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Physical Review Letters

Endoscopy in the Paul Trap: Measurement of the Vibratory Quantum State of a Single Ion

P. J. Bardroff, C. Leichtle, G. Schrade, Wolfgang P. Schleich

Published Sep 9, 1996• 99 citations
Publisher

Abstract

We reconstruct the density operator of the center-of-mass motion of an ion stored in a Paul trap by mapping the dynamics of the motion onto the internal dynamics of the ion. Our technique takes into account the explicit time dependence of the trap potential, operates outside the Lamb-Dicke limit, and is not restricted to pure states. We demonstrate the feasibility of this method using the example of a damped Schr\"odinger cat state.

DOI
10.1103/physrevlett.77.2198

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•Jan 1, 1997•0 citations
We reconstruct the density operator of the center-of-mass motion of an ion stored in a Paul trap by mapping the dynamics of the motion onto the internal dynamics of the ion. Our technique takes into account the explicit time dependence of the trap potential, operates outside the Lamb-Dicke limit, and is not restricted to pure states. We demonstrate the feasibility of this method using the example of a damped Schr"odinger cat state.
Summary form only given. The recent experimental generation of nonclassical states of the motion of an ion in a Paul trap has propelled the field of quantum state preparation into a new era. But how to prove that the ion is indeed in a Fock, a squeezed, or a Schrodinger cat state? How can we measure a motional state which, due to the explicit time dependence of the binding force of the Paul trap, displays a complicated time dependence? In this paper we present the first method that measures a vibratory state of an ion taking into account the complete time dependence of the Paul trap. Moreover our technique operates outside the Lamb-Dicke regime and is not limited to pure states only.
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.
Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.
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.