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Bound states in the continuum (BICs) defy conventional wisdom that assumes a spectral separation between propagating waves, that carry energy away, and spatially localized waves corresponding to discrete frequencies. They can be described as resonance states with infinite lifetime, i.e., leaky modes with zero leakage. The advent of metamaterials and nanophotonics allowed the creation of BICs in a variety of systems. Mainly, BICs have been realized by destructive interference between outgoing resonant modes or exploiting engineered global symmetries that enforce the decoupling of a symmetry-incompatible bound mode from the surrounding radiation modes. Here, we introduce theoretically BICs relying on a different mechanism, namely local symmetries that enforce a field concentration on a part of a complex system without implying any global symmetry. We experimentally implement such BICs using microwaves in a compact one-dimensional photonic network and show that they emerge from the annihilation of two topological singularities, a zero and a pole, of the measured scattering matrix. Our alternative for achieving BICs in complex wave systems may be useful for applications like sensing, lasing, and enhancement of nonlinear interactions that require high-$Q$ modes., Comment: 11 pages, 6 figures

This article is dedicated to the following class of problems. Start with an $N\times N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1\le t \le N$, we study the difference between the spectra of the perturbed and the reference matrices as a function of $t$ and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes $t$ diagonal elements and a stronger perturbation randomizing successively $t$ rows and columns. In the first case we derive universal expressions in the scaled parameter $\tau=t/N$ for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dyson's three Gaussian ensembles. In the second case we find an additional dependence on the matrix size $N$., Comment: 23 pages, 6 figures. Slightly revised version following referee comments

Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schr\"odinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by $\{ E_n\}_{n=1}^{\infty}$ and $\{\widetilde E_n\}_{n=1}^{\infty}$ correspondingly, are level-2 interlaced, so that $E_{n-2}\le \widetilde E_n\le E_{n+2}$. The proofs are guided by considerations of the quantum graphs' discrete analogs.

We apply the framework developed in the preceding paper in this series (Smilansky 2017 J. Phys. A: Math. Theor. 50, 215301) to compute the time-delay distribution in the scattering of ultra short radio frequency pulses on complex networks of transmission lines which are modeled by metric (quantum) graphs. We consider wave packets which are centered at high wave number and comprise many energy levels. In the limit of pulses of very short duration we compute upper and lower bounds to the actual time-delay distribution of the radiation emerging from the network using a simplified problem where time is replaced by the discrete count of vertex-scattering events. The classical limit of the time-delay distribution is also discussed and we show that for finite networks it decays exponentially, with a decay constant which depends on the graph connectivity and the distribution of its edge lengths. We illustrate and apply our theory to a simple model graph where an algebraic decay of the quantum time-delay distribution is established.

We employ Random Matrix Theory in order to investigate coherent perfect absorption (CPA) in lossy systems with complex internal dynamics. The loss strength $\gamma_{\rm CPA}$ and energy $E_{\rm CPA}$, for which a CPA occurs are expressed in terms of the eigenmodes of the isolated cavity -- thus carrying over the information about the chaotic nature of the target -- and their coupling to a finite number of scattering channels. Our results are tested against numerical calculations using complex networks of resonators and chaotic graphs as CPA cavities., Comment: Supplementary material is included. Updated version with minor modifications

We report on a hitherto unnoticed type of resonances occurring in scattering from networks (quantum graphs) which are due to the complex connectivity of the graph - its topology. We consider generic open graphs and show that any cycle leads to narrow resonances which do not fit in any of the prominent paradigms for narrow resonances (classical barriers, localization due to disorder, chaotic scattering). We call these resonances `topological' to emphasize their origin in the non-trivial connectivity. Topological resonances have a clear and unique signature which is apparent in the statistics of the resonance parameters (such as e.g., the width, the delay time or the wave-function intensity in the graph). We discuss this phenomenon by providing analytical arguments supported by numerical simulation, and identify the features of the above distributions which depend on genuine topological quantities such as the length of the shortest cycle (girth). These signatures cannot be explained using any of the other paradigms for narrow resonances. Finally, we propose an experimental setting where the topological resonances could be demonstrated, and study the stability of the relevant distribution functions to moderate dissipation.

