Two-Particle Scattering Matrix of Two Interacting Mesoscopic Conductors

We consider two quantum coherent conductors interacting weakly via long range Coulomb forces. We describe the interaction in terms of two-particle collisions described by a two-particle scattering matrix. As an example we determine the transmission probability and correlations in a two-particle scattering experiment and find that the results can be expressed in terms of the density-of-states matrices of the noninteracting scatterers.

Figure 1

Two quantum dots coupled via the interaction $\lambda {\widehat{Q}}^{\mathrm{I}}{\widehat{Q}}^{\mathrm{II}}$. The operators $a^{†}$ ($c^{†}$) create incoming electrons in the scattering states in the left ($L$) and right ($R$) leads of dot I (II), while $b^{†}$ ($d^{†}$) are similar operators for outgoing electrons.

Figure 2

Quantum dot (system I) coupled to an edge state (system II). The quantum point contact (with transmission probability $T$ and reflection probability $R$) is outside the interacting region.