  We investigate the possibility to use transient absorption of a coherent
bound electron wave packet in hydrogen as an attosecond pulse characterization
technique. In recent work we have shown that photoionization of such a coherent
bound electron wave packet opens up for pulse characterization with
unprecedented temporal accuracy --- independent of the atomic structure ---
with maximal photoemission at all kinetic energies given a wave packet with
zero relative phase [Pabst and Dahlstr\"om, Phys. Rev. A, 94, 13411 (2016)].
Here, we perform numerical propagation of the time-dependent Schr\"odinger
equation and analytical calculations based on perturbation theory to show that
the energy-resolved maximal absorption of photons from the attosecond pulse
does not uniquely occur at zero relative phase of the initial wave packet.
Instead, maximal absorption occurs at different relative wave packet phases,
distributed as a non-monotonous function with a smooth $-\pi/2$ shift across
the central photon energy (given a Fourier-limited Gaussian pulse). Similar
results are found also in helium. Our finding is surprising because it implies
that the energy-resolved photoelectrons are not mapped one-to-one with the
energy-resolved absorbed photons of the attosecond pulse.
