Two-ion superradiance theory.

Abstract

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 \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} from the one-ion value \ensuremath{\gamma}. A master equation is derived, giving a fractional change in the upper state lifetime \ensuremath{\gamma}/\ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}-1=sinkR/2kR [${\mathit{J}}_{0}^{2}$(z)-2${\mathit{J}}_{1}^{2}$(z)+...]cos\ensuremath{\Phi}, where k=2\ensuremath{\pi}/\ensuremath{\lambda}, \ensuremath{\lambda} 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 \ensuremath{\Phi} is the two-ion relative phase due to the preparation. In the Lamb-Dicke regime, a, ${\mathit{J}}_{0}^{2}$(z)\ensuremath{\approxeq}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.