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Physical Review A

Elliptical ion traps and trap arrays for quantum computation

Ralph G. DeVoe

Published Aug 1, 1998• 52 citations
Publisher

Abstract

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/${\mathrm{cm}}^{2},$ making possible arrays containing $>{10}^{6}$ ions in the Lamb-Dicke regime for precision spectroscopy or quantum computation.

DOI
10.1103/physreva.58.910

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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≈100\ensuremath{\approx}100\ensuremath≈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.
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.
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.
The quadrupole linear Paul trap is one of the key instruments in building highly stable atomic clocks. However, a frequency reference based on a single trapped ion is limited in stability due to the time needed for the interrogation cycle which cannot be further shortened. A promising strategy is the utilization of multiple trapped ions. The ions of the same kind then repulse each other with the Coulomb force, which is countered by the ponderomotive force of the time depended field in the trap. A few ions form a chain along the axis of a linear Paul trap. Adding more ions (a few tens or hundreds) gives rise to Coulomb crystals. We created an efficient simulation code which calculates the motion of such collections of ions in quasi-static radiofrequency fields of real linear quadrupole traps (including the micromotion). We include a model for laser cooling of the ions. The simulation tool can be used to study the formation and the dynamics of Coulomb crystals under conditions corresponding to various experimental setups.
Qubits based on ions trapped in linear radio-frequency traps form a successful platform for quantum computing, due to their high fidelity of operations, all-to-all connectivity, and degree of local control. In principle, there is no fundamental limit to the number of ion-based qubits that can be confined in a single 1D register. However, in practice, there are two main issues associated with long trapped-ion crystals, that stem from the “softening” of their modes of motion, upon scaling up: high heating rates of the ions’ motion and a dense motional spectrum; both impede the performance of high-fidelity qubit operations. Here, we propose a holistic, scalable architecture for quantum computing with large ion crystals that overcomes these issues. Our method relies on dynamically operated optical potentials that instantaneously segment the ion crystal into cells of a manageable size. We show that these cells behave as nearly independent quantum registers, allowing for parallel entangling gates on all cells. The ability to reconfigure the optical potentials guarantees connectivity across the full ion crystal and also enables efficient midcircuit measurements. We study the implementation of large-scale parallel multiqubit entangling gates that operate simultaneously on all cells and present a protocol to compensate for crosstalk errors, enabling full-scale usage of an extensively large register. We illustrate that this architecture is advantageous both for fault-tolerant digital quantum computation and for analog quantum simulations. Published by the American Physical Society 2024