Laser-cooled arrays or crystals of $^{171}\mathrm{Yb}$${}^{+}$ ions containing a single impurity, $^{172}\mathrm{Yb}$${}^{+}$ isotope, are confined in a linear radio-frequency Paul trap. Site-to-site hopping of the impurity ion, distinguished by a lack of fluorescence, is studied as a function of the $^{171}\mathrm{Yb}$${}^{+}$ laser-cooling parameters and as a function of the anisotropy of the trapping potential. Imaging of the independently resolved crystal sites permits the extraction of the impurity's hopping trajectory, from which the dwell times at a given site can be obtained as well as the spatial distribution of hopping rate and hopping destination. The onset of rapid hopping is found to occur at average thermal energies approaching a significant fraction of the Coulomb potential energy. Furthermore, the hopping rate is enhanced at trap anisotropies near the critical value for the structural phase transition to a two-dimensional zigzag phase. Finally, the hopping mobility of the impurity ion is observed to be highest near the center of the crystal, which may have an intrinsic cause related to the crystal structure and dynamics near the zigzag transition.
Laser-cooled arrays or crystals of ^{171}\mathrm{Yb}$${}^{+} ions containing a single impurity, ^{172}\mathrm{Yb}$${}^{+} isotope, are confined in a linear radio-frequency Paul trap. Site-to-site hopping of the impurity ion, distinguished by a lack of fluorescence, is studied as a function of the ^{171}\mathrm{Yb}$${}^{+} laser-cooling parameters and as a function of the anisotropy of the trapping potential. Imaging of the independently resolved crystal sites permits the extraction of the impurity's hopping trajectory, from which the dwell times at a given site can be obtained as well as the spatial distribution of hopping rate and hopping destination. The onset of rapid hopping is found to occur at average thermal energies approaching a significant fraction of the Coulomb potential energy. Furthermore, the hopping rate is enhanced at trap anisotropies near the critical value for the structural phase transition to a two-dimensional zigzag phase. Finally, the hopping mobility of the impurity ion is observed to be highest near the center of the crystal, which may have an intrinsic cause related to the crystal structure and dynamics near the zigzag transition.
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.
With two ions in a spheroidal Paul trap, the harmonic trapping potential turns, by Coulomb repulsion, into a bistable well. Two Ba ions have been confined, laser-cooled, and observed, via their laser-excited resonance scattering, by a spatially resolving photomultiplier, or intensifying CCD camera. A repump laser releases the ions from a metastable state. Well-cooled ions are found localized. When the laser is detuned, the Raman cooling rate and the ions’ temperature vary. A transition from the crystallized state to a toroidal gas takes place. The ions are discriminated when one (i) gets excited into a non-fluorescing metastable state, or (ii) is of another isotopic species. Hopping rates of 1 s −1 are found at high thermal excitation. Elsewhere, the rates drop by two orders of magnitude, and vary resonantly with temperature near potential energy/kinetic energy = 15, where the bright ion’s hopping upstream of the laser beam is five times more likely than downstream. Two discernible ions in a trap represent a microscopic model system for the study of reaction kinetics.
The nonequilibrium dynamics of an ion chain in a highly anisotropic trap is studied when the transverse trap frequency is quenched across the value at which the chain undergoes a continuous phase transition from a linear to a zigzag structure. Within Landau theory, an equation for the order parameter, corresponding to the transverse size of the zigzag structure, is determined when the vibrational motion is damped via laser cooling. The number of structural defects produced during a linear quench of the transverse trapping frequency is predicted and verified numerically. It is shown to obey the scaling predicted by the Kibble-Zurek mechanism, when extended to take into account the spatial inhomogeneities of the ion chain in a linear Paul trap.
Structural defects in ion crystals can be formed during a linear quench of the transverse trapping frequency across the mechanical instability from a linear chain to a zigzag structure. The density of defects after the sweep can be conveniently described by the Kibble–Zurek mechanism (KZM). In particular, the number of kinks in the zigzag ordering can be derived from a time-dependent Ginzburg–Landau equation for the order parameter, here the zigzag transverse size, under the assumption that the ions are continuously laser cooled. In a linear Paul trap, the transition becomes inhomogeneous, since the charge density is larger in the center and more rarefied at the edges. During the linear quench, the mechanical instability is first crossed in the center of the chain, and a front, at which the mechanical instability is crossed during the quench, is identified that propagates along the chain from the center to the edges. If the velocity of this front is smaller than the sound velocity, the dynamics become adiabatic even in the thermodynamic limit and no defect is produced. Otherwise, the nucleation of kinks is reduced with respect to the case in which the charges are homogeneously distributed, leading to a new scaling of the density of kinks with the quenching rate. The analytical predictions are verified numerically by integrating the Langevin equations of motion of the ions, in the presence of a time-dependent transverse confinement. We argue that the non-equilibrium dynamics of an ion chain in a Paul trap constitutes an ideal scenario to test the inhomogeneous extension of the KZM, which lacks experimental evidence to date.
Structural defects in ion crystals can be formed during a linear quench of the transverse trapping frequency across the mechanical instability from a linear chain to a zigzag structure. The density of defects after the sweep can be conveniently described by the Kibble–Zurek mechanism (KZM). In particular, the number of kinks in the zigzag ordering can be derived from a time-dependent Ginzburg–Landau equation for the order parameter, here the zigzag transverse size, under the assumption that the ions are continuously laser cooled. In a linear Paul trap, the transition becomes inhomogeneous, since the charge density is larger in the center and more rarefied at the edges. During the linear quench, the mechanical instability is first crossed in the center of the chain, and a front, at which the mechanical instability is crossed during the quench, is identified that propagates along the chain from the center to the edges. If the velocity of this front is smaller than the sound velocity, the dynamics become adiabatic even in the thermodynamic limit and no defect is produced. Otherwise, the nucleation of kinks is reduced with respect to the case in which the charges are homogeneously distributed, leading to a new scaling of the density of kinks with the quenching rate. The analytical predictions are verified numerically by integrating the Langevin equations of motion of the ions, in the presence of a time-dependent transverse confinement. We argue that the non-equilibrium dynamics of an ion chain in a Paul trap constitutes an ideal scenario to test the inhomogeneous extension of the KZM, which lacks experimental evidence to date.
We experimentally study two-dimensional (2D) Coulomb crystals in the "radial-2D" phase of a linear Paul trap. This phase is identified by a 2D ion lattice aligned entirely with the radial plane and is created by imposing a large ratio of axial to radial trapping potentials. Using arrays of up to 19 ^{171}Yb^{+} ions, we demonstrate that the structural phase boundaries of such crystals are well described by the pseudopotential approximation, despite the time-dependent ion positions driven by intrinsic micromotion. We further observe that micromotion-induced heating of the radial-2D crystal is confined to the radial plane. Finally, we verify that the transverse motional modes, which are used in most ion-trap quantum simulation schemes, are well-predictable numerically and remain decoupled and cold in this geometry. Our results establish radial-2D ion crystals as a robust experimental platform for realizing a variety of theoretical proposals in quantum simulation and computation.