Abstract In the mean-field framework, using variational analysis and numerical simulation, we investigate the effect of anharmonic trap and atomic interaction on the ground-state phases of spin-orbit (SO) coupled Bose–Einstein condensates (BECs) in the harmonic plus quartic potential. Then, the Gaussian wave function is selected to predict the analytical conditions of the phase transition boundary of the SO coupled BECs by using the variational method. We found that the anharmonicity of the external potential induces the SO coupled BECs to undergo a phase transition between the zero-momentum phase and plane-wave phase, which is more pronounced in the cases of weak harmonic potential or strong interspecies interaction. Since the potential energy of the system modified by anharmonicity competes with other energies of the system, the anharmonicity changes the critical SO coupling strength and Raman coupling strength when the phase transition occurs. At the same time, the critical anharmonic coefficients are also affected by interspecies interaction and harmonic potential. Finally, the correctness of the theoretical results is verified by numerical simulation of the Gross-Pitaevskii equation.
Abstract In the mean-field framework, using variational analysis and numerical simulation, we investigate the effect of anharmonic trap and atomic interaction on the ground-state phases of spin-orbit (SO) coupled Bose–Einstein condensates (BECs) in the harmonic plus quartic potential. Then, the Gaussian wave function is selected to predict the analytical conditions of the phase transition boundary of the SO coupled BECs by using the variational method. We found that the anharmonicity of the external potential induces the SO coupled BECs to undergo a phase transition between the zero-momentum phase and plane-wave phase, which is more pronounced in the cases of weak harmonic potential or strong interspecies interaction. Since the potential energy of the system modified by anharmonicity competes with other energies of the system, the anharmonicity changes the critical SO coupling strength and Raman coupling strength when the phase transition occurs. At the same time, the critical anharmonic coefficients are also affected by interspecies interaction and harmonic potential. Finally, the correctness of the theoretical results is verified by numerical simulation of the Gross-Pitaevskii equation.
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.
We study the ground-state phases, the stability phase diagram and collapse dynamics of Bose—Einstein condensates (BECs) with tunable spin—orbit (SO) coupling in the two-dimensional harmonic potential by variational analysis and numerical simulation. Here we propose the theory that the collapse stability and collapse dynamics of BECs in the external trapping potential can be manipulated by the periodic driving of Raman coupling (RC), which can be realized experimentally. Through the high-frequency approximation, an effective time-independent Floquet Hamiltonian with two-body interaction in the harmonic potential is obtained, which results in a tunable SO coupling and a new effective two-body interaction that can be manipulated by the periodic driving strength. Using the variational method, the phase transition boundary and collapse boundary of the system are obtained analytically, where the phase transition between the spin-nonpolarized phase with zero momentum (zero momentum phase) and spin-polarized phase with non-zero momentum (plane wave phase) can be manipulated by the external driving and sensitive to the strong external trapping potential. Particularly, it is revealed that the collapsed BECs can be stabilized by periodic driving of RC, and the mechanism of collapse stability manipulated by periodic driving of RC is clearly revealed. In addition, we find that the collapse velocity and collapse time of the system can be manipulated by periodic driving strength, which also depends on the RC, SO coupling strength and external trapping potential. Finally, the variational approximation is confirmed by numerical simulation of Gross—Pitaevskii equation. Our results show that the periodic driving of RC provides a platform for manipulating the ground-state phases, collapse stability and collapse dynamics of the SO coupled BECs in an external harmonic potential, which can be realized easily in current experiments.
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 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.
We consider a SU(3) spin–orbit coupled Bose–Einstein condensate confined in a harmonic plus quartic trap. The ground-state wave functions of such a system are obtained by minimizing the Gross–Pitaevskii energy functional, and the effects of the spin-dependent interaction and spin–orbit coupling are investigated in detail. For the case of ferromagnetic spin interaction, the SU(3) spin–orbit coupling induces a threefold-degenerate plane wave ground state with nontrivial spin texture. For the case of antiferromagnetic spin interaction, the system shows phase separation for weak SU(3) spin–orbit coupling, where three discrete minima with unequal weights in momentum space are selected, while hexagonal honeycomb lattice structure for strong SU(3) SOC, where three discrete minima with equal weights are selected.
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.