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

Abstract

We study the probability distribution of an atomic ion being laser-cooled in a periodically-driven Paul trap using a Floquet approach to the semiclassical photon scattering dynamics. We show that despite the microscopic nonequilibrium forces, a stationary thermal-like exponential distribution can be obtained in the Hamiltonian action, or equivalently in the number of quanta (phonons) of the motion linearized about the zero of the potential. At the presence of additional stray electric fields, the ion is pushed from the origin of the potential and set into a large-amplitude driven oscillation, and above a threshold amplitude of such "excess micromotion", the action distribution of excitations about the driven oscillation broadens and becomes distinctly nonthermal. We find that by a proper choice of the laser detuning the distribution can be made exponential again, with a mean phonon number close to that of the Doppler cooling limit. We derive a relation allowing to deduce just from the experimentally observable photon scattering rate both the required detuning for optimal cooling and the final mean phonon number. These results are important for quantum information processing and other applications, and in particular the derived approach can be applied to crystals of trapped ions in planar configurations, where the driven motion of ions is unavoidable.

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

“…The resulting absorption and emission rates are equal on average, even if there are finite delays between these events. A more detailed derivation of the photon absorption probability from the optical Bloch equations will be presented separately in [31] using a Floquet approach, which allows one to improve the accuracy of accounting for the micromotion drive. Let us consider an ion moving in the trap, and at some arbitrary time t a when it is at position r a with momentum p a (velocity v a ), it absorbs a laser photon of energy (ω L + ∆).…”

Can a periodically driven particle resist laser cooling and noise?

“…with S(I, t) a probability flux, Π I (I) an action drift coefficient and Π II (I) a diffusion coefficient. The calculation of Π I and Π II proceeds in a straightforward by using the formulas derived for a linear Floquet system in [28]. 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 mean photon scattering rate is 1.2 × 10 6 s −1 × s L , with s L the saturation parameter. This rate can be contrasted with the rate at the effective trap minimum [28], (Γs L /2)/ 1 + [2∆/Γ] 2 , which gives 66 × 10 6 s −1 × s L for the optimal detuning ∆ = −Γ/2, but 0.3 × 10 6 s −1 × s L for ∆ = −10.6Γ. The photons can be detected and a Fourier transform of the fluoresence would show peaks at the s-subharmonics of the rf drive frequency.…”

“…This provides an example of a system driven far-out-of-equilibrium that is nevertheless described by effective dynamics (coarse-grained over the angle), which approximately obey detailed-balance [since in steady state, the probability flux in action, S(I, t) of Eq. (6), must vanish [28], a constraint obeyed by Eq. (7)].…”

“…In the limit of small amplitudes, the coordinate u(t) linearized about the island centres corresponds to a quantum parametric oscillator [24,28]. For the presented parameters, the mean action with ∆ = −10.6Γ corresponds (by using the semiclassical relation n ≈ I/ , with n the phonon number), to n ≈ 170 phonons, somewhat higher than the standard Doppler cooling limit n ≈ 100 of an identical ion whose secular frequency is similar, 2π × 150 kHz.…”