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arXiv:quant-ph/9704016 · Physical Review A

Measured quantum dynamics of a trapped ion

Lorenza Viola, Roberto Onofrio

Published May 1, 1997• 20 citations
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Abstract

The measurement process is taken into account in the dynamics of trapped ions prepared in nonclassical motional states. The induced decoherence is shown to manifest itself both in the inhibition of the internal population dynamics and in a damping of the vibrational motion without classical counterpart. Quantitative comparison with present experimental capabilities is discussed, leading to a proposal for the verification of the predicted effects.

DOI
10.1103/physreva.55.r3291

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The measurement process is taken into account in the dynamics of trapped ions prepared in nonclassical motional states. The induced decoherence is shown to manifest itself both in the inhibition of the internal population dynamics and in a damping of the vibrational motion without classical counterpart. Quantitative comparison with present experimental capabilities is discussed, leading to a proposal for the verification of the predicted effects.
The decoherence of nonclassical motional states of a trapped 9Be+{}^{9}{\mathrm{Be}}^{+}9Be+ ion in a recent experiment is investigated theoretically. Sources of decoherence considered here destroy the characteristic coherent quantum dynamics of the system, but do not cause energy dissipation. Here they are first introduced phenomenologically, and then described using a microscopic Hamiltonian formulation. Theoretical predictions are compared to experimental results.
We present a nonlinear decoherence model which models decoherence effects caused by various decohering sources in a quantum system through a nonlinear coupling between the system and its environment, and apply it to investigating decoherence in nonclassical motional states of a single trapped ion. We obtain an exactly analytic solution of the model and find very good agreement with experimental results for the population decay rate of a single trapped ion observed in the NIST experiments by Meekhof and co-workers [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)].
Abstract When a quantum state is subjected to a measurement and the state is not an eigenstate of the dynamical variable being measured, the outcome is unpredictable. Only the probabilities of the various possible outcomes are predicted by theory. This phenomenon is sometimes discussed in terms of “wavefunction collapse.” Trapped ions can be used for real, as opposed to gedanken demonstrations of this basic process. In the experiments described here, a single ion, or a few identical ions, were prepared in well defined superpositions of two internal energy eigenstates. The populations of the energy levels were then measured. When the state amplitudes were equal, the population fluctuations were greater than when one of the amplitudes was nearly zero, in agreement with the predictions of quantum mechanics. In other experiments, such as those with atomic beams, the number of atoms under observation fluctuates, and this obscures the fluctuations from other sources. However, if the number of atoms is small and constant, the fundamental quantum mechanical fluctuations can be observed.
We present a simple method that can be used to directly measure the squeezing of the quantum motional states of a trapped ion. Through the use of the interaction between the trapped ion and classical lasers, one can design a required coupling between the internal electronic and external vibrational degrees of freedom of the ion and can transfer information of the expectation value of a vibrational operator to the atomic internal populations. Thus measurement of squeezing on the quantum motional state can directly be realized. By adjusting the phases of the interacting lasers, one can measure the squeezing of both position and momentum quadratures.
The Lindblad approach to open quantum systems is introduced to study the dynamics of a single trapped ion prepared in nonclassical motional states and subjected to continuous measurement of its internal population. This results in an inhibition of the dynamics similar to the one occurring in the quantum Zeno effect. In particular, modifications to the Jaynes-Cummings collapses and revivals arising from an initial coherent state of motion in various regimes of interaction with the driving laser are dealt with in detail.
We investigate the dynamics of a cold trapped ion coupled to the quantized field inside a high-finesse cavity, considering exact resonance between the ionic internal levels and the field (carrier transition). We derive an intensity-dependent Hamiltonian in which terms proportional to the square of the Lamb-Dicke parameter $(\ensuremath{\eta})$ are retained. We show that different nonclassical effects arise in the dynamics of the ionic population inversion, depending on the initial states of the vibrational motion and field and on the values of $\ensuremath{\eta}.$