Ionic crystals in a linear Paul trap

Efficiently loading 40Ca+-27Al+ ion crystal using sympathetic cooling and pulsed laser ablation

We demonstrate a method to efficiently load a pair of ⁴⁰Ca⁺-²⁷Al⁺ ion crystals with sympathetic cooling and pulsed laser ablation, serving as a starting step for the ²⁷Al⁺ clock. We achieved a technique to rapidly detect the loading of hot ions by monitoring the ²S_1/2 → ²D_5/2 narrow transition of ⁴⁰Ca⁺ that couples to the shared motional modes between the two ions. The sympathetic cooling time of the ⁴⁰Ca⁺-²⁷Al⁺ ion pair is measured. Two traps are employed to compare the loading time from two directions and it was found that the loading from the axial direction takes much shorter time than loading from the radial direction of the trap. With the help of adaptively controlled trap potential, our method reduced the average loading time of a ⁴⁰Ca⁺-²⁷Al⁺ pair from 26 to 1 min. This research is an important step for increasing the uptime ratio of the ²⁷Al⁺ optical clock and is useful for other mixed-species ion crystals based on sympathetic cooling.

Square Lattice Formation in a Monodisperse Complex Plasma

We present the first observations of a square lattice formation in a monodisperse complex plasma system, a configurational transition phenomenon that has long been an experimental challenge in the field. The experiments are conducted in a tabletop L-shaped dusty plasma experimental device in a dc glow discharge Argon plasma environment. By a careful control of the vertical potential confining the charged particles as well as the strength of the ion wake charge interactions with the dust particles, we are able to steer the system toward a crystalline phase that exhibits a square lattice configuration. The transition occurs when the vertical confinement strength is slightly reduced below a critical value leading to a buckling of the monodisperse hexagonal 2D dust crystal to form a narrowly separated bilayer state (a quasi-2D state). Some theoretical insights into the transition process are provided through molecular dynamics simulations carried out for the parameters relevant to our experiment.

Two-dimensional supersolidity in a dipolar quantum gas

Supersolid states simultaneously feature properties typically associated with a solid and with a superfluid. Like a solid, they possess crystalline order, manifesting as a periodic modulation of the particle density; but unlike a typical solid, they also have superfluid properties, resulting from coherent particle delocalization across the system. Such states were initially envisioned in the context of bulk solid helium, as a possible answer to the question of whether a solid could have superfluid properties^1-5. Although supersolidity has not been observed in solid helium (despite much effort)⁶, ultracold atomic gases provide an alternative approach, recently enabling the observation and study of supersolids with dipolar atoms^7-16. However, unlike the proposed phenomena in helium, these gaseous systems have so far only shown supersolidity along a single direction. Here we demonstrate the extension of supersolid properties into two dimensions by preparing a supersolid quantum gas of dysprosium atoms on both sides of a structural phase transition similar to those occurring in ionic chains^17-20, quantum wires^21,22 and theoretically in chains of individual dipolar particles^23,24. This opens the possibility of studying rich excitation properties^25-28, including vortex formation^29-31, and ground-state phases with varied geometrical structure^7,32 in a highly flexible and controllable system.

Dynamic characteristics of three-dimensional strongly coupled plasmas

The dynamic structure factor and other dynamic characteristics of strongly coupled one-component plasmas have been studied [Yu. V. Arkhipov et al., Phys. Rev. Lett. 119, 045001 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.045001] using the self-consistent version of the method of moments. Within any version of the latter, the system dielectric function satisfies all involved sum rules and other exact relations automatically, and the advantage of this version is that, in addition, the dynamic characteristics (the dynamic structure factor, the dispersion, and decay parameters of the collective modes) are all expressed in terms of the static ones (the static structure factor) without any adjustment to the simulation data. The approach outlined in the aforementioned Letter is justified in detail and applied mainly to the classical Coulomb systems achieving satisfactory agreement with new numerical simulation data. It is shown how the realm of applicability of the method can be extended to partly degenerate and multicomponent systems, even to simple liquids. Some additional theoretical results are presented in the Supplemental Material.

