  We consider the dissipative single-qubit circuit QED architecture in which
the atomic transition frequency undergoes a weak external time-modulation. For
sinusoidal modulation with linearly varying frequency we derive effective
Hamiltonians that resemble the Landau-Zener problem of finite duration
associated to a two- or multi-level systems. The corresponding off-diagonal
coupling coefficients originate either from the rotating or the
counter-rotating terms in the Rabi Hamiltonian, depending on the values of the
modulation frequency. It is demonstrated that in the dissipation less case one
can accomplish almost complete transitions between the eigenstates of the bare
Rabi Hamiltonian even for relatively short duration of the frequency sweep. To
assess the experimental feasibility of our scheme we solved numerically the
phenomenological and the microscopic quantum master equations in the Markovian
regime at zero temperature. Both models exhibit qualitatively similar behavior
and indicate that photon generation from vacuum via effective Landau-Zener
transitions could be implemented with the current technology on the timescales
of a few microseconds. Moreover, unlike the harmonic dynamical Casimir effect
implementations, our proposal does not require the precise knowledge of the
resonant modulation frequency to accomplish meaningful photon generation.
