Abstract
The tunable band-gap structure is fundamentally important in the dynamics of both linear and nonlinear modes trapped in a lattice because Bloch modes can only exist in the bands of the periodic system and nonlinear modes associating with them are usually confined to the gaps. We reveal that when a momentum operator is introduced into the Gross-Pitaevskii equation (GPE), the bandgap spectra of the periodic system can be shifted upward parabolically by the growth of the constant momentum coefficient. During this process, the band edges become asymmetric, in sharp contrast to the standard GPE with an external periodic potential. Extended complex Bloch modes with asymmetric profiles can be derived by applying a phase transformation to the symmetric profiles. We find that the inherent parity-time symmetry of the complex system is never broken with increasing momentum coefficient. Under repulsive interactions, solitons with different numbers of peaks bifurcating from the band edges are found in finite gaps. We also address the existence of embedded solitons in the generalized two-dimensional GPE. Linear stability analysis corroborated by direct evolution simulations demonstrates that multi-peaked solitons are almost completely stable in their entire existence domains.
Highlights
The Bloch band is crucial to our understanding of periodic systems
While most current studies focusing on the topologically protected unidirectional edge states in honeycomb lattices are based on the combination of spin-orbit coupling (SOC) and Zeeman splitting, little attention has been paid to the dynamics of nonlinear excitations in a scalar Gross-Pitaevskii equation (GPE) with a momentum operator, which can be reduced from the coupled
For a nonzero momentum coefficient, the band edges are no longer symmetric about k = 0, which is in sharp contrast to the band spectrum of the conventional periodic systems
Summary
We start our analysis from the 1D coupled Gross-Pitaevskii equations or nonlinear Shrödinger equations:[18,19] i ∂Ψ ∂t. Where the spinor Ψ = (ψ1, ψ2)T describes the quasi-1D spin-orbit-coupled Bose-Einstein condensates trapped in an optical potential V′(x), σ1,3 are the Pauli matrices, and Ω denotes the Zeeman splitting. V(x) is an external trapping potential and g = ±1 so-called momentum operator with γ(x) being stand for the attractive and repulsive nonlinear interactions. Eq (2) describes a BEC cloud loaded into an optical lattice in the mean-field approach This model equation is made dimensionless by using the characteristic scales of the lattice, length aL = d/π, energy Erec = 2/2 mal[2] and time ωL−1 = /Erec, where d is the lattice period and m the mass of the trapped atoms. Stationary solutions of nonlinear modes can be solved numerically either by the relaxation method or by the Newton-conjugate-gradient method[31]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