For two-dimensional quantum billiards we derive the partial Weyl law, i.e. the average density of states, for a subset of eigenstates concentrating on an invariant region $\Gamma$ of phase space. The leading term is proportional to the area of the billiard times the phase-space fraction of $\Gamma$. The boundary term is proportional to the fraction of the boundary where parallel trajectories belong to $\Gamma$. Our result is numerically confirmed for the mushroom billiard and the generic cosine billiard, where we count the number of chaotic and regular states, and for the elliptical billiard, where we consider rotating and oscillating states., Comment: 5 pages, 3 figures, derivation extended, cosine billiard added

The distribution of scattering delay times is analyzed for classical electrons which are transmitted through a finite waveguide. For non-zero magnetic field the distribution shows a regular pattern of maxima (logarithmic singularities). Although their location follows from a simple commensurability condition, there is no straightforward geometric explanation of this self-pulsing effect. Rather it can be understood as a time-dependent analog of transverse magnetic focusing, in terms of the stationary points of the delay time. We also discuss the possibility of singularities in the delay-time distribution for generic 2D scattering systems.

We study the conductance of chaotic or disordered wires in a situation where equilibrium transport decomposes into biased diffusion and a counter-moving regular current. A possible realization is a semiconductor nanostructure with transversal magnetic field and suitably patterned surfaces. We find a non-trivial dependence of the conductance on the wire length which differs qualitatively from Ohm's law by the existence of a characteristic length scale and a finite saturation value.

In generic Hamiltonian systems with a mixed phase space chaotic transport may be directed and ballistic rather than diffusive. We investigate one particular model showing this behaviour, namely a spatially periodic billiard chain in which electrons move under the influence of a perpendicular magnetic field. We analyze the phase-space structure and derive an explicit expression for the chaotic transport velocity. Unlike previous studies of directed chaos our model has a parameter regime in which the dispersion of an ensemble of chaotic trajectories around its moving center of mass is essentially diffusive. We explain how in this limit the deterministic chaos reduces to a biased random walk in a billiard with a rough surface. The diffusion constant for this simplified model is calculated analytically.

We make a distinction between the spectroscopic and the mesoscopic conductance of closed systems. We show that the latter is not simply related to the Landauer conductance of the corresponding open system. A new ingredient in the theory is related to the non-universal structure of the perturbation matrix which is generic for quantum chaotic systems. These structures may created bottlenecks that suppress the diffusion in energy space, and hence the rate of energy absorption. The resulting effect is not merely quantitative: For a ring-dot system we find that a smaller Landauer conductance implies a smaller spectroscopic conductance, while the mesoscopic conductance increases. Our considerations open the way towards a realistic theory of dissipation in closed mesoscopic ballistic devices., Comment: 18 pages, 5 figures, published version with updated refs

It is shown that the Husimi representations of chaotic eigenstates are strongly correlated along classical trajectories. These correlations extend across the whole system size and, unlike the corresponding eigenfunction correlations in configuration space, they persist in the semiclassical limit. A quantitative theory is developed on the basis of Gaussian wavepacket dynamics and random-matrix arguments. The role of symmetries is discussed for the example of time-reversal invariance., Comment: Published version with minor corrections to version 1

We present a comprehensive account of directed transport in one-dimensional Hamiltonian systems with spatial and temporal periodicity. They can be considered as Hamiltonian ratchets in the sense that ensembles of particles can show directed ballistic transport in the absence of an average force. We discuss general conditions for such directed transport, like a mixed classical phase space, and elucidate a sum rule that relates the contributions of different phase-space components to transport with each other. We show that regular ratchet transport can be directed against an external potential gradient while chaotic ballistic transport is restricted to unbiased systems. For quantized Hamiltonian ratchets we study transport in terms of the evolution of wave packets and derive a semiclassical expression for the distribution of level velocities which encode the quantum transport in the Floquet band spectra. We discuss the role of dynamical tunneling between transporting islands and the chaotic sea and the breakdown of transport in quantum ratchets with broken spatial periodicity., Comment: 22 pages

The amount of charge which is pushed by a moving scatterer is $dQ = -G dX$, where $dX$ is the displacement of the scatterer. The question is what is $G$. Does it depend on the transmission $g_0$ of the scatterer? Does the answer depend on whether the system is open (with leads attached to reservoirs) or closed? In the latter case: what are the implications of having ``quantum chaos" and/or coupling to to the environment? The answers to these questions illuminate some fundamental aspects of the theory of quantum pumping. For the analysis we take a network (graph) as a model system, and use the Kubo formula approach., Comment: 4 pages, 2 figures, minor changes, to be published in PRE (Rapid)

We study the quantum probability to survive in an open chaotic system in the framework of the van Vleck-Gutzwiller propagator and present the first such calculation that accounts for quantum interference effects. Specifically we calculate quantum deviations from the classical decay after the break time for both broken and preserved time-reversal symmetry. The source of these corrections is identified in interfering pairs of correlated classical trajectories. In our approach the quantized chaotic system is modelled by a quatum graph., Comment: minor changes

