Absence of Chaos in Bohmian Dynamics

Origin of chaos near three-dimensional quantum vortices: A general Bohmian theory

We provide a general theory for the structure of the quantum flow near 3-d nodal lines, i.e. one-dimensional loci where the 3-d wavefunction becomes equal to zero. In suitably defined co- ordinates (co-moving with the nodal line) the generic structure of the flow implies the formation of 3-d quantum vortices. We show that such vortices are accompanied by nearby invariant lines of the co-moving quantum flow, called X-lines, which are normally hyperbolic. Furthermore, the stable and unstable manifolds of the X-lines produce chaotic scatterings of nearby quantum (Bohmian) trajectories, thus induc…

Chaos in 2-d Bohmian Trajectories

We make a short review of the most general mechanism for the generation of chaos in 2-d Bohmian trajectories, the so called `nodal point-X-point complex' (NPXPC) mechanism. The presentation is based on numerical calculations made with Maple and is enriched with new results on the details of the generation of chaos, and the form of the potential around the NPXPC.

The simplest non-trivial model of chaotic causal dynamics

The simplest non-trivial model of chaotic Bohmian dynamics is identified. We argue that its most important features can be observed in more complex models, above all, the presumable mechanism of the appearance of chaos in the Bohmian-type dynamical systems.

Wave Packet under Continuous Measurement via Bohmian Mechanics

A new quantum mechanical description of the dynamics of wave packet under continuous measurement is formulated via Bohmian mechanics. The solution to this equation is found through a wave packet approach which establishes a direct correlation between a classical variable with a quantum variable describing the dynamics of the center of mass and the width of the wave packet. The approach presented in this paper gives a comparatively clearer picture than approaches using restrited path integrals and master equation approaches. This work shows how the extremely irregular character of classical cha…

Chaotic trajectories in complex Bohmian systems

We consider the Bohmian trajectories in a 2-d quantum harmonic oscillator with non commensurable frequencies whose wavefunction is of the form $Ψ=aΨ_{m_1,n_1}(x,y)+bΨ_{m_2,n_2}(x,y)+cΨ_{m_3,n_3}(x,y)$. We first find the trajectories of the nodal points for different combinations of the quantum numbers $m,n$. Then we study, in detail, a case with relatively large quantum numbers and two equal $m's$. We find %We find first the nodal points where $Ψ=0$. The nodes can be found analytically only if $m$ and $n$ are small. If two $m's$ (or two $n's$ are equal we can find explicitly the nodal points ,…

Motion of vortices implies chaos in Bohmian mechanics

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.

两量子比特系统中的玻姆混沌与纠缠

Bohmian Chaos and Entanglement in a Two-Qubit System

本文研究两纠缠量子比特系统中玻姆流临界点在混沌发生中的作用，发现增加纠缠程度可缩短混沌轨迹的有限时间LCN收敛至最终正值的时间。

Chaos and ergodicity in an entangled two-qubit Bohmian system

We study in detail the onset of chaos and the probability measures formed by individual Bohmian trajectories in entangled states of two-qubit systems for various degrees of entanglement. The qubit systems consist of coherent states of 1-d harmonic oscillators with irrational frequencies. In weakly entangled states chaos is manifested through the sudden jumps of the Bohmian trajectories between successive Lissajous-like figures. These jumps are succesfully interpreted by the `nodal point-X-point complex' mechanism. In strongly entangled states, the chaotic form of the Bohmian trajectories is ma…

Nodal points and the transition from ordered to chaotic Bohmian trajectories

We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are: a) A proof of the existence of bounded domains in configuration space which are devoid of nodal points, b) An analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability. c) A detail…

Bohmian mechanics and consistent histories

The interpretations of a particular quantum gedanken experiment provided by Bohmian mechanics and consistent histories are shown to contradict each other, both in the absence and in the presence of a measuring device. The consistent history result seems closer to standard quantum mechanics, and shows no evidence of the mysterious nonlocal influences present in the Bohmian description.

