Bulletin of the American Physical Society
2005 APS March Meeting
Monday–Friday, March 21–25, 2005; Los Angeles, CA
Session S23: Quantum Chaos |
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Sponsoring Units: GSNP Chair: Nicholas Cerruti, Washington State University Room: LACC 410 |
Wednesday, March 23, 2005 2:30PM - 2:42PM |
S23.00001: Local Level Velocity Variances and the Fidelity in Integrable Systems Nicholas R. Cerruti, Steven Tomsovic In a chaotic system, a wave packet's Wigner transform will ergodically explore the entire energy surface on an extremely short time scale. However, for integrable systems the Wigner transform will remain on constant action surfaces and can access only a fraction of the available phase space. Thus, the local properties of integrable systems can be more important than the global properties. An example occurs when a system is perturbed and the energy levels are redistributed. Using first order quantum perturbation theory, this redistribution is largely responsible for changes in the evolution of the wave packet. More specifically, it is the local variance of the level velocities, which are defined by the changes in the eigenenergies due to the perturbation, that defines the rate of decoherence between the same initial wave packet evolved through the unperturbed and perturbed systems. We derive a semiclassical expression for these local variances and show an application of the variances to the fidelity. The results are demonstrated in the rectangle billiard which is fully integrable. [Preview Abstract] |
Wednesday, March 23, 2005 2:42PM - 2:54PM |
S23.00002: Nonlinear $\sigma$ -model for a ballistic quantum dot with random boundary absorption Igor Rozhkov, Ganpathy Murthy The problem of evaluation of a two-point function in an integrable ballistic billiard (circular quantum dot) is formulated in terms of the supersymmetric nonlinear $\sigma$ -model. The dot is assumed to be slightly open. It is modeled by attaching the closed circular billiard to the leads with random coupling constants between the eigenstates of the dot and the outgoing states. In the limit of large number of uniformly distributed leads with an infinitesimal couplings to a single channel in each lead we are able to derive a nonlinear $\sigma$ for interacting angular harmonics of the supersymmetric field. Our procedure is done within saddle point approximation without introduction of a disorder potential or diffusive boundary scattering; no energy averaging was performed. Supported by: NSF DMR 0311761 [1]B. A. Muzikantskii and D. E. Khmelnitskii, JETP Lett. 62, 76 (1995). [2] A. V. Andreev, O. Agam, B. D. Simons, and B. L. Altshuler, Phys. Rev. Lett. 76, 3947 (1996). [3] K. B. Efetov and V. R. Kogan, Phys. Rev. B 67, 245312 (2003). [Preview Abstract] |
Wednesday, March 23, 2005 2:54PM - 3:06PM |
S23.00003: Persistent currents for interacting electrons in ballistic/chaotic billiards Oleksandr Zelyak, Ganpathy Murthy We study persistent currents in a quantum billiard enclosing a magnetic flux $\phi$ by analytical and numerical methods. We concentrate on the family of Robnik-Berry billiards generated by conformal maps of the unit disk. We study the persistent current as a function of magnetic flux and parameters of the billiard in the chaotic regime. We include Fermi-liquid interactions in a mean-field approach, justified by the recent large-$N$ approach[1] for ballistic/chaotic quantum dots. [1] G. Murthy, R. Shankar, D. Herman, and H. Mathur, Phys. Rev. B 69, 075321 (2004); G. Murthy, R. Shankar, and H. Mathur, cond-mat/0411280. [Preview Abstract] |
Wednesday, March 23, 2005 3:06PM - 3:18PM |
S23.00004: Weak Localization In Periodical Structures Without Disorders Chushun Tian, Anatoly Larkin he dynamics of a moving particle in some periodic structures exhibits (normal) diffusion at the classical level [1]. We find weak localization phenomena in such structures. Remarkably, no random quantum potentials are introduced so that the analytical treatments do not involve calculations with the regularizer. In periodic structures, an additional random quantum potential does not affect the perturbative regime of localization phenomena (loop expansion). In sharp contrast, in the quantum limit, it leads to strong localization (in the $1$D and $2$D cases), but quantum diffractions result in the (extended) Bloch state due to spatial periodicity. At the semiclassical level, we find the one loop (frequency- dependent) quantum correction to the diffusion constant has the same functional form as chaotic systems [2]. However, the Ehrenfest time, which signals the crossover between a classical and quantum picture is found without any random quantum potentials. The predicted classical-to-quantum crossover may be studied experimentally in periodic quantum dot systems. The results may be helpful for understanding the crossover between ray and wave optics in photonic crystals. [1] P. Gaspard, Phys. Rev. E {\bf 53}, 4379 (1995). [2] I. Aleiner and A. Larkin, Phys. Rev. B {\bf 54}, 14423 (1996). [Preview Abstract] |
