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arXiv:1809.10519 · Physical Review A

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

Haggai Landa

Published Jul 16, 2019• 5 citations
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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.

DOI
10.1103/physreva.100.013413

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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.
We study the probability distribution of an atomic ion being laser-cooled in a radio-frequency Paul trap, in terms of Hamiltonian action-angle coordinates appropriate for the time-dependent potential. Close to the effective minimum in many Paul traps, the periodically-driven potential is approximately quadratic in the coordinates. We find that despite the microscopic nonequilibrium forces, a stationary thermal-like exponential distribution in the action is reached after coarse-graining the angles, with a mean action corresponding to the Doppler cooling limit. However, residual electric fields can push the ion from the origin of the quadrupole potential and set it into a large-amplitude oscillation. Above a threshold amplitude of such excess micromotion, the action distribution for motion expanded 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 action close to the Doppler cooling limit. This is important for quantum information processing and other applications, and in particular the derived approach can be immediately applied to crystals of trapped ions in planar configurations, where the driven motion of ions is unavoidable.
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.
Laser cooling of a single trapped ion in a Paul trap is discussed theoretically in the Lamb-Dicke limit, with full consideration of the time dependence of the trapping potential. Resulting mean kinetic energies are defined as time averages over one period of the micromotion and are compared with final temperatures expected from the laser-cooling treatment with harmonic traps. For laser-atom detunings close to the micromotion frequency the results differ significantly from those expected for a harmonic trap potential. A physical interpretation is given and simple formulas are derived for the strong confinement case.
We study the time-dependent fluorescence of an initially hot, multilevel, single atomic ion trapped in a radio-frequency Paul trap during Doppler cooling. We have developed an analytical model that describes the fluorescence dynamics during Doppler cooling which is used to extract the initial energy of the ion. While previous models of Doppler cooling thermometry were limited to atoms with a two-level energy structure and neglected the effect of the trap oscillating electric fields, our model applies to atoms with multilevel energy structure and takes into account the influence of micromotion on the cooling dynamics. This thermometry applies to any initial energy distribution. We experimentally test our model with an ion prepared in coherent, thermal, and Tsallis energy distributions.
An experimental and theoretical study of laser sideband cooling of a trapped ion in a radiofrequency trap is presented. The influence of the micromotion in the time-dependent potential of a Paul trap can lead to a counterintuitive situation, where sideband cooling is possible for positive values of the laser detuning, i.e., for a laser frequency that is higher than the resonance frequency of the ion at rest. The cooling rate and the capture range are calculated in a semiclassical model. Experimental results of laser sideband cooling at positive detuning are demonstrated with a single trapped In+{\mathrm{In}}^{+}In+ ion stored in a miniature Paul trap.
The optimization of the parameters of laser cooling and the investigation of the heating rate in ion traps require the measurement of the temperature of ion chains for which the Lamb–Dicke regime is satisfied. A novel method based on the investigation of the dynamics of Rabi oscillations at a narrow electron transition in an individual ion of the chain has been suggested for such measurement. An analytical expression for the population of the upper state as a function of the excitation time is derived taking into account the thermal distribution of phonons over the vibrational modes of a chain with an arbitrary number of ions. The method is tested experimentally for the chain of five 171Yb+^{{171}}{\text{Y}}{{{\text{b}}}^{ + }}171Yb+ ions using the quadrupole transition at 435 nm, as well as for a single ion. The heating rate measured for the axial vibrational mode in the implemented trap is 8 × 10 3 phonons/s.