Quantum brachistochrone problem for spin-1/2 mixed states with decoherence
We study passage time needed for the evolution of the initial mixed spin-1/2 state into final one in magnetic field. The time of evolution depends only on the angle between the Bloch vectors of initial and final states and the angle of precession. Minimal time of evolution that corresponds to the quantum brachistochrone is achieved when magnetic field is perpendicular to the Bloch vector. We also take into account decoherence, which is caused by the fluctuations of the absolute value of magnetic field with fixed direction. Considering the quantum brachistochrone problem we characterize initial and final states only by the direction of the Bloch vector. It is interesting that when magnetic field is perpendicular to the initial Bloch vector the decoherence does not effect on the passage time. It is shown that in linear approximation over the parameter of decoherence the passage time is lager than without decoherence. In this approximation the minimal time is equal to the minimal time of evolution without decoherence and is achieved when the magnetic field is perpendicular to the initial Bloch vector.