Unifying S\o rensen-M\o lmer gate and Milburn gate with an optomechanical example

TL;DR

This paper unifies the Sørensen-Mølmer and Milburn geometric phase gates by showing the former as a continuous limit of the latter, providing a comprehensive framework for their analysis and application in optomechanical systems.

Contribution

It demonstrates that Sørensen-Mølmer gate is the continuous limit of Milburn gate, unifying their mathematical descriptions and offering a new approach to analyze their dynamics in optomechanical systems.

Findings

Fidelities differ between the gates under parameter errors.

Purities increase with interaction strength and multiple phase space traversals.

The unified framework simplifies deriving dynamics without differential equations.

Abstract

S\o rensen-M\o lmer gate and Milburn gate are two geometric phase gates, generating nonlinear self-interaction of a target mode via its interaction with an auxiliary mechanical mode, in the continuous and pulsed interaction regime, respectively. In this paper, we aim at unifying the two gates by demonstrating that S\o rensen-M\o lmer gate is the continuous limit of Milburn gate, emphasising the geometrical interpretation in the mechanical phase space. We explicitly consider imperfect gate parameters, focusing on relative errors in time for S\o rensen-M\o lmer gate and in phase angle increment for Milburn gate. We find that, although the purities of the final states increase for the two gates upon reducing the interaction strength together with traversing the mechanical phase space multiple times, the fidelities behave differently. We point out that, the difference exists because the interaction strength depends on the relative error when taking the continuous limit from the pulsed regime, thereby unifying the mathematical framework of the two gates. We demonstrate this unification in the example of an optomechanical system, where mechanical dissipation is also considered. We highlight that, the unified framework facilitates the new method of deriving the dynamics of the continuous interaction regime without solving differential equations.