Abstract Recent experimental observations of spin–orbit coupling (SOC) in Bose–Einstein condensates (BECs) open the way for investigating novel physics of nonlinear waves with promising applications in atomic physics and condensed matter physics. The interplay between atomic interactions and SOC are crucial for the understanding of the dynamics of nonlinear waves in BECs with SOC. Here, in the small linear coupling regime, an approach is presented which allows us to derive an infinite number of novel approximate solutions of the Gross–Pitaevskii equations (GPEs) in one and two dimensions including SOCs, time-dependent external potentials, and nonlinearities leading to breathers and periodic as well as quasiperiodic nonlinear waves. To verify the theoretical predictions we perform numerical simulations which show for several cases a very good agreement with the analytics. For the case of one spatial dimension, it is shown that functions describing the external potential and nonlinearities cannot be chosen independently. The management of the solutions is clarified along with some important physical properties such as Josephson oscillations and Rosen–Zener oscillations.
Abstract Recent experimental observations of spin–orbit coupling (SOC) in Bose–Einstein condensates (BECs) open the way for investigating novel physics of nonlinear waves with promising applications in atomic physics and condensed matter physics. The interplay between atomic interactions and SOC are crucial for the understanding of the dynamics of nonlinear waves in BECs with SOC. Here, in the small linear coupling regime, an approach is presented which allows us to derive an infinite number of novel approximate solutions of the Gross–Pitaevskii equations (GPEs) in one and two dimensions including SOCs, time-dependent external potentials, and nonlinearities leading to breathers and periodic as well as quasiperiodic nonlinear waves. To verify the theoretical predictions we perform numerical simulations which show for several cases a very good agreement with the analytics. For the case of one spatial dimension, it is shown that functions describing the external potential and nonlinearities cannot be chosen independently. The management of the solutions is clarified along with some important physical properties such as Josephson oscillations and Rosen–Zener oscillations.
We present a unified description of different types of matter-wave solitons that can emerge in quasi one-dimensional spin-orbit coupled (SOC) Bose-Einstein condensates (BECs). This description relies on the reduction of the original two-component Gross-Pitaevskii SOC-BEC model to a single nonlinear Schrödinger equation, via a multiscale expansion method. This way, we find approximate bright and dark soliton solutions, for attractive and repulsive interatomic interactions respectively, for different regimes of the SOC interactions. Beyond this, our approach also reveals "negative mass" regimes, where corresponding "negative mass" bright or dark solitons can exist for repulsive or attractive interactions, respectively. Such a unique opportunity stems from the structure of the excitation spectrum of the SOC-BEC. Numerical results are found to be in excellent agreement with our analytical predictions.
Spin-orbit coupling (SOC), the interaction between the spin and momentum of a quantum particle, is crucial for many important condensed matter phenomena. The recent experimental realization of SOC in neutral bosonic cold atoms provides a new and ideal platform for investigating spin-orbit coupled quantum many-body physics. In this Letter, we derive a generic Gross-Pitaevskii equation as the starting point for the study of many-body dynamics in spin-orbit coupled Bose-Einstein condensates. We show that different laser setups for realizing the same SOC may lead to different mean-field dynamics. Various ground state phases (stripe, phase separation, etc.) of the condensate are found in different parameter regions. A new oscillation period induced by the SOC, similar to the Zitterbewegung oscillation, is found in the center-of-mass motion of the condensate.