Long-Range Multibody Interactions and Three-Body Antiblockade in a Trapped Rydberg Ion Chain

Trapped Rydberg ions represent a flexible platform for quantum simulation and information processing that combines a high degree of control over electronic and vibrational degrees of freedom. The possibility to individually excite ions to high-lying Rydberg levels provides a system where strong interactions between pairs of excited ions can be engineered and tuned via external laser fields. We show that the coupling between Rydberg pair interactions and collective motional modes gives rise to effective long-range and multibody interactions consisting of two, three, and four-body terms. Their shape, strength, and range can be controlled via the ion trap parameters and strongly depends on both the equilibrium configuration and vibrational modes of the ion crystal. By focusing on an experimentally feasible quasi one-dimensional setup of ^{88}Sr^{+} Rydberg ions, we demonstrate that multibody interactions are enhanced by the emergence of soft modes associated with, e.g., a structural phase transition. This has a striking impact on many-body electronic states and results-for example-in a three-body antiblockade effect that can be employed as a sensitive probe to detect structural phase transitions in Rydberg ion chains. Our study unveils the possibilities offered by trapped Rydberg ions for studying exotic phases of matter and quantum dynamics driven by enhanced multibody interactions.

Estimation of the ion-trap assisted electrical loads and resulting BBR shift

Capacitive, inductive and resistive loads of an ion-trap system, which can be modelled as LCR circuits, are important to know for building a high accuracy experiment. Accurate estimation of these loads is necessary for delivering the desired radio frequency (RF) signal to an ion trap via an RF resonator. Of particular relevance to the trapped ion optical atomic clock, determination of these loads lead to accurate evaluation of the Black-Body Radiation (BBR) shift resulting from the inaccurate machining of the ion-trap itself. We have identified different sources of these loads and estimated their values using analytical and finite element analysis methods, which are found to be well in agreement with the experimentally measured values. For our trap geometry, we obtained values of the effective inductive, capacitive and resistive loads as: 3.1 μH, 3.71 (1) μH, 3.68 (6) μH; 50.4 pF, 51.4 (7) pF, 40.7 (2) pF; and 1.373 Ω, 1.273 (3) Ω, 1.183 (9) Ω by using analytical, numerical and experimental methods, respectively. The BBR shift induced by the excess capacitive load arising due to machining inaccuracy in the RF carrying parts has been accurately estimated, which results to a fractional frequency shift of 6.6 × 10⁻¹⁷ for an RF of 1 kV at 2π × 15 MHz and with ±10 μm machining inaccuracy. This needs to be incorporated into the total systematic uncertainty budget of a frequency standard as it is about one order of magnitude higher than the present precision of the trapped ion optical clocks.

Stability Analysis of an Array of Magnets: When Will It Jump?

A bidimensional array of magnets whose magnetic moments share the same vertical orientation, and lying on a planar surface, can be gradually compacted. As the density reaches a threshold, the assembly becomes unstable, and the magnets violently pop out of plane. In this Letter, we investigate experimentally and theoretically the maximum packing fraction (or density) of a bidimensional planar assembly of identical cylindrical magnets. We show that the instability can be attributed to local fluctuations of the altitude of the magnets on the planar surface. The maximum density is theoretically predicted assuming dipolar interactions between the magnets and is in excellent agreement with experimental results using a variety of cylindrical magnets.

Revealing Quantum Statistics with a Pair of Distant Atoms

Quantum statistics have a profound impact on the properties of systems composed of identical particles. At the most elementary level, Bose and Fermi quantum statistics differ in the exchange phase, either 0 or π, which the wave function acquires when two identical particles are exchanged. In this Letter, we demonstrate that the exchange phase can be directly probed with a pair of massive particles by physically exchanging their positions. We present two protocols where the particles always remain spatially well separated, thus ensuring that the exchange contribution to their interaction energy is negligible and that the detected signal can only be attributed to the exchange symmetry of the wave function. We discuss possible implementations with a pair of trapped atoms or ions.

Direct Determination of Dynamic Properties of Coulomb and Yukawa Classical One-Component Plasmas

Dynamic characteristics of strongly coupled classical one-component Coulomb and Yukawa plasmas are obtained within the nonperturbative model-free moment approach without any data input from simulations so that the dynamic structure factor (DSF) satisfies the first three nonvanishing sum rules automatically. The DSF, dispersion, decay, sound speed, and other characteristics of the collective modes are determined using exclusively the static structure factor calculated from various theoretical approaches including the hypernetted chain approximation. A good quantitative agreement with molecular dynamics simulation data is achieved.

