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arXiv:1202.0145 · Physical Review Letters

Stable Periodic Density Waves in Dipolar Bose-Einstein Condensates Trapped in Optical Lattices

Aleksandra Maluckov, Goran Gligorić, Ljupčo R. Hadžievski, Boris A. Malomed, Tilman Pfau

Published Apr 3, 2012• 36 citations
arXivHTMLPDF

Abstract

Density-wave patterns in discrete media with local interactions are known to be unstable. We demonstrate that stable double- and triple-period patterns (DPPs and TPPs), with respect to the period of the underlying lattice, exist in media with nonlocal nonlinearity. This is shown in detail for dipolar Bose-Einstein condensates, loaded into a deep one-dimensional optical lattice. The DPP and TPP emerge via phase transitions of the second and first kind, respectively. The emerging patterns may be stable if the dipole-dipole interactions are repulsive and sufficiently strong, in comparison with the local repulsive nonlinearity. Within the set of the considered states, the TPPs realize a minimum of the free energy. A vast stability region for the TPPs is found in the parameter space, while the DPP stability region is relatively narrow. The same mechanism may create stable density-wave patterns in other physical media featuring nonlocal interactions.

DOI
10.1103/physrevlett.108.140402

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Density-wave patterns in discrete media with local interactions are known to be unstable. We demonstrate that stable double- and triple-period patterns (DPPs and TPPs), with respect to the period of the underlying lattice, exist in media with nonlocal nonlinearity. This is shown in detail for dipolar Bose-Einstein condensates, loaded into a deep one-dimensional optical lattice. The DPP and TPP emerge via phase transitions of the second and first kind, respectively. The emerging patterns may be stable if the dipole-dipole interactions are repulsive and sufficiently strong, in comparison with the local repulsive nonlinearity. Within the set of the considered states, the TPPs realize a minimum of the free energy. A vast stability region for the TPPs is found in the parameter space, while the DPP stability region is relatively narrow. The same mechanism may create stable density-wave patterns in other physical media featuring nonlocal interactions.
Periodic patterns with doubled lattice periodicity (DPP) that originate from the modulationally unstable continuous-wave (CW)-type state are found in dipolar Bose-Einstein condensates loaded into a deep one-dimensional optical lattice. The DPP can be created in the presence of any type of contact and/or dipole-dipole (DD) interaction in the system. The main finding is the possibility of creating the stable DPP branch from the CW solution via supercritical pitchfork bifurcation in a condensate with the repulsive contact and certain values of the repulsive DD interaction parameters. In all other combinations of interaction types, we showed that close to the anticontinuum limit DPPs are long-lived, while the instability grows with an increase of the inter-site coupling.
We study the stability of the continuous waves in the pancake shaped dipolar Bose-Einstein condensate trapped in the strong optical lattice potential with the coexisting local (the short-range s-wave) interaction and nonlocal (the dipole-dipole) interactions between the condensate atoms.The system is modeled by two two-dimensional discrete models derived from the Gross-Pitaevskii equation accounting the dipole-dipole interactions: discrete nonlinear Schrödinger equation with cubic nonlinearity and nonpolynomial Schrödinger equation.The corresponding dispersion relations are calculated analytically and the regions of the modulation instability in the parametric space are summarized into the stability diagrams.
Modulational instability and discrete matter wave solitons in dipolar BECs, loaded into a deep optical lattice, are investigated analytically and numerically. The process of modulational instability of nonlinear plane matter waves in a dipolar nonlinear lattice is studied and the regions of instability are established. The existence and stability of bulk discrete solitons are analyzed analytically and confirmed by numerical simulations. In marked contrast with the usual discrete nonlinear Schr\"odinger behavior (no dipolar interactions), we found a region where the two fundamental modes are simultaneously unstable, allowing enhanced mobility across the lattice for large norm values. To study the existence and properties of surface discrete solitons, an analysis of the dimer configuration is performed. The properties of symmetric and antisymmetric modes including stability diagrams and bifurcations are investigated in closed form. For the case of a bulk medium, properties of fundamental on-site and intersite localized modes are analyzed. On-site and intersite surface localized modes are studied, and we find that they do not exist when nonlocal interactions predominate with respect to local ones.
We analyze the formation and dynamics of bright unstaggered solitons in the disk-shaped dipolar Bose-Einstein condensate, which features the interplay of contact (collisional) and long-range dipole-dipole (DD) interactions between atoms. The condensate is assumed to be trapped in a strong optical-lattice potential in the disk's plane, hence it may be approximated by a two-dimensional (2D) discrete model, which includes the on-site nonlinearity and cubic long-range (DD) interactions between sites of the lattice. We consider two such models, which differ by the form of the on-site nonlinearity, represented by the usual cubic term, or more accurate nonpolynomial one, derived from the underlying three-dimensional Gross-Pitaevskii equation. Similar results are obtained for both models. The analysis is focused on the effects of the DD interaction on fundamental localized modes in the lattice (2D discrete solitons). The repulsive isotropic DD nonlinearity extends the existence and stability regions of the fundamental solitons. New families of on-site, inter-site, and hybrid solitons, built on top of a finite background, are found as a result of the interplay of the isotropic repulsive DD interaction and attractive contact nonlinearity. By themselves, these solutions are unstable, but they evolve into robust breathers which exist on an oscillating background. In the presence of the repulsive contact interactions, fundamental localized modes exist if the DD interaction (attractive isotropic or anisotropic) is strong enough. They are stable in narrow regions close to the anticontinuum limit, while unstable solitons evolve into breathers. In the latter case, the presence of the background is immaterial.
We analyse the existence, stability and dynamics of localized discrete modes with intrinsic vorticity S = 1 and S = 2 in the disc-shaped dipolar Bose-Einstein condensate loaded into a deep two-dimensional optical lattice. The condensate, which features the interplay of local contact and nonlocal dipole-dipole (DD) interactions between atoms, is modelled by the 2D discrete Gross-Pitaevskii equation which includes the long-range DD term. Various species of discrete vortex solitons, which are known in the model of the condensate with local interactions, are found to exist in the presence of the DD interaction too. In locally self-attractive condensates, the isotropic DD repulsion, which corresponds to the orientation of atomic dipoles perpendicular to the confinement plane, helps to extend the region of the vortex stability, while in the case of anisotropic DD interactions, corresponding to the in-plane orientation of the dipoles, vortices are unstable. In the former case, those vortices which are unstable may evolve into robust ring-shaped breathers. The attractive isotropic DD interaction can create localized vortices in the condensate with the local self-repulsion, but they all are unstable, evolving into single-peak asymmetric structures.
We study the stabilization properties of dipolar Bose–Einstein condensate in a deep one-dimensional optical lattice with an additional external parametrically modulated harmonic trap potential. Through both analytical and numerical methods, we solve a dimensionless nonlocal nonlinear discrete Gross–Pitaevskii equation with both the short-range contact interaction and the long-range dipole–dipole interaction. It is shown that, the stability of dipolar condensate in modulated deep optical lattice can be controled by coupled effects of the contact interaction, the dipolar interaction and the external modulation. The system can be stabilized when the dipolar interaction, the contact interaction, the average strength of potential and the ratio of amplitude to frequency of the modulation satisfy a critical condition. In addition, the breather state, the diffused state and the attractive-interaction-induced-trapped state are predicted. The dipolar interaction and the external modulation of the lattice play important roles in stabilizing the condensate.