Mode Coupling in Electromechanical Systems: Recent Advances and Applications

Mode interactions have recently become the focus of intense research in micro/nanoelectromechanical systems (M/NEMS) due to their ability to improve device performance and explore the frontiers of fundamental physics. Understanding and controlling coupling between vibrational modes are critical for the development of advanced M/NEMS devices. This review summarizes the recent advances in studies of coupling between multiple modes in M/NEMS resonators, focusing especially on experimental developments and practical applications. First, depending on spatial distribution of interacting modes, this review divides the coupled mechanism in three types: intermode, near‐neighbor, and long‐range coupling. Then, several fundamental physics approaches based on mechanical‐mode coupling, including linear and nonlinear coupling, modal localization, synchronization, and phonon manipulation are reviewed. This review also introduces sensing applications by using coupled mechanical modes. Finally, the state‐of‐the‐art open questions and challenges are reviewed.


Introduction
Micro/nanoelectromechanical systems (M/NEMS) are ubiquitous in modern information technology that exhibit excellent performance over a wide range of applications in both applied (e.g., sensing, imaging, timing, and signal processing) and fundamental sciences (e.g., quantum problems, such as mode cooling, [1] mechanically induced transparency, [1] Rabi oscillation, [2] two-mode squeezing, and phonon lasing [3] ) (see Figure 1). Many of these outstanding applications benefit from mode coupling by employing micro/ nanoscale mechanical resonators that are often parts of beam-, membrane-or platetype structures.
M/NEMS can be widely used for energy/ information transduction between different physical domains, as the corresponding mechanical modes can be coupled with a variety of physical degrees of freedom. For instance, conversion of mechanical vibration into electrical signal has been achieved in M/NEMS, which form the backbones of today's various stateof-the-art sensing devices. [4] Moreover, the most sensitive probes in exploring limits of measurement/experiments in cavity optomechanics [5] have demonstrated the ability to entangle propagating photons and phonons via radiation pressure forces. [6] In the quantum regime, mechanical resonators can be coupled to "artificial atom" systems such as superconducting qubits or semiconducting (charge or spin) qubits through Coulomb interactions or magnetic dipole forces, respectively. [7] Given a large number of vibrational modes supported by a mechanical resonator, the coupling of multiple vibrational modes provides multiple physical dimensions. This coupling of resonator modes has become an important issue in design and analysis of M/NEMS, where appropriate engineering of the mode coupling mechanisms opens roadmap for spectacular physical phenomena and enabling flexible manipulation tools within classical and quantum information domains.
A simple model to describe mode coupling of multiple mechanical resonators with effective masses m i , spring constants k i , and complex coupling constants κ ij , can be generalized as follows [8]  where u i is the deflection of the ith resonator, i ≠ j and ij = ji for complex coupling constants κ ij . Meanwhile, we often use the coupling strength g ij to express κ ij g m m ij ij i j i j κ ω ω = (2) where ω i is the angular eigenfrequencies of the ith resonator.
In the weak coupling regime, g < γ, κ, ω, and ω 0 . The strong coupling regime is when γ, κ < g < ω, ω 0 .The ultrastrong coupling regime is when g > γ, κ and g ≳ 0.1 ω 0 . Resonators can be coupled in different ways. In a linear context, coupling of the two mechanical resonators, which are geometrically or electrostatic interconnected, has been the common approach in MEMS studies from the early development stage. [9] However, a key obstacle for the further development of this method is the ability to coherently manipulate the coupling between different mechanical resonators. This limitation arises as a consequence of the usually weak vibration coupling between the constituent nanomechanical elements as typically only short-range coupling is supported. Recently theoretical and experimental studies have demonstrated the potential of a long-range (e.g., optical, microwave, phonon cavity assisted, etc.) model coupling. [10] This new approach of coupling can be used to artificially enhance the performance of the next generation sensors and electronics, to facilitate energy/information transfer between coupled modes and to open new possibilities in studies of coupled quantum systems.
The mode coupling combined with the dynamic tuning of mechanical resonators opens up a path to coherently control the coupled mechanical oscillations. Although the initial studies were based mainly on mode coupling of one or two mechanical resonators only, the recent theoretical and experimental developments has demonstrated the potential of coupling of multi ple resonators. [11] As a turning point, in the early 2000s, mode localization phenomenon was observed in a mechanically coupled microcantilever array. [12] Later an electrostatically coupled microbeam array was experimentally studied demonstrating complex nonlinear dynamics. [13] Those studies draw a lot of attention and motivated theoretical and experimental activity in this field. Coupled resonator array provides a platform for future studies of multiphysical coupling and information transduction, the scalable phonon-based information processing and the tunable phonon-mediated transferring and processing of information between distant phonon modes. Moreover, in the quantum regime, it is possible to manipulate and entangle the energy quanta between different mechanical modes. [14] Keeping in mind these remarkable advantages, we decided to restrict the scope of this review solely to the mode coupling in the field of M/NEMS. Although there have been some reviews focusing on the research related to M/NEMS, [15] such as linear and nonlinear dynamics, [16] nonlinear couplings and energy transfers [17] and sensing applications utilizing mode localiza- Figure 1. The mechanism, fundamental dynamical phenomenon, and applications of mechanical mode coupling. The parameters g and γ describing the coupling strength and dissipation. Applications of mode coupling of electromechanical systems are typically related to band-pass filters. Reproduced with permission. [19] Copyright 2010, Elsevier, self-sustained oscillation. Reproduced with permission. [20] Copyright 2023, American Physical Society sensor. Reproduced with permission. [4b] Copyright 2023, American Physical Society. Reproduced with permission. [21] Copyright 2010, AIP Publishing logics. Reproduced with permission. [22] Copyright 2008, Springer Nature amplifiers. Reproduced with permission. [23] Copyright 2023, American Physical Society acoustic switches. Reproduced with permission. [24] Copyright 2014, Springer Nature memory. Reproduced with permission. [25] Copyright 2014, Springer Nature and so on.
www.advelectronicmat.de tion, [18] to our best knowledge so far none has been focused on mechanisms of mode coupling. Hence, it is necessary to provide a comprehensive review of previous and ongoing research related to the mode coupling in electromechanical systems from coupling mechanism to application, which may be helpful to people in the fields of M/NEMS, optomechanics, phononic science and technology, and hybrid quantum systems with nanomechanics. We organized this review as follows (see Figure 1). The first part will focus on reviewing the types of mode coupling in M/NEMS, while we classify the coupling based on the spatial mediated channel by three types: intermode, near-neighbor, and long-range coupling. The second part will focus on several fundamental dynamic phenomena emerging from the mode coupling, including the linear and the nonlinear coupling, modal localization, synchronization, and phonon manipulation in M/NEMS. The third part of this paper will review the recent sensing technology based on the constructive utilization of mode coupling in M/NEMS. In the last part, we will discuss our views on some of the open questions and important challenges in this field.

