Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet

Abstract

Controllable arrays of ions and ultra-cold atoms can simulate complex many-body phenomena and may provide insights into unsolved problems in modern science. To this end, experimentally feasible protocols for quantifying the buildup of quantum correlations and coherence are needed, as performing full state tomography does not scale favorably with the number of particles. Here we develop and experimentally demonstrate such a protocol, which uses time reversal of the manybody dynamics to measure out-of-time-order correlation functions (OTOCs) in a long-range Ising spin quantum simulator with more than 100 ions in a Penning trap. By measuring a family of OTOCs as a function of a tunable parameter we obtain fine-grained information about the state of the system encoded in the multiple quantum coherence spectrum, extract the quantum state purity, and demonstrate the buildup of up to 8-body correlations. Future applications of this protocol could enable studies of many-body localization, quantum phase transitions, and tests of the holographic duality between quantum and gravitational systems.Time-reversal has fascinated and puzzled physicists for centuries. In an iconic example, Josef Loschmidt argued that the second law of thermodynamics would be violated by time-reversing an entropy-increasing collision [1]. Ludwig Boltzmann responded by formulating the probabilistic definition of entropy, one of the cornerstones of statistical mechanics, and, now a fundamental concept in quantum information. Since the days of Boltzmann and Loschmidt, the notion of time-reversal has moved from the arena of thought experiments into the laboratory, with time-reversal of non-interacting quantum systems in the form of Hahn spin echoes [2] forming an essential part of nuclear magnetic resonance (NMR) [3] and magnetic resonance imaging.Recently, the experimental implementation of manybody time-reversal protocols [4,5] in atomic quantum systems have attracted attention [6][7][8][9] for their potential to quantify the flow of quantum information in time and set bounds on thermalization times [10][11][12][13], which might also enable experimental tests of the holographic duality between quantum and gravitational systems [6,[14][15][16][17]. The key quantities sought after are special types of outof-time-order correlation (OTOC) functions,whereŴ (τ ) = e iĤτŴ e −iĤτ , withĤ an interacting many-body Hamiltonian andŴ andV two commuting unitary operators. Physically, F (τ ) measures the "scrambling" of quantum information across the system's manybody degrees of freedom, for example, how fast an initial * These authors contributed equally.† john.bollinger@nist.gov ‡ arey@jila.colorado.edu local perturbation becomes inaccessible to local probesencapsulates the degree by which the initially commuting operatorsŴ andV fail to commute at later times due to the interactions generated byĤ, which we adopt as an operational definition of scrambling.Most theoretical studies of scrambling have focused on so-called fast scramblers in thermal states [10,11,16]...

Holographic butterfly effect at quantum critical points

“…Recently, new protocols and methods, that are versatile to simulate diverse many-body systems and achievable with state-of-the-art technology, have been proposed to measure the OTOC [14,32]. Furthermore, experimental measurements of the OTOC have also been implemented [33,34]. All these progresses provide test beds for our proposal.…”

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

“…One can also investigate quantum quenches as a possible probe of SYK physics by disconnecting or reconnecting the dot and wires to effectively freeze the zero modes or restore their coupling. Finally, much work has been done regarding measuring out-of-time-order correlators in cold atoms and qubit systems (see, e.g., [60][61][62][63]). Our setup offers the exciting prospect of exploiting Majorana hardware and topological quantum information ideas to measure such quantities in pursuit of the SYK model's hallmark maximal chaos.…”

Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator

“…For instance, small numbers of qubits stored in trapped atomic ions [2,3] and superconducting circuits [4] have been used to simulate various magnetic spin or Hubbard models with individual qubit state preparation and measurement. Large numbers of atoms have simulated similar models, but with global control and measurements [5] or with correlations that only appear over a few atom sites [6]. An outstanding challenge is to increase qubit number while maintaining individual qubit control and measurement, with the goal of performing simulations or algorithms that cannot be efficiently solved classically.…”