Narrow Results

Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at the vertices. A question that is frequently raised is the extent to which these models are the same or complementary. In particular, are all energy-independent unitary vertex scattering matrices associated with a self-adjoint Hamiltonian? Here we review results related to this issue. In addition, we observe that a self-adjoint Dirac operator with four component spinors produces a secular equation for the graph spectrum that matches the secular equation associated with wave propagation on the graph when the Dirac operator describes particles with zero mass and the vertex conditions do not allow spin rotation at the vertices.

The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

We show how semiclassical black holes can be reinterpreted as an effective geometry, composed of a large ensemble of horizonless naked singularities (eventually smoothed at the Planck scale). We call these new items frizzy-balls, which can be rigorously defined by Euclidean path integral approach. This leads to interesting implications about information paradoxes. We demonstrate that infalling information will chaotically propagate inside this system before going to the full quantum gravity regime (Planck scale).

This paper is a physicist’s review of the major conceptual issues concerning the problem of spectral universality in quantum systems. Here, we present a unified, graph-based view of all archetypical models of such universality (billiards, particles in random media, interacting spin or fermion systems). We find phenomenological relations between the onset of ergodicity (Gaussian-random delocalization of eigenstates) and the structure of the appropriate graphs, and we construct a heuristic picture of summing trajectories on graphs that describes why a generic interacting system should be ergodic. We also provide an operator-based discussion of quantum chaos and propose criteria to distinguish bases that can usefully diagnose ergodicity. The result of this analysis is a rough but systematic outline of how ergodicity changes across the space of all theories with a given Hilbert space dimension. As a particular example, we study the SYK model and report on the transition from maximal to partial ergodicity as the disorder strength is decreased.

We show the relation between the Heisenberg averaging of regularized 2-point out-of-time ordered correlation function and the 2-point spectral form factor in bosonic quantum mechanics. The generalization to all even-point is also discussed. We also do the direct extension from the bosonic quantum mechanics to the noninteracting scalar field theory. Finally, we find that the coherent state and large-$N$ approaches are useful in the late-time study. We find that the computation of the coherent state can be simplified by the Heisenberg averaging. Therefore, this provides a simplified way to probe the late-time quantum chaos through a coherent state. The large-$N$ result is also comparable to the $N=3$ numerical result in the large-$N$ quantum mechanics. This can justify that large-$N$ technique in bosonic quantum mechanics can probe the late time, not the early time. Because the quantitative behavior of large-$N$ can be captured from the $N=3$ numerical result, the realization in experiments should be possible.

In this paper, we give a general review on the application of ergodic theory to the investigation of dynamics of the Yang–Mills gauge fields and of the gravitational systems, as well as its application in the Monte Carlo method and fluid dynamics. In ergodic theory the maximally chaotic dynamical systems (MCDS) can be defined as dynamical systems that have nonzero Kolmogorov entropy. The hyperbolic dynamical systems that fulfill the Anosov C-condition belong to the MCDS insofar as they have exponential instability of their phase trajectories and positive Kolmogorov entropy. It follows that the C-condition defines a rich class of MCDS that span over an open set in the space of all dynamical systems. The large class of Anosov–Kolmogorov MCDS is realized on Riemannian manifolds of negative sectional curvatures and on high-dimensional tori. The interest in MCDS is rooted in the attempts to understand the relaxation phenomena, the foundations of the statistical mechanics, the appearance of turbulence in fluid dynamics, the nonlinear dynamics of Yang–Mills field and gravitating $N$-body systems as well as black hole thermodynamics. Our aim is to investigate classical- and quantum-mechanical properties of MCDS and their role in the theory of fundamental interactions.

A simple semiclassical Hénon–Heiles model is constructed based on Dirac’s time-dependent variational principle. We obtain an effective semiclassical Hamiltonian using a Hartree-type two-body trial wave function in the Jackiw–Kerman form. Numerical results show that quantum effects can in fact induce chaos in the nonchaotic regions of the classical Hénon–Heiles model.

