Efficient algorithms for the dynamics of large and infinite classical central spin models

Abstract

We investigate the time dependence of correlation functions in the central spin model, which describes the electron or hole spin confined in a quantum dot, interacting with a bath of nuclear spins forming the Overhauser field. For large baths, a classical description of the model yields quantitatively correct results. We develop and apply various algorithms in order to capture the longtime limit of the central spin for bath sizes from 1000 to infinitely many bath spins. Representing the Overhauser field in terms of orthogonal polynomials, we show that a carefully reduced set of differential equations is sufficient to compute the spin correlations of the full problem up to very long times, for instance up to 10 5 /JQ where JQ is the natural energy unit of the system. This technical progress renders an analysis of the model with experimentally relevant parameters possible. We benchmark the results of the algorithms with exact data for a small number of bath spins and we predict how the long-time correlations behave for different effective numbers of bath spins.

Nuclear frequency focusing in periodically pulsed semiconductor quantum dots described by infinite classical central spin models

“…In Ref. 27, a scaling of time with √ γ has been established for h = 0 and without periodic pulsing. The inset in Fig.…”

“…While these equations can be numerically solved by standard algorithms such as the Runge-Kutta algorithm of various orders, the direct simulation of 10 5 equations, let alone of an infinite number of them, is not an option. Thus, we resort to the spectral density approach introduced previously 27 . The ensemble of bath spins parametrized according to (2) can be represented by the linear weight function W (ε) = (ε/γ)θ( √ 2γ − ε) where we set the energy scale J Q to unity.…”

“…The deviations decrease quadratically upon increasing N tr . It is this approach which we employ in the present paper 27 .…”

Nonequilibrium nuclear spin distribution function in quantum dots subject to periodic pulses

“…Alternatively, one can map the dynamics onto a set of classical equations of motion [5,12,23] which shows remarkably good agreement with the full quantum mechanical treatment [10] but is easily extendable to a large number of spins. Below, we address the key challenge of how to combine quantum and classical calculations in a systematic way to incorporate the formation on a nonequilibrium density distribution of the Overhauser field [18] which is the origin of the self-focusing experimentally observed by Greilich et al [8].…”

“…The path integral for expectation values uses spin coherent states for each spin which are parameterized by the solid angle. The saddle point approximation leads to (N + 1) coupled Euler-Lagrange equations [10,12,23] …”