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 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 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.
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 revisit the problem of the reduction of the three-dimensional (3D) dynamics of Bose-Einstein condensates, under the action of strong confinement in one direction (z), to a 2D mean-field equation. We address this problem for the confining potential with a singular term, viz., V_{z}(z)=2z^{2}+ζ^{2}/z^{2}, with constant ζ. A quantum phase transition is induced by the latter term, between the ground state (GS) of the harmonic oscillator and the 3D condensate split in two parallel noninteracting layers, which is a manifestation of the "superselection" effect. A realization of the respective physical setting is proposed, making use of resonant coupling to an optical field, with the resonance detuning modulated along z. The reduction of the full 3D Gross-Pitaevskii equation (GPE) to the 2D nonpolynomial Schrödinger equation (NPSE) is based on the factorized ansatz, with the z -dependent multiplier represented by an exact GS solution of the 1D Schrödinger equation with potential V_{z}(z). For both repulsive and attractive signs of the nonlinearity, the 2D NPSE produces GS and vortex states, that are virtually indistinguishable from the respective numerical solutions provided by full 3D GPE. In the case of the self-attraction, the threshold for the onset of the collapse, predicted by the 2D NPSE, is also virtually identical to its counterpart obtained from the 3D equation. In the same case, stability and instability of vortices with topological charge S=1, 2, and 3 are considered in detail. Thus, the procedure of the spatial-dimension reduction, 3D → 2D, produces very accurate results, and it may be used in other settings.
Abstract Numerically we investigate the ground state of effective one-dimensional spin–orbit (SO) and Rabi coupled two pseudo-spinor Bose–Einstein condensates (BECs) under the effect of harmonic traps. For both signs of the interaction, density profiles of SO and Rabi coupled BECs in harmonic potentials, which simulate a real experimental situation are obtained. The harmonic trap causes a strong reduction of the multi-peak nature of the condensate and it increases its density. For repulsive interactions, the increase of SO coupling results in an uncompressed less dense condensate and with increased multi-peak nature of the density. The increase of Rabi coupling leads to a density increase with an almost constant number of multi-peaks. For both signs of the interaction and negative values of Rabi coupling, the condensate develops a notch in the central point and it seems to a dark-in-bright soliton. In the case of the attractive nonlinearity, an interesting result is the increase of the collapse threshold under the action of the SO and Rabi couplings.
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.