  When the level separation of a qubit is modulated periodically across an
avoided crossing, tunneling to the excited state - and consequently
Landau-Zener-St\"uckelberg interference - can occur. The types of modulation
studied so far correspond to a continuous change of the level separation. Here
we study periodic latching modulation, in which the level separation is
switched abruptly between two values and is kept constant otherwise. In this
case, the conventional approach based on the asymptotic Landau-Zener (LZ)
formula for transition probabilities is not applicable. We develop a novel
adiabatic-impulse model for the evolution of the system and derive the
resonance conditions. Additionally, we derive analytical results based on the
rotating-wave approximation (RWA). The adiabatic-impulse model and the RWA
results are compared with those of a full numerical simulation. These
theoretical predictions are tested in an experimental setup consisting of a
transmon whose flux bias is modulated with a square wave form. A rich spectrum
is observed, with distinctive features correspoding to two regimes:
slow-modulation and fast-modulation. These experimental results are shown to be
in very good agreement with the theoretical models. Also, differences with
respect to the well known case of sinusoidal modulation are discussed, both
theoretically and experimentally.
