Third-quantization based method for studying quantum transport of interacting bosons

Due to the Bose-Einstein statistics, the Fock space of a single site bosonic system can be infinite dimensional. For open quantum systems, the Lindblad master equation provides a reasonable description. When the Hamiltonian is quadratic, a method known as third quantization can be used to find the steady state. For interacting bosons, we apply the Hartree approximation to the third quantization method and to study the steady state density profiles and particle current. Compared to other numerical methods that truncate the local Fock space, the Hartree-third quantization method keeps the full Fock space but treats the interaction at the mean field level. To illustrate the power of this method, we demonstrate interaction based rectification and geometry based internal circulation of currents. Our method is suitable for studying quantum transport of ultracold atoms in engineered potentials.