Nonlinear dispersive waves in repulsive lattices

The propagation of nonlinear waves in a lattice of repelling particles is studied theoretically and experimentally. A simple experimental setup is proposed, consisting of an array of coupled magnetic dipoles. By driving harmonically the lattice at one boundary, we excite propagating waves and demonstrate different regimes of mode conversion into higher harmonics, strongly influenced by dispersion and discreteness. The phenomenon of acoustic dilatation of the chain is also predicted and discussed. The results are compared with the theoretical predictions of the α-Fermi-Pasta-Ulam equation, describing a chain of masses connected by nonlinear quadratic springs and numerical simulations. The results can be extrapolated to other systems described by this equation.

Preparation of Entangled States through Hilbert Space Engineering

We apply laser fields to trapped atomic ions to constrain the quantum dynamics from a simultaneously applied global microwave field to an initial product state and a target entangled state. This approach comes under what has become known in the literature as "quantum Zeno dynamics" and we use it to prepare entangled states of two and three ions. With two trapped ^{9}Be^{+} ions, we obtain Bell state fidelities up to 0.990_{-5}^{+2}; with three ions, a W-state fidelity of 0.910_{-7}^{+4} is obtained. Compared to other methods of producing entanglement in trapped ions, this procedure can be relatively insensitive to certain imperfections such as fluctuations in laser intensity.

Stability Diagrams for Paul Ion Traps Driven by Two-Frequencies

In this paper, we present and discuss stability diagrams for Paul traps driven by two ac voltages. In contrast to a typical Paul trap, here we suggest a secondary ac voltage whose frequency is twice the frequency of the primary one. The ratio between their amplitudes can be used to expand the region of stability and to access different states of motion of trapped ions. This provides a further mechanism to trap, cool, and manipulate single ions and also to improve the experimental framework where ion clouds and crystals can be prepared and controlled. Such approach opens the possibility of designing more sophisticated trapping architectures, leading to a wide variety of applications on ion trap research and mass analysis techniques.

Active stabilization of ion trap radiofrequency potentials

We actively stabilize the harmonic oscillation frequency of a laser-cooled atomic ion confined in a radiofrequency (rf) Paul trap by sampling and rectifying the high voltage rf applied to the trap electrodes. We are able to stabilize the 1 MHz atomic oscillation frequency to be better than 10 Hz or 10 ppm. This represents a suppression of ambient noise on the rf circuit by 34 dB. This technique could impact the sensitivity of ion trap mass spectrometry and the fidelity of quantum operations in ion trap quantum information applications.

Rotational dynamics of a diatomic molecular ion in a Paul trap

We present models for a heteronuclear diatomic molecular ion in a linear Paul trap in a rigid-rotor approximation, one purely classical and the other where the center-of-mass motion is treated classically, while rotational motion is quantized. We study the rotational dynamics and their influence on the motion of the center-of-mass, in the presence of the coupling between the permanent dipole moment of the ion and the trapping electric field. We show that the presence of the permanent dipole moment affects the trajectory of the ion and that it departs from the Mathieu equation solution found for atomic ions. For the case of quantum rotations, we also evidence the effect of the above-mentioned coupling on the rotational states of the ion.

Note: Measuring capacitance and inductance of a helical resonator and improving its quality factor by mutual inductance alteration

Narrow bandwidth and high voltage radio frequency (RF) is an essential requirement for stable confinement of ions within a RF trap and helical resonators are commonly used for that purpose. Effective capacitance and inductance of a helical resonator are estimated by measuring resonant frequencies for different external loads. Load capacitance of an ion trap can be estimated from this method and a resonator can be constructed for desired resonant frequency. We demonstrate a very simple method to achieve higher Q-factor of a resonator by optimizing mutual separation between the primary antenna and helical coil. We also formulate a set of analytical equations for calculating overall inductance, resistance, and Q-factor of a loaded helical resonator.

On-demand electrostatic coupling of individual precharacterized nano- and microparticles in a segmented Paul trap

We present a novel versatile method for one-by-one coupling of single nano- and microparticles. The particles are levitated in a segmented linear Paul trap, which is ideal for fast particle characterization and assembly of two or more preselected particles by electrostatic attraction. The final compound particles remain in the trap or can be deposited on other structures. We present the assembly protocol with a theoretical background of particle stability. Results for different particle combinations showing electromagnetic coupling are presented as well as a method for the deposition on optical fibers.

