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Journal of Physics B Atomic Molecular and Optical Physics

Quantisation of the elliptical Penning trap

Frances Crimin, B. M. Garraway, J. Verdú

Published May 7, 2021• 1 citation
Publisher

Abstract

Abstract We present the quantum theory of the elliptical Penning trap, i.e. the general case where the cylindrical symmetry of the electrostatic trapping potential around the trapping magnetic field axis is broken. The theory applies to both slightly and highly elliptical traps, where it is shown that the difference between the quantum states of particles in these traps corresponds to a variation of the degree of squeezing of their motional modes in the xy-plane. In a trap with tunable ellipticity, such as the Geonium Ghip planar Penning trap, it follows that control of the ellipticity via the trapping voltages enables squeezing of the quantum states of the particle. We discuss the adiabatic preparation of such squeezed states, which follows naturally from the appearance of an avoided crossing between the diabatic levels of the coupled motional states of the particle.

DOI
10.1088/1361-6455/abfeda

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Abstract We present the quantum theory of the elliptical Penning trap, i.e. the general case where the cylindrical symmetry of the electrostatic trapping potential around the trapping magnetic field axis is broken. The theory applies to both slightly and highly elliptical traps, where it is shown that the difference between the quantum states of particles in these traps corresponds to a variation of the degree of squeezing of their motional modes in the xy-plane. In a trap with tunable ellipticity, such as the Geonium Ghip planar Penning trap, it follows that control of the ellipticity via the trapping voltages enables squeezing of the quantum states of the particle. We discuss the adiabatic preparation of such squeezed states, which follows naturally from the appearance of an avoided crossing between the diabatic levels of the coupled motional states of the particle.
The objective of this thesis is to develop the quantum theory of the motional degrees of freedom of a charged particle in a Penning trap. The theory is treated within the formalism of quantum optics, and explores the use of dressed-atom methods by exploiting the threefold SU(N) algebraic structure of the problem. The quantum form of the experimental techniques of sideband coupling and driving to the ultra-elliptical regime are examined in this context, and resulting future applications considered. Interpretation of the quantum dynamics of the separate x and y motions of an electron is discussed, motivated by the desire to modify the trapping potential without changing the basic experimental configuration. A detailed discussion of operator methods which exploit the algebraic structure of the problem is given. This results in a clearer understanding of the physical manifestations of a range of unitary transformations upon a general three-dimensional system, and a novel interpretation of the mapping between canonical angular momentum components of isotropic and anisotropic trapping systems. The results highly promote future use of these methods in Penning trap theory, detailing a robust formulation of unitary operations which can be used to prepare the quantum state of a charged particle. The majority of the results can be applied to any Penning trap, but the theory is based throughout upon the “Geonium Chip" trap at Sussex; the scalability and planar design of this trap promotes it as natural candidate in experimental quantum optics and Gaussian quantum information studies. The work in this thesis aims to provide framework for such future applications.
A novel planar Penning trap is presented, which results from the projection of the well-known three-dimensional cylindrical trap onto the surface of a chip. The introduced trap is also a coplanar-waveguide cavity, similar to those used in circuit quantum electrodynamics experiments with superconducting two-level systems. It opens up the possibility of integrating a single trapped electron, or geonium atom, into quantum circuits. The trap is an elliptical Penning trap, with the magnetic field parallel to the chip's surface. A design procedure is described, which permits the compensation of electric anharmonicities up to sixth order. This should render possible the observation of a single trapped electron and the accurate measurement of its eigenfrequencies, a sine qua non requirement for a useful planar geonium technology.
We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under the breaking of its axial symmetry (inclination of the magnetic field) and tuning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation of the voltage on the trap electrode, there exists an avoided crossing effect and a corresponding effect of bilocalization of quantum states on pairs of closed trajectories belonging to common energy levels. This quantum tunneling happens on the symplectic leaves of the symmetry algebra, and hence it generates a tunneling of quantum states of the electron between the 3D-tori in the whole 6D-phase space. We present a geometric formula for the leading term of asymptotics of the tunnel energy-splitting in terms of symplectic area of membranes bounded by invariantly defined instantons.
We discuss the potential of the Penning ion trap for quantum information processing. An ion plasma in a Penning trap rotates about the trap (or magnetic field) axis. We have demonstrated phase-locked control of this rotation by a rotating electric field perturbation (a “rotating wall”). This precise control enables the formation of planar arrays of ions that could potentially be used for quantum computation with the Cirac-Zoller scheme, mimicking a “hard disk” geometry. Current experiments with miniature rf traps observe significant heating and decoherence which is believed to be due to fluctuating patch fields on the trap electrodes. This heating should be several orders of magnitude less in the larger Penning trap.
The quantum harmonic oscillator in a nonuniformly rotating frame is studied. It is shown that squeezed states and particles are generated by inertial effects. The analytic forms of the squeezing and Bogoliubov coefficients are given explicitly in terms of the solutions of the classical equations of motion. The squeezed states have the minumum dispersion allowed by the Robertson uncertainty relations for an arbitrary number of observables (the explicit form of these relations is obtained). The problem of an ``abrupt twist'' is worked out in detail as an example. The formalism can also be applied to the problem of a charged particle in a Penning trap, where it is shown that squeezed states are produced, with a squeezing frequency of the order of the magnetron frequency. In both examples, analytic expressions for the variances of the position and momentum are obtained. \textcopyright{} 1996 The American Physical Society.