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Chinese Journal of Quantum Electronics

Quantum mechanical problem for two ions in a Paul trap

Wenhua Hai

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

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.

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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.
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 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 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.
Based on quantum coding way,we have analyzed a theoretical method of the two imprisoned ion quantum-gate system(single-bit logic non-gate,two-bit logic non-gate) by quantum theory and the exact solution of Paul Wells two-line system.We made use of the vibration mode of mass center and two ion to build the quantum theory of and gate(without third-ion).With a linear superposition a single bit of quantum non-gate together with quantum and gate,it proved that the trapped two ions system in the relatively quantum number l=1.Preparation can be completed in parallel quantum logic operation.Quantum non-gate system and gate are the basic structural unit to quantum computer system configuration.It is worthy of consideration at experimental study of quantum information.