We study quantum localization phenomena in chaotic systems with a parameter. The parametric motion of energy levels proceeds without crossing any other and the defined avoided crossings quantify the interaction between states. We propose the elimination of avoided crossings as the natural mechanism to uncover localized structures. We describe an efficient method for the elimination of avoided crossings in chaotic billiards and apply it to the stadium billiard. We find many scars of short periodic orbits revealing the skeleton on which quantum mechanics is built. Moreover, we have observed strong interaction between similar localized structures.
We study quantum localization phenomena in chaotic systems with a parameter. The parametric motion of energy levels proceeds without crossing any other and the defined avoided crossings quantify the interaction between states. We propose the elimination of avoided crossings as the natural mechanism to uncover localized structures. We describe an efficient method for the elimination of avoided crossings in chaotic billiards and apply it to the stadium billiard. We find many scars of short periodic orbits revealing the skeleton on which quantum mechanics is built. Moreover, we have observed strong interaction between similar localized structures.
We study chaotic quantum billiards using both microwave cavities and numerical simulations. For the same geometry, viz., a Sinai billiard, agreement to remarkable precision is found for both the eigenvalue magnitudes and the spatial detail of the eigenfunctions. The association of the eigenfunctions with classical periodic orbits is demonstrated, and scarred states are identified. Desymmetrizing the Sinai billiard by slightly moving the central disk is shown to lead to strong localization of the eigenfunction. The calculated eigenstates of the symmetric billiard show an even- and odd-parity pair whose linear combination gives the localized state.
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium-a chaotic billiard-level statistics described by a Brody distribution with parameter delta similar or equal to 0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)similar or equal to 1150 compared to the rectangular billiard (regular) case, where the average PR similar or equal to 1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter δ≃0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)≃1150 compared to the rectangular billiard (regular) case, where the average PR≃1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.
Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.
The surprisingly long-lasting oscillations observed in the dynamics of highly excited states of chains of Rydberg atoms defy the expectation that interacting systems should thermalize fast. The phenomenon is reminiscent of wavepackets in quantum billiards that trace classical periodic orbits. While analogs of the associated scarred eigenfunctions have been found for Rydberg chains, an underlying classical limit hosting periodic orbits has remained elusive. Here we generalize the Rydberg (pseudospin S=1/2) system to a chain of arbitrary spin S. Its classical limit features unexpectedly stable periodic orbits that are essential to understand the emergence of robust, parametrically suppressed quantum chaoticity, with semi-classical coherence times diverging as S. The classical limit successfully explains several empirical features of the quantum limit.