Quantum computation under micromotion in a planar ion crystal

We propose a scheme to realize scalable quantum computation in a planar ion crystal confined by a Paul trap. We show that the inevitable in-plane micromotion affects the gate design via three separate effects: renormalization of the equilibrium positions, coupling to the transverse motional modes, and amplitude modulation in the addressing beam. We demonstrate that all of these effects can be taken into account and high-fidelity gates are possible in the presence of micromotion. This proposal opens the prospect to realize large-scale fault-tolerant quantum computation within a single Paul trap.

Longitudinal and Transverse Single File Diffusion in Quasi-1D Systems

We review our recent results on Single File Diffusion (SFD) of a chain of particles that cannot cross each other, in a thermal bath, with long ranged interactions, and arbitrary damping. We exhibit new behaviors specifically associated to small systems and to small damping. The fluctuation dynamics is explained by the decomposition of the particles' motion in the normal modes of the chain. For longitudinal fluctuations, we emphasize the relevance of the soft mode linked to the translational invariance of the system to the long time SFD behavior. We show that close to the zigzag threshold, the transverse fluctuations also exhibit the SFD behavior, characterized by a mean square displacement that increases as the square root of time. This cannot be explained by the single file ordering, and the SFD behavior results from the strong correlation of the transverse displacements of neighbouring particles near the bifurcation. Extending our analytical modelization, we demonstrate the existence of this subdiffusive regime near the zigzag transition, in the thermodynamic limit. The zigzag transition is a supercritical pitchfork bifurcation, and we show that the transverse SFD behavior is closely linked to the vanishing of the frequency of the zigzag transverse mode at the bifurcation threshold.

[Formula: see text] Special Issue Comments: This article presents mathematical results on the dynamics in files with longitudinal movements. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics,28 expanding files,63 and files with force and advanced formulations.29

Control of quantum thermodynamic behavior of a charged magneto-oscillator with momentum dissipation

In this work we expose the role of environment, confinement, and external magnetic field B in determining the low-temperature thermodynamic behavior in the context of cyclotron motion of a charged oscillator with anomalous dissipative coupling involving momentum instead of the much studied coordinate coupling. Explicit expressions for different quantum thermodynamic functions (QTFs) are obtained at low temperatures for different quantum heat baths characterized by the spectral density function μ(ω). The power-law fall of different QTFs is in conformity with the third law of thermodynamics; however, the sensitivity of decay, i.e., the power of the power-law decay, explicitly depends on μ(ω). We also discuss separately the influence of confinement and magnetic field on the low-temperature behavior of different QTFs. In this process we demonstrate how to control the low-temperature behavior of anomalous dissipative quantum systems by varying the confining length a, B, and the temperature T. Momentum dissipation reduces the effective mass of the system and we also discuss its effect on different QTFs at low temperatures.

Off-resonance energy absorption in a linear Paul trap due to mass selective resonant quenching

Linear Paul traps (LPT) are used in many experimental studies such as mass spectrometry, atom-ion collisions, and ion-molecule reactions. Mass selective resonant quenching (MSRQ) is implemented in LPT either to identify a charged particle's mass or to remove unwanted ions from a controlled experimental environment. In the latter case, MSRQ can introduce undesired heating to co-trapped ions of different mass, whose secular motion is off resonance with the quenching ac field, which we call off-resonance energy absorption (OREA). We present simulations and experimental evidence that show that the OREA increases exponentially with the number of ions loaded into the trap and with the amplitude of the off-resonance external ac field.

Noisy zigzag transition, fluctuations, and thermal bifurcation threshold

We study the zigzag transition in a system of particles with screened electrostatic interaction, submitted to a thermal noise. At finite temperature, this configurational phase transition is an example of noisy supercritical pitchfork bifurcation. The measurements of transverse fluctuations allow a complete description of the bifurcation region, which takes place between the deterministic threshold and a thermal threshold beyond which thermal fluctuations do not allow the system to flip between the symmetric zigzag configurations. We show that a divergence of the saturation time for the transverse fluctuations allows a precise and unambiguous definition of this thermal threshold. Its evolution with the temperature is shown to be in good agreement with theoretical predictions from noisy bifurcation theory.