Wednesday, March 23, 2005 3:18PM - 3:30PM |
S23.00005: Spectroscopic Evidence of Discrete Energy Levels in Nanosize Clusters of Metal Atoms using a Low Temperature STM Laura Adams, Brian Lang, Allen M. Goldman A new method of obtaining spectroscopic information about clusters is realized through the interplay between the surface states at a metal-semiconductor interface and the discrete electronic energy levels of nanosize ($\sim $ 30 {\AA} in diameter) Pb clusters. When these surface states come into registry with the energy levels of the cluster, resonant peaks emerge in the I(V) characteristics. The histograms of the peak intensities and spacings are consistent with Porter Thomas and Wigner Distributions, respectively. Metallic clusters were fabricated in situ by a buffer layer assisted growth technique developed by Huang, Chey and J. H. Weaver$^{1}$. [1] L. Huang, S. Jay Chey, and J. H. Weaver, ``Buffer-Layer-Assisted Growth of Nanocrystals: Ag-Xe-Si (111)'', PRL 80, 4095 (1998). [Preview Abstract] |
Wednesday, March 23, 2005 3:30PM - 3:42PM |
S23.00006: Geometric Phase of Phase Space Trajectories:Mobius Strip and Nonlinear Oscillators Radha Balakrishnan, Indubala Satija We present a gauge invariant formulation of associating a geometric phase with classical phase space trajectories. This geometric phase which depends upon the integrated torsion of the trajectory, bears a close analogy to the generalized Berry phase associated with the time evolution of the quantum wave functions. This topological quantity serves as an order parameter signalling phase transitions including novel geometrical transitions. One of the interesting aspects seen in Duffing and other nonlinear oscillators is the sudden jumps in the geometric phase which is accompanied by the divergence of the local torsion and the vanishing of the local curvature. Intriguingly, the analogous phenomenon was seen in a mobius strip when the ratio of the width to the length of the strip exceeds beyound a critical value. [Preview Abstract] |
Wednesday, March 23, 2005 3:42PM - 3:54PM |
S23.00007: Spectrum of Geometric Phases in a Driven Oscillator Indu Satija, Radha Balakrishnan We study geometric phases underlying the time evolution of the quantum wave function of a driven nonlinear oscillator exhibiting periodic, quasiperiodic as well as chaotic dynamics. In the asymptotic limit, irrespective of the classical dynamics, the geometric phases are found to increase linearly with time. However, the fingerprints of classical motion are present in the bounded fluctuations that are superimposed on the monotonically growing phases, as well as in the difference in phases between two neighboring quantum states. [Preview Abstract] |
Wednesday, March 23, 2005 3:54PM - 4:06PM |
S23.00008: Topological Singularities and Transport in Kicked Harper Model Indubala Satija Quasienergy spectrum of kicked Harper model is found to exhibit a series of diabolic crossings. These conical degeneracies reside mostly on the symmetry line of its two-dimensional parameter space and their location is found to coincide with the location of maxima of the kinetic energy of the kicked system. Additionally, there are also branch point singularities, the exceptional points, that are associated with avoided crossings and are obtained by analytically continuing the kicking parameter in the complex plane. The location of these singularities also appear to be closely correlated with the maxima and minima of the kinetic energy, suggesting a correlation between the transport and the topological characteristics of the system. [Preview Abstract] |
Wednesday, March 23, 2005 4:06PM - 4:18PM |
