Quantum calculus of Fibonacci divisors and Fermion–Boson entanglement for infinite hierarchy of \(N=2\) supersymmetric golden oscillators

Abstract

The quantum calculus with two bases, represented by powers of the golden and silver ratios, relates the Fibonacci divisor derivative with Binet formula for the Fibonacci divisor number operator, acting in the Fock space of quantum states. It provides a tool to study the hierarchy of golden oscillators with energy spectrum in the form of Fibonacci divisor numbers. We generalize this model to the supersymmetric number operator and corresponding Binet formula for the supersymmetric Fibonacci divisor number operator. The operator determines Hamiltonian of the 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 introducing the supersymmetric Fibonacci divisor annihilation operator, we construct the hierarchy of supersymmetric coherent states as eigenstates of this operator. The entanglement of fermions with bosons in these states is calculated by the concurrence, represented as the Gram determinant and expressed in terms of the hierarchy of golden exponential functions. We show that the reference states and the corresponding von Neumann entropy measuring the fermion–boson entanglement are characterized completely by powers of the golden ratio. We give a geometrical classification of entangled states by the Frobenius ball and interpret the concurrence as the double area of a parallelogram in a Hilbert space.