Electronic transport through chaotic quantum dots exhibits universal, system independent, properties, consistent with random matrix theory. The quantum transport can also be rooted, via the semiclassical approximation, in sums over the classical scattering trajectories. Correlations between such trajectories can be organized diagrammatically and have been shown to yield universal answers for some observables. Here, we develop the general combinatorial treatment of the semiclassical diagrams, through a connection to factorizations of permutations. We show agreement between the semiclassical and random matrix approaches to the moments of the transmission eigenvalues. The result is valid for all moments to all orders of the expansion in inverse channel number for all three main symmetry classes (with and without time reversal symmetry and spin-orbit interaction) and extends to nonlinear statistics. This finally explains the applicability of random matrix theory to chaotic quantum transport in terms of the underlying dynamics as well as providing semiclassical access to the probability density of the transmission eigenvalues.
We consider quantum systems with a chaotic classical limit that depend on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor $K(τ,x)$ which depends on a scaled parameter difference $x$. For parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the small $τ$ expansion of the form factor agrees with Random Matrix Theory for systems with and without time reversal symmetry.
We sketch the semiclassical core of a proof of the so-called Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical chaos has a quantum energy spectrum with universal fluctuations on the scale of the mean level spacing. We show how in the semiclassical limit all system specific properties fade away, leaving only ergodicity, hyperbolicity, and combinatorics as agents determining the contributions of pairs of classical periodic orbits to the quantum spectral form factor. The small-time form factor is thus reproduced semiclassically. Bridges between classical orbits and (the n…
The statistics of quantum transport through chaotic cavities with two leads is encoded in transport moments $M_m={\rm Tr}[(t^†t)^m]$, where $t$ is the transmission matrix, which have a known universal expression for systems without time-reversal symmetry. We present a semiclassical derivation of this universality, based on action correlations that exist between sets of long scattering trajectories. Our semiclassical formula for $M_m$ holds for all values of $m$ and arbitrary number of open channels. This is achieved by mapping the problem into two independent combinatorial problems, one involv…
We present a semiclassical approach to eigenfunction statistics in chaotic and weakly disordered quantum systems which goes beyond Random Matrix Theory, supersymmetry techniques, and existing semiclassical methods. The approach is based on a generalization of Berry's Random Wave Model, combined with a consistent semiclassical representation of spatial two-point correlations. We derive closed expressions for arbitrary wavefunction averages in terms of universal coefficients and sums over classical paths, which contain, besides the supersymmetry results, novel oscillatory contributions. Their ph…
Quantum chaos of many-body systems has been swiftly developing into a vibrant research area at the interface between various disciplines, ranging from statistical physics to condensed matter to quantum information and to cosmology. In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between classically chaotic dynamics and corresponding universal features at the quantum level. Recently, single-particle techniques dealing with ergodic wave interference in the usual semiclassical limit $\hbar \rightarrow 0$ have begun to be transformed into the fiel…
We address the quantum-classical correspondence for chaotic systems with a crossover between symmetry classes. We consider the energy level statistics of a classically chaotic system in a weak magnetic field. The generating function of spectral correlations is calculated by using the semiclassical periodic-orbit theory. An explicit calculation up to the second order, including the non-oscillatory and oscillatory terms, agrees with the prediction of random matrix theory. Formal expressions of the higher order terms are also presented. The nonlinear sigma (NLS) model of random matrix theory, in …
The Loschmidt echo (LE) is a magnitude that measures the sensitivity of quantum dynamics to perturbations in the Hamiltonian. For a certain regime of the parameters, the LE decays exponentially with a rate given by the Lyapunov exponent of the underlying classically chaotic system. We develop a semiclassical theory, supported by numerical results in a Lorentz gas model, which allows us to establish and characterize the universality of this Lyapunov regime. In particular, the universality is evidenced by the semiclassical limit of the Fermi wavelength going to zero, the behavior for times longe…
We examine spectral equilibration of quantum chaotic spectra to universal statistics, in the context of the Brownian motion model. Two competing time scales, proportional and inversely proportional to the classical relaxation time, jointly govern the equilibration process. Multiplicity of quantum systems having the same semiclassical limit is not sufficient to obtain equilibration of any spectral modes in two-dimensional systems, while in three-dimensional systems equilibration for some spectral modes is possible if the classical relaxation rate is slow. Connections are made with upper bounds …
We consider the semiclassical ballistic sigma-model as an effective theory describing the quantum mechanics of classically chaotic systems. Specifically, we elaborate on close analogies to the recently developed semiclassical theory of quantum interference in chaotic systems and show how semiclassical 'diagrams' involving near action degenerate sets of periodic orbits emerge in the field theoretical description. We further discuss how the universality phenomenon (i.e. the fact that individual chaotic systems behave according to the prescriptions of random matrix theory) can be understood from …
The classification of universality classes of random-matrix theory has recently been extended beyond the Wigner-Dyson ensembles. Several of the novel ensembles can be discussed naturally in the context of superconducting-normal hybrid systems. In this paper, we give a semiclassical interpretation of their spectral form factors for both quantum graphs and Andreev billiards.
We construct a unified semiclassical theory of charge and spin transport in chaotic ballistic and disordered diffusive mesoscopic systems with spin-orbit interaction. Neglecting dynamic effects of spin-orbit interaction, we reproduce the random matrix theory results that the spin conductance fluctuates universally around zero average. Incorporating these effects in the theory, we show that geometric correlations generate finite average spin conductances, but that they do not affect the charge conductance to leading order. The theory, which is confirmed by numerical transport calculations, allo…
We present a semiclassical approach to n-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary n. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality.
