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

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.

Quantum simulation of extended polaron models using compound atom-ion systems

“…A convenient approach to calculate the phonon spectrum in the general case has been provided, e.g., in Ref. [63] (see Appendix A for more details). Each ion can be associated with its local harmonic oscillator frequency…”

“…A convenient approach to calculate the phonon spectrum for an arbitrary array of ions has been provided by Bissbort et al [63]. The procedure begins by finding the classical equilibrium positions of the ions and expansion of the potential energy around the equilibrium up to the second order…”

Quantum simulation of extended polaron models using compound atom-ion systems

“…A convenient approach to calculate the phonon spectrum in the general case has been provided, e.g., in Ref. [57] (see Appendix A for more details). Each ion can be associated with its local harmonic oscillator frequency defined as Ω j = Vjj M , where…”

“…Appendix A: Hamiltonian derivation details 1. Phonon modes A convenient approach to calculate the phonon spectrum for an arbitrary array of ions has been provided by Bissbort et al [57]. The procedure begins by finding the classical equilibrium positions of the ions, and expansion of the potential energy around the equilibrium up to the second order…”