Relaxation of flying spin qubits in quantum wires by hyperfine interaction

We consider the relaxation of a spin qubit in a quantum dot propagating as a whole in a one-dimensional semiconductor with hyperfine coupling. We show that this motion leads to qualitatively new features in this process compared to static quantum dots. For a fast straightforward motion, the initial spin density decreases to zero with the relaxation rate independent of the spatial spread of the electron wave function and inversely proportional to the electron speed. However, for the oscillatory motion, the qubit acquires memory, and the dephasing becomes Gaussian rather than exponential. After some time, one third of the initial spin polarization is restored, as it happens for static qubits. This revival can occur either through periodic peaks or through a monotonous increase in the polarization, after a minimum, until a plateau has been reached. Our results can be useful for the understanding of the spin dynamics and decoherence in quantum wires.