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Physical Review A

Reconstructing the vibrational state of a trapped ion

Cassius D'Helon, Gerard J Milburn

Published Jul 1, 1996• 67 citations
Publisher

Abstract

We show that probe resonance fluorescence measurements on a single trapped ion at the node of a pulsed optical standing wave can be used to tomographically reconstruct the initial vibrational state of the ion. The Wigner function describing the density matrix of the initial vibrational state is determined by measuring the population inversion of the ion at the end of the standing-wave pulse over a range of free evolution times and pulse strengths.

DOI
10.1103/physreva.54.r25

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We show that probe resonance fluorescence measurements on a single trapped ion at the node of a pulsed optical standing wave can be used to tomographically reconstruct the initial vibrational state of the ion. The Wigner function describing the density matrix of the initial vibrational state is determined by measuring the population inversion of the ion at the end of the standing-wave pulse over a range of free evolution times and pulse strengths.
A protocol is provided to reconstruct the Wigner function for the motional state of a trapped ion via fluorescence detection on another ion in the same trap. This ``sympathetic tomography'' of a dark ion without optical transitions suitable for state measurements is based on the mapping of its motional state onto one of the collective modes of the ion pair. The quantum state of this vibrational eigenmode is subsequently measured through sideband excitation of the bright ion. Physical processes to implement the desired state transfer are derived and the accomplishment of the scheme is evaluated for different mass ratios.
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.
Starting from the Jaynes-Cummings (J-C) model describing the interaction of a trapped ion with a standing-wave laser field, we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excitated resonantly with 1-quantum vibrational mode. After solving the time-dependent states of this model, we study the evolution of population inversion for the ion numerically. It is shown that the period of the quantum collapse and revival of the population inversionis related with the Lamb-Dick parameter and the position of the ion in the standing-wave laser field. With the increasing of the parameter, it will decrease the period of the collapse and revival. When moving the ion from the node of the standing-wave laser to the antinode, the period of the collapse and revival becomes longer.
We reconstruct the density matrices and Wigner functions for various quantum states of motion of a harmonically bound 9Be+{}^{9}{\mathrm{Be}}^{+}9Be+ ion. We apply coherent displacements of different amplitudes and phases to the input state and measure the number state populations. Using novel reconstruction schemes we independently determine both the density matrix in the number state basis and the Wigner function. These reconstructions are sensitive indicators of decoherence in the system.
A single ion trapped in a harmonic potential and laser cooled at the node of a standing wave can be described in terms of the Jaynes-Cummings Hamiltonian known in quantum optics. Similar to nonclassical states of light we propose to prepare nonclassical states of the ion motion in a harmonic trap. In particular, several schemes for the preparation of Fock states are presented, and techniques for the preparation of squeezed states and coherent states of the ion motion are indicated. For an observation of these nonclassical states we propose to measure quantum collapse and revival in the population inversion of a single trapped ion.