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Chinese Physics B

Analytical treatment of Anderson localization in a chain of trapped ions experiencing laser Bessel beams

Jun Wen, Jian‐Qi Zhang, L.-L. Yan, Mang Feng

Published Jan 1, 2019• 1 citation
Publisher

Abstract

Trapped ions, under electromagnetic confinement and Coulomb repulsion, can behave as non-interacting particles in one-dimensional lattices. Here we explore analytically the possible effects regarding Anderson localization in a chain of trapped ions experiencing laser Bessel beams. Under an experimentally feasible condition, we predict an analytical form of the energy-dependent mobility edges, which is verified to be in good agreement with the exact numerical results except for the top band. Some other important properties regarding the phonon localization in the ion chain are also discussed both analytically and numerically. Our results are relevant to experimental observation of localization–delocalization transition in the ion trap and helpful for deeper understanding of the rich phenomena due to long-range phonon hopping.

DOI
10.1088/1674-1056/28/1/010306

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Trapped ions, under electromagnetic confinement and Coulomb repulsion, can behave as non-interacting particles in one-dimensional lattices. Here we explore analytically the possible effects regarding Anderson localization in a chain of trapped ions experiencing laser Bessel beams. Under an experimentally feasible condition, we predict an analytical form of the energy-dependent mobility edges, which is verified to be in good agreement with the exact numerical results except for the top band. Some other important properties regarding the phonon localization in the ion chain are also discussed both analytically and numerically. Our results are relevant to experimental observation of localization–delocalization transition in the ion trap and helpful for deeper understanding of the rich phenomena due to long-range phonon hopping.
Trapped ions offer an excellent platform to investigate fascinating phenomena in quantum physics. Here we propose an experimental scheme to achieve Anderson localization of phonons using an ion chain under irradiation of a laser Bessel beam. The quasiperiodicity comes from the characteristic of the Bessel beam, which is only relevant to the positions of the individual ions with respect to the standing wave of the Bessel beam, rather than the internal degrees of freedom of the ions. We demonstrate, by the inverse participation ratio, the delocalization-localization transition, which is sensitive to the strength and the phase of the Bessel beam laser. Our model involving long-range phonon hopping could present richer phenomena than the Aubry-Andr\'e model including only nearest-neighbor coupling.
We propose and analyze a scheme to observe topological phenomena with ions in microtraps. We consider a set of trapped ions forming a regular structure in two spatial dimensions and interacting with lasers. We find phonon bands with nontrivial topological properties, which are caused by the breaking of time-reversal symmetry induced by the lasers. We investigate the appearance of edge modes, as well as their robustness against perturbations. Long-range hopping of phonons caused by the Coulomb interaction gives rise to flat bands which, together with induced phonon-phonon interactions, can be used to produce and explore strongly correlated states. Furthermore, some of these ideas can also be implemented with cold atoms in optical lattices.
Energy transport in low-dimensional systems or systems with long-range interactions shows unusual properties not observed in conventional three-dimensional materials. Trapped linear ion chains have been proposed as an ideal platform to study such phenomena, but many predicted features have not been demonstrated yet and the existing experiments mainly focus on the weak excitation limit where the dynamics can be described by linearized phonon modes. Here, we study the energy transport in a chain of 174Yb+^{174}\mathrm{Yb}^{+}174Yb+ ions using a focused heating laser beam. Nontrivial effects are observed such as a nonmonotonic energy distribution with reflection symmetry, oscillatory heating dynamics, and a change in the heating pattern due to the competition between the external driving and interion coupling. Our work deepens the understanding about transport in low-dimensional systems and also provides guidelines for the sympathetic cooling of long ion chains, with applications in quantum information processing and precision measurement.
Driving an ion at a motional sideband transition induces the Jaynes--Cummings (JC) interaction. This JC interaction creates an anharmonic ladder of JC eigenstates, resulting in the suppression of phonon hopping due to energy conservation. Here, we report the confinement of multiple local phonons to a single ion site using this nonlinear quantum effect. By using phonon-number-resolving detection, we observe the dynamics of multiple phonons in an ion chain. In addition to simply confining local phonons, by turning the JC interaction on and off dynamically, we can confine a single local phonon to a single ion site and subsequently retrieve it out of the site. Our work establishes a key technological component for quantum simulation with multiple bosonic particles, which can simulate classically intractable problems.
The electronic and motional degrees of freedom of trapped ions can be controlled and coherently coupled on the level of individual quanta.Assembling complex quantum systems ion by ion while keeping this unique level of control remains a challenging task.For many applications, linear chains of ions in conventional traps are ideally suited to address this problem.However, driven motion due to the magnetic or radio-frequency electric trapping fields sometimes limits the performance in one dimension and severely affects the extension to higher-dimensional systems.Here, we report on the trapping of multiple barium ions in a single-beam optical dipole trap without radio-frequency or additional magnetic fields.We study the persistence of order in ensembles of up to six ions within the optical trap, measure their temperature, and conclude that the ions form a linear chain, commonly called a one-dimensional Coulomb crystal.As a proof-of-concept demonstration, we access the collective motion and perform spectrometry of the normal modes in the optical trap.Our system provides a platform that is free of driven motion and combines advantages of optical trapping, such as state-dependent confinement and nanoscale potentials, with the desirable properties of crystals of trapped ions, such as long-range interactions featuring collective motion.Starting with small numbers of ions, it has been proposed that these properties would allow the experimental study of many-body physics and the onset of structural quantum phase transitions between one-and two-dimensional crystals.
We analyze the behavior of edge states in long-range (LR) interacting systems. In terms of lattice model Hamiltonian with the LR coupling, we determine analytically the condition of existence of edge states within the transfer matrix method (TMM). The expressions we obtain are general and hold for any choice of the LR hopping. The reason why edge states can appear is the transfer matrix in the bulk different from that in the boundary layers. Our predictions are in good agreement with numerical results by exact diagonalization. Our result is helpful in solving novel edge states in one- and two-dimensional (2D) superconductors with LR hopping and pairing.