Emergent Prethermalization Signatures in Out-of-Time Ordered Correlations

How a many-body quantum system thermalizes—or fails to do so—under its own interaction is a fundamental yet elusive concept. Here we demonstrate nuclear magnetic resonance observation of the emergence of prethermalization by measuring out-of-time ordered correlations. We exploit Hamiltonian engineering techniques to tune the strength of spin-spin interactions and of a transverse magnetic field in a spin chain system, as well as to invert the Hamiltonian sign to reveal out-of-time ordered correlations. At large fields, we observe an emergent conserved quantity due to prethermalization, which can be revealed by an early saturation of correlations. Our experiment not only demonstrates a new protocol to measure out-of-time ordered correlations, but also provides new insights in the study of quantum thermodynamics.

Figure 1

(a) Distinct behavior for transverse ($Y$) and longitudinal ($Z$) magnetization: $⟨Y(t)Y{⟩}_{\beta =0}$ at $g/J=0.25$ shows a fast decay as a function of time, indicating erasure of initial memory. $⟨Z(t)Z{⟩}_{\beta =0}$ at $g/J=1$ shows instead slow nonergodic dynamics with periodic oscillations. In the opposite regimes (dashed lines) both correlations quickly decay to zero. Experimentally measured OTO commutator, $C_{YZ}$, as a function of transverse field strength (b) and normalized time (c). We observe the fastest growth around $g/J=0.5$ [marked by a dashed line in (b)]. Here and in the rest of the Letter, error bars are determined from the noise in the free induction decay (see Supplemental Material [50] for details on the experimental scheme).

Figure 2

(a) Experimentally averaged $C_{YY}$ (dashed) and $C_{ZZ}$ (solid) as a function of transverse field strength. (b) Experimentally measured $\mathrm{Tr}({\tilde{Z}}^2)$ (dashed) and $⟨|[\tilde{Z},Z(0)]{|}^2{⟩}_{\beta =0}$ (solid) versus transverse field strength. The time average is taken over the values $Jt=3.77$, 5.02, 6.28, 7.54, 8.80, 10.05, with the longest time corresponding to 16 cycles (1.54 ms).

Figure 3

Numerical characterization of the prethermal Hamiltonian. (a) Eigenvalue difference $r$ with respect to maximum order $n_M$ for different values of $g/J$. $r$ shows a divergence, up to a maximum field value $g/J=0.5$. (b) Decomposition of $Z(t)$ (obtained by exact diagonalization) at $Jt={10}^3$ according to the operator Hamming weight: $f_k$ is the contribution of all possible spin correlations with Hamming weight $k$. For small fields, $g/J=0.05$ (dashed line), the result follows closely the distribution (triangles) obtained randomly sampling all possible operators. For large fields, $g/J=5$ (solid line) there is a significant contribution of single-body terms, related to the quasiconserved quantity $Z_{\mathrm{pre}}$. In the inset: $f_1$ as a function of $g$. $f_1$ for $Z(t)$ (c) and $Y(t)$ (d) as a function of normalized time, showing the nonthermal behavior of $Z$ at large $g/J$, while $Y$ is always thermal even for small $g/J$. The system size is $L=12$ for (a) and 13 for (b)–(d).