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.
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.
Using a sum rule approach we investigate the dipole oscillation of a spin-orbit coupled Bose-Einstein condensate confined in a harmonic trap. The crucial role played by the spin polarizability of the gas is pointed out. We show that the lowest dipole frequency exhibits a characteristic jump at the transition between the stripe and spin-polarized phase. Near the second-order transition between the spin-polarized and the single minimum phase the lowest frequency is vanishingly small for large condensates, reflecting the divergent behavior of the spin polarizability. We compare our results with recent experimental measurements as well as with the predictions of effective mass approximation.
Interaction between matter field and gauge field plays a fundamental role in many areas of physics. Recently, a non-abelian gauge field has been successfully synthesized in laboratory for ultracold neutral bosonic atoms, yielding the coupling between the pseudo-spin of an atom and its orbital motion. This new superfluid has many unconventional properties remaining to be explored. Here we study the collective oscillation of a spin-orbit coupled Bose-Einstein condensate in a harmonic trap. We find that the frequency of dipole oscillation deviates from the harmonic trap frequency and depends on the oscillation amplitude. Such an unharmonic behavior reveals the nonabelain character of the gauge field. We also observe that a magnetization oscillation is induced by the dipole motion, as a manifestation of the absence of the Galilean invariance of this system. We develop a variational wave-function method that well describes the dipole oscillation, except in the regime with two degenerate single-particle energy minima separated by a low barrier. In this regime, since the harmonic trapping potential plays an analogous role as kinetic energy in a real-space double-well potential, macroscopic quantum tunneling in momentum space is observed in the dipole oscillation.
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 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 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.
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.