Locality of dynamics in general harmonic quantum systems
M. Cramer, A. Serafini, J. Eisert
备注：
19 pages latex, contribution to proceedings of "Quantum Information and Many-Body Quantum Systems" Workshop, 26-31 Mar 2007, Pisa. (v2) minor changes
发表信息：
Quantum information and many body quantum systems, Eds. M. Ericsson, S. Montangero, Pisa: Edizioni della Normale, pp 51-72, 2008 (Publications of the Scuola Normale Superiore. CRM Series, 8)
The Lieb-Robinson theorem states that locality is approximately preserved in the dynamics of quantum lattice systems. Whenever one has finite-dimensional constituents, observables evolving in time under a local Hamiltonian will essentially grow linearly in their support, up to exponentially suppressed corrections. In this work, we formulate Lieb-Robinson bounds for general harmonic systems on general lattices, for which the constituents are infinite-dimensional, as systems representing discrete versions of free fields or the harmonic approximation to the Bose-Hubbard model. We consider both local interactions as well as infinite-ranged interactions, showing how corrections to locality are inherited from the locality of the Hamiltonian: Local interactions result in stronger than exponentially suppressed corrections, while non-local algebraic interactions result in algebraic suppression. We derive bounds for canonical operators, Weyl operators and outline generalization to arbitrary operators. As an example, we discuss the Klein-Gordon field, and see how the approximate locality in the lattice model becomes the exact causality in the field limit. We discuss the applicability of these results to quenched lattice systems far from equilibrium, and the dynamics of quantum phase transitions.
Motivated by the development of on-going optomechanical experiments aimed at constraining non-local effects inspired by some quantum gravity scenarios, the Hamiltonian formulation of a non-local harmonic oscillator, and its coupling to a cavity field mode(s), is investigated. In particular, we consider the previously studied model of non-local oscillators obtained as the non-relativistic limit of a class of non-local Klein-Gordon operators, $f(\Box)$, with $f$ an analytical function. The results of previous works, in which the interaction was not included, are recovered and extended by way of …
We perform a Bohmian-trajectory analysis of high-order harmonic generation (HHG), focusing on the fact that typical HHG spectra are best reproduced by the Bohmian trajectory starting at the innermost part of the core [Phys. Rev. A \textbf{88}, 023415 (2013)]. Using ensemble averages around this central trajectory, we show that, for the high-plateau and cutoff harmonics, small ensembles of Bohmian trajectories are sufficient for a quantitative agreement with the numerical solution of the time-dependent Schrödinger equation (TDSE), while larger ensembles are necessary in the low-plateau region. …
Quantum non-locality based on the well-known Bell inequality is of kinematic nature. A different type of quantum non-locality, the non-locality of the quantum equation of motion, is recently put forward with connection to the Aharonov-Bohm effect [Nature Phys. 6, 151 (2010)]. Evolution of the displacement operator provides an example to manifest such dynamic quantum non-locality. We propose an experiment using single-atom interferometry to test such dynamic quantum non-locality. We show how to measure evolution of the displacement operator with clod atoms in a spin-dependent optical lattice po…
The Lieb-Robinson bound shows the existence of a maximum speed of signal propagation in discrete quantum mechanical systems with local interactions. This generalizes the concept of relativistic causality beyond field theory, and provides a powerful tool in theoretical condensed matter physics and quantum information science. Here, we extend the scope of this seminal result by considering general Markovian quantum evolution, where we prove that an equivalent bound holds. In addition, we use the generalized bound to demonstrate that correlations in the stationary state of a Markov process decay …
The nonrelativistic limit of nonlocal modifications to the Klein Gordon operator is studied, and the experimental possibilities of casting stringent constraints on the nonlocality scale via planned and/or current optomechanical experiments are discussed. Details of the perturbative analysis and semianalitical simulations leading to the dynamical evolution of a quantum harmonic oscillator in the presence of non locality reported in [1], together with a comprehensive account of the experimental methodology with particular regard to sensitivity limitations related to thermal decoherence time and …
We consider open many-body systems governed by a time-dependent quantum master equation with short-range interactions. With a generalized Lieb-Robinson bound, we show that the evolution in this very generic framework is quasi-local, i.e., the evolution of observables can be approximated by implementing the dynamics only in a vicinity of the observables' support. The precision increases exponentially with the diameter of the considered subsystem. Hence, the time-evolution can be simulated on classical computers with a cost that is independent of the system size. Providing error bounds for Trott…
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The quantum theoretical concepts of modular momentum and dynamical non-locality, which were introduced four decades ago, have recently been used to explain single particle quantum interference phenomena. Although the non-local exchange of modular momentum associated with such phenomena cannot be directly observed, it has been suggested that effects induced by this exchange can be measured experimentally using weak measurements of pre- and post-selected ensembles of particles. This paper reports on such an optical experiment that yielded measured weak values that were consistent with the theore…
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We review some recent results that express or rely on the locality properties of the dynamics of quantum spin systems. In particular, we present a slightly sharper version of the recently obtained Lieb-Robinson bound on the group velocity for such systems on a large class of metric graphs. Using this bound we provide expressions of the quasi-locality of the dynamics in various forms, present a proof of the Exponential Clustering Theorem, and discuss a multi-dimensional Lieb-Schultz-Mattis Theorem.
Structured Non-Locality and Emergent Locality in Cavity-QED Many-Body Dynamics
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学技术研究院）；ICREA-Institució Catalana de Recerca i Estudis Avançats（ICREA-加泰罗尼亚高级研究学院）
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Quantum mechanics predicts correlation between spacelike separated events which is widely argued to violate the principle of Local Causality. By contrast, here we shall show that the Schrödinger equation with Born's statistical interpretation of wave function and uncertainty relation can be derived from a statistical model of microscopic stochastic deviation from classical mechanics which is selected uniquely, up to a free parameter, by the principle of Local Causality. Quantization is thus argued to be physical and Planck constant acquires an interpretation as the average stochastic deviation…
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A simple classical non-local dynamical system with random initial conditions and an output projecting the state variable on selected axes has been defined to mimic a two-channel quantum coincidence experiment. Non-locality is introduced by a parameter connecting the initial conditions to the selection of the projection axes. The statistics of the results shows violations up to 100% of the Bell's inequality, in the form of Clauser-Horne- Shimony-Holt (CHSH), strongly depending on the non-locality parameter. Discussions on the parallelism with Bohmian mechanics are given.
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Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔理工学院、信息学实验室）；University of Warwick（华威大学）；University of Copenhagen（哥本哈根大学）
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