Feedback Cooling of a Single Trapped Ion

Based on a real-time measurement of the motion of a single ion in a Paul trap, we demonstrate its electromechanical cooling below the Doppler limit by homodyne feedback control (cold damping). The feedback cooling results are well described by a model based on a quantum mechanical master equation.

Figure 1

A single ${}^{138}\mathrm{Ba}^{+}$ ion in a Paul trap (parabola) is laser excited and cooled on its $S_{1/2}$ to $P_{1/2}$ transition at 493 nm. A retro-reflecting mirror 25 cm away from the trap and a lens (not shown) focus back the fluorescence onto the ion. The resulting interference fringes with up to 73% contrast are observed by a photomultiplier (PMT 1). The ion’s oscillation in the trap creates an intensity modulation of the PMT signal which is observed as a sideband on a spectrum analyzer (rfsa) [16]. For feedback cooling, the sideband signal is filtered, phase-shifted, and applied to the ion as a voltage on a trap electrode.

Figure 2

Feedback cooling spectra. The vibrational sideband around $\nu =1\mathrm{MHz}$ is shown on top of the spectrally flat shot-noise background. The solid line represents a Lorentzian fitting function. The upper curves (a), (b), (c) are measured outside the feedback loop, while the lower curves (d), (e), (f) are the in-loop results. Spectra (a) and (d) are for laser cooling only, the other curves are recorded with feedback at the indicated gain values. The feedback phase is set to $(-\pi /2)$. The gain values indicate the settings of the final amplifier in the feedback loop.

Figure 3

Steady-state energy of the cooled oscillator: measured sideband area, normalized to the value without feedback, versus gain of the feedback loop, for $(-\pi /2)$ (main plot) and $\pi$ (inset) feedback phase. The curves are the model calculations. The gain axis is scaled to the experimental values of the electronic gain.