Two-ion superradiance theory

A superradiance theory is developed for two identical hydrogenic ions (four states each) in a microtrap, as in recent experiments. The ions oscillate (micromotion) due to the trap’s radio frequency (rf) electric quadrupole field. One signature of superradiance is a deviation of the two-ion average upper state decay rate γ¯ from the one-ion value γ. A master equation is derived, giving a fractional change in the upper state lifetime γ/γ¯-1=sinkR/2kR [${\mathit{J}}_0^2$(z)-2${\mathit{J}}_1^2$(z)+...]cosΦ, where k=2π/λ, λ is the emission wavelength, R is the ion-ion distance, ${\mathit{J}}_{\mathit{n}}$(z) is a Bessel function of integer order n, and z=ka, a is the ion amplitude of motion, and Φ is the two-ion relative phase due to the preparation. In the Lamb-Dicke regime, a<λ, ${\mathit{J}}_0^2$(z)≊1 and thus superradiance is not influenced significantly by ion motion. This damped sinusoid is diluted by the factor of 1/2 due to destructive interference. Superradiance vanishes in the absence of coherent preparation, e.g., with inversion, as indicated by the time evolution of the two-ion dipole correlation function. Fringes and a beat at the rf are predicted in forward scattering, the most elementary form of optical free induction decay.