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.