Half-integer Shapiro-steps in superconducting qubit with a $pi$-Josephson junction

A superconducting quantum interference device (SQUID) comprising 0- and $pi$-Josephson junctions (JJs), called $pi$-SQUID, is studied by the resistively shunted junction model. The $pi$-SQUID shows half-integer Shapiro-steps (SS) under microwave irradiation at the voltage $V$ = $(hbar/2e)Omega (n/2)$, with angular frequency $Omega$ and half-integer $n$/2 in addition to integer $n$. We show that the $pi$-SQUID can be a $pi$-qubit with spontaneous loop currents by which the half-integer SS are induced. Making the 0- and $pi$-JJs equivalent is a key for the half-integer SS and realizing the $pi$-qubit.