  We introduce a family of quantized field states that can perform exact
(entanglement- and error-free) rotations of a two-level atom starting from a
specific state on the Bloch sphere. We discuss the similarities and differences
between these states and the recently-introduced "transcoherent states." Our
field states have the property that they are left unchanged after the rotation,
and we find they are the asymptotic states obtained when a field interacts with
a succession of identically prepared ancillary atoms. Such a scheme was
recently proposed [npj Quantum Information 3:17 (2017)] as a way to "restore" a
field state after its interaction with a two-level atom, so as to reuse it
afterwards, thus reducing the energy requirements for successive quantum
logical operations. We generalize this scheme to find optimal pulses for
arbitrary rotations, and also study analytically what happens if the ancillas
are in a mixed, rather than a pure state. Consistent with the numerical results
in the original proposal, we find that as long as the ancilla preparation error
is small (of the order of $1/\bar n$, where $\bar n$ is the average number of
atoms in the pulses considered) it will introduce only higher-order errors in
the performance of the restored pulse.
