  We formulate a theory of nonequilibrium quantum solvation in which parameters
of the solvent are explicitly depending on time. We assume in a simplest
approach a spherical molecular Onsager cavity with a time-dependent radius. We
analyze the relaxation properties of a test molecular point dipole in a
dielectric solvent and consider two cases: (i) a shrinking Onsager sphere, and,
(ii) a breathing Onsager sphere. Due to the time-dependent solvent, the
frequency-dependent response function of the dipole becomes time-dependent. For
a shrinking Onsager sphere, the dipole relaxation is in general enhanced. This
is reflected in a temporally increasing line width of the absorptive part of
the response. Furthermore, the effective frequency-dependent response function
shows two peaks in the absorptive part which are symmetrically shifted around
the eigenfrequency. In contrast, a breathing sphere reduces damping as compared
to the static sphere. Interestingly, we find a non-monotonous dependence of the
relaxation rate on the breathing rate and a resonant suppression of damping
when both rates are comparable. Moreover, the line width of the absorptive part
of the response function is strongly reduced for times when the breathing
sphere reaches its maximal extension.
