**Quantum walks on circles in phase space via cavity or circuit quantum electrodynamics**

Quantum Walks on Circles in Phase Space via Cavity or Circuit Quantum Electrodynamics
ABSTRACT: We show how a quantum walk in phase space can be implemented via cavity or circuit quantum electrodynamics (CQED) where only the resonator field (i.e. the walker) needs to be driven and measured. The atom or Cooper pair box (i.e. the coin) is controlled indirectly via Jaynes-Cummings coupling. Decoherence can be tuned so that the transition from quantum to classical walk can be controlled, which confirms the quantum nature of the walk. In contrast to previous proposals for CQED realizations, the walker is not confined to one circle in phase space (fixed mean energy) but rather leaps to other circles in phase space. Despite this complication, the quantum enhanced diffusion of walker's phase can be cleanly observed and rigorously explained, thereby enabling the first experimental realization of a single-walker quantum walk without the need for direct control of the coin state.