Experimental and numerical investigation of parametric spectral properties of quantum graphs with unitary or symplectic symmetry

Coupled unidirectional chaotic microwave graphs

We experimentally investigate the undirected open microwave network Γ with internal absorption composed of two coupled directed halves, unidirectional networks Γ_{+} and Γ_{-}, corresponding to two possible directions of motion on their edges. The two-port scattering matrix of the network Γ is measured and the spectral statistics and the elastic enhancement factor of the network are evaluated. The comparison of the number of experimental resonances with the theoretical one predicted by the Weyl's law shows that within the experimental resolution the resonances are doubly degenerate. This conclusion was also corroborated by the numerical calculations. Though the network is characterized by the time-reversal symmetry, the missing level spectral statistics and the elastic enhancement factor are rather close to the Gaussian unitary ensemble predictions in random matrix theory. We used numerical calculations for the open nondissipative quantum graph possessing the same structure as the microwave network Γ to investigate the doublet structures in the spectrum which otherwise would not be experimentally resolved. We show that the doublet size distribution is close to the Poisson distribution.

Closed and open superconducting microwave waveguide networks as a model for quantum graphs

We report on high-precision measurements that were performed with superconducting waveguide networks with the geometry of a tetrahedral and a honeycomb graph. They consist of junctions of valency three that connect straight rectangular waveguides of equal width but incommensurable lengths. The experiments were performed in the frequency range of a single transversal mode, where the associated Helmholtz equation is effectively one-dimensional and waveguide networks may serve as models of quantum graphs with the joints and waveguides corresponding to the vertices and bonds. The tetrahedral network comprises T junctions, while the honeycomb network exclusively consists of Y junctions, that join waveguides with relative angles 90^{∘} and 120^{∘}, respectively. We demonstrate that the vertex scattering matrix, which describes the propagation of the modes through the junctions, strongly depends on frequency and is nonsymmetric at a T junction and thus differs from that of a quantum graph with Neumann boundary conditions at the vertices. On the other hand, at a Y junction, similarity can be achieved in a certain frequency range. We investigate the spectral properties of closed waveguide networks and fluctuation properties of the scattering matrix of open ones and find good agreement with random matrix theory predictions for the honeycomb waveguide graph.

Loss-Induced Violation of the Fundamental Transmittance-Asymmetry Bound in Nonlinear Complex Wave Systems

Nonlinearity-induced asymmetric transport (AT) can be utilized for on-chip implementation of nonreciprocal devices that do not require odd-vector biasing. This scheme, however, is subject to a fundamental bound dictating that the maximum transmittance asymmetry is inversely proportional to the asymmetry intensity range (AIR) over which AT occurs. Contrary to the conventional wisdom, we show that the implementation of losses can lead to an increase of the AIR without deteriorating the AT. We develop a general theory that provides a new upper bound for AT in nonlinear complex systems and highlights the importance of their structural complexity and of losses. Our predictions are confirmed numerically and experimentally using a microwave complex network of coaxial cables.

Distributions of the Wigner reaction matrix for microwave networks with symplectic symmetry in the presence of absorption

We report on experimental studies of the distribution of the reflection coefficients, and the imaginary and real parts of Wigner's reaction (K) matrix employing open microwave networks with symplectic symmetry and varying size of absorption. The results are compared to analytical predictions derived for the single-channel scattering case within the framework of random-matrix theory (RMT). Furthermore, we performed Monte Carlo simulations based on the Heidelberg approach for the scattering (S) and K matrix of open quantum-chaotic systems and the two-point correlation function of the S-matrix elements. The analytical results and the Monte Carlo simulations depend on the size of absorption. To verify them, we performed experiments with microwave networks for various absorption strengths. We show that deviations from RMT predictions observed in the spectral properties of the corresponding closed quantum graph and attributed to the presence of nonuniversal short periodic orbits does not have any visible effects on the distributions of the reflection coefficients and the K and S matrices associated with the corresponding open quantum graph.

Implications of Spectral Interlacing for Quantum Graphs

Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed.

Eigenfunction and eigenmode-spacing statistics in chaotic photonic crystal graphs

The statistical properties of wave chaotic systems of varying dimensionalities and realizations have been studied extensively. These systems are commonly characterized by the statistics of the eigenmode spacings and the statistics of the eigenfunctions. Here, we propose photonic crystal (PC) defect waveguide graphs as a physical setting for chaotic graph studies. Photonic crystal waveguides possess a dispersion relation for the propagating modes, which is engineerable. Graphs constructed by joining these waveguides possess junctions and bends with distinct scattering properties. We present numerically determined statistical properties of an ensemble of such PC graphs including both eigenfunction amplitude and eigenmode-spacing studies. Our proposed system is compatible with silicon nanophotonic technology and opens chaotic graph studies to a new community of researchers.

