I. Cycle averaged amplification and dampingConcerning the origin of equation ( 1) in the main text, and referring to figure 1, the respective scattering forces 1 are F c,a (v, i c,a ) = αi c,a f γ (∆ω c,a − kv) where i c,a is the cooling/amplification beam intensity at the ion position; ∆ω c,a ≡ ω c,a − ω o (ω o is the optical transition frequency); and α ≡ hkγ/2i sat (i sat is the saturation intensity of the dipole transition, and k = ω o /c). f γ (ω) is

The 1S-2S two-photon transition in singly ionized helium is a highly interesting candidate for precision tests of bound-state quantum electrodynamics ͑QED͒. With the recent advent of extreme ultraviolet frequency combs, highly coherent quasi-continuous-wave light sources at 61 nm have become available, and precision spectroscopy of this transition now comes into reach for the first time. We discuss quantitatively the feasibility of such an experiment by analyzing excitation and ionization rates, propose an experimental scheme, and explore the potential for QED tests.

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We report on injection locking of optically excited mechanical oscillations of a single, trapped ion. The injection locking dynamics are studied by analyzing the oscillator spectrum with a spatially selective Fourier transform technique and the oscillator phase with stroboscopic imaging. In both cases we find excellent agreement with theory inside and outside the locking range. We attain injection locking with forces as low as 5ð1Þ Â 10 À24 N so this system appears promising for the detection of ultraweak oscillating forces. [5]. In this Letter we report injection locking of an almost ultimately simple and well-controlled system, the motion of a single, harmonically bound ion.A Mg þ ion, held in a linear rf trap, is addressed by two laser beams tuned below (red detuned) and above (blue detuned) an atomic resonance, respectively. While the reddetuned laser damps the motion of the ion, the bluedetuned laser provides gain by amplifying an existing motion. For appropriate settings of laser detunings and intensities, regenerative oscillations with a stable amplitude start from noise. The forces due to the laser beams saturate as periodic Doppler shifts are induced by the ion motion. Recently, we could show that the amplification is due to the stimulated generation of vibrational quanta (phonons) and that the system represents a mechanical analogue of an optical laser, a phonon laser [6].We study the action of a weak, harmonic rf signal tuned close to the motional resonance on the dynamics of this oscillator using two different techniques, which both rely on the good spatial and temporal resolution of our imaging system. With the first method, a spatially selective Fourier transform technique, we obtain the motional spectrum of the oscillator. The second method, stroboscopic imaging, allows us to retrieve the phase of the ion relative to the drive. All our observations agree very well with theory, further substantiate the analogy to an optical laser, and extend its applicability. Our analysis reveals that we attain injection locking with minute forces as small as 5(1) yN (yocto 10 À24 ). This great sensitivity might allow the detection of nuclear spin flips of single atomic or molecular ions.Injection locking is a well-understood phenomenon [7,8]. Let us review some basic features for later reference: consider an oscillator of free-running frequency ! 0 , mass m, and linewidth . A weak oscillation induced by an injected auxiliary rf signal of frequency ! i and voltage U i is amplified by the blue-detuned laser. The gain of this auxiliary frequency component is proportional to [7] jgð Þj 2 % 2 2( 1) for sufficiently large detunings ¼ ! i À ! 0 . As the detuning decreases, the injected signal takes up more of the limited gain available from the blue-detuned laser. The gain at the free oscillation reduces accordingly; once it drops below threshold only the oscillation at ! i prevailsthe oscillator is injection locked to the external source [7]. Assuming the injected signal is weak, the total oscillation amplitude of the...