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Chinese Physics Letters

Coherence-Like States of Two Coulomb-Correlated Ions Confined in a Paul Trap

Hui Li, Hai Wen-Hua, Chen Wen-Qin, Xu Jun

Published Apr 1, 2007• 2 citations
Publisher

Abstract

We report the n × n ' coherence-like state solutions in the cases of n , n ' = 1,2,... for the system including two Coulomb-correlated ions confined in a one-dimensional Paul trap with a time-dependent harmonic potential. One of the n ' exact solutions of the centre-of-mass motion describes a generalized coherent state. For a small driving strength the n approximate solutions of relative motion are constructed, which describe the coherent oscillations of the two ions around the classical equilibrium position.

DOI
10.1088/0256-307x/24/4/002

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We report the n × n ' coherence-like state solutions in the cases of n , n ' = 1,2,... for the system including two Coulomb-correlated ions confined in a one-dimensional Paul trap with a time-dependent harmonic potential. One of the n ' exact solutions of the centre-of-mass motion describes a generalized coherent state. For a small driving strength the n approximate solutions of relative motion are constructed, which describe the coherent oscillations of the two ions around the classical equilibrium position.
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 exact quantum motions of two ions in a two-dimensional Paul trap. When the Coulomb-correlation of the system is considered, we obtain a set of exact solutions of the system and the corresponding perturbed solutions, containing the exact discrete eigenstates and eigenenergies, and the approximate piecewise continuous spectra and the band structure. Taking several quantum states with lower quantum numbers as examples, we compute the average distances between two ions and give the plots of probability distribution of the two ions.
Making use of the well-known truncated-series method, we give a numerical program to calculate the exact solutions of two Coulomb-correlated ions confined in a linear Paul trap. The discrete motion state functions and energy spectra are obtained for the trap frequencies approaching the experimentally realizable limiting values, which correspond to a denumerable infinite point-set of the frequencies. Continuously varying the trap frequency between two neighboring points of the denumerable infinite set, from the quantum perturbation technique we find the energy-band structure based on the exact discrete spectra in which the bands are narrower and the gaps wider. The band structure can influence the laser cooling of Paul trapped ions and the quantum logic operation based on the system that should be considered in the corresponding experiments thereby.
Following the paper by Combescure [Ann. Phys. (N.Y.) 204, 113 (1990)], we apply the quantum singular time-dependent oscillator model to describe the relative one-dimensional motion of two ions in a trap. We argue that the model can be justified for low-energy excited states with the quantum numbers $n\ensuremath{\ll}{n}_{\mathrm{max}}\ensuremath{\sim}100$, provided the dimensionless constant characterizing the strength of the repulsive potential is large enough ${g}_{*}\ensuremath{\sim}{10}^{5}$. Time-dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center-of-mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential since the large value of g∗{g}_{*}g∗​ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.
By using Husimi’s test-function method, we construct n exact solutions of a single Paul trapped ion modelled by a quantum harmonic oscillator with a time-dependent ‘spring constant’. Any n th solution is a combination of the Fock state and coherent state, which describes a wavepacket train consisting of n + 1 packets. Its centre oscillates like the classical harmonic oscillator with variable frequency, while the width and height of each packet change simultaneously. When the deformation is small, it behaves like a propagated and non-deformed train, and the large deformation is identified with breathing of the wavepacket train.