[en] We started this work with a description of two devices that were recently developed in the context of quantum information processing. These devices are used as read-out for superconducting quantum bits based on Josephson junctions. The classical description has to be extended to the quantum regime. As the main result we calculate the leading order corrections in ℎ on the escape rate. We took into account a standard metastable potential with a static energy barrier and showed how to derive an extension of the classical diffusion equation. We did this within a systematic semiclassical formalism starting from a quantum mechanical master equation. This master equation contains an extra term for the loss of population due to tunneling through the barrier and, in contrast to previous approaches, finite barrier transmission which also affects the transition probabilities between the states. The escape rate is obtained from the stationary non-equilibrium solution of the diffusion equation. The quantum corrections on the escape rate are captured by two factors, the first one describes zero-point fluctuations in the well, while the second one describes the impact of finite barrier transmission close to the top. Interestingly, for weak friction there exists a temperature range, where the latter one can actually prevail and lead to a reduction of the escape compared to the classical situation due to finite reflection from the barrier even for energies above the barrier. Only for lower temperatures does the quantum result exceed the classical one. The approach can not strictly be used for the Duffing oscillator because of the time dependent term in its Hamiltonian. But it is possible to move in a frame rotating with a frequency equal to the response frequency of the Duffing oscillator in order to obtain a time-independent Hamiltonian. Therefore a system plus reservoir model was applied to consistently derive in the weak coupling limit the master equation for the reduced density in the rotating frame. We found that a position-position interaction between system and bath in the laboratory frame translates into additional momentum-momentum couplings in the rotating frame. We introduced the concept of an effective temperature to analyze the energy exchange between system and bath. For a structured environment we found a negative effective temperature physically corresponding to the fact that absorption becomes more probable than emission and a population inversion is induced. This effect can explain recent experimental observations of enhanced relaxation in a quantronium circuit coupled to a cavity bifurcation amplifier. Finally we applied the approach on the new master equation in the rotating frame to calculate the quantum escape rate for the Duffing oscillator. We discovered that there is an additional quantum effect compared to the case of a static barrier. In the rotating frame the quantum fluctuations that accompany relaxation of the system coupled to a bath lead to diffusion away from one stationary state and to a transition over the dynamic barrier to the second stationary state. This mechanism is due to the particular form of the interaction between system and bath. We found also that by tuning the bifurcation parameter we can change the effective friction. So, we easily can move from the underdamped regime, studied in this work, to the classical overdamped regime and finally to the quantum overdamped regime. (orig.)