Title: Qubit dynamics at tunneling Fermi-edge singularity in $\it{a.c.}$ response,
Abstract: We consider tunneling of spinless electrons from a single-channel emitter
into an empty collector through an interacting resonant level of the quantum
dot. When all Coulomb screening of sudden charge variations of the dot during
the tunneling is realized by the emitter channel, the system is described with
an exactly solvable model of a dissipative qubit. To study manifestations of
the coherent qubit dynamics in the collector $\it{a.c.}$ response we derive
solution to the corresponding Bloch equation for the model quantum evolution in
the presence of the oscillating voltage of frequency $% \omega$ and calculate
perturbatively the $\it{a.c.}$ response in the voltage amplitude. We have shown
that in a wide range of the model parameters the coherent qubit dynamics
results in the non-zero frequencies resonances in the amplitudes dependence of
the $\it{a.c.}$ harmonics and in the jumps of the harmonics phase shifts across
the resonances. In the first order the $\it{a.c.}$ response is directly related
to the spectral decomposition of the corresponding transient current and
contains only the first $\omega$ harmonic, whose amplitude exhibits resonance
at $\omega =\omega_I $, where $\omega_I$ is the qubit oscillation frequency. In
the second order we have obtained the $2 \omega$ harmonic of the $\it{a.c.}$
response with resonances in the frequency dependence of its amplitude at
$\omega_I$, $\omega_I/2$ and zero frequency and also have found the frequency
dependent shift of the average steady current.