Spin noise at an arbitrary spin temperature

TL;DR

This paper extends the understanding of spin noise to arbitrary spin temperatures by incorporating zero-point fluctuations, showing that quantum uncertainty limits can be viewed as thermal bath zero-point fluctuations.

Contribution

It introduces a generalized relation for spin noise at any temperature, linking quantum uncertainty with thermal fluctuations through the fluctuation-dissipation theorem.

Findings

Zero-point fluctuations extend the spin noise-temperature relation to all spin temperatures.

The uncertainty principle-limited spin projection noise is linked to zero-point fluctuations in the thermal bath.

The generalized relation applies to any spin S, not just spin 1/2 in high temperature limit.

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 $S_{x}$ can be calculated by ensemble average of the expectation value of $S_{x}^{2}$ with respect to an equilibrium density matrix $\rho =e^{\beta S_{z}}/Z$. 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.