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We study the distinctive features of the phase pumping effect in Josephsontransport through a three-layered ferromagnet F$_1$/F/F$_2$ with non-coplanarmagnetization. Using Gor'kov and Bogoliubov-de Gennes formalisms we go beyondthe quasiclassical approximation and analyze the dependence of the spontaneousJosephson phase $\psi$ on the exchange field $h$ in the F layer and details ofmagnetization profile. The pumping of the Josephson phase can be generated bythe mutual rotation of magnetizations in F$_1$ and F$_2$ layers resulting inthe 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 gainof the Josephson phase at the pumping period switches from $0$ to $2\pi$. Westudy the scenario of these switchings originating from the interplay betweenseveral competing local minima of the free energy of the junction versus thesuperconducting phase difference. Our analysis provides the basis for thesearch of experimental setup realizing the phase pumping phenomenon.

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