Stability of Floquet Majorana box qubits

A topological superconductor in one dimension can host Majorana zero modes at its edge. By driving the system periodically, so-called π modes (also named Floquet-Majoranas) can arise. These are topologically protected modes with the quasi-energy π/T, where T is the period of the drive. We consider the role of π modes in the presence of long-range Coulomb interactions and therefore study a Cooper pair box made of two Josephson coupled superconducting topological quantum wires. Time-dependent gate voltages periodically drive the system and can induce π modes. The presence of four Majoranas and four π Majoranas in our setup allows us to define three topological qubits in a fixed fermion parity sector within one single box. We investigate how to obtain and control the π modes and study their stability in the presence of interactions. The stability of the Floquet-Majorana box qubit depends crucially on the initialization of the Floquet state. If the system is prepared by adiabatically increasing the amplitude of the oscillating gate voltage, the topological Floquet phase is always inherently unstable. The instability arises due to resonant quasi-particle creation mediated by interactions. However, a stable Floquet phase can be reached by using a two-step protocol. First, the amplitude of the oscillating gate voltage is adiabatically increased, while the frequency of the oscillation is small. Then, the oscillation frequency is increased slowly. With this frequency-sweep protocol, we can achieve a stable Floquet device despite interactions.