Out-of-time-order operators and the butterfly effect

Abstract

Out-of-time-order (OTO) operators have recently become popular diagnostics of quantum chaos in many-body systems. The usual way they are introduced is via a quantization of classical Lyapunov growth, which measures the divergence of classical trajectories in phase space due to the butterfly effect. However, it is not obvious how exactly they capture the sensitivity of a quantum system to its initial conditions beyond the classical limit. In this paper, we analyze sensitivity to initial conditions in the quantum regime by recasting OTO operators for many-body systems using various formulations of quantum mechanics. Notably, we utilize the Wigner phase space formulation to derive an -expansion of the OTO operator for spatial degrees of freedom, and a large spin 1/sexpansion for spin degrees of freedom. We find in each case that the leading term is the Lyapunov growth for the classical limit of the system and argue that quantum corrections become dominant at around the scrambling time, which is also when we expect the OTO operator to saturate. We also express the OTO operator in terms of propagators and see from a different point of view how it is a quantum generalization of the divergence of classical trajectories.

Exact relaxation dynamics and quantum information scrambling in multiply quenched harmonic chains

“…The one body momentum distribution, also obtained from the RDM, carries the signature of the thermal behavior of the system [2]. Recently, the out of time order correlator (OTOC) [29] has gained prominence in the context of scrambling of quantum information in non-equilibrium systems [30][31][32][33][34][35][36][37][38][39]. Although information scrambling is usually a property of chaotic systems, the OTOC of certain non-local operators exhibit scrambling even in an integrable Ising chain [34].…”

Tunable quantum chaos in the Sachdev-Ye-Kitaev model coupled to a thermal bath

“…(1.8) which diagnoses the quantum butterfly effect [22][23][24][25] and has been applied to various models recently [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. There are also experimental measurements of OTOC in different systems [43,44].…”

Does Scrambling Equal Chaos?

“…This differs from the exponential growth rate of an OTOC, λ OTOC , which is rather the log of the phase space average of sensitivity squared. Since the log of the average is larger than the average of the log, we have [22][23][24][25][26][27][28]:…”