Types of Mode Coupling
Recently the mode coupling of mechanical resonators has emerged as the subject of intense interest. Multimodal functionality of M/NEMS can be achieved by coupling of two or more vibrational modes. In what follows, we shall present the types of mode coupling, which mainly include three options (see Figure  2): intermode (activated, e.g., piezoelectrically, by microwave pumping, by photothermal effect, etc.), near-neighbor (e.g., electrostatic, direct mechanical connection, etc.), and long-range coupling (e.g., optical, microwave, phonon cavity assisted, etc.).

Intermode Coupling
While two or more modes are excited simultaneously in a single resonator system, the coupling type of these internally coupled modes is called intermode coupling. In particular, achieving intermode coupling between multiple modes can be challenging. The most common method of intermode coupling is parametric coupling. [26] In parametric coupling, two modes are coupled through a pump tone at the frequency difference, further the modes experience each other's oscillations as resonant forces. The internal resonance, the nonlinearities coupling two or more vibrational modes with an integer ratio of their resonance frequencies, is a special kind of intermode coupling that can be achieved via parametric pump [27] or the other method. [28] In such a situation the energy imported to one mode can be transferred to the other mode or cycled between the two modes involved. As a result, the motion of one mode strongly affects the other (and vice versa) through nonlinear interactions. [26a] As a nonlinear dynamical phenomenon, it will be reviewed in detail in Section 3.1. The strain induced intermode coupling in an electromechanical resonator can be realized by piezoelectric periodic modulation, [9,29] microwave pumping, [26d,e,30] electrostatic [31] or dielectric interaction [32] (see Figure 2). The strong coupling of the two mechanical modes can be also realized by using the photothermal effect. [33] Owing to the large stress modulation produced by laser irradiation, the coupling rate of the optically induced mechanical modes is an order of magnitude larger than the electrically induced coupling. [34] If the (large) coupling rate of the mechanical resonators significantly exceeds the damping rate, it can result in strong coupling, which is the signature of rapid energy exchange between the mechanical modes. Such approach enables the realization of coherent mechanical mode Figure 2. The types of mode coupling, which mainly include three options: intermode, near-neighbor, and long-range coupling. Different coupling systems can be controlled via the same method. It has been demonstrated that microwave can be used for both intermode coupling and long-rang coupling. Intermode coupling and near-neighbor coupling can be mediated by piezoelectric effect, electrostatic and dielectric force. www.advelectronicmat.de coupling even at room temperature, and permits the arbitrary manipulation of mechanical motions at a wide temperature range. Moreover, in contrast to other methods of mode coupling, the photothermally induced mode coupling eliminates electric crosstalk and simplifies the device design, as it does not require existence of any electric gates or magnetic coils, [35] which is the big advantage for such applications as mechanical logic circuits, transducers, and simulators.

