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Reviews in Mathematical Physics

Light cones for open quantum systems in the continuum

Sébastien Breteaux, Jérémy Faupin, Marius Lemm, Dong Hao Ou Yang, Israel Michael Sigal, Jingxuan Zhang

Published Feb 14, 2024• 10 citations
Publisher

Abstract

We consider Markovian open quantum dynamics (MOQD) in the continuum. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann–Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound for the slope of this light cone.

DOI
10.1142/s0129055x24600043

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We consider Markovian open quantum dynamics (MOQD) in the continuum. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann–Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound for the slope of this light cone.
We consider Markovian open quantum dynamics (MOQD). We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann-Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound for the slope of this light cone.
We prove a maximal velocity bound for the dynamics of Markovian open quantum systems. The dynamics is described by one-parameter semigroups of quantum channels satisfying the von Neumann–Lindblad equation. Our result says that dynamically evolving states are contained inside a suitable light cone up to polynomial errors. We also give a bound on the slope of the light cone, i.e. the maximal propagation speed. The result implies an upper bound on the speed of propagation of local perturbations of stationary states in open quantum systems.
Memory effects in many-body quantum systems interacting with an environment significantly impact their behavior. Nevertheless, much of our understanding of many-body open quantum systems is based on models within the Born-Markov approximation where these effects are neglected. In this paper, we consider a general class of spatially local open quantum lattice models with a non-Markovian environment that is approximated as Gaussian. Our main result is a set of Lieb-Robinson bounds for such models. For models where the environment has independent local baths, we show that the Lieb-Robinson velocity is controlled by one-norm of the memory kernel within the evolution time-interval, yielding a linear light cone whenever the memory kernel has a bounded one-norm. For models where excitations can propagate through the environment, we prove a polynomial light cone that captures bath-mediated acceleration of information spreading. In both of these cases, we show that the derived Lieb-Robinson bounds are tight by also constructing lattice models saturating these bounds. Finally, we consider the problem of approximating a non-Markovian model by a larger Markovian model, which contains the system spins together with only a finite number of bosonic environment modes. We establish that as a consequence of our Lieb-Robinson bounds, the number of environment modes {per system site} needed to accurately capture local observables is independent of the size of the system.
The speed limit of information propagation is one of the most fundamental features in non-equilibrium physics. The region of information propagation by finite-time dynamics is approximately restricted inside the effective light cone that is formulated by the Lieb-Robinson bound. To date, extensive studies have been conducted to identify the shape of effective light cones in most experimentally relevant many-body systems. However, the Lieb-Robinson bound in the interacting boson systems, one of the most ubiquitous quantum systems in nature, has remained a critical open problem for a long time. This study reveals a tight effective light cone to limit the information propagation in interacting bosons, where the shape of the effective light cone depends on the spatial dimension. To achieve it, we prove that the speed for bosons to clump together is finite, which in turn leads to the error guarantee of the boson number truncation at each site. Furthermore, we applied the method to provide a provably efficient algorithm for simulating the interacting boson systems. The results of this study settle the notoriously challenging problem and provide the foundation for elucidating the complexity of many-body boson systems.
Understanding the ultimate rate at which information propagates is a pivotal issue in nonequilibrium physics. Nevertheless, the task of elucidating the propagation speed inherent in quantum bosonic systems presents challenges due to the unbounded nature of their interactions. In this study, we tackle the problem of macroscopic particle transport in a long-range generalization of the lattice Bose-Hubbard model through the lens of the quantum speed limit. By developing a unified approach based on optimal transport theory, we rigorously prove that the minimum time required for macroscopic particle transport is always bounded by the distance between the source and target regions, while retaining its significance even in the thermodynamic limit. Furthermore, we derive an upper bound for the probability of observing a specific number of bosons inside the target region, thereby providing additional insights into the dynamics of particle transport. Our results hold true for arbitrary initial states under both long-range hopping and long-range interactions, thus resolving an open problem of particle transport in generic bosonic systems.
In locally interacting quantum many-body systems, the velocity of information propagation is finitely bounded, and a linear light cone can be defined.Outside the light cone, the amount of information rapidly decays with distance.When systems have long-range interactions, it is highly nontrivial whether such a linear light cone exists.Herein, we consider generic long-range interacting systems with decaying interactions, such as R -α with distance R. We prove the existence of the linear light cone for α > 2D þ 1 (D, the spatial dimension), where we obtain the Lieb-Robinson bound as k½O i ðtÞ; O j k ≲ t 2Dþ1 ðR -vtÞ -α with v ¼ Oð1Þ for two arbitrary operators O i and O j separated by a distance R.Moreover, we provide an explicit quantum-state transfer protocol that achieves the above bound up to a constant coefficient and violates the linear light cone for α < 2D þ 1.In the regime of α > 2D þ 1, our result characterizes the best general constraints on the information spreading.