经典量子混沌与玻姆量子混沌的比较

A comparison between classical and Bohmian quantum chaos

本研究针对含两个相互作用谐振子的二维系统，比较不同相互作用强度下经典轨迹与玻姆量子轨迹的混沌特性，发现玻姆轨迹长期均会出现混沌，且混沌出现时间与振子相互作用强度相关。

Chaotic Bohmian trajectories for stationary states

In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the possibility of chaos in the case of stationary nodes. We address this question by considering stationary states, which provide the simplest examples of wave functions with stationary nodes. We provide examples of stationary wave functions for which there is chaos, as demonstrated by numerical computations, for one particle moving in 3 spatial dimensions and for two and three entangled particles in two …

双稳势井中的玻姆轨迹

Bohmian Trajectories in a Bistable Potential Well

研究双稳势井中量子粒子的动力学，揭示玻姆轨迹在单维系统中可出现周期性、准周期性和混沌运动，推翻文献中关于混沌行为不可能的论断。

Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can't contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is ``random numbers'', as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-info…

Some clarifications on the relation between Bohmian quantum potential and Mach's principle

Mach's principle asserts that the inertial mass of a body is related to the distribution of other distant bodies. This means that in the absence of other bodies, a single body has no mass. In this case, talking about motion is not possible, because the detection of motion is possible only relative to other bodies. But in physics we are faced with situations that are not fully Machian. As in the case of general theory of relativity where geodesics exist in the absence of any matter, the motion has meaning. Another example which is the main topic of our discussion, refers to Bohmian quantum mech…

可积与不可积系统中的经典轨迹与玻姆轨迹

Classical and Bohmian trajectories in integrable and non integrable systems

本文通过比较可积与不可积Hénon-Heiles系统的经典轨迹和玻姆轨迹，发现玻姆轨迹在两种情形下均呈现混沌，但混沌出现时间不同。

Bohmian trajectories in an entangled two-qubit system

In this paper we examine the evolution of Bohmian trajectories in the presence of quantum entanglement. We study a simple two-qubit system composed of two coherent states and investigate the impact of quantum entanglement on chaotic and ordered trajectories via both numerical and analytical calculations.

玻姆轨迹系综统计中的有限样本偏差与收敛性

Finite-sample deviations and convergence in the statistics of Bohmian trajectory ensembles

本文研究玻姆轨迹的有限样本统计，发现规则流中样本统计与玻恩规则一致，而简并叠加和混沌动力学可导致明显偏差，为数值玻姆模拟提供指导。

Can Bohmian trajectories account for quantum recurrences having classical periodicities?

Quantum systems in specific regimes display recurrences at the period of the periodic orbits of the corresponding classical system. We investigate the excited hydrogen atom in a magnetic field -- a prototypical system of 'quantum chaos' -- from the point of view of the de Broglie Bohm (BB) interpretation of quantum mechanics. The trajectories predicted by BB theory are computed and contrasted with the time evolution of the wavefunction, which shows pronounced features at times matching the period of closed orbits of the classical hydrogen in a magnetic field problem. Individual BB trajectories…

Justifying Born's rule $P_α=|Ψ_α|^2$ using deterministic chaos, decoherence, and the de Broglie-Bohm quantum theory

In this work we derive Born's rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to a environement made of "qubits" (i.e., Bohmian pointers) we show that entanglement together with deterministic chaos lead to a fast relaxation from any statistitical distribution $ρ(x)$ (of finding a particle at point $x$) to the Born probability law $|Ψ(x)|^2$. Our model is discussed in the context of Boltzmann's kinetic theory and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.

到达时间与探测时间的比较

Arrival Times Versus Detection Times

针对量子探测时间概率分布的争议问题，本文指出玻姆力学中无探测器的到达时间与有探测器的到达时间、探测时间未必一致，并纠正了Das和Dürr相关研究中基于三者重合假设的分布结论偏差。

A dynamical systems approach to Bohmian trajectories

Vortices are known to play a key role in the dynamics of the quantum trajectories defined within the framework of the de Broglie-Bohm formalism of quantum mechanics. It has been rigourously proved that the motion of a vortex in the associated velocity field can induce chaos in these trajectories, and numerical studies have explored the rich variety of behaviors that due to their influence can be observed. In this paper, we go one step further and show how the theory of dynamical systems can be used to construct a general and systematic classification of such dynamical behaviors. This should co…

Are Bohmian trajectories real? On the dynamical mismatch between de Broglie-Bohm and classical dynamics in semiclassical systems

