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We consider a class of discrete-time one-dimensional quantum walks,associated with CMV unitary matrices, in the presence of a local field. Thisclass is parametrized by a transmission parameter $t\in[0,1]$. We show that fora certain range for $t$, the corresponding asymptotic velocity can be madearbitrarily small by introducing a periodic local field with a sufficientlylarge period. In particular, we prove an upper bound for the velocity of the$n$-periodic quantum walk that is decaying exponentially in the period length$n$. Hence, localization-like effects are observed even after a long number ofquantum walk steps when $n$ is large.

Exponentially decaying velocity bounds of quantum walks in periodic fields