Motional frequency shifts of trapped ions in the Lamb-Dicke regime

TL;DR

This paper analyzes how vibronic transitions influence motional frequency shifts in trapped ions within the Lamb-Dicke regime, providing analytical expressions for these shifts in Rabi and Ramsey schemes, independent of initial vibrational states.

Contribution

It offers the first detailed quantification of vibronic transition effects on frequency shifts in the Lamb-Dicke regime, including analytical formulas for Rabi and Ramsey schemes.

Findings

Frequency shifts are independent of initial vibrational states.

Analytical expressions for frequency shifts are derived.

Vibronic transitions cause measurable shifts in the carrier absorption band.

Abstract

First order Doppler effects are usually ignored in laser driven trapped ions when the recoil frequency is much smaller than the trapping frequency (Lamb-Dicke regime). This means that the central, carrier excitation band is supposed to be unaffected by vibronic transitions in which the vibrational number changes. While this is strictly true in the Lamb-Dicke limit (infinitely tight confinement), the vibronic transitions do play a role in the Lamb-Dicke regime. In this paper we quantify the asymptotic behaviour of their effect with respect to the Lamb-Dicke parameter. In particular, we give analytical expressions for the frequency shift, ``pulling'' or ``pushing'', produced in the carrier absorption band by the vibronic transitions both for Rabi and Ramsey schemes. This shift is shown to be independent of the initial vibrational state.