The problem of characterizing complexity of quantum dynamics - in particular of locally interacting chains of quantum particles - will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested.
We extract the information of a quantum motion and decode it into a certain orbit via a single measurable quantity. Such that a quantum chaotic system can be reconstructed as a chaotic attractor. Two configurations for reconstructing this certain orbit are illustrated, which interpret quantum chaotic motions from the perspectives of probabilistic nature and the uncertainty principle, respectively. We further present a strategy to import classical chaos to a quantum system, revealing a connection between the classical and quantum worlds.
This article tackles a fundamental long-standing problem in quantum chaos, namely, whether quantum chaotic systems can exhibit sensitivity to initial conditions, in a form that directly generalizes the notion of classical chaos in phase space. We develop a linear response theory for complexity, and demonstrate that the complexity can exhibit exponential sensitivity in response to perturbations of initial conditions for chaotic systems. Two immediate significant results follows: i) the complexity linear response matrix gives rise to a spectrum that fully recovers the Lyapunov exponents in the c…
We investigate circuit complexity to characterize chaos in multiparticle quantum systems. In the process, we take a stride to analyze open quantum systems by using complexity. We propose a new diagnostic of quantum chaos from complexity based on the reduced density matrix by exploring different types of quantum circuits. Through explicit calculations on a toy model of two coupled harmonic oscillators, where one or both of the oscillators are inverted, we demonstrate that the evolution of complexity is a possible diagnostic of chaos.
Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wavefunctions localized along periodic orbits we reveal the existence of an oscillatory behavior, that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit.
We propose a new diagnostic for quantum chaos. We show that time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behaviour.
Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator.
Entropy production in quantum (field) systems requiring environment-induced decoherence is described in a Gaussian variational approximation. The new phenomenon of Semiquantum Chaos is reported. (Presented at the International Conference on Nonlinear Dynamics, Chaotic and Complex Systems, Zakopane (Poland), 7-12.11.95.)
We have identified ultra-cold atoms in magneto-optical double-well potentials as a very clean setting in which to study the quantum and classical dynamics of a nonlinear system with multiple degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from nonseparable couplings between the atomic spin and its center of mass motion. The main features of the chaotic dynamics are analyzed using action-angle variables and Poincare surfaces of section. We show that for the initial state prepared in current experiments [D. J. Haycock et al., Phys. Rev…
We have found stable chaotic solutions for optomechanical systems coupled with a Two-Level System or qubit. In this system methods have been found which can be used to Tune in and out of Chaos as well as various n-period motions. This includes achieving chaos by changing the detuning, coupling parameters, and Power of the driving laser. This allows us to manipulate chaos using either the qubit or the optical cavity. Chaotic motion was also observed in both the qubit and cavity by only changing the relative phase between of driving fields of the two. This gives us the prospect of creating and e…
We apply a notion of quantum complexity, called "Krylov complexity", to study the evolution of systems from integrability to chaos. For this purpose we investigate the integrable XXZ spin chain, enriched with an integrability breaking deformation that allows one to interpolate between integrable and chaotic behavior. K-complexity can act as a probe of the integrable or chaotic nature of the underlying system via its late-time saturation value that is suppressed in the integrable phase and increases as the system is driven to the chaotic phase. We furthermore ascribe the (under-)saturation of t…
Krylov complexity is a novel approach to study how an operator spreads over a specific basis. Recently, it has been stated that this quantity has a long-time saturation that depends on the amount of chaos in the system. Since this quantity not only depends on the Hamiltonian but also on the chosen operator, in this work we study the level of generality of this hypothesis by studying how the saturation value varies in the integrability to chaos transition when different operators are expanded. To do this, we work with an Ising chain with a transverse-longitudinal magnetic field and compare the …
This is the contribution to Quarks'2018 conference proceedings. This contribution is devoted to the holographic description of chaos and quantum complexity in the strongly interacting systems out of equilibrium. In the first part of the talk we present different holographic complexity proposals in out-of-equilibrium CFT following the local perturbation. The second part is devoted to the chaotic growth of the local operator size at a finite chemical potential. There are numerous results stating that the chemical potential may lead to the chaos disappearance, and we confirm these results from ho…
Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambli…
We discuss the behavior of fidelity for a classically chaotic quantum system in the metallic regime. We show the existence of a critical value of the perturbation below which the exponential decay of fidelity is determined by the width of the Breit-Wigner distribution and above which the quantum decay follows the classical one which is ruled by the Lyapunov exponent. The independence of the decay {\it rate} from the perturbation strength derives from the similarity of the quantum and classical relaxation process inside the Heisenberg time scale.
