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arXiv:1407.6937 · Physical Review B

Nonequilibrium spin noise and noise of susceptibility

Pablo Schad, B. N. Narozhny, Gerd Schön, Alexander Shnirman

Published Nov 17, 2014• 15 citations
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Abstract

We analyze out-of-equilibrium fluctuations in a driven spin system and relate them to the noise of spin susceptibility. In the spirit of the linear response theory we further relate the noise of susceptibility to a 4-spin correlation function in equilibrium. We show that, in contrast to the second noise (noise of noise), the noise of susceptibility is a direct measure of non-Gaussian fluctuations in the system. We develop a general framework for calculating the noise of susceptibility using the Majorana representation of spin-1/2 operators. We illustrate our approach by a simple example of noninteracting spins coupled to a dissipative (Ohmic) bath.

DOI
10.1103/physrevb.90.205419

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We analyze out-of-equilibrium fluctuations in a driven spin system and relate them to the noise of spin susceptibility. In the spirit of the linear response theory we further relate the noise of susceptibility to a 4-spin correlation function in equilibrium. We show that, in contrast to the second noise (noise of noise), the noise of susceptibility is a direct measure of non-Gaussian fluctuations in the system. We develop a general framework for calculating the noise of susceptibility using the Majorana representation of spin-1/2 operators. We illustrate our approach by a simple example of noninteracting spins coupled to a dissipative (Ohmic) bath.
Per the fluctuation-dissipation theorem, the information obtained from spin fluctuation studies in thermal equilibrium is necessarily constrained by the system's linear response functions. However, by including weak radio frequency magnetic fields, we demonstrate that intrinsic and random spin fluctuations even in strictly unpolarized ensembles can reveal underlying patterns of correlation and coupling beyond linear response, and can be used to study nonequilibrium and even multiphoton coherent spin phenomena. We demonstrate this capability in a classical vapor of (41)K alkali atoms, where spin fluctuations alone directly reveal Rabi splittings, the formation of Mollow triplets and Autler-Townes doublets, ac Zeeman shifts, and even nonlinear multiphoton coherences.
Using the Majorana fermion representation of spin-1/2 local moments, we show how the dynamic spin correlation and susceptibility are obtained directly from the one-particle Majorana propagator. We illustrate our method by applying it to the spin dynamics of a nonequilibrium quantum dot, computing the voltage-dependent spin relaxation rate and showing that, at weak coupling, the fluctuation-dissipation relation for the spin of a quantum dot is voltage dependent. We confirm the voltage-dependent Curie susceptibility recently found by Parcollet and Hooley [Phys. Rev. B 66, 085315 (2002)]].
We present results for the dynamics of an impurity spin coupled to a magnetic field and to two ohmic baths that are out of equilibrium due to the application of a bias voltage. Both the nonequilibrium steady state and the rate constants describing the approach to the steady state are found to depend sensitively on the relative strengths of a magnetic field and a voltage dependent decoherence rate. Computation of physical quantities, including the frequency dependent ratio of the response to the correlation functions and the probabilities of the two spin states, allows the extraction of voltage dependent effective temperatures. The temperatures extracted from different quantities differ from one another in magnitude and in their dependence on parameters, and, in general, are nonmonotonic.
We study the fluctuations of the autocorrelation and autoresponse functions and, in particular, their variances and co-variance. In a first general part of the Article, we show the equivalence of the variance of the response function with the second-order susceptibility of a composite operator, and we derive an equilibrium fluctuation-dissipation theorem beyond-linear order relating it to the other variances. In a second part of the paper we apply the formalism to the study to non-disordered ferromagnets, in equilibrium or in the coarsening kinetics following a critical or sub-critical quench. We show numerically that the variances and the non-linear susceptibility obey scaling with respect to the coherence length ξξξ in equilibrium, and with respect to the growing length L(t)L(t)L(t) after a quench, similarly to what is known for the autocorrelation and the autoresponse functions.
It is shown that the measurable nonlinear susceptibilities of third and higher orders probe only a fraction of equilibrium fluctuations, and in general can give only lower bounds on the equilibrium correlation functions. This contrasts with the linear and quadratic response, which completely determines the corresponding correlations by general fluctuation–dissipation theorems. The exactly solvable one-dimensional kinetic Ising model illustrates which fluctuations give rise to dissipation; only two-magnon processes of short wavelength and opposite momenta contribute to the third-order nonlinear susceptibilities, while two- and three-magnon processes contribute to the corresponding equilibrium correlation functions.
The Kubo fluctuation-dissipation theorem relates the current fluctuations of a system in an equilibrium state with the linear AC-conductance. This theorem holds also out of equilibrium provided that the system is in a stationary state and that the linear conductance is replaced by the (dynamic) conductance with respect to the nonequilibrium state. We provide a simple proof for that statement and then apply it in two cases. We first show that in an excess noise measurement at zero temperature, in which the impedance matching is maintained while driving a mesoscopic sample out of equilibrium, it is the nonsymmetrized noise power spectrum which is measured, even if the bare measurement, i.e. without extracting the excess part of the noise, obtains the symmetrized noise. As a second application we derive a commutation relation for the two components of fermionic or bosonic currents which holds in every stationary state and which is a generalization of the one valid only for bosonic currents. As is usually the case, such a commutation relation can be used e.g. to derive Heisenberg uncertainty relationships among these current components.