Phase-space study of surface-electrode Paul traps: Integrable, chaotic, and mixed motions

Abstract

We present a comprehensive phase-space treatment of the motion of charged particles in electrodynamic traps. Focusing on five-wire surface-electrode Paul traps, we study the details of integrable and chaotic motion of a single ion. We introduce appropriate phase-space measures and give a universal characterization of the trap effectiveness as a function of the parameters. We rigorously derive the commonly used (time-independent) pseudopotential approximation, quantify its regime of validity and analyze the mechanism of its breakdown within the time-dependent potential. The phase space approach that we develop gives a general framework for describing ion dynamics in a broad variety of surface Paul traps. To probe this framework experimentally, we propose and analyze, using numerical simulations, an experiment that can be realized with an existing four-wire trap. We predict a robust experimental signature of the existence of trapping pockets within a mixed regular and chaotic phase-space structure. Intricately rich escape dynamics suggest that surface traps give access to exploring microscopic Hamiltonian transport phenomena in phase space.

Far-from-equilibrium noise-heating and laser-cooling dynamics in radio-frequency Paul traps

“…Beyond the integrable motion, some phase-space regions may become chaotic, and from some chaotic regions the ion can escape the trap on a very fast timescale [11]. Even for chaotic motion that is bounded, the ion explores arXiv:1808.07816v3 [physics.atom-ph] 16 May 2019 a non-zero volume in phase-space whose dimension is not reduced by conservation laws, and typically does so in an apparently random manner.…”

Can a periodically driven particle resist laser cooling and noise?

“…The dynamical mechanism we explored is formally close to models of Hamiltonian and Brownian ratchets [37], which are basic models of transport [38,39]. Transport in a mixed phase-space is especially complex [40][41][42][43][44][45][46] and the ability to control and accurately measure motion in complex time-dependent potentials make ion-trap experiments suitable for quantitative tests of such ideas [21], extended even to many particles [47,48]. Our results can be generalized to frequency-locked limit-cycles in the relative coordinate of two interacting particles [49,50], or in the rotation of a macroscopic particle, with V 2 a periodic function of the rotation angle [51,52].…”

“…Since the periodic drive frequency has been rescaled to 2, V (z, t) is π-periodic. In numerical simulations we take V 2 (z) = 4 π arctan 1 2z − arctan 3 2z to be the potential of a model surface trap [21] along an axis perpendicular to the electrode plane (defined by z=0, with d the width of two electrodes carrying the rf-modulated voltage). V 2 vanishes at z s = √ 3/2 ≈ 0.866.…”

“…If we find a region of action where the ion remains bounded for a very long time (as determined by the Fokker-Planck dynamics), we can assume an approximately stationary probability distribution in the action, which then takes the form Taking concrete physical parameters we consider a 24 Mg + ion. The nondimensional parameters are (see [21]) a z = 4eU DC /mΩ 2 c z and q = 2eU rf /md 2 Ω 2 , with m, e the ion's mass and charge, U DC , U rf the voltages on the static and rf-modulated electrodes respectively, c z a geometric constant, and we set d = 150 µm and Ω = 2π × 20 MHz. The resulting oscillation frequency at the effective minimum is ω z ≈ 2π ×2.3 MHz.…”

“…We have also simulated the Hamiltonian motion within the full time-dependent potential of the five-wire trap in three spatial dimensions for the parameters analyzed above, verifying that this mechanism is robust with large amplitudes of motion in the remaining coordinates. Island chains frequently develop in surface traps with voltages in the upper range of experimentally relevant values, and they can have very large relative sizes [21]. This is true also for other trap types, since the potential often attains some anharmonic contributions which become relevant above some energy or spatial scale.…”