The de Broglie-Bohm interpretation of quantum mechanics aims to give a realist description of quantum phenomena in terms of the motion of point-like particles following well-defined trajectories. This work is concerned by the de Broglie-Bohm account of the properties of semiclassical systems. Semiclassical systems are quantum systems that display the manifestation of classical trajectories: the wavefunction and the observable properties of such systems depend on the trajectories of the classical counterpart of the quantum system. For example the quantum properties have a regular or disordered …

Chaos in de Broglie - Bohm quantum mechanics and the dynamics of quantum relaxation

We discuss the main mechanisms generating chaotic behavior of the quantum trajectories in the de Broglie - Bohm picture of quantum mechanics, in systems of two and three degrees of freedom. In the 2D case, chaos is generated via multiple scatterings of the trajectories with one or more `nodal point - X-point complexes'. In the 3D case, these complexes form foliations along `nodal lines' accompanied by `X-lines'. We also identify cases of integrable or partially integrable quantum trajectories. The role of chaos is important in interpreting the dynamical origin of the `quantum relaxation' effec…

玻姆力学作为一种实用工具

Bohmian Mechanics as a Practical Tool

本章展示玻姆力学可作为计算工具用于预测现象学信息，通过条件波函数构建量子框架（如高频电子输运模型），并从玻姆轨迹数值性质导出可观测量算符，在弱值实验中操作性确定，从而实用地表征量子系统。

导波理论中轨迹的（不可）探测性

The (un)detectability of trajectories in pilot-wave theory

研究导波理论中粒子轨迹的探测性问题，针对‘弱速度测量’和‘超现实’轨迹实验这两个质疑进行分析，指出它们均不能构成支持或反对导波理论的证据，谜题源于对量子力学测量意义缺乏连贯解释。

玻姆力学与量子引力有效作用量的推广

Generalization of Bohmian Mechanics and Quantum Gravity Effective Action

本文利用有效作用量将玻姆力学推广至量子引力，以PFQG构建有效作用量并近似为QFT形式，从而获得场位形与粒子轨迹。

Dynamical chaos versus quantum interference

We discuss the dephasing induced by the internal classical chaotic motion in the absence of any external environment. To this end a new extension of fidelity for mixed states is introduced, which we name {\it allegiance}. Such quantity directly accounts for quantum interference and is measurable in a Ramsey interferometry experiment. We show that in the semiclassical limit the decay of the allegiance is exactly expressed, due to the dephasing, in terms of an appropriate classical correlation function. Our results are derived analytically for the case of a nonlinear driven oscillator and then n…

Quantum mechanics without statistical postulates

The Bohmian formulation of quantum mechanics is used in order to describe the measurement process in an intuitive way without a reduction postulate in the framework of a deterministic single system theory. Thereby the motion of the hidden classical particle is chaotic during almost all nontrivial measurement processes. For the correct reproduction of experimental results, it is further essential that the distribution function $P(x)$ of the results of a position measurement is identical with $|Ψ|^2$ of the wavefunction $Ψ$ of the single system under consideration. It is shown that this feature …

利用退相干历史的熵探测量子混沌

Probing quantum chaos with the entropy of decoherent histories

该研究针对量子混沌缺乏普适判据的问题，以退相干历史为量子轨迹对应物提出新的量子混沌定义，通过开放量子受击陀螺模型验证了退相干历史熵的产生在可积与混沌区存在显著差异，可作为量子混沌的特征标识。

玻色子采样作为混沌和可积量子动力学的探针

Boson Sampling as a Probe of Chaotic and Integrable Quantum Dynamics in a Photonic Chip

利用多光子干涉对单粒子幺正动力学随机矩阵性质的敏感性，通过可编程硅基光子芯片实现玻色子采样，实验验证了三种互补探针（Porter-Thomas统计距离、香农熵、时序关联函数等效观测量）能够有效区分混沌与可积动力学。

Quantum chaos and the double-slit experiment

We report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two correspondi…

Quantum dephasing and decay of classical correlation functions in chaotic systems

We discuss the dephasing induced by the internal classical chaotic motion in the absence of any external environment. To this end we consider a suitable extension of fidelity for mixed states which is measurable in a Ramsey interferometry experiment. We then relate the dephasing to the decay of this quantity which, in the semiclassical limit, is expressed in terms of an appropriate classical correlation function. Our results are derived analytically for the example of a nonlinear driven oscillator and then numerically confirmed for the kicked rotor model.

