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

Abstract

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.

Tuning nonthermal distributions to thermal ones in time-dependent Paul traps

“…Thus we find that ∆ can be used as a control parameter to counteract, to a high extent, the effect of excess micromotion. As shown in [15], increasing ∆ is also efficient for cooling the ion from the regime of approximately integrable high amplitude motion in the anharmonic potential of a surface-electrode trap with a high micromotion frequency, wherein a low-detuning laser (with ∆ ≈ −Γ/2) may more easily heat the ion past the trap barrier. We also show in Sec.…”

“…In this work we use a recently developed semiclassical framework for studying laser cooling dynamics of ions driven by micromotion up to large amplitudes of motion [15]. We focus on a single ion's distribution in the final stage of the cooling.…”

“…When v can be assumed to evolve slowly on the scale of Γ/2, the steady-state of the coupled OBE can be found by setting the time-derivatives to 0, and σ ee , which is the mean probability of the electron to be found in the excited level, obtains a well-known Lorentzian form in the velocity [15] with the saturation accounted for).…”

Can a periodically driven particle resist laser cooling and noise?

“…Laser cooling is widely used in ion trapping [6][7][8][9][10][11][12][13][14][15][16]. With the cooling beam turned on, the ion may be expected to be damped to the minimum of the effective potential or, in some cases [17], heat up or diffuse to a larger amplitude where it may escape from the trap. However, even if the ion is cooled by the laser, the nonequilibrium nature of the dynamics implies in general that the peaks of its spatial probability distribution may not coincide with the minima of the potential.…”

“…By the linearity of the expanded motion, the transformation can be obtained in analytic closed form, with the coefficients of the Fourier expansion calculated numerically [24,26,27]. Averaging over the angle θ, we obtain an effective Fokker-Planck equation for the probability distribution P (I, t), which is a probability density function that depends on time and action only [17],…”