Quantum Langevin Equations for the Brune Multiport Hamiltonian

Current models of superconducting qubit dynamics rely on Lindblad master equations expressed in the Schrödinger picture. Traditionally the Lindblad terms are lumped into relaxation and dephasing contributions, T₁ and T_φ, respectively. The system is described as the qubit and electromagnetic environment bilinearly coupled to a thermal bath, most often as a collection of harmonic oscillators. We present a methodology for including multiple baths with linear and non-linear coupling between the system and bath operators in the Heisenberg picture. Quantum Langevin equations describe the evolution of the canonically conjugate variables of flux and charge in a superconducting circuit of qubits and linear passive circuit elements. We derive Langevin equations for the special case of a Brune multiport Hamiltonian bilinearly coupled to a harmonic bath and discuss our approach to couple other baths to this system, including the two-bath model where ensembles of two level systems are coupled to the system and lose their energy to a phonon bath.