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arXiv:1707.04107 · New Journal of Physics

Spin–orbit coupling and nonlinear modes of the polariton condensate in a harmonic trap

Hidetsugu Sakaguchi, Boris A. Malomed, Dmitry V. Skryabin

Published Aug 16, 2017• 12 citations
arXivHTMLPDF

Abstract

We consider a model of the exciton-polariton condensate based on a system of two Gross–Pitaevskii equations coupled by the second-order differential operator, which represents the spin–orbit coupling in the system. Also included are the linear gain, effective diffusion, nonlinear loss, and the standard harmonic-oscillator trapping potential, as well as the Zeeman splitting. By means of combined analytical and numerical methods, we identify stable two-dimensional modes supported by the nonlinear system. In the absence of the Zeeman splitting, these are mixed modes, which combine zero and nonzero vorticities in each of the two spinor components, and vortex–antivortex complexes. We have also found a range of parameters where the mixed-mode and vortex–antivortex states coexist and are stable. Sufficiently strong Zeeman splitting creates stable semi-vortex states, with vorticities 0 in one component and 2 in the other.

DOI
10.1088/1367-2630/aa7fb0

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We consider a model of the exciton-polariton condensate based on a system of two Gross–Pitaevskii equations coupled by the second-order differential operator, which represents the spin–orbit coupling in the system. Also included are the linear gain, effective diffusion, nonlinear loss, and the standard harmonic-oscillator trapping potential, as well as the Zeeman splitting. By means of combined analytical and numerical methods, we identify stable two-dimensional modes supported by the nonlinear system. In the absence of the Zeeman splitting, these are mixed modes, which combine zero and nonzero vorticities in each of the two spinor components, and vortex–antivortex complexes. We have also found a range of parameters where the mixed-mode and vortex–antivortex states coexist and are stable. Sufficiently strong Zeeman splitting creates stable semi-vortex states, with vorticities 0 in one component and 2 in the other.
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 elaborate a mechanism for the formation of stable solitons of the semivortex type (with vorticities 0 and 1 in their two components), populating a finite band gap in the spectrum of the spin-orbit-coupled binary Bose-Einstein condensate with the Zeeman splitting, in the two-dimensional (2D) free space, under conditions which make the kinetic-energy terms in the respective coupled Gross-Pitaevskii equations negligible. Unlike a recent work which used long-range dipole-dipole interactions to construct stable gap solitons in a similar setting, we here demonstrate that stable solitons are supported by generic local interactions of both attractive and repulsive signs, provided that the relative strength of the cross- and self-interactions in the two-component system does not exceed a critical value $\ensuremath{\approx}0.77$. A boundary between stable and unstable fundamental 2D gap solitons is precisely predicted by the Vakhitov-Kolokolov criterion, while all excited states of the 2D solitons, with vorticities (m,1+m)\left(m,1+m\right)(m,1+m) in the two components, m=1,2,...m=1,2,...m=1,2,..., are unstable. The analysis of the one-dimensional (1D) reduction of the system produces an exact analytical solution for the family of gap solitons which populate the entire band gap, the family being fully stable. Motion of the 1D solitons in the trapping potential is considered too, showing that their effective mass is positive or negative if the cubic nonlinearity is attractive or repulsive, respectively.
It is commonly known that two-dimensional mean-field models of optical and matter waves with cubic self-attraction cannot produce stable solitons in free space because of the occurrence of collapse in the same setting. By means of numerical analysis and variational approximation, we demonstrate that the two-component model of the Bose-Einstein condensate with the spin-orbit Rashba coupling and cubic attractive interactions gives rise to solitary-vortex complexes of two types: semivortices (SVs, with a vortex in one component and a fundamental soliton in the other), and mixed modes (MMs, with topological charges 0 and ±1 mixed in both components). These two-dimensional composite modes can be created using the trapping harmonic-oscillator (HO) potential, but remain stable in free space, if the trap is gradually removed. The SVs and MMs realize the ground state of the system, provided that the self-attraction in the two components is, respectively, stronger or weaker than the cross attraction between them. The SVs and MMs which are not the ground states are subject to a drift instability. In free space (in the absence of the HO trap), modes of both types degenerate into unstable Townes solitons when their norms attain the respective critical values, while there is no lower existence threshold for the stable modes. Moving free-space stable solitons are also found in the present non-Galilean-invariant system, up to a critical velocity. Collisions between two moving solitons lead to their merger into a single one.
The Gross–Pitaevskii equation for two-component rotating Bose–Einstein condensates with the Rashba-type spin–orbit (SO) coupling is studied with numerical simulations and variational analyses. A multiquantum vortex state becomes a ground state in a harmonic potential when mutual interaction is absent. When the attractive interaction is strong, the multiquantum vortex state exhibits modulational instability in the azimuthal direction, and a soliton-like state appears. When the repulsive interaction is strong, a vortex lattice state with a multiquantum vortex at the center is created. We find that the vortex lattice state is approximated at a linear combination of multiquantum vortex states.
Solitons are self-trapped modes existing in various nonlinear systems. Creating stable solitons in two- and three-dimensional settings is a challenging goal in various branches of physics. Several methods have been developed theoretically and experimentally to achieve this, but few of them can support stable multi-dimensional solitons in free space. Recently, a new scheme using spin-orbit-coupling (SOC) has been proposed to create stable 2D solitons in Bose–Einstein condensates (BECs). This paper reviews recent theoretical progress on creating stable 2D solitons in spinor dipolar BEC with SOC, combined with long-range dipole-dipole interaction (DDI), Zeeman splitting (ZS) and contact nonlinearity, in free space. The continuous family of stable symmetric vortex solitons (SVS), asymmetric vortex solitons (AVS), as well as gap solitons (GS) is found via different settings. Their existence and stability conditions are summarized and discussed in detail. The mobility properties of these types of solitons are also addressed. For SVS, a potential method to manipulate its shape and mobility is investigated. These results are supposed to enrich our understanding of 2D solitons and help create multi-dimensional solitons in experiments.
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.