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Origins of Spin Noise

机译：自旋噪声的起源

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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.
机译：像任何多粒子系统一样，自旋系统也会显示波动。波动是讨论自旋弛豫的关键，但是当自旋系统处于磁场中时，这些波动也可以作为自旋噪声来测量。不需要脉冲；我们只需要收集一个长的连续数据集。没有相位相干性，因此幅度数据的积累会适得其反，但是如果将数据集分成多个块，则在标准光谱仪上相对容易看到信号。块的长度应大致为自旋-自旋弛豫时间T 2 的长度。这些块分别进行傅立叶变换，并计算其功率谱，以消除任何相位影响。然后可以累积这些功率谱以提供可测量的信号。例如，通常使用密度矩阵来计算自旋系统的动力学，例如遵循Abragam的“核磁性原理”（A. Abragam的《核磁性原理》，克拉伦登出版社，牛津，1961年）。但是，就密度矩阵而言，通常的公式预测信号将呈指数衰减至零，并且无法解决自旋噪声。在本文中，我们借鉴了随机介质电磁散射理论中的相关数学思想，并将现代随机微积分方法与详细的量子力学描述相结合，以包括自旋噪声。工作的核心涉及自旋的纯态的定义，以及在制定密度矩阵时如何组合这些纯态。我们保留原始的（而不是集合平均的）密度矩阵，该矩阵是根据对应于系统组成纯状态的投影算子的概率加权总和构造的。我们成功地描述了自旋噪声，并进一步讨论了它如何照亮自旋弛豫的某些过程。 Abragam使用的近似值是合理的，并且我们展示了标准密度矩阵如何适合这种更一般的公式。

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