Experimental Detection of the Correlation Rényi Entropy in the Central Spin Model

We propose and experimentally measure an entropy that quantifies the volume of correlations among qubits. The experiment is carried out on a nearly isolated quantum system composed of a central spin coupled and initially uncorrelated with 15 other spins. Because of the spin-spin interactions, information flows from the central spin to the surrounding ones forming clusters of multispin correlations that grow in time. We design a nuclear magnetic resonance experiment that directly measures the amplitudes of the multispin correlations and use them to compute the evolution of what we call correlation Rényi entropy. This entropy keeps growing even after the equilibration of the entanglement entropy. We also analyze how the saturation point and the timescale for the equilibration of the correlation Rényi entropy depend on the system size.

Figure 1

Schematic illustration of the flow of information initially contained in the central spin (orange circle) to the surrounding 15 spins. Each shaded area indicates a cluster of correlated spins.

Figure 2

Free induction decay (a) and entanglement entropy (b) for a central spin coupled with 15 surrounding spins via the ZZ interaction in Eq. (1). Thin lines are for each realization of ${\omega }_j$, and a thick line gives the average of over 300 random orientations of the molecules. The dotted lines in (b) also give the ensemble average for $N=10$ and $N=20$. Inset of panel (b): averaged entangled entropy for $N=15,16,\dots ,20$ from bottom to top.

Figure 3

Evolution of the coherence orders’ intensities $|C_n^x(T){|}^2$ in (a) and for a longer time in the inset, and the first and second order correlation Rényi entropies in (b). Symbols represent the experimental data, and solid lines give the numerical results for $N=15$ averaged over 300 orientations of the molecule.

Figure 4

(a) Evolution of the correlation Rényi entropy $S_2$ for $N=5,\dots ,30$ averaged over various orientations of the molecules. The thick red lines correspond to bath sizes that are multiples of 5. Inset in (a): solid lines correspond to the curves of $S_2(T)$ up to saturation and the dashed lines indicate $\overline{S_2}$. (b) Scaling analysis of the factor $\beta$ in $S_2(T)\sim -4.18+\beta {\log }_2(T)$ (symbols) fitted with $0.65+0.07\ln (N)$ (solid line). (c) Scaling analysis of the saturation values of $S_2$ (symbols) fitted with $1.34+0.71\ln (N)$ (solid line).