I. Classical and stochastic chaos

The main focus of my research has been so far on systems showing chaotic behavior, meaning unpredictability of the dynamical evolution, due to high sensitivity to initial conditions [1]. Examples range from turbulence in fuids to coupled electric circuits, to atoms in highly excited states interacting with an electromagnetic feld. Such sensitivity to initial conditions makes it pointless to integrate the equations of motion for a long time, as roundoff errors grow exponentially with the time of iteration. As a consequence, it becomes important to develop alternative methods to direct numerical simulations for the evaluation of long-time averages of observables, such as decay of correlations, diffusion coeffcients, energy spectra, or escape rates [2]. To realize that in practice, the main idea is to understand which regions of the phase space are more often visited by the trajectories and thus more important for the dynamics, and which ones are less, to then weigh the averages mentioned above accordingly [3]. In chaos jargon this process is called partitioning the phase space. My first contribution to the feld came in this framework, the specifc task being that to recognize and classify different degrees of instability in two-dimensional chaotic lattice systems ("coupled map lattices" [4]). Making a partition means separating regions of a Poincaré section of the phase space where the dynamics is qualitatively different, and associate a symbol, like "0", "1", etc., to each region. By doing that, every trajectory on the surface of section is characterized by a symbolic sequence, say "...010001110110..." for a binary symbolic dynamics. That means for example, that there is an infinite number of points that occupy a finite area of the section which have sequence "01" when iterated once, and there is a density of points that occupy a smaller area of the section that bear sequence "010" when iterated twice, and an even smaller subset of that area which has sequence "010001", etc.. In other words, in an everywhere unstable chaotic dynamical system, consecutive Poincaré section returns subdivide the phase space into an exponentially growing number of regions, each one indicating trajectories with a different history, and labeled by a distinct symbol sequence. In the unstable directions these regions stretch, while in the stable directions they shrink exponentially. As this process has no end, a chaotic phase space can be divided into smaller and smaller subsets with no limit in its resolution, and turns out to have a fractal structure [5]. In this sense, chaos has no scale. Yet, every physical system is inevitably affected by a certain amount of noise, coming for example from uncertainty in the measurements, roundoff errors in the computations, or neglected degrees of freedom in the models [6]. Although weak noise does not change the chaotic nature of the phase space, it introduces uncertainty in the borders separating regions of different symbols. The uncertainty grows linearly with iteration time, according to the diffusion law, whereas as said, the area of the regions with longer and longer memory (symbolic sequences) shrinks exponentially. As a consequence, at some critical time the uncertainty on the borders is comparable to the size of the neighborhoods, and nearby trajectories become indistinguishable due to noise. In other words, fractals are destroyed, and the phase space acquires a finite resolution. The main achievement of my PhD work has been, in this context, that to determine the finest possible scale at which the phase space can be resolved [7]. As it turns out, such resolution is not represented by a single critical time, as mentioned above, (the analog of Heisenberg time in quantum mechanics). Instead, the resolution changes from region to region in the phase space, depending on the local interaction of the deterministic dynamics with noise. The following methodology is employed to realize that: the phase space is divided according to the deterministic partition, but in each neighborhood we solve a discrete-time version of Fokker-Planck equation [8], whose solutions are "periodic orbits" of the stochastic fow, meaning densities that map into themselves locally. Unlike periodic orbits of the deterministic fow, these solutions (or eigenfunctions of the evolution operator) have a support of finite size, and, as we find more and more of them, they end up covering the whole phase space, until they overlap with each other, and the resolution is set at that point. The effect of background noise on a chaotic system turns out to be advantageous to the computation of the dynamical averages mentioned at the beginning of this section, as I now explain. Every average on the phase space is to be weighted by the invariant measure, a density expressing the rate at which each point is visited by the dynamics, and therefore how important it is. The invariant measure is the first eigenfunction of the spectrum of the Fokker-Planck operator (or just the ground-state solution to Fokker-Planck equation). A finite resolution of the phase space turns this infinite-dimensional operator into a finite-, often low-dimensional matrix, definitely easier to diagonalize. Thus, averages can be computed by numerical diagonalization of such matrix or alternative, more sophisticated methods such as periodic orbit expansions. The latter technique consists of writing the evolution matrix as a Markov graph [9], associating a weight to each closed loop in the graph (an unstable periodic orbit of the dynamics), which accounts for both the stability of the orbit and its sensitivity to noise, and fnally computing the desired average as a weighted sum over all loop contributions. Although conceptually and computationally harder than numerical diagonalization, periodic orbit theory can be straightforwardly extended to higher-dimensional systems, that is why it was employed in this framework. Initially developed and tested for one-dimensional discrete-time dynamical systems, the formalism for our "optimal partition hypothesis" has been generalized to arbitrary dimensions [10]. The problem of finding local eigenfunctions to the discrete-time Fokker-Planck equation is recast into the resolution of the well-known Lyapunov equation for symmetric matrices. Moreover, an extension to the method is proposed to deal with stickiness in the dynamics, that is parts of the phase space where correlations are stronger and decay as a power law instead of exponentially in time.

