Controllable arrays of ions and ultracold 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 favourably with the number of particles. Here we develop and experimentally demonstrate such a protocol, which uses time reversal of the many-body 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. Characterizing the correlations of quantum many-body systems is known to be hard, but there are ways around: for example, a new method for measuring out-of-time correlations demonstrated in a Penning trap quantum simulator with over 100 ions.

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

Microwave spectroscopy was used to probe the superfluid–Mott insulator transition of a Bose-Einstein condensate in a three-dimensional optical lattice. By using density-dependent transition frequency shifts, we were able to spectroscopically distinguish sites with different occupation numbers and to directly image sites with occupation numbers from one to five, revealing the shell structure of the Mott insulator phase. We used this spectroscopy to determine the onsite interaction and lifetime for individual shells.

Imaging the Mott insulator shells using atomic clock shifts

In this review we summarize and amplify recent investigations of coupled quantum dynamical systems with few degrees of freedom in the short-wavelength, semiclassical limit. Focusing on the correspondence between quantum and classical physics, we mathematically formulate and attempt to answer three fundamental questions. (i) How can one drive a small dynamical quantum system to behave classically? (ii) What determines the rate at which two single-particle quantum-mechanical subsystems become entangled when they interact? (iii) How does irreversibility occur in quantum systems with few degrees of freedom? These three questions are posed in the context of the quantum-classical correspondence for dynamical systems with few degrees of freedom, and we accordingly rely on two short-wavelength approximations to quantum mechanics to answer them: the trajectory-based semiclassical approach on the one hand, and random matrix theory on the other hand. We construct novel investigative procedures towards decoherence an...

Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom

Out-of-time-order correlations (OTOCs) characterize the scrambling, or delocalization, of quantum information over all the degrees of freedom of a system and thus have been proposed as a proxy for chaos in quantum systems. Recent experimental progress in measuring OTOCs calls for a more thorough understanding of how these quantities characterize complex quantum systems, most importantly in terms of the buildup of entanglement. Although a connection between OTOCs and entanglement entropy has been derived, the latter only quantifies entanglement in pure systems and is hard to access experimentally. In this work, we formally demonstrate that the multiple-quantum coherence spectra, a specific family of OTOCs well known in NMR, can be used as an entanglement witness and as a direct probe of multiparticle entanglement. Our results open a path to experimentally testing the fascinating idea that entanglement is the underlying glue that links thermodynamics, statistical mechanics, and quantum gravity.

Relating Out-of-Time-Order Correlations to Entanglement via Multiple-Quantum Coherences.

Small perturbations to systems near critical points of quantum phase transitions can induce drastic changes in the system properties. Here I show that this sensitivity can be exploited for weak-signal detection applications. This is done by relating a widely studied signature of quantum chaos and quantum phase transitions known as the Loschmidt echo to the minimum error probability for a quantum detector and noting that the echo, and therefore the error, can be significantly reduced near a critical point. Three examples, namely, the quantum Ising model, the optical parametric oscillator model, and the Dicke model, are presented to illustrate the concept. For the latter two examples, the detectable perturbation can exhibit a Heisenberg scaling with respect to the number of detectors, even though the detectors are not entangled and no special quantum state preparation is specified.

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Quantum transition-edge detectors

Time reversal is an oft-used technique to probe and measure a quantum system. In particular, it is key for some theoretical information approaches to quantum criticality like fidelity decay. Here proposed is how to reverse the dynamics of cold bosonic atoms in an optical lattice in the regime described by the Bose-Hubbard model. The proposed experiment uses proven techniques—a linear phase imprint on the lattice, and a change in magnetic field to tune the boson-boson scattering length through a Feshbach resonance. The sensitivity of quantum dynamics across the insulator-superfluid quantum phase transition of the Bose-Hubbard model is studied, and sensing applications of dynamic reversals such as measuring the intensity of external potentials (e.g., gravity) is discussed.

Fernando M. Cucchietti

Papers

Time reversal in an optical lattice