Exact quantum dynamics of XXZ central spin problems

We obtain analytically close forms of benchmark quantum dynamics of the collapse and revival, reduced density matrix, Von Neumann entropy, coherence factor, 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. The effective magnetic field $B$, homogenous 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. In particular, the presence of a finite longitudinal interaction $\Delta$ allows for quantum revivals even at a very small number of bath spins $N$, facilitating experimental control of entangled states. 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 quantum collapse and revival recovers the known expression found in the Jaynes-Cummings model, thus building up an exact dynamical connection between the central spin problem and the light-matter interacting system in quantum nonlinear optics, and revealing the statistical nature of Holstein-Primakoff transformation.