We consider universal shot noise in ballistic chaotic cavities from a semiclassical point of view and show that it is due to action correlations within certain groups of classical trajectories. Using quantum graphs as a model system we sum these trajectories analytically and find agreement with random-matrix theory. Unlike all action correlations which have been considered before, the correlations relevant for shot noise involve four trajectories and do not depend on the presence of any symmetry., Comment: 4 pages, 2 figures (a mistake in version 1 has been corrected)

We show that enhanced wavefunction localization due to the presence of short unstable orbits and strong scarring can rely on completely different mechanisms. Specifically we find that in quantum networks the shortest and most stable orbits do not support visible scars, although they are responsible for enhanced localization in the majority of the eigenstates. Scarring orbits are selected by a criterion which does not involve the classical Lyapunov exponent. We obtain predictions for the energies of visible scars and the distributions of scarring strengths and inverse participation ratios., Comment: 5 pages, 2 figures

We connect quantum compact graphs with infinite leads, and turn them into scattering systems. We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph. The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads. Finally, we analyze the statistical properties of the resonance widths and compare our results with the predictions of Random Matrix Theory. Deviations appearing due to the dynamical nature of the system are pointed out and explained., Comment: 17 pages, 7 figures. submitted to Waves in Random Media, special issue for graphs

The reaction matrix of a cavity with attached waveguides connects scattering properties to properties of a corresponding closed billiard for which the waveguides are cut off by straight walls. On the one hand this matrix is directly related to the S-matrix, on the other hand it can be expressed by a spectral sum over all eigenfunctions of the closed system. However, in the physically relevant situation where these eigenfunctions vanish on the impenetrable boundaries of the closed billiard, the spectral sum for the reaction matrix, as it was used before, fails to converge and does not reliably reproduce the scattering properties. We derive here a convergent representation of the reaction matrix in terms of eigenmodes satisfying Dirichlet boundary conditions and demonstrate its validity in the rectangular and the Sinai billiards.

For certain types of quantum graphs we show that the random-matrix form factor can be recovered to at least third order in the scaled time $\tau$ from periodic-orbit theory. We consider the contributions from pairs of periodic orbits represented by diagrams with up to two self-intersections connected by up to four arcs and explain why all other diagrams are expected to give higher-order corrections only. For a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time-reversal symmetry sum to zero. The mechanism for this cancellation is rather general which suggests that it may also apply at higher-orders in the expansion. This expectation is in full agreement with the fact that in this case the linear-$\tau$ contribution, the diagonal approximation, already reproduces the random-matrix form factor for $\tau<1$. For systems with time-reversal symmetry there are more diagrams which contribute at third order. We sum these contributions for quantum graphs with uniformly hyperbolic dynamics, obtaining $+2\tau^{3}$, in agreement with random-matrix theory. As in the previous calculation of the leading-order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes., Comment: 23 pages (including 4 fig.) - numerous typos corrected

We report the failure of the semiclassical eigenfunction hypothesis if regular classical transport coexists with chaotic dynamics. All eigenstates, instead of being restricted to either a regular island or the chaotic sea, ignore these classical phase-space structures. We argue that this is true even in the semiclassical limit for extended systems with transporting regular islands such as the standard map with accelerator modes., Comment: color figures

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, $K(\tau)$, of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution $-2\tau^2$ to the form factor which agrees with random-matrix theory., Comment: 5 pages (2 figs). Version to appear in PRL [RevTeX4] - with a bonus appendix for nlin.CD users

We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard which is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.

We explain the mechanism leading to directed chaotic transport in Hamiltonian systems with spatial and temporal periodicity. We show that a mixed phase space comprising both regular and chaotic motion is required and derive a classical sum rule which allows to predict the chaotic transport velocity from properties of regular phase-space components. Transport in quantum Hamiltonian ratchets arises by the same mechanism as long as uncertainty allows to resolve the classical phase-space structure. We derive a quantum sum rule analogous to the classical one, based on the relation between quantum transport and band structure., Comment: published version

We present a few combinatorial identities which were encountered in our work on the spectral theory of quantum graphs. They establish a new connection between the theory of random matrix ensembles and combinatorics., Comment: 16 pages, RevTeX, 1 figure

