Quantum Calculus of Fibonacci Divisors and Fermion-Boson Entanglement for Infinite Hierarchy of N = 2 Supersymmetric Golden Oscillators

TL;DR

This paper develops a quantum calculus framework based on Fibonacci divisors and Golden ratios to analyze supersymmetric Golden oscillators, revealing entanglement properties and geometric classifications of states.

Contribution

It introduces a supersymmetric Fibonacci divisor number operator and constructs a hierarchy of supersymmetric Golden oscillators with novel entanglement measures.

Findings

Eigenstates are double degenerate and characterized by the super-Bloch sphere.

Fermion-boson entanglement is quantified by concurrence linked to Golden ratios.

States are geometrically classified using Frobenius ball and parallelogram areas.

Abstract

The quantum calculus with two bases, as powers of the Golden and the Silver ratio, relates Fibonacci divisor derivative with Binet formula of Fibonacci divisor number operator, acting in Fock space of quantum states.It provides a tool to study the hierarchy of Golden oscillators with energy spectrum in form of Fibonacci divisor numbers. We generalize this model to supersymmetric number operator and corresponding Binet formula for supersymmetric Fibonacci divisor number operator. The operator determines the Hamiltonian of hierarchy of supersymmetric Golden oscillators, acting in fermion-boson Hilbert space and belonging to N=2 supersymmetric algebra. The eigenstates of the super Fibonacci divisor number operator are double degenerate and can be characterized by a point on the super-Bloch sphere. By the supersymmetric Fibonacci divisor annihilation operator, we construct the hierarchy of supersymmetric coherent states as eigenstates of this operator. Entanglement of fermions with bosons in these states is calculated by the concurrence, represented by the Gram determinant and hierarchy of Golden exponential functions. We show that the reference states and corresponding von Neumann entropy, measuring fermion-boson entanglement, are characterized completely by the powers of the Golden ratio. The simple geometrical classification of entangled states by the Frobenius ball and meaning of the concurrence as double area of parallelogram in Hilbert space are given.