Heating and thermal squeezing in parametrically driven oscillators with added noise

In this paper we report a theoretical model based on Green's functions, Floquet theory, and averaging techniques up to second order that describes the dynamics of parametrically driven oscillators with added thermal noise. Quantitative estimates for heating and quadrature thermal noise squeezing near and below the transition line of the first parametric instability zone of the oscillator are given. Furthermore, we give an intuitive explanation as to why heating and thermal squeezing occur. For small amplitudes of the parametric pump the Floquet multipliers are complex conjugate of each other with a constant magnitude. As the pump amplitude is increased past a threshold value in the stable zone near the first parametric instability, the two Floquet multipliers become real and have different magnitudes. This creates two different effective dissipation rates (one smaller and the other larger than the real dissipation rate) along the stable manifolds of the first-return Poincaré map. We also show that the statistical average of the input power due to thermal noise is constant and independent of the pump amplitude and frequency. The combination of these effects causes most of heating and thermal squeezing. Very good agreement between analytical and numerical estimates of the thermal fluctuations is achieved.

Cryogenic linear Paul trap for cold highly charged ion experiments

Storage and cooling of highly charged ions require ultra-high vacuum levels obtainable by means of cryogenic methods. We have developed a linear Paul trap operating at 4 K capable of very long ion storage times of about 30 h. A conservative upper bound of the H(2) partial pressure of about 10(-15) mbar (at 4 K) is obtained from this. External ion injection is possible and optimized optical access for lasers is provided, while exposure to black body radiation is minimized. First results of its operation with atomic and molecular ions are presented. An all-solid state laser system at 313 nm has been set up to provide cold Be(+) ions for sympathetic cooling of highly charged ions.

An apparatus for immersing trapped ions into an ultracold gas of neutral atoms

We describe a hybrid vacuum system in which a single ion or a well-defined small number of trapped ions (in our case Ba(+) or Rb(+)) can be immersed into a cloud of ultracold neutral atoms (in our case Rb). This apparatus allows for the study of collisions and interactions between atoms and ions in the ultracold regime. Our setup is a combination of a Bose-Einstein condensation apparatus and a linear Paul trap. The main design feature of the apparatus is to first separate the production locations for the ion and the ultracold atoms and then to bring the two species together. This scheme has advantages in terms of stability and available access to the region where the atom-ion collision experiments are carried out. The ion and the atoms are brought together using a moving one-dimensional optical lattice transport which vertically lifts the atomic sample over a distance of 30 cm from its production chamber into the center of the Paul trap in another chamber. We present techniques to detect and control the relative position between the ion and the atom cloud.

Enhanced fluctuations of interacting particles confined in a box

We study the position fluctuations of interacting particles aligned in a finite cell that avoid any crossing in equilibrium with a thermal bath. The focus is put on the influence of the confining force directed along the cell length. We show that the system may be modeled as a 1D chain of particles with identical masses, linked with linear springs of varying spring constants. The confining force may be accounted for by linear springs linked to the walls. When the confining force range is increased toward the inside of the chain, a paradoxical behavior is exhibited. The outermost particles fluctuations are enhanced, whereas those of the inner particles are reduced. A minimum of fluctuations is observed at a distance of the cell extremities that scales linearly with the confining force range. Those features are in very good agreement with the model. Moreover, the simulations exhibit an asymmetry in their fluctuations which is an anharmonic effect. It is characterized by the measurement of the skewness, which is found to be strictly positive for the outer particles when the confining force is short ranged.

Electronically excited cold ion crystals

The laser excitation of an ion crystal to high-lying and long-lived electronic states is a genuine many-body process even if in fact only a single ion is excited. This is a direct manifestation of the strong coupling between internal and external dynamics and becomes most apparent in the vicinity of a structural phase transition. Here we show that utilizing highly excited states offers a new approach to the coherent manipulation of ion crystals. This opens up a new route towards the creation of nonclassical motional states in a Paul trap and permits the study of quantum phenomena that rely on a strong coupling between electronic and vibrational dynamics.

