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Physical Review A

Dynamics of noise-induced heating in atom traps

Michael E. Gehm, Kenneth M. O’Hara, T. A. Savard, John E. Thomas

Published Nov 1, 1998• 208 citations
Publisher

Abstract

A Fokker-Planck equation is derived for the energy distribution of atoms in a three-dimensional harmonic trap with fluctuations in the spring constant and the equilibrium position. Using this model, we predict trap lifetimes based on the measurable noise spectra of the fluctuations. The energy distributions evolve into a single eigenmode where the apparent temperature of the distribution remains constant while the population decays as a consequence of the energy input. The method of analysis and the corresponding results are applicable to any optical, magnetic, or ion trap that is approximately harmonic, and offer useful insights into both noise-induced and optical heating processes.

DOI
10.1103/physreva.58.3914

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A Fokker-Planck equation is derived for the energy distribution of atoms in a three-dimensional harmonic trap with fluctuations in the spring constant and the equilibrium position. Using this model, we predict trap lifetimes based on the measurable noise spectra of the fluctuations. The energy distributions evolve into a single eigenmode where the apparent temperature of the distribution remains constant while the population decays as a consequence of the energy input. The method of analysis and the corresponding results are applicable to any optical, magnetic, or ion trap that is approximately harmonic, and offer useful insights into both noise-induced and optical heating processes.
Motional heating of ions in microfabricated traps is one of the open challenges hindering experimental realizations of large-scale quantum processing devices. Recently, a series of measurements of the heating rates in surface-electrode ion traps characterized their frequency, distance, and temperature dependencies, but our understanding of the microscopic origin of this noise remains incomplete. In this work we develop a theoretical model for the electric field noise which is associated with a random distribution of adsorbed atoms on the trap electrode surface. By using first-principles calculations of the fluctuating dipole moments of the adsorbed atoms we evaluate the distance, frequency, and temperature dependence of the resulting electric field fluctuation spectrum. Our theory reproduces correctly the ${d}^{\ensuremath{-}4}$ dependence with distance of the ion from the electrode surface and calculates the noise spectrum beyond the standard scenario of two-level fluctuators by incorporating all the relevant vibrational states. Our model predicts a regime of 1/f1/f1/f noise which commences at roughly the frequency of the fundamental phonon transition rate and a thermally activated noise spectrum which for higher temperatures exhibits a crossover as a function of frequency.
We consider two different kinds of fluctuations in an ion trap potential: external fluctuating electrical fields, which cause statistical movement (``wobbling'') of the ion relative to the center of the trap, and fluctuations of the spring constant, which are due to fluctuations of the ac component of the potential applied in the Paul trap for ions. We write down master equations for both cases and, averaging out the noise, obtain expressions for the heating of the ion. We compare our results to previous results for far-off resonance optical traps and heating in ion traps. The effect of fluctuating external electrical fields for a quantum gate operation (controlled-NOT) is determined and the fidelity for that operation derived.
The motion of a particle in a finite-depth confining potential, such as that of an electromagnetically trapped atom or ion, is approximately harmonic at low energy and softens as the escape energy is approached. When the particle is driven by a stochastic force, the heating and escape times depend sensitively on the noise power spectrum. We study escape times of particles in softening potentials driven by power-law colored noise. We identify two nested conditions for weak noise: the particle must execute many oscillations before escape, and noise forces must be small relative to forces exerted by the trap so as not to tilt the potential appreciably. In this regime, we derive expressions for the mean escape time using stochastic averaging techniques and compare them with Monte Carlo simulations of the full equations of motion. We also examine the crossover to regimes where these conditions are not met and large quasi-static forces give rise to additional escape mechanisms. Applying these results to an experimental measurement of lifetimes in a microfabricated ion trap, we find that spectral color alone cannot account for the rapid escape rates observed, whereas quasi-static barrier tilting by a stray field of plausible magnitude could. These results are relevant to the heating and escape of trapped atoms due to noisy force fields, as well as to other nonlinear oscillators subject to colored noise.
We study the stochastic dynamics of a particle in a periodically driven potential. For atomic ions trapped in radio-frequency Paul traps, noise heating and laser cooling typically act slowly in comparison with the unperturbed motion. These stochastic processes can be accounted for in terms of a probability distribution defined over the action variables, which would otherwise be conserved within the regular regions of the Hamiltonian phase space. We present a semiclassical theory of low-saturation laser cooling applicable from the limit of low-amplitude motion to large-amplitude motion, accounting fully for the time-dependent and anharmonic trap. We employ our approach to a detailed study of the stochastic dynamics of a single ion, drawing general conclusions regarding the nonequilibrium dynamics of laser-cooled trapped ions. We predict a regime of anharmonic motion in which laser cooling becomes diffusive (i.e., it is equally likely to cool the ion as it is to heat it), and can also turn into effective heating. This implies that a high-energy ion could be easily lost from the trap despite being laser cooled; however, we find that this loss can be counteracted using a laser detuning much larger than Doppler detuning.