Experimental study of closed and open microwave waveguide graphs with preserved and partially violated time-reversal invariance

We report on experiments that were performed with microwave waveguide systems and demonstrate that in the frequency range of a single transversal mode they may serve as a model for closed and open quantum graphs. These consist of bonds that are connected at vertices. On the bonds, they are governed by the one-dimensional Schrödinger equation with boundary conditions imposed at the vertices. The resulting transport properties through the vertices may be expressed in terms of a vertex scattering matrix. Quantum graphs with incommensurate bond lengths attracted interest within the field of quantum chaos because, depending on the characteristics of the vertex scattering matrix, its wave dynamic may exhibit features of a typical quantum system with chaotic counterpart. In distinction to microwave networks, which serve as an experimental model of quantum graphs with Neumann boundary conditions, the vertex scattering matrices associated with a waveguide system depend on the wave number and the wave functions can be determined experimentally. We analyze the spectral properties of microwave waveguide systems with preserved and partially violated time-reversal invariance, and the properties of the associated wave functions. Furthermore, we study properties of the scattering matrix describing the measurement process within the framework of random matrix theory for quantum chaotic scattering systems.

Fluctuation properties of the eigenfrequencies and scattering matrix of closed and open unidirectional graphs with chaotic wave dynamics

We present experimental and numerical results for the fluctuation properties in the eigenfrequency spectra and of the scattering matrix of closed and open unidirectional quantum graphs, respectively. Unidirectional quantum graphs, that are composed of bonds connected by reflectionless vertices, were introduced by Akila and Gutkin [Akila and Gutkin, J. Phys. A: Math. Theor. 48, 345101 (2015)1751-811310.1088/1751-8113/48/34/345101]. The nearest-neighbor spacing distribution of their eigenvalues was shown to comply with random-matrix theory predictions for typical chaotic systems with completely violated time-reversal invariance. The occurrence of short periodic orbits confined to a fraction of the system, that lead in conventional quantum graphs to deviations of the long-range spectral correlations from the behavior expected for typical chaotic systems, is suppressed in unidirectional ones. Therefore, we pose the question whether such graphs may serve as a more appropriate model for closed and open chaotic systems with violated time-reversal invariance than conventional ones. We compare the fluctuation properties of their eigenvalues and scattering matrix elements and observe especially in the long-range correlations larger deviations from random-matrix theory predictions for the unidirectional graphs. These are attributed to a loss of complexity of the underlying dynamic, induced by the unidirectionality.

Spectral duality in graphs and microwave networks

Quantum graphs and their experimental counterparts, microwave networks, are ideally suited to study the spectral statistics of chaotic systems. The graph spectrum is obtained from the zeros of a secular determinant derived from energy and charge conservation. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The first ones are realized in experiments, since the standard junctions connecting the bonds obey Neumann boundary conditions due to current conservation. On average, the corresponding Neumann and Dirichlet eigenvalues alternate as a function of the wave number, with the consequence that the Neumann spectrum is described by random matrix theory only locally, but adopts features of the interlacing Dirichlet spectrum for long-range correlations. Another spectral interlacing is found for the Green's function, which in contrast to the secular determinant is experimentally accessible. This is illustrated by microwave studies and numerics.

Delay-time distribution in the scattering of short Gaussian pulses in microwave networks

We investigate delay-time distributions in the scattering of short Gaussian pulses in microwave networks which simulate quantum graphs. We show that in the limit of short delay times the delay-time distribution is very sensitive to the internal structure of the networks. Therefore, it can be used to reveal their local structure including the boundary conditions at the vertices of the networks. In the frequency domain the pulses comprise many resonance frequencies of the networks. Furthermore, we show that the time-delay distribution averaged over different internal configurations of a finite network decays exponentially. Our experimental results for four-vertex and isoscattering microwave networks are in very good agreement with the theoretical ones obtained from the modified theory of U. Smilansky and H. Schanz [J. Phys. A 51, 075302 (2018)1751-811310.1088/1751-8121/aaa0df]. We modified the theory to account for internal absorption of microwave networks.

A new spectral invariant for quantum graphs

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic [Formula: see text], with [Formula: see text] denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic [Formula: see text] of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic [Formula: see text] can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic [Formula: see text] a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.

