Seeing the Quantum Butterfly Effect

Xiao-Liang Qi

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Seeing the Quantum Butterfly Effect

Leinweber Institute for Theoretical Physics, Stanford University, Stanford, CA, US

February 9, 2026• Physics 19, 12

A combined experimental and theoretical study reveals the emergence of quantum chaos in a complex system, suggesting that it can be described with a universal theoretical framework.

Figure caption — **Figure 1:** Illustration of the experimental setup. A system of coupled nuclear spins evolves forward in time (blue horizontal arrow), undergoes a spin rotation by an angle $𝜑$ , and then evolves backward. Under perfect time reversal, the system would follow the blue dashed trajectory, deviating exponentially from the original state solely owing to intrinsic quantum chaos (the butterfly effect). In practice, experimental errors in the reversed dynamics lead to a deviated trajectory (red solid curve) and to a distorted final state (fuzzy butterfly).

Consider the following thought experiment: Take all the air molecules in a thunderstorm and evolve them backward in time for an hour, effectively rewinding a molecular movie. Then slightly perturb the velocity directions of a few molecules and evolve the system forward again to the current moment. Because such systems are chaotic, microscopic perturbations in the past will lead to dramatically different futures. This “butterfly effect” also occurs in quantum systems. To observe it, researchers measure a mathematical entity called the out-of-time-ordered correlator (OTOC). Loosely speaking, the OTOC measures how quickly a system “forgets” its initial state. Unfortunately, the OTOC is notoriously difficult to measure because it typically requires experimental protocols that implement an effective many-body time reversal. For example, a nuclear magnetic resonance (NMR) quantum simulator measured OTOCs by engineering forward and backward evolutions of the target dynamics using pulse-based control sequences [1], while a trapped-ion quantum magnet measured OTOCs using a protocol based on reversing the sign of the engineered nearest-neighbor interaction [2]. In both platforms, however, the time-reversal operation is intrinsically imperfect—limited by control errors and decoherence. Now Yu-Chen Li of the University of Science and Technology of China and his collaborators have surmounted this hurdle [3]. Harnessing a theoretical framework known as scramblon theory, they successfully separated genuine quantum chaos from the imperfections of time reversal in a macroscopic solid-state sample manipulated with NMR.

To grasp the OTOC, consider quantum operators U and V and apply them in two different sequences to the same system. First, evolve the system, apply U, reverse time, and apply V. Next, take the system from the previous starting point and first apply V, evolve the system, apply U, and reverse time. Last, determine the OTOC by comparing the system at the two end points. A large OTOC (meaning a large deviation between the two states prepared above) indicates that sensitivity to perturbations such as U and V makes the system susceptible to the quantum butterfly effect.

The difficulty in measuring the OTOC arises because small control errors or sources of environmental noise always arise when we try to reverse time. As a result, the realized time-reversed evolution inevitably deviates exponentially from the ideal inverse dynamics. These errors meddle with the signal of quantum chaos measured by the operators U and V that we intend to apply and make it nearly impossible to distinguish between intrinsic chaotic scrambling and extrinsic experimental artifacts.

Li and his collaborators realized they could overcome the difficulty by applying the techniques of solid-state NMR to a powder of adamantane—a colorless, crystalline solid with the formula C₁₀H₁₆. This system serves as an excellent test bed for quantum many-body physics because it contains a macroscopic number of hydrogen nuclei, which interact exclusively via simple dipolar forces. Using a technique called Floquet engineering, the researchers subjected the system to carefully timed sequences of radio pulses. Such pulses had the effect of synthesizing Hamiltonians operators that generated chaotic dynamics in a controllable way. Repeating the pulses backward had the effect of reversing time.

To measure quantum chaos, the researchers prepared the system in a slightly spin-polarized state and evolved it forward for a time t. They then slightly rotated the spin polarization by an angle $𝜑$ and evolved it backward to the initial time (Fig. 1). Lastly, they measured the total magnetization of the spins for the final state. This magnetization decays thanks to a combination of two effects. The first is the intrinsically chaotic dynamics, which causes the effect of the small rotation to become an increasingly nonlocal operator—that is, an operator that applies simultaneously to many nuclear spins. After the backward evolution, the initial state becomes difficult to recover. The second effect is the deviation of the backward evolution from the strict time reversal of the forward evolution, which is caused by imperfections in switching the Hamiltonian’s sign.

