Quantum scar is an intriguing phenomenon in quantum or wave dynamics that the wavefunction takes an exceptionally large value around an unstable periodic orbit. It has attracted much attention and advances the understanding of the semiclassical quantization. Most of previous researches involving quantum scars focus on hard-wall quantum billiards. Here we investigate the quantum billiard with a smooth confinement potential which possesses complex classical dynamics. We demonstrate that the semiclassical quantization approach works well for both the stable and unstable classical periodic orbit, besides the fact that the shape of the orbits varies as the energy increases or even the stability switches. The recurrence rule of the quantum scars in this complex solf-wall billiard differs from that of the hard-wall nonrelativistic quantum billiard, such as being equally spaced in energy instead of being equally spaced in the square root of energy. These results implement the previous knowledge and may be used for understanding the measurements of density of states and transport properties in two-dimensional electron systems with random long-range impurities.
Quantum scar is an intriguing phenomenon in quantum or wave dynamics that the wavefunction takes an exceptionally large value around an unstable periodic orbit. It has attracted much attention and advances the understanding of the semiclassical quantization. Most of previous researches involving quantum scars focus on hard-wall quantum billiards. Here we investigate the quantum billiard with a smooth confinement potential which possesses complex classical dynamics. We demonstrate that the semiclassical quantization approach works well for both the stable and unstable classical periodic orbit, besides the fact that the shape of the orbits varies as the energy increases or even the stability switches. The recurrence rule of the quantum scars in this complex solf-wall billiard differs from that of the hard-wall nonrelativistic quantum billiard, such as being equally spaced in energy instead of being equally spaced in the square root of energy. These results implement the previous knowledge and may be used for understanding the measurements of density of states and transport properties in two-dimensional electron systems with random long-range impurities.
We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.
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.
Quantum scars are tracks of enhanced probability densities along unstable classical periodic orbits in quantum chaotic systems. They are a visual representation of classical-quantum correspondence as well as an example of quantum suppression of chaos. A new type of quantum scarring was recently discovered in radially symmetric two-dimensional potential wells perturbed by localized impurities. These perturbation-induced scars show potential to be utilized in controlled applications in classically chaotic quantum systems due to the ability to efficiently propagate wave packets along the scars. In this thesis, the focus is on special linearly shaped perturbation-induced quantum scars which correspond to the classical motion of a ball bouncing between two walls. These bouncing ball quantum scars have potential to be used as efficient channels in quantum transport, but their behavior is still mostly unexplored. This thesis studies how bouncing ball scars could be controlled by changing the perturbation parameters and how stable these scars are against external noise. Both of these features are relevant when utilization in experimental applications is considered. Bouncing ball scars are studied in a radially symmetric two-dimensional (2D) potential well perturbed by a singular Gaussian-like potential bump or a dip, which are placed into the same location. For the stability calculations, randomly placed smaller Gaussian bumps are also placed into the potential with a constant uniform mean density in addition to the main perturbation. Imaginary time propagation algorithm is then applied to solve the 2D Schrödinger equation in our model systems. Bouncing ball scars are detected from the solved eigenstates by integrating the probability densities of the states inside narrow detection boxes. The results indicate that the singular potential bump and the dip affect the same states of the clean system in different ways. The bump seems to pin the bouncing ball scars onto itself, when the dip causes the scars to avoid it. This means that the different perturbation types have the opposite scar orientation preferences. The abundance of the bouncing ball scars seems to also be connected to the perturbation parameters, and the results indicate that the parameters could be optimized in order to maximize the abundance. The stability results show impressive robustness against external noise. The number of bouncing ball scars remain nearly constant with weaker noise and even when the noise bumps were 10% from the main perturbation size, a notable amount of bouncing ball scars persist. It also seems that the scars retain their orientations even in the noisy systems.
Quantum many-body scars (QMBSs) represent a weak ergodicity-breaking phenomenon that defies the common scenario of thermalization in closed quantum systems. They are often regarded as a many-body analog of quantum scars (QSs)—a single-particle phenomenon in quantum chaos—due to their superficial similarities. However, unlike QSs, a clear connection between QMBSs and classical chaos has remained elusive. It has nevertheless been speculated that in an appropriate semiclassical limit, QMBSs should have a correspondence to weakly unstable periodic orbits. In this paper, I present a counterexample to this conjecture by studying a bosonic model with a large number of flavors. The dynamics of out-of-time-ordered correlators (OTOCs) suggest that QMBS do not display chaotic behavior in the semiclassical limit. In contrast, chaotic dynamics are expected for initial states not associated with QMBS. Interestingly, the anomalous OTOC dynamics persist even under weak perturbations that eliminate the scarred eigenstates, suggesting a certain robustness in the phenomenon.
We extend the asymptotic boundary layer (ABL) method, originally developed for stable resonator modes, to the description of individual wave functions localized around unstable periodic orbits. The formalism applies to the description of scar states in fully or partially chaotic quantum systems, and also allows for the presence of smooth and sharp potentials, as well as magnetic fields. We argue that the separatrix wave function provides the largest contribution to the scars on a single wave function. This agrees with earlier results on the wave-function asymptotics and on the quantization condition of the scar states. Predictions of the ABL formalism are compared with the exact numerical solution for a strip resonator with a parabolic confinement potential and a magnetic field.
A way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SWF's as well as to calculate energy spectra for the circular and rectangular billiards as well as to construct the simplest SWF's and the respective spectrum for the Bunimovich stadium. The scar phenomena are considered and a possibility of their description by the skeleton method is discussed. PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv Key Words: Schrödinger equation, semiclassical expansion, Lagrange manifolds, classical trajectories, chaotic dynamics, quantum chaos, scars