Two-dimensional Coulomb Crystals Research Articles

Dynamics of vortex defect formation in two-dimensional Coulomb crystals

We study the non-equilibrium dynamics of two dimensional planar ion Coulomb crystals undergoing a structural buckling transition to a three plane configuration, driven by a reduction of the transverse confining frequency. This phase transition can be theoretically modeled using a mapping to a two dimensional Ginzburg-Landau theory with complex order parameter field. We demonstrate that finite rate quenches result in creation of stable topological vortices, which are localized point regions around which the phase of the order parameter field winds a multiple of 2{\pi}. The density of the defects as a function of quench rate is investigated using molecular dynamics simulations and its scaling is shown to be consistent with Kibble-Zurek theory of defect formation. Following the quench, the annihilation of vortex and anti-vortex pairs results in the relaxation of defect density that follows a diffusive scaling with a logarithmic correction. This work highlights the potential for investigating complex non-equilibrium statistical physics of topological defects in an experimentally accessible ion trap setting.

Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals.

Fully controllable ultracold atomic systems are creating opportunities for quantum sensing, yet demonstrating a quantum advantage in useful applications by harnessing entanglement remains a challenging task. Here, we realize a many-body quantum-enhanced sensor to detect displacements and electric fields using a crystal of ~150 trapped ions. The center-of-mass vibrational mode of the crystal serves as a high-Q mechanical oscillator, and the collective electronic spin serves as the measurement device. By entangling the oscillator and collective spin and controlling the coherent dynamics via a many-body echo, a displacement is mapped into a spin rotation while avoiding quantum back-action and thermal noise. We achieve a sensitivity to displacements of 8.8 ± 0.4 decibels below the standard quantum limit and a sensitivity for measuring electric fields of 240 ± 10 nanovolts per meter in 1 second. Feasible improvements should enable the use of trapped ions in searches for dark matter.

Verification of a Many-Ion Simulator of the Dicke Model Through Slow Quenches across a Phase Transition.

We use a self-assembled two-dimensional Coulomb crystal of ∼70 ions in the presence of an external transverse field to engineer a simulator of the Dicke Hamiltonian, an iconic model in quantum optics which features a quantum phase transition between a superradiant (ferromagnetic) and a normal (paramagnetic) phase. We experimentally implement slow quenches across the quantum critical point and benchmark the dynamics and the performance of the simulator through extensive theory-experiment comparisons which show excellent agreement. The implementation of the Dicke model in fully controllable trapped ion arrays can open a path for the generation of highly entangled states useful for enhanced metrology and the observation of scrambling and quantum chaos in a many-body system.

Buckling Transitions and Clock Order of Two-Dimensional Coulomb Crystals

Crystals of repulsively interacting ions in planar traps form hexagonal lattices, which undergo a buckling instability towards a multi-layer structure as the transverse trap frequency is reduced. Numerical and experimental results indicate that the new structure is composed of three planes, whose separation increases continuously from zero. We study the effects of thermal and quantum fluctuations by mapping this structural instability to the six-state clock model. A prominent implication of this mapping is that at finite temperature, fluctuations split the buckling instability into two thermal transitions, accompanied by the appearance of an intermediate critical phase. This phase is characterized by quasi-long-range order in the spatial tripartite pattern. It is manifested by broadened Bragg peaks at new wave vectors, whose line-shape provides a direct measurement of the temperature dependent exponent $\eta(T)$ characteristic of the power-law correlations in the critical phase. A quantum phase transition is found at the largest value of the critical transverse frequency: here the critical intermediate phase shrinks to zero. Moreover, within the ordered phase, we predict a crossover from classical to quantum behavior, signifying the emergence of an additional characteristic scale for clock order. We discuss experimental realizations with trapped ions and polarized dipolar gases, and propose that within accessible technology, such experiments can provide a direct probe of the rich phase diagram of the quantum clock model, not easily observable in condensed matter analogues. Therefore, this works highlights the potential for ionic and dipolar systems to serve as simulators for complex models in statistical mechanics and condensed matter physics.

