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Acta Physica Sinica

Energy eigenstates of two ions in a two-dimensional paul trap

Wu Yun-Wen, Hai Wen-Hua, 湖南吉首大学物理系，吉首 416000;湖南师范大学物理系，长沙 410081

Published Jan 1, 2006• 2 citations
Publisher

Abstract

Taking into consideration the Coulomb interaction between two ions trapped in the coplanar Paul trap,the exact solutions of the Schrdinger equations of the two-ions system are obtained.The energy level degeneracy for center-of mass is obtained using the exact solutions. We also calculate the equilibrium distance of the ions and several state functions of the low energy levels. The radial probability distribution graph for the center of mass is plotted using a computer program.

DOI
10.7498/aps.55.3315

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Taking into consideration the Coulomb interaction between two ions trapped in the coplanar Paul trap,the exact solutions of the Schrdinger equations of the two-ions system are obtained.The energy level degeneracy for center-of mass is obtained using the exact solutions. We also calculate the equilibrium distance of the ions and several state functions of the low energy levels. The radial probability distribution graph for the center of mass is plotted using a computer program.
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.
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.
By use of the well-known truncating series method, A program for calculating the Schrodinger exact solutions of quantum gate by two ions confined in a linear Paul trap is presented. The lower energy's state function for center of mass is analyzed. Probability distributing grawing is given. We calculate quantum state of variance in relatively position△r2 h/2 and quantum state of variance in relatively momentum△p2 h/2, the variance in position and momentum given by the Heisenberg uncertainty principle△r·△p h/2. It is a condition for squeezed states. We conclude that with center of mass in ground state, the quantum states of two ions in relatively position are squeezed states, and the quantum state of two ions is an entanglement state. Uncertainty of quantum measurement and entanglement must be consider in experiment.
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.
Ion cloud and crystal are recognized as two distinct states in ion traps, but there is no theory that can describe the energy-level structures of these two states from the perspective of quantum theory. In this paper, we construct a model describing ions in a linear Paul trap to investigate the energy-level structure with the linear response function, which is often used in the field of condensed matter to compute the energy-level of electrons or plasmas. We support a method to calculate the energy-level of both the crystal and cloudy states of trapped ions in a linear Paul trap. Furthermore, we present the energy-level diagram of two trapped ions and give a quantum interpretation for the fluorescence hysteresis loops that were observed in our laboratory. This is a fundamental theory that will help us to understand the crystallization and cooling processes of ions in a linear Paul trap.