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Effect of a Coulombic dot-lead coupling on the dynamics of a quantum dot

The effect of a Coulombic coupling on the dynamics of a quantum dot hybridized to leads is determined. The calculation treats the interaction between charge fluctuations on the dot and the dynamically generated image charge in the leads. A formally exact solution is presented for a dot coupled to a Luttinger liquid and an approximate solution, equivalent to treating the lead dynamics within a random phase approximation, is given for a dot coupled to a two- or three-dimensional metallic lead. The leading divergences arising from the long-ranged Coulomb interaction are found to cancel, so that in the two- and three-dimensional cases the quantum-dot dynamics is equivalent to that obtained by neglecting both the dot-lead Coulomb coupling and the Coulomb renormalization of the lead electrons, while in the one-dimensional case the dot-lead mixing is enhanced relative to the non-interacting case. Explicit results are given for the short-time dynamics.