Memory effects in a sequence of measurements of non-commuting observables

We use continuous, stochastic quantum trajectories within a framework of quantum state diffusion (QSD) to describe alternating measurements of two non-commuting observables. Projective measurement of an observable completely destroys memory of the outcome of a previous measurement of the conjugate observable. In contrast, measurement under QSD is not projective and it is possible to vary the rate at which information about previous measurement outcomes is lost by changing the strength of measurement. We apply our methods to a spin 1/2 system and a spin 1 system undergoing alternating measurements of the $S_{z}$ and $S_{x}$ spin observables. Performing strong $S_{z}$ measurements and weak $S_{x}$ measurements on the spin 1 system, we demonstrate return to the same eigenstate of $S_{z}$ to a degree beyond that expected from projective measurements and the Born rule. Such a memory effect appears to be greater for return to the $\pm1$ eigenstates than the $0$ eigenstate. Furthermore, the spin 1 system follows a measurement cascade process where an initial superposition of the three eigenstates of the observable evolves into a superposition of just two, before finally collapsing into a single eigenstate, giving rise to a distinctive pattern of evolution of the spin components.