Out-of-Time-Ordered Correlators

Updated 9 November 2025

OTOC is a quantum diagnostic tool defined by four-point correlation functions that reveal operator scrambling and chaos in many-body systems.
The semiclassical approach connects OTOCs with classical chaos via Lyapunov exponents and phase-space flow, clarifying early-time exponential dynamics.
Experimental and numerical protocols validate unique OTOC behaviors in various models, highlighting differences in quantum, open, and non-Hermitian systems.

Out-of-time-ordered correlators (OTOCs) are four-point quantum correlation functions defined via the squared commutator of a time-evolved and a stationary observable. Originally introduced to quantify the spread and scrambling of quantum information, OTOCs provide a powerful diagnostic of chaos, operator growth, and delocalization in quantum many-body systems. Their early-time exponential growth governed by the classical Lyapunov exponent, long-time saturation, and sensitivity to quantum interference underpin much of the modern understanding of thermalization, localization, and quantum chaos in both theoretical and experimental settings.

1. Mathematical Definition and Physical Interpretation

Given two Hermitian operators $A$ and $B$ , with $A$ evolved to time $t$ in the Heisenberg picture ( $A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}$ ), the OTOC is defined as

$C(t) \equiv \langle |[A(t), B]|^2 \rangle = \langle [A(t), B]^\dagger [A(t), B] \rangle.$

For Hermitian $A$ and $B$ , this reduces to $C(t) = - \langle [A(t), B]^2 \rangle$ . In ensemble language (density matrix $\rho$ ),

$C(t) = - \mathrm{Tr}\{ \rho [A(t), B]^2 \}.$

$C(t)$ quantifies how much a local operator $A$ at time $t$ fails to commute with $B$ at time zero—a direct measure of operator spreading and information scrambling. In chaotic systems, OTOCs typically display an early-time exponential ramp, $C(t) \sim e^{2\lambda t}$ , where $\lambda$ is the classical Lyapunov exponent, followed by saturation at a plateau value that reflects the system's ergodicity, many-body level statistics, and quantum interference effects.

2. Semiclassical and Quasiclassical Approaches

The connection between OTOCs and classical chaos is formalized through semiclassical methods. The van Vleck–Gutzwiller propagator yields the semiclassical evolution kernel,

$K(q_f, q_i; t) \simeq (2\pi i \hbar)^{-d/2} \sum_{\gamma: q_i \to q_f} \sqrt{|\det(-\partial^2 S_\gamma / \partial q_f \partial q_i)|} \, \exp\left[\frac{i}{\hbar} S_\gamma - i \mu_\gamma \frac{\pi}{2}\right],$

where $S_\gamma$ is the action along classical trajectory $\gamma$ and $\mu_\gamma$ the Maslov index. Calculating the quantum OTOC requires inserting four such propagators and integrating over initial and final coordinates.

The "diagonal approximation," valid in the semiclassical ( $\hbar \to 0$ ) limit for mixing/chaotic dynamics, retains only contributions from pairs of nearly identical trajectories. This yields a quasiclassical OTOC,

$C_{\mathrm{qcl}}(t) = \int dp\, dq\, W(p, q) \, \{ A[\Phi^t(p, q)], B(p, q)\}^2,$

where $W(p, q)$ is the Wigner function, $\Phi^t$ the classical phase-space flow, and $\{\cdot, \cdot\}$ the Poisson bracket. Physically, this replaces the quantum four-point function with the square of the classical sensitivity of $A$ to $B$ , averaged over initial phase-space points.

In the short-time regime, this approach recovers the exponentiation of classical chaos: for strongly chaotic flow,

$C_{\mathrm{qcl}}(t) \sim \langle \{ A(t), B \}^2 \rangle \sim e^{2\lambda t}, \qquad t \lesssim t_E,$

with Ehrenfest time $t_E \approx (1/\lambda)\, \ln(1/\hbar)$ setting the semiclassical-quantum crossover.

For $t \gg t_{\mathrm{mix}}$ , where $t_{\mathrm{mix}}$ is the classical mixing time, but still $t \ll t_{\mathrm{Heis}} \sim \hbar^{-d}$ , the OTOC approaches a quasiclassical plateau,

$C_{\infty}^{(\mathrm{cl})} = \int dp\, dq\, W(p, q) \, \overline{ \{ A, B \}^2 },$

with the overbar denoting energy-shell or microcanonical average.

3. Quantum Corrections, Saturation, and Limitations of the Diagonal Approximation

Numerical investigations in few-site Bose–Hubbard models, both static and periodically driven, directly test the quasiclassical predictions. For chains with $L=3$ or $4$ sites and $N=20$ –$30$ bosons, comparison between exact diagonalization of the quantum OTOC and truncated-Wigner (semiclassical) evaluation reveals:

For $t \lesssim t_E$ , the quantum and quasiclassical OTOCs coincide, showing the expected exponential rise.
For $t \gtrsim t_E$ , the quantum OTOC overshoots the semiclassical plateau, reaching a higher saturation value ( $C^{(\mathrm{qm})}_\infty$ ) by factors of $2$–$5$ depending on system parameters. This confirms that the diagonal approximation fully accounts for classical early-time scrambling but fails at late times, underestimating the true quantum plateau (Michel et al., 8 Oct 2024).

