Dynamics of a Single Spin-1/2 Coupled to x- and y-Spin Baths: Algorithm and Results

The real-time dynamics of a single spin-1/2 particle, called the central spin, coupled to the x(y)-components of the spins of one or more baths is simulated. The bath Hamiltonians contain interactions of x(y)-components of the bath spins only but are general otherwise. An efficient algorithm is described which allows solving the time-dependent Schr’odinger equation for the central spin, even if the x(y) baths contain hundreds of spins. The algorithm requires storage for 2 × 2 matrices only, no matter how many spins are in the baths. We calculate the expectation value of the central spin, as well as its von Neumann entropy S(t), the quantum purity P(t), and the off-diagonal elements of the quantum density matrix. In the case of coupling the central spin to both x- and y- baths the relaxation of S(t) and P(t) with time is a power law, compared to an exponential if the central spin is only coupled to an x-bath. The effect of different initial states for the central spin and bath is studied. Comparison with more general spin baths is also presented.