Operator-based derivation of phonon modes and characterization of correlations for trapped ions at zero and finite temperature

TL;DR

This paper introduces an operator-based method to derive phonon spectra and analyze correlations in trapped ion chains at various temperatures, providing insights into their quantum and thermal properties.

Contribution

It presents a novel operator-based formalism for deriving the phonon spectrum and characterizing correlations in trapped ions, applicable at zero and finite temperatures.

Findings

Derived the complete phonon spectrum and normal form of the Hamiltonian.

Computed spatial correlations, heat capacity, and susceptibility for different regimes.

Quantified energy reduction due to quantum correlations in the ground state.

Abstract

We present a self-contained operator-based approach to derive the spectrum of trapped ions. This approach provides the complete normal form of the low energy quadratic Hamiltonian in terms of bosonic phonons, as well as an effective free particle degree of freedom for each spontaneously broken spatial symmetry. We demonstrate how this formalism can directly be used to characterize an ion chain both in the linear and the zigzag regimes. In particular we compute, both for the ground state and finite temperature states, spatial correlations, heat capacity and dynamical susceptibility. Last, for the ground state which has quantum correlations, we analyze the amount of energy reduction compared to an uncorrelated state with minimum energy, thus highlighting how the system can lower its energy by correlations.