The excess in $C^{(\mathrm{qm})}_\infty$ is due to "nondiagonal," genuinely quantum contributions: pairs of classical trajectories with small but nonzero action differences $\Delta S \lesssim \hbar$ (off-diagonal in the semiclassical sum). In spectral statistics, these encode Sieber-Richter diagrams and similar encounter structures. Such terms are negligible at short times, but at $t \sim t_E$ and beyond, their interference contributions dominate, leading to complete quantum saturation as required by unitary dynamics and random-matrix theory.

4. OTOCs in Paradigmatic and Integrable Models

OTOCs have been investigated across a range of integrable and nonintegrable models:

In the quantum baker’s map, explicit analytic results show exponential OTOC growth at the Lyapunov rate up to the Ehrenfest time, followed by saturation at a universal random-matrix plateau (Lakshminarayan, 2018).
For single-particle billiards and harmonic oscillators, OTOCs do not necessarily exhibit exponential growth even when the classical system is chaotic (Hashimoto et al., 2017). In these cases, OTOCs are periodic or saturate with $T$ -linear scaling and reflect the lack of many-body effects.
In the Aubry-André model, OTOCs in the extended phase equilibrate to zero at infinite time (operator spreading), but local saturation persists in the localized regime. The critical point is marked by anomalously large momentum-space OTOCs and slow scrambling (Riddell et al., 2019).
In the XY chain, OTOCs display polynomial early-time growth, power-law saturation, and operator-dependent spatiotemporal profiles. Even in integrable models, universal wavefront forms and butterfly velocities emerge (Bao et al., 2019).

5. Experimental and Numerical Measurement Protocols

Several protocols have been developed for measuring OTOCs:

In digital quantum simulators, finite-temperature OTOCs are accessed by preparing thermofield double states and using forward/backward Trotterized evolution to extract correlations as a function of time and temperature. Postselection on symmetry sectors can substantially reduce error rates (Green et al., 2021).
Ancilla-based deterministic quantum computation with one clean qubit (DQC1) circuits enable measurement of OTOCs at infinite temperature using only a single pure qubit and a maximally mixed register, providing exponential speedup over classical algorithms for efficiently simulable Hamiltonians (PG et al., 2020).
In artificial neural network approaches, restricted Boltzmann machines can variationally represent many-body wavefunctions and time evolution to accurately compute early-time OTOCs even in large, highly entangled two-dimensional systems (Wu et al., 2019).

6. OTOCs Beyond Closed, Unitary Dynamics: Open, Non-Hermitian, and Classical Limits

The phenomenology of OTOCs fundamentally changes in open, finite, or non-Hermitian systems:

In finite open quantum systems coupled to dissipative baths, OTOCs decay exponentially and saturate to a value determined by coupling strengths and relaxation rates. There is no exponential operator growth; rather, decoherence dominates, with distinct timescales for dephasing and inelastic processes (Syzranov et al., 2017).
In non-Hermitian PT-symmetric quantum systems, OTOCs become sensitive probes of both static and dynamical exceptional points; at the ground-state exceptional point, short-time OTOC amplitude and period diverge, while at the dynamical exceptional point, OTOCs grow exponentially at a rate set by the imaginary part of the leading excited-state eigenvalue (Zhai et al., 2019).
In classical, spatially extended deterministic or stochastic PDEs (Kuramoto-Sivashinsky or KPZ equations), the OTOC framework, reformulated in terms of linear response to infinitesimal perturbations, characterizes light-cone propagation of chaos, butterfly velocities, and Lyapunov spectra (Roy et al., 2023).
In systems with mixed regular and chaotic phase space, the late-time approach of the OTOC to equilibrium is governed by classical generalized Ruelle-Pollicott resonances, providing a direct link between quantum scrambling and classical mixing rates even in nonergodic regimes (Notenson et al., 2023).

7. Structural and Theoretical Aspects: Path Integrals, Quasiprobabilities, and Generalizations

The formal structure of OTOCs is illuminated by their representation on multidimensional timefolded (k-OTO) contours, generalizing the Schwinger-Keldysh path integral to arbitrary operator orderings. The "OTO number" classifies the degree of time-ordering violation, with chaos-diagnostic four-point OTOCs corresponding to proper 2-OTO correlators (Haehl et al., 2017).

Quasiprobability perspectives—specifically, the Kirkwood-Dirac distribution and its higher-moment generalizations—reveal that OTOCs are moments of complex-valued quasiprobabilities. Their negativity and complexity are intertwined with nonclassical operator growth and scrambling. Weak-measurement and interference protocols have been proposed to reconstruct these underlying quasiprobabilities experimentally, connecting OTOCs to broader notions of contextuality and diagnostic power in quantum information theory (Halpern et al., 2017).

Thermodynamic analogies further link OTOCs to fluctuation relations via "Jarzynski-like" equalities, in which the OTOC is expressed as a second derivative of the averaged exponential of a generalized (quasiprobabilistic) work variable. This connection offers new experimental pathways and highlights the deep structural relation between scrambling, energy fluctuations, and nonequilibrium quantum thermodynamics (Halpern, 2016).

In summary, OTOCs are now foundational observables for probing quantum chaos, dynamical phase transitions, localization phenomena, and quantum information spreading across a wide variety of quantum, classical, open, and non-Hermitian contexts. Their universal early-time exponential growth in strongly chaotic regimes, sensitivity to the structure of phase space and system symmetries, and dependence on genuinely quantum interference at long times underscore their central role in the modern theory of quantum dynamics.