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 examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general threedimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the ('Floquet-Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small 'peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.numerical study of a small crystal in a linear Paul trap, and conclude the paper in section 6 with comments on directions for further applications and research. Overview of Paul trappingThe first crystallization of a cloud of charged aluminum microparticles has been reported in [42]. Wigner crystals of 2 to approximately 100 trapped ions in a Paul trap were reported in [9,10] and further investigated in [43] both experimentally and numerically. The simulations included rf trapping, Coulomb interaction, laser cooling and random noise. Depending on trap parameters, the ions were found to equilibrate either as an apparently chaotic cloud or in an ordered structure. The latter is defined as the 'crystal' solution when it is a simple limit cycle, with the ions oscillating at the rf frequency about well-defined average points. The transition between the two phases has been investigated, it was shown that both phases can coexist and hysteresis in the transition has been observed [43].The motion of two ions in the Paul trap has been investigated in detail in various publications [44][45][46][47][48][49][50][51][52][53]. In addition to the aforementioned phases, frequency-locked periodic attractors (where the nonlinearity pulls the motional frequencies into integral fractions of the external rf frequency) were found in numerical simulations and experiments. These solutions are different from the crystal in that the ions move in extended (closed) orbits in the trap, whose period is a given multiple of the rf period. However, many of these frequency-locked solutions are unstable, especially those of a large period, and perturbations such as those coming from the nonlinearity of the laser-cooling mechanism tend to destroy them.Despite the large amplitude motion, the frequency-locked solutions, being periodic, are of course not chaotic. However, even in the presence of cooling, some solutions in the ion trap may behave chaotically for exponentially long times. The authors of [54] suggest that, eventually, all trajectories settle at frequency-locked attractors, at least for two ions at a = 0. Numerical simulations and experiments with more ion...

We propose to realize quantized discrete kinks with cold trapped ions. We show that long-lived solitonlike configurations are manifested as deformations of the zigzag structure in the linear Paul trap, and are topologically protected in a circular trap with an odd number of ions. We study the quantum-mechanical time evolution of a high-frequency, gap separated internal mode of a static kink and find long coherence times when the system is cooled to the Doppler limit. The spectral properties of the internal modes make them ideally suited for manipulation using current technology. This suggests that ion traps can be used to test quantum-mechanical effects with solitons and explore ideas for the utilization of the solitonic internal-modes as carriers of quantum information.Solitons are localized configurations of nonlinear systems which are nonperturbative and topologically protected [1]. Quantum-mechanical properties of solitons, such as squeezing, have been predicted and measured in optical systems [2]. Quantum dynamics has been observed with a single Josephson junction soliton [3]. In waveguide arrays [4,5] and Bose-Einstein condensates [6] solitons are mean field solutions, localized to a few sites of a periodic potential. In chains of coupled particles, solitons are discrete spatial configurations, as in the Frenkel-Kontorova (FK) model [7,8].Discrete solitons of the FK model and its generalizations are referred to as kinks. An important property of kinks is the existence of localized modes. One mode is the kink's translational 'zero-mode', whose frequency generally rises above zero. Other localized modes are known as 'internal modes' [9,10]. Physically they describe 'shapechange' excitations of the kink and typically they are separated by an energy gap from other long-wavelength phononic modes. It was suggested to use the internal mode as a carrier of quantum information [11].Quantum information processing in ion traps [12] has dramatically improved over the last decades [13,14]. Recently there has been a considerable interest in using trapped ions for quantum simulation of various systems such as spin-chains [15][16][17] In this Letter we demonstrate that quantum coherence in static discrete kinks can be observed with ordinary Paul traps without external additions. We explore quasi-2D discrete kinks resembling those of the zigzag model [23]. In the linear trap we find local metastable deformations of the zigzag structure [24], as depicted in Fig. 1, which are long-lived already with a moderate number of ions, N 20. In a circular trap with an odd number of ions, similar configurations form the ground state. We study the robustness of a high-frequency internal mode of the kink against decoherence in the thermal environment of all the other modes. With all nonlinear interactions accounted for, we numerically integrate a non-Markovian master equation, which leads us to our main result: already at the standard Doppler cooling limit coherence persists in the internal mode for many oscillations. This could allow ...

We expand the solutions of linearly coupled Mathieu equations in terms of infinite continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the ('Floquet-Lyapunov') transformation to coordinates in which the motion is that of decoupled linear oscillators. We use this transformation to solve the Heisenberg equations of the corresponding quantum-mechanical problem, and find the quantum wavefunctions for stable oscillations, expressed in configuration-space. The obtained transformation and quantum solutions can be applied to more general linear systems with periodic coefficients (coupled Hill equations, periodically driven parametric oscillators), and to nonlinear systems as a starting point for convenient perturbative treatment of the nonlinearity.

The accurate characterization of eigenmodes and eigenfrequencies of two-dimensional ion crystals provides the foundation for the use of such structures for quantum simulation purposes. We present a combined experimental and theoretical study of two-dimensional ion crystals. We demonstrate that standard pseudopotential theory accurately predicts the positions of the ions and the location of structural transitions between different crystal configurations. However, pseudopotential theory is insufficient to determine eigenfrequencies of the two-dimensional ion crystals accurately but shows significant deviations from the experimental data obtained from resolved sideband spectroscopy. Agreement at the level of 2.5×10 −3 is found with the full time-dependent Coulomb theory using the Floquet-Lyapunov approach and the effect is understood from the dynamics of two-dimensional ion crystals in the Paul trap. The results represent initial steps towards an exploitation of these structures for quantum simulation schemes. Accurate control of ion crystals is of major importance for spectroscopy, quantum simulation, or quantum computing with such experimental platform. Since the invention of dynamical trapping by Paul [1], this versatile instrument has been adapted and optimized for specific purposes. Charged particles, more specifically singly charged ions, are confined in a radio frequency (rf) potential, which is formed by tailored electrode structures. In the case of the linear Paul trap, one aims for a quadrupole field along one z axis, such that a harmonic pseudopotential in x and y direction is formed. This radial potential strongly confines the ions, while an additional weaker axial potential in z direction is generated with static (dc) voltages applied to end cap electrodes. Trapped ions are cooled by laser radiation [2] in the potential described by three trap frequencies ω x,y,z eventually forming a crystalized structure.The conditions of operation are characterized by two anisotropy parameters where the radial confinement ω (x,y) typically exceeds the axial dc confinement ω z . For sufficiently small values of α (x,y) ≡ ω 2 z /ω 2 (x,y) , the ion crystal is linear and aligned along the weakest axis, the z trap axis; all ions are placed in the node of the rf potential. Spectacular highlights using linear crystals of cold ions are the demonstration of quantum logic operations [3,4], the generation of entangled states [5,6], sympathetic cooling of ions of different species [7,8], or the quantum-logic clock [9]. To reach the level of quantum control, as required in the experiments listed above, the first precondition was a complete understanding of eigenmodes and eigenfrequencies for such stored linear ion crystals [10][11][12].For larger numbers of ions, or for larger values of α, the linear crystal undergoes a transition to a zigzag structure and eventually to a fully crystalline two-or threedimensional structure [13,14]. Especially interesting are planar ion crystals, where usually one of the confining radial potentia...