Near-Neighbor Coupling
So far, considerable efforts have been made to achieve the strong near-neighbor mode coupling in M/NEMS, [4,9,30,36] when the rate of coherent energy exchange between the two modes is larger than the dissipation. The different modes can be activated by direct mechanic, electrostatic, or dielectric interactions (see Figure 2).
The mechanical motion of the two resonators can be coupled even if they are geometrically interconnected. Such direct mechanical coupling has been demonstrated in paired pendulums already in the mid-seventeenth century, and has been further studied in M/NEMS for their novel dynamics. [4a,37] Although such coupling can be achieved simply by providing direct mechanical link between the two resonators, the fabrication of corresponding structures is rather challenging. [38] The electrostatic coupling [13,36,39] eliminates the need for any distinct physical linking elements in between the resonators and demonstrates the highest degrees of efficiency, tunability, and design flexibility. The concept of electric coupling was first developed in MEMS structures by implementation of a higher order band pass filter by Pourkamali and Ayazi, [40] and was first used in a mode-localized coupled resonant sensor by Thiruvenkatanathan et al. [41] Furthermore, Agrawal et al. [39c] reported the first experimental realization of electrostatically driven coupled autonomous micromechanical oscillators. Two doubleended tuning-fork silicon microresonators were electrostatically coupled and their synchronized response was observed. The approach can be scaled to larger-scale arrays, offering a platform for investigation of various dynamic effects. The particularly intriguing direction for further research is the combination of electrostatic-mediated parametric coupling and the dynamic manipulation of mechanical resonators, which can enable the significant level of dynamically tuned coupling strength. [34,36,39b] Such strong dynamical coupling paves a way to coherently control coupled mechanical resonators, and has the significant potential for practical applications such as sensing. In contrast to the strain force coupling, electrostatic parametric coupling between resonators can be achieve in programmable lattices of nanomechanical systems [42] and an on-chip reconfigurable networks, [43] where the coupling strength can be tuned by application of external voltage leading to perfect coherent interaction.
The direct mechanical coupling is generally weak, which leads to a key obstacle of manipulating coupling. Another dynamic coupling mechanism has been suggested [9,44] via piezoelectrically induced parametric pumping. The technique opens door for coherent phonon manipulation and enables strong coupling between two resonators with significant difference of resonance frequencies.
Any polarizable object placed in a nonuniform electric field experiences an electrostatic (dielectric) force. Extending this fundamental mechanism to NEMS, one can obtain a low dissipation, broadband and scalable method for mode coupling. [32a,b] Moreover, combination of the dielectric coupling and high mechanical quality factors, offers an excellent opportunity for phase noise dynamics research without introduction of any extra noise to a system. [45] The mode coupling, based on dielectric gradient forces, can be utilized in various applications in signal processing and sensing, and also can be used for exploration of certain fundamental problems. [30] Recently, Zhang et al. [46] realized strong dynamic coupling between two adjacent high-Q mechanical resonators demonstrating the coherent transfer of mechanical excitations in artificial nanomechanical lattice, where the coupling is controlled by the dynamic dielectric force generated from the applied static and dynamic voltages. This scalable scheme opens the door to study complicated dynamics in 1D topological or multilevel systems.
Another interesting method of coupling is through the Casimir force. The corresponding physics, grew out of Casimir's remarkable discovery and describes the force of attraction due to zero-point fluctuations of vacuum, e.g., between two parallel uncharged conducting plates separated by vacuum. As the most famous mechanical effect related to vacuum fluctuations, Casimir force has been extensively studied [47] in the last two decades due to its fundamental interest in quantum field theory and practical importance for M/NEMS research. [48] Its role in thermodynamics has been predicted recently, enabling phonon transport between nearby objects induced by Casimir effect and the corresponding heat transfer through a vacuum gap. [49] Ezzahri and Joulain [49b] specifically investigated how such induced phonon transfer could be mediated by considering the coupling mechanism due to Casimir force in the framework of a local dielectric permittivity function theory. However, owing to stringent experimental requirements, this intriguing quantum phenomenon has not been yet observed. In 2019, Fong et al. [50] experimentally demonstrated heat transfer between two objects separated by a vacuum gap induced by quantum fluctuations (rather than electrostatic interactions and near-field thermal radiation), where Casimir force effectively acts as a coupling mediator that connects the two objects. The method they presented for achieving and controlling strong Casimir phonon coupling opens the door to study quantum thermodynamics using nanomechanical devices and provides a promising platform for implementing coherent phonon processing including phonon state transfer and entanglement through a quantum vacuum. Recently, Xu et al. [51] realized nonreciprocal energy transfer between the two quantum-vacuum-mediated strong coupled micromechanical oscillators via parametrically modulated Casimir interaction.

Long-Range Coupling
Entanglement, synchronization, and other phenomena emerging from interactions between mode couplings in mechanical resonators are of significantly scientific and technological importance. However, existing approaches to observe such behavior are not scalable above certain distance. It is www.advelectronicmat.de rather challenging to couple long-range (distant) mechanical resonators. Some methods have been proposed as the coupling media enables low losses, and the link can be extended over long distances by means of light, microwave, magnetic, and phonon-cavity interactions (see Figure 2).
In addition to several near-neighbor coupling approaches, mentioned above, Pai et al. [10a] successfully demonstrated the feasibility of magnetic coupling between the two resonators separated by a large distance (≈8 mm). Grinberg et al. [38] investigated the magnetostatic spring and coupling effects induced by the interaction of magnetized mechanical resonators. In contrast to mechanically and electrically coupled resonators, these new techniques avoid the undesired influence of structural nonuniformities on coupling and, hence, can be considered of