Engineering $\mathrm{SU}(1,1) \otimes \mathrm{SU}(1,1)$ vibrational states

TL;DR

This paper presents a scheme to generate and manipulate entangled and squeezed vibrational states in a two-dimensional ion trap using specific sideband driving, enabling quantum state engineering for interferometry.

Contribution

It introduces a novel method for preparing $ ext{SU}(1,1) imes ext{SU}(1,1)$ vibrational states with reversible dynamics in ion traps, expanding quantum state engineering capabilities.

Findings

Generation of $ ext{SU}(1,1)$ Perelomov coherent states.

Engineering of lossless 50/50 $ ext{SU}(2)$ beam splitter states.

Potential use of generated states as resources for interferometry.

Abstract

We propose an ideal scheme for preparing vibrational $\mathrm{SU(1,1)} \otimes \mathrm{SU(1,1)}$ states in a two-dimensional ion trap using red and blue second sideband resolved driving of two orthogonal vibrational modes. Symmetric and asymmetric driving provide two regimes to realize quantum state engineering of the vibrational modes. In one regime, we show that time evolution synthesizes so-called $\mathrm{SU}(1,1)$ Perelomov coherent states, that is separable squeezed states and their superposition too. The other regime allows engineering of lossless 50/50 $\mathrm{SU}(2)$ beam splitter states that are entangled states. These ideal dynamics are reversible, thus, the non-classical and entangled states produced by our schemes might be used as resources for interferometry.