Manipulation of half-integer flux quanta*

Because of the different pairing symmetry in high-T$_{c}$ cuprates ($d$-wave) and low-T$_{c}$ superconductors ($s$-wave) it is possible to realize hybrid superconducting rings with a built-in $\pi $-phase shift. These rings have a twofold degenerate groundstate characterized by spontaneously generated circulating currents corresponding to a half flux quantum. We have realized such rings by connecting the cuprate YBa$_{2}$Cu$_{3}$O$_{7-\delta }$ to the low-T$_{c}$ superconductor Nb via ramp-type Josephson junctions. We present the integration of $\pi $-rings in a superconducting quantum electronics device, a toggle flip-flop, in which the fractional flux polarity is toggled by applying single flux quantum pulses. We will also present experiments in which the half flux quanta at the discontinuities of 0-$\pi $-0 Josephson corner junctions are manipulated.