Ion cloud model for a linear quadrupole ion trap

If large numbers of ions are stored in a linear quadrupole ion trap, space charge causes the oscillation frequencies of ions to decrease. Ions then appear at higher apparent masses when resonantly ejected for mass analysis. In principle, to calculate mass shifts requires calculating the positions of all ions, interacting with each other, at all times, with a self-consistent space charge field. Here, we propose a simpler model for the ion cloud in the case where mass shifts and frequency shifts are relatively small (ca 0.2% and 0.4%, respectively), the trapping field is much stronger (ca × 10(2)) than the space charge field and space charge only causes small perturbations to the ion motion. The self-consistent field problem need not be considered. As test ions move with times long compared to a cycle of the trapping field, the motion of individual ions can be ignored. Static positions of the ions in the cloud are used. To generate an ion cloud, trajectories of N (ca 10,000) ions are calculated for random times between 10 and 100 cycles of the trapping radio frequency field. The ions are then distributed axially randomly in a trap four times the field radius, r(0) in length. The potential and electric field from the ion cloud are calculated from the ion positions. Near the trap center (distances r< 1r(0)), the potential and electric fields from space charge are not cylindrically symmetric, but are quite symmetric for greater values of r. Trajectories of test ions, oscillation frequencies and mass shifts can then be calculated in the trapping field, including the space charge field. Mass shifts are in good agreement with experiments for reasonable values of the initial positions and speeds of the ions. Agreement with earlier analytical models for the ion cloud, based on a uniform occupation of phase space, or a thermal (Boltzmann) distribution of ions trapped in the effective potential [D. Douglas and N.V. Konenkov, Rapid Commun. Mass Spectrom. 26, 2105 (2012)] is quite good. All three models give similar electric fields to match experimental mass shifts.

Signal-to-noise ratio in parametrically driven oscillators

We report a theoretical model based on Green's functions and averaging techniques that gives analytical estimates to the signal-to-noise ratio (SNR) near the first parametric instability zone in parametrically driven oscillators in the presence of added ac drive and added thermal noise. The signal term is given by the response of the parametrically driven oscillator to the added ac drive, while the noise term has two different measures: one is dc and the other is ac. The dc measure of noise is given by a time average of the statistically averaged fluctuations of the displacement from equilibrium in the parametric oscillator due to thermal noise. The ac measure of noise is given by the amplitude of the statistically averaged fluctuations at the frequency of the parametric pump. We observe a strong dependence of the SNR on the phase between the external drive and the parametric pump. For some range of the phase there is a high SNR, while for other values of phase the SNR remains flat or decreases with increasing pump amplitude. Very good agreement between analytical estimates and numerical results is achieved.

Quantum zigzag transition in ion chains

A string of trapped ions at zero temperature exhibits a structural phase transition to a zigzag structure, tuned by reducing the transverse trap potential or the interparticle distance. The transition is driven by transverse, short wavelength vibrational modes. We argue that this is a quantum phase transition, which can be experimentally realized and probed. Indeed, by means of a mapping to the Ising model in a transverse field, we estimate the quantum critical point in terms of the system parameters, and find a finite, measurable deviation from the critical point predicted by the classical theory. A measurement procedure is suggested which can probe the effects of quantum fluctuations at criticality. These results can be extended to describe the transverse instability of ultracold polar molecules in a one-dimensional optical lattice.

Dynamic properties of one-component strongly coupled plasmas: the sum-rule approach

The dynamic characteristics of strongly coupled one-component plasmas are studied within the moment approach. Our results on the dynamic structure factor and the dynamic local-field correction satisfy the sum rules and other exact relations automatically. A quantitative agreement is obtained with numerous simulation data on the plasma dynamic properties, including the dispersion and decay of collective modes. Our approach allows us to correct and complement the results previously found with other treatments.

Weak chaos and the "melting transition" in a confined microplasma system

We present results demonstrating the occurrence of changes in the collective dynamics of a Hamiltonian system which describes a confined microplasma characterized by long-range Coulomb interactions. In its lower energy regime, we first detect macroscopically the transition from a "crystallinelike" to a "liquidlike" behavior, which we call the "melting transition." We then proceed to study this transition using a microscopic chaos indicator called the smaller alignment index (SALI), which utilizes two deviation vectors in the tangent dynamics of the flow and is nearly constant for ordered (quasiperiodic) orbits, while it decays exponentially to zero for chaotic orbits as exp[-(lambda(1)-lambda(2))t], where lambda(1)>lambda(2)>0 are the two largest Lyapunov exponents. During the melting phase, SALI exhibits a peculiar stairlike decay to zero, reminiscent of "sticky" orbits of Hamiltonian systems near the boundaries of resonance islands. This alerts us to the importance of the Deltalambda=lambda(1)-lambda(2) variations in that regime and helps us identify the energy range over which "melting" occurs as a multistage diffusion process through weakly chaotic layers in the phase space of the microplasma. Additional evidence supporting further the above findings is given by examining the GALI(k) indices, which generalize SALI (=GALI(2)) to the case of k>2 deviation vectors and depend on the complete spectrum of Lyapunov exponents of the tangent flow about the reference orbit.

