Self-consistent microscopic derivation of Markovian master equations for open quadratic quantum systems

TL;DR

This paper rigorously derives Markovian master equations for quadratic quantum systems coupled to thermal baths, applicable to fermionic and bosonic models of any size or dimension, ensuring thermodynamic consistency and explicit normal mode coupling.

Contribution

It provides a general, rigorous derivation of Lindblad master equations for quadratic open quantum systems without symmetry restrictions, including explicit normal mode coupling and steady-state analysis.

Findings

Normal modes serve as effective Lindblad operators.

Steady states are unique and can be computed efficiently.

Particle and energy currents follow Landauer's formula, ensuring thermodynamic consistency.

Abstract

We provide a rigorous construction of Markovian master equations for a wide class of quantum systems that encompass quadratic models of finite size, linearly coupled to an environment modeled by a set of independent thermal baths. Our theory can be applied for both fermionic and bosonic models in any number of physical dimensions, and does not require any particular spatial symmetry of the global system. We show that, for non-degenerate systems under a full secular approximation, the effective Lindblad operators are the normal modes of the system, with coupling constants that explicitly depend on the transformation matrices that diagonalize the Hamiltonian. Both the dynamics and the steady-state (guaranteed to be unique) properties can be obtained with a polynomial amount of resources in the system size. We also address the particle and energy current flowing through the system in a minimal two-bath scheme and find that they hold the structure of Landauer's formula, being thermodynamically consistent.