Newton series expansion of bosonic operator functions

TL;DR

This paper introduces a Newton series expansion method for functions of bosonic operators, providing a normal-ordered, systematic approach that overcomes limitations of Taylor series, with applications in spin representations and quantum counting statistics.

Contribution

It develops a novel Newton series expansion framework for bosonic operator functions, enabling exact and systematic operator expansions and expectation value calculations.

Findings

Provides an exact finite-term series expansion for spin operators.

Facilitates calculation of expectation values in coherent states.

Connects normal ordering with Newton and Taylor series via Mellin transform.

Abstract

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.