(1)Bifurcations and the transition to chaos for the resonant tunneling diode, Monteiro T.S., Delande D.,Fisher A J, Boebinger G S Phys.Rev B 56 3913 (1997); also Monteiro TS and Dando P, Phys.Rev.E 53 3369(1996).
(2)Quantum Chaos with Non-periodic, Complex orbits , Saraga D S and Monteiro T S., Phys.Rev Lett 81 5796 (1998)
(3)Semiclassical Gaussian matrix elements for chaotic quantum wells , Saraga D S and Monteiro T S., Nonlinearity 13 1613 (2000).
(4)Quantum wells in tilted fields: semiclassical amplitudes and phase coherence times (PDF) Monteiro T S and Saraga D S, Foundations of Physics, Volume 31 p355-70 (2001) (Festschrift issue for Martin Gutzwiller).
|
CHAOS AND TUNNELING DIODES
A 1994 experiment in Nottingham found that Resonant Tunneling Diodes (RTD) in tilted fields exhibited oscillations in the tunnelling current which corresponded to classical periodic orbits which survive in the chaotic regime. This correspondence between classical chaos and quantum behaviour is predicted by Gutzwiller theory, a well-known tool of quantum chaos based on Feynman's sum-over-paths quantum theory. In the RTD, as the magnetic field is tilted away from the applied electric field, the classical dynamics become increasingly chaotic. Thus the RTD provided an interesting new paradigm of chaos in a mesoscopic system. The short periodic orbits underwent a number of interesting period-doubling and saddle-point bifurcations, all of which gave a clear signature in the corresponding quantum spectrum. They were also gave rise to nice "quantum scars". An example is shown on the right. A large study of this system was undertaken by the Boebinger group in Bell-Labs and we were fortunate to be given access to this data. Most experimental data corresponded to the mixed phase-space regime where there are cascades of bifurcations. The Classical Phase-space plot on the right (from Ref(3)) shows the two of the most important short Periodic Orbits (POs)) In Ref[1] we found that this system has a scaling property which makes it especially amenable to quantitative comparison between quantum theory and classical dynamics: you can identify stretches of thousands of energy states all corresponding to the same classical regime. We found that the main PO (t0) is the Figure on the right) underwent Saddle-point bifurcations, which anihilate the orbit: it becomes a "ghost" (an orbit which is only periodic in complex phase-space ie if position and momentum can take imaginary values). |
|
|
GHOST OR NOT?? However a quantitative description of the quantum current using the usual form of the sum-over-classical paths (sums over isolated Periodic Orbits) approach was shown not to be sufficient; it did not give quantitative agreement with the observed current- even if ghosts were included. The reason is that the current corresponds to a weighted spectrum (ie how much do the states overlap with the input to the well and are thus able to contribute to the current). In the Feynmann sum-over-paths theory, this weighting selects different trajectories. On the right we show some of Greg Boebinger's data. There are oscillations below the scaled field value of 6500, where there is NO real Periodic Orbit. There is a "ghost" here, but the theory predicts that its oscillations would be too weak to explain oscillations in this and other data. A whole "zoo" of different types of paths were proposed (Normal Orbits, Closed orbits etc) by ourselves and others. Ref[3] gives a review of these and compares them with the experiment and the numerical quantum spectra. None were entirely satisfactory. Finally, a theory (Ref[2]) based on a new type of complex (ie with imaginary time and coordinates), non-periodic 'Saddle Orbit' provided an analytical formula which acurately predicted the phase and amplitude of observed oscillations in the tunnelling current: it did not break down at PO bifurcations/ ghosts and is nearly always "isolated" (in the sense of its action not being close to another periodic orbit's: that can complicate the analysis). |
|
