Numerical experiments on quantum chaotic billiards

A recently proposed numerical technique for generation of high-quality unstructured meshes is combined with a finite-element method to solve the Helmholtz equation that describes the quantum mechanics of a particle confined in two-dimensional cavities. Different shapes are treated on equal footing, including Sinai, stadium, annular, threefold symmetric, mushroom, cardioid, triangle, and coupled billiards. The results are shown to be in excellent agreement with available measurements in flat microwave resonator counterparts with nonintegrable geometries.     

Observation of chaotic and regular dynamics in atom-optics billiards

We report on experimental observations of chaotic and regular motion of ultracold atoms confined by a billiard-shaped optical dipole potential induced by a rapidly scanning laser beam. To investigate the dynamics of the atoms confined by such an "atom-optics" billiard we measure the decay of the number of trapped atoms through a hole on the boundary. A fast and purely exponential decay, the clear signature of chaotic motion, is found for a stadium billiard, but not for a circular or an elliptical billiard, in agreement with theory. We also investigated the effects of decoherence, velocity spread, and gravity on regular and chaotic motion.     

Stable quantum resonances in atom optics

A theory for stabilization of quantum resonances by a mechanism similar to one leading to classical resonances in nonlinear systems is presented. It explains recent surprising experimental results, obtained for cold cesium atoms when driven in the presence of gravity, and leads to further predictions. The theory makes use of invariance properties of the system allowing for separation into independent kicked rotor problems. The analysis relies on a fictitious classical limit where the small parameter is not Planck's constant, but rather the detuning from the frequency that is resonant in the absence of gravity.     

Classical and quantum chaos in atom optics

The interaction of an atom with an electro-magnetic field is discussed in the presence of a time periodic external modulating force. It is explained that a control on atom by electro-magnetic fields helps to design the quantum analog of classical optical systems. In these atom optical systems chaos may appear at the onset of external fields. The classical and quantum chaotic dynamics is discussed, in particular in an atom optics Fermi accelerator. It is found that the quantum dynamics exhibits dynamical localization and quantum recurrences.     

Correlation decay in quantum chaotic billiards with bulk or surface disorder

We study the decay properties of correlation functions in quantum billiards with surface or bulk disorder. The quantum system is modeled by means of a tight-binding Hamiltonian with diagonal disorder, solved on LxL clusters of the square lattice. The correlation function is calculated by launching the system at t=0 into a wave function of the regular (clean) system and following its time evolution. The results show that the correlation function decays exponentially with a characteristic correlation time (inverse of the Lyapunov exponent lambda). For small enough disorder the Lyapunov exponent is approximately given by the imaginary part of the self-energy induced by disorder. On the other hand, if the scaling of the Lyapunov exponent with L is investigated by keeping constant l/L, where l is the mean free path, the results show that lambda is proportional to 1/L.     

Decay of correlations in a chaotic measure-preserving transformation

For a chaotic, area-preserving map on the torus, we study the decay of correlations in detail. Taking as observables the square-integrable functions, we find examples of decay rates which are algebraic, exponential, and faster than exponential. For correlations that decay exponentially the rate is sensitive to the choice of function. The implications for numerical experiments of this nonuniformity in the decay are discussed.     

Decay of a thermofield-double state in chaotic quantum systems

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite N for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.     

A new model of quantum chaotic billiards: application to granular metals

We discuss the properties of a recently proposed model of quantum chaotic billiards in two and three dimensions. The model is based on a tight-binding Hamiltonian in which the energies of the atomic levels at the boundary sites are chosen at random between -W/2 and W/2. The energy spectra show a complex behavior with regions that obey Wigner-Dyson statistics, and regions with localized and quasi-ideal states distributed according to Poisson statistics. Whereas at low energies long-range energy fluctuations follow Random Matrix Theory (RMT) for all W, at high energies fluctuations are below (above) RMT for small (large) W. For small W, the mean free path l is proportional to L/W ², L being the system size, and reaches a minimum for W of the order of the band width, at which l ≈ L/2. In 3D we found that the energy fluctuations of the highest occupied level are much larger than the average interlevel spacing. This provides an explanation for autoionization effects of the grains in granular metals.     

Entanglement in quantum integrable and chaotic systems with Markov dynamics

Entanglement evolution is studied in open systems represented by rings of qubits with the Ising interaction and variously oriented external field and with Markov environments. The effect of thermal or dephasing environment is manifested as exponential decrease of the entanglement superposed on its dynamics in the isolated system.     

