Transition from order to chaos in reduced quantum dynamics.

We study a damped kicked top dynamics of a large number of qubits (N→∞) and focus on an evolution of a reduced single-qubit subsystem. Each subsystem is subjected to the amplitude damping channel controlled by the damping constant r∈[0,1], which plays the role of the single control parameter. In the parameter range for which the classical dynamics is chaotic, while varying r we find the universal period-doubling behavior characteristic to one-dimensional maps: period-2 dynamics starts at r_{1}≈0.3181, while the next bifurcation occurs at r_{2}≈0.5387. In parallel with period-4 oscillations observed for r≤r_{3}≈0.5672, we identify a secondary bifurcation diagram around r≈0.544, responsible for a small-scale chaotic dynamics inside the attractor. The doubling of the principal bifurcation tree continues until r≤r_{∞}∼0.578, which marks the onset of the full scale chaos interrupted by the windows of the oscillatory dynamics corresponding to the Sharkovsky order. Finally, for r=1 the model reduces to the standard undamped chaotic kicked top.

Fidelity susceptibility in Gaussian random ensembles.

The fidelity susceptibility measures the sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here, we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system sizes. The results are verified by a comparison with numerical data.

Randomness in quantum mechanics: philosophy, physics and technology.

This progress report covers recent developments in the area of quantum randomness, which is an extraordinarily interdisciplinary area that belongs not only to physics, but also to philosophy, mathematics, computer science, and technology. For this reason the article contains three parts that will be essentially devoted to different aspects of quantum randomness, and even directed, although not restricted, to various audiences: a philosophical part, a physical part, and a technological part. For these reasons the article is written on an elementary level, combining simple and non-technical descriptions with a concise review of more advanced results. In this way readers of various provenances will be able to gain while reading the article.

Resonances and poles in isoscattering microwave networks and graphs.

Can one hear the shape of a graph? This is a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such as quantum graphs and microwave networks. It addresses an important mathematical problem whether scattering properties of such systems are uniquely connected to their shapes? Recent experimental results based on a characteristics of graphs such as the cumulative phase of the determinant of the scattering matrices indicate a negative answer to this question [O. Hul, M. Lawniczak, S. Bauch, A. Sawicki, M. Kus, and L. Sirko, Phys. Rev. Lett. 109, 040402 (2012)]. In this paper we consider important local characteristics of graphs such as structures of resonances and poles of the determinant of the scattering matrices. Using these characteristics we study experimentally and theoretically properties of graphs and directly confirm that the pair of graphs considered in the cited paper is isoscattering. The experimental results are compared to the theoretical ones for a broad frequency range from 0.01 to 3 GHz. In the numerical calculations of the resonances of the graphs absorption present in the experimental networks is taken into account.

Extremal spacings between eigenphases of random unitary matrices and their tensor products.

Extremal spacings between eigenphases of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N=4. We study ensembles of tensor product of k random unitary matrices of size n which describe independent evolution of a composite quantum system consisting of k subsystems. In the asymptotic case, as the total dimension N=n(k) becomes large, the nearest neighbor distribution P(s) becomes Poissonian, but statistics of extreme spacings P(s(min)) and P(s(max)) reveal certain deviations from the Poissonian behavior.

Quantum chaos in SU(3) models with trapped ions.

A scheme to generate long-range spin-spin interactions between three-level ions in a chain is presented, providing a feasible experimental route to the rich physics of well-known SU(3) models. In particular, we demonstrate different signatures of quantum chaos which can be controlled and observed in experiments with trapped ions.

Are scattering properties of graphs uniquely connected to their shapes?

The famous question of Kac "can one hear the shape of a drum?" addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator, can be legitimately extended to scattering systems. In the modified version, one asks whether the geometry of a vibrating system can be determined by scattering experiments. We present the first experimental approach to this problem in the case of microwave graphs (networks) simulating quantum graphs. Our experimental results strongly indicate a negative answer. To demonstrate this we consider scattering from a pair of isospectral microwave networks consisting of vertices connected by microwave coaxial cables and extended to scattering systems by connecting leads to infinity to form isoscattering networks. We show that the amplitudes and phases of the determinants of the scattering matrices of such networks are the same within the experimental uncertainties. Furthermore, we demonstrate that the scattering matrices of the networks are conjugated by the so-called transplantation relation.

Concurrence of mixed multipartite quantum States.

We propose generalizations of concurrence for multipartite quantum systems that can distinguish qualitatively distinct quantum correlations. All introduced quantities can be evaluated efficiently for arbitrary mixed sates.

Concurrence of mixed bipartite quantum states in arbitrary dimensions.

We derive a lower bound for the concurrence of mixed bipartite quantum states, valid in arbitrary dimensions. As a corollary, a weaker, purely algebraic estimate is found, which detects mixed entangled states with a positive partial transpose.

Global entangling properties of the coupled kicked tops.

We study global entangling properties of the system of coupled kicked tops testing various hypotheses and predictions concerning entanglement in quantum chaotic systems. In order to analyze the averaged initial entanglement production rate and the averaged asymptotic entanglement, various ensembles of initial product states are evolved. Two different ensembles with natural probability distribution are considered: product states of independent spin-coherent states and product states of random states. It appears that the choice of either of these ensembles results in significantly different averaged entanglement behavior. We investigate also a relation between the averaged asymptotic entanglement and the mean entanglement of eigenvectors of the evolution operator. Lower bound on the averaged asymptotic entanglement is derived, expressed in terms of the eigenvector entanglement.