Connection between the GKSL master equation and the Landauer formula

Technical Summary: Connection between the GKSL Master Equation and the Landauer Formula

Problem Statement
Electron transport in mesoscopic systems, such as molecular junctions and quantum dots, is typically analyzed using two distinct frameworks: the Green's function approach (well-suited for coherent transport) and the master equation approach (commonly used in weak coupling regimes involving dissipation and dephasing). While both are widely employed, the theoretical connection between them remains not fully understood. Previous work by Jin et al. established a link for specific system-bath models under high-temperature and large chemical potential limits, showing convergence to the Meir–Wingreen and Landauer formulas. However, a derivation of the current formula within the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation formalism for a general non-interacting system, and a precise identification of the conditions under which it reduces to the Landauer formula, is required to clarify this connection under broader conditions.

Methodology
The authors analyze a non-interacting electron system connected to left (L) and right (R) electrodes, characterized by Fermi distribution functions. The analysis proceeds within the GKSL master equation framework under the following assumptions:

1. Markov Approximation & Wide-Band Limit Approximation (WBLA): The coupling between each orbital and the electrodes is sufficiently weak, allowing the Fermi distribution to be approximated as energy-independent within the relevant energy window.
2. Hamiltonian: The system is described by a general electronic Hamiltonian $\hat{H} = \sum_{ij} h_{ij}\hat{c}^\dagger_i \hat{c}_j$.
3. Dynamics: The time evolution of the density matrix $\rho$ is governed by the GKSL equation, incorporating Lindblad superoperators that account for electron injection and extraction from the electrodes. The coefficients $\alpha_{a,i}$ and $\beta_{a,i}$ are derived from the coupling strengths $\gamma_{a,i}$ and the Fermi distributions $f_{a,i}$.
4. Steady State: The authors solve for the steady-state single-electron density matrix $n$ by setting the time derivative to zero. This leads to a Sylvester equation of the form $AX - XB = Y$.
5. Mathematical Derivation: Using the general integral solution for the Sylvester equation involving contour integration, the authors express the steady-state density matrix $n$ in terms of retarded ($G^r$) and advanced ($G^a$) Green's functions. This expression is then substituted into the current definition to derive a current formula expressed via Green's functions.

Key Contributions and Results
The paper derives an explicit expression for the electric current flowing from the left electrode to the system within the GKSL formalism. The resulting current $I_L$ is given by:
$$I_L = \frac{q}{h} \int_{-\infty}^{\infty} d\epsilon \, \text{Tr} \left[ f_L \gamma_L G^a(\epsilon) \gamma_R G^r(\epsilon) - f_R \gamma_R G^a(\epsilon) \gamma_L G^r(\epsilon) \right]$$
This formulation allows for a direct comparison with the standard Landauer formula:
$$I = \frac{q}{h} \int_{-\infty}^{\infty} d\epsilon \, \text{Tr} \left[ \gamma_L G^r(\epsilon) \gamma_R G^a(\epsilon) \right] (f_L(\epsilon) - f_R(\epsilon))$$

The authors identify two specific conditions under which the GKSL-derived formula coincides with the Landauer formula:

1. Diagonal Hamiltonian (Independent Modes): If the system Hamiltonian is diagonal (representing independent multichannel modes), the GKSL current reduces to a form identical to the Landauer formula in the weak coupling limit.
2. Energy-Independent Fermi Distributions: If the Fermi distribution functions $f_L$ and $f_R$ can be treated as energy-independent constants within the energy range of interest (as illustrated in the model's schematic), both the Landauer and GKSL expressions reduce to the same simplified form:
   $$I = \frac{q}{h} (f_L - f_R) \int_{-\infty}^{\infty} d\epsilon \, \text{Tr} \left[ \gamma_L G^r(\epsilon) \gamma_R G^a(\epsilon) \right]$$
   The authors note that the condition of energy-independent distributions is physically justified when all single-particle orbital energies lie strictly between the chemical potentials of the electrodes ($\mu_R < \epsilon_i < \mu_L$) and the temperature is sufficiently low ($k_B T \ll \mu_L - \mu_R$), such that $f_L \approx 1$ and $f_R \approx 0$.

Significance
The paper provides a rigorous derivation linking the GKSL master equation approach to the Landauer formula for non-interacting systems. By utilizing the general solution to the Sylvester equation, the authors demonstrate that the two frameworks are not inherently contradictory but become equivalent under specific physical constraints. Specifically, the equivalence holds when the energy dependence of the Fermi distributions can be neglected within the relevant energy window. This work clarifies the theoretical boundary conditions where the dissipative, sequential tunneling picture of the master equation converges with the coherent transport picture of the Landauer formalism.