SU(2) hyper-clocks with programmable composite phase-shifts

In 1949, Ramsey's method of separated oscillating fields was proposed boosting over many decades metrological performances of atomic clocks and becoming the standard technique for very high precision spectroscopic measurements. A full generalization of this interferometric method is presented replacing the two single coherent excitations by arbitrary composite laser pulses. The rotation of the state vector of a two-level system under the effect of a single pulse is described using the Pauli-spin matrices basis of the SU(2) group. It is then generalized to multiple excitation pulses by a recursive Euler-Rodrigues-Gibbs algorithm describing a composition of rotations with different rotation axes. A general analytical formula for the phase-shift associated with the clock's interferometric signal is derived. As illustrations, hyper-clocks based on three-pulse and five-pulse interrogation protocols are studied and shown to exhibit nonlinear cubic and quintic sensitivities to residual probe-induced light-shifts. The presented formalism is well suited to conceive programmable composite phase-shifts produced by tailored quantum algorithms in order to design a new generation of optical frequency standards, robust qubit and qudit rotations for quantum computation and address new challenges in AMO physics with ultra-high precision laser spectroscopy on cold matter and anti-matter.