The classical and quantum theory of thermal magnetic noise, with applications in spintronics and quantum microscopy

Abstract

Thermal fluctuations generate magnetic noise in the vicinity of any conductive and/or magnetically permeable solid. This magnetic noise plays a fundamental role in the design of spintronic devices: namely, it sets the time scale during which electron spins retain their coherence. This paper presents a rigorous classical and quantum analysis of thermal magnetic noise, together with prac- tical engineering examples. Starting with the fluctuation-dissipation theorem and Maxwell's equations, a closed-form expression for the spectral density of thermal magnetic noise is derived. Quantum decoherence, as induced by thermal magnetic noise, is analyzed via the independent oscillator heat bath model of Ford et al. The resulting quantum Langevin equations yield closed-form expressions for the spin relaxation times. For realistic experiments in spin-tronics, magnetic resonance force microscopy, Bose-Einstein condensates, atomic physics, and solid-state quantum computing, the predicted relaxation rates are rapid enough that substantial experimental care must be taken to minimize them. At zero temperature, the quantum entanglement between a spin state and a thermal reservoir is computed. The same Hamiltonian matrix elements that govern fluctuation and dissipation are shown to also govern entanglement and renormalization, and a specific example of a fluctuation-dissipation-entanglement theorem is constructed. We postulate that this theorem is independent of the detailed structure of thermal reservoirs, and therefore expresses a general thermodynamic principle.

Magnetostatic field noise near metallic surfaces

“…Results along this line have been computed and experimentally verified for planar metallic layers by Varpula and Poutanen in 1984 [4]. Sidles and co-workers give an extensive discussion with applications for magnetic resonance microscopy and quantum computing [6].…”

“…Accurate calculations of magnetic near field noise are clearly needed for this purpose. Magnetic fluctuations are also relevant in other contexts, for example in biophysics where they impose ultimate limits on the sensitivity of SQUID detectors [4], and in magnetic resonance force microscopy, a near-field variant of magnetic resonance imaging [5,6].…”

Entanglement between a qubit and the environment in the spin-boson model

“…The entropy vanishes as α → 0 and approaches its maximum value as α → 1 − (see also Ref. [19] for weak dissipation results for E). For α > 1, we are in the ferromagnetic sector of the AKM where σ z = 1, σ x = 0, and the reduced density matrix eigenvalues p ± = 0, 1 giving E = 0, i.e., E(α) drops discontinuously at the quantum critical point α = 1 [20].…”

Magnetic noise around metallic microstructures

“…These fields are relevant for many applications like high-precision measurements of biomagnetic signals 1 , nuclear magnetic resonance microscopy 2 , and miniaturized traps for ultra cold atoms 3,4 . The original purpose of Purcell's influential 1946 paper 5 was to point out that these magnetic fields have a spectral density that by far exceeds the Planck law for blackbody radiation at low frequencies.…”