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arXiv:cond-mat/0603730 · Journal of Physics A Mathematical and General

Solving spin quantum master equations with matrix continued-fraction methods: application to superparamagnets

José Luis García-Palacios, David Zueco

Published Oct 4, 2006• 24 citations
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Abstract

We implement continued-fraction techniques to solve exactly quantum master equations for a spin with arbitrary S coupled to a (bosonic) thermal bath. The full spin density matrix is obtained, so that along with relaxation and thermoactivation, coherent dynamics is included (precession, tunnel, etc). The method is applied to study isotropic spins and spins in a bistable anisotropy potential (superparamagnets). We present examples of static response, the dynamical susceptibility including the contribution of different relaxation modes, and of spin resonance in transverse fields.

DOI
10.1088/0305-4470/39/42/005

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We implement continued-fraction techniques to solve exactly quantum master equations for a spin with arbitrary S coupled to a (bosonic) thermal bath. The full spin density matrix is obtained, so that along with relaxation and thermoactivation, coherent dynamics is included (precession, tunnel, etc). The method is applied to study isotropic spins and spins in a bistable anisotropy potential (superparamagnets). We present examples of static response, the dynamical susceptibility including the contribution of different relaxation modes, and of spin resonance in transverse fields.
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The continued-fraction method of solving classical Fokker–Planck equations has been adapted to tackle quantum master equations of the Caldeira–Leggett type. This was done taking advantage of the phase-space (Wigner) representation of the quantum density matrix. The approach differs from those in which some continued-fraction expression is found for a certain quantity, in that the full solution of the master equation is obtained by continued-fraction methods. This allows us to study in detail the effects of the environment (fluctuations and dissipation) on several classes of nonlinear quantum systems. We apply the method to the canonical problem of quantum Brownian motion in periodic potentials both for cosine and ratchet potentials (lacking inversion symmetry).
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.
We address the problem of spin dynamics in the presence of a thermal bath, by solving exactly the appropriate quantum master equations with continued-fraction methods. The crossover region between the quantum and classical domains is studied by increasing the spin value , and the asymptote for the classical absorption spectra is eventually recovered. Along with the recognized relevance of the coupling strength, we show the critical role played by the structure of the system-environment interaction in the emergence of classical phenomenology.
Abstract. The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n × 4n 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 and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbor Heisenberg XY spin 1/2 chain in a transverse magnetic field. PACS numbers: 02.30.Ik, 03.65.Yz, 05.30.Fk, 75.10.PqA general method to solve master equations for quadratic open Fermi systems 2 1.
We consider a scenario where interacting electrons confined in quantum dots (QDs) are either too close to be resolved, or we do not wish to apply measurements that resolve them. Then, the physical observable is an electron spin only and the system state is fully described by the spin density matrix. Accounting for the spatial degrees of freedom, we examine to what extent a Hamiltonian description of the spin-only degrees of freedom is valid. We show that as long as there is no coupling between singlet and triplet states, this is indeed the case, but when there is such a coupling, there are open systems effects, i.e., the dynamics is nonunitary even without interaction with a true bath. Our primary focus is an investigation of nonunitary effects, based on exact master equations we derive for the spin density matrix in the Lindblad and time-convolutionless forms, and the implications for quantum computation. In particular, we demonstrate that the Heisenberg interaction does not affect the unitary part (apart from a Lamb shift) but does affect the nonunitary contributions to the time evolution of the spin density matrix. In a sequel paper, we present a detailed analysis of an example system of two quantum dots, including spin-orbit effects.