Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator (1609.01246v3)

Abstract: The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, but also diagnoses the chaotic behavior of many-body quantum systems and characterizes the information scrambling. Based on the OTOCs, three different concepts -- quantum chaos, holographic duality, and information scrambling -- are found to be intimately related to each other. Despite of its theoretical importance, the experimental measurement of the OTOC is quite challenging and so far there is no experimental measurement of the OTOC for local operators. Here we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and non-integrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for non-intgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.

• The paper demonstrates an experimental measurement of out-of-time-order correlators on a one-dimensional Ising spin chain using an NMR quantum simulator.
• The paper reveals that integrable systems show revival phenomena while non-integrable systems exhibit persistent scrambling linked to entanglement entropy dynamics.
• The paper extracts the butterfly velocity to quantify the speed of information propagation, aligning with theoretical predictions on Lieb-Robinson bounds.

An Analysis of Out-of-Time-Order Correlators in Many-Body Quantum Systems Using NMR Quantum Simulation

The concept of out-of-time-order correlators (OTOCs) has recently been pivotal in advancing our understanding of quantum chaos, holographic duality, and information scrambling in many-body systems. This paper reports on an experimental approach using a nuclear magnetic resonance (NMR) quantum simulator to measure OTOCs in a controlled setting, specifically focusing on a one-dimensional Ising spin chain. This research bridges the gap between theoretical predictions and experimental realizations, offering a methodological avenue to investigate these complex quantum phenomena.

Experimental Setup and Methodology

The paper focuses on measuring OTOCs for an Ising spin chain, characterized by the Hamiltonian $$H = -\sum_{i} \hat\sigma^z_{i} \hat\sigma^z_{i+1} + g\hat\sigma^x_i + h\hat\sigma^z_i$$, where integrability and non-integrability of the system can be tuned through parameters $g$ and $h$. The experimental apparatus employs an NMR quantum simulator, leveraging the spin dynamics in the ^{13}C and ^{19}F nuclear spins of iodotrifluoroethylene molecules. Control over spin interactions is achieved through the careful design of RF pulse sequences, allowing for precise simulation of the quantum dynamics prescribed by the Ising model.

Key Findings

1. OTOC Measurement and Dynamics: The measurement of OTOCs reveals distinct behavior in integrable versus non-integrable systems. For integrable cases, the system exhibits revival phenomena where OTOCs return to their initial values at long times, reinforcing the presence of well-defined quasi-particles and conserved quantities. Conversely, non-integrable cases demonstrate persistent scrambling, characterized by sustained deviation from initial OTOC values, signifying chaotic behavior.
2. Entanglement Entropy: The paper employs OTOC measurements to reconstruct the 2nd Rényi entropy growth after a quench. In integrable systems, entropy oscillates, while in chaotic systems, it saturates, highlighting different entanglement dynamics underpinning these two regimes. These observations are consistent with theoretical predictions correlating OTOCs with entanglement entropy growth.
3. Butterfly Velocity: By analyzing the spatial spread of local perturbations, the research extracts the butterfly velocity, which quantifies the speed of information propagation across the system. This provides insight into the Lieb-Robinson bounds and the causal structure of quantum interactions in the chaotic regime.

Implications and Future Directions

This work's experimental approach and results have significant implications for quantum simulations, offering a viable technique for studying elusive quantum phenomena beyond traditional thermodynamic measures. The methodology can be extended to other quantum systems with full local control, like superconducting circuits and trapped ions, thereby enriching our experimental toolkit for probing quantum chaos and holography.

Moreover, the findings have theoretical implications for understanding the boundary conditions of chaos characterized by Lyapunov exponents and their potential saturation in systems with holographic duals. Future research could capitalize on this work by increasing system size and exploring finite temperature effects, which are crucial for studying systems conjectured to have holographic duality.

Within the current quantum technology landscape, this paper represents a critical step toward realizing practical systems capable of illustrating theoretical constructs of quantum chaos and complexity, thus paving the way for deeper insights into the foundational aspects of quantum mechanics and gravitational duality.