PHYSICAL REVIEW APPLIED 19, 014031 (2023) Generation and Evolution of Phononic Frequency Combs via Coherent Energy Transfer between Mechanical Modes Jiangkun Sun , 1,2 Sheng Yu, 2 Hemin Zhang , 1 Dongyang Chen, 1 Xin Zhou , 2 Chun Zhao, 1 Dustin D. Gerrard, 3 Ryan Kwon, 3 Gabrielle Vukasin , 3 Dingbang Xiao , 2, * Thomas W. Kenny, 3 Xuezhong Wu, 2 and Ashwin Seshia 1, † 1 Nanoscience Centre, University of Cambridge, Cambridge CB3 0FF, United Kingdom 2 College of Intelligence Science, National University of Defense Technology, Changsha 410073, China 3 Department of Mechanical Engineering, Stanford University, Stanford, California 94305, USA (Received 12 November 2021; revised 12 August 2022; accepted 5 December 2022; published 10 January 2023) Phononic frequency combs represent the mechanical analog of optical frequency combs. Several inde- pendent experimental studies have demonstrated the onset and evolution frequency comb response in a variety of micro- and nanoelectromechanical devices in recent years. A theoretical basis for exploring and understanding the conditions for comb generation and evolution with varying driving parameters is essential to enable future practical applications. Here, we present the comparison between modeling and experimental results on the generation and evolution mechanism of phononic frequency combs in a nonlin- ear micromechanical resonator from the perspective of coherent energy transfer between two mechanical modes. Phononic frequency combs emerge in a strong coupling regime involving nonlinear resonances when the amplitudes and phases of two coupled mechanical modes are modulated via coherent energy transfer. The spacing and number of comb teeth can be analytically estimated based on modeling the nature of the interaction of the coupled modes under the specified driving conditions. As the driving con- ditions are varied, the phononic frequency comb evolves into different forms and a phenomenological model for the system is established to accurately predict the evolution of phononic frequency combs. The alignment between experiment and model provides a basis for the engineering of this phenomenon in future device applications. DOI: 10.1103/PhysRevApplied.19.014031 I. INTRODUCTION A frequency comb is composed of a series of discrete, equally spaced lines in the spectrum. In the optical domain, photonic frequency combs have generated widespread interest due to their diverse applications spanning from frequency metrology [1–4], ultraprecision clocks [5,6], optical communications to quantum information process- ing [7–9]. To enable miniaturization and integration of photonic comb devices, dissipative Kerr solitons in coher- ently pumped high quality-factor optical microresonators have been extensively studied [10–15]. More recently, micromechanical devices have emerged as a platform for the generation of a phononic frequency comb (PFC). Multifrequency parametric mixing mediated via nonlin- ear mode coupling as a path towards PFC generation has previously been experimentally demonstrated in mechan- ical resonators [16–20]. Specifically, the phenomenon of * dingbangxiao@nudt.edu.cn † aas41@cam.ac.uk internal resonance, which satisfies the commensurate rela- tionship between resonant modes and can enable strong mode interaction with weaker signal levels has proved to be an effective way to generate PFCs [21]. For example, some previous studies have demonstrated the phenomenon of PFCs and explored the boundaries of Hopf bifurca- tions in a micromachined resonator [22,23]. However, a theoretical treatment of the experimental observations that accurately describe and predict the generation and evolution of PFCs demonstrating good alignment for the devices under test is essential to extend these observa- tions towards practical applications. This paper establishes a key milestone towards this objective by developing a suitable dynamical model for an established microelec- tromechanical device that accurately captures experimen- tal observations of internal resonance and PFC formation and evolution with varying drive conditions. In this paper, we demonstrate the generation and evo- lution mechanism of a PFC in a micromachined disk resonator based on the strong nonlinear coupling of two mechanical modes through 1:2 internal resonance [24,25]. When one mode is driven by a single actuation signal, 2331-7019/23/19(1)/014031(13) 014031-1 © 2023 American Physical Society