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We generalize the representation of the real time Green's functionsintroduced by Langreth and Nordlander [Phys. Rev. B 43 2541 (1991)] and Meirand Wingreen [Phys. Rev. Lett. 68 2512 (1992)] in stationary quantum transportin order to study problems with hybrid structures containing normal (N) andsuperconducting (S) pieces. We illustrate the treatment in a S-N junction undera stationary bias and investigate in detail the behavior of the equilibriumcurrents in a normal ring threaded by a magnetic flux with attachedsuperconducting wires at equilibrium. We analyze the flux sensitivity of theAndreev states and we show that their response is equivalent to the onecorresponding to the Cooper pairs with momentum q=0 in an isolatedsuperconducting ring.

physical review. b, condensed matter and materials physics

Stationary transport in mesoscopic hybrid structures with contacts to  superconducting and normal wires. A Green's function approach for  multiterminal setups