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arXiv:1408.2256 · Physical Review A

Localized modes in quasi-two-dimensional Bose-Einstein condensates with spin-orbit and Rabi couplings

Luca Salasnich, Wesley Bueno Cardoso, Boris A. Malomed

Published Sep 30, 2014• 102 citations
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Abstract

We consider a two-component pancake-shaped, i.e., effectively two-dimensional (2D), Bose-Einstein condensate coupled by the spin-orbit (SO) and Rabi terms. The SO coupling adopted here is of the mixed Rashba-Dresselhaus type. For this configuration, we derive a system of two 2D nonpolynomial Schr\"odinger equations (NPSEs), for both attractive and repulsive interatomic interactions. In the low- and high-density limits, the system amounts to previously known models, namely, the usual 2D Gross-Pitaevskii equation, or the Schr\"odinger equation with the nonlinearity of power $7/3$. We present simple approximate localized solutions, obtained by treating the SO and Rabi terms as perturbations. Localized solutions of the full NPSE system are obtained in a numerical form. Remarkably, in the case of the attractive nonlinearity acting in free space (i.e., without any 2D trapping potential), we find parameter regions where the SO and Rabi couplings make 2D fundamental solitons dynamically stable.

DOI
10.1103/physreva.90.033629

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We consider a two-component pancake-shaped, i.e., effectively two-dimensional (2D), Bose-Einstein condensate coupled by the spin-orbit (SO) and Rabi terms. The SO coupling adopted here is of the mixed Rashba-Dresselhaus type. For this configuration, we derive a system of two 2D nonpolynomial Schr"odinger equations (NPSEs), for both attractive and repulsive interatomic interactions. In the low- and high-density limits, the system amounts to previously known models, namely, the usual 2D Gross-Pitaevskii equation, or the Schr"odinger equation with the nonlinearity of power 7/37/37/3. We present simple approximate localized solutions, obtained by treating the SO and Rabi terms as perturbations. Localized solutions of the full NPSE system are obtained in a numerical form. Remarkably, in the case of the attractive nonlinearity acting in free space (i.e., without any 2D trapping potential), we find parameter regions where the SO and Rabi couplings make 2D fundamental solitons dynamically stable.
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.
We consider a binary Bose-Einstein condensate with linear and nonlinear interactions between its components, which emulate the spinor system with spin-orbit (SO) and Rabi couplings. For a relatively dense condensate, one-dimensional coupled equations with the nonpolynomial nonlinearity of both repulsive and attractive signs are derived from the three-dimensional Gross-Pitaevskii equations. Profiles of modes confined in an external potential under the action of the self-repulsion, and self-trapped solitons in the case of the self-attraction, are found in a numerical form and by means of analytical approximations. In the former case, the interplay of the SO and Rabi couplings with the repulsive nonlinearity strongly distorts shapes of the trapped modes, adding conspicuous sidelobes to them. In the case of the attractive nonlinearity, the most essential result is reduction of the collapse threshold under the action of the SO and Rabi couplings.
We study fundamental and vortical solitons in disk-morphed Bose-Einstein condensates (BECs) subject to strong confinement along the axial direction. Starting from the three-dimensional (3D) Gross-Pitaevskii equation (GPE), we proceed to an effective two-dimensional (2D) nonpolynomial Schr\"odinger equation (NPSE) derived by means of the integration over the axial coordinate. Results produced by the latter equation are in very good agreement with those obtained from the full 3D GPE, including cases when the formal 2D equation with the cubic nonlinearity is unreliable. The 2D NPSE is used to predict the density profiles and dynamical stability of repulsive and attractive BECs with zero and finite topological charges in various planar trapping configurations, including the axisymmetric harmonic confinement and one-dimensional periodic potential. In particular, we find a stable dynamical regime that was not reported before, viz., periodic splitting and recombination of trapped vortices with topological charges 2 or 3 in the self-attractive BEC.
We study two-dimensional (2D) matter-wave solitons in spinor Bose–Einstein condensates under the action of the spin–orbit coupling and opposite signs of the self- and cross-interactions. Stable 2D two-component solitons of the mixed-mode type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case, under the action of competing self-attraction and cross-repulsion. The solitons exist with the total norm taking values below a collapse threshold. Further, in the case of the repulsive self-interaction and inter-component attraction, stable 2D self-trapped modes, which may be considered as quantum droplets (QDs), are created if the beyond-mean-field Lee–Huang–Yang terms are added to the self-repulsion in the underlying system of coupled Gross–Pitaevskii equations. Stable QDs of the mixed-mode type, of a large size with an anisotropic density profile, exist with arbitrarily large values of the norm, as the Lee–Huang–Yang terms eliminate the collapse. The effect of the spin–orbit coupling term on characteristics of the QDs is systematically studied. We also address the existence and stability of QDs in the case of SOC with mixed Rashba and Dresselhaus terms, which makes the density profile of the QD more isotropic. Thus, QDs in the spin-orbit-coupled binary Bose–Einstein condensate are for the first time studied in the present work.
We present an analysis of two-dimensional (2D) matter-wave solitons, governed by the pseudospinor system of Gross-Pitaevskii equations with self- and cross attraction, which includes the spin-orbit coupling (SOC) in the general Rashba-Dresselhaus form, and, separately, the Rashba coupling and the Zeeman splitting. Families of semivortex (SV) and mixed-mode (MM) solitons are constructed, which exist and are stable in free space, as the SOC terms prevent the onset of the critical collapse and create the otherwise missing ground states in the form of the solitons. The Dresselhaus SOC produces a destructive effect on the vortex solitons, while the Zeeman term tends to convert the MM states into the SV ones, which eventually suffer delocalization. Existence domains and stability boundaries are identified for the soliton families. For physically relevant parameters of the SOC system, the number of atoms in the 2D solitons is limited by ∼1.5×10^{4}. The results are obtained by means of combined analytical and numerical methods.
We introduce one- and two-dimensional (1D and 2D) continuum and discrete models for the two-component BEC, with the spin-orbit (SO) coupling of the Rashba type between the components, and attractive cubic interactions, assuming that the condensate is fragmented into a quasidiscrete state by a deep optical-lattice potential. In 1D, it is demonstrated, in analytical and numerical forms, that the ground states of both the discrete system and its continuum counterpart switch from striped bright solitons, featuring deep short-wave modulations of its profile, to smooth solitons, as the strength ratio of the inter- and intracomponent attraction, γ, changes from γ<1 to γ>1. At the borderline, γ=1, there is a continuous branch of stable solitons, which share a common value of the energy and interpolate between the striped and smooth ones. Unlike the 2D system, the 1D solitons, which do not represent the ground state at given γ, are nevertheless stable against small perturbations, and they remain stable too in collisions with other solitons. In 2D, a transition between two different types of discrete solitons, which represent the ground state, viz., semivortices and mixed modes, also takes place at γ=1. A specific property of 2D discrete solitons of both types is their discontinuous transition into a delocalized state at a critical value of the SO-coupling strength. We also address the continuum 2D model in the borderline case of γ=1, which was not studied previously, and demonstrate the existence of an energy-degenerate branch of dynamically stable solitons connecting the semivortex and the mixed mode. Last, it is demonstrated that 1D and 2D discrete solitons are mobile, in a limited interval of velocities.