Abstract
Classical chaotic systems exhibit exponentially diverging trajectories due to small differences in their initial state. The analogous diagnostic in quantum many-body systems is an exponential growth of out-of-time-ordered correlation functions (OTOCs). These quantities can be computed for various models, but their experimental study requires the ability to evolve quantum states backward in time, similar to the canonical Loschmidt echo measurement. In some simple systems, backward time evolution can be achieved by reversing the sign of the Hamiltonian; however in most interacting many-body systems, this is not a viable option. Here we propose a new family of protocols for OTOC measurement that do not require backward time evolution. Instead, they rely on ordinary time-ordered measurements performed in the thermofield double (TFD) state, an entangled state formed between two identical copies of the system. We show that, remarkably, in this situation the Lyapunov chaos exponent $\lambda_L$ can be extracted from the measurement of an ordinary two-point correlation function. As an unexpected bonus, we find that our proposed method yields the so-called "regularized" OTOC -- a quantity that is believed to most directly indicate quantum chaos. According to recent theoretical work, the TFD state can be prepared as the ground state of two weakly coupled identical systems and is therefore amenable to experimental study. We illustrate the utility of these protocols on the example of the maximally chaotic Sachdev-Ye-Kitaev model and support our findings by extensive numerical simulations.
Highlights
A key characteristic of chaotic quantum many-body systems is rapid dispersal of quantum information deposited among a small number of elementary degrees of freedom
We introduced and extensively tested the concept of entanglement in the thermofield double state as a tool to measure out-of-time ordered correlators in quantum many-body systems
While previous work has implemented ordered correlation functions (OTOCs) measurements in small-scale and highly controllable quantum systems [10,11,12,13], these approaches do not lend themselves to the analysis of large, complex many-body systems that realize quantum chaos in solid-state platforms
Summary
A key characteristic of chaotic quantum many-body systems is rapid dispersal of quantum information deposited among a small number of elementary degrees of freedom. We address this pressing challenge by introducing an approach to diagnosing chaos in quantum many-body systems which does not require backward time evolution during the measurement. We first review the concept of the TFD state and demonstrate how it can be used to diagnose quantum chaotic behavior via an OTOC measurement that does not require explicit backward time evolution of quantum states (Sec. II). We discuss in detail how the TFD state can be prepared and used to extract the chaos exponent λL from an equilibrium measurement of a two-point correlation function We apply these general ideas to a pair of coupled SYK Hamiltonians, recently argued to be holographically dual to a traversable wormhole, and known to admit a TFD ground state (Sec. III).
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