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Journal of Physics A Mathematical and Theoretical

Quantum control of a Paul-trapped ion via double radio-frequency driving

Qiong Chen, Kuo Hai, Wenhua Hai

Published Oct 19, 2010• 4 citations
Publisher

Abstract

Due to an infinite series solution of the classical Mathieu equation, using single-radio-frequency (rf) driving makes it difficult to manipulate motional states of a Paul-trapped ion. Here, we apply double-rf driving consisting of two external fields to generate the squeezed coherent states of simple forms. Stability parameter regions of the system are found in which both the first and second rf fields may be very strong but only the latter can be weak. Based on exact solutions, we investigate the time evolutions of expectation energies and controllable transitions between not only two stationary states, but also two nonstationary states. Such a quantum-control scenario may be tested experimentally.

DOI
10.1088/1751-8113/43/45/455302

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Due to an infinite series solution of the classical Mathieu equation, using single-radio-frequency (rf) driving makes it difficult to manipulate motional states of a Paul-trapped ion. Here, we apply double-rf driving consisting of two external fields to generate the squeezed coherent states of simple forms. Stability parameter regions of the system are found in which both the first and second rf fields may be very strong but only the latter can be weak. Based on exact solutions, we investigate the time evolutions of expectation energies and controllable transitions between not only two stationary states, but also two nonstationary states. Such a quantum-control scenario may be tested experimentally.
An exact solution of the Schrodinger equation for the system is constructed.It is found that when the ratio of the driving frequency to the radio frequency is equal to certain discrete values,the double-rf Paul trap exists stable spatially localized states.This phenomenon is similar to the dynamic localization of a particle on a discrete lattice reported by Dunlap and Kenkre in 1986.However,the conditions for the latter case deal with the ratio of the field magnitude to the field frequency.The results of our calculations demonstrate a new category concerning the dynamic localization.
We have studied the quantum and classical motions of a single Paul trapped ion interacting with a time-periodic laser field. By using the test-function method, we construct n exact solutions of quantum dynamics that describe the generalized squeezed coherent states with the expectation orbits being the corresponding classical ones. The space-time evolutions of the exact probability densities show some wavepacket trains. It is demonstrated analytically that by adjusting the laser intensity and frequency, we can control the center motions of the wavepacket trains. We also discuss the other physical properties such as the expectation value of energy, the widths and heights of the wavepackets, and the resonance loss of stability.
Single trapped ions represent elementary quantum systems that are well isolated from the environment. They can be brought nearly to rest by laser cooling, and both their internal electronic states and external motion can be coupled to and manipulated by light fields. This makes them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.
We investigate the quantum motion of two ions stored in a Paul trap and interacting with a time-periodic laser field. In the pseudopotential approximation and large detuning condition, we find that the relative motion is independent of the laser field, but the exact centre-of-mass motion is closely related to the laser field. By adjusting the laser intensity and frequency, we can well control the quantum motion of the centre-of-mass. We illustrate some physical properties described by the centre-of-mass states, such as the squeezed coherent property, the widths and heights of the wavepackets of probability density, the classical-quantum correspondence, the resonance ladders of expectation energy and the transition probabilities between time-dependent quantum levels.
We investigate the quantum dynamics of a single ion confined in a radio-frequency (RF) Paul trap in the presence of weak octupole-induced anharmonicity, as can occur in realistic ion-trap potential. The explicitly time-dependent quadrupole potential is treated in a reference-oscillator Fock basis, allowing the effects of the RF drive and the quartic correction to be resolved directly at the level of motional-state couplings. While the quadratic RF interaction generates Δn=±2Δn=\pm2Δn=±2 transitions and Gaussian squeezing, the quartic contribution opens additional Δn=±4Δn=\pm4Δn=±4 coupling pathways. Starting from the reference-oscillator vacuum, we show that the octupole interaction broadens the even-parity Fock-state distribution and enhances the phonon-number fluctuations more strongly than the mean occupation. At the same time, the RF-induced sub-vacuum quadrature squeezing remains robust, although its magnitude and orientation are modified by the quartic interaction. The number statistics and quadrature fluctuations therefore respond differently to the same nonlinear perturbation, reflecting the distinct roles of Fock-space redistribution and phase coherence. The resulting dynamics provide a quantum-mechanical characterization of weak octupole corrections beyond the ideal quadrupole approximation and clarify how realistic anharmonicities modify nonclassical trapped-ion motion. Importantly, the motional nonclassicality remains robust against the weak octupole perturbation, as evidenced by persistent sub-vacuum quadrature squeezing and Klyshko-inequality violation. These results characterize how realistic anharmonic corrections reshape trapped-ion motion beyond the ideal quadrupole approximation.
We investigate how to apply a high-frequency driving field to the quantum control of a single particle in an open double-well system. The linear stability analysis points out that the stability depends on the external-field parameters and the loss (or gain) coefficients of the system, and the instability leads to a transition of the Floquet quasienergy from real to complex values and results in decaying probabilities for the particle to be in the double well. By combining analytical solutions in the high-frequency approximation with numerical calculations based on an accurate model, we exhibit quantum-dynamical behavior of the particle such as Floquet oscillation, coherent destruction of tunneling, quasi-NOON-state population, partial one-particle tunneling, and the decay of the probabilities of occupation, which are due to the competition and balance between the quantum coherence and the loss (or gain) effect. The results suggest an experimental method for testing quantum motion in an open system by adjusting the driving field.