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arXiv:2311.15567 · Physical Review A

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

Hongyi Huang, Kunal K. Das

Published Nov 22, 2021• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.104.053320

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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 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 analyze the dynamics of Bose-Einstein condensates loaded in rotating ring lattices made of a few sites, and show how rotation maps the states found in this finite system onto those belonging to a static infinite lattice. Ring currents and soliton states in the absence of a lattice find their continuation in the presence of the lattice as nonlinear Bloch waves and soliton-like states connecting them. Both bright gap solitons and dark-soliton trains are shown to connect continuously to linear solutions. The existence of adiabatic paths upon varying rotation frequency between states with quantized supercurrents suggests highly controllable methods for the experimental generation of persistent currents.
We consider ultracold atoms trapped in a toroidal trap with an azimuthal lattice for utility as a macroscopic simulator of quantum optics phenomena. We examine the dynamics induced by the adiabatic introduction of the lattice that serves to couple the normal modes as an analog of a laser field coupling electronic states. The system is found to display two distinct behaviors, manifest in the angular momentum---coherent oscillation and self-trapping---reminiscent of nonlinear dynamics yet not requiring interatomic interactions. The choice is set by the interplay of discrete parameters, the specific initial mode, and the periodicity of the lattice. However, rotation can cause continuous transition between the two regimes, causing periodic quenches and revivals in the oscillations as a function of the angular velocity. Curiously, the impact of rotation is determined entirely by the energy spectrum in the absence of the lattice, a feature that can be attributed to adiabaticity. We assess the effects of varying the lattice parameters and consider applications in rotation sensing.
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 demonstrate that all of the salient features of the Harper-Hofstadter model can be implemented with ultracold atoms trapped in a bichromatic ring-shaped lattice. Using realistic sinusoidal lattice potentials rather than assume the idealized tight-binding picture, we determine the optimal conditions necessary to realize the critical point where the spectrum becomes fractal, and we identify the nature and cause of the departures from the discrete model predictions. We also show that even with a commensurate ring with a few lattice sites, the Aubry-Andr\'e localization transition can be realized. Localized states that behave like edge states with energies that reside in the band gaps can be generated by introducing a surprisingly small local perturbation within the ring. Spectrum oscillation arising from complex coupling can be implemented by uniform rotation of the ring, but with certain significant differences that are explained.
Properties of bosonic atoms in small systems with a periodic quasi one-dimensional circular toroidal lattice potential subjected to rotation are examined by performing exact diagonalization in a truncated many body space. The expansion of the many-body Hamiltonian is considered in terms of the first band Bloch functions, and no assumption regarding restriction to nearest-neighbor hopping (tight-binding approximation) is involved. A finite size version of the zero temperature phase diagrams of Fisher et al. \cite{Fisher} is obtained and the results, in remarkable quantitative correspondence with the results available for larger systems, discussed. Ground state properties relating to superfluidity are examined in the context of two-fluid phenomenology. The basic tool, consisting of the intrinsic inertia associated with small rotation angular velocities in the lab frame, is used to obtain ground state `superfluid fractions' numerically. They are analytically associated with one-body, uniform solenoidal currents in the case of the adopted geometry. These currents are in general incoherent superpositions of contributions from each eigenstate of the associated reduced one-body densities, with the corresponding occupation numbers as weights. Full coherence occurs therefore only when only one eigenstate is occupied by all bosons. The obtained numerical values for the superfluid fractions remain small throughout the parameter region corresponding to the `Mott insulator to superfluid' transition, and saturate at unity only as the lattice is completely smoothed out.