S23.00009: Universal Statistics of the Scattering Coefficient of Chaotic Microwave Cavities. Sameer Hemmady, Xing Zheng, Thomas Antonsen, Edward Ott, Steven Anlage We experimentally investigate theoretical statistical predictions [X.Zheng, \textit{et al.} cond-mat/0408327] for the universal scattering coefficient in wave chaotic systems using a microwave analog of a quantum chaotic infinite square well potential [S. Hemmady, \textit{et al. }submitted to Phys.Rev.E]. We consider the statistics of the scattering coefficient $S$ of a two-dimensional chaotic microwave cavity coupled to a single port. The non-universal effects of the coupling in the experimental $S$ data are removed using the radiation impedance of the port, obtained directly from the experiments [S. Hemmady, \textit{et al.} Submitted to Phys. Rev. Lett., cond-mat/0403225]. A normalized scattering coefficient is obtained, and its Probability Density Function (PDF) is predicted to be universal in that it depends only on the loss (quality factor) of the cavity. We compare experimental PDFs of the normalized scattering coefficients for different degrees of quantified loss with those obtained from Random Matrix Theory (RMT), and find excellent agreement. We will discuss how these results apply to scattering measurements on other quantum chaotic systems including those with broken Time Reversal Symmetry. [Preview Abstract] |
Wednesday, March 23, 2005 4:18PM - 4:30PM |
S23.00010: Spectral Statistics of Pointer States in an Open Stadium Billiard Richard Akis, David Ferry In quantum-measurement theory, a point of discussion has been the manner in which the quantum states of a system evolve into classical states. The interaction of a system with the environment has been suggested to lead to einselection[1], the selection of a discrete set of pointer states that remain robust while their superposition with other states is reduced by decoherence. It has been predicted[1] that pointer states are the basis of the transition to classical behavior, and actually possess classical properties. In the case of open quantum dots, such pointer states yield measurable conductance resonances[2]. In our presentation, we shall discuss the results of a new study of the energy level spacing statistics in a stadium quantum dot cavity perturbed by attached leads. When the leads are \textit{closed,} the eigenstates follow the Wigner distribution associated with chaos. However, when the stadium is sufficiently opened to the external environment so that only the pointer states remain resolved, the distribution becomes Poissonian, indicating that these states are intimately associated with the \textit{regular} classical orbits. [1] W. H. Zurek, Rev. Mod. Phys. \textbf{75}, 715 (2003). [2] D. K. Ferry, R. Akis, J.P. Bird, Phys. Rev. Lett. \textbf{93}, 026803 (2004). [Preview Abstract] |
Wednesday, March 23, 2005 4:30PM - 4:42PM |
S23.00011: Complexity of Quantum Spectra Yuri Dabaghian It has been long recognized that the problem of semiclassical evaluation of quantum spectra is fundamentally more difficult for classically chaotic systems than for the classically integrable ones. It appears now that the quantum spectra of the chaotic systems may also differ among themselves by level of their complexity. This is indicated by the hierarchy of the explicit spectral solutions for 1D quantum networks. [Preview Abstract] |
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S23.00012: Chaos Beyond Linearized Stability Analysis: Folding of the Phase
Space and Distribution of Lyapunov Exponents Peter Silvestrov, I.V. Ponomarev We have found~\cite{Reference} a universal mechanism leading to the enhanced probability, $P(\lambda ,t)$, to find small values of the finite time Lyapunov exponent, $\lambda$. In our investigation of chaotic dynamical systems we go beyond the linearized stability analysis of nearby divergent trajectories and consider folding of the phase space in the course of chaotic evolution. We show that the spectrum of the Lyapunov exponents $F(\lambda)= \lim_{t\rightarrow\infty} t^{-1}\ln P(\lambda ,t)$ at the origin has a finite value $F(0)=-\tilde{\lambda}$ and a slope $F'(0)\le 1$. This means that all negative moments of the distribution $\langle e^{-m\lambda t}\rangle$ are saturated by rare events with $\lambda\rightarrow 0$. Extensive numerical simulations confirm the results. Among the practical applications of our findings are the problem of a gap in spectrum of a semiclassical Andreev billiard, conductance fluctuations in a smooth quantum dot and stability to perturbations in chaotic wave-packet dynamics. \begin{thebibliography}{99} \bibitem{Reference} P. G. Silvestrov, I.V.Ponomarev, preprint nlin.CD/0409053. \end{thebibliography} [Preview Abstract] |
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