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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Trapped ions, entanglement, and quantum computing

Christopher J. Myatt, B. E. King, D. Kielpinski, Dietrich G. Leibfried, Q. A. Turchette, C. S. Wood, Wayne M. Itano, Christopher R. Monroe, David J. Wineland

Published May 15, 1998• 2 citations
Publisher

Abstract

A miniature, elliptical ring rf ion trap has been sued in recent experiments toward realizing a quantum computer in a trapped ion system. With the combination of small spatial dimensions and high rf drive potentials, around 500 V amplitude, we have achieved secular oscillation frequencies in the range of 5-20 MHz. The equilibrium positions of pairs of ions that are crystallized in this trap lie along the long axis of the ellipse. By adding a static potential to the trap, the micromotion of two crystallized ions may be reduced relative to the case of pure rf confinement. The presence of micromotion reduces the strength of internal transitions in the ion, an effect that is characterized by a Debye-Waller factor, in analogy with the reduction of Bragg scattering at finite temperature in a crystal lattice. We have demonstrated the dependence of the rates of internal transitions on the amplitude of micromotion, and we propose a scheme to use this effect to differentially address the ions.

DOI
10.1117/12.308371

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A miniature, elliptical ring rf (Paul) ion trap has been used in recent esperiments toward realizing a quantum computer in a trapped ion system. With the combination of small spatial dimensions and high rf drive potentials, around 500 V amplitude, we have achieved secular oscillation frequencies in the range of 5-20 MHz. The equilibrium positions of pairs of ions that arc crystallized in this trap lie along the long &\\is of the ellipse. By adding a static potential to the trap, the micromotion of two crystallized ions may be reduced relative to the case of pure rf confinement. The presence of micromotion reduces the strength of internal transitions in the ion, an effect that is characterized by a Debye-Waller factor, in analogy with the reduction of Bragg scattering at finite temperature in a cvstal lattice. We have demonstrated the dependence of the rates of internal transitions on the amplitude of micromotion, and we propose a scheme to use this effect to differentially address the ions.