Hypersensitivity and chaos signatures in the quantum baker's maps

Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has lead to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unita…

二维Clifford电路中局域化的缺失

Absence of localization in two-dimensional Clifford circuits

本文通过随机图和渗流理论证明二维随机Clifford Floquet电路中局域化缺失、算符弹道扩散，而一维电路则出现强局域化，并揭示二维谱形状因子具有混沌准自由费米子特征。

具有量子群对称性的系统中的混沌

Chaos in Systems with Quantum Group Symmetry

研究了具有量子群对称性的自旋链系统，发现其在非可积情况下表现出混沌行为，并通过本征值统计验证，同时表现出本征值变为复数的转变。

量子场论中的纠缠抑制：全息、混沌与混态动力学

Entanglement Suppression in Quantum Field Theories: Holography, Chaos, and Mixed-State Dynamics

本提案围绕CFTs中的纠缠抑制现象，规划高维推广、OTOC混沌、可积模型、纠缠负性及AdS散射几何五条研究路线。

Quantum chaos in triangular billiards

We present an extensive numerical study of spectral statistics and eigenfunctions of quantized triangular billiards. We compute two million consecutive eigenvalues for six representative cases of triangular billiards, three with generic angles with irrational ratios with $π$, whose classical dynamics is presumably mixing, and three with exactly one angle rational with $π$, which are presumably only weakly mixing or even only non-ergodic in case of right-triangles. We find excellent agreement of short and long range spectral statistics with the Gaussian orthogonal ensemble of random matrix theo…

预热化、影子破裂以及非积分化量子电路中无Trotter化转变

Prethermalization, shadowing breakdown, and the absence of Trotterization transition in quantum circuits

研究量子电路中离散时间模拟代表连续时间哈密顿演化的持续时间，揭示预热化现象及非积分化系统中无Trotter化转变，通过截断算子传播器分析能量稳定性及其他可观测量。

Chaotic Einstein-Podolsky-Rosen pairs, measurements and time reversal

We consider a situation when evolution of an entangled Einstein-Podolsky-Rosen (EPR) pair takes place in a regime of quantum chaos being chaotic in the classical limit. This situation is studied on an example of chaotic pair dynamics described by the quantum Chirikov standard map. The time evolution is reversible even if a presence of small errors breaks time reversal of classical dynamics due to exponential growth of errors induced by exponential chaos instability. However, the quantum evolution remains reversible since a quantum dynamics instability exists only on a logarithmically short Ehr…

Quantum chaos and random matrix theoryfor fidelity decay in quantum computationswith static imperfections

We determine the universal law for fidelity decayin quantum computations of complex dynamics in presenceof internal static imperfections in a quantum computer. Our approach is based on random matrix theory applied toquantum computations in presence of imperfections.The theoretical predictions are tested and confirmed in extensive numerical simulations of a quantum algorithm for quantum chaos in the dynamical tent map with up to 18 qubits. The theory developed determines the time scales forreliable quantum computations in absence of the quantum error correction codes. These time scales are rela…

量子混沌的Wasserstein空间

Wasserstein Space of Quantum Chaos

研究发现量子系统越混沌，能量本征态的Wasserstein空间有效维度越小，通过Husimi Q-表示研究量子耦合谐振子系统，并利用Gram-spectrum方法构建嵌入几何，发现量子 scrambling 引起的折叠结构可能解释Wasserstein维度的混沌减少，且分离曲线处Wasserstein距离能正确捕捉李雅普诺夫指数，同时发现Wasserstein空间中的分支结构指示量子疤痕态，为量子混沌、量子 scrambling、量子疤痕和量子李雅普诺夫指数提供了新诊断方法。

Early-Time Exponential Instabilities in Non-Chaotic Quantum Systems

The vast majority of dynamical systems in classical physics are chaotic and exhibit the butterfly effect: a minute change in initial conditions can soon have exponentially large effects elsewhere. But this phenomenon is difficult to reconcile with quantum mechanics. One of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts. In isolated systems in the absence of decoherence, there is such a correspondence in dynamics, but it usually persists only over a short time window, after which quan…

开放量子电路中量子混沌的鲁棒性与异常弛豫

Robustness of quantum chaos and anomalous relaxation in open quantum circuits

研究耗散对混沌量子多体动力学的影响，通过精确计算通用Floquet量子电路的耗散形式因子，发现系统弛豫存在能隙闭合与否的两种区间，证明量子混沌在能隙非零时辅助弛豫且具鲁棒性。