Quantum walks on circles in phase space via superconducting circuit QED

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.\r\n\r\nAlthough the quantum walk has been one of the most important recent advances in quantum information science, we only have idealized, unrealistic proposals for a quantum quincunx to demonstrate a quantum walk in the laboratory. Unless a major advance comes along, the separation between ideal and practical quantum walks will not disappear. Our paper presents this advance so that a quantum walk is feasible experimentally. The realization is in cavity quantum electrodynamics where preparation of a vacuum state is commonplace. The obstacle of having to drive the atom directly in order to effect a Hadamard oin flip is eliminated by effecting the coin flip by directly driving the cavity. Typically cavity quantum electrodynamics in both the optical and microwave domains involves driven cavities, not driven atoms, so our generalization from driven atom to driven cavity is exactly what is needed to make the quantum quincunx feasible.\r\nHowever driving the cavity breaks the photon number conservation condition. As the quantum walk occurs on a circle in phase space with constant photon number, breaking this symmetry would, intuitively, be expected to completely destroy the fragile quantum walk features being sought experimentally. However we have shown that, not only does the quantum walk survive this violation of number conservation, but also the quantum quincunx is feasible under existing conditions. \r\n\r\nOur proposal is for any cavity quantum electrodynamics realization.\r\n