Fig. 4

Entanglement buildup. a Absolute values of the central spin density-matrix elements \(|\rho _{m_S,n_S}^{\mathrm{c}}|\) with \(\rho _{m_s,n_S}^{\mathrm{c}} \equiv \langle m_S|\hat \rho _0|n_S\rangle\), extracted from GDTWA simulations with the same parameters as in Fig. 2. Off-diagonal single-site coherences are destroyed as the spins become entangled during the quantum dynamics (left two panels: t = 0, after the tilt, right two panels: after t = 30 ms evolution). For small rotation angles (upper panels: θ = 0.2π), the system evolves locally into a partially mixed state (uneven spin-state population). For larger rotation angles (lower panels: θ = 0.5π), the local state resembles a maximally mixed state of the form \(\hat \rho _0 \propto \frac{1}{7}\mathop {\sum}\nolimits_{m_S = - 3}^3 |m_S\rangle \langle m_S\rangle\). b Evolution of \(S_2^{\mathrm{c}}\), the value of the second-order Renyi entropy \(S_0^{(2)}\) for the central spin density matrix. For larger rotation angles, the entanglement entropy increases with time, almost reaching the maximum value (\(S_0^{(2){\mathrm{max}}} = {\mathrm{log}}_2(7)\), black dashed line). The red dotted line shows the upper bound \(S_2^{\mathrm{D}}\), computed only from the diagonal elements (populations) of the average single-site density matrix \(\hat \rho _S\) (see text) for θ = 0.5π. c Comparison of the theoretically computed diagonal entropy (GDTWA: thick dotted line; mean-field: thin dashed line) with the one reconstructed from the measured populations. Error bars correspond to statistical standard deviations