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is an integer, the electron orbits change fromlocalised Bloch-like trajectories to unbounded stochastic orbits, which difiuserapidly through intricate web patterns in phase space. To quantify how thesewebs afiect electron transport, we make drift-difiusion calculations of thecurrent{voltage curves including the efiects of space-charge build up. Whenthe magnetic fleld is tilted, our simulations reveal a large resonant peak,which originates from stochastic delocalisation of the electron orbits. Weshow that the corresponding quantised eigenstates change discontinuouslyfrom a highly localised character when the system is ofi resonance to a fullydelocalised form when the resonance condition is satisfled.PACS numbers: 73.20.At, 73.21.Cd

We present experimental results demonstrating that fractal conductance fluctuations are remarkably robust to deviations from the semiclassical conduction regime. We find that the fractal dimension depends solely on an empirical parameter that quantifies the resolution of the billiard's discrete energy level spectrum. Non-fractal fluctuations are obtained only in the extreme classical and quantum-mechanical conduction regimes.

We report a new type of quantum chaotic system in which the classical Hamiltonian originates from the intrinsically quantum mechanical nature of the device. The system is a semiconductor superlattice in a magnetic field. The energy–momentum dispersion curves can be used to calculate semi- classical orbits for electrons confined to a single miniband. When a magnetic field is applied along the superlattice axis (x-direction), the electrons perform Bloch oscillations along the axis with cyclotron motion in the orthogonal plane. But when the magnetic field is tilted away from the x-direction, the orbits are chaotic, and have a spatial width along the superlattice axis, which is much larger than the amplitude of the Bloch oscillations. This is because the tilted field transfers momentum between the x- and z-directions, thereby delocalizing the electron path. This type of chaotic dynamics is fundamentally different to that identified in our previous studies of double–barrier resonant tunneling diodes. We investigate the relation between the orbits of the effective Hamiltonian, and the quantum states of the superlattice. In the regime of strong chaos, the wave functions have a highly diffuse structure which extends across many periods of the superlattice, just like the corresponding classical orbits. This chaos-induced delocalization increases the current flow through real devices. By contrast, in the stable domain the electron orbits remain localized along the paths of particular quasi-periodic orbits. We use theoretical and experimental current–voltage curves to show how the onset of chaos manifests itself in the transport properties of two- and three-terminal superlattice structures, and identify current oscillations associated with classical resonances. We also consider analogies with ultra-cold atoms in an optical lattice with a tilted harmonic trap.

Fractal behaviour in the magneto-conductance fluctuations of mesoscopic systems has been predicted using semiclassical theory and observed experimentally in semiconductor billiards. Surface-gate billiards provide an ideal environment for investigations of fractal behaviour due to the ease with which parameters in the experimental system can be tuned. In this study we vary temperature and billiard size to investigate the classical and quantum limits of the semiclassical theory for fractal magneto-conductance fluctuations.

We introduce semiconductor superlattices as a new type of quantum chaotic system that is accessible to experiment. In contrast to previous physical systems that have been used to study quantum chaos, the classical Hamiltonian for the superlattices has an intrinsically quantum–mechanical origin. For low electric fields, superlattices have well-defined minibands which originate from the quantum–mechanical coupling of electron states in a periodic array of potential wells. The energy-wave-vector dispersion curves define an effective Hamiltonian that can be used to calculate semiclassical orbits for electrons confined to a single miniband. We show that applying a tilted magnetic field induces a controllable transition from stable regular motion to strong classical chaos. The onset of chaos delocalizes both the classical orbits and the corresponding quantum wave functions, and is therefore expected to produce a sharp increase in the current flow measured in electron transport experiments. We show that magnetically confined atoms in a periodic optical potential exhibit similar chaotic dynamics at ultra-low (μK) temperatures.

The semiclassical motion of an electron along the axis of a superlattice may be calculated from the miniband dispersion curve. Under a weak electric field the electron executes Bloch oscillations which confines the motion along the superlattice axis. When a magnetic field, tilted with respect to the superlattice axis, is applied the electron orbits become chaotic. The onset of chaos is characterised by a complex mixed stable-chaotic phase space and an extension of the orbital trajectories along the superlattice axis. This delocalisation of the orbits is also reflected in the quantum eigenstates of the system some of which show well-defined patterns of high probability density whose shapes resemble certain semiclassical orbits. This suggests that the onset of chaos will be manifest in electron transport through a finite superlattice. We also propose that these phenomena may be observable in the motion of trapped ultra-cold atoms in an optically induced superlattice potential and magnetic quadrupole potential whose axis is tilted relative to the superlattice axis.

