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The time evolution and the asymptotic outcome of aLandau-Zener-Stueckelberg-Majorana (LZ) process under continuous weaknon-selective measurement is analyzed. We compare two measurement protocols inwhich the populations of either the adiabatic or the non-adiabatic levels are(continuously and weakly) monitored. The weak measurement formalism, describedusing a Gaussian Kraus operator, leads to a time evolution characterized by aMarkovian dephasing process, which, in the non-adiabatic measurement protocolis similar to earlier studies of LZ dynamics in a dephasing environment.Casting the problem in the language of measurement theory makes it possible forus to compare diabatic and adiabatic measurement scenarios, to considerengineered dephasing as a control device and to examine the manifestation ofthe Zeno effect under the different measurement protocols. In particular, undermeasurement of the non- adiabatic populations, the Zeno effect is manifestednot as a freezing of the measured system in its initial state, but rather as anapproach to equal asymptotic populations of the two diabatic states. Thisbehavior can be traced to the way by which the weak measurement formalismbehaves in the strong measurement limit, with a built-in relationship betweenmeasurement time and strength.

Landau-Zener evolution under weak measurement: Manifestation of the Zeno effect under diabatic and adiabatic measurement protocols