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Physical Review A

Externally induced metastability of an electron in a Penning trap: Analytical results

S. Brouard, Jesús Plata

Published Nov 14, 2001• 2 citations
Publisher

Abstract

The effect of a driving field on the cyclotron mode of a relativistic electron in a Penning trap is studied analytically. The Hamiltonian dynamics of this driven nonlinear oscillator is analyzed by using linearization techniques and displaced squeezed-state formalism. With the approximate analytical expressions obtained for the eigenstates in this approach, a simplified treatment of the dissipative dynamics is carried out and some of the nontrivial features found in a recent numerical study [D. Enzer and G. Gabrielse, Phys. Rev. Lett. 78, 1211 (1997)] are unraveled. The emergence of different time scales and the generation of a metastable statistical mixture are understood in terms of the changes induced in the structure of the master equation by the nonuniform characteristics of the eigenstates; the partial revivals of specific coherent states are accounted for by the evolution of particular coherences. The control of these effects by a proper choice of the driving parameters is discussed.

DOI
10.1103/physreva.64.063405

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The effect of a driving field on the cyclotron mode of a relativistic electron in a Penning trap is studied analytically. The Hamiltonian dynamics of this driven nonlinear oscillator is analyzed by using linearization techniques and displaced squeezed-state formalism. With the approximate analytical expressions obtained for the eigenstates in this approach, a simplified treatment of the dissipative dynamics is carried out and some of the nontrivial features found in a recent numerical study [D. Enzer and G. Gabrielse, Phys. Rev. Lett. 78, 1211 (1997)] are unraveled. The emergence of different time scales and the generation of a metastable statistical mixture are understood in terms of the changes induced in the structure of the master equation by the nonuniform characteristics of the eigenstates; the partial revivals of specific coherent states are accounted for by the evolution of particular coherences. The control of these effects by a proper choice of the driving parameters is discussed.
Recent numerical work has demonstrated how one of the bistable states of a driven, damped, anharmonic classical oscillator is metastable in the corresponding quantum oscillator [D. Enzer and G. Gabrielse, Phys. Rev. Lett. 78, 1211 (1997)]. Based on an analysis of the classical oscillator, we present a simple analytical treatment of the quantum oscillator. Calculated fluctuations in the quantum observables and the estimated decay rate of the metastable state are in an excellent agreement with the numerical results of Enzer and Gabrielse.
The phase damping induced in the cyclotron mode of a trapped electron by the coupling to the axial fluctuations is studied analytically. This system, described as a nonlinear oscillator stochastically driven in frequency, allows testing the generality of some elements present in the phenomenology of decoherence. In our approach, the reduced density matrix is obtained by performing a statistical average from the propagator for each noise realization. For short times, the decay of the coherences presents a nonexponential form, rooted in the non-Gaussian character of the stochastic driving. For large times, the decay becomes purely exponential, the rate showing a complex dependence on the difference between the Fock indices. As the populations do not change, the asymptotic state corresponds to a nonthermalized statistical mixture.
Parametrically driven Kerr nonlinear quantum oscillator plays a central role in quantum optics, superconducting circuits, and in engineered light-matter interactions. For the development of future quantum technologies, it is crucial to understand the behavior of this quantum system under realistic dissipative conditions. The objective of this thesis is to investigate the phase-space dynamics of a dissipative Kerr nonlinear oscillator under different driving regimes and corresponding dissipation channels, with a particular focus on the emergence of nonclassical features in phase space. The aim is also to understand how nonlinearity and different driving mechanisms influence quantum properties such as coherence, nonclassicality, and phase-space structure. The analysis was carried out using the Wigner phase-space formalism and numerical simulations of the Lindblad master equation. The results show that in the case of finite nonlinearity the coherence time is prolonged compared to the harmonic case. In the single photon regime, the interplay between the photon drive and nonlinearity can lead to coherence revivals after which the system relaxes into a squeezed steady state. In the two photon and dissipation regime, the system exhibits parity-protected dynamics, leading to very long coherence times.
A detailed quantum-mechanical description of an electron in a Penning trap is developed in which nonlinear relativistic effects as well as stochastic effects arising from radiative damping and the continuous measurement of the electronic motion are taken into account. A master equation is derived that describes the influence of these environmental effects on the bistable dynamics of the electron. Manifestations of measurement-induced fluctuations on the bistable dynamics of individual continuous measurement processes are investigated with the help of the quantum state diffusion model.
We present a detailed account of the design choices required for planar Penning traps that feature a highly harmonic confining potential. High harmonicity is indispensable for a number of applications, particularly for confinement of single electrons as qubits in quantum-information processing. The present work extends previous studies [D. Goldmann and G. Gabrielse, Phys. Rev. A 81, 052335 (2010)] by a fully analytic treatment of finite electrode gaps and their relevance for small traps on the millimetre size scale and below, when relative gap sizes are non-negligible. We derive the overall trap potential with a particular focus on different models of finite-gap potentials and show how to find the optimum trap geometry and electrode voltages to minimize anharmonicities. The analytic calculations are compared with detailed finite-element simulations.
Recently [F. Krumm and W. Vogel, Phys. Rev. A 97, 043806 (2018)], the detuned and nonlinear Jaynes-Cummings model describing the quantized motion of a trapped ion was introduced and its corresponding dynamics was solved via considering the driving laser in a quantized manner. For the dynamics with a classical driving laser an approximation in terms of neglected time ordering was considered to obtain analytical expressions and compared to numerically obtained exact dynamics, which revealed the insufficiency of this approximation. In the present work we reconsider this model and show that it can likewise be solved analytically, i.e., including time ordering, with a classical driving laser field. Using the analytical solution, we investigate the quantum time-ordering effects of the system with respect to nonclassicality and nonsimulatability of the motional states of the ion. Furthermore, we use the Magnus expansion to analyze the impact of certain orders of the time ordering. We derive an exact radius of convergence for these time-ordering corrections beyond the established and only sufficient criterion. Finally, the differences between the solution derived here and the previously found one using a quantized pump are discussed.