Creation of two-dimensional Coulomb crystals of ions in oblate Paul traps for quantum simulations

We develop the theory to describe the equilibrium ion positions and phonon modes for a trapped ion quantum simulator in an oblate Paul trap that creates two-dimensional Coulomb crystals in a triangular lattice. By coupling the internal states of the ions to laser beams propagating along the symmetry axis, we study the effective Ising spin-spin interactions that are mediated via the axial phonons and are less sensitive to ion micromotion. We find that the axial mode frequencies permit the programming of Ising interactions with inverse power law spin-spin couplings that can be tuned from uniform to with DC voltages. Such a trap could allow for interesting new geometrical configurations for quantum simulations on moderately sized systems including frustrated magnetism on triangular lattices or Aharonov-Bohm effects on ion tunneling. The trap also incorporates periodic boundary conditions around loops which could be employed to examine time crystals.

Shell models of two-dimensional Coulomb crystals: Assessment and comparison with the three-dimensional case

Three shell models, differing in accuracy and computational cost, are formulated for two-dimensional Coulomb crystals. Offering a new means of predicting and analyzing properties of these species, the new models also provide new insights into their previously derived three-dimensional counterparts. In particular, analysis of the individual components of the energy error points out to the neglect of the positional relaxation as the main source of the differences between the approximate and exact energies. Within the realm of shell models, the two-dimensional case turns out to be somewhat more challenging than the three-dimensional one. Due to the lack of exact closed-form expressions for the optimal shell radii, it is computationally more expensive and the energy predictions at the same level of approximation are less accurate (as indicated by the maximum relative energy error of 0.15% vs. that of 0.03% found for spherical Coulomb crystals).

Packing and melting of mesoscopically confined two-dimensional Coulomb crystals in straight narrow channels

The microstructure and melting dynamics of the two-dimensional mesoscopic Coulomb crystal with 1/r-type mutual interaction force and parabolic transverse confining potential, under different degrees of incommensurability, are investigated through molecular-dynamics simulation. To tune the degree of incommensurability, N(a) extra particles are added into the commensurate uniform triangular lattice which has seven-layer structure and 40 particles in each layer with the periodic longitudinal boundary condition, until the system reaches another commensurate packing with eight-layer structure at N(a)=40. It is found that the increasing incommensurability with the increasing N(a) or 40-N(a) gradually deteriorates the structural order with the presence of intrinsic defects and the anisotropic bond-length distribution, except for the defect-free configurations at a few magic N(a)'s. The system prefers the seven- and the eight-layer single structures through the entire crystal for the low- and the high-N(a) regimes, respectively, and the configurations with seven- and eight-layer domain mixtures for 18≤N(a)≤24. The increasing strain or the worse local particle interlocking around the intrinsic defects with the increasing incommensurability also causes the easier structural rearrangement associated with the easier particle hopping and the earlier onset of melting transition. The transverse confinement suppresses the transverse motion, induces nonuniform melting, and sustains the layered structure after melting.

Formation and rotation of two-dimensional Coulomb crystals in magnetized complex plasma

The formation and rotation of two-dimensional (2D) dust Coulomb crystals with finite number of particles are simulated in a self-consistent manner, based on a dynamical model for radio-frequency sheath over a concave electrode with an axial magnetic field. The dust particles are initially placed at random positions inside the sheath, and then their motions are tracked by a molecular-dynamics method until stable, ordered 2D structures are formed, which rotate in a plane perpendicular to the magnetic field, driven by the azimuthal ion flow due to the E×B effect. The rotation patterns are found to depend, in a complex way, on the particle number, the crystal structure, and the discharge parameters, such as the discharge pressure and the magnetic field. In particular, a rigid rotation is typically observed for those clusters with high symmetry structures, for example, the cluster 19 (1,6,12), with the rotation angular frequency of the order of 10−2Hz, which is proportional to the magnetic field magnitude, but decreases with the increasing discharge pressures. However, the intershell rotations are also observed for those structures of the dust crystals with less symmetry, e.g., 20 (1,7,12), especially for strong magnetic fields.

Dispersion properties of the out-of-plane transverse wave in a two-dimensional Coulomb crystal.

