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arXiv:2206.03412 · Physical Review A

Localization of ultracold atoms in Zeeman lattices with incommensurate spin-orbit coupling

Dmitry A. Zezyulin, V. V. Konotop

Published Jun 22, 2022• 18 citations
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Abstract

We consider a particle governed by a one-dimensional Hamiltonian in which artificial periodic spin-orbit coupling and the Zeeman lattice have incommensurate periods. Using the best rational approximations to such a quasiperiodic Hamiltonian, the problem is reduced to a description of spinor states in a superlattice. In the absence of constant Zeeman splitting, the system acquires an additional symmetry, which hinders the localization. However, if the lattices are deep enough, then localized states can appear even for Zeeman field with a zero or small mean value. Spatial distribution of localized modes is nearly uniform and is directly related to the topological properties of the effective superlattice: center-of-mass coordinates of modes are determined by Zak phases computed from the superlattice band structure. The best rational approximations feature the ``memory'' effect: Each rational approximation holds the information about the energies and spatial distribution of the modes obtained under preceding, less accurate approximations. Dispersion of low-energy initial wave packets is characterized by the law $\ensuremath{\propto}{t}^{\ensuremath{\beta}}$, with $\ensuremath{\beta}$ varying between $1/2$ at the initial stage and 1 at longer, but still finite-time, evolution. The dynamics of initial wave packets, exciting mainly localized modes, manifests quantum revivals.