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A miniature, elliptical ring rf ion trap has been sued in recent experiments toward realizing a quantum computer in a trapped ion system. With the combination of small spatial dimensions and high rf drive potentials, around 500 V amplitude, we have achieved secular oscillation frequencies in the range of 5-20 MHz. The equilibrium positions of pairs of ions that are crystallized in this trap lie along the long axis of the ellipse. By adding a static potential to the trap, the micromotion of two crystallized ions may be reduced relative to the case of pure rf confinement. The presence of micromotion reduces the strength of internal transitions in the ion, an effect that is characterized by a Debye-Waller factor, in analogy with the reduction of Bragg scattering at finite temperature in a crystal lattice. We have demonstrated the dependence of the rates of internal transitions on the amplitude of micromotion, and we propose a scheme to use this effect to differentially address the ions.
A miniature, elliptical ring rf (Paul) ion trap has been used in recent esperiments toward realizing a quantum computer in a trapped ion system. With the combination of small spatial dimensions and high rf drive potentials, around 500 V amplitude, we have achieved secular oscillation frequencies in the range of 5-20 MHz. The equilibrium positions of pairs of ions that arc crystallized in this trap lie along the long &\\is of the ellipse. By adding a static potential to the trap, the micromotion of two crystallized ions may be reduced relative to the case of pure rf confinement. The presence of micromotion reduces the strength of internal transitions in the ion, an effect that is characterized by a Debye-Waller factor, in analogy with the reduction of Bragg scattering at finite temperature in a cvstal lattice. We have demonstrated the dependence of the rates of internal transitions on the amplitude of micromotion, and we propose a scheme to use this effect to differentially address the ions.
The properties of a rf quadrupole trap, the elliptical ion trap, are derived. Elliptical traps can confine large numbers of ions in the Lamb-Dicke regime due to a hitherto unrecognized mechanism unique to one-dimensional Coulomb crystals, implicit in the theories of Dubin and Schiffer. This follows from a linear crystal stability condition, which uniquely relates the crystal size to ellipticity, and a micromotion relation, which reveals a 1/5-root dependence on the number of trapped ions. Elliptical traps offer several advantages over linear traps in the Cirac-Zoller model of quantum computing, both for initial tests and as a potential method of creating a full-scale quantum computer. Numerical solutions of a one-electrode structure show that microscopic elliptical traps, each containing $\ensuremath{\approx}100$ ions, can be constructed at a density of 100 traps/cm2,{\mathrm{cm}}^{2},cm2, making possible arrays containing >106>{10}^{6}>106 ions in the Lamb-Dicke regime for precision spectroscopy or quantum computation.
Summary form only given. The Cirac-Zoller model of quantum computation assumes that ions are confined in a linear ion trap. However, no experimental test of quantum computation or even of Raman cooling has yet been performed in a linear trap; instead the NIST group has used a miniature Paul trap in which a slight asymmetry or ellipticity is introduced to define an axis of crystallization. In this paper the case of arbitrary ellipticity is considered and two results for the crystal size and micromotion amplitude are derived. The first relation is a linear crystal stability condition, which shows that the maximum number of ions which can be confined is uniquely related to the ellipticity by a 7/3 power law. The second result shows that the micromotion in elliptical traps is far smaller than one would expect naively, specifically that the micromotion amplitude depends approximately on the 1/5 power of the number of trapped ions.
Trapping and cooling large numbers of ions, each confined to much less than an optical wavelength, is of great interest for high accuracy spectroscopy, improved frequency standards and atom-radiation field experiments. This can be achieved in a linear rf ion trap where the rf electric fields go to zero along a line rather than at only a point. As a step toward this goal we have constructed a linear trap and have observed simple crystallized linear structures of up to tens of 199Hg+ ions. By varying the axial well depth (determined by a static potential) we have also observed more complex crystallized structures. The ion positions in these structures are in close agreement with a calculation that used as input the measured secular frequencies of the trap. We have also observed the 40.5-GHz ground-state hyperfine transition of trapped 199Hg+ ions by microwave/optical double resonance.
Trapped-ion quantum information processors are among the leading candidates to realise large-scale quantum computing. In trying to scale the number of confined ions in radio-frequency (rf) traps, several challenges arise due to the use of the rf fields. Ion placement is restricted to the vicinity of the rf null in order to minimise excess micromotion. Miniaturisation of trap geometries is desirable for increased ion density, but is at odds with the need to deliver high-voltage rf and tolerate the resulting power dissipation. We demonstrate a platform based on Penning traps, where ions are radially confined using a strong magnetic field. Static electric fields are then sufficient to provide confinement along the axis. We confine a single9Be + ion in a 3 T magnetic field using a micro-fabricated trap and realise transport to arbitrary positions above the trap chip. Furthermore, we show full quantum control of the spin and motion of the trapped ion. Ground state cooling of all motional modes is achieved and we measure the rate of motional heating due to electric field noise in the absence of an rf drive, obtaining noise levels below those observed in similar rf traps. Since only static fields are required for ion confinement, we can temporarily electrically isolate the trap electrodes from technical noise using switches, further reducing the observed electric field noise.
Ion traps are a foundational technology in quantum information processing, precision spectroscopy and foundational physics tests. At the heart of their performance is the inhibition of micromotion, an oscillation of ions through imperfect trapping potentials, that restricts the laser cooling efficiency, coherence times, and the accuracy of measurements. The paper provides a systematic computational analysis of four different electrode geometries (Linear Paul, Segmented, Surface and Ring) in 24 different RF / DC configurations (C01-C24) and 13 performance parameters that comprise trap depth (0.62 1.48 eV), radial secular frequency (1,100 1 Multi-objective optimization with weighted scoring of these metrics indicates that Ring trap geometry scores the highest total (80.2/100), followed by Linear Paul (72.0), Segmented (67.0) and Surface (55.1) configurations. ANOVA (one-way) statistically verified the presence of statistically significant inter-geometry differences in micro-motions amplitude (F = 18.43, p < .001) and cooling efficiency (F = 22.17, p < .001), with Tukey HSD post-hoc revealed that the most significant difference was between the Ring and the Surface. The Pearson correlation analysis shows that RF drive voltage and micromotion amplitude (r = -0.94 to -0.99) and cooling efficiency and micromotion (r = -1.00) have near-perfect inverse correlations, which prove that the quality of confinement is regulated by electrode geometry, RF amplitude, and spacing. These results give quantitative design principles of next-generation ion traps in quantum computing and atomic clock. Keywords: ion trap, micromotion minimization, RF/DC electrode optimization, laser cooling, Mathieu stability, secular frequency, quantum computing