Quantum entanglement and chaos in kicked two-component Bose-Einstein condensates

We study two-component Bose-Einstein condensates that behave collectively as a spin system obeying the dynamics of a quantum kicked top. Depending on the nonlinear interaction between atoms in the classical limit, the kicked top exhibits both regular and chaotic dynamical behavior. The quantum entanglement is physically meaningful if the system is viewed as a bipartite system, where the subsystem is any one of the two modes. The dynamics of the entanglement between the two modes in this classical chaotic system has been investigated. The chaos leads to rapid rise and saturation of the quantum entanglement. Furthermore, the saturated values of the entanglement fall short of its maximum. The mean entanglement has been used to clearly display the close relation between quantum entanglement and underlying chaos.     

Spectral fluctuations and 1/f noise in the order-chaos transition regime

Level fluctuations in a quantum system have been used to characterize quantum chaos using random matrix models. Recently time series methods were used to relate the level fluctuations to the classical dynamics in the regular and chaotic limit. In this, we show that the spectrum of the system undergoing order to chaos transition displays a characteristic f(-gamma) noise and gamma is correlated with the classical chaos in the system. We demonstrate this using a smooth potential and a time-dependent system modeled by Gaussian and circular ensembles, respectively, of random matrix theory. We show the effect of short periodic orbits on these fluctuation measures.     

Chaos in atomic physics

The influence of quantum chaos on small atomic systems is reviewed. It now seems clear that chaos in the usually understood sense (e.g. exponential sensitivity to perturbations) is not found in isolated quantum systems. However, there are phenomena which appear only when the corresponding classical system is chaotic. The stability of the classical dynamics, in other words whether it is regular or chaotic, has a profound effect on the character of the corresponding quantum spectrum and wavefunctions.     

Dissipative death of quantum coherences in a spin system

We investigate the influence of weak dissipation on the dynamics of a kicked spin system. Compared to its classical limit the quantum system is strongly affected by small dissipation. We present the attenuation of two different intrinsically quantum phenomena, recurrencies and tunneling. We find agreement of analytical perturbative estimates and numerical results. We further show that dissipation acts quite differently on the quantum evolution depending on whether we are in a classically chaotic or regular domain.     

The effect of measurement on the quantum features of a chaotic system

We investigate the quantum dynamics of a periodically kicked nonlinear spin system which exhibits regular and chaotic dynamics in the classical regime. The quantum behaviour is characterised by the evolving eigenvalue distributions for the angular momentum components and the features, including recurrences in the quantum means and the presence of quantum tunneling, are discussed. We employ the evolution operator eigenvalue distribution to prove that coherent quantum tunneling occurs between the fixed points in the regular regions of phase space. Continual quantum measurement is included in the model: the classical dynamics are unchanged but a destruction of coherences occurs in the quantum system. Recurrences in the means are destroyed and quantum tunneling is suppressed by measurement, a manifestation of the quantum Zeno effect.     

Evolution of order and chaos across a first-order quantum phase transition

We study the evolution of the dynamics across a generic first-order quantum phase transition in an interacting boson model of nuclei. The dynamics inside the phase coexistence region exhibits a very simple pattern. A classical analysis reveals a robustly regular dynamics confined to the deformed region and well separated from a chaotic dynamics ascribed to the spherical region. A quantum analysis discloses regular bands of states in the deformed region, which persist to energies well above the phase-separating barrier, in the face of a complicated environment. The impact of kinetic collective rotational terms on this intricate interplay of order and chaos is investigated.     

Quantum chaos and tunneling in the kicked top

We analyze the dynamics of the kicked top in a deeply quantum regime. Signatures of classical chaos in the quantum dynamics that can be identified from a semiclassical treatment persist in a deeply quantum regime. Structures in the classical-phase space can also be identified in the tunneling dynamics of the quantum system. Our results show that quantum chaos is observable in the regime that is accessible to future experiments with trapped ions or cold atoms.     

Nonmonotonicity in the quantum-classical transition: chaos induced by quantum effects

The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.     

Dynamical entanglement as a signature of chaos in the semiclassical limit

The relationship between classically chaotic dynamics and the entanglement properties of the corresponding quantum system is examined in the semiclassical limit. Numerical results are computed for a modified kicked top, keeping the classical dynamics constant while investigating the entanglement for several versions of the corresponding quantum system characterized by a different value of the effective Planck constant ℏ _eff. Our findings indicate that as ℏ _eff → 0, the apparent signatures of classical chaos in the entanglement properties, such as characteristic oscillations in the time-dependence of the linear entropy, can also be obtained in the regular regime. These results suggest that entanglement is not a universal marker of chaotic dynamics of the corresponding classical system.     

