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arXiv:2311.15479 · Physical Review A

Effects of a rotating periodic lattice on coherent quantum states in a ring topology: The case of negative nonlinearity

Jonathan Tekverk, Christopher Siebor, Kunal K. Das

Published Feb 9, 2024• 6 citations
arXivHTMLPDF

Abstract

We study the spectrum and stationary states in a ring-shaped lattice potential in the context of ultracold atoms with attractive interatomic interactions. We determine analytical solutions in the absence of a lattice by mapping them to those for repulsive interactions, and then we numerically follow the transformation of those solutions as the lattice is introduced and strengthened. Several features emerge that are specific to negative nonlinearity, which include soliton branches detaching to create new ground states, gaps opening up at the bottom of the primary spectral branch, and multiple splitting and rejoining of some branches. We correlate the spectral features with the behavior of the density and phase of the corresponding eigenstates, and track them along branches and as various system parameters change. We find that the phase is sensitive to how a specific point in the spectrum is approached, particularly relevant at certain persistent gaps in the spectrum. The symmetry and stability properties are generally found to be opposite of that found for repulsive interactions.

DOI
10.1103/physreva.109.023315

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We study the spectrum and stationary states in a ring-shaped lattice potential in the context of ultracold atoms with attractive interatomic interactions. We determine analytical solutions in the absence of a lattice by mapping them to those for repulsive interactions, and then we numerically follow the transformation of those solutions as the lattice is introduced and strengthened. Several features emerge that are specific to negative nonlinearity, which include soliton branches detaching to create new ground states, gaps opening up at the bottom of the primary spectral branch, and multiple splitting and rejoining of some branches. We correlate the spectral features with the behavior of the density and phase of the corresponding eigenstates, and track them along branches and as various system parameters change. We find that the phase is sensitive to how a specific point in the spectrum is approached, particularly relevant at certain persistent gaps in the spectrum. The symmetry and stability properties are generally found to be opposite of that found for repulsive interactions.
We study the landscape of solutions of the coherent quantum states in a ring-shaped lattice potential in the context of ultracold atoms with an effective positive nonlinearity induced by interatomic interactions. The exact analytical solutions in the absence of lattice are used as a starting point, and the transformation of those solutions is mapped as the lattice is introduced and strengthened. This approach allows a simple classification of all the solutions into states with periods commensurate or incommensurate with the lattice period and those with or without nodes. Their origins are traced to the primary dispersion curve and the swallowtail branches of the lattice-free spectrum. The commensurate states tend to remain delocalized with increasing lattice depth, whereas the incommensurate ones may be localized. The symmetry and stability properties of the solutions are examined and correlated with branch energies. The crucial importance of rotation is highlighted by its utility in continuously transforming solutions and accessing in a finite ring with a few sites the full spectrum of nonlinear Bloch waves on an infinite lattice.
We report on results of a systematic analysis of two-dimensional solitons and localized vortices in models including a rotating periodic potential and the cubic nonlinearity, with the latter being both self-attractive and self-repulsive. The models apply to Bose-Einstein condensates stirred by rotating optical lattices and to twisted photonic-crystal fibers, or bundled arrays of waveguides, in nonlinear optics. In the case of the attractive nonlinearity, we construct compound states in the form of vortices, quadrupoles, and supervortices, all trapped in the slowly rotating lattice, and identify their stability limits (fundamental solitons in this setting were studied previously). In rapidly rotating potentials, vortices decouple from the lattice in the azimuthal direction and assume an annular shape. In the model with the repulsive nonlinearity, which was not previously explored in this setting, gap solitons and vortices are found in both cases of the slow and rapid rotations. It is again concluded that the increase in the rotation frequency leads to the transition from fully trapped corotating vortices to ring-shaped ones. We also study ``crater-shaped'' vortices in the attraction model, which, unlike their compound counterparts, are trapped, essentially, in one cell of the lattice. Previously, only unstable vortices of this type were reported. We demonstrate that they have a certain stability region. Solitons and vortices are found here in the numerical form, and, in parallel, by means of the variational approximation.
We investigate the formation of steady states in one-dimensional Bose-Einstein condensates of repulsively interacting ultracold atoms loaded into a quasiperiodic potential created by two incommensurate periodic lattices. We study the transformations between linear and nonlinear modes and describe the general patterns that govern the birth of nonlinear modes emerging in spectral gaps near band edges. We show that nonlinear modes in a symmetric potential undergo both symmetry-breaking pitchfork bifurcations and saddle-node bifurcations, mimicking the prototypical behaviors of symmetric and asymmetric double-well potentials. The properties of the nonlinear modes differ for bifurcations occurring below and above the mobility edge. In the generic case, when the quasiperiodic potential consists of two incommensurate lattices with a nonzero phase shift between them, the formation of localized modes in the spectral gaps occurs through a cascade of saddle-node bifurcations. Because of the analogy between the Gross-Pitaevskii equation and the nonlinear Schr\"odinger equation, our results can also be applied to optical modes guided by quasiperiodic photonic lattices.
Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with nr=0,1{n}_{r}=0,1nr​=0,1, and 2 radial nodes, respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge mmm, i.e., vorticity. It is found that for repulsive interatomic interactions only the ground state is linearly stable throughout the parameter range examined. Furthermore, this is true for topological charges m=0m=0m=0 or 1; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing (attractive) case the ground state with nr=0{n}_{r}=0nr​=0 and m=0m=0m=0 can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.
We show how the chiral properties of Bose-Einstein condensates subject to current-density interactions and loaded in optical lattices can be observed in the realization of nonlinear Bloch states, whose spectrum lacks the usual periodic structure. Chirality is also manifested by spatially localized states, or gap solitons, which are found for positive rotation rates of the lattice at the energy gaps between the linear energy bands, whereas for negative rotations they appear in the semi-infinite gap of the linear spectrum. The stability of extended and localized states is checked through the spectrum of linear excitations and nonlinear time evolution of perturbed states, and the phenomenon of Bloch oscillations is explored. Our results are obtained in quasi-one-dimensional ring geometries with feasible experimental parameters.
We systematically study the properties of lattice solitons in Bose-Einstein condensates with either attractive or repulsive atom interactions. This is done, by exactly solving the mean-field Gross-Pitaevskii equation in the presence of a periodic potential. We find new families of lattice soliton solutions that are characterized by the position of the energy eigenvalue within the associated band structure. These include lattice solitons in condensates with either attractive or repulsive atom interactions that exist in finite or semi-infinite gaps, as well as nonlinear modes that exhibit atomic population cutoffs.