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Physical Review Letters

Experimental Determination of the Motional Quantum State of a Trapped Atom

D. Leibfried, D. M. Meekhof, B. E. King, Christopher R. Monroe, W. M. Itano, D. J. Wineland

Published Nov 18, 1996• 682 citations
Publisher

Abstract

We reconstruct the density matrices and Wigner functions for various quantum states of motion of a harmonically bound ${}^{9}{\mathrm{Be}}^{+}$ 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.

DOI
10.1103/physrevlett.77.4281

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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.
We report the creation and full determination of several quantum states of motion of a 9Be+ ion bound in a RF (Paul) trap. The states are coherently prepared from an ion which has been initially laser cooled to the zero-point of motion. We create states having both classical and non-classical character including thermal, number, coherent, squeezed, and ‚Schrödinger cat‘ states. The motional quantum state is fully reconstructed using two novel schemes that determine the density matrix in the number state basis or the Wigner function. Our techniques allow well controlled experiments decoherence and related phenomena on the quantum-classical borderline.
Abstract : We report the creation and full determination of several quantum states of motion of a 9Be ion bound in a RF (Paul) trap. The states are coherently prepared from an ion which has been initially laser cooled to the zero-point of motion. We create states having both classical and non-classical character including thermal, number, coherent, squeezed, and Schrodinger cat states. The motional quantum state is fully reconstructed using two novel schemes that determine the density matrix in the number state basis or the Wigner function. Our techniques allow well controlled experiments decoherence and related phenomena on the quantum-classical borderline.
We report the creation and full determination of several quantum states of motion of a 9Be+ ion bound in a RF (Paul) trap. The states are coherently prepared from an ion which has been initially laser cooled to the zero-point of motion. We create states having both classical and non-classical character including thermal, number, coherent, squeezed, and ‚Schrödinger cat‘ states. The motional quantum state is fully reconstructed using two novel schemes that determine the density matrix in the number state basis or the Wigner function. Our techniques allow well controlled experiments decoherence and related phenomena on the quantum-classical borderline.
Summary form only given, as follows. The ability to create and completely characterize a variety of fundamental quantum states has long been sought after in the laboratory since it brings to the forefront issues involving the relationship between quantum and classical physics. The interaction of a trapped atom with classical laser light constitutes a simple and well-isolated quantum system that makes experiments along these lines possible. The motion of an ion trapped in a rf-trap can be described as a quantum harmonic oscillator. If the ion is driven by a classical light field, several interactions similar to and beyond the simple Jaynes-Cummings coupling can be realized. These couplings allow for precisely controlled engineering and measurement of quantum states. We create states having both classical and nonclassical character including thermal, number, coherent, squeezed, and Schrodinger cat states. The motional quantum state is fully reconstructed with use of two novel schemes that determine the density matrix in the number state basis and the Wigner function. Our techniques allow well-controlled experiments on decoherence and related phenomena on the quantum-classical borderline.
Trapped atomic ions are a versatile platform for studying interactions between spins and bosons by coupling the internal states of the ions to their motion. Measurement of complex motional states with multiple modes is challenging, because all motional state populations can only be measured indirectly through the spin state of ions. Here we present a general method to determine the Fock state distributions and to reconstruct the density matrix of an arbitrary multimode motional state. We experimentally verify the method using different entangled states of multiple radial modes in a five-ion chain. This method can be extended to any system with Jaynes-Cummings-type interactions.
A quantum state is fully characterized by its density matrix or equivalently by its quasiprobabilities in phase space. A scheme to identify the quasiprobabilities of a quantum state is an important tool in the recent development of quantum technologies. One of the most fundamental interaction models in quantum optics is the so-called Jaynes-Cummings model (JCM), which has been massively studied theoretically and experimentally. However, the expected essential dynamics of the field states under the resonant JCM has not been observed experimentally due to the lack of a proper reconstruction scheme. In this paper, we further develop a highly efficient vacuum measurement scheme and study the JCM dynamics in a trapped ion system with the capability of the vacuum measurement to reconstruct its quasiprobability QQQ function, which is a preferred choice to study the core of the dynamics of a quantum state in phase space. During the JCM dynamics, the Gaussian peak of the initial coherent state bifurcates and rotates around the origin of phase space. They merge at the so-called revival time at the other side of phase space. The measured QQQ function agrees with the theoretical prediction. Moreover, we reconstruct the Wigner function by deconvoluting the QQQ function and observe the quantum interference in the Wigner function at half of the revival time, where the vibrational state becomes nearly disentangled from the internal energy states and forms a superposition of two composite states. The scheme can be applied to other physical setups including cavity or circuit-QED and optomechanical systems.