Fundamental concepts of quantum chaos

We review the fundamental concepts of quantum chaos in Hamiltonian systems. The quantum evolution of bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, whereas the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. The effects of quantum localization of the chaotic eigenstates are treated phenomenologically, resulting in Brody-like level statistics, which can be found also at very high-lying levels, while the coupling between the regular and the irregular eigenstates due to tunneling, and of the corresponding levels, manifests itself only in low-lying levels.     

Chaos-assisted tunneling and 1/falpha spectral fluctuations in the order-chaos transition

It has been shown that the spectral fluctuations of different quantum systems are characterized by 1/falpha noise, with 1< or =alpha< or =2, in the transition from integrability to chaos. This result is not well understood. We show that chaos-assisted tunneling gives rise to this power-law behavior. We develop a random matrix model for intermediate quantum systems, based on chaos-assisted tunneling, and we discuss under which conditions it displays 1/falpha noise in the transition from integrability to chaos. We conclude that the variance of the elements that connect regular with chaotic states must decay with the difference of energy between them. We compare the characteristics of the transition modeled in this way with what is obtained for the Robnik billiard.     

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The dynamics of cold atoms in conservative optical lattices obviously depends on the geometry of the lattice. But very similar lattices may lead to deeply different dynamics. In a 2D optical lattice with a square mesh, it is expected that the coupling between the degrees of freedom leads to chaotic motions. However, in some conditions, chaos remains marginal. The aim of this paper is to understand the dynamical mechanisms inhibiting the appearance of chaos in such a case. As the quantum dynamics of a system is defined as a function of its classical dynamics – e.g. quantum chaos is defined as the quantum regime of a system whose classical dynamics is chaotic – we focus here on the dynamical regimes of classical atoms inside a well. We show that when chaos is inhibited, the motions in the two directions of space are frequency locked in most of the phase space, for most of the parameters of the lattice and atoms. This synchronization, not as strict as that of a dissipative system, is nevertheless a mechanism powerful enough to explain that chaos cannot appear in such conditions.     

Squeezed states and quantum chaos

We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos. Zh. éksp. Teor. Fiz. 113, 111–127 (January 1998)     

光与物质相互作用系统中的量子Fisher信息和自旋压缩

Fisher信息是估计理论中的重要概念,最近发现与量子信息中的纠缠判据具有密切联系.非旋波近似条件下,Dicke模型经典相空间表现为混沌动力学特征.本文详细考察了Dicke模型描述的光与物质相互作用系统中量子Fisher信息和自旋压缩动力学特性.结果表明:在短时瞬态情况下,无论初态处于规则区域还是混沌区域系统均表现为纠缠性质;但在长时稳态情况下,初态处于规则区域时系统纠缠消失,而初态处于混沌区域时系统则一直存在纠缠.通过与系统自旋压缩动力学性质相比较,发现量子Fisher信息可以更有效地刻画量子混沌.进一步考察初态处于规则和混沌区域时系统密度矩阵和纯度的动力学演化特性,发现混沌导致系统退相干现象发生,说明量子Fisher信息对混沌更敏感.     

Dicke模型的量子混沌和单粒子相干动力学特性

量子化的Dicke模型在非旋波近似条件下表现为量子混沌动力学特征.利用单粒子一阶时间关联函数,通过数值计算详细考察了Dicke模型中单粒子相干动力学特性.结果表明:当初始相干态处在混沌区域时,一阶时间关联函数曲线衰减较快,而当初始相干态处在规则区域时,一阶时间关联函数曲线衰减较慢,单粒子相干动力学对初态具有较强的敏感性,经典混沌抑制量子相干.考察单粒子相干动力学在相空间的平均演化性质,得到一种较好的量子经典对应关系.最后研究了相空间单粒子相干的整体动力学性质,更好地揭示了相空间的混沌和规则结构.     

Border between regular and chaotic quantum dynamics

We identify a border between regular and chaotic quantum dynamics. The border is characterized by a power-law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. For example, the overlap decay for the quantum kicked top is well fitted with [1+(q-1)(t/tau)2](1/(1-q)) (with the nonextensive entropic index q and tau depending on perturbation strength) in the region preceding the emergence of quantum interference effects. This region corresponds to the edge of chaos for the classical map from which the quantum chaotic dynamics is derived.     

Classical chaos with Bose-Einstein condensates in tilted optical lattices

A widely accepted definition of "quantum chaos" is "the behavior of a quantum system whose classical limit is chaotic." The dynamics of quantum-chaotic systems is nevertheless very different from that of their classical counterparts. A fundamental reason for that is the linearity of Schr?dinger equation. In this paper, we study the quantum dynamics of an ultracold quantum degenerate gas in a tilted optical lattice and show that it displays features very close to classical chaos. We show that its phase space is organized according to the Kolmogorov-Arnold-Moser theorem.     

Regular-to-chaotic tunneling rates using a fictitious integrable system

We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.