Adiabatic phase pumping in S/F/S hybrids with non-coplanar magnetization

We study the distinctive features of the phase pumping effect in Josephson transport through a three-layered ferromagnet F$_1$/F/F$_2$ with non-coplanar magnetization. Using Gor'kov and Bogoliubov-de Gennes formalisms we go beyond the quasiclassical approximation and analyze the dependence of the spontaneous Josephson phase $\psi$ on the exchange field $h$ in the F layer and details of magnetization profile. The pumping of the Josephson phase can be generated by the mutual rotation of magnetizations in F$_1$ and F$_2$ layers resulting in the nontrivial phase gain at the rotation period (Berry phase). The increase in $h$ is shown to cause changes in the topology of the phase evolution: the gain of the Josephson phase at the pumping period switches from $0$ to $2\pi$. We study the scenario of these switchings originating from the interplay between several competing local minima of the free energy of the junction versus the superconducting phase difference. Our analysis provides the basis for the search of experimental setup realizing the phase pumping phenomenon.