Scar functions in the Bunimovich Stadium billiard

In the context of the semiclassical theory of short periodic orbits, scar functions play a crucial role. These wavefunctions live in the neighbourhood of the trajectories, resembling the hyperbolic structure of the phase space in their immediate vicinity. This property makes them extremely suitable for investigating chaotic eigenfunctions. On the other hand, for all practical purposes reductions to Poincare sections become essential. Here we give a detailed explanation of resonances and scar functions construction in the Bunimovich stadium billiard and the corresponding reduction to the boundary. Moreover, we develop a method that takes into account the departure of the unstable and stable manifolds from the linear regime. This new feature extends the validity of the expressions.Comment: 21 pages, 10 figure

The scar mechanism revisited

Unstable periodic orbits are known to originate scars on some eigenfunctions of classically chaotic systems through recurrences causing that some part of an initial distribution of quantum probability in its vicinity returns periodically close to the initial point. In the energy domain, these recurrences are seen to accumulate quantum density along the orbit by a constructive interference mechanism when the appropriate quantization (on the action of the scarring orbit) is fulfilled. Other quantized phase space circuits, such as those defined by homoclinic tori, are also important in the coherent transport of quantum density in chaotic systems. The relationship of this secondary quantum transport mechanism with the standard mechanism for scarring is here discussed and analyzed.Comment: 6 pages, 6 figure

Beyond the First Recurrence in Scar Phenomena

The scarring effect of short unstable periodic orbits up to times of the order of the first recurrence is well understood. Much less is known, however, about what happens past this short-time limit. By considering the evolution of a dynamically averaged wave packet, we show that the dynamics for longer times is controlled by only a few related short periodic orbits and their interplay.Comment: 4 pages, 4 Postscript figures, submitted to Phys. Rev. Let

Semiclassical basis sets for the computation of molecular vibrational states

In this paper, we extend a method recently reported [F. Revuelta et al., Phys. Rev. E 87, 042921 (2013)] for the calculation of the eigenstates of classically highly chaotic systems to cases of mixed dynamics, i.e., those presenting regular and irregular motions at the same energy. The efficiency of the method, which is based on the use of a semiclassical basis set of localized wave functions, is demonstrated by applying it to the determination of the vibrational states of a realistic molecular system, namely, the LiCN moleculeWe acknowledge financial support of the Spanish Ministry of Economy and Competitiveness (MINECO) under Contract Nos. MTM2012-39101 and MTM2015-63914-P and ICMAT Severo Ochoa under Contract No. SEV-2015-055

Using basis sets of scar functions

We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two-dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the system using a small basis set. An estimate of the basis size is obtained from the mean participation ratio. A thorough analysis of the results using different indicators, such as eigenstate reconstruction in the local representation, scar intensities, participation ratios, and error bounds, is also presentedThis work was supported by MINECO (Spain), under projects MTM2009-14621 and ICMAT Severo Ochoa SEV-2011-0087, and by CEAL Banco de Santander–UAM. F.R. is grateful for the support from a doctoral fellowship from UPM and the hospitality of the members of the Departamento de Física in the Laboratorio TANDAR–Comisión Nacional de la Energía Atómica, where part of this work was don

Superscars in the LiNC=LiCN isomerization reaction

We demonstrate the existence of superscarring in the LiNC=LiCN isomerization reaction described by a realistic potential interaction in the range of readily attainable experimental energies. This phenomenon arises as the effect of two periodic orbits appearing "out of the blue"in a saddle--node bifurcation taking place in the dynamics of the system. Potential practical consequences of this superlocalization in the corresponding wave functions are also considered.Comment: 6 pages, 5 figures. to appear in EP

Sensitivity to perturbations in a quantum chaotic billiard

The Loschmidt echo (LE) measures the ability of a system to return to the initial state after a forward quantum evolution followed by a backward perturbed one. It has been conjectured that the echo of a classically chaotic system decays exponentially, with a decay rate given by the minimum between the width $\Gamma$ of the local density of states and the Lyapunov exponent. As the perturbation strength is increased one obtains a cross-over between both regimes. These predictions are based on situations where the Fermi Golden Rule (FGR) is valid. By considering a paradigmatic fully chaotic system, the Bunimovich stadium billiard, with a perturbation in a regime for which the FGR manifestly does not work, we find a cross over from $\Gamma$ to Lyapunov decay. We find that, challenging the analytic interpretation, these conjetures are valid even beyond the expected range.Comment: Significantly revised version. To appear in Physical Review E Rapid Communication

Parametric Evolution for a Deformed Cavity

We consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where (Q,P) describes a particle moving inside a cavity, and x controls a deformation of the boundary. The quantum-eigenstates of the system are |n(x)>. We describe how the parametric kernel P(n|m) = , also known as the local density of states, evolves as a function of x-x0. We illuminate the non-unitary nature of this parametric evolution, the emergence of non-perturbative features, the final non-universal saturation, and the limitations of random-wave considerations. The parametric evolution is demonstrated numerically for two distinct representative deformation processes.Comment: 13 pages, 8 figures, improved introduction, to be published in Phys. Rev.

Dynamics of quantum trajectories in chaotic systems

Quantum trajectories defined in the de Broglie--Bohm theory provide a causal way to interpret physical phenomena. In this Letter, we use this formalism to analyze the short time dynamics induced by unstable periodic orbits in a classically chaotic system, a situation in which scars are known to play a very important role. We find that the topologies of the quantum orbits are much more complicated than that of the scarring and associated periodic orbits, since the former have quantum interference built in. Thus scar wave functions are necessary to analyze the corresponding dynamics. Moreover, these topologies imply different return routes to the vicinity of the initial positions, and this reflects in the existence of different contributions in each peak of the survival probability function.Comment: 7 pages, 4 figures. Accepted for publication in Europhysics Letter

Computationally efficient method to construct scar functions

Phys. Rev. E 85, 026214-026219 (2012) Desarrollo de un nuevo y eficiente método para la construcción de funciones de scar a lo largo de las órtbitas periódicas inestables de sistemas clásicamente caótico