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An ensemble of spins oriented along the $z$ direction exhibits nonzerofluctuation in the transverse ($x$- and $y$-) components of the spin angularmomentum in accordance with the uncertainty principle. When the spins obey aspin temperature distribution, the mean square fluctuation in $S_{x}$ can becalculated by ensemble average of the expectation value of $S_{x}^{2}$ withrespect to an equilibrium density matrix $\rho =e^{\beta S_{z}}/Z$. Thefluctuation can also be calculated from the fluctuation-dissipation theorem ashas been done in literature in the context of NMR spin noise. For spin 1/2particles in the high temperature limit, appropriate for many NMR experiments,the two methods are known to produce the same, temperature-independent spinnoise. We show that inclusion of the zero-point fluctuation term in theoriginal Nyquist relation extends this correspondence to an arbitrary spintemperature for any spin $S$. This indicates that the uncertaintyprinciple-limited spin projection noise can be viewed as a result of thezero-point fluctuation in the thermal bath coupled to the spins.

Spin noise at an arbitrary spin temperature