From bosons and fermions to spins: A multi-mode extension of the Jordan-Schwinger map

TL;DR

This paper extends the Jordan-Schwinger map to multi-mode systems, enabling a faithful spin representation of bosonic and fermionic states, and reveals connections to Gaussian polynomials with applications to entangled states.

Contribution

It provides a systematic method to construct multi-mode spin states from bosonic or fermionic modes, generalizing the Jordan-Schwinger map beyond two modes.

Findings

Explicit relations for three-mode Fock and spin states

New interpretations of GHZ and W entangled states

Link between degeneracy of multi-mode spin states and Gaussian polynomials

Abstract

The Jordan-Schwinger map is widely employed to switch between bosonic or fermionic mode operators and spin observables, with numerous applications ranging from quantum field theories of magnetism and ultracold quantum gases to quantum optics. While the construction of observables obeying the algebra of spin operators across multiple modes is straightforward, a mapping between bosonic or fermionic Fock states and spin states has remained elusive beyond the two-mode case. Here, we generalize the Jordan-Schwinger map by algorithmically constructing complete sets of spin states over several bosonic or fermionic modes, allowing one to describe arbitrary multi-mode systems faithfully in terms of spins. As a byproduct, we uncover a deep link between the degeneracy of multi-mode spin states in the bosonic case and Gaussian polynomials. We demonstrate the feasibility of our approach by deriving explicit relations between arbitrary three-mode Fock and spin states, which provide novel interpretations of the genuinely tripartite entangled GHZ and W state classes.