Von Neumann spin measurements with Rashba fields

Abstract

We show that dynamics in spin-orbit coupling field simulates the von Neumann measurement of a particle spin. We demonstrate how the measurement influences the spin and coordinate evolution of a particle by comparing two examples of such a procedure. First example is a simultaneous measurement of spin components, $\sigma _{x}$ and $\sigma _{y}$, corresponding to non-commuting operators, which cannot be accurately obtained together at a given time instant due to the Heisenberg uncertainty ratio. By mapping spin dynamics onto a spatial walk such a procedure determines measurement-time averages of $\sigma _{x}$ and $\sigma _{y}$, which already can be precisely evaluated in a single short-time measurement. The other, qualitatively different, example is the spin of a one-dimensional particle in a magnetic field. Here the outcome depends on the angle between the spin-orbit coupling and magnetic fields. These results can be applied to studies of spin-orbit coupled cold atoms and electrons in solids.