  The time evolution and the asymptotic outcome of a
Landau-Zener-Stueckelberg-Majorana (LZ) process under continuous weak
non-selective measurement is analyzed. We compare two measurement protocols in
which the populations of either the adiabatic or the non-adiabatic levels are
(continuously and weakly) monitored. The weak measurement formalism, described
using a Gaussian Kraus operator, leads to a time evolution characterized by a
Markovian dephasing process, which, in the non-adiabatic measurement protocol
is similar to earlier studies of LZ dynamics in a dephasing environment.
Casting the problem in the language of measurement theory makes it possible for
us to compare diabatic and adiabatic measurement scenarios, to consider
engineered dephasing as a control device and to examine the manifestation of
the Zeno effect under the different measurement protocols. In particular, under
measurement of the non- adiabatic populations, the Zeno effect is manifested
not as a freezing of the measured system in its initial state, but rather as an
approach to equal asymptotic populations of the two diabatic states. This
behavior can be traced to the way by which the weak measurement formalism
behaves in the strong measurement limit, with a built-in relationship between
measurement time and strength.

Quantum Physics;Statistical Mechanics