Nuclear Magnetic Resonance (NMR) experiments involve coherent and incoherent dynamics. Incoherent dynamics give rise to nuclear spin relaxation typically accounted for by a differential equation known as a quantum master equation. Out of the existing master equations the Lindblad equation is particularly important. A central theorem of open quantum theory states that a relaxation process represents a quantum Markovian process if and only if it can be described by a Lindblad master equation. This in turn raises serious questions regarding the compatibility of conventional NMR relaxation theory and one of the most fundamental results of open quantum theory. The NMR treatment of important Markovian relaxation mechanisms such as Brownian motion or chemical exchange events do in general not result in a master equation in Lindblad form and would therefore be at odds with the mathematically rigorous theory of open quantum systems. We explore this apparent conflict for two important chemical exchange phenomena and show that their conventional treatment may be recast into a suitable Lindblad equation. Similarities between spatial diffusion and multi-site exchange events allow us to extend our results to nuclear spin relaxation due to translational diffusion. As a result we are able to show that the NMR treatment of intramolecular exchange and translational diffusion represent valid quantum Markovian processes compatible with the rigorous definitions of open quantum theory.
We present a general master equation, correct to second order in the system-environment coupling strength, that takes into account the initial system-environment correlations. We assume that the system and its environment are in a joint thermal equilibrium state, and thereafter, a unitary operation is performed to prepare the desired initial system state, with the system Hamiltonian possibly changing thereafter as well. We show that the effect of the initial correlations shows up in the second-order master equation as an additional term, similar in form to the usual second-order term describing relaxation and decoherence in quantum systems. We apply this master equation to a generalization of the paradigmatic spin-boson model, namely, a collection of two-level systems interacting with a common environment of harmonic oscillators, as well as a collection of two-level systems interacting with a common spin environment. We demonstrate that, in general, the initial system-environment correlations need to be accounted for in order to accurately obtain the system dynamics.
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of
N sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the
N=2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.
The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4 n × 4 n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states (NESS) and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbour Heisenberg XY spin-1/2 chain in a transverse magnetic field.
A spin system with unperturbed Hamiltonian ${\mathcal{H}}_{0}=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}\ensuremath{-}\ensuremath{\gamma}H\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{x}}^{j}$ relaxing via the spin-lattice coupling $G=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{C}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}({{\ensuremath{\sigma}}_{y}}^{k}+{{\ensuremath{\sigma}}_{z}}^{k})$ is studied by means of the general density-matrix theory of magnetic relaxation. By making some assumptions about the magnitude and time constants of the lattice correlation functions
〈Cij(t)Ckl(0)〉, a master equation is obtained. It agrees at high temperatures with a master equation previously suggested by Glauber for the one-dimensional nearest-neighbor case. At high temperatures the magnetic moment relaxes with a single relaxation time, and the spin pair-correlation functions satisfy a closed set of equations. At low temperatures, however, the equations for the magnetization and the correlation functions are coupled to higher-order moments.
The present paper presents a derivation of the "master" or Boltzmann "gain-loss" equation from the Schr\"odinger equation, i.e., a derivation of the equation for the evolution in time of the probabilities of finding a physical system in its various states from the equation for the corresponding probability amplitudes. The "master" equation is derived for an, in effect completely self-enclosed, "supersystem," [
A+B], consisting of a "system of interest," [
A], and a "surroundings," [
B], in relatively weak mutual interaction. A discussion is given of the range of validity of the "master" equation for [
A+B] and it is shown that the random phase assumption is required for the state vector of [
A+B] at the initial time only. The normally microcanonical character of the equilibrium statistical configuration of [
A+B] is demonstrated and a treatment is given of exceptional, "extremely quantal-coherent," initial statistical distributions of [
A+B] which may evolve away from equilibrium. Derivations are also presented of the "master" equation for [
A] and of the "master" equation for an individual particle or quasi-particle [
q], within [
A]; a discussion of the range of validity of these "master" equations is given and the normally canonical character of the equilibrium statistical configuration of [
A] is deduced. General solutions of the "master" equations for [
A+B], [
A], and [
q] are worked out and the relation between the principles of "microscopic reversibility" and "detailed balance" and the nonoscillatory character of the approach to equilibrium are exhibited. A theorem is presented regarding the time variation of the entropy of [
A].As illustrations of the general methods developed two important processes in magnetic resonance---the time variation of the longitudinal magnetization, ${〈\ensuremath{\mu}〉}_{t}$, and the time variation of the transverse magnetization, ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$---are discussed in some detail. It is shown that the variation of ${〈\ensuremath{\mu}〉}_{t}$ with
t and of ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$ with
t for a "nonrigid" lattice can be described by means of the "master" equation for an individual spin [
q] and several special cases are discussed on the basis of the evaluation of the appropriate transition probabilities; a comparison with the "spin-temperature" procedure is also appended. On the other hand, it is demonstrated that for a "rigid" lattice no description of the variation of ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$ with
t can be given on the basis of a "master" equation; in this case, quantal coherence effects neglected in the derivation of the "master" equation from the Schr\"odinger equation are vital and ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$ must be evaluated by a rigorous calculation of Trace {[appropriate time dependent density matrix] ${\ensuremath{\mu}}^{\ensuremath{'}}$}.