Recently, we have considered a quantum system dynamics caused by successive selective and non-selective measurements of the probe coupled to the system. For the finite measurement rate \tau^-1 and the system-probe interaction strength \gamma we derive analytical evolution equations in the stroboscopic limit \tau -> 0 and \gamma^2 \tau = const, which can be considered as a deviation from the Zeno subspace dynamics on a longer timescale T ~ (\gamma^2 \tau)^-1 >> \gamma^-1. Though the repeatedly measured probe evolution is frozen in the case of projective rank-1 selective measurements, the system evolution is not frozen! In fact, the system dynamics is non-linear in this case, and we derive the effective non-Hermitian Hamiltonian for such an evolution. The deduced non-linear dynamics may find applications in quantum amplifiers. Moreover, the system dynamics is analyzed for selective stroboscopic projective measurements of an arbitrary rank, when both the system and the probe evolve non-trivially. In the case of non-selective measurements, we derive the semigroup dynamics of the system-probe aggregate and find the particular form of the dissipator. Both non-linear and decoherent effects become significant at the timescale T ~ (\gamma^2 \tau)^-1, which is illustrated by a number of physical examples (Heisenberg interaction of spin particles, partial SWAP operations in quantum optics).
I. A. Luchnikov, S. N. Filippov. Quantum evolution in the stroboscopic limit of repeated measurements. Phys. Rev. A 95, 022113 (2017), E-print arXiv:1609.05501 [quant-ph]