Out-of-Time-Ordered Correlators

• Out-of-time-ordered correlators (OTOCs) are four-point dynamical correlation functions that measure how initially commuting operators evolve to signal quantum chaos and information scrambling.
• They capture operator spreading in systems ranging from harmonic oscillators to chaotic billiards, highlighting the roles of spectral structure and quantum interference.
• OTOCs can be computed via exact diagonalization or semiclassical approximations, offering practical insights into the impacts of temperature, system size, and coherence on dynamics.

Out-of-time-ordered correlators (OTOCs) are four-point dynamical correlation functions that quantify the degree to which two initially commuting operators fail to commute at later times under quantum evolution. As such, OTOCs are widely regarded as diagnostic tools for quantum chaos, information scrambling, and operator spreading in both single-particle and many-body quantum systems. Their temporal behavior reveals deep connections between quantum coherence, classical chaos, and the emergent statistical dynamics of complex quantum systems.

1. Mathematical Formulation and Theoretical Basis

The canonical OTOC for a system with time-independent Hamiltonian $H$, position operator $x$, and momentum operator $p$ is

$C_T(t) = - \langle [x(t), p(0)]^2 \rangle$

where $x(t) = e^{iHt} x e^{-iHt}$ denotes time evolution in the Heisenberg picture, and the angle brackets denote the thermal average with respect to the Boltzmann weight $e^{-\beta H}$, $\beta = 1/T$ (Hashimoto et al., 2017).

The thermal OTOC can be computed for general time-independent Hamiltonians by expanding in the energy eigenbasis $|n\rangle$, giving

$C_T(t) = \frac{1}{Z} \sum_n e^{-\beta E_n} c_n(t)$

where $c_n(t) = -\langle n| [x(t), p]^2 |n\rangle$ and $Z$ is the partition function. Through algebraic manipulation and use of the commutation relation $[H, x] = -2i p$ for systems of the natural form $H = p^2 + U(x)$, the calculation of OTOCs reduces to summations over products of energy eigenvalues and position operator matrix elements.

For systems with discrete energy spectra and full knowledge of eigenfunctions and operator matrix elements, OTOCs can be computed exactly or via controlled numerics. The OTOC can also be interpreted semiclassically: replacing commutators $[A,B]$ with $i\hbar\{A,B\}_{\rm PB}$ (Poisson brackets), the OTOC becomes, in the classical limit,

$$C_{\rm cl}(t) = \langle \{x(t), p(0)\}^2_{\rm PB} \rangle$$

which grows exponentially if $x(t)$ is exponentially sensitive to $p(0)$, as in classical chaos.

2. Characteristic Temporal Behavior Across Exemplars

Explicit evaluation of OTOCs in various paradigmatic systems showcases a spectrum of possible dynamical behavior (Hashimoto et al., 2017):

• Harmonic Oscillator: Exhibits exact periodicity in OTOCs due to its commensurate energy spectrum. For the Hamiltonian $H = p^2 + \frac{\omega^2}{4} x^2$, $C_T(t) = \cos^2(\omega t)$ is strictly periodic.
• Particle in a 1D Box: The OTOC is also periodic but with non-uniform, state-dependent oscillations. The energy spectrum $E_n = \pi^2 n^2$ and analytic structure of $x_{nm}$ yield recurrences with period $\Delta t = 1/\pi$.
• Circle Billiard (Integrable System): The spectrum is not strictly commensurate; microcanonical OTOCs are non-periodic, exhibit characteristic dips, and represent a quasiperiodic response reflecting integrable dynamics.
• Stadium Billiard (Chaotic in Classical Mechanics): Despite classical chaos (positive Lyapunov exponent, $λ \sim v/\sqrt{A}$), quantum OTOCs for the stadium billiard do not display the anticipated early-time exponential growth. Instead, after a transient, the OTOC saturates: $C_T(\infty) \propto mTA_{\rm sys}$, i.e., linearly in temperature and system size squared, lacking periodicity or recurrences.

