Related papers: Hear $\pi$ from Quantum Galperin Billiards
In "Playing Pool with $\pi$", Galperin invented an extraordinary method to learn the digits of $\pi$ by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's…
In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number $\pi$. This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of $\pi$ in a base…
This paper studies a well-known $\pi$ machine illustrated by Fig.(1). It is shown that the $\pi$ machine can compute digits of $\pi$ if the ratio of block weights, $m_2/m_1$, satisfies certain conditions, and that dynamics of the $\pi$…
We study finite two dimensional spin lattices with definite geometry (spin billiards) demonstrating the display of collective integrable or chaotic dynamics depending on their shape. We show that such systems can be quantum simulated by…
Have you ever played or watched a game of pool? If so, you have already seen a billiard system in action. In mathematics and physics, a billiard system describes a ball that moves in straight lines and bounces off walls. Despite these…
Quantum billiards are a key focus in quantum mechanics, offering a simple yet powerful model to study complex quantum features. While the development of algebras for quantum systems is traced from one-dimensional integrable models to…
Integrability of a square billiard is spontaneously broken as it rotates about one of its corners. The system becomes quasi-integrable where the invariant tori are broken with respect to a certain parameter, $\lambda = 2E/\omega^{2}$ where…
The Halperin $(m,m',n)$ bilayer quantum Hall states are studied on thin cylinders. In this limit, charge density wave patterns emerge that are characteristic of the underlying quantum Hall state. The general patterns are worked out from a…
Quantum billiards have been simulated so far in many ways, but in this work a new aproximation is considerated. This study is based on the quantum billiard already obtained by others authors via a tensor product of two 1-D quantum walks .…
We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various…
We consider a billiard model of a self-bound, interacting three-body system in two spatial dimensions. Numerical studies show that the classical dynamics is chaotic. The corresponding quantum system displays spectral fluctuations that…
A method of obtaining the number pi is considered, which derives pi from the number of elastic collisions between two blocks and a wall.
We study the problem of arithmetic billiards from a new perspective. We first raise a similar problem about reflecting lights inside grids. For the solution to this problem, we will give three proofs. Next, we consider a similar problem in…
The spectral properties of a two-dimensional microwave billiard showing threefold symmetry have been studied with a new experimental technique. This method is based on the behavior of the eigenmodes under variation of a phase shift between…
Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in $d$-dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B.…
Generic one-parameter billiards are studied both classically and quantally. The classical dynamics for the billiards makes a transition from regular to fully chaotic motion through intermediary soft chaotic system. The energy spectra of the…
We study the quantum behaviour of chaotic billiards which exhibit classically diffusive behaviour. In particular we consider the stadium billiard and discuss how the interplay between quantum localization and the rich structure of the…
We study the quantum behaviour of the stadium billiard. We discuss how the interplay between quantum localization and the rich structure of the classical phase space influences the quantum dynamics. The analysis of this model leads to new…
By making use of the greatest common divisor's ($gcd$) properties we can highlight some connections between playing billiard inside a unit square and the Fibonacci sequence as well as the Euclidean algorithm. In particular by defining two…
One of the key challenges in quantum information is coherently manipulating the quantum state. However, it is an outstanding question whether control can be realized with low error. Only gates from the Clifford group -- containing $\pi$,…