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Physical Review A

Nonlinear coherent states of trapped-atom motion

Z. Kis, W. Vogel, L. Davidovich

Published Aug 1, 2001• 79 citations
Publisher

Abstract

The general character of nonlinear coherent states (NCS) is considered. A method is introduced by which any pure state of the quantum harmonic oscillator can be represented in a limiting sense as a NCS. The representation of a Fock state as a NCS is discussed in detail. As a physical example we show how to prepare a highly excited Fock state in an ion trap based on the concept of NCS.

DOI
10.1103/physreva.64.033401

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The general character of nonlinear coherent states (NCS) is considered. A method is introduced by which any pure state of the quantum harmonic oscillator can be represented in a limiting sense as a NCS. The representation of a Fock state as a NCS is discussed in detail. As a physical example we show how to prepare a highly excited Fock state in an ion trap based on the concept of NCS.
Some properties of the nonlinear coherent states (NCS), recognized by Vogel and de Matos Filho as dark states of a trapped ion, are extended to NCS on a circle, for which the Wigner functions are presented. These states are obtained by applying a suitable displacement operator ${D}_{h}(\ensuremath{\alpha})$ to the vacuum state. The unity resolutions in terms of the projectors $|\ensuremath{\alpha},h〉〈\ensuremath{\alpha}{,h}^{\ensuremath{-}1}|,|\ensuremath{\alpha}{,h}^{\ensuremath{-}1}〉〈\ensuremath{\alpha},h|$ are presented together with a measure allowing a resolution in terms of $|\ensuremath{\alpha},h〉〈\ensuremath{\alpha},h|.$ ${D}_{h}(\ensuremath{\alpha})$ is also used for introducing the probability distribution funtion ${\ensuremath{\rho}}_{A,h}(z)$ while the existence of a measure is exploited for extending the P representation to these states. The weight of the nthn\mathrm{th}\mathrm{}nth Fock state of the NCS relative to a trapped ion with Lamb-Dicke parameter $\ensuremath{\eta},$ oscillates so wildly as n grows up to infinity that the normalized NCS fill the open circle ${\ensuremath{\eta}}^{\ensuremath{-}1}$ in the complex $\ensuremath{\alpha}$ plane. In addition, this prevents the existence of a measure including normalizable states only. This difficulty is overcome by introducing a family of deformations that are rational functions of n, each of them admitting a measure. By increasing the degree of these rational approximations, the deformation of a trapped ion can be approximated with any degree of accuracy and the formalism of the P representation can be applied.
The preparation of coherent states of the harmonic oscillator on a sphere is considered. Based on the nonlinear coherent states of the atomic center-of-mass motion of a trapped ion, one may control the laser-ion interaction in such a way that sphere coherent states are obtained as motional dark states of the system. The curvature can be controlled by the Rabi frequencies of the driving lasers. The nonclassical properties of the resulting quantum states are analyzed. By variation of the curvature of the sphere, the states can be tuned continuously between spin-squeezed states and motional Fock states.
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.
Nonclassical states of light are of fundamental importance in quantum optics. The properties of these states are due to the quantal nature of the electromagnetic field. In recent years there have been many experimental demonstrations of the realizability of nonclassical states in various physical schemes such as resonance fluorescence, four-wave mixing, centre-of-mass motion of a trapped ion and manipulation of field in a cavity. It is of interest to introduce new classes of nonclassical states and investigate their properties. In this context, generalization of the concept of coherent states (CSs) has played a major role. New concepts such as the nonlinear coherent states (NCSs) and interference in phase space have emerged from such studies. In this tutorial review we are concerned with the construction of new classes of nonclassical states by generalizing the notion of CSs. Our investigations on the photon-added coherent states (PACSs) indicate that these states can be interpreted as NCSs. Also, we introduce a new class of nonclassical states related to the PACSs. Having introduced a realizable example of NCSs, we extend the notion of even and odd CSs to the case of NCSs by introducing even and odd NCSs. With this new definition we interrelate some of the well known states of light. We suggest a scheme to generate a class of even and odd NCSs in the centre-of-mass motion of a trapped, laser-cooled, two-level ion.