3. Quantum vs Classical Limits, Ehrenfest Time, and Wavepacket Dynamics

Comparisons between quantum and classical OTOCs stress the fundamental limitations of semiclassical correspondence. In the 1D box, the classical OTOC ($C_{\rm cl}(t) = 1$) fails to describe the quantum, temperature-dependent OTOC. Analysis of a localized Gaussian wavepacket

$$\phi(x) = (2\pi\sigma^2)^{-1/4} \exp\left[-\frac{(x-x_0)^2}{4\sigma^2} + ik_0(x-x_0)\right]$$

shows that the two-point commutator $b_\phi = -i\langle\phi|[x(t),p(0)]|\phi\rangle$ closely follows classical predictions (including deterministic “bounces” off hard walls) with smooth crossovers described by error functions. In contrast, the four-point correlator $c_\phi = -\langle\phi|[x(t),p(0)]^2|\phi\rangle$ develops sharp quantum-induced spikes at wall collision times. Furthermore, the time scale $t_s$ at which quantum deviations from classical OTOC emerge

$t_s \sim 0.175\sqrt{\sigma/k_0}$

is parametrically shorter than the Ehrenfest time $t_E \sim \sigma$, evidencing that quantum interference effects can become relevant for OTOCs much earlier than for expectation values or local observables.

4. Periodicity, Recurrences, and Saturation: Role of Spectral Structure

The structure of the energy spectrum determines the generic behavior of OTOCs:

• Commensurate spectra (harmonic oscillator, box): lead to exact or quasiperiodic OTOCs with complete recurrences.
• Asymptotically commensurate/non-commensurate spectra (circle billiard): yield nonperiodic OTOCs with aperiodic recurrences or characteristic dip features.
• Stadium billiard (mixed/chaotic spectra): OTOCs demonstrate initial growth and then saturate without recurrences, and the late-time OTOC value scales as $C_T(\infty) \sim mTL^2_{\rm sys}$.

The absence of early-time exponential growth in quantum OTOCs for the stadium indicates that, in single-particle systems, the quantum manifestations of chaos (i.e., scrambling as probed by OTOCs) can be considerably more subtle than classical analogies would suggest.

5. Quantum Chaos and OTOC as a Scrambling Diagnostic

While in many-body or large-$N$ systems (e.g., SYK model) OTOCs display clear early-time exponential growth, with a quantum Lyapunov exponent $\lambda_{\rm L}$, this behavior is not generic for all systems. The present paper demonstrates that:

• Integrable quantum systems produce periodic or recurrent OTOCs.
• Classically chaotic single-particle systems (e.g., the quantum stadium) may lack early-time exponential behavior in OTOCs, with quantum interference and finite-size effects dominating.
• Late-time saturation of OTOCs is governed by temperature, system size, and geometry, rather than by any classical Lyapunov instability.

This nuanced behavior underscores that OTOCs must be interpreted with care as chaos indicators outside many-body, semiclassical, or large-$N$ limits. The deviation of the full four-point OTOC from classical expectations, especially at timescales shorter than the Ehrenfest time, further complicates simple correspondences.

6. Implications, Limitations, and Practical Computation

The developed formalism—diagonalizing the Hamiltonian, exploiting commutator algebra, and expressing momentum matrix elements through energy and position operator matrices—enables explicit computation of OTOCs in arbitrary stationary quantum systems with known spectra and operator elements. While analyticity permits closed-form results in some cases (harmonic oscillator), general systems (box, billiards) require controlled numerical summation over eigenstates due to the complexity of the spectra and matrix elements.

A salient result is that the quantum OTOC can substantially deviate from the classical value at times much shorter than the Ehrenfest time, meaning that quantum interference and coherent dynamics can dominate the scrambling diagnostic before the system exhibits classical ergodic behavior. Additionally, since OTOC saturation scales linearly with temperature and quadratically with system size, the late-time value provides a dimensional estimate of information capacity but does not directly encode classical chaos indicators.

These results clarify the limitations of OTOCs as universal chaos measures in single-particle quantum contexts and highlight their dependence on spectral structure, system size, and temperature, while establishing practical computational techniques for a broad class of quantum systems.