We show how semiconductor superlattices can be used to provide a new type of quantum chaotic system that is accessible to experiment. When a magnetic field is tilted relative to the superlattice axis, the semiclassical orbits of electrons confined to a single miniband undergo a controllable transition from stable Bloch oscillations to chaotic orbits. The onset of chaos delocalizes the electron paths and generates complex trajectories that extend across many periods of the superlattice. We show that the chaotic motion has a fundamentally different origin to that reported for previous semiconductor structures.

We have investigated ballistic transport through a sub-micron-sized Sinai billiard fabricated using a high-mobility two-dimensional electron gas. The geometry of the billiard is defined by a circular surface gate surrounded by an outer square gate with two quantum point contacts. When both gates are biased negatively, the Sinai billiard is formed and collisions with the confining potential walls generate chaotic classical motion. Recent experiments have revealed novel ‘self-similar’ fluctuation patterns in the conductance of the Sinai billiard, measured as a function of magnetic field perpendicular to the plane of the electron gas. We have undertaken quantum mechanical calculations of the device conductance using a detailed model for the potential landscape under the surface gates. We compare these results with predictions for a hard-walled billiard and show that the surface gate geometry can have a large effect on the observed transport properties.

We describe a gate-defined semiconductor billiard which undergoes an evolution from a square to Sinai geometry. We present experimental and theoretical investigations of clusters of magnetoconductance structure which emerge during the transition and originate from ‘quantum chaos’ processes.

We show theoretically that the dynamics of cold atoms in the lowest-energy band of a stationary optical lattice can be transformed and controlled by a second, weaker, periodic potential moving at a constant speed along the axis of the stationary lattice. The atom trajectories exhibit complex behavior, which depends sensitively on the amplitude and speed of the propagating lattice. When the speed and amplitude of the moving potential are low, the atoms are dragged through the static lattice and perform drifting orbits with frequencies an order of magnitude higher than that corresponding to the moving potential. Increasing either the speed or amplitude of the moving lattice induces Bloch-like oscillations within the energy band of the static lattice, which exhibit complex resonances at critical values of the system parameters. In some cases, a very small change in these parameters can reverse the atom's direction of motion. In order to understand these dynamics we present an analytical model, which describes the key features of the atom transport and also accurately predicts the positions of the resonant features in the atom's phase space. The abrupt controllable transitions between dynamical regimes, as well as the associated set of resonances, provide a mechanism for transporting atoms between precise locations in a lattice, as required for using cold atoms to simulate condensed matter or as a stepping stone to quantum information processing. The system also provides a direct quantum simulator of acoustic waves propagating through semiconductor nanostructures in sound analogs of the optical laser (saser).

The transition from regular to chaotic classical electron dynamics in a wide potential well with a high tilted magnetic field is investigated using Poincare sections. The corresponding quantized energy level spectrum for the well is calculated as a function of the tilt angle π. For values of π where the system exhibits strong classical chaos, the distribution of nearest-neighbor level spacings obeys universal Wigner statistics. Regular long-range fluctuations in the density of levels are identified and related to distinct unstable closed classical orbits in accordance with the Gutzwiller trace formula. These orbits are found to produce regions of high probability density (scars) in the wavefunctions associated with subsets of almost equally-spaced energy levels. The energies of these scarred states can be located using a simple semiclassical quantization rule. This periodic scarring of individual wavefunctions is shown to have a pronounced influence on the tunneling characteristics of double barrier structures. Tunneling transitions into the scarred states dominate the current-voltage curves and generate a series of resonant peaks as observed in recent magnetotunneling experiments. Regimes in which resonant tunneling spectroscopy might provide experimental evidence for the existence of scarred states are identified.

Recent work on electron transport through classically chaotic heterostructure systems is briefly reviewed. The use of tunnel-current spectroscopy to map the energy levels of the classically chaotic electron motion in a quantum well at high magnetic field is described in detail. This technique provides the first experimental evidence for the periodic scarring of individual wavefunctions in a chaotic system.

The energy levels and eigenfunctions of the electronic motion in a wide potential well with a large tilted magnetic field have been calculated in the regime of strong classical chaos. Periodic modulations of the density of states due to level clustering are identified and related to the occurrence of distinct unstable periodic classical orbits. Individual periodic orbits are found to “scar” the wave functions corresponding to regular subsets of energy levels determined by quantizing the classical action. Resonant tunnelling spectroscopy provides a mean of investigating the link between the classical and quantum pictures of the motion. The tunnelling rates into scarred states are found to be much greater than the rates into nearby unscarred states. Owing to level broadening present experimental results do not distinguish between tunnelling into a discrete scarred state and tunnelling into a cluster of closely spaced levels. Further tunnelling experiments on double-barrier structures with narrower wells in higher magnetic fields will provide an accessible means of studying wave function scarring.

