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Physical Review Letters

Quantum Nondemolition Measurement of the Motional Energy of a Trapped Atom

R. L. de Matos Filho, W. Vogel

Published Jun 10, 1996• 82 citations
Publisher

Abstract

We propose a quantum nondemolition measurement of the motional energy of an atom confined in a trap. The quantum nondemolition coupling arises from the nonlinear vibronic dynamics of the system which is resonantly driven on a weak electronic transition. Applications are considered, such as laser cooling and preparations of nonclassical states of atomic motion. Examples are vibrational number states and quantum superpositions of coherent states undergoing a Kerr-type nonlinear interaction.

DOI
10.1103/physrevlett.76.4520

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We propose a quantum nondemolition measurement of the motional energy of an atom confined in a trap. The quantum nondemolition coupling arises from the nonlinear vibronic dynamics of the system which is resonantly driven on a weak electronic transition. Applications are considered, such as laser cooling and preparations of nonclassical states of atomic motion. Examples are vibrational number states and quantum superpositions of coherent states undergoing a Kerr-type nonlinear interaction.
The quantum nondemolition measurement is applied to a two-dimensional (2D) trapped-ion model in which two laser beams drive the corresponding vibrational motions and are carrier resonant with the two-level system of the ion. The information about the ionic vibrational energy can be detected by the occupation probability of the internal electronic level. The substantial difference of the 2D model from the one-dimensional one is that two orthogonal beams have a fixed phase shift instead of statistical independence. As a result, the atomic Rabi oscillation is involved in the coherent superposition of two sub-Rabi oscillations induced by the corresponding driving beams. This means that, in the bidimensional case, the phase difference between the two laser beams plays the role of an adjustable external parameter which allows us to optimize the measurement scheme itself, in terms of precision and sensitivity. As in the one-dimensional case, our proposal leads to interesting applications as, for example, the cooling and preparation of arbitrary bimodal Fock states. The model can be considered as a cooling method for the trapped ion from the vibrational thermal state to the vacuum state. Due to the coherent superposition of two sub-Rabi oscillations, the Rabi frequency degeneration and offset may occur in this model. This provides the possibility of generating various entangled superposition of Fock states, the pair coherent state, and entangled pair coherent states.
We propose a method for measuring entangled vibronic quantum states of a trapped atom. It is based on the nonlinear dynamics of the system that appears by resonantly driving a weak electronic transition. The proposed technique allows the direct sampling of a Wigner-function matrix, displaying all knowable information on the quantum correlations of the motional and electronic degrees of freedom of the atom. It opens possibilities for testing fundamental predictions of the quantum theory concerning interaction phenomena.
We discuss a simple implementation of a quantum nondemolition measurement of the vibronic state of a trapped ion. This scheme allows the production of Fock states associated to the center-of-mass motion and the determination of the population distribution of the trap levels, and is based on the vibronic level dependence of the frequency of Rabi oscillations between two internal states of the ion, induced by a resonant carrier field. The same scheme can be used to produce a Schr\"odinger-cat-like state after a single interaction between the trapped ion and the carrier pulse, followed by the detection of the ionic internal state. \textcopyright{} 1996 The American Physical Society.