Exact quantum dynamics of XXZ central spin problems

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 $\mathrm{\Delta }$ 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 $B=\mathrm{\Delta }=A$, the location of the $m\mathrm{th}$ revival peak in time reaches a simple relation $t_r\simeq \frac{\pi N}{A}m$ for a large $N$. For $\mathrm{\Delta }=0$, $N\to \infty$, 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.

Figure 1

The central spin polarization evolves in time under different values of the magnetic field $B$ and longitudinal interaction $\mathrm{\Delta }$. In contrast to a two-level atom coupled to a cavity [1], here we demonstrate that at resonance $\mathrm{\Delta }=A=B=1$, the quantum CR can be observed even for a small system size $N=8$ [for $N=4,6$; see the Supplemental Material (SM) 58]. Such a small number of bath spins are experimentally accessible, for example by superconducting circuits [23, 24].

Figure 2

(a) Time evolution of the central spin polarization for different bath spin sizes $N\in [50$–250], with particle number step $\delta N=20$. All the curves are shifted upward by 1 for $N>50$. We use the circles and dashed lines to mark the revival peaks. The times of the revival peaks depend linearly on the bath size $N$; see the main text. Time evolution of the central spin polarization for a smaller value of $N=15$ (b), together with the corresponding quantum purity (c) and Von Neumann entropy (d). Here $B=\mathrm{\Delta }=A=1$.

Figure 3

Contour plots of fidelity vs the phase $\varphi$ for different values of $\theta$. (a)–(c) Time evolution of the fidelity of the reduced density matrix of the central spin ${\rho }_{\text{cs}}$ against the state $|\varphi \rangle =\frac{1}{\sqrt{2}}{\left[\right|\uparrow \rangle }_0+e^{-i\varphi }{|\downarrow \rangle }_0]$. Panels (d) and (e) show time evolution of the fidelity for the phase angles $\varphi =\pi /2$ and $\varphi =3\pi /2$. In all panels, $B=\mathrm{\Delta }=A=1$.

Figure 4

Effect of inhomogeneity on the CR dynamics. (a) Coupling strengths $A_j=Aexp[-\alpha (j-1)/N]$, with $\alpha =0,1,5$, corresponding to homogeneous, intermediate, and strongly inhomogeneous couplings, respectively. (b)–(d) Time evolution of $S_0^z$ obtained from Eq. (1) using ${\mathrm{\Delta }}_j=A_j,A=1,N=12$, $B=1$, and the three chosen values of $\alpha$. (e)–(g) Expectation values of the collective bath projector.