Abstract
We present theoretical and experimental investigations of higher order correlations of mechanical motion in the recently demonstrated optical tweezer phonon laser, consisting of a silica nanosphere trapped in vacuum by a tightly focused optical beam [R. M. Pettit et al., Nature Photonics 13, 402 (2019)]. The nanoparticle phonon number probability distribution is modeled with the master equation formalism in order to study its evolution across the lasing threshold. Up to fourth-order equal-time correlation functions are then derived from the probability distribution. Subsequently, the master equation is transformed into a nonlinear quantum Langevin equation for the trapped particle's position. This equation yields the non-equal-time correlations, also up to fourth order. Finally, we present experimental measurements of the phononic correlation functions, which are in good agreement with our theoretical predictions. We also compare the experimental data to existing analytical Ginzburg-Landau theory where we find only a partial match.
Highlights
In addition to the usual center-of-mass mechanical motion, degrees of freedom such as spin [17,18,19] and charge [20,21] can be accommodated by the levitated nanoparticles
We have investigated equal-time and non-equal-time higher order correlations for the levitated nanoparticle phonon laser using theoretical as well as experimental methods
The modulation-evolution of phonon number probability distribution was obtained by numerical methods
Summary
Based on their extreme isolation from the environment, simplicity, and ready manipulability, optically levitated nanoparticle systems are promising platforms for investigating the fundamental principles of quantum mechanics [1,2,3,4], statistical physics [5,6,7,8,9], nonlinear science [10,11,12,13] and precision measurement [14,15,16]. We have proposed and demonstrated a phonon laser based on a nanosphere levitated in an optical tweezer by applying linear and nonlinear feedback on a center-of-mass mode of oscillation [27]. The uses of the phonon laser demonstrated earlier by us are expected to be practical (generating nonclassical mechanical states starting with coherent phonons) as well as conceptual (exploring analogies to the optical laser). The higher order correlation functions of the phonon number operator are considered up to fourth order to quantify the degree of coherence throughout the transition.
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