Hartree approximation of bosonic transport via third quantization of the Lindblad master equation

The third quantization in the Liouville space of many-body open bosonic systems provides an exact steady-state solution for the Lindblad master equation with quadratic Hamiltonian and linear Lindbladian operators. Using the Hartree approximation, we reduce the onsite interaction of a one dimensional Bose Hubbard model (BHM) to quadratic terms and present a self-consistent steady state solution of the open bosonic system. This approach spans the infinite Fock space allowed by the Bose-Einstein statistics while scaling polynomially with lattice size. We analyze three BHM examples: the uniform BHM, interaction induced diode effect, and the Su-Schrieffer-Heeger Hubbard model. When compared to small lattice simulations with truncated Fock states, third quantization with the Hartree approximation captures the qualitative behavior. Our approach provides a feasible way for investigating large-scale interacting bosonic transport relevant to cold-atom experiments.