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We study the transport properties of generic out-of-equilibrium quantumsystems connected to fermionic reservoirs. We develop a new method, based on anexpansion of the current in terms of the inverse system size and out ofequilibrium formulations such as the Keldysh technique and the Meir-Wingreenformula. Our method allows a simple and compact derivation of the current for alarge class of systems showing diffusive/ohmic behavior. In addition, we obtainexact solutions for a large class of quantum stochastic Hamiltonians (QSHs)with time and space dependent noise, using a self consistent Born diagrammaticmethod in the Keldysh representation. We show that these QSHs exhibit diffusiveregimes which are encoded in the Keldysh component of the single particleGreen's function. The exact solution for these QSHs models confirms thevalidity of our system size expansion ansatz, and its efficiency in capturingthe transport properties. We consider in particular three fermionic models: i)a model with local dephasing ii) the quantum simple symmetric exclusion processmodel iii) a model with long-range stochastic hopping. For i) and ii) wecompute the full temperature and dephasing dependence of the conductance of thesystem, both for two- and four-points measurements. Our solution gives accessto the regime of finite temperature of the reservoirs which could not beobtained by previous approaches. For iii), we unveil a novelballistic-to-diffusive transition governed by the range and the nature (quantumor classical) of the hopping. As a by-product, our approach equally describesthe mean behavior of quantum systems under continuous measurement.

Exact description of quantum stochastic models as quantum resistors