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

TL;DR

This work extends the Landauer-Büttiker framework to open quantum systems that exchange particles with gain and loss by developing a Lindblad-Keldysh formalism-based extended LB formula. The currents split into a direct Landauer term and reservoir-mediated contributions, with a bosonic extension using Bose distributions and Bose stimulation, and explicit treatment of gain/loss via matrices $\mathbf{P}$ and $\mathbf{Q}$. Key findings include current generation from inversion-symmetry breaking, disorder-induced currents, and the smoothing of transmission that enables the Wiedemann–Franz law to hold across and beyond the energy band; it also analyzes the non-Hermitian skin effect's impact on transport. The framework provides a versatile, non-equilibrium transport description relevant for monitored quantum platforms and non-Hermitian open systems, with concrete demonstrations in two-site models and disorder settings.

Abstract

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.

Figures (10)

• Figure 1: Model scheme: A central system (yellow region) is coupled to two leads (left and right) with temperatures $T_{L(R)}$ and chemical potentials $\mu_{L(R)}$. It also connects to Markovian reservoirs (green region) for particle transfer. The current splits into three components (Eq. (\ref{['current2']})): $J_{LR}$, $J_{LS}$, and $J_{RS}$. $J_{LR}$ is described by the Landauer-Büttiker formula, while the others arise from particle exchange with the reservoirs.
• Figure 2: (a) $J^0$ as a function of $\gamma_{g}/\gamma_{l}$ and $\mu/k_BT$, in units of $e^2/h$, with $k_BT=20$, $\gamma_l=0.2$, and the system size $N=40$. (b1) Scheme of a two-site system with local monitoring. (b2) $J^0$ as a function of monitoring strength $\gamma_0$. The blue and red curves represent monitoring at the first and second sites, respectively, while the green curve represents monitoring at both sites. (c) $J^0$ as a function of disorder strength $V$ with $200$ samples and $N=100$. The black dashed line represents the envelope curve of the current. (d) The sample average of $(J^0)^2$. The red dashed line is the fitting function $\langle(J^0)^2\rangle\sim V^{1.91}$. Other parameters are (b-d) $\mu=0.1$, $k_BT=10^{-4}$, $\gamma_{l}=0.1$, and (a-d) $t_S=1$, $(\bm{\Gamma}_L)_{11}=(\bm{\Gamma}_R)_{NN}=1.1$.
• Figure 3: (a) $G$ and $K$ as functions of the chemical potential $\mu$ at temperature $k_B T = 0.2$, where $K$ is expressed in units of the thermal conductance quantum $\frac{\pi^2 k_B^2 T}{3h}$. (b) The ratio $K/G$ as a function of $T$. $\mu = 0$, $\mu = 2$, and $\mu = 4$ are located inside the band, at the band edge, and outside the band, respectively. The black dotted line represents the Wiedemann-Franz law. Here we take $t_S=1$ and $N = 80$.
• Figure 4: (a) Schematic of the setup to study transport in a system with NHSE caused by bond loss. The transmission functions (b) $\tau_0$ and (c) $\tau_1$ as a function of $t_S$ and $\omega$ for a system with size $N = 80$. (d) The variation of $J_0^0$, $\delta J_1^0$, and $J^0 = J_0^0 + \delta J_1^0$ with system size, with $t_S=3$ fixed. Other parameters: (b-d) $\gamma_{-}=1$, $\mu=0.12$; (c-d) $\delta\mu=0.6$.
• Figure 5: (a) Zero-temperature conductance as a function of system size. (b) Spatial profile of the bond current $J^b_{j}$ at zero temperature. Parameters: $t_S=1$, $\gamma_l=0.2$, $\gamma_g=0.1$, $\mu_L=0.2$, and $\mu_R=0.1$.