Soft-landing and optical characterization of a preselected single fluorescent particle on a tapered optical fiber

Tapered optical fibers offer easy access to the evanescent field of their guided modes which is ideal for sensing applications. We introduce a soft-landing technique utilizing a linear Paul trap to select and place a single microparticle on the surface of a tapered optical fiber. This approach allows on-demand functionalization of fragile nanophotonic components with arbitrary particles, e.g., for advanced nanosensors.

Sympathetic cooling of complex molecular ions to millikelvin temperatures

Gas-phase singly protonated organic molecules of mass 410 Da (Alexa Fluor 350) have been cooled from ambient temperature to the hundred millikelvin range by Coulomb interaction with laser-cooled barium ions. The molecules were generated by an electrospray ionization source, transferred to and stored in a radio-frequency trap together with the atomic ions. Observations are well described by molecular dynamics simulations, which are used to determine the spatial distribution and thermal energy of the molecules. In one example, an ensemble of 830 laser-cooled 138Ba+ ions cooled 200 molecular ions to less than 115 mK. The demonstrated technique should allow a large variety of protonated molecules to be sympathetically cooled, including molecules of much higher mass, such as proteins.

Sympathetic cooling of 4He+ ions in a radio-frequency trap

We have generated Coulomb crystals of ultracold 4He+ ions in a linear radio-frequency trap, by sympathetic cooling via laser-cooled 9Be+. Stable crystals containing up to 150 localized He+ ions at approximately 20 mK were obtained. Ensembles or single ultracold He+ ions open up interesting perspectives for performing precision tests of QED and measurements of nuclear radii. This Letter also indicates the feasibility of cooling and crystallizing highly charged atomic ions using 9Be+ as coolant.

Dynamics of an ion chain in a harmonic potential

Cold ions in anisotropic harmonic potentials can form ion chains of various sizes. Here, the density of ions is not uniform, and thus the eigenmodes are not phononic-like waves. We study chains of N>>1 ions and evaluate analytically the long-wavelength modes and the density of states in the short-wavelength limit. These results reproduce with good approximation the dynamics of chains consisting of dozens of ions. Moreover, they allow one to determine the critical transverse frequency required for the stability of the linear structure, which is found to be in agreement with results obtained by different theoretical methods [Phys. Rev. Lett. 71, 2753 (1993)]] and by numerical simulations [Phys. Rev. Lett. 70, 818 (1993)]]. We introduce and explore the thermodynamic limit for the ion chain. The thermodynamic functions are found to exhibit deviations from extensivity.

Eigenmodes and thermodynamics of a Coulomb chain in a harmonic potential

The density of ions trapped in a harmonic potential in one dimension is not uniform. Consequently the eigenmodes are not phononlike waves. We calculate the long-wavelength modes in the continuum limit, and evaluate the density of states in the short-wavelength limit for chains of N>>1 ions. Remarkably, the results that are found analytically in the thermodynamic limit provide a good estimate of the spectrum of excitations of small chains down to few tens of ions. The spectra are used to compute the thermodynamic functions of the chain. Deviations from the extensivity of the thermodynamic quantities are found. An analytic expression for the critical transverse frequency determining the stability of a linear chain is derived.

Lyapunov exponent of ion motion in microplasmas

Dynamical chaos is studied in the Hamiltonian motion of ions confined in a Penning trap and forming so-called microplasmas. The dynamical chaos of the ion motion is characterized by the maximum Lyapunov exponent. Results are reported on the dependence of this exponent on the energy of the system, on the number of ions, as well as on the geometry of the trap. Different dynamical regimes are characterized from the crystalline state to a strongly chaotic regime, and to quasiharmonic motion in the external potential of the trap. Across these regimes, the Lyapunov exponent increases, reaches a maximum value, and decreases as a function of energy. Besides, the maximum value of the Lyapunov exponent increases as a function of the number of ions.