We use semiclassical Hamiltonian optics to investigate the propagation of light rays through two-dimensional photonic crystals when slow spatial modulation of the lattice parameters induces mixed stable-chaotic ray dynamics. This modulation changes both the shape and frequency range of the allowed frequency bands, thereby bending the resulting semiclassical ray trajectories and confining them within particular regions of the crystal. The curved boundaries of these regions, combined with the bending of the orbits themselves, creates a hierarchy of stable and unstable chaotic trajectories in phase space. For certain lattice parameters and electromagnetic wave frequencies, islands of stable orbits act as a dynamical barrier, which separates the chaotic trajectories into two distinct regions of the crystal, thereby preventing the rays propagating through the structure. We show that changing the frequency of the light strongly affects the distribution of stable and unstable orbits in both real and phase space. This switches the dynamical barriers on and off and thus modulates the transmission of rays through the crystal. We propose microwave analogues of the photonic crystals as a route to the experimental study of the transport effects that we predict., 10 pages, 8 figures

We use supercomputer simulations to show that inter-atomic interactions can strongly affect the phase evolution of Bose-Einstein condensates that are diffracted from atom chips, thereby explaining recent experiments. Interactions broaden and depopulate non-central diffraction orders and so, counter-intuitively, they actually reduce the spatial width of the diffracted cloud. This means, more generally, that the experiments cannot be fully described by non-interacting theories and, consequently, interferometers that use many interacting atoms to enhance sensitivity require complex calibration. We identify device geometries required to approximate non-interacting behavior., 5 pages, 4 figures

We show that quantum-mechanical reflection can severely disrupt the structure of Bose-Einstein condensates and explain anomalously low reflection coefficients recently observed for condensates reflected from a Si surface.

QUALITATIVE insight into the properties of a quantum-mechanical system can be gained from the study of the relationship between the system's classical newtonian dynamics, and its quantum dynamics as described by the Schrodinger equation. The Bohr–Sommerfeld quantization scheme—which underlies the historically important Bohr model for hydrogen-like atoms—describes the relationship between the classical and quantum-mechanical regimes, but only for systems with stable, periodic or quasi-periodic orbits1. Only recently has progress been made in understanding the quantization of systems that exhibit non-periodic, chaotic motion. The spectra of quantized energy levels for such systems are irregular, and show fluctuations associated with unstable periodic orbits of the corresponding classical system1–3. These orbits appear as 'scars'—concentrations of probability amplitude—in the wavefunction of the system4. Although wavefunction scarring has been the subject of extensive theoretical investigation5–10, it has not hitherto been observed experimentally in a quantum system. Here we use tunnel-current spectroscopy to map the quantum-mechanical energy levels of an electron confined in a semiconductor quantum well in a high magnetic field10–13. We find clear experimental evidence for wavefunction scarring, in full agreement with theoretical predictions10.

We investigate the effects of solitons and vortices on the dynamics of Bose-Einstein condensates in an optical lattice. The geometry of these excitations has a pronounced effect on the evolution of the atom cloud

Recent investigations of fractal conductance fluctuations (FCF) in electron billiards reveal crucial discrepancies between experimental behavior and the semiclassical Landauer-Buttiker (SLB) theory that predicted their existence. In particular, the roles played by the billiard's geometry, potential profile and the resulting electron trajectory distribution are not well understood. We present measurements on two custom-made devices - a 'disrupted' billiard device and a 'bilayer' billiard device - designed to probe directly these three characteristics. Our results demonstrate that intricate processes beyond those proposed in the SLB theory are required to explain FCF., 17 pages, 4 figures, in press for Physical Review B

Employing three geometries, we use correlation analysis to investigate the relationship between exact and statistical self-similarity in the magneto-conductance of semiconductor billiards.

We demonstrate that self-similar magnetoconductance fluctuations observed in semiconductor Sinai billiards originate from quantum interference processes associated with classical trajectories enclosing discrete areas. We identify the exact areal distributions and use a simple model to reproduce the self-similarity with remarkable accuracy.

Experimental and theoretical studies of mixed stable-chaotic electron dynamics in the quantum well of a resonant tunneling diode are reported. The classical orbits of electrons injected into the well from an emitter accumulation layer change from regular to chaotic when an applied magnetic field is tilted away from the normal to the well walls. In the regime of mixed stable-chaotic electron dynamics, we identify a period-tripling bifurcation, which, in contrast to previous experiments, is directly accessible to the tunneling electrons. Stable classical orbits associated with this bifurcation are related directly to the wave functions of the quantum well using a Wigner function analysis. There is a striking similarity between the form of the electron Wigner functions and the corresponding classical Poincar\'e sections. This allows us to identify subsets of quantum well states associated with stable orbits around the period-tripling bifurcation. These states control the flow of electrons through the device and produce a series of strong resonant peaks in the measured current-voltage characteristics. We also identify resonant features associated with ``scarred'' quantum well states in which the probability density is concentrated along an unstable but periodic classical path.