Abstract We report results of systematic analysis of confined steadily rotating patterns in the two-component BEC including the spin–orbit coupling (SOC) of the Rashba type, which acts in the interplay with the attractive or repulsive intra-component and inter-component nonlinear interactions and confining potential. The analysis is based on the system of the Gross–Pitaevskii equations (GPEs) written in the rotating coordinates. The resulting GPE system includes effective Zeeman splitting. In the case of the attractive nonlinearity, the analysis, performed by means of the imaginary-time simulations, produces deformation of the known two-dimensional SOC solitons (semi-vortices and mixed-modes). Essentially novel findings are reported in the case of the repulsive nonlinearity. They demonstrate patterns arranged as chains of unitary vortices which, at smaller values of the rotation velocity Ω, assume the straight (single-string) form. At larger Ω, the straight chains become unstable, being spontaneously replaced by a trilete star-shaped array of vortices. At still larger values of Ω, the trilete pattern rebuilds itself into a star-shaped one formed of five and, then, seven strings. The transitions between the different patterns are accounted for by comparison of their energy. It is shown that the straight chains of vortices, which form the star-shaped structures, are aligned with boundaries between domains populated by plane waves with different wave vectors. A transition from an axisymmetric higher-order (multiple) vortex state to the trilete pattern is investigated too.
This mini-review collects results predicting the creation of matter-wave solitons by the spinor system of Gross-Pitaevskii equations (GPEs) with the self-attractive cubic nonlinearity and linear first-order-derivative terms accounting for the spin-orbit coupling (SOC). In 1D, the so-predicted bright solitons are similar to usual ones, supported by the GPE in the absence of SOC. Essentially new results were recently obtained for 2D and 3D systems: SOC suppresses the collapse instability in the multidimensional GPE, creating 2D ground-state solitons and metastable 3D ones of two types: semi-vortices (SVs), with vorticities m = 1 in one component and m = 0 in the other, and mixed modes (MMs), with m = 0 and present in both components. With the Galilean invariance broken by SOC, moving solitons exist up to a certain critical velocity. The latest result predicts stable 2D “quantum droplets” of the MM type in the presence of the Lee-Huang-Yang corrections to the GPE system, induced by quantum fluctuations, in the case when the inter-component attraction dominates over the self-repulsion in each component.
We investigate the one-dimensional (1D) and two-dimensional (2D) reduction of a quantum field theory starting from the three-dimensional (3D) many-body Hamiltonian of interacting bosons with spin–orbit (SO) and Rabi couplings. We obtain the effective time-dependent 1D and 2D nonpolynomial Heisenberg equations for both the repulsive and attractive signs of the inter-atomic interaction. Our findings show that in the case in which the many-body state coincides with the Glauber coherent state, the 1D and 2D Heisenberg equations become 1D and 2D nonpolynomial Schrödinger equations (NPSEs). These models were derived in a mean-field approximation from 3D Gross-Pitaevskii equation (GPE), describing a Bose–Einstein condensate (BEC) with SO and Rabi couplings. In the present work self-repulsive and self-attractive localized solutions of the 1D NPSE and the 1D GPE are obtained in a numerical form. The combined action of SO and Rabi couplings produces conspicuous sidelobes on the density profile, for both signs of the interaction. In the case of the attractive nonlinearity, an essential result is the possibility of getting an unstable condensate by the increasing of SO coupling.
In this paper, we study mathematically and numerically the ground states of rotating spin-orbit coupled two-component Bose-Einstein condensates modeled by the coupled Gross-Pitaevskii equations. Firstly, we rigorously prove the existence and nonexistence results for the ground states and derive some analytical properties, including the virial identity and negativity of SOC energy. Secondly, we propose an efficient and accurate preconditioned nonlinear conjugate gradient (PCG) algorithm to compute the ground states. We truncate the whole space into a bounded rectangular domain and readily apply the Fourier spectral method to approximate the wave function. The PCG method is successfully adapted with appropriate modifications to the adaptive step size control strategy for the one-parameter energy minimization problem and to the choice of preconditioners, achieving great performance in terms of accuracy and efficiency. Lastly, we carry out extensive numerical experiments to confirm the spatial spectral accuracy by traversing the most commonly-used initial guesses for each component. Moreover, we investigate the effects of local interaction, angular velocity, spin-orbit coupling and external trapping potential on the ground states, and predict some interesting physical phenomena, such as vortex lattices.