We study the statistical properties of the scattering matrix associated with generic quantum graphs. The scattering matrix is the quantum analogue of the classical evolution operator on the graph. For the energy-averaged spectral form factor of the scattering matrix we have recently derived an exact combinatorial expression. It is based on a sum over families of periodic orbits which so far could only be performed in special graphs. Here we present a simple algorithm implementing this summation for any graph. Our results are in excellent agreement with direct numerical simulations for various graphs. Moreover we extend our previous notion of an ensemble of graphs by considering ensemble averages over random boundary conditions imposed at the vertices. We show numerically that the corresponding form factor follows the predictions of random-matrix theory when the number of vertices is large---even when all bond lengths are degenerate. The corresponding combinatorial sum has a structure similar to the one obtained previously by performing an energy average under the assumption of incommensurate bond lengths., Comment: 8 pages, 3 figures. Contribution to the conference on Dynamics of Complex Systems, Dresden (1999)

We present the first quantum system where Anderson localization is completely described within periodic-orbit theory. The model is a quantum graph analogous to an a-periodic Kronig-Penney model in one dimension. The exact expression for the probability to return of an initially localized state is computed in terms of classical trajectories. It saturates to a finite value due to localization, while the diagonal approximation decays diffusively. Our theory is based on the identification of families of isometric orbits. The coherent periodic-orbit sums within these families, and the summation over all families are performed analytically using advanced combinatorial methods., Comment: 4 pages, 3 figures, RevTeX

We consider the Schroedinger operator on graphs and study the spectral statistics of a unitary operator which represents the quantum evolution, or a quantum map on the graph. This operator is the quantum analogue of the classical evolution operator of the corresponding classical dynamics on the same graph. We derive a trace formula, which expresses the spectral density of the quantum operator in terms of periodic orbits on the graph, and show that one can reduce the computation of the two-point spectral correlation function to a well defined combinatorial problem. We illustrate this approach by considering an ensemble of simple graphs. We prove by a direct computation that the two-point correlation function coincides with the CUE expression for 2x2 matrices. We derive the same result using the periodic orbit approach in its combinatorial guise. This involves the use of advanced combinatorial techniques which we explain., Comment: 25 pages, 2 figures, REVTeX (slightly extended with respect to original submission)

Chaotic systems that decompose into two cells connected only by a narrow channel exhibit characteristic deviations of their quantum spectral statistics from the canonical random-matrix ensembles. The equilibration between the cells introduces an additional classical time scale that is manifest also in the spectral form factor. If the two cells are related by a spatial symmetry, the spectrum shows doublets, reflected in the form factor as a positive peak around the Heisenberg time. We combine a semiclassical analysis with an independent random-matrix approach to the doublet splittings to obtain the form factor on all time (energy) scales. Its only free parameter is the characteristic time of exchange between the cells in units of the Heisenberg time., Comment: 37 pages, 15 figures, changed content, additional author

The semiclassical description of billiard spectra is extended to include the diffractive contributions from orbits which are nearly tangent to a concave part of the boundary. The leading correction for an unstable isolated orbit is of the same order as the standard Gutzwiller expression itself. The importance of the diffraction corrections is further emphasized by an estimate which shows that for any large fixed k almost all contributing periodic orbits are affected. The theory is tested numerically using the annulus and the Sinai billiard. For the Sinai billiard the investigation of the spectral density is complemented by an analysis which is based on the scattering approach to quantization. The merits of this approach as a tool to investigate refined semiclassical theories are discussed and demonstrated., Comment: LaTex using IOP style files, 43 pages including 20 Postscript figures

The relation between the dynamical properties of a coupled quasiparticle-oscillator system in the mixed quantum-classical and fully quantized descriptions is investigated. The system is considered to serve as a model system for applying a stepwise quantization. Features of the nonlinear dynamics of the mixed description such as the presence of a separatrix structure or regular and chaotic motion are shown to be reflected in the evolution of the quantum state vector of the fully quantized system. In particular it is demonstrated how wave packets propagate along the separatrix structure of the mixed description and that chaotic dynamics leads to a strongly entangled quantum state vector. Special emphasis is given to view the system from a dynamical Born-Oppenheimer approximation defining integrable reference oscillators and elucidating the role of the nonadiabatic couplings which complements this approximation into a rigorous quantization scheme., Comment: RevTex, 35 pages including 11 Postscript figures, submitted to Phys. Rev A

