Stefan Heusler

Periodic-orbit theory of level correlations.

We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of periodic orbits. The asymptotic expansions of both the nonoscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.

Periodic-Orbit Theory of Universality in Quantum Chaos

We argue semiclassically, on the basis of Gutzwillers 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(tau) as power series in the time tau. Each term tau(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 calculated by invoking ergodicity. Combinatorial rules for building pairs involve nontrivial properties of permutations. We show our series to be equivalent to perturbative implementations of the nonlinear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs have a one-to-one relationship with Feynman diagrams known from the sigma model.

Periodic-orbit theory of universal level correlations in quantum chaos

Using Gutzwillers semiclassical periodic-orbit theory, we demonstrate universal behavior of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time-reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the periodic-orbit expansion in terms of a matrix integral, like the one known from the sigma model of random matrix theory.

Semiclassical approach to chaotic quantum transport

We describe a semiclassical method to calculate universal transport properties of chaotic cavities. While the energy-averaged conductance turns out governed by pairs of entrance-to-exit trajectories, the conductance variance, shot noise and other related quantities require trajectory quadruplets; simple diagrammatic rules allow us to find the contributions of these pairs and quadruplets. Both pure symmetry classes and the crossover due to an external magnetic field are considered.

Semiclassical prediction for shot noise in chaotic cavities

We show that in clean chaotic cavities the power of shot noise takes a universal form. Our predictions go beyond previous results from random-matrix theory, in covering the experimentally relevant case of few channels. Following a semiclassical approach we evaluate the contributions of quadruplets of classical trajectories to shot noise. Our approach can be extended to a variety of transport phenomena as illustrated for the crossover between symmetry classes in the presence of a weak magnetic field.

The semiclassical origin of the logarithmic singularity in the symplectic form factor

Sieber and Richter achieved a breakthrough towards a proof of the universality of spectral fluctuations of chaotic quantum systems conjectured by Bohigas, Giannoni and Schmidt by calculating semiclassically the first term beyond the diagonal approximation of the orthogonal form factor. In this letter, the semiclassical origin of the logarithmic singularity of the symplectic form factor is deduced perturbatively by combining this result with the contribution that arises due to the spin. This approach stands in contrast to the duality approach introduced by Bogomolny and Keating, which is essentially non-perturbative, and where the structure around the Heisenberg time is related to the structure for very small time which can be deduced using the diagonal approximation.

Semiclassical theory for parametric correlation of energy levels

Parametric energy-level correlation describes the response of the energy-level statistics to an external parameter such as the magnetic field. Using semiclassical periodic-orbit theory for a chaotic system, we evaluate the parametric energy-level correlation depending on the magnetic field difference. The small-time expansion of the spectral form factor K(τ) is shown to be in agreement with the prediction of parameter dependent random matrix theory to all orders in τ.

Visualization of the Invisible: The Qubit as Key to Quantum Physics.

Quantum mechanics is one of the pillars of modern physics, however rather difficult to teach at the introductory level due to the conceptual difficulties and the required advanced mathematics. Nevertheless, attempts to identify relevant features of quantum mechanics and to put forward concepts of how to teach it have been proposed.1–8 Here we present an approach to quantum physics based on the simplest quantum mechanical system—the quantum bit (qubit).1 Like its classical counterpart—the bit—a qubit corresponds to a two-level system, i.e., some system with a physical property that can admit two possible values. While typically a physical system has more than just one property or the property can admit more than just two values, in many situations most degrees of freedom can be considered to be fixed or frozen. Hence a variety of systems can be effectively described as a qubit. For instance, one may consider the spin of an electron or atom, with spin up and spin down as two possible values, and where other...

Action correlation of orbits through non-conventional time reversal

Recently, Sieber and Richter calculated semiclassically a first off-diagonal contribution to the orthogonal form factor for a billiard on a surface of constant negative curvature. Following prior suggestions from the theory of disordered systems, they considered orbit pairs with almost the same action. For a generalization to systems invariant under non-conventional time reversal, which also belong to the orthogonal symmetry class, we show here that it is necessary to redefine the configuration space by a suitable canonical transformation; the distinction of this space is that it lets time reversal look conventional.

An acoustic teaching model illustrating the principles of dynamic mode magnetic force microscopy

Abstract Magnetic force microscopy (MFM) represents a versatile technique within the manifold methods of scanning probe microscopy (SPM), focusing on the investigation of magnetic phenomena at the nanoscale. Although magnetism is a fundamental element of physics education, educational content at the cutting edge of actual scientific topics and techniques in magnetism, like MFM, is lacking. Therefore, we present a scaled teaching model imparting the core principles of MFM, implementing a macroscopic model operating in dynamic mode. The experimental configuration of the model is based on popular bricks by LEGO and drivers based on LEGO Mindstorms (Lego, Billund, Denmark), as well as on further off the shelf components being easily accessible for schools and universities. Investigations of macroscopic magnetic structures reveal numerical, visual and auditory information based on magnetic forces between an oscillating cantilever and ferromagnetic samples allowing a sensual experience of force microscopy for students. Along these lines, students obtain multiple representations to study the precision measurement process of SPM in general and MFM in particular at a scale that allows experiencing micro- and nanoscopic effects. The magnetic force gradients and spatial resolution of the macroscopic model are in agreement with those of an authentic microscopic magnetic force microscope.