Fast partial decoherence of a superconducting flux qubit in a spin bath

The superconducting flux qubit has two quantum states with opposite magnetic flux. Environment of nuclear spins can find out the direction of the magnetic flux after a decoherence time $\tau_0$ inversely proportional to the magnitude of the flux and the square root of the number of spins. When the Hamiltonian of the qubit drives fast coherent Rabi oscillations between the states with opposite flux, then flux direction is flipped at a constant rate $\omega$ and the decoherence time $\tau=\omega\tau_0^2$ is much longer than $\tau_0$. However, on closer inspection decoherence actually takes place on two timescales. The long time $\tau$ is a time of full decoherence but a part of quantum coherence is lost already after the short time $\tau_0$. This fast partial decoherence biases coherent flux oscillations towards the initial flux direction and it can affect performance of the superconducting devices as qubits.