Long time tail correlations in discrete chaotic dynamics

A scaling relation is derived connecting the exponent of the algebraically decaying correlation and response functions with the degree of intermittency and the order of the maximum. It is universal, i.e. within a large class independent of the correlated variables. This implies universal 1/f-like spectra. The corrections to scaling are investigated, too.     

弱相互作用玻色-爱因斯坦凝聚体中涡旋动力学与量子混沌轨道简

利用变分法和数值模拟方法,我们分别从解析上和数值上研究了弱相互作用玻色-爱因斯坦凝聚体中不同涡旋蔟的动力学性质.借助玻姆量子力学中的方法,我们定义了相应量子流体中的量子轨道,并且研究了由于不同涡旋结构的存在而导致量子轨道出现混沌的性质.当存在一单涡旋,我们发现混沌轨道出现与否和涡旋轨道的形状紧密相关.此外,玻色凝聚体原子中的两体相互作用对各向异性谐振子势下涡旋对出现时量子轨道混沌的发生也具有重要的作用.因为这一非线性相互作用会破坏相应速度场的时间周期性.最后,在涡旋极子情形下,我们还讨论了由于涡旋相互激发或淹没的作用而导致规则岛膨胀的性质.这些规则区域镶嵌在一定的混沌海中.     

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.     

Effect of noise in open chaotic billiards

We investigate the effect of white-noise perturbations on chaotic trajectories in open billiards. We focus on the temporal decay of the survival probability for generic mixed-phase-space billiards. The survival probability has a total of five different decay regimes that prevail for different intermediate times. We combine new calculations and recent results on noise perturbed Hamiltonian systems to characterize the origin of these regimes and to compute how the parameters scale with noise intensity and billiard openness. Numerical simulations in the annular billiard support and illustrate our results.     

The local geometry of chaotic billiards

For a billiard of a general shape a transformation is introduced which projects the boundary on the unit circle. This introduces a non-Euclidean metric on the plane which contains all relevant information of the shape of the boundary. Classically the straight lines of the free motion correspond to geodesics and quantum mechanically the energy spectrum is that of Laplace–Beltrami operator with Dirichlet boundary conditions on the unit circle. The geodesic equations are highly non-linear. Nevertheless for the interval between two consecutive scatterings we have two integrals of motion, the kinetic energy and the angular momentum. This fact helps to solve explicitly the geodesic equations. These solutions can be used to derive interesting properties for the classical scattering. Quantum mechanically the spectrum of the above billiards is obtained for certain parameter values both perturbatively for small values of the parameter and also using a diagonalization procedure. This method is applicable to any particular form of a billiard for which the transformation is invertible and can be used on one hand as a quick method of approximate spectral determination and as a theoretical tool to analyse specific properties of integrability and chaos through the associated connection form and the Laplace–Beltrami operator. Finally as a first indication of the potentiality of this method we present a graphical test where for very small deviations from the circular billiard an integrable and two non-integrable billiards can be distinguished by the distribution of the differences of the first order corrections while this distinction is not evident by the usual test for the nearest neighbor level spacings.     

Regular and chaotic quantum dynamics in atom-diatom reactive collisions

A new microirreversible 3D theory of quantum multichannel scattering in the three-body system is developed. The quantum approach is constructed on the generating trajectory tubes which allow taking into account influence of classical nonintegrability of the dynamical quantum system. When the volume of classical chaos in phase space is larger than the quantum cell in the corresponding quantum system, quantum chaos is generated. The probability of quantum transitions is constructed for this case. The collinear collision of the Li + (FH) → (LiF) + H system is used for numerical illustration of a system generating quantum (wave) chaos. The text was submitted by the authors in English.     

Generalized measures of quantum correlations for mixed states

The exponential speedup achieved in certain quantum algorithms based on mixed states with negligible entanglement has renewed the interest on alternative measures of quantum correlations. Here we discuss a general measure of quantum correlations for composite systems based on generalized entropic functions, defined as the minimum information loss due to a local measurement. For pure states, the present measure becomes an entanglement entropy, i.e., it reduces to the generalized entropy of the reduced state. However, for mixed states it can be nonzero in separable states, vanishing just for states diagonal in a general product basis, like the quantum discord. Quadratic measures of quantum correlations can be derived as particular cases of the present formalism. The minimum information loss due to a joint local measurement is also considered. The evaluation of these measures in a simple yet relevant case is also discussed.