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Physical Review Letters

Reconstruction of the Quantum Mechanical State of a Trapped Ion

Sascha Wallentowitz, W. Vogel

Published Oct 16, 1995• 174 citations
Publisher

Abstract

It is shown that the full information on the quantum state of the vibrational center-of-mass motion of a trapped ion can be transferred to its electronic dynamics by appropriately irradiating a long-living (e.g., quadrupole) electronic transition by laser light. This allows us to determine the quantum mechanical state of the ion with high quantum efficiency by probing a strong (dipole) transition for the appearance of resonance fluorescence.

DOI
10.1103/physrevlett.75.2932

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It is shown that the full information on the quantum state of the vibrational center-of-mass motion of a trapped ion can be transferred to its electronic dynamics by appropriately irradiating a long-living (e.g., quadrupole) electronic transition by laser light. This allows us to determine the quantum mechanical state of the ion with high quantum efficiency by probing a strong (dipole) transition for the appearance of resonance fluorescence.
We develop a method for reconstructing the quantum mechanical state of a trapped ion by bichromatically irradiating it on a weak electronic transition and subsequently probing a strong electronic transition for resonance fluorescence. Based on this recently proposed scheme [S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995)] the density matrix of the vibrational motion can be readily obtained either in a generalized position representation or in the number-state representation. The method allows to uniquely define an ultimate classical noise level by a reference measurement with an ion cooled to its vibrational ground-state. Disturbations of the measurement and their suppression are considered. Moreover, we study the effect of the back action of the measurement principle on the motional quantum state. It consists in the splitting of the state to be measured into two substates, giving rise to quantum interference effects. \textcopyright{} 1996 The American Physical Society.
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.
It is well known that electronic transitions driven by nearly resonant travelling- or standing- wave lasers are often accompanied with vibrational sideband transitions of a single trapped ion undergoing quantized center-of-mass motion [1], Our previous studies showed that quantum statistics of radiation fields can be used to manipulate the vibronic motion of a trapped ion localised within the Lamb-Dicke limit (LDL) [2]. The present paper addresses quantum manipulating vibronic motion of an ion trapped far beyond the LDL.
The laser excitation of an ion crystal to high-lying and long-lived electronic states is a genuine many-body process even if in fact only a single ion is excited. This is a direct manifestation of the strong coupling between internal and external dynamics and becomes most apparent in the vicinity of a structural phase transition. Here we show that utilizing highly excited states offers a new approach to the coherent manipulation of ion crystals. This opens up a new route towards the creation of nonclassical motional states in a Paul trap and permits the study of quantum phenomena that rely on a strong coupling between electronic and vibrational dynamics.