We solve for the dynamics of a pair of spins, coupled to each other and also to an environmental sea of oscillators. The environment mediates an indirect interaction between the spins, causing both mutual coherence effects and dissipation. This model describes a wide variety of physical systems, ranging from 2 coupled microscopic systems (e.g. magnetic impurities, chromophores, etc.), to 2 coupled macroscopic quantum systems. We obtain analytic results for 3 regimes, viz., (i) The "locked" regime, where the 2 spins lock together; (ii) The "correlated relaxation" regime (mutually correlated incoherent relaxation); and (iii) The "mutual coherence" regime, with correlated damped oscillations. These results cover most of the parameter space of the system.
We solve for the dynamics of a pair of spins, coupled to each other and also to an environmental sea of oscillators. The environment mediates an indirect interaction between the spins, causing both mutual coherence effects and dissipation. This model describes a wide variety of physical systems, ranging from 2 coupled microscopic systems (e.g. magnetic impurities, chromophores, etc.), to 2 coupled macroscopic quantum systems. We obtain analytic results for 3 regimes, viz., (i) The "locked" regime, where the 2 spins lock together; (ii) The "correlated relaxation" regime (mutually correlated incoherent relaxation); and (iii) The "mutual coherence" regime, with correlated damped oscillations. These results cover most of the parameter space of the system.
We consider the quantum dynamics of two spin-1/2 systems, each coupled to a bath of oscillators, so that a bath-mediated coupling is generated between the spins. We find that the interactions destroys any coherent motion of the 2 spins, even if the coupling of each spin to the bath is quite weak, unless the interaction is extremely small. This is because the dynamic mutual bias between the spins blocks any coherent transitions between nearly degenerate states. In many quantum measurement operations this means that decoherence effects will be much stonger during the actual measurement
We consider here the problem of a "central spin", with spin quantum number S≫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: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., TbFe3 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.
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.
This paper presents the results of a functional-integral approach to the dynamics of a two-state system coupled to a dissipative environment. It is primarily an extended account of results obtained over the last four years by the authors; while they try to provide some background for orientation, it is emphatically not intended as a comprehensive review of the literature on the subject. Its contents include (1) an exact and general prescription for the reduction, under appropriate circumstances, of the problem of a system tunneling between two wells in the presence of a dissipative environment to the "spin-boson" problem; (2) the derivation of an exact formula for the dynamics of the latter problem; (3) the demonstration that there exists a simple approximation to this exact formula which is controlled, in the sense that we can put explicit bounds on the errors incurred in it, and that for almost all regions of the parameter space these errors are either very small in the limit of interest to us (the "slow-tunneling" limit) or can themselves be evaluated with satisfactory accuracy; (4) use of these results to obtain quantitative expressions for the dynamics of the system as a function of the spectral density $J(\ensuremath{\omega})$ of its coupling to the environment. If $J(\ensuremath{\omega})$ behaves as ${\ensuremath{\omega}}^{s}$ for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For s<1 the system is localized at zero temperature, and at finite T relaxes incoherently at a rate proportional to $\mathrm{exp}\ensuremath{-}{(\frac{{T}_{0}}{T})}^{1\ensuremath{-}s}$. For s>2 it undergoes underdamped coherent oscillations for all relevant temperatures, while for 1<s<2 there is a crossover from coherent oscillation to overdamped relaxation as T increases. Exact expressions for the oscillation and/or relaxation rates are presented in all these cases. For the "ohmic" case, s=1, the qualitative nature of the behavior depends critically on the dimensionless coupling strength $\ensuremath{\alpha}$ as well as the temperature T: over most of the ($\ensuremath{\alpha}$,T) plane (including the whole region $\ensuremath{\alpha}>1$) the behavior is an incoherent relaxation at a rate proportional to ${T}^{2\ensuremath{\alpha}\ensuremath{-}1}$, but for low T and $0<\ensuremath{\alpha}<\frac{1}{2}$ the authors predict a combination of damped coherent oscillation and incoherent background which appears to disagree with the results of all previous approximations. The case of finite bias is also discussed.
In this thesis, a scheme for calculating the dynamics of two coupled dissipative spins is developed, where each of the spins is coupled to its own boson bath. We derive analytic path sum results both in the Markov-regime and in the one-boson exchange regime. The analysis is also performed with the Bloch-Redfield method. It is shown that the two different approaches lead to identical results for the dynamics. This is not obvious a priori because both methods are based on different procedures. The path sum method gives detailed insight into the internal dynamics of two coupled spins because we consider every path sequence that contributes to the time evolution. While most studies of coupled spins are restricted to one type of coupling, e.g., Ising type, we will generalize here to linear combinations of possible couplings. Especially interesting is the occurrence of a frustration of decoherence, if the spins are interacting via a linear combination of longitudinal and transverse coupling and, when in addition, some of the eigenfrequencies become degenerate. Our analysis shows that degenerate but mutual exclusive ground states lead to increased coherence times. Maximization of coherence is one of the crucial goals of quantum state engineering. Another topic of interest is the impact of non-linear quantum environments, formed by surrounding dissipative spins. A distribution of bistable background charges is known to be responsible for 1/f noise in solid state devices, like the superconducting quantum interference devices. Since 1/f noise is seemingly the dominating source of decoherence at very low temperature, there is a profound theoretical interest in modeling it. Therefore, we analyze two interacting spins, where one of them is coupled to a boson bath and thereby represents a resonant non-linear quantum environment. We study the crossover from a non-linear to a linear bath and study the corresponding time scale for the relevant bath correlations. Interestingly, for large and increasing temperature we find a decreasing decoherence for the central spin.
Applications of quantum devices and simulators require an accurate understanding of their dynamics as well as the effects of environmental noise. While some sources of noise can be modeled as Markovian dissipation, coupling to resonators or vibrational modes often produces non-Markovian dynamics as well, making it difficult to characterize the system, particularly for interacting many-body systems. Here, we identify a broad class of exactly solvable many-body quantum systems with both non-Markovian dynamics and Markovian dissipation. We consider a prototypical model of spins coupled to lossy bosonic modes that mediate Ising-type interactions. We exclude a transverse field but allow local spin dissipation that need not commute with the interactions, including dephasing, relaxation, pumping, and even processes that generate coherences. Using an exact stochastic representation, we decouple the many-body dynamics into the evolution of individual spins, yielding exact spin observables and correlations without adiabatically eliminating the bosons. In the Markovian limit, our solutions describe Ising models with correlated dephasing and nonreciprocal interactions. An example of spin squeezing shows that both the achievable squeezing and its optimal preparation time depend strongly on the interplay between non-Markovian dynamics and Markovian dissipation.