The formation of a two-dimensional (2D) Coulomb crystal in a typical experimental environment was simulated with a computer code called BOX_TREE. The dispersion properties of a novel dust lattice wave (DLW) mode, the out-of-plane transverse wave, were obtained. The dispersion relation was determined to be an opticlike inverse dispersion when wave number k is lower than a critical value k(critical), and a positive dispersion when k>k(critical). The negative group velocity of the wave for k<k(critical) depends on the kappa value (with kappa=a/lambda(D), where a is the interparticle spacing and lambda(D) is the Debye length) and the positive group velocity for k>k(critical) depends on the propagation direction. The value of k(critical) depends on both kappa and propagation direction, but changes very little for all propagation directions and the range of kappa investigated. An analytical method has also been used to derive the dispersion relations assuming a hexagonal 2D lattice and Yukawa interparticle potential. These dispersion relations compare favorably with the simulation results. The dispersion relation for a 1D string was also obtained via BOX_TREE simulation and shown to agree with the analytical result given by Vladimirov [Physica A 315, 222 (2002)]. Comparison shows that the out-of-plane transverse DLW in a 2D lattice when k<k(critical) has a negative group velocity much larger than that of the 1D string, given the same particle parameters and operating environment. Again k(critical) for 1D string and 2D lattice are in the same range.

Mach Cones in a Coulomb Lattice and a Dusty Plasma

Mach cones, or V-shaped disturbances created by supersonic objects, have been detected in a two-dimensional Coulomb crystal. Electrically charged microspheres levitated in a glow-discharge plasma formed a dusty plasma, with particles arranged in a hexagonal lattice in a horizontal plane. Beneath this lattice plane, a sphere moved faster than the lattice sound speed. Mach cones were double, first compressive then rarefactive, due to the strongly coupled crystalline state. Molecular dynamics simulations using a Yukawa potential also show multiple Mach cones.

Two-Dimensional Melting in a Coulomb Crystal of Dusty Plasmas

Two-dimensional melting was observed in a simple hexagonal Coulomb crystal of dusty plasmas. When gas pressure was increased, the particle arrangement of crystalline structure changed from the ordered to the disordered phase. In the change, translational and bond-orientational orders of the arrangement were disturbed. On the other hand, Monte Carlo simulation was employed to examine the experimental results. In the Yukawa system observed in the simulation, where the interaction between the charged particles became weaker, the translational order became short range at first. After that, the bond-orientational order also became short range. According to both experimental and simulation results, melting transition was induced by decreasing the Coulomb interaction between particles with increasing pressure in the laboratory dusty plasma. The dynamics of two-dimensional Coulomb crystals, especially two-dimensional melting, was partly clarified in this work.

Damage and annealing in two-dimensional Coulomb crystals

Experiments have shown that the attenuation of shear modes in a 2D classical Coulomb crystal, formed from helium ions trapped below the free surface of superfluid helium, is sensitive to damage in the crystal, and it can therefore be used to monitor the annealing process that follows deliberate damage to the crystal. The characteristic time required for annealing in a circular crystal of radius 13 mm is found to be of order 500 s, and it rises only slowly with decreasing temperature. It is suggested that such behaviour requires the production of configurations of dislocations that can disappear only through dislocation climb, which requires the absorption and emission of vacancies or interstitials, and that the relatively small temperature dependence implies that there must be some source of vacancies or interstitials within the crystal even at the lowest temperatures.

Damage and annealing in two-dimensional Coulomb crystals

We have demonstrated experimentally that strong excitation of shear modes, exceeding the shear strength of the crystal, results in severe damage in a 2D crystal of positive ions trapped below the free surface of superfluid helium. Annealing takes place in times that are of order 500 s and not strongly temperature dependent.

Shear modes in two-dimensional ionic Coulomb crystals

We report an experimental investigation of shear modes propagating in two-dimensional Coulomb crystals of positive ions trapped below the surface of superfluid helium. The dependence of the shear-mode resonant frequencies and linewidths on external drive, temperature, magnetic field, and pool radius has been investigated.