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Holger Schanz

Publications and source records attributed to Holger Schanz.

At least 37 records · Page 2Linked to original sources
arXiv2003

Scars on quantum networks ignore the Lyapunov exponent

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.

nlin.CD↗
arXiv2003

Statistical properties of resonance widths for open Quantum Graphs

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.

nlin.CD↗
arXiv2002

Reaction matrix for Dirichlet billiards with attached waveguides

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.

cond-mat.mes-hall↗
arXiv2002

Eigenstates Ignoring Regular and Chaotic Phase-Space Structures

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.

nlin.CD↗
arXiv2002

The Leading Off-Diagonal Correction to the Form Factor of Large Graphs

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, $K(τ)$, 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 $-2τ^2$ to the form factor which agrees with random-matrix theory.

nlin.CD↗
arXiv2001

Classical and Quantum Hamiltonian Ratchets

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.

nlin.CD↗
arXiv2001

Effective Coupling for Open Billiards

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.

nlin.CD↗
arXiv2000

Quantum Graphs: A model for Quantum Chaos

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.

nlin.CD↗
arXiv1999

Periodic-Orbit Theory of Anderson Localization on Graphs

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.

chao-dyn↗
arXiv1999

Spectral Statistics for Quantum Graphs: Periodic Orbits and Combinatorics

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.

chao-dyn↗
arXiv1999

Spectral Statistics in Chaotic Systems with Two Identical Connected Cells

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.

chao-dyn↗
arXiv1996

Penumbra diffraction in the quantization of concave billiards

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.

chao-dyn↗
arXiv1996

Mixed Quantum-Classical versus Full Quantum Dynamics: Coupled Quasiparticle-Oscillator System

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.

chao-dyn↗
arXiv1996

Transfer and Decay of an Exciton Coupled to Vibrations in a Dimer

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.

cond-mat↗
arXiv1996

Nonadiabatic couplings and incipience of quantum chaos

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.

chao-dyn↗
arXiv1995

The Role of Diffraction in the Quantization of Dispersing Billiards

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.

chao-dyn↗
arXiv1995

Excitonic - vibronic coupled dimers: A dynamic approach

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.

chem-ph↗