Conservation laws protect dynamic spin correlations from decay: Limited role of integrability in the central spin model

Abstract

Mazur's inequality renders statements about persistent correlations possible. We generalize it in a convenient form applicable to any set of linearly independent constants of motion. This approach is used to show rigorously that a fraction of the initial spin correlations persists indefinitely in the isotropic central spin model unless the average coupling vanishes. The central spin model describes a major mechanism of decoherence in a large class of potential realizations of quantum bits. Thus the derived results contribute significantly to the understanding of the preservation of coherence. We will show that persisting quantum correlations are not linked to the integrability of the model, but caused by a finite operator overlap with a finite set of constants of motion. PACS numbers: 78.67.Hc, 72.25.Rb, 03.65.Yz, 02.30.Ik a. Introduction. The two-time correlation function of two observables reveals important information about the dynamics of a system in and out of equilibrium: The noise spectra are obtained from symmetric combinations of correlation functions, while the causal, antisymmetric combination determines the susceptibilities required for the theory of linear response.The two-time correlation function only depends on the time difference if at t = 0 the system of interest is prepared in a stationary state whose density operator commutes with the time-independent Hamiltonian. This is what will be considered in this work. Since correlations generically decay for t → ∞, important information about the system dynamics is gained if a non-decaying fraction of correlations prevails at infinite times. Such non-decaying correlations are clearly connected to a limited dynamics in certain subspaces of the Hilbert space. The question arises if such a restricted dynamics is always linked to the integrability of the Hamiltonian. Here integrability means that the Hamiltonian can be diagonalized by Bethe ansatz which implies that there is an extensive number of constants of motion. Identifying and understanding those non-decaying correlations can be potentially exploited in applications for persistent storage of (quantum) information.In this Letter we first prove that persisting correlations are not restricted to integrable systems by using a generalized form of Mazur's inequality 1,2 . This is in contrast to the behavior of the Drude weight in the frequency-dependent conductivity of one-dimensional systems which appears to vanish abruptly once the integrability is lost, even if only by including an arbitrarily small perturbation. So far, the Drude weight has been the most common application of Mazur's inequality, see for instance Refs. 3-6 and references therein. Second, we apply this approach to the central spin model (CSM) 7 describing the interaction of a single spin, e.g., an electronic spin in a quantum dot 8,9 , an effective two-level model in a NV center in diamond 10 , or a 13 C nuclear spin 11 , coupled to a bath of surrounding nuclear spins inducing decoherence.Persisting spin correlations have been found in ...

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

“…We recall that the lower bounds as discussed in Refs. [25,26] result from the existence of conserved quantities such as the total angular momentum and the total energy. These quantities are conserved also for the classical model and the choices of couplings we are considering here.…”

“…Density-matrix renormalization group (DMRG) can tackle much larger systems up to about 1000 spins, but is restricted to short times up to about 40 /J Q [22][23][24]. The strict limit of infinite times, i.e., of persisting correlations has been tackled by mathematically rigorous bounds [25,26]. Techniques based on non-Markovian master equations give access to large bath sizes, but are well justified only for sufficiently strong external fields [27][28][29][30][31][32][33][34].…”

Spin noise of electrons and holes in (In,Ga)As quantum dots: Experiment and theory

“…Adding the quadrupolar couplings H µ N has two effects on the nuclear spin dynamics: firstly it enhances the nuclear Larmor frequency, and, secondly, the breaking of the total spin conservation law translates into a change of an effective precession axis which is not longer determined by the carrier spin direction only. Both effects yields to an addition dephasing of the Overhauser field and consequently to a dephasing of the non-decaying fraction of the spin-correlation function [25] on a time scale dominated by the quadrupolar coupling strength.…”

Influence of the Nuclear Electric Quadrupolar Interaction on the Coherence Time of Hole and Electron Spins Confined in Semiconductor Quantum Dots

“…The separation of time scales [7] -a fast electronic precession around an effective nuclear magnetic field, and slow nuclear spin precessions around the fluctuating electronic spin -has motivated various semiclassical approximations [1,7,20,[23][24][25] which describe the short-time dynamics of the central spin polarization very well. As can be shown rigorously [26] the CSM predicts a finite non-decaying spin polarization [7,27,28] whose lower bound depends on the distribution function of the hyperfine couplings and is only linked to conservation laws. In semi-classical theories [7,27] it is given by a third of the initial spin polarization leading to a large spectral weight at zero-frequency in the spin-noise spectrum.…”