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arXiv:0804.3585

Spin noise at an arbitrary spin temperature

S. K. Lee

Published Apr 22, 2008• 0 citations
arXivHTMLPDF

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 $ρ=e^{βS_{z}}/Z$. 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.

DOI
10.48550/arxiv.0804.3585

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The Callen-Welton formula (fluctuation-dissipation theorem) of voltage and current noise of a resistance are the sum of Nyquist's classical Johnson noise equations and a (quantum) zero-point term with power density spectrum proportional to frequency and independent of temperature. At zero temperature, the classical Nyquist term vanishes however the zero-point term produces non-zero noise voltage and current. We show that the claim of zero-point noise directly contradicts to the Fermi-Dirac distribution, which defines the thermodynamics of electrons according to quantum-statistical physics. As a consequence, the Johnson noise must be zero at zero temperature, which is in accordance with Nyquist's original formula. Further investigation shows that the Callen-Welton derivation has conceptual errors such as neglecting phonon scattering, disregarding the Pauli principle during calculating the transition probabilities and using bosonic (linear oscillator) energies leading to the zero-point noise artifact. Following Kleen's proposal, the possible origin of the heterodyne (Koch - van Harlingen - Clark) experimental results are also discussed in terms of Heffner theory of quantum noise of frequency/phase-sensitive linear amplifiers. Experiments that failed to see the zero-point noise term are also mentioned.
Abstract There is a longstanding debate about the zero-point term in the Johnson noise voltage of a resistor. This term originates from a quantum-theoretical treatment of the fluctuation-dissipation theorem (FDT). Is the zero-point term really there, or is it only an experimental artifact, due to the uncertainty principle, for phase-sensitive amplifiers? Could it be removed by renormalization of theories? We discuss some historical measurement schemes that do not lead to the effect predicted by the FDT, and we analyse new features that emerge when the consequences of the zero-point term are measured via the mean energy and force in a capacitor shunting the resistor. If these measurements verify the existence of a zero-point term in the noise, then two types of perpetual motion machines can be constructed. Further investigation with the same approach shows that, in the quantum limit, the Johnson–Nyquist formula is also invalid under general conditions even though it is valid for a resistor-antenna system. Therefore we conclude that in a satisfactory quantum theory of the Johnson noise, the FDT must, as a minimum, include also the measurement system used to evaluate the observed quantities. Issues concerning the zero-point term may also have implications for phenomena in advanced nanotechnology.