The problem of existence and constructing of integrals of motion in stationary quantum mechanics and its connection with quantum chaoticity is discussed. It is shown that the earlier suggested quantum chaoticity criterion characterizes destruction of initial symmetry of regular system and of basis quantum numbers under influence of perturbation. The convergent procedure allowing to construct approximate integrals of motion in the form of non-trivial combinations depending on operators $(q,p)$ is suggested. Properties of the obtained integrals with complicated structure and the consequences of …
We propose a measure of quantum state complexity defined by minimizing the spread of the wave-function over all choices of basis. Our measure is controlled by the "survival amplitude" for a state to remain unchanged, and can be efficiently computed in theories with discrete spectra. For continuous Hamiltonian evolution, it generalizes Krylov operator complexity to quantum states. We apply our methods to the harmonic and inverted oscillators, particles on group manifolds, the Schwarzian theory, the SYK model, and random matrix models. For time-evolved thermofield double states in chaotic system…
Achieving full control of the time-evolution of a many-body quantum system is currently a major goal in physics. In this work we investigate the different ways in which the controllability of a quantum system can be influenced by its complexity, or even its chaotic properties. By using optimal control theory, we are able to derive the control fields necessary to drive various physical processes in a spin chain. Then, we study the spectral properties of such fields and how they relate to different aspects of the system complexity. We find that the spectral bandwidth of the fields is, quite gene…
We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the "frame potential," which is minimized by unitary $k$-designs and measures the $2$-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order $2k$-point correlators is proportional to the $k$th frame potential, providing a quantitative connection between chaos and pseudor…
In this paper, we show how the restriction of the Quantum Geometric Tensor to manifolds of states that can be generated through local interactions provides a new tool to understand the consequences of locality in physics. After a review of a first result in this context, consisting in a geometric out-of-equilibrium extension of the quantum phase transitions, we argue the opportunity and the usefulness to exploit the Quantum Geometric Tensor to geometrize quantum chaos and complexity.
Unfolded Krylov complexity: universal chaotic dynamics without false positives
Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Shanghai Jiao Tong University（上海交通大学）；Pohang University of Science and Technology（韩国科学技术院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
We examine the multifold complexity and Loschmidt echo for an inverted harmonic oscillator. We give analytic expressions for any number of precursors, implementing multiple backward and forward time evolutions of the quantum state, at the leading order in the perturbation. We prove that complexity is dominated by the longest permutation of the given time combination in an alternating ``zig-zag'' order, the exact same result obtained with holography. We conjecture that the general structure for multifold complexity should hold true universally for generic quantum systems, in the limit of a larg…
In recent years, analysis and control of quantum chaos are increasingly important, but the lack of the concept of trajectory makes it impossible to analyze quantum chaos by the methods used in classical chaos. This research aims to connect Newton's world to the quantum world by the complex mechanics so that quantum chaos can be analyzed and controlled by the complex-extended Newtonian mechanics. Through the bridge of complex mechanics, in this article, we model quantum motions for 2D charged anisotropic harmonic oscillator by complex-valued dynamic equations, based on which quantum chaos can b…
We establish the emergence of chaotic motion in optomechanical systems. Chaos appears at negative detuning for experimentally accessible values of the pump power and other system parameters. We describe the sequence of period doubling bifurcations that leads to chaos, and state the experimentally observable signatures in the optical spectrum. In addition to the semi-classical dynamics we analyze the possibility of chaotic motion in the quantum regime. We find that quantum mechanics protects the optomechanical system against irregular dynamics, such that simple periodic orbits reappear and repl…
We compute the circuit complexity of scalar curvature perturbations on FLRW cosmological backgrounds with fixed equation of state $w$ using the language of squeezed vacuum states. Backgrounds that are accelerating and expanding, or decelerating and contracting, exhibit features consistent with chaotic behavior, including linearly growing complexity. Remarkably, we uncover a bound on the growth of complexity for both expanding and contracting backgrounds $λ\leq \sqrt{2} \ |H|$, similar to other bounds proposed independently in the literature. The bound is saturated for expanding backgrounds wit…
The chaotic dynamics in nuclear collective motion is studied in the framework of a schematic shell model which has only monopole and quadrupole degrees of freedom. The model is shown to reproduce the experimentally observed global trend toward less chaotic motion in heavier nuclei. The relation between current approach and the earlier studies with bosonic models is discussed.
University of the Witwatersrand（威特沃特斯兰德大学）；Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）；International Solvay Institutes（国际索尔维研究所）；Kyoto University（京都大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心（iTHEMS））
