The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. A particular area of interest is in the optical response of solid state systems, where dissipation is introduced phenomenologically through the relaxation time approximation and its effects are usually gauged perturbatively. Analytical exact results for driven systems under this approximation are scarce and typically pertain only to the stationary regime. Here, we obtain the analytical solution for the current response of general single band tight-binding system driven by a uniform electric field with generic time dependence under the relaxation time approximation. We explore the effects of dissipation in two limiting cases: a monochromatic field, where we analytically deduce the effect of dissipation on high harmonic generation, and a constant electric field, where a generalization for the Esaki-Tsu equation is presented for any single band tight-binding system. We specify the results for a simple two-dimensional nearest-neighbor tight-binding lattice to emphasize the effect of the scale competition introduced by the two different neighbors in both the monochromatic and constant field cases. Finally, we compare our exact result for the constant field to the one obtained from the usual perturbation theory calculation to probe the validity of the latter.

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• Charge dynamics
• Dissipative dynamics
• Nonlinear optics
• Quantum transport
• Metals
• Density matrix methods
• Tight-binding model