Spin noise at an arbitrary spin temperature
Abstract
An ensemble of spins oriented along the z direction exhibits nonzero fluctuation in the transverse (x- and y-) components of the spin angular momentum in accordance with the uncertainty principle. When the spins obey a spin temperature distribution, the mean square fluctuation in Sx can be calculated by ensemble average of the expectation value of Sx2 with respect to an equilibrium density matrix =eβ Sz/Z. The fluctuation can also be calculated from the fluctuation-dissipation theorem as has been done in literature in the context of NMR spin noise. For spin 1/2 particles in the high temperature limit, appropriate for many NMR experiments, the two methods are known to produce the same, temperature-independent spin noise. We show that inclusion of the zero-point fluctuation term in the original Nyquist relation extends this correspondence to an arbitrary spin temperature for any spin S. This indicates that the uncertainty principle-limited spin projection noise can be viewed as a result of the zero-point fluctuation in the thermal bath coupled to the spins.