The Bell Based Super Coherent States. Uncertainty Relations, Golden Ratio and Fermion-Boson Entanglement

TL;DR

This paper introduces Bell super-coherent states combining fermion-boson entanglement, explores their properties, and reveals a connection between uncertainty relations and Fibonacci numbers converging to the Golden ratio.

Contribution

It presents a novel class of maximally entangled fermion-boson super-coherent states and analyzes their entanglement, orthogonality, and uncertainty relations, highlighting a link to Fibonacci numbers and the Golden ratio.

Findings

States are characterized by displaced Fock states and photon added coherent states.

Entanglement is independent of the coherent state parameter and time.

Uncertainty ratios converge to the Golden ratio as sequence index increases.

Abstract

The set of maximally fermion-boson entangled Bell super-coherent states is introduced. A superposition of these states with separable bosonic coherent states, represented by points on the super-Bloch sphere, we call the Bell based super-coherent states. Entanglement of bosonic and fermionic degrees of freedom in these states is studied by using displacement bosonic operator. It acts on the super-qubit reference state, representing superposition of the zero and the one super-number states, forming computational basis super-states. We show that the states are completely characterized by displaced Fock states, as a superposition with non-classical, the photon added coherent states, and the entanglement is independent of coherent state parameter $\alpha$ and of the time evolution. In contrast to never orthogonal Glauber coherent states, our entangled super-coherent states can be orthogonal. The uncertainty relation in the states is monotonically growing function of the concurrence and for entangled states we get non-classical quadrature squeezing and representation of uncertainty by ratio of two Fibonacci numbers. The sequence of concurrences, and corresponding uncertainties $\hbar F_n/F_{n+1}$, in the limit $n \rightarrow \infty $, convergent to the Golden ratio uncertainty $\hbar/\varphi$, where $\varphi = \frac{1 + \sqrt{5}}{2}$ is found.