Transmission through a quantum dot in a four-terminal phase-coherent system

Motivated by the recent experiment by Schuster et al (Schuster R, Buks E, Heiblum M, Mahalu D, Umansky V and Shtrikman H 1997 Nature 385 417), we study a four-terminal phase-coherent system, each arm having a quantum dot embedded in it (dot 1 for studying and dot 0 for reference). Using the nonequilibrium-Green-function method, the open-circuit collector voltage v(4) is derived. We find the following features. (1) The phase behaviours are similar for all of the resonance peaks. (2) In a single resonance peak, the phase phi increases by pi on a scale of about the half-peak-width Gamma(omega). (3) An abrupt phase drop, by pi, occurs near the point half way between two consecutive peaks. These results agree well with experiment. We attribute the characteristic (3) to the off-diagonal linewidth of dot `, which is a single-electron effect. In addition, the crossover of the phase behaviour in going from the four-terminal system to a two-terminal system is studied. Finally, another manifestation of this off-diagonal linewidth is also discussed.