Missing-level statistics in a dissipative microwave resonator with partially violated time-reversal invariance

We report on the experimental investigation of the fluctuation properties in the resonance frequency spectra of a flat resonator simulating a dissipative quantum billiard subject to partial time-reversal-invariance violation (TIV) which is induced by two magnetized ferrites. The cavity has the shape of a quarter bowtie billiard of which the corresponding classical dynamics is chaotic. Due to dissipation it is impossible to identify a complete list of resonance frequencies. Based on a random-matrix theory approach we derive analytical expressions for statistical measures of short- and long-range correlations in such incomplete spectra interpolating between the cases of preserved time-reversal invariance and complete TIV and demonstrate their applicability to the experimental spectra.

Universal S-matrix correlations for complex scattering of wave packets in noninteracting many-body systems: Theory, simulation, and experiment

We present an in-depth study of the universal correlations of scattering-matrix entries required in the framework of nonstationary many-body scattering of noninteracting indistinguishable particles where the incoming states are localized wave packets. Contrary to the stationary case, the emergence of universal signatures of chaotic dynamics in dynamical observables manifests itself in the emergence of universal correlations of the scattering matrix at different energies. We use a semiclassical theory based on interfering paths, numerical wave function based simulations, and numerical averaging over random-matrix ensembles to calculate such correlations and compare with experimental measurements in microwave graphs, finding excellent agreement. Our calculations show that the universality of the correlators survives the extreme limit of few open channels relevant for electron quantum optics, albeit at the price of dealing with large-cancellation effects requiring the computation of a large class of semiclassical diagrams.

Missing-level statistics in classically chaotic quantum systems with symplectic symmetry

We present experimental and theoretical results for the fluctuation properties in the incomplete spectra of quantum systems with symplectic symmetry and a chaotic dynamics in the classical limit. To obtain theoretical predictions, we extend the random-matrix theory (RMT) approach introduced in Bohigas and Pato [O. Bohigas and M. P. Pato, Phys. Rev. E 74, 036212 (2006)PLEEE81539-375510.1103/PhysRevE.74.036212] for incomplete spectra of quantum systems with orthogonal symmetry. We validate these RMT predictions by randomly extracting a fraction of levels from complete sequences obtained numerically for quantum graphs and experimentally for microwave networks with symplectic symmetry and then apply them to incomplete experimental spectra to demonstrate their applicability. Independently of their symmetry class, quantum graphs exhibit nongeneric features which originate from nonuniversal contributions. Part of the associated eigenfrequencies can be identified in the level dynamics of parameter-dependent quantum graphs and extracted, thereby yielding spectra with systematically missing eigenfrequencies. We demonstrate that, even though the RMT approach relies on the assumption that levels are missing at random, it is possible to determine the fraction of missing levels and assign the appropriate symmetry class by comparison of their fluctuation properties with the RMT predictions.

Application of topological resonances in experimental investigation of a Fermi golden rule in microwave networks

We investigate experimentally a Fermi golden rule in two-edge and five-edge microwave networks with preserved time reversal invariance. A Fermi golden rule gives rates of decay of states obtained by perturbing embedded eigenvalues of graphs and networks. We show that the embedded eigenvalues are connected with the topological resonances of the analyzed systems and we find the trajectories of the topological resonances on the complex plane.

Experimental investigation of distributions of the off-diagonal elements of the scattering matrix and Wigner's K[over ̂] matrix for networks with broken time reversal invariance

We present an extensive experimental study of the distributions of the real and imaginary parts of the off-diagonal elements of the scattering matrix S[over ̂] and the Wigner's reaction K[over ̂] matrix for open microwave networks with broken time (T) reversal invariance. Microwave Faraday circulators were applied in order to break T invariance. The experimental distributions of the real and imaginary parts of the off-diagonal entries of the scattering matrix S[over ̂] are compared with the theoretical predictions from the supersymmetry random matrix theory [A. Nock, S. Kumar, H.-J. Sommers, and T. Guhr, Ann. Phys. (NY) 342, 103 (2014)10.1016/j.aop.2013.11.006]. Furthermore, we show that the experimental results are in very good agreement with the recent predictions for the distributions of the real and imaginary parts of the off-diagonal elements of the Wigner's reaction K[over ̂] matrix obtained within the framework of the Gaussian unitary ensemble of random matrix theory [S. B. Fedeli and Y. V. Fyodorov, J. Phys. A: Math. Theor. 53, 165701 (2020)1751-811310.1088/1751-8121/ab73ab]. Both theories include losses as tunable parameters and are therefore well adapted to the experimental verification.