Exact quantum dynamics of XXZ central spin problems

TL;DR

This paper provides exact analytical solutions for the quantum dynamics of the XXZ central spin problem, revealing how parameters influence decoherence, entanglement, and revival phenomena, with implications for quantum metrology and optics.

Contribution

It introduces exact analytical forms for key quantum dynamical quantities in the XXZ central spin model, connecting it to light-matter systems and exploring parameter effects.

Findings

Revival times depend on system parameters and scale with system size.

Analytical results recover known models like Jaynes-Cummings under specific conditions.

Moderate inhomogeneity in couplings preserves revival dynamics.

Abstract

We obtain analytically close forms of benchmark quantum dynamics of the collapse and revival (CR), 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 few to many 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 $\Delta$ 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 $B=\Delta=A$, the location of the $m$-th revival peak in time reaches a simple relation $t_{r} \simeq\frac{\pi N}{A} m$ for a large $N$. For $\Delta =0$, $N\to \infty$ 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.