Stochastic Models of Population and Phase Relaxation

Abstract

The time constants T 1 and T 2 were introduced many years ago to describe relaxation in nuclear magnetic resonance.1,2 To be explicit, let us consider a collection of spin 1/2 particles in a static magnetic field in the z: direction. Each of the spins can of course be found in either of two quantum states, up or down. In thermal equilibrium, this produces a nonzero z-component of the magnetization. If the system is prepared in a nonequilibrium state, the longitudinal (z) and transverse (x or y) magnetizations relax in time to the appropriate equilibrium values. In the simple phenomenological model of Bloch, these longitudinal and transverse components decay exponentially with time constants T 1 and T 2 respectively. Focusing instead on the 2 × 2 density matrix for the two spin states, T 1 and T 2 also describe the exponential decay to equilibrium of the diagonal and off-diagonal elements respectively. For that reason T 1 and T 2 are called the population and phase relaxation times, respectively.