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# Fermi's golden rule

About: Fermi's golden rule is a research topic. Over the lifetime, 674 publications have been published within this topic receiving 13615 citations.

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TL;DR: The eigenstate thermalization hypothesis (ETH) as discussed by the authors is a natural extension of quantum chaos and random matrix theory (RMT) that allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath.

Abstract: This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the i...

1,536 citations

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TL;DR: In this paper, the frequency of the A1g(Γ) phonon mode monotonically decreases with the number of layers and the excitation wavelength in the visible range (488, 514 and 647 nm).

Abstract: The Raman scattering of single- and few-layered WS2 is studied as a function of the number of S-W-S layers and the excitation wavelength in the visible range (488, 514 and 647 nm). For the three excitation wavelengths used in this study, the frequency of the A1g(Γ) phonon mode monotonically decreases with the number of layers. For single-layer WS2, the 514.5 nm laser excitation generates a second-order Raman resonance involving the longitudinal acoustic mode (LA(M)). This resonance results from a coupling between the electronic band structure and lattice vibrations. First-principles calculations were used to determine the electronic and phonon band structures of single-layer and bulk WS2. The reduced intensity of the 2LA mode was then computed, as a function of the laser wavelength, from the fourth-order Fermi golden rule. Our observations establish an unambiguous and nondestructive Raman fingerprint for identifying single- and few-layered WS2 films.

1,213 citations

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TL;DR: In this article, the harmonic and anharmonic force constants extracted from density functional calculations within a supercell were used to compute thermodynamic and thermal properties of any crystal, including the bulk thermal conductivity.

Abstract: Using harmonic and anharmonic force constants extracted from density functional calculations within a supercell, we have developed a relatively simple but general method to compute thermodynamic and thermal properties of any crystal. First, from the harmonic, cubic, and quartic force constants, we construct a force field for molecular dynamics. It is exact in the limit of small atomic displacements and thus does not suffer from inaccuracies inherent in semiempirical potentials such as Stillinger-Weber's. By using the Green-Kubo formula and molecular dynamics simulations, we extract the bulk thermal conductivity. This method is accurate at high temperatures where three-phonon processes need to be included to higher orders, but may suffer from size scaling issues. Next, we use perturbation theory (Fermi golden rule) to extract the phonon lifetimes and compute the thermal conductivity $\ensuremath{\kappa}$ from the relaxation-time approximation. This method is valid at most temperatures, but will overestimate $\ensuremath{\kappa}$ at very high temperatures, where higher-order processes neglected in our calculations also contribute. As a test, these methods are applied to bulk crystalline silicon, and the results are compared and differences are discussed in more detail. The presented methodology paves the way for a systematic approach to model heat transport in solids using multiscale modeling, in which the relaxation time due to anharmonic three-phonon processes is calculated quantitatively, in addition to the usual harmonic properties such as phonon frequencies and group velocities. It also allows the construction of an accurate bulk interatomic potentials database.

678 citations

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TL;DR: In this paper, the authors consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations and show that all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate.

Abstract: We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field.

369 citations

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TL;DR: In this article, a quantum theory of photoinduced electron transfer, based on the Redfield theory of relaxation, is developed and applied to the standard two state-one mode system interacting with a thermal bath.

Abstract: A quantum mechanical theory of photoinduced electron transfer, based on the Redfield theory of relaxation, is developed and applied to the standard two state–one mode system interacting with a thermal bath. Quantum mechanical treatment of the reaction coordinate allows incorporation of both finite vibrational dephasing and energy flow rates into the description of electron transfer dynamics. The field–matter interaction is treated explicitly to properly incorporate the total energy and magnitude of the vibrational coherence present in the initially prepared state. Calculation of the reduced density matrix of the system is carried out in a vibronic basis that diagonalizes the electron exchange coupling so that the method is valid for arbitrarily large coupling strength. For weak electronic coupling, we demonstrate the equivalence between the results from Redfield theory and those obtained from the standard perturbative expression (golden rule) for nonadiabatic electron transfer. We then discuss quantitatively the breakdown of the Fermi golden rule with increasing electronic coupling strength. The failure of the golden rule is seen to result from either slow energy equilibration in the reactant or product well or from quantum interference effects resulting from finite dephasing rates. For cases where the reorganization energy is large compared to the frequency of reactive motion, such that we may ignore nuclear tunneling, results from the theory show good agreement with those from the semiclassical Landau–Zener theory when motion of the reaction coordinate through the surface crossing region can be considered to be ballistic. Finally results are shown in the weak damping (coherent) limit that demonstrate interference effects between phase coherences involving states in both wells.

289 citations