Extension to the quantum Langevin equation in the incoherent hopping regime

An extension to the quantum Langevin equation is derived, that is valid in the incoherent hopping regime, and which allows one to incorporate quantum tunneling events. This is achieved by the inclusion of additional stochastic variables in the Langevin equation representing the tunneling events. A systematic derivation of this extension and of its regime of validity is presented. The study is motivated by efforts to determine the error in reading the state of a superconducting quantum bit.