Wang Mingkang, Perez-Morelo Diego J, Lopez Daniel, Aksyuk Vladimir A
Microsystems and Nanotechnology Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA.
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA.
Phys Rev X. 2022;12(4). doi: 10.1103/physrevx.12.041025.
Many nonlinear systems are described by eigenmodes with amplitude-dependent frequencies, interacting strongly whenever the frequencies become commensurate at internal resonances. Fast energy exchange via the resonances holds the key to rich dynamical behavior, such as time-varying relaxation rates and signatures of nonergodicity in thermal equilibrium, revealed in the recent experimental and theoretical studies of micro- and nanomechanical resonators. However, a universal yet intuitive physical description for these diverse and sometimes contradictory experimental observations remains elusive. Here we experimentally reveal persistent nonlinear phase-locked states occurring at internal resonances and demonstrate that they are essential for understanding the transient dynamics of nonlinear systems with coupled eigenmodes. The measured dynamics of a fully observable micromechanical resonator system are quantitatively described by the lower-frequency mode entering, maintaining, and exiting a persistent phase-locked period-tripling state generated by the nonlinear driving force exerted by the higher-frequency mode. This model describes the observed phase-locked coherence times, the direction and magnitude of the energy exchange, and the resulting nonmonotonic mode energy evolution. Depending on the initial relative phase, the system selects distinct relaxation pathways, either entering or bypassing the locked state. The described persistent phase locking is not limited to particular frequency fractions or types of nonlinearities and may advance nonlinear resonator systems engineering across physical domains, including photonics as well as nanomechanics.
许多非线性系统由频率与振幅相关的本征模描述，每当频率在内部共振时变得可公度，它们就会强烈相互作用。通过共振的快速能量交换是丰富动力学行为的关键，比如时变弛豫率以及热平衡中非遍历性的特征，这些已在近期关于微纳机械谐振器的实验和理论研究中有所揭示。然而，对于这些多样且有时相互矛盾的实验观测，一个通用且直观的物理描述仍然难以捉摸。在此，我们通过实验揭示了在内部共振时出现的持续非线性锁相状态，并证明它们对于理解具有耦合本征模的非线性系统的瞬态动力学至关重要。一个完全可观测的微机械谐振器系统的测量动力学，可通过低频模式进入、维持并退出由高频模式施加的非线性驱动力所产生的持续锁相三倍周期状态来定量描述。该模型描述了观测到的锁相相干时间、能量交换的方向和大小以及由此产生的非单调模式能量演化。根据初始相对相位，系统会选择不同的弛豫路径，要么进入锁定状态，要么绕过锁定状态。所描述的持续锁相并不局限于特定的频率分数或非线性类型，并且可能推动跨物理领域的非线性谐振器系统工程发展，包括光子学以及纳米力学。
