Dr. Perotti is currently working in collaboration with Dr. D. Bessis on the development of a new method in the spectral analysis of noisy time-series data for damped oscillators which makes use of J-matrices to evaluate Padé Approximations built on the time-series Z-transform. Applications of the method range from the detection of Gravitational Wave Bursts to the analysis of Magnetic Resonance data.
While in In Atlanta Dr. Perotti perfected codes which -starting from a suitable Pade’ approximant of the desired transmittance- reconstructed, through inverse scattering techniques, the corresponding monodimensional effective self-consistent potential having no bound states. In collaboration with Dr. A. Mezincescu he also wrote codes to calculate and subtract from the potential thus obtained the self-consistent field for a given density of donor dopant ions and finally –trough a change of coordinates- obtain the chemical composition profile (solution of the variable mass BenDaniel and Duke’s equation). The technique allows to efficiently design semiconductor heterostructures given the desired electron reflectance. They furthermore developed codes that, using double Darboux transformations on a given monodimensional potential with one bound state, maximize the asymmetry of the continuum function at a given energy having nonzero dipole matrix element with the bound state. The potential thus obtained optimizes the probability for an electron excited by incident light of a given frequency to be emitted in the desired direction. While In Munich Dr. Perotti developed -for comparison with laboratory experiments- semiclassical and full quantum Monte Carlo simulations of an ion moving in a one-dimensional optical lattice and a weak confining static electric field while at the same time being subject to a periodic driving force. The solution of the Fokker-Planck equation for the system in the diffusion limit, found in collaboration with Dr. V. Alekseev of the Lebedev Physical Institute in Moscow, allows to explain the apparent rise of the spatial diffusion coefficient when increasing the periodic force amplitude. In Pittsburgh Dr. Perotti worked on Classical numerical simulations and calculation of the Lyapunov exponents for a system consisting of an ensemble of (two-level) Rydberg atoms collectively interacting with its own selfconsistent field and an external microwave field in a cavity. By changing the strength of the external field and/or its detuning from the chosen atomic transition, the behavior of the system can be made to change from regular to weakly chaotic and finally to strongly chaotic. The system moreover offers a convenient way to change the total action by varying the number of interacting atoms. It is therefore considered a convenient system to explore the different dependence on the total action of the time at which quantum evolution begins to significantly deviate from the classical one in the case of regular and chaotic systems [G. P. Berman, E. N. Bulgakov and D. D. Holm, Los Alamos Report LA-UR-93_2187 (1993)]. In the first case this time -called “quantum break-time”- is expected to be a polynomial function of the action, in the latter a logarithmic one. His simulations were aimed at identifying the parameter ranges for the different regimes (regular, strongly chaotic, weakly chaotic) and finding their signature in the (averaged) quantities to be measured in the laboratory. He also conducted Classical and quantum simulations of the monodimensional hydrogen atom in a pulsed microwave field and a collinear constant static electric field, both with and without noise. These were compared with experimental results for atoms prepared in extreme Stark states in a static electric field and made to interact with a pulse of microwaves polarized collinear to the static field used to align the atoms. Furthermore he calculated the instantaneous quantum quasienergy (Floquet) levels for the above system and the Husimi functions of the corresponding states and of the projections on those states of the pulsed wavefunction to help interpret the results of the above quantum numerical simulations. In particular he was able to show the relevance of small classical phase space structures to pulsed quantum evolution. In Milano Dr. Perotti helped develop the codes used to numerically investigate the quantum dynamical localization in action (principal quantum number) of the electron probability for a mono (and bi-) dimensional hydrogen atom in a microwave field when the corresponding classical system is chaotic. He has applied similar techniques to the study of the stochastic ionisation of Alkali-metal atoms: microwave experiments with Alkali atoms are easier to perform than with Hydrogen; on the other hand, core effects cause significant deviations from the hydrogenic behaviour. The numerical simulations published by A. Buchleitner and his collaborators are in very good agreement with the laboratory experiment but are very time consuming; moreover only very qualitative arguments and no model has yet been advanced to interpret the regimes where Alkali ionisation deviates from that of Hydrogen. For regime II of A. Buchleitner and coworkers, where quantum effects are still dominant, Dr. Perotti developed an evaluation of the ionisation threshold based on the similarities between the phase space structures of Alkali in monochromatic microwave fields and Hydrogen in a bichromatic field. He has also shown that in the regime of very low microwave frequencies (regime III of A. Buchleitner and coworkers), a classical explanation is possible in terms of Chirikov’s criterion for transition to chaos. Recent research interests of Dr. Perotti also include the quantum double pendulum: a low dimensional autonomous system which in the classical limit displays chaotic behavior . It is simple enough to allow an extensive study which will be able to shed light on more complex chaotic autonomous systems being studied currently: e.g. the Hydrogen atom in crossed static electric and magnetic fields and the Helium atom. He has performed a study of energy curves and compared Husimi functions of eigenstates at important avoided crossings with the corresponding classical surfaces of section. He is now studying the behavior of suitable packets.