We investigate how fractals evolve into nonfractal behavior as the generation process is gradually suppressed. Fractals observed in the conductance of semiconductor billiards are of particular interest because the generation process is semiclassical and can be suppressed by transitions towards either fully classical or fully quantum-mechanical conduction. Investigating a range of billiards, we identify a ``universal'' behavior in the changeover from fractal to nonfractal conductance, which is described by a smooth evolution rather than deterioration in the fractal scaling properties.

Considerable progress has been made over the last ten years in understanding the thermoelectric behaviour of low dimensional transport systems. There are two regimes to consider. At low temperatures (below about 1K) the effect of phonons is negligible. This is the “diffusion” thermopower regime which we discuss briefly for 2D electron gases (2DEGS) in Section 2. In Section 3 we discuss a particularly interesting aspect of the formalism for 2DEGS which is that it may be modified easily to apply to the quantisation effects observed recently in quantum point contacts (van Wees et al, 1988; Wharam et al, 1988; van Houten et al, 1992; Molenkamp et al 1992).

A Comment on the Letter by A. S. Sachrajda et al., Phys. Rev. Lett. 80, 1948 (1998). The authors of the Letter offer a Reply.

Fromhold et al. reply — Unstable but periodic classical orbits are of fundamental importance in quantum chaology6,7. They produce regular clustering of the energy levels6 and certain states exhibit regions of enhanced probability density or ‘scars’ in the neighbourhood of the classical periodic trajectories2. The scarring effect seems to have been unexpected in view of the instability of the periodic orbits8. The typical trajectory is chaotic and irregular, which would be expected to correspond to wavefunctions exhibiting irregular and diffuse patterns of probability density. We recently showed1 that scarred and unscarred states can make very different contributions to a physically measurable quantity, in our case the tunnel current through a semiconductor device.

We have used resonant tunnelling spectroscopy to investigate the energy level spectrum of a wide 60 nm potential well with a strong magnetic field applied at an angle θ to the normal to the barriers. In this geometry, the current-voltage characteristic I(V) reveal distinct series of resonances which change dramatically with both V and θ. In a classical picture, the electron orbits in the well are strongly chaotic for 15° ≤ θ ≤ 75° and voltages V ≤ 1.2 V. However, we have identified a hierarchy of unstable but periodic orbits whose calculated properties explain the origin and θ-dependence of the resonant structure. We show that the classical motion becomes non-chaotic at high bias voltages and that the changeover to stable orbits is characteriaed by the appearance of widely-spaced resonances in I(V) .

Resonant tunneling spectroscopy is used to study the electronic energy levels of a wide potential well when a strong magnetic field is applied at an angle θ to the normal to the plane of the well. Even though the classical motion of electrons in the potential well is chaotic in the tilted field geometry, the current-voltage characteristics reveal distinct series of quasi-periodic resonances. The voltage range and period of each series of resonances change dramatically with θ. We explain this behavior by considering the unstable periodic orbits within the chaotic sea of the potential well.

Resonant tunneling spectroscopy is used to study the energy-level spectrum of a new chaotic dynamical system, an electron in a trapezoidal potential well in the presence of a high magnetic field tilted relative to the confining barriers. Distinct series of quasi-periodic resonances are observed in the current-voltage chracteristics which change dramatically with tilt angle. These resonances are related to unstable closed orbits within the chaotic domain. The experimental results are explained by identifying and studying the properties of periodic orbits accessible to the tunneling electrons.

Self-similarity ~the invariance of an intrinsic pattern under scaling ! in the fractal magnetotransport properties of mesoscopic billiards is highly topical. Employing three billiard geometries, we investigate the relationship between the two classes of observed scaling behavior—exact and statistical self-similarity. We employ a correlation analysis to determine the conditions required for their observation and to show that the two forms of self-similarity can be understood within a common framework. @S0163-1829~98!04934-0#

Recent studies of quantum transport through semiconductor structures with chaotic electron dynamics are briefly reviewed. The use of tunnel-current spectroscopy to probe the quantum states corresponding to chaotic electron orbits in a quantum well with a high tilted magnetic field is described in detail. Periodic orbit theory is used to provide a detailed quantitative analysis of recent experimental studies of quantum wells in tilted fields. This analysis reveals the crucial role played by ‘scarred’ wavefunctions.