Recently, a novel measure for the complexity of operator growth is proposed based on Lanczos algorithm and Krylov recursion method. We study this Krylov complexity in quantum mechanical systems derived from some well-known local toric Calabi–Yau geometries, as well as some nonrelativistic models. We find that for the Calabi–Yau models, the Lanczos coefficients grow slower than linearly for small $n$’s, consistent with the behavior of integrable models. On the other hand, for the nonrelativistic models, the Lanczos coefficients initially grow linearly for small $n$’s, then reach a plateau. Although this looks like the behavior of a chaotic system, it is mostly likely due to saddle-dominated scrambling effects instead, as argued in the literature. In our cases, the slopes of linearly growing Lanczos coefficients almost saturate a bound by the temperature. During our study, we also provide an alternative general derivation of the bound for the slope.

In this paper, the quantum system with two particles is analyzed and the energy level spacing statistics distribution and Δ₃-statistic are given. The results show that hard quantum chaos appear in the system with a certain potential. Tunnelling effect develops quantum chaos.

In this paper, the dynamical behavior of a non-symmetric double potential well in a tilted magnetic field is studied. The classical Poincare section is given to exhibit the chaotic behavior of the system, and non-linear resonant lead to chaos. The paper has also given the energy spectral statistics which satisfies Brody's distribution, tunnelling effect develops quantum chaos and also holds back the development of chaos.

The quantum-classical limit together with the associated onset of chaos is employed here in order to illustrate the importance of a proper choice of distance in probability space if one wishes to describe dynamical properties from the information theory viewpoint.

In this paper, the classical and quantum chaos characteristic of single electronic motion in the double quantum well with external magnetic field are studied. The system can be regarded as the linear coupling of a harmonic oscillator and a Duffing oscillator. The study shows that because of the interaction of two oscillators, the system demonstrates the characteristic of quasi-periodicity, multi-chaos coexisting attractors, chaos, super-chaos, etc., with different energy. Furthermore, as shown in the corresponding analysis of spectrum distribution statistics, the system in most energy fields demonstrates the coexisting of the integrable and non-integrable characteristic, which means that there is a close corresponding relation in classical and quantum behavior.

We study generic effects on the quantum dynamics of classical trapping-leaking mechanism by investigating in detail the 2δ-kicked rotors whose classical phase space is partitioned into momentum cells separated by trapping regions which slow down the motion. We focus on a range of parameters where the dynamics is generic, namely, the phase space has no stable islands. As a consequence of the trapping-leaking mechanism, we show that the classical motion is described by a process of anomalous diffusion. We investigate in detail the impact of the underlying classical anomalous diffusion on the quantum dynamics with special emphasis on the phenomenon of dynamical localization. Based on the study of the quantum density of probability, its second moment and the return probability, we identify a region of weak dynamical localization where the quantum diffusion is still anomalous but the diffusion rate is slower than in the classical case. Moreover, we examine how other relevant time scales, such as the quantum-classical breaking time and the one related to the beginning of full dynamical localization, are modified by the classical anomalous diffusion. Finally, we discuss the relevance of our results for understanding the role of classical cantori in quantum mechanics.

We introduce a characteristic function method to describe charge-counting statistics (CCS) in phase coherent systems that directly connects the three most successful approaches to quantum transport: random-matrix theory (RMT), the nonlinear σ-model and the trajectory-based semiclassical method. The central idea is the construction of a generating function based on a multivariate hypergeometric function, which can be naturally represented in terms of quantities that are well-defined in each approach. We illustrate the power of our scheme by obtaining exact analytical results for the first four cumulants of CCS in a chaotic quantum dot coupled ideally to electron reservoirs via perfectly conducting leads with arbitrary number of open scattering channels.

