  We investigate the role of a time and spin-dependent phase shift on the
evolution of one-dimensional discrete-time quantum walks. By employing Floquet
engineering, a time and spin-dependent phase shift ($\phi$) is imprinted onto
the walker's wave function while it shifts from one lattice site to another.
For a quantum walk driven by the standard protocol we show with our numerical
simulations that complete revivals with equal periods occur in the probability
distribution of the walk for rational values of the phase factor, i.e.,
$\phi/2\pi = p/q$. For an irrational value of $\phi/2\pi$ our results show
partial revivals in the probability distribution with unpredictable periods,
and the walker remains localized in a small region of the lattice. We further
investigate revivals in a split-step quantum walk with a time and
spin-dependent phase shift for rational values of $\phi/2\pi$. In contrast to
the case of standard protocol, the split-step quantum walk shows partial
revivals in the probability distribution. Furthermore, in view of an
experimental realization we investigate the robustness of revivals against
noise in the phase shift. Our results show that signatures of revivals persist
for smaller values of the noise parameter, while revivals vanish for larger
values. Our work is important in the context of quantum computation where
quantum walks with a time and spin-dependent phase shift can be used to control
and manipulate a desired quantum state on demand.
