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arXiv:quant-ph/0004106

Fluctuation, Dissipation, and Entanglement: the Classical and Quantum Theory of Thermal Magnetic Noise

John A. Sidles, Joseph L. Garbini, W. Dougherty, S. H. Chao

Published Apr 28, 2000• 2 citations
arXivHTMLPDF

Abstract

A general theory of thermal magnetic fluctuations near conductive materials is developed; such fluctuations are the magnetic analog of Johnson noise. For realistic experiments in quantum computing and magnetic resonance force microscopy, the predicted relaxation can be rapid enough that substantial experimental care should be taken to minimize it. 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.

DOI
10.48550/arxiv.quant-ph/0004106

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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 practical 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 spintronics, 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.

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A general theory of thermal magnetic fluctuations near conductive materials is developed; such fluctuations are the magnetic analog of Johnson noise. For realistic experiments in quantum computing and magnetic resonance force microscopy, the predicted relaxation can be rapid enough that substantial experimental care should be taken to minimize it. 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.
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 practical 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 spintronics, 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.
Johnson noise is a small random voltage that appears between terminals of any resistor interacting with its thermal bath at temperature T. It looks like continuous, but the discreteness of the electrical charge suggests its discrete origin coming from the charge noise due to random translocations of individual electrons between terminals. The capacitance allowing these translocations would quantize the energy entering the resistor in this way, thus acting as the antenna of the resistor to pick up thermal energy in the form of charge unbalances (fluctuations of energy) between its terminals. The subsequent relaxations of these fluctuations by the conductance G=1/R of the resistor (the collective reaction of all its carriers) would give rise to its Johnson noise. This collective reaction to dissipate fluctuations of energy caused by individual electrons, agrees with the Fluctuation-Dissipation framework that Callen and Welton proposed in 1951 for noisy processes.
Abstract : A key step in the construction of quantum computing technologies is the identification and elimination of unwanted sources of noise and its resulting quantum decoherence. In solid state devices, one important source of noise is the fluctuating electric and magnetic fields that come from the device elements themselves. The ultimate source of these fluctuations is the random electric currents that are strongest in the metallic electrodes and leads. This random motion has both thermal and quantum components, though in most devices presently under investigation, the thermal part dominates. In the past, this important physical effect has generally been modeled by picturing the device using an equivalent RC circuit and applying the Nyquist theorem to this circuit. This approach has limitations, since it depends on global quantities, rather than on the detailed geometry and the local electromagnetic properties of the components. Our aim in this research project was to compute the effect from first principles. We considered the underlying physical law (quantum electrodynamics) governing the devices in question. Given the position- and frequency-dependent dielectric function, the electromagnetic noise spectrum can be calculated, and the decohering effect can be deduced. Semiconducting quantum dot and superconducting flux qubits were investigated.
We analytically describe the decay to equilibrium of generic observables of a nonintegrable system after a perturbation in the form of a random matrix. We further obtain an analytic form for the time-averaged fluctuations of an observable in terms of the rate of decay to equilibrium. Our result shows the emergence of a fluctuation-dissipation theorem corresponding to a classical Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions can be tested in quantum simulation experiments, thus helping to bridge the gap between theoretical and experimental research in quantum thermalization. We test our analytic results by exact numerical experiments in a spin chain. We argue that our fluctuation-dissipation relation can be used to measure the density of states involved in the nonequilibrium dynamics of an isolated quantum system.
All metal objects support fluctuating currents that are responsible for evanescent-wave Johnson noise in their vicinity due to both thermal and quantum effects. The noise fields can decohere qubits. It is quantified by the average value of $\mathbf{B}(\mathbit{x},t)\phantom{\rule{0.28em}{0ex}}\mathbf{B}({\mathbit{x}}^{\ensuremath{'}},{t}^{\ensuremath{'}})$ and its time Fourier transform. We develop the formalism particularly for objects whose dimensions are small compared with the skin depth, which is the appropriate regime for nanoscale devices. This leads to a general and surprisingly simple formula for the two-point noise correlation function of an object of arbitrary shape. This formula has a clear physical interpretation in terms of induced currents in the object, and it can be the basis for straightforward numerical evaluation. We discuss its experimental implications. For a sphere, a solution is given in closed form in terms of a generalized multipole expansion. Plots of the solution illustrate the physical principles involved. We give examples of how the spatial pattern of noise can affect quantum information processing in nearby qubits. The theory implies that if the qubit system is miniaturized to a scale DDD, then decoherence rates of qubits scale as 1/D1/D1/D.
Abstract General concepts in the theory of fluctuations and dissipation are reviewed and applied to examples in quantum optics. Brownian motion, Fokker-Planck and Langevin equations, and the Wiener-Khintchine theorem are reviewed, followed by a derivation and discussion of the fluctuation-dissipation theorem. The general problem of an oscillator coupled to a heat bath is revisited, as is the nonrelativistic theory of radiation reaction. The general ideas about fluctuations and dissipation developed in the first part of the chapter are then applied to the theory of the fundamental laser linewidth, the photon statistics of linear amplifiers and attenuators, the noise figure, amplified spontaneous emission, and the quantum theory of the beam slitter and homodyne detection.