Precise Experimental Investigation of Eigenmodes in a Planar Ion Crystal

Abstract

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...

Ab initiocharacterization of the quantum linear-zigzag transition using density matrix renormalization group calculations

“…The high degree of control they offer makes them an ideal platform for quantum information processing [7][8][9] , quantum communication devices [10][11][12] and quantum simulators [13][14][15][16][17][18][19][20] . Moreover, they constitute a perfect playground for studying general, distinctive features of condensed-phase systems; above all, phase transitions and critical phenomena 3,[20][21][22][23][24][25][26][27][28] . One prominent example is the linear ion chain 2,3,29 , which results from the interplay between long-range Coulomb repulsion and a highly anisotropic confinement due to an ion trap 30 .…”

“…We calculate G(∆j) from the numerical data summing over j sites in (25) such that we disregard sites sitting too close to the boundary. Namely, we include only pairs {j, j + ∆j} located within the central third of the chain.…”

Investigation of two-frequency Paul traps for antihydrogen production

“…The validity of the pseudopotential approximation is discussed in detail in Refs. [28,29]. The pseudopotential for particle B in (5) can be arrived at by alternately considering the limiting cases where q B 1,2 go to zero.…”

“…[30], and application of Floquet theory to Paul traps in Refs. [28,29,31]. Here we simply state that the boundary of stability regions may be found by identifying parameters for which the solution x(τ ) has periodicity T or 2T .…”

Mode shaping in mixed ion crystals of 40Ca2+ and 40Ca+

“…The Rydberg excitation from the 3 2 D 5/2 state, together with the single ion addressing of the 4 2 S 1/2 → 3 2 D 5/2 transition, may allow for a full control over the radial mode structure. In future, one may extend the mode shaping to two-dimensional ion crystals with an even more complex mode structure [24].…”