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arXiv:1211.6867 · Physical Review Letters

Trapping of Topological-Structural Defects in Coulomb Crystals

Manuel Mielenz, J. Brox, S. Kahra, Günther Leschhorn, Magnus Albert, Tobias Schaetz, Haggai Landa, Benni Reznik

Published Mar 28, 2013• 110 citations
arXivHTMLPDF

Abstract

We study experimentally and theoretically structural defects which are formed during the transition from a laser cooled cloud to a Coulomb crystal, consisting of tens of ions in a linear radio frequency trap. We demonstrate the creation of predicted topological defects ("kinks") in purely two-dimensional crystals and also find kinks which show novel dynamical features in a regime of parameters not considered before. The kinks are always observed at the center of the trap, showing a large nonlinear localized excitation, and the probability of their occurrence saturates at ∼0.5. Simulations reveal a strong anharmonicity of the kink's internal mode of vibration, due to the kink's extension into three dimensions. As a consequence, the periodic Peierls-Nabarro potential experienced by a discrete kink becomes a globally confining potential, capable of trapping one cooled defect at the center of the crystal.

DOI
10.1103/physrevlett.110.133004

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We study experimentally and theoretically structural defects which are formed during the transition from a laser cooled cloud to a Coulomb crystal, consisting of tens of ions in a linear radio frequency trap. We demonstrate the creation of predicted topological defects ("kinks") in purely two-dimensional crystals and also find kinks which show novel dynamical features in a regime of parameters not considered before. The kinks are always observed at the center of the trap, showing a large nonlinear localized excitation, and the probability of their occurrence saturates at ∼0.5. Simulations reveal a strong anharmonicity of the kink's internal mode of vibration, due to the kink's extension into three dimensions. As a consequence, the periodic Peierls-Nabarro potential experienced by a discrete kink becomes a globally confining potential, capable of trapping one cooled defect at the center of the crystal.
We propose and numerically demonstrate a deterministic protocol for preparing and tailoring topological defects in two-dimensional single-species Coulomb crystals confined in radio-frequency traps. In conventional finite-rate quenches through the linear-to-zigzag transition, defects form stochastically from independently chosen broken-symmetry domains, and the quench rate controls the resulting domain structure. We instead use an initial three-dimensional helix structure and planarize it into a deterministic defected state. We demonstrate that the number and approximate axial positions of kinks are encoded by the projected nodes of the initial helix. We find that selected helices can remain on metastable finite-winding branches during planarization and yield reproducible single- or multi-kink states, even in the adiabatic limit. Using the quench rate as an additional control parameter substantially enlarges the set of precursor helices that result in defected states, as a fast planarization can prevent reordering during the quench and freeze in the pre-existing domain pattern. After planarization, the survival and mutual interaction of the defects are governed by their effective Peierls--Nabarro potential landscape. By numerically isolating the interaction energy for the two defect types, we find that the interaction of extended kinks is attractive over the investigated parameter range, while odd kinks exhibit a short-range repulsive barrier tunable by the anisotropy of the trapping potential ααα. Finally, we introduce an experimentally motivated post-selection scheme in which ααα is used to selectively destabilize, annihilate, or convert targeted defects. Our results provide a route to deterministic single- and multi-kink preparation in single-species Coulomb crystals, enabling controlled studies of defect interactions, transport, and nanofriction.
Laser-cooled ions held in a linear Paul trap with strong transverse confinement organize into a one-dimensional (1-D) linear crystal. If the transverse confinement is relaxed, the linear ion crystal undergoes a continuous, structural phase transition to a 2-D zigzag configuration. We study the dynamics near the critical point of the linear-zigzag transition. In the first part of this thesis, we study the spontaneous nucleation and dynamics of topological kink defects, formed as a result of a rapid quench across the linear-zigzag transition. The experimental results are compared to the Kibble-Zurek mechanism, which provides an intuitive model of defect formation and predicts a power-law scaling for the number of defectsformed as a function of transition quench rate. The second part of this thesis is focused on one of the key requirements for investigations of the near-transition dynamics in the quantum regime. To achieve an efficient ground state cooling of the zigzag vibrational mode, we demonstrate 3-D polarization-gradient cooling of strings of 1-4 trapped ions as an intermediate step between Doppler and sideband cooling, and study the polarization-gradient cooling rate and cooling limit as a function of the cooling beam intensity in and near the Lamb-Dicke regime. The results of this thesis pave the way towards our future experiments aimed at assessing the coherence time of a zigzag superposition state through measurements of tunneling oscillations.
Laser-cooled and trapped ions can crystallize and feature discrete solitons that are nonlinear, topologically protected configurations of the Coulomb crystal. Such solitons, as their continuum counterparts, can move within the crystal, while their discreteness leads to the existence of a gap-separated, spatially localized motional mode of oscillation above the spectrum. Suggesting that these unique properties of discrete solitons can be used for generating entanglement between different sites of the crystal, we study a detailed proposal in the context of state-of-the-art experimental techniques. We analyze the interaction of periodically driven planar ion crystals with optical forces, revealing the effects of micromotion in radio-frequency traps inherent to such structures, as opposed to linear ion chains. The proposed method requires Doppler cooling of the crystal and sideband cooling of the soliton's localized modes alone. Since the gap separation of the latter is nearly independent of the crystal size, this approach could be particularly useful for producing entanglement and studying system-environment interactions in large, two- and possibly three-dimensional systems.
Trapped ion systems constitute a well controllable scenario for the study and emulation of nanofriction, and in particular of Frenkel-Kontorova-like models. This is in particular the case when a topological defect is created in a zigzag ion Coulomb crystal, which results in an Aubry transition from free sliding to pinned phase as a function of the trap aspect ratio. We explore the quantum effects of the Aubry transition by means of an effective simplified model, in which the defect is treated like a single quantum particle that experiences an effective Peierls-Nabarro potential and a position-dependent mass. We demonstrate the relevance of quantum tunneling in a finite range of aspect ratios close the critical point, showing that the quantum effects may be observed in the kink dynamics for sufficiently low temperatures. Finally, we discuss the requirements to reveal quantum effects at the Aubry transition in future experiments on trapped ions.
Ion Coulomb crystals are ordered structures formed by laser-cooled ions in traps that are characterized by interparticle distances of several micrometers and energy scales on the order of μμμeV. Their crystalline structure emerges from the interplay between Coulomb repulsion and the external confining potential, which can be readily tuned. Moreover, individual ions can be precisely manipulated with lasers and imaged via resonance fluorescence. These unusual and unique properties make ion crystals a powerful platform for studying phases of matter in the strongly correlated regime and at low temperatures where their dynamics is manifestly quantum mechanical. This review examines the theoretical framework and experimental characterization of ion Coulomb crystals from a condensed-matter perspective. We discuss their dynamical and thermodynamic properties in one, two, and three dimensions, and review recent investigations into their out-of-equilibrium behavior. We provide outlooks on future directions for exploring novel condensed matter phenomena with trapped ion crystals, as well as for exploiting these features for scientific and technical applications.
The spontaneous nucleation and dynamics of topological kink defects have been studied in trapped arrays of 41--43 Yb ions. The number of kinks formed as a function of quench rate across the linear-zigzag transition is measured in the underdamped regime of the inhomogeneous Kibble-Zurek theory. The experimental results agree well with molecular dynamics simulations, which show how losses mask the intrinsic nucleation rate. Simulations indicate that doubling the ion number and optimization of laser cooling can help reduce the effect of losses. A range of kink dynamics is observed including configural change, motion and lifetime, and behavioral sensitivity to ion number.