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arXiv:cond-mat/9511009

Decoherence in the Spin Bath

Nikolay V. Prokof'ev, Philip C. E. Stamp

Published Nov 2, 1995• 0 citations
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Abstract

We develop a mathematical description of the decoherence caused by "spin baths", such as nuclear spins or magnetic impurities. In contrast to the usual oscillator bath models of quantum environments, decoherence in the spin bath can occur without any dissipation. Given the almost ubiquitous presence of nuclear spins in nature, our results have important consequences for quantum measurement theory, particularly as the decoherence mechanisms in spin baths work very differently from those in oscillator baths.

DOI
10.48550/arxiv.cond-mat/9511009

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We develop a mathematical description of the decoherence caused by "spin baths", such as nuclear spins or magnetic impurities. In contrast to the usual oscillator bath models of quantum environments, decoherence in the spin bath can occur without any dissipation. Given the almost ubiquitous presence of nuclear spins in nature, our results have important consequences for quantum measurement theory, particularly as the decoherence mechanisms in spin baths work very differently from those in oscillator baths.
The quantum dynamics of mesoscopic or macroscopic systems is always complicated by their coupling to many `environmental' modes. At low T these environmental effects are dominated by localized modes, such as nuclear and paramagnetic spins, and defects (which also dominate the entropy and specific heat). This environment, at low energies, maps onto a `spin bath' model. This contrasts with `oscillator bath' models (originated by Feynman and Vernon) which describe delocalized environmental modes such as electrons, phonons, photons, magnons, etc. The couplings to N spin bath modes are independent of N (rather than the ~O(1/( N ) 1/2 ) dependence typical of oscillator baths), and often strong. One cannot in general map a spin bath to an oscillator bath (or vice versa); they constitute distinct `universality classes' of quantum environment. We show how the mapping to spin bath models is made, and then discuss several examples in detail, including moving particles, magnetic solitons, nanomagnets, and SQUIDs, coupled to nuclear and paramagnetic spin environments. We then focus on the `central spin' model, which couples a central two-level system to a background spin bath. It is the spin bath analogue of the famous `spin-boson' oscillator model, and describes, e.g., the tunnelling dynamics of nanoscopic and mesoscopic magnets and superconductors. We show how to average over (or `integrate out') spin bath modes, using an operator instanton technique, to find the central spin dynamics. The formal manouevres involve four separate averages - each average corresponds physically to a different `decoherence' mechanism acting on the central spin dynamics. Each environmental spin has its own topological `spin phase', which by interacting with the phase of the central system, decoheres it - this can happen even without dissipation. We give analytic results for the central spin correlation functions, under various conditions. We then describe the application of this theory to magnetic and superconducting systems. Particular attention is given to recent work on tunnelling magnetic macromolecules, where the role of the nuclear spin bath in controlling the tunnelling is very clear; we also discuss other magnetic systems in the quantum regime, and the influence of nuclear and paramagnetic spins on flux dynamics in SQUIDs. Finally, we discuss decoherence mechanisms and coherence experiments in superconductors and magnets. We show that a spin bath environment causes decoherence even in the T 0 limit. Control of this decoherence will be essential in the effort to construct `qubits' for quantum computers.
We study decoherence of central spins by a spin bath, focusing on the difference between measurement of a single central spin and measurement of a large number of central spins (as found in typical spin-resonance experiments). For a dilute spin bath, the single spin demonstrates Gaussian free-induction decay, in contrast to exponential decay characteristic of spin ensembles. A strong difference between a single spin and a spin ensemble also exists for the Rabi oscillation decay: for a repeated Rabi oscillation experiment, suppression of decoherence happens for a single spin while acceleration takes place for a spin ensemble. The mathematical origin of such behavior is similar to quantum Zeno/anti-Zeno effects.
We consider here the problem of a "central spin", with spin quantum number S≫1S \gg 1S≫1, interacting with a set of microscopic spins. Interactions between the microscopic spins are ignored. This model describes magnetic grains or magnetic macromolecules (ferromagnetically or antiferromagnetically ordered) interacting with nuclear spins and with any surrounding paramagnetic electronic spins. It has also been used to describe Si:PSi:PSi:P near the metal-insulator transition, and quantum spin glasses. In this paper we consider the possible coherent motion of the central spin. We reduce the model to an effective low-energy Hamiltonian, and then calculate the correlation function using instanton methods. The general solution is given for the unbiased case, for all relevant values of the couplings. Under certain conditions, nuclear spin diffusion also plays a role. The results are then used to calculate the spectral absorption for 2 experimental systems, viz., TbFe3TbFe_3TbFe3​ grains, and ferritin molecules. Our results have general implications for the observation of mesoscopic and macroscopic quantum coherence, and for the foundations of quantum mechanics. They show that a spin environment usually has a very destructive effect on coherence, which cannot be understood using the conventional "oscillator bath" models of quantum environments.
Decoherence of electron spins in nanoscale systems is important to quantum technologies such as quantum information processing and magnetometry. It is also an ideal model problem for studying the crossover between quantum and classical phenomena. At low temperatures or in light-element materials where the spin-orbit coupling is weak, the phonon scattering in nanostructures is less important and the fluctuations of nuclear spins become the dominant decoherence mechanism for electron spins. Since the 1950s, semi-classical noise theories have been developed for understanding electron spin decoherence. In spin-based solid-state quantum technologies, the relevant systems are in the nanometer scale and nuclear spin baths are quantum objects which require a quantum description. Recently, quantum pictures have been established to understand the decoherence and quantum many-body theories have been developed to quantitatively describe this phenomenon. Anomalous quantum effects have been predicted and some have been experimentally confirmed. A systematically truncated cluster-correlation expansion theory has been developed to account for the many-body correlations in nanoscale nuclear spin baths that are built up during electron spin decoherence. The theory has successfully predicted and explained a number of experimental results in a wide range of physical systems. In this review, we will cover this recent progress. The limitations of the present quantum many-body theories and possible directions for future development will also be discussed.
The decohering environment of a quantum bit is often described by the coupling to a large bath of spins. The quantum bit itself can be seen as a spin S=12S=\frac{1}{2}S=21​ which is commonly called the central spin. The resulting central spin model describes an important mechanism of decoherence. We provide mathematically rigorous bounds for a persisting magnetization of the central spin in this model with and without magnetic field. In particular, we show that there is a well-defined limit of infinite number of bath spins. Only if the fraction of very weakly coupled bath spins tends to 100% does no magnetization persist.