To disentangle these two effects and obtain an intrinsic measure of quantum chaos—the OTOC—the researchers turned to scramblon theory, a framework originally developed to model the rapid, chaotic dispersal, or scrambling, of quantum information [4–7]. Scramblon theory postulates that information scrambling is mediated by collective excitations called scramblons. Unlike sound waves and other ordinary collective modes, which only oscillate or damp, scramblon modes can grow exponentially in time. A generic quantum chaotic system is difficult to describe microscopically, but if a single-scramblon mode is responsible for the OTOC’s behavior, it leads to simple predictions that can be compared with experimental results. Generically, a system can have many scramblon modes, but, interestingly, the new experimental results seem to be consistent with the simplest hypothesis of a single scramblon, as I will discuss more below.

On the basis of scramblon theory, Li and his collaborators derived an ansatz, or educated guess, that embodied the interplay between intrinsic chaos and experimental error. Although the two effects are conflated in any single run, varying the rotation angle $𝜑$ allowed the researchers to disentangle them mathematically from the experimental data. From this analysis the researchers extracted two fundamental parameters. The first is the Lyapunov exponent $𝜘$ , which governs the rate of short-time exponential growth of chaos; its inverse is the timescale over which the butterfly effect becomes significant. The second parameter is a dimensionless factor $Δ$ . These parameters govern the long-time exponential decay through the product 2 $Δ 𝜘$ . Crucially, the ansatz predicts that the error-induced decay follows the same exponential rate as the chaotic growth and that the exponential rate is independent of the spin-polarization direction. This is a fingerprint of a single-scramblon mode.

When Li and his collaborators applied their ansatz to their adamantane data, they obtained a remarkably precise fit, satisfying strict mathematical constraints that previous models could not. The agreement validates scramblon theory for realistic, power-law interacting systems. That said, scramblon dynamics can be more generic. Future experiments on different systems could conceivably see different behavior.

This result is significant not only because it provides the first reliable measurement of the quantum Lyapunov exponent in a macroscopic many-body system, but it also verifies that the simplest effective description—involving a single scramblon—appears to hold. The finding suggests universality in quantum chaotic systems. Verifying this universality by conducting similar experiments in other physical systems will be valuable.

This work has also built new links between tabletop experiments and high-energy physics, as the concept of the scramblon arises from holographic duality models relating quantum chaos to gravitational dynamics [8]. Validating these predictions in crystals of adamantane suggests that universal laws of chaos may provide a common language linking NMR physics to black holes. Through their experiment and theory, Li and his collaborators have provided a new lens through which to view the chaotic quantum world, making the butterfly’s wings clearly visible even through the haze of experimental imperfection.

References

J. Li et al., “Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator,” Phys. Rev. X 7, 031011 (2017).
M. Gärttner et al., “Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet,” Nat. Phys. 13, 781 (2017).
Y.-C. Li et al., “Error-resilient reversal of quantum chaotic dynamics enabled by scramblons,” Phys. Rev. Lett. 136, 060403 (2026).
A. Kitaev and S. J. Suh, “The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,” J. High Energ. Phys. 2018, 183 (2018).
Y. Gu and A. Kitaev, “On the relation between the magnitude and exponent of OTOCs,” J. High Energ. Phys. 2019, 75 (2019).
D. Stanford et al., “Subleading Weingartens,” J. High Energ. Phys. 2022, 200 (2022).
Y. Gu et al., “A two-way approach to out-of-time-order correlators,” J. High Energ. Phys. 2022, 133 (2022).
S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” J. High Energ. Phys. 2014, 67 (2014).

About the Author

Xiao-Liang Qi is a professor of physics at Stanford University. He received his PhD in 2007 from Tsinghua University in China and did postdoctoral research at Stanford University and the University of California, Santa Barbara, before joining the Stanford faculty. His current research focuses on the relations between quantum entanglement, quantum chaos, and quantum gravity.