Extended Landauer-Büttiker Formula for Current through Open Quantum Systems with Gain or Loss

The Landauer-Büttiker formula, which characterizes the current flowing through a finite region connected to leads, has significantly advanced our understanding of transport. We extend this formula to describe particle and energy currents with gain or loss in the intermediate region by using the Lindblad-Keldysh formalism. Based on the derived formula, several novel effects induced by gain or loss in the current are discussed: the breaking of inversion symmetry in the gain and loss terms or in the system can lead to current generation; the anomalous phenomenon that disorder can induce current generation; the presence of gain and loss makes the thermal and electrical conductances continuous and ensures they follow the Wiedemann-Franz law even outside the energy band; the effect of bond loss-induced skin effect on current. This work deepens and extends our understanding of transport phenomena in open systems.

💡 Research Summary

The paper presents a comprehensive extension of the Landauer‑Büttiker formalism to quantum transport through a finite region that experiences particle gain and loss. Starting from a one‑dimensional tight‑binding chain coupled to two metallic leads, the authors incorporate Markovian reservoirs via Lindblad operators that describe loss (L₁) and gain (L₂) processes. By casting the problem into the Keldysh non‑equilibrium Green‑function framework and embedding the Lindblad dissipators, they derive an effective dynamical matrix X = hₛ − iP − iQ (with P and Q the loss and gain matrices) and a non‑Hermitian “imbalance” matrix Y = 2(P − Q). The retarded/advanced Green functions are g^{R/A} = (ω − X)^{-1}, while the Keldysh component reads g^{K}=−(ω − X)^{-1}Y(ω − X†)^{-1}. Lead self‑energies Σ_{L,R} are introduced in the usual way, leading to full Green functions G^{R/A}=