DOI
10.1103/physreva.105.063323

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We consider a particle governed by a one-dimensional Hamiltonian in which artificial periodic spin-orbit coupling and the Zeeman lattice have incommensurate periods. Using the best rational approximations to such a quasiperiodic Hamiltonian, the problem is reduced to a description of spinor states in a superlattice. In the absence of constant Zeeman splitting, the system acquires an additional symmetry, which hinders the localization. However, if the lattices are deep enough, then localized states can appear even for Zeeman field with a zero or small mean value. Spatial distribution of localized modes is nearly uniform and is directly related to the topological properties of the effective superlattice: center-of-mass coordinates of modes are determined by Zak phases computed from the superlattice band structure. The best rational approximations feature the ``memory'' effect: Each rational approximation holds the information about the energies and spatial distribution of the modes obtained under preceding, less accurate approximations. Dispersion of low-energy initial wave packets is characterized by the law \ensuremath∝t\ensuremathβ\ensuremath{\propto}{t}^{\ensuremath{\beta}}\ensuremath∝t\ensuremathβ, with \ensuremathβ\ensuremath{\beta}\ensuremathβ varying between 1/21/21/2 at the initial stage and 1 at longer, but still finite-time, evolution. The dynamics of initial wave packets, exciting mainly localized modes, manifests quantum revivals.
The dynamic localization of a two-level atom in a periodic potential under the action of spin-orbit coupling and a weak harmonically varying linear force is studied. We consider optical and Zeeman potentials that are either in phase or out of phase in two spinor components, respectively. The expectation value for the position of the atom after one oscillation period of the linear force is recovered in authentic resonances or in pseudoresonances. The frequencies of the linear force corresponding to authentic resonances are determined by the band structure of the periodic potential and are affected by the spin-orbit coupling. The width or dispersion of the wave packet in authentic resonances is usually minimal. The frequencies corresponding to pseudoresonances do not depend on the type of potential and on the strength of the spin-orbit coupling, while the evolution of excitations at the corresponding frequencies is usually accompanied by significant dispersion. Pseudoresonances are determined by the initial phase of the linear force and by the quasimomentum of the wave packet. Due to the spinor nature of the system, the motion of the atom is accompanied by periodic, but not harmonic, spin oscillations. Under the action of spin-orbit coupling the oscillations of the wave packet can be nearly completely suppressed in optical lattices. Dynamic localization in Zeeman lattices is characterized by doubling of the resonant oscillation periods due to band crossing at the boundary of the Brillouin zone. We also show that higher harmonics in the Fourier expansion of the energy band lead to effective dispersion, which can be strong enough to prevent dynamic localization of the Bloch wave packet.
We theoretically investigate two-particle and ground-state many-particle Anderson localizations of a spin-orbit coupled ultracold atomic Fermi gas trapped in a quasiperiodic potential and subjected to an out-of-plane Zeeman field. We solve exactly the two-particle problem in a finite length system by exact diagonalization and solve approximately the ground-state many-particle problem within the mean-field Bogoliubov-de Gennes approach. At a small Zeeman field, the localization properties of the system are similar to that of a Fermi gas with conventional sss-wave interactions. As the disorder strength increases, the two-particle binding energy increases and the fermionic superfluidity of many particles disappears above a threshold. At a large Zeeman field, where the interatomic interaction behaves effectively like a ppp-wave interaction, the binding energy decreases with increasing disorder strength, and the resulting topological superfluidity shows a much more robust stability against disorder than the conventional sss-wave superfluidity. We also analyze the localization properties of the emergent Majorana fermions in the topological phase. Our results could be experimentally examined in future cold-atom experiments, where the spin-orbit coupling can be induced artificially by using two Raman lasers, and the quasiperiodic potential can be created by using bichromatic optical lattices.
Flat bands and dynamical localization of binary mixtures of Bose-Einstein condensates, with spin-orbit coupling subjected to a deep optical lattice which is shaking in time and to a periodic time modulation of the Zeeman field, are investigated. In contrast with usual dynamical localization in the absence of spin-orbit coupling, we find that to fully suppress the tunneling in the system the optical lattice shaking is not enough, and a proper tuning of the spin-orbit term, achievable via the Zeeman field modulation, is also required. This leads to a sequence of Zeeman parameter values where energy bands become flat, the tunneling in the system is suppressed, and the dynamical localization phenomenon occurs. Exact wave functions at the dynamical localization points show that the binary mixture localizes on a dimer with the two components occupying different sites. This type of localization occurs in exact form also for the ground state of the system at the dynamical localization points in the presence of nonlinearity and remains valid, although in approximate form, for a wide range of the Zeeman parameter around these points. The possibility of observing the above phenomena in real experiments is also briefly discussed.
We study the localization phenomena in a one-dimensional lattice system with a uniformly moving disordered potential. At a low-moving velocity, we find a sliding localized phase in which the initially localized matter wave adiabatically follows the moving potential without diffusion, thus resulting in an initial state memory in the many-body dynamics. Such an intriguing localized phase distinguishes itself from the standard Anderson localization in two aspects: it is not robust against interaction, but persists in the presence of slowly varying perturbations. Such a sliding localized phase can be understood as a consequence of interference between the wave-packet paths under moving quasiperiodic potentials with various periods that are incommensurate with the lattice constant. The experimental realization and detection are also discussed.
We investigate numerically the zero-temperature physics of the one-dimensional Bose-Hubbard model in an incommensurate cosine potential, recently realized in experiments with cold bosons in optical superlattices [L. Fallani et al., Phys. Rev. Lett. 98, 130404 (2007)]. An incommensurate cosine potential has intermediate properties between a truly periodic and a fully random potential, displaying a characteristic length scale (the quasiperiod) which is shown to set a finite lower bound to the excitation energy of the system at special incommensurate fillings. This leads to the emergence of gapped incommensurate band-insulator (IBI) phases along with gapless Bose-glass (BG) phases for strong quasiperiodic potential for both hard-core and soft-core bosons. Enriching the spatial features of the potential by the addition of a second incommensurate component appears to remove the IBI regions, stabilizing a continuous BG phase over an extended parameter range. Moreover, we discuss the validity of the local-density approximation in the presence of a parabolic trap, clarifying the notion of a local BG phase in a trapped system; we investigate the behavior of first- and second-order coherence upon increasing the strength of the quasiperiodic potential; and we discuss the ab initio derivation of the Bose-Hubbard Hamiltonian with quasiperiodic potential starting from the microscopic Hamiltonian of bosons in an incommensurate superlattice.
One-dimensional superlattices with periodic spatial modulations of onsite potentials or tunneling coefficients can exhibit a variety of properties associated with topology or symmetry. Recent developments of ring-shaped optical lattices allow a systematic study of those properties in superlattices with or without boundaries. While superlattices with additional modulating parameters are shown to have quantized topological invariants in the augmented parameter space, we also found localized or zero-energy states associated with symmetries of the Hamiltonians. Probing those states in ultracold atoms is possible by utilizing recently proposed methods analyzing particle depletion or the local density of states. Moreover, we summarize feasible realizations of configurable optical superlattices using currently available techniques.