that the three quantum phases possess distinct features in their collective excitation properties. In particular, the spin dipole and the spin breathing modes can be used to clearly map out the phase boundaries. We confirm these predictions by direct numerical simulations of the quench dynamics that excites the relevant collective modes.
We studied the collective modes of a Bose-Einstein condensate (BEC) with spin-orbit coupling. We developed the hydrodynamic equations for spin-orbit coupled BECs and used them to study collective modes in the plane-wave phase and large Rabi coupling regime for both a uniform BEC and a BEC in a harmonic trap. In the homogeneous situation, we obtained energy spectra of elementary excitations and found that the spin-orbit coupling can increase the effective mass of the atoms, which will suppress the sound velocity. The spin-orbit coupling can also change the spin mixing, which will modify the interaction energy, and may lead to an enhancement of sound velocity. The competition between these two effects gives the behavior of sound velocity. In a harmonic trap, we found that the dipole mode and the breathing mode are coupled together in the plane-wave phase, and these two modes have a $\ensuremath{\pi}/2$ phase difference, because the spin-orbit coupling and the interaction are not invariant under spin rotation. However, in the large Rabi coupling regime, the dipole mode and the breathing mode are decoupled due to the symmetry restriction.
We study the collective dynamics of the spin-orbit coupled two pseudospin components of a Bose-Einstein condensate trapped in a quasi-one-dimensional harmonic potential, by using variational and directly numerical approach of binary mean-field Gross-Pitaevskii equations. The results show that, because of strong coupling of spin-orbit coupling (SOC), Rabi coupling, and atomic interaction, the collective dynamics of the system behave as complex characters. When the Rabi coupling is absent, the density profiles of the system preserve the Gauss type and the wave packets do harmonic oscillations. The amplitude of the collective oscillations increases with SOC. Furthermore, when the SOC strength increases, the dipole oscillations of the two pseudospin components undergo a transition from in-phase to out-of-phase oscillations. When the Rabi coupling present, there will exist a critical value of SOC strength (which depends on the Rabi coupling and atomic interaction). If the SOC strength is less than this critical value, the density profiles of the system can preserve the Gauss type and the wave packets do anharmonic (the frequency of dipole oscillations depends on SOC) oscillations synchronously (i.e., in-phase oscillations). However, if the SOC strength is larger than this critical value, the wave packets are dynamically fragmented and the stable dipole oscillations of the system can not exist. The collective dynamics of the system can be controlled by adjusting the atomic interaction, SOC, and Rabi-coupling strength.
In this Letter, we present an experimental study of the collective dipole oscillation of a spin-orbit coupled Bose-Einstein condensate in a harmonic trap. The dynamics of the center-of-mass dipole oscillation is studied in a broad parameter region as a function of spin-orbit coupling parameters as well as the oscillation amplitude. The anharmonic properties beyond the effective-mass approximation are revealed, such as the amplitude-dependent frequency and finite oscillation frequency at a place with a divergent effective mass. These anharmonic behaviors agree quantitatively with variational wave-function calculations. Moreover, we experimentally demonstrate a unique feature of the spin-orbit coupled system predicted by a sum-rule approach, stating that spin polarization susceptibility--a static physical quantity--can be measured via the dynamics of dipole oscillation. The divergence of polarization susceptibility is observed at the quantum phase transition that separates the magnetic nonzero-momentum condensate from the nonmagnetic zero-momentum phase. The good agreement between the experimental and theoretical results provides a benchmark for recently developed theoretical approaches.