Out-of-time-order correlators in quantum mechanics

TL;DR

This work develops a general framework to compute out-of-time-order correlators ($C_T(t)$) in single-particle quantum mechanics and applies it to integrable (harmonic oscillator, particle in a box, circle billiard) and chaotic (stadium billiard) systems. The authors show that integrable systems yield periodic or quasi-periodic OTOCs, while the circle and stadium billiards exhibit saturation rather than exponential growth, with the late-time OTOCs scaling linearly with temperature and system size in the chaotic case. Classical statistics fail to reproduce the high-temperature quantum behavior, and substantial quantum–classical divergence appears even for wavepackets at times well before the Ehrenfest time, due to pre-Ehrenfest-time interference effects. Overall, the results clarify the limits of single-particle OTOCs as universal chaos probes and motivate further study in many-body or decohering settings where quantum Lyapunov growth may emerge, as in models like the SYK system.

Abstract

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time.

Figures (13)

• Figure 1: The OTOC $C_T(t) = -\langle [x(t),p(0)]^2\rangle$ of a particle in a 1D box (a) and that of a billiard (b). $T$ represents the temperature of the systems. We find a clear distinction between the two: The OTOC for the particle in a box periodically comes back to its initial value ($=1$), while that for stadium billiard saturates to a constant value. The asymptotic value grows with the temperature. In the inset of the right panel, we show the early time evolution of the OTOC. We find no clear exponential growth of the OTOC.
• Figure 2: Microcanonical out-of-time-order correlators for the particle in a 1D box ($L=1$).
• Figure 3: Out-of-time-order correlators for the circle billiard.
• Figure 4: A typical trajectory of a classical particle in the stadium billiard with $a/R=1$. The maximum Lyapunov exponent vs deformation parameter $a/R$ for fixed area of the stadium and velocity of the particle, $A=v=1$.
• Figure 5: Eigenvalues of the quantum stadium billiard with $a/R=1$.