Reentrant transitions in colloidal or dusty plasma bilayers

The phase diagram of crystalline bilayers of particles interacting via a Yukawa potential is calculated for arbitrary screening lengths and particle densities. Staggered rectangular, square, rhombic, and triangular structures are found to be stable including a first-order transition between two different rhombic structures. For varied screening length at fixed density, one of these rhombic phases exhibits both a single and even a double reentrant transition. Our predictions can be verified experimentally in strongly confined charged colloidal suspensions or dusty plasma bilayers.

Quantum information with neutral atoms as qubits

One of the essential features of a quantum computer is a quantum 'register' of well-characterized qubits. Neutral atoms in optical lattices are a natural candidate for such a register. We have demonstrated a patterned-loading technique that can be used to load atoms into large arrays of tightly confined but optically resolvable lattice sites. We have also seen preliminary indications of the Mott-insulator transition, which provides a route for single-atom initialization of the individual sites. Combining the two experiments should allow for large arrays of individually addressable single atoms, a system which provides a starting point for further quantum computation studies.

Charged particle layers in the Debye limit

We develop an equivalent of the Debye-Hückel weakly coupled equilibrium theory for layered classical charged particle systems composed of one single charged species. We consider the two most important configurations, the charged particle bilayer and the infinite superlattice. The approach is based on the link provided by the classical fluctuation-dissipation theorem between the random-phase approximation response functions and the Debye equilibrium pair correlation function. Layer-layer pair correlation functions, screened and polarization potentials, static structure functions, and static response functions are calculated. The importance of the perfect screening and compressibility sum rules in determining the overall behavior of the system, especially in the r--> infinity limit, is emphasized. The similarities and differences between the quasi-two-dimensional bilayer and the quasi-three-dimensional superlattice are highlighted. An unexpected behavior that emerges from the analysis is that the screened potential, the correlations, and the screening charges carried by the individual layers exhibit a marked nonmonotonic dependence on the layer separation.

Crystalline ion beams

By freezing out the motion between particles in a high-energy storage ring, it should be possible to create threads of ions, offering research opportunities beyond the realm of standard accelerator physics. The usual heating due to intra-beam collisions should completely vanish, giving rise to a state of unprecedented brilliance. Despite a continuous improvement of beam cooling techniques, such as electron cooling and laser cooling, the ultimate goal of beam crystallization has not yet been reached in high-energy storage rings. Electron-cooled dilute beams of highly charged ions show liquid-like order with unique applications. An experiment using laser cooling suggested a reduction of intra-beam heating, although the results were ambiguous. Here we demonstrate the crystallization of laser-cooled Mg+ beams circulating in the radiofrequency quadrupole storage ring PALLAS at a velocity of 2,800 m s-1, which corresponds to a beam energy of 1 eV. A sudden collapse of the transverse beam size and the low longitudinal velocity spread clearly indicate the phase transition. The continuous ring-shaped crystalline beam shows exceptional stability, surviving for more than 3,000 revolutions without cooling.

From the cover: temperature, ordering, and equilibrium with time-dependent confining forces

The concepts of temperature and equilibrium are not well defined in systems of particles with time-varying external forces. An example is a radio frequency ion trap, with the ions laser cooled into an ordered solid, characteristic of sub-mK temperatures, whereas the kinetic energies associated with the fast coherent motion in the trap are up to 7 orders of magnitude higher. Simulations with 1,000 ions reach equilibrium between the degrees of freedom when only aperiodic displacements (secular motion) are considered. The coupling of the periodic driven motion associated with the confinement to the nonperiodic random motion of the ions is very small at low temperatures and increases quadratically with temperature.

Decoherence of quantum superpositions through coupling to engineered reservoirs

The theory of quantum mechanics applies to closed systems. In such ideal situations, a single atom can, for example, exist simultaneously in a superposition of two different spatial locations. In contrast, real systems always interact with their environment, with the consequence that macroscopic quantum superpositions (as illustrated by the 'Schrodinger's cat' thought-experiment) are not observed. Moreover, macroscopic superpositions decay so quickly that even the dynamics of decoherence cannot be observed. However, mesoscopic systems offer the possibility of observing the decoherence of such quantum superpositions. Here we present measurements of the decoherence of superposed motional states of a single trapped atom. Decoherence is induced by coupling the atom to engineered reservoirs, in which the coupling and state of the environment are controllable. We perform three experiments, finding that the decoherence rate scales with the square of a quantity describing the amplitude of the superposition state.