P.B. Wilkinson

Chaotic electron diffusion through stochastic webs enhances current flow in superlattices.

Understanding how complex systems respond to change is of fundamental importance in the natural sciences. There is particular interest in systems whose classical newtonian motion becomes chaotic as an applied perturbation grows. The transition to chaos usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov–Arnold–Moser (KAM) theorem—a cornerstone of nonlinear dynamics that explains, for example, gaps in the asteroid belt. By contrast, ‘non-KAM’ chaos switches on and off abruptly at critical values of the perturbation frequency. This type of dynamics has wide-ranging implications in the theory of plasma physics, tokamak fusion, turbulence, ion traps, and quasicrystals. Here we realize non-KAM chaos experimentally by exploiting the quantum properties of electrons in the periodic potential of a semiconductor superlattice with an applied voltage and magnetic field. The onset of chaos at discrete voltages is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns in phase space. Non-KAM chaos therefore provides a mechanism for controlling the electrical conductivity of a condensed matter device: its extreme sensitivity could find applications in quantum electronics and photonics.

Chaotic ray dynamics in slowly varying two-dimensional photonic crystals.

We use Hamiltonian optics to investigate chaotic ray dynamics in a photonic crystal whose lattice parameters vary slowly with position. The ray dynamics are chaotic even in regimes where only stable motion has been found in previous studies of energy band transport. Stable ray paths provide dynamical barriers that localize chaotic motion to certain regions of the photonic crystal.

Manifestations of quantum chaos in resonant tunnelling

Abstract 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.

A physical explanation for the origin of self-similar magnetoconductance fluctuations in semiconductor billiards

Abstract We report quantum-mechanical calculations which replicate the self-similar magnetoconductance fluctuations observed in recent experiments on semiconductor Sinai billiards. We interpret these fluctuations by considering the mixed stable-chaotic classical dynamics of electrons in the billiard. In particular, we show that the fluctuation patterns are dominated by individual stable orbits. The scaling characteristics of the self-similar fluctuations depend on the geometry of the associated stable orbit. We find that our analysis is insensitive to the details of the potential landscape, and is applicable to real devices with a wide range of soft-wall profiles. We show that our analysis also provides a possible explanation for the distinct series of magnetoconductance fluctuations observed in recent experiments on carbon nanotubes.

Chaos in quantum wells and analogous optical systems

Abstract The quantized states of a 60 nm wide potential well in a large tilted magnetic field are investigated using scaled field resonant tunnelling spectroscopy. In contrast to previous experiments on this type of system, the tunnelling characteristics are measured by changing both the magnetic field strength B and the applied bias voltage V such that V/B2 is approximately constant. This ensures that the classical phase space for electrons in the potential well has the same mixed stable-chaotic character for all fields. As a consequence of this scaling, each closed orbit in the potential well produces many periodic resonant peaks in plots of d2I/dB2 versus B. This type of scaled field experiment can be used to probe quantum states corresponding to dynamical regimes which are inaccessible to fixed field resonant tunnelling studies. We also consider analogies between the electron orbits and light rays in gradient refractive index lenses.

Quantum chaos for cold atoms in an optical lattice with a tilted harmonic trap

We show how ultracold (sub-µK temperature) atoms in a one-dimensional periodic optical potential with a harmonic trap can provide a new quantum chaotic system that is accessible to experiment. Provided that the harmonic potential varies slowly over a lattice period, the optical lattice has well defined energy bands. The energy-wavevector dispersion curves define an effective Hamiltonian that can be used to calculate classical orbits for atoms confined to a single band. We show that tilting the harmonic trap relative to the optical lattice induces a controllable transition from stable regular motion to classical chaos. The onset of chaos strongly delocalizes the atom orbits and should be manifest in both the classical and quantum properties of the trapped atoms. We also consider how the transition to chaos might affect the collective time-dependent dynamics of interacting atoms in Bose-Einstein condensates.

Quantum chaotic transport in double barrier tunnel structures

Abstract 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.

Using Stochastic Webs to Control the Quantum Transport of Electrons in Semiconductor Superlattices

We show that electrons in a semiconductor superlattice can be used to realize and exploit the unique dynamics of the driven harmonic oscillator that were discovered and explored by George Zaslavsky and colleagues. Under the action of an electric and tilted magnetic field, the semiclassical dynamics of electrons in an energy band of the superlattice exhibit non-KAM chaos, which strongly affects the electrical conductivity. At certain critical field parameters, the electron trajectories change abruptly from fully localized to completely unbounded, and map out intricate stochastic webs in phase space, which act as conduction channels for the electrons. Delocalization of the electron paths produces a series of strong resonant peaks in the electron drift velocity versus electric field curves. We use these drift velocity characteristics to make self-consistent drift-diffusion calculations of the current-voltage and differential conductance-voltage curves of the superlattices, which agree well with our experimental data and reveal strong resonant features originating from the sudden delocalization of the stochastic single-electron paths. We show that this delocalization has a pronounced effect on the distribution of space charge and electric field domains within the superlattices. Inter-miniband tunneling greatly reduces the amount of space-charge buildup, thus enhancing the domain structure and both the strength and number of the current resonances.

The transition to chaos in a wide quantum well

Abstract Experimental and theoretical studies of electron transport in a resonant tunnelling diode with a tilted magnetic field are reported. We map the transition to chaotic dynamics for electrons injected into the potential well from an emitter accumulation layer. In the regime of mixed stable-chaotic electron dynamics, we have identified a stable-orbit trifurcation that leads to a sudden three-fold reduction in the voltage spacing of resonant peaks in the calculated and measured current–voltage curves. We explain this effect by considering how the dynamical properties of the stable orbits affect the quantized states of the potential well.

Chaotic ray dynamics and fast optical switching in micro-cavities with a graded refractive index

Abstract We report a new type of non-linear optical system that can be used to study electromagnetic wave interference effects analogous to quantum chaos, and has potential for applications in lasers and fast optical switches. The system consists of an optical cavity formed from a short planar waveguide with a graded refractive index and obliquely angled reflective ends. We show that the classical ray paths exhibit mixed stable-chaotic dynamics and consider how the intrinsic instability of the chaotic rays could be used to provide ultra-fast optical switching. We investigate how chaotic ray dynamics affect the transmission of electromagnetic waves through the system and find that, for certain cavity modes, the electric field is strongly enchanced along classical periodic ray paths. This effect is analogous to wave function `scarring’ in quantum chaotic systems.