Strong electron tunneling through mesoscopic metallic grains

We describe electron transport through small metallic grains with Coulomb blockade effects beyond the perturbative regime. For this purpose we study the real-time evolution of the reduced density matrix of the system. In the first part of the paper we present a diagrammatic expansion for not too high junction conductance, $h/4\mathrm{}{\pi }^2{e}^2{R}_t\lesssim 1,$ in a basis of charge states. Quantum fluctuations renormalize system parameters and lead to finite lifetime broadening in the gate-voltage-dependent differential conductance. We derive analytic results for the spectral density and the conductance in the limit where only two charge states play a role. In the second part of the paper we consider junctions with large conductance, ${h/4e}^2{R}_t\gtrsim 1.$ In this case contributions from all charge states, which broaden and overlap, become important. We analyze the problem in a quasiclassical approximation. The two complementary approaches cover the essential features of electron tunneling for all parameters.