Third quantization: a general method to solve master equations for quadratic open Fermi systems

TL;DR

This paper introduces a general third quantization method to explicitly solve Lindblad master equations for quadratic fermionic systems, enabling analysis of non-equilibrium steady states and relaxation in quantum transport.

Contribution

It presents a novel third quantization approach that diagonalizes a 4n x 4n matrix to solve master equations for quadratic fermionic systems, extending analytical capabilities.

Findings

Explicit solutions for non-equilibrium steady states

Calculation of relaxation rates in quantum transport

Application to Heisenberg XY spin chain

Abstract

The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n x 4n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbor Heisenberg XY spin 1/2 chain in a transverse magnetic field.