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Entropy

Many-Body Dynamics and Decoherence of the XXZ Central Spin Model in External Magnetic Field

Xu Zhou, Qing-Kun Wan, Xiao-Hui Wang

Published Dec 23, 2019• 5 citations
Publisher

Abstract

The many-body dynamics of an electron spin−1/2 qubit coupled to a bath of nuclear spins by hyperfine interactions, as described by the central spin model in two kinds of external field, are studied in this paper. In a completely polarized bath, we use the state recurrence method to obtain the exact solution of the X X Z central spin model in a constant magnetic field and numerically analyze the influence of the disorder strength of the magnetic field on fidelity and entanglement entropy. For a constant magnetic field, the fidelity presents non-attenuating oscillations. The anisotropic parameter λ and the magnetic field strength B significantly affect the dynamic behaviour of the central spin. Unlike the periodic oscillation in the constant magnetic field, the decoherence dynamics of the central spin act like a damping oscillation in a disordered field, where the central spin undergoes a relaxation process and eventually reaches a stable state. The relaxation time of this process is affected by the disorder strength and the anisotropic parameter, where a larger anisotropic parameter or disorder strength can speed up the relaxation process. Compared with the constant magnetic field, the disordered field can regulate the decoherence over a large range, independent of the anisotropic parameter.

DOI
10.3390/e22010023

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The many-body dynamics of an electron spin−1/2 qubit coupled to a bath of nuclear spins by hyperfine interactions, as described by the central spin model in two kinds of external field, are studied in this paper. In a completely polarized bath, we use the state recurrence method to obtain the exact solution of the X X Z central spin model in a constant magnetic field and numerically analyze the influence of the disorder strength of the magnetic field on fidelity and entanglement entropy. For a constant magnetic field, the fidelity presents non-attenuating oscillations. The anisotropic parameter λ and the magnetic field strength B significantly affect the dynamic behaviour of the central spin. Unlike the periodic oscillation in the constant magnetic field, the decoherence dynamics of the central spin act like a damping oscillation in a disordered field, where the central spin undergoes a relaxation process and eventually reaches a stable state. The relaxation time of this process is affected by the disorder strength and the anisotropic parameter, where a larger anisotropic parameter or disorder strength can speed up the relaxation process. Compared with the constant magnetic field, the disordered field can regulate the decoherence over a large range, independent of the anisotropic parameter.
In spin-based nanosystems for quantum information processing, electron spin qubits are subject to decoherence due to their interactions with nuclear spin environments. In this paper, we present an exact master equation for a central spin-1/2 system in time-dependent external fields and coupled to a spin-half bath in terms of hyperfine interaction. The master equation provides a unified description for free and controlled dynamics of the central spin and is formally independent of the details and size of spin environments. Different from the previous approaches, the master equation remains exact even in the presence of external control fields. Using the parameters for realistic nanosystems with nonzero nuclear spins, such as GaAs, we investigate the Overhauser's effect on the decoherence dynamics of the central spin under different distributions of bath-spin frequencies and system-bath coupling strengths. Furthermore, we apply the leakage elimination operator, in a nonperturbative manner, to this system to suppress the decoherence induced by hyperfine interaction.
The dynamics and decoherence of an electronic spin-1/2 qubit coupled to a bath of nuclear spins via hyperfine interactions in a quantum dot is studied. We show how exact results from the integrable solution can be used to understand the dynamic behavior of the qubit. It is possible to predict the main frequency contributions and their broadening for relatively general initial states analytically, leading to an estimate of the corresponding decay times, which are related to T1{T}_{1}T1​ of the electron. Furthermore, for a small bath polarization, a distinct low-frequency time scale is observed.
This present issue is an extension of the work of Y. Xiao-Zhong et al. who investigated the influence of constant external magnetic field on the decoherence of a central electron spin of atom coupled to an anti-ferromagnetic environment. We have shown in this work that the character variability of the field induces oscillations amongst the eigen modes of the environment. This observation is made via the derivation of the transition probability density of state, a manner by which critical parameters (parameters where transition occur) of the system could be obtained as it shows resonance peak. We equally observed that the two different magnons modes resulting from the frequency splitting via the application of the time-varying external B-Field, exhibit each a resonant peak of similar amplitude at different temperature ranges. This additional information shows that the probability for the central spin system to remain in its initially prepared diabatic state is enhanced for some temperature ranges for the corresponding two magnon modes. Hence, these temperature ranges where the probability density is maximum could save as decoherence free environment; an important requirement for the implementation of quantum computation and information processing in solid state circuitry. The theoretical and numerical results presented for the decoherence time and the probability density are that of a decohered central electron spin coupled to an anti-ferromagnetic spin bath. The theory is based on a spin wave approximation and on the density matrix using both transformations of Bloch, Primakov and Bogoliobuv in the adiabatic limit.
We study the dynamics of a single spin-1/2 coupled to a bath of spins-1/2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. This formula is evaluated explicitly for initial conditions such that the bath spins are completely polarized at the beginning. For this case we find, after an initial decay, a persistent oscillatory behaviour of the central spin. For a large number of bath spins NbN_bNb​, the oscillation frequency is proportional to NbN_bNb​, whereas the amplitude behaves like 1/Nb1/N_b1/Nb​, to leading order. No asymptotic decay due to the non-uniform couplings is observed, in contrast to some recent studies.
We study the dynamics of a single spin 1∕21∕21∕2 coupled to a bath of spins 1∕21∕21∕2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. For initially completely polarized bath spins and small magnetic field, we find persistent oscillations of the central spin about a nonzero mean value. For a large number of bath spins Nb{N}_{b}Nb​, the oscillation frequency is proportional to Nb{N}_{b}Nb​, whereas the amplitude behaves as 1∕Nb1∕{N}_{b}1∕Nb​, to leading order. No asymptotic decay of the oscillations due to the nonuniform couplings is observed, in contrast to some recent studies.
We obtain analytically close forms of benchmark quantum dynamics of the collapse and revival (CR), the reduced density matrix, Von Neumann entropy, and fidelity for the XXZ central spin problem. These quantities characterize the quantum decoherence and entanglement of the system with any number of bath spins, and for a short to infinitely long time evolution. For the homogeneous central spin problem, the effective magnetic field BBB, coupling constant AAA, and longitudinal interaction $\mathrm{\ensuremath{\Delta}}$ significantly influence the time scales of the quantum dynamics of the central spin and the bath, providing a tunable resource for quantum metrology. Under the resonance condition $B=\mathrm{\ensuremath{\Delta}}=A$, the location of the mthm\mathrm{th}mth revival peak in time reaches a simple relation ${t}_{r}\ensuremath{\simeq}\frac{\ensuremath{\pi}N}{A}m$ for a large NNN. For $\mathrm{\ensuremath{\Delta}}=0$, $N\ensuremath{\rightarrow}\ensuremath{\infty}$, and a small polarization in the initial spin coherent state, our analytical result for the CR recovers the known expression found in the Jaynes-Cummings model, thus building up an exact dynamical connection between the central spin problems and the light-matter interacting systems in quantum nonlinear optics. In addition, the CR dynamics is robust to a moderate inhomogeneity of the coupling amplitudes while disappearing at strong inhomogeneity.