Numerical test of few-qubit clock protocols

The stability of several clock protocols based on 2 to 20 entangled atoms is evaluated numerically by a simulation that includes the effect of decoherence due to classical oscillator noise. In this context the squeezed states discussed by Andr\'{e}, S{\o}rensen and Lukin [PRL 92, 239801 (2004)] offer reduced instability compared to clocks based on Ramsey's protocol with unentangled atoms. When more than 15 atoms are simulated, the protocol of Bu\v{z}ek, Derka and Massar [PRL 82, 2207 (1999)] has lower instability. A large-scale numerical search for optimal clock protocols with two to eight qubits yields improved clock stability compared to Ramsey spectroscopy, and for two to three qubits performance matches the analytical protocols. In the simulations, a laser local oscillator decoheres due to flicker-frequency (1/f) noise. The oscillator frequency is repeatedly corrected, based on projective measurements of the qubits, which are assumed not to decohere with one another.