Transfer and decay dynamics of an exciton coupled to a polarization vibration in a dimer is investigated in a mixed quantum-classical picture with the exciton decay incorporated by a sink site. Using a separation of time scales, it is possible to explain analytically the most important characteristics of the model. If the vibronic subsystem is fast, these are the enhancement of nonlinear self trapping due to the sink and the slowing down of the exciton decay for large coupling or sink strength. Numerical results obtained recently for the DST approximation to the model are quantitatively explained and new dynamic effects beyond this approximation are found. If the vibronic subsystem is slow, the behavior of the system follows closely the predictions of the adiabatic approximation. In this regime, the exciton decay crucially depends on the initial conditions of the vibronic subsystem. In the transition regime between adiabatic and DST approximation, complex dynamics is observed by numerical computation. We discuss the correspondence to the chaotic behavior of the excitonic-vibronic coupled dimer without trap., Comment: RevTex, 41 pages including 8 Postscript figures

The quantization of the electronic two site system interacting with a vibration is considered by using as the integrable reference system the decoupled oscillators resulting from the adiabatic approximation. A specific Bloch projection method is applied which demonstrates how besides some regular regions in the fine structure of the spectrum and the associated eigenvectors irregularities appear when passing from the low to the high coupling case. At the same time even for strong coupling some of the regular structure of the spectrum rooted in the adiabatic potentials is kept intact justifying the classification of this situation as incipience of quantum chaos., Comment: 16 pages including figures

We study diffraction corrections to the semiclassical spectral density of dispersing (Sinai) billiards. They modify the contributions of periodic orbits (PO's), with at least one segment which is almost tangent to the concave part of the boundary. Given a wavenumber $k$, all the PO's with length up to the Heisenberg length $O(k)$ are required for quantization. We show that most of the contributions of PO's which are longer than a limit $O(k^{2/3})$ must be corrected for diffraction effects. For orbits which just miss tangency, the corrections are of the same magnitude as the semiclassical contributions themselves. Orbits which bounce at extreme forward angles give very small terms in the standard semiclassical theory. The diffraction corrections increase their amplitude substantially., Comment: 12 pages, self-extracting uuencoded-tar-compressed PostScript file. In case of extraction problems, please write to Harel Primack at fnharel1@wicc.weizmann.ac.il

We obtained the spectrum of the Sinai billiard as the zeroes of a secular equation, which is based on the scattering matrix of a related scattering problem. We show that this quantization method provides an efficient numerical scheme, and its implementation for the present case gives a few thousands of levels without encountering any serious difficulty. We use the numerical data to check some approximations which are essential for the derivation of a semiclassical quantization method based also on this scattering approach., Comment: uuencoded compressed Postscript file containing text and figures

We study the interplay between nonlinearity in exciton transport and trapping due to a sink site for the dimer and the trimer with chain configuration by a numerical integration of the discrete nonlinear Schroedinger equation. Our results for the dimer show, that the formation of a self trapped state due to the nonlinear coupling increases the life time of the exciton substantially. Self trapping can be enhanced by the sink for short times, but for long times it disappears. In the trimer consisting of a subdimer extended by a sink site exists a transition between states localized on the two sites of the subdimer before for larger nonlinear coupling self trapping on one site of the subdimer is observed. For large trapping rates the fear of death effect leads to an increasing life time of the excitation on both, the dimer and the trimer. The sink site is then effectively decoupled. We explain this effect using an asymptotic theory for strong trapping and demonstrate it by direct numerical computation., Comment: uuencoded compressed Postscript file containing text and figures. In case of questions, please, write to schanz@summa.physik.hu-berlin.de

The dynamical properties of exciton transfer coupled to polarization vibrations in a two site system are investigated in detail. A fixed point analysis of the full system of Bloch - oscillator equations representing the coupled excitonic - vibronic flow is performed. For overcritical polarization a bifurcation converting the stable bonding ground state to a hyperbolic unstable state which is basic to the dynamical properties of the model is obtained. The phase space of the system is generally of a mixed type: Above bifurcation chaos develops starting from the region of the hyperbolic state and spreading with increasing energy over the Bloch sphere leaving only islands of regular dynamics. The behaviour of the polarization oscillator accordingly changes from regular to chaotic., Comment: uuencoded compressed Postscript file containing text and figures. In case of questions, please, write to schanz@summa.physik.hu-berlin.de .

The semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived. The parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards. It is also used to check in detail some recently proposed parametric spectral statistics. The methods are illustrated in the analysis of the spectrum of the Sinai billiard and its parametric dependence on the boundary condition on the dispersing arc., Comment: Self-extracting uuencoded compressed tar file. In case of extraction problems please write to: fnharel1@wicc.weizmann.ac.il