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Progress of Theoretical Physics

Model for Trapped Ion Interacting with Standing Wave via Gauge-Like Transformations

Sher Alam, Cleo L. Bentley

Published Apr 1, 1998• 0 citations
Publisher

Abstract

We treat the interaction of a trapped ion with a standing-wave light field, relaxing the previous condition of the trapping ion being on the node of the standing wave. In addition we work to second order in the Lamb-Dicke parameter η the extension to higher orders in η is also indicated. Only a simple transformation to the “dressed basis” (subject to the constraint of Pauli algebra) is required in order to show that the ion interacting with a standing light-wave may be described by the Jaynes-Cummings Hamiltonian. The ansatz of leaving invariant the Pauli algebra after the transformations amounts to a linear transformation in the su(2) algebra describing the two level system. The outcome is a Jaynes-Cummings like Hamiltonian with effective coupling dependent on the parameter Χ aimed to figure the trapping location in the standing wave. The main idea, namely leaving the Hamiltonian form invariant while changing the physical parameters can pave the way to a new understanding of gauge like transformations in the present context.

DOI
10.1143/ptp.99.577

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We treat the interaction of a trapped ion with a standing-wave light field, relaxing the previous condition of the trapping ion being on the node of the standing wave. In addition we work to second order in the Lamb-Dicke parameter η the extension to higher orders in η is also indicated. Only a simple transformation to the “dressed basis” (subject to the constraint of Pauli algebra) is required in order to show that the ion interacting with a standing light-wave may be described by the Jaynes-Cummings Hamiltonian. The ansatz of leaving invariant the Pauli algebra after the transformations amounts to a linear transformation in the su(2) algebra describing the two level system. The outcome is a Jaynes-Cummings like Hamiltonian with effective coupling dependent on the parameter Χ aimed to figure the trapping location in the standing wave. The main idea, namely leaving the Hamiltonian form invariant while changing the physical parameters can pave the way to a new understanding of gauge like transformations in the present context.
We treat the interaction of a trapped ion with a standing-wave light field relaxing the previous condition of the trapping ion being on the node of the standing wave. In addition we work to second order in the Lamb-Dicke parameter ηηη, the extension to higher orders in ηηη is also indicated. Only a simple transformation to the ``dressed basis'' [subject to the constraint of Pauli algebra] is required in order to show that the ion interacting with a standing light-wave may be described by the Jaynes-Cummings Hamiltonian.
Starting from the Jaynes-Cummings (J-C) model describing the interaction of a trapped ion with a standing-wave laser field, we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excitated resonantly with 1-quantum vibrational mode. After solving the time-dependent states of this model, we study the evolution of population inversion for the ion numerically. It is shown that the period of the quantum collapse and revival of the population inversionis related with the Lamb-Dick parameter and the position of the ion in the standing-wave laser field. With the increasing of the parameter, it will decrease the period of the collapse and revival. When moving the ion from the node of the standing-wave laser to the antinode, the period of the collapse and revival becomes longer.
We have investigated the position-dependent dynamics of a trapped ion in a standing wave laser by transforming it to the Jaynes-Cummings-type system under the Lamb-Dicke limit. A variety of novel phenomena are exhibited, e.g. periodic collapse and revival features and long-time scaled revivals of the ionic inversion, depending on its position in the standing wave. Our result provides a way of producing a system equivalent to the two-photon Jaynes-Cummings model in the trapped ion system, with its exact periodicities.
We present a second-order perturbative analysis of the model describing a two-level trapped ion interacting with a travelling laser field, in the Lamb–Dicke regime. Unlike the customary approach, based on the interaction picture and the rotating wave approximation, we reduce the original Hamiltonian to a time-independent model, unitarily equivalent to a spin-boson model, which features a form well suited to apply the perturbation theory. We determine the first- and second-order corrections of eigenstates and eigenvalues. By varying the interaction parameters we identify four regimes characterized by different eigenvalue distributions (possibly including level doublets) and suggest the presence of an intermediate regime where the spectrum undergoes macroscopic changes. In this regime, where the perturbation approach does not hold, our reduced Hamiltonian is shown to reproduce the Jaynes–Cummings model.
We have investigated the position-dependent dynamics of a trapped ion in a standing wave laser by transforming it to the Jaynes-Cummings-type system under the Lamb-Dicke limit.A variety of novel phenomena are exhibited.e.g.periodic collapse and revival features and long-time scaled revivals of the ionic inversion,depending on its position in the standing wave.Our result provides a way of producing a system equivalent to the two-photon Jaynes-Cummings model in the trapped ion system,with its exact periodicities.