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Bulletin of the American Physical Society

Full analytical Expressions of the Quantum Mechanical Energy Eigenstates of charged Particles confined in a Penning Trap at non-relativistic Energies.

Eric Steinfelds, Keith Andrew

Published Nov 20, 2015• 0 citations
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We present a thorough analysis of the electronic detection of charged particles, confined in a Penning trap, via image charges induced in the trap electrodes. Trapping of charged particles in an electrode structure leads to frequency shifts, which are due to image charge and space charge effects. These effects are of importance for Penning trap experiments which involve high charge densities or require high precision in the motional frequencies. Our analysis of image charges shows that only (higher order) odd powers of the particle displacement lead to induced charge differences, giving rise to a signal. This implies that, besides the centre-of-mass frequency of a trapped particle cloud, also higher order individual particle frequencies induce a signal, which can be picked up by an electronic detection circuit attached to the trap electrodes. We also derive analytic expressions for the image charge and space charge induced frequency shifts and perform simulations of space charge effects. In relation to this, we discuss the consequences of the shifted particle frequencies for resistive cooling of the particle motion.
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.
Abstract We discuss the confinement of an electron in a Penning trap as the realisation of a one-dimensional quantum-mechanical oscillator. In such a situation, the non-zero ground-state energy of the electron in the confining electric and magnetic fields of the trap (as compared to the classical zero of the energy of a free electron) plays a role in certain kinds of investigations, and can be directly measured with high precision in photo-detachment spectroscopy of confined atomic anions such as H − . We detail a concrete implementation of such a measurement and discuss its implications.