Abstract
The Schrödinger equation for a quantum particle in a two-dimensional triangular billiard can be written as the Helmholtz equation with a Dirichlet boundary condition. We numerically explore the quantum entanglement of the eigenfunctions of the triangle billiard and its relation to the irrationality of the triangular geometry. We also study the entanglement dynamics of the coherent state with its center chosen at the centroid of the different triangle configuration. Using the von Neumann entropy of entanglement, we quantify the quantum entanglement appearing in the eigenfunction of the triangular domain. We see a clear correspondence between the irrationality of the triangle and the average entanglement of the eigenfunctions. The entanglement dynamics of the coherent state shows a dependence on the geometry of the triangle. The effect of quantum squeezing on the coherent state is analyzed and it can be utilize to enhance or decrease the entanglement entropy in a triangular billiard.
Highlights
IntroductionThere are many interesting studies concerning the quantum and classical properties of the two-dimensional geometries like the Robnik billiard [1,2] and the Bunimovich stadium [3,4]
There are many interesting studies concerning the quantum and classical properties of the two-dimensional geometries like the Robnik billiard [1,2] and the Bunimovich stadium [3,4].A quantum particle inside a triangular potential is another interesting problem studied by several researchers [5,6,7,8,9]
We explain the quantum entanglement in the reduced two-particle system, which is identical to the single particle trapped in a two-dimensional triangular domain
Summary
There are many interesting studies concerning the quantum and classical properties of the two-dimensional geometries like the Robnik billiard [1,2] and the Bunimovich stadium [3,4]. We explain the quantum entanglement in the reduced two-particle system, which is identical to the single particle trapped in a two-dimensional triangular domain. Generalizing our geometry from the simple equilateral triangle to the irrational triangle, the boundary of the domain. In the usual sense, building triangular billiards can be done by assigning rational or irrational values to the ratio between the inner angles of the triangle and π. In order to construct the irrational triangular billiard, he has considered acute triangles with sides N, N + 1, and N + 2, where N is an integer. This scheme has the advantage that each triangle can be solely identified with the parameter N.
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