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Obtaining correlation functions in quantum impurity models, which are often to describe charge and spin transport through molecules and quantum dots, is a matter of major importance in condensed phase materials science. In particular, in the language of these functions a rigorous mapping exists between bulk strongly correlated electron systems (such as transition metal oxides) and interacting impurities embedded within a non-interacting effective bath, via the "dynamical mean-field theory" (DMFT). The extraction of dynamical properties like correlation functions from the imaginary-time Monte Carlo methods commonly used within DMFT is an ill-posed problem, and reliable results both for transport in molecular electronics and DMFT require real-time methods. Unfortunately, until now such methods have only addressed single-time properties such as state populations and transport, while correlation functions are two-time observables.

We have developed a numerically exact real time quantum Monte Carlo method for computing correlation functions of impurity models in equilibrium and nonequilibrium. We show that with this tool we can reliably resolve the spectral function of weakly and strongly correlated impurities at all frequencies. We go on to consider an impurity in a junction, where we show how the correspondence between the spectral function and the differential conductance breaks down when nonequilibrium effects are taken into account. Finally, a long-standing dispute regarding this model has involved the voltage splitting of the Kondo peak, an effect which was predicted over two decades ago by approximate analytical methods but was never successfully confirmed by reliable numerics. We finally settle this issue by demonstrating that the splitting indeed occurs.