Testing the upper bound on the speed of scrambling with an analogue of Hawking radiation using trapped ions

The Lyapunov exponent of a quantum system has been predicted to be bounded by λ _L≤ 2π T/ħ , where T is its temperature, as established by Maldacena, Shenker, and Stanford (MSS). This bound plays an important role in studying very diverse topics of physics, ranging from the dynamics of interacting many-body systems to the black hole information problem, and it is saturated when the system under consideration is the exact holographic dual of a black hole. Based on the fact that an inverted harmonic oscillator (IHO) may exhibit the behavior of thermal energy emission, in close analogy to the Hawking radiation emitted by black holes, we propose using a trapped ion as an implementation of the IHO to verify, in a concrete analogue-gravity system, whether the MSS bound can be identically saturated. To this end, we provide prescriptions for experimentally observing the scattering process at the IHO potential, which yields an analogue of Hawking radiation, as well as for how to measure the corresponding out-of-time-ordered correlation function (OTOC), diagnosing quantum chaos, in this thermally excited semiclassical system. We theoretically show, for an experimentally realizable analogue-gravity setup, that the effective Hawking temperature of the trapped-ion-IHO indeed matches the upper MSS bound for the speed of scrambling.