Convergent semiclassical trace formulae for the density of states and cohesive force of a narrow constriction in an electron gas, whose classical motion is either chaotic or integrable, are derived. It is shown that mode quantization in a metallic point contact or nanowire leads to universal oscillations in its cohesive force: the amplitude of the oscillations depends only on a dimensionless quantum parameter describing the crossover from chaotic to integrable motion, and is of order 1 nano-Newton, in agreement with recent experiments. Interestingly, quantum tunneling is shown to be described …
We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical e…
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor $K(τ)$ as power series in the time $τ$. Each term $τ^n$ of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is c…
We show that in the semiclassical limit, classically chaotic systems have universal spectral statistics. Concentrating on short-time statistics, we identify the pairs of classical periodic orbits determining the small-$τ$ behavior of the spectral form factor $K(τ)$ of fully chaotic systems. The two orbits within each pair differ only by their connections inside close self-encounters in phase space. The frequency of occurrence of these self-encounters is determined by ergodicity. Permutation theory is used to systematically sum over all topologically different families of such orbit pairs. The …
We investigate the universality of correlation functions of chaotic and disordered quantum systems as an external parameter is varied. A new, general scaling procedure is introduced which makes the theory invariant under reparametrizations. Under certain general conditions we show that this procedure is unique. The approach is illustrated with the particular case of the distribution of eigenvalue curvatures. We also derive a semiclassical formula for the non-universal scaling factor, and give an explicit expression valid for arbitrary deformations of a billiard system.
The Wigner time delay, defined by the energy derivative of the total scattering phase shift, is an important spectral measure of an open quantum system characterising the duration of the scattering event. It is related to the trace of the Wigner-Smith matrix Q that also encodes other time-delay characteristics. For chaotic cavities, these exhibit universal fluctuations that are commonly described within random matrix theory. Here, we develop a new semiclassical approach to the time-delay matrix which is formulated in terms of the classical trajectories that connect the exterior and interior re…
Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories - for both the Ber…
We present experimental and numerical results for the long-range fluctuation properties in the spectra of quantum graphs with chaotic classical dynamics and preserved time-reversal invariance. Such systems are generally believed to provide an ideal basis for the experimental study of prob- lems originating from the field of quantum chaos and random matrix theory. Our objective is to demonstrate that this is true only for short-range fluctuation properties in the spectra, whereas the observation of deviations in the long-range fluctuations is typical for quantum graphs. This may be attributed t…
We review properties of open chaotic mesoscopic systems with a finite Ehrenfest time tau_E. The Ehrenfest time separates a short-time regime of the quantum dynamics, where wave packets closely follow the deterministic classical motion, from a long-time regime of fully-developed wave chaos. For a vanishing Ehrenfest time the quantum systems display a degree of universality which is well described by random-matrix theory. In the semiclassical limit, tau_E becomes parametrically larger than the scattering time off the boundaries and the dwell time in the system. This results in the emergence of a…
We present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a Hamiltonian quantum system defined by two coupled quartic oscillators, which on the classical level exhibits mixed phase space dynamics. We show that the signature of a pitchfork bifurcation is two-fold: Beside the known effect of an enhanced periodic orbit contribution due to its peculiar $\hbar$-dependence at the bifurcation, we demonstrate that the orbit pair born {\em at} the bifurcation gives rise to distinct deviations from universality slightly {\em…
We present a semiclassical theory for the scattering matrix ${\cal S}$ of a chaotic ballistic cavity at finite Ehrenfest time. Using a phase-space representation coupled with a multi-bounce expansion, we show how the Liouville conservation of phase-space volume decomposes ${\cal S}$ as ${\cal S}={\cal S}^{\rm cl} \oplus {\cal S}^{\rm qm}$. The short-time, classical contribution ${\cal S}^{\rm cl}$ generates deterministic transmission eigenvalues T=0 or 1, while quantum ergodicity is recovered within the subspace corresponding to the long-time, stochastic contribution ${\cal S}^{\rm qm}$. This …
Using semiclassical periodic orbit theory for a chaotic system in a weak magnetic field, we obtain the form factor predicted by Pandey and Mehta's two matrix model up to the third order. The third order contribution has a peculiar term which exists only in the intermediate crossover domain between the GOE (Gaussian Orthogonal Ensemble) and the GUE (Gaussian Unitary Ensemble) universality classes. The exact expression is obtained by taking account of the contribution from encounter regions where orbit loops are connected.
Experimentally, the phase of the amplitude for electron transmission through a quantum dot (transmission phase) shows the same pattern between consecutive resonances. Such universal behavior, found for long sequences of resonances, is caused by correlations of the signs of the partial-width amplitudes of the resonances. We investigate the stability of these correlations in terms of a statistical model. For a classically chaotic dot, the resonance eigenfunctions are assumed to be Gaussian distributed. Under this hypothesis, statistical fluctuations are found to reduce the tendency towards unive…
In searching for the manifestations of sensitivity of the eigenfunctions in quantum billiards (with Dirichlet boundary conditions) with respect to the boundary data (the normal derivative) we have performed instead various numerical tests for the Robnik billiard (quadratic conformal map of the unit disk) for 600 shape parameter values, where we look at the sensitivity of the energy levels with respect to the shape parameter. We show the energy level flow diagrams for three stretches of fifty consecutive (odd) eigenstates each with index 1,000 to 2,000. In particular, we have calculated the (un…
We review the main ideas and results in the stationary problems of quantum chaos in generic (mixed) systems, whose classical dynamics has regular (invariant tori) and chaotic regions coexisting in the phase space. First we discuss the universality classes of spectral fluctuations (GOE/GUE for ergodic systems, and Poissonian for integrable systems). We explain the problems in the calculation of the invariant (Liouville) measure of classically chaotic components, which has recently been studied by Robnik et al (1997) and by Prosen and Robnik (1998). Then we describe the Berry-Robnik (1984) pictu…
The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of th…
