Origins of Spin Noise

Abstract

Spin systems, like any multiparticle systems, show fluctuations. Fluctuations are central to the discussion of spin relaxation, but when the spin system is in a magnetic field, these fluctuations are also measurable as spin noise. No pulse is required; we just need to collect a long continuous dataset. There is no phase coherence, so accumulation of amplitude data is counter-productive, but if the dataset is broken into blocks, the signal is relatively easy to see on standard spectrometers. The blocks should be roughly the length of T 2, the spin–spin relaxation time. These blocks are each Fourier transformed, and their power spectrum is calculated, to remove any phase effects. These power spectra can then be accumulated to provide a measurable signal. The dynamics of a spin system is usually calculated using the density matrix, following Abragam’s “The Principles of Nuclear Magnetism” (A. Abragam, The Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1961), for instance. However, the usual formulation in terms of the density matrix predicts that the signal will decay exponentially to zero, and does not address the spin noise. In this article, we draw on related mathematical ideas in the theory of electromagnetic scattering from random media and apply the modern methods of stochastic calculus in combination with a detailed quantum mechanical description to include spin noise. The core of the work concerns the definition of a pure state for a spin, and how these pure states are combined when the density matrix is formulated. We retain the raw, as opposed to ensemble averaged, density matrix which is constructed in terms of the probability-weighted sum of projection operators corresponding to the constituent pure states of the system. We successfully describe spin noise and further discuss how this illuminates some of the processes of spin relaxation. The approximations that Abragam uses are justified, and we show how the standard density matrix fits within this more general formulation.