Detecting non-Abelian geometric phases with superconducting nanocircuits

We propose a feasible scheme to detect the noncommutative feature of non-Abelian geometric phases (NAGP) with superconducting nanocircuits. The induced NAGP associated with two-fold degenerate states naturally arises in the chosen four-level subsystem interacting with the microwave pulses. We explicitly show the noncommutative effect of the NAGP by considering the difference between the level populations at the end of the composed evolution loops l2l1l2l1 and the counter-ordered one l1l2l1l2. The scheme opens the new possibility for detecting the fundamental characteristics of the NAGP with superconducting circuit devices.