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Quantum Electronics and Laser Science Conference

Elliptical ion traps for quantum computation

Ralph G. DeVoe

Published Jan 20, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1109/qels.1999.807382

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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.
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.
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.
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.
Two- or three-dimensional Paul traps can confine a large number of ions forming a Wigner crystal, which would provide an ideal architecture for scalable quantum computation except for the micromotion, an issue that is commonly believed to be the obstacle for any high-fidelity quantum gate. Here we show that the obstacle of micromotion can be overcome with current technology, even though the magnitude of the micromotion is way outside the Lamb-Dicke region. Through exact solution of the quantum Mathieu equations, we demonstrate the principle of the gate design under micromotion using two ions in a quadrupole Paul trap as an example. The proposed micromotion quantum gates can be extended to the many-ion case and may pave a new way for scalable trapped-ion quantum computation.
Ion trap is one of the most promising candidates for quantum computing. High-fidelity gates have been demonstrated in small ion crystals and schemes like ion shuttling have been proposed for larger systems. This thesis discusses the possibility of direct quantum computing on a large ion crystal in a Paul trap, without any shuttling of the ions. We first review a scheme to entangle two ions in a small ion crystal mediated by the collective phonon modes and analyze the gate errors. The generalization to larger systems is divided into three parts. (1) We present numerical methods to solve all the normal modes of the ion crystal, including the micromotion, up to arbitrary precision. The stability of the crystal under infinitesimal perturbation is ensured when all the normal modes have real frequencies. For finite disturbance, direct molecular dynamics simulation will be needed; after discussing some potential problems in the simulation, we give a rule of thumb for the ion crystal to be stable at a given temperature. (2) We show that when designing an entangling gate between two nearby ions, all the ions far away can be neglected, so that only a small number of normal modes are relevant. Similarly, distant entangling gates can be applied in parallel and the crosstalk is shown to decay cubically with the distance between these gates. (3) Then we present numerical methods to include the solved micromotion into the design of the entangling gates efficiently, again up to arbitrary precision. Thus we conclude that the design and the implementing of entangling gates are scalable in a large ion crystal. Finally we consider a near-term application to simulate an all-to-all coupled Ising model in a small to medium-sized 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.