The Langevin equation for a quantum heat bath

We compute the quantum Langevin equation (or more exactly, the quantum stochastic differential equation) representing the action of a quantum heat bath at thermal equilibrium on a simple quantum system. These equations are obtained by taking the continuous limit of the Hamiltonian description for repeated quantum interactions with a sequence of quantum systems at a given density matrix state. In particular we specialise these equations to the case of thermal equilibrium states. In the process, new quantum noises are appearing: thermal quantum noises. We discuss the mathematical properties of these thermal quantum noises. We compute the Lindblad generator associated with the action of the heat bath on the small system. We exhibit the typical Lindblad generator that provides thermalization of a given quantum system.