II. Opening-induced localization in quantum chaotic systems

Quantum dynamics is regulated by linear, wave-like equations of motion, meaning the evolution of any state can be written as a superposition of the solutions to these equations. However, the quantization of a classically chaotic Hamiltonian often shows signatures of "nonlinear" behavior, such as strong mixing [11]. As a result, the wave function intensities are typically distributed as random variables in both real and phase space. More recently, open quantum systems have become of great interest from a theoretical standpoint, as well as for a wide range of applications, spanning from electrical conduction problems, or quantum communication, to optical devices such as dielectric microcavities. In the framework of quantum chaos, the validity of Random Matrix Theory (RMT) to describe the statistics of the wave functions intensity, and possible deviations from it due to localization phenomena are still open issues [12]. In the context of open systems, my current task is to investigate the effects of the opening on the random wave approximation, in particular, looking for opening-induced localization in the wave functions (precisely, in their Husimi representations). The model in consideration is the kicked rotor (which has been also experimentally realized through cold atoms in a modulated optical lattice [13]), a kind of pendulum whose amplitude is modulated via pulses. Aside from looking at Husimi projections, deviations from RMT are investigated by theoretically estimating and numerically computing the energy and loss dependence of the decay of correlations in the scattering process. Some results have already been obtained in this direction [14]. The long-term goal is a comprehensive understanding of the mechanisms of such localization.

[1] E. Ott, "Chaos in Dynamical Systems", Cambridge University Press, 2002.- S.H. Strogatz, "Nonlinear Dynamics and Chaos", Westview Press, 2001.- P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, "Chaos: Classical and Quantum", ChaosBook.org, Niels Bohr Institute, Copenhagen 2009.

[2] D. Ruelle, "Thermodynamic Formalism", Cambridge University Press, 2004. -P. Gaspard, "Chaos, Scattering, and Statistical Mechanics", Cambridge University Press, 1998.

[3] B.P. Kitchen, "Symbolic Dynamics", Springer, 1998

[4] C.P. Dettmann and D. Lippolis, Chaos, Solitons and Fractals, 23, 43-54 (Jan 2005)

[5] B. Mandelbrot, "The fractal geometry of nature", W.H. Freeman and Co., 1982

[6] N.G. van Kampen, "Stochastic Processes in Physics and Chemistry", Elsevier 2007.

[7] D. Lippolis and P. Cvitanovic, Phys. Rev. Lett. 104, 014101 (Jan 2010)

[8] H. Risken, "The Fokker-Planck equation", Srpinger 1996.

[9] Given the exponential decay of correlations in time, and the fact that the noise is also uncorrelated in time, the dynamics turns out to be Markovian.

[10] P. Cvitanovic and D. Lippolis, in M. Robnik and V.G. Romanovski, eds., Let's Face Chaos through Nonlinear Dynamics, pp. 82-126 (Am. Inst. of Phys., Melville, New York, July 2012).

[11] F. Haake, "Quantum signatures of chaos", Springer 2004 .

[12] H. J. Stoeckmann, "Quantum Chaos", Cambridge 1999.

[13] S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and P. S. Jessen, Nature 461, p. 768 .

[14] D. Lippolis, J.W. Ryu, S.Y. Lee, and S.W. Kim, On-manifold localization in open quantum maps, Phys. Rev. E 86,066213 (2012).

updated: May 21 2013