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Physical Review A

Scheme for direct measurement of the Weyl characteristic function for the motion of a trapped ion

Shi-Biao Zheng, Xiwen Zhu, Mang Feng, Lei Shi

Published Aug 15, 2000• 15 citations
Publisher

Abstract

We propose a scheme for the reconstruction of the motional state of an ion trapped in a one-dimensional harmonic potential. In the scheme the ion is multichromatically excited by three lasers. Then the measurement of the population of the lower internal state directly yields the Weyl characteristic function for the motional state. The scheme is easily generalized to the two-dimensional case. The scheme operates in the Lamb-Dicke limit.

DOI
10.1103/physreva.62.035801

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We propose a scheme for the reconstruction of the motional state of an ion trapped in a one-dimensional harmonic potential. In the scheme the ion is multichromatically excited by three lasers. Then the measurement of the population of the lower internal state directly yields the Weyl characteristic function for the motional state. The scheme is easily generalized to the two-dimensional case. The scheme operates in the Lamb-Dicke limit.
We propose a simplified scheme for measuring the entangled state of the motional and electronic degrees of freedom of an ion trapped in a one-dimensional harmonic potential. The scheme is based on the polychromatic excitation of the ion by three lasers tuned to the carrier, the first lower, and the upper vibrational sidebands, respectively. The method directly yields the elements of the Wigner characteristic function matrix in the electronic basis. The scheme can be easily generalized to the two-dimensional case.
We proposed a scheme for the reconstruction of the quantum states for the center-of-mass vibrational mode of two trapped ions. In the scheme the ions are multichromatically excited by three lasers. Then measurement of the difference between probabilities of the ions being both in electronic ground and excited states directly yields the Wigner characteristic function for the center-of-mass vibrational state. The scheme can also be used to prepare entangled coherent states for the center-of-mass and relative vibrational modes.
We proposed a scheme for the reconstruction of the quantum states for the center-of-mass vibrational mode of two trapped ions. In the scheme the ions are multichromatically excited by three lasers. Then measurement of the difference between probabilities of the ions being both in electronic ground and excited states directly yields the Wigner characteristic function for the center-of-mass vibrational state. The scheme can also be used to prepare entangled coherent states for the center-of-mass and relative vibrational modes.
In this paper, a procedure is proposed for measuring Wigner characteristic function for motion of a single trapped ion. In the scheme, the trapped ion is driven by two laser beams with different frequencies tuned to the first lower and upper vibrational sidebands, respectively. Unlike other previous scheme, our scheme does not require the displacement of the external state before the ion-laser interaction and thus our scheme might be more simple and feasible. The scheme can also be used to prepare the motional Schrodinger cat states and entangled coherent states.
A method for the experimental reconstruction of the quantum state of motion for a single trapped ion is proposed. It is based on the measurement of the ground-state population of the trap after a sudden change of the trapping potential. In particular, we show how the $Q(\ensuremath{\alpha})$ function and the quadrature distribution $P(x, \ensuremath{\theta})$ can be measured directly. In an example we demonstrate the principle and analyze the sensitivity of the reconstruction process to experimental uncertainties as well as to finite grid limitations. Our method is not restricted to the Lamb-Dicke Limit and works in one or more dimensions.