In this paper we investigate the local and global spectral properties of the triaxial rigid rotator. We demonstrate that, for a fixed value of the total angular momentum, the energy spectrum can be divided into two sets of energy levels, whose classical analogs are librational and rotational motions. By using diagonalization, semiclassical and algebric methods, we show that the energy levels follow the anomalous spectral statistics of the one-dimensional harmonic oscillator.

The statistics of quantum Poincaré recurrences in Hilbert space for diamagnetic hydrogen atom in strong magnetic field has been investigated. It has been shown that quantities characterizing classical chaos are in good agreement with the ones that are used to describe quantum chaos. The equality of classical and quantum Poincaré recurrences has been shown. It has been proved that one of the signs of the emergence of quantum chaos is the irreversible transition from a pure quantum mechanical state to a mixed one.

Polyatomic molecules can perform internal rotational motion of two types: torsional oscillation and free rotation of one part of the molecule with respect to the other part. On the phase plane, these two types of motion are separated by the separatrix. Phase trajectories, originated as a result of periodical external force action on the system, have stochastic nature. For quantum consideration, regarding the motion near to the classical separatrix, transition from the pure quantum-mechanical state to the mixed one takes place. Originating at that mixed state, this must be considered as the quantum analog of the classical dynamic stochasticity and is named as the quantum chaos. This work is devoted to the investigation of the quantum chaos manifestation in the polyatomic molecules, which have a property that performs internal rotation. For the molecule of ethane C₂H₆, the emergence of quantum chaos and possible ways of its experimental observation has been studied. It is shown that radio-frequency field can produce the non-direct transitions between rotational and oscillatory states. These transitions, being the sign of the existence of quantum chaos, are able to change population levels sizeably. Due to this phenomenon, experimental observation of the infrared absorption is possible.

This work is devoted to the investigation of the possibility of controlling of ion motion inside the Paul trap. It has been shown that by proper selection of the parameters controlling the electric fields, stable localization of ions inside the Paul trap is possible. Quantum consideration of this problem is reduced to the investigation of the Mathieu–Schrodinger equation. It has been shown that quantum consideration is appreciably different from the classical one that leads to stronger limitations of the values of the parameters of stable motion. Connection between the problem under study and the possibility of experimental observation of quantum chaos has been shown.

In this paper, the statistical properties of energy levels are studied numerically for atom in parallel electric and magnetic fields, which is an ideal system to examine the contributions of external fields and ionic core to quantum chaos. The Stark maps of diamagnetic spectra and nearest neighbor spacing (NNS) distributions are obtained by diagonalization method incorporating core effect. We identify obvious level anti-crossing and large value of $q$ for barium, indicating that core effect has predominant contribution to chaotic dynamics in barium. To study the core effect in detail, we sweep the quantum defect artificially and find that larger core effect will undoubtedly induce stronger chaotic dynamics.

First I derive the power spectrum (of the cumulative or integrated level density) for the regular (Poissonian) energy spectrum which is 1/t², and for the fully chaotic one, which is 1/t, as was observed numerically by Relano et al. [2002], and recently discussed independently by Faleiro et al. [2004]. The statement refers to small values of t ≤ 1, i.e. for times smaller than Heisenberg time. For t ≫ 1 it is always 1/t². Then I analyze the autocorrelation function for spectrum which is a superposition of statistically independent spectral sequences and derive the exact additivity formula for the autocorrelation function, and consequently for the form factor of the density of energy levels and of the cumulative (integrated) density of energy levels (the form factor is the Fourier transform of the corresponding autocorrelation function). Therefore this theory provides prediction for the deep semiclassical regime of sufficiently small effective Planck constant ℏ_eff. However, in not sufficiently deep semiclassical regime (not sufficiently small ℏ_eff) we see a deviation from this behavior, as recently demonstrated numerically by Gomez et al. [2004] in a billiard system [Robnik, 1983] where a power law behavior 1/t^α is found, and the exponent α goes continuously from 2 to 1, as the system's dynamics goes from complete integrability (Poisson) to full chaoticity (GOE and GUE), respectively.