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arXiv:1603.01320 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Stochastic Boltzmann equation for magnetic relaxation in high-spin molecules

Daniel M. Packwood, Helmut G. Katzgraber, Winfried Teizer

Published Mar 1, 2016• 1 citation
arXivHTMLPDF

Abstract

We introduce the stochastic Boltzmann equation (SBE) as an approach for exploring the spin dynamics of magnetic molecules coupled to a stochastic environment. The SBE is a time-evolution equation for the probability density of the spin density matrix of the system. This probability density is relevant to experiments which take measurements on single molecules, in which probabilities of observing particular spin states (rather than ensemble averages) are of interest. By analogy with standard treatments of the regular Boltzmann equation, we propose a relaxation-time approximation for the SBE and show that solutions to the SBE under the relaxation-time approximation can be obtained by performing simple trajectory simulations for the case of a boson gas environment. Cases where the relaxation-time approximation are satisfied can therefore be investigated by careful choice of the parameters for the boson gas environment, even if the actual environment is quite different from a boson gas. The application of the SBE approach is demonstrated through an illustrative example.

DOI
10.1098/rspa.2015.0699

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We introduce the stochastic Boltzmann equation (SBE) as an approach for exploring the spin dynamics of magnetic molecules coupled to a stochastic environment. The SBE is a time-evolution equation for the probability density of the spin density matrix of the system. This probability density is relevant to experiments which take measurements on single molecules, in which probabilities of observing particular spin states (rather than ensemble averages) are of interest. By analogy with standard treatments of the regular Boltzmann equation, we propose a relaxation-time approximation for the SBE and show that solutions to the SBE under the relaxation-time approximation can be obtained by performing simple trajectory simulations for the case of a boson gas environment. Cases where the relaxation-time approximation are satisfied can therefore be investigated by careful choice of the parameters for the boson gas environment, even if the actual environment is quite different from a boson gas. The application of the SBE approach is demonstrated through an illustrative example.
An alternative to the standard Metropolis-based approach for evaluating quantum statistical mechanical ensemble averages is introduced. The method is based on the close similarity between the Bloch equation and the Smoluchowski diffusion equation. Stochastic ‘‘Bloch trajectories’’ are constructed from local solutions to the Bloch equation for the density matrix and used to evaluate Boltzmann ensemble averages and partition functions. This is analogous to the use of Brownian dynamics trajectories in diffusion calculations. Sample results for selected one-dimensional systems are presented and the extension to multidimensional problems is discussed.
Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.
Extracting macroscopic properties of a system from microscopic interactions has always been an interesting topic with the most diverse applications. Here, we use the quantum Boltzmann equation to investigate the density matrix evolution of a system of nucleons. Using the quantum field theory tools for constructing the density matrix operators and calculating the interactions is the main advantage of this equation. The right-hand side of this equation involves forward scattering and usual collision terms. As examples of application, we calculate the standard Bloch equations for the nucleon system in the presence of a constant and an oscillating magnetic field from the forward scattering term. We find the longitudinal and transverse relaxation times from the collision term by considering the nucleon-nucleon scattering.
Extracting macroscopic properties of a system from microscopic interactions has always been an interesting topic with the most diverse applications. Here, we use the quantum Boltzmann equation to investigate the density matrix evolution of a system of nucleons. Using the quantum field theory tools for constructing the density matrix operators and calculating the interactions is the main advantage of this equation. The right-hand side of this equation involves forward scattering and usual collision terms. As examples of application, we calculate the standard Bloch equations for the nucleon system in the presence of a constant and an oscillating magnetic field from the forward scattering term. We find the longitudinal and transverse (decoherence) relaxation times from the collision term by considering the nucleon-nucleon scattering.
Spin-boson models involving many interacting spins and bosons are ubiquitous in quantum simulation platforms. At the same time, characterizing the dynamics of these quantum systems represents a significant challenge. Here, we consider general spin-boson models where bosons are subject to Markovian dissipation (e.g., due to cavity loss). We present an exact hybrid quantum-classical stochastic approach where the solution of a classical stochastic equation—mimicking the bosonic modes—is input into a quantum stochastic equation for the spins. Furthermore, the spins are effectively decoupled for each stochastic realization, which nevertheless comes at the expense of sampling over unphysical states. In contrast with existing stochastic approaches based on the influence functional formalism, we place no restriction (factorizability or Gaussianity) on the initial state, or the spin-boson coupling (except that it be linear in the bosonic operator). Markovian dissipation, being at the heart of our approach, renders the stochastic equations Markovian even in the strong coupling regime. Furthermore, it ensures hermiticity (though not positivity) of the density matrix for each realization, thus improving the convergence of stochastic sampling. Interestingly, we find a condition on the classical simulability of the system based solely on the single atom cooperativity even in a many-body setting. We benchmark and showcase the utility of our approach in several examples, specifically in cases where a direct numerical computation is unfeasible.
The multiplicative stochastic process treatment of the time development of the density matrix for a subsystem in contact with a heat reservoir is applied to the specific problem of the relaxation of a nuclear magnetic moment which is interacting with a fluctuating magnetic environment. A model for the fluctuating interaction Hamiltonian, appropriate for the magnetic moment case, is presented, and the Bloch equations for nuclear magnetic relaxation are constructed as a consequence. Agreement with empirical observations is noted.