Quantum and Classical Dynamics of the Out-of-Time-Ordered Correlator (OTOC) Operator

The Out-of-Time-Ordered Correlator (OTOC) operator exhibits exponential growth in classical systems. The magnitude of the exponent corresponds to the Lyapunov exponent, a measure of chaotic behavior. In the quantum domain, the OTOC operator displays analogous exponential growth within the Ehrenfest time $t_E$ /cite{adagideli2003ehrenfest}, marking a transition from quantum to classical dynamics. Beyond $t_E$, the diffusion of the OTOC operator transitions to a power-law regime. /n/nThis study considers the momentum operators $p(t)$ and $p(0)$ as the equivalent of $W(t)$ and $V(0)$ within the OTOC operator, respectively. This allows for expressing the OTOC as $C_T(t) = -//left< //left[ p(t), p(0) //right]^2 //right>$. Expanding this expression and transforming it to the Schrodinger representation enables further analysis of the operator's dynamics.