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arXiv:1810.03012 · Physical review. B./Physical review. B

Exact quantum dynamics of XXZ central spin problems

Wenbin He, Stefano Chesi, Hai‐Qing Lin, Xi‐Wen Guan

Published May 20, 2019• 34 citations
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Abstract

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 $B$, coupling constant $A$, 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 $m\mathrm{th}$ revival peak in time reaches a simple relation ${t}_{r}\ensuremath{\simeq}\frac{\ensuremath{\pi}N}{A}m$ for a large $N$. 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.

DOI
10.1103/physrevb.99.174308

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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 \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ 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=\ensuremathΔ=AB=\mathrm{\ensuremath{\Delta}}=AB=\ensuremathΔ=A, the location of the mthm\mathrm{th}mth revival peak in time reaches a simple relation tr\ensuremath≃\ensuremathπNAm{t}_{r}\ensuremath{\simeq}\frac{\ensuremath{\pi}N}{A}mtr​\ensuremath≃A\ensuremathπN​m for a large NNN. For \ensuremathΔ=0\mathrm{\ensuremath{\Delta}}=0\ensuremathΔ=0, N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, 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.
Central spin dynamics and relaxation of antiferromagnetic order in a central-spin−-−XXZ-chain system.
Using an equations-of-motion method based on analytical representations of spin-operator matrix elements in the XX chain, we obtain exact long-time dynamics of a composite system consisting of a spin-SSS central spin and an XXZ chain, with the two interacting via inhomogeneous XXZ-type hyperfine coupling. Three types of initial bath states, namely, the Neel state, the ground state, and the spin coherent state are considered. We study the reduced dynamics of both the central spin and the XXZ bath. For the Neel state, we find that strong hyperfine couplings slow down the initial decay but facilitate the long-time relaxation of the antiferromagnetic order. Moreover, for fixed hyperfine coupling a larger SSS leads to a faster initial decay of the antiferromagnetic order. We then study the purity dynamics of an S=1S=1S=1 central spin coupled to an XXZ chain prepared in the ground state. The time-dependent purity is found to reach the highest values at the critical point. We finally study the polarization dynamics of the central spin homogeneously coupled to a bath prepared in the spin coherent state. Under the resonant condition, the polarization dynamics for S>12S>\frac{1}{2}S>21​ exhibits collapse-revival behaviors with fine structures. However, the collapse-revival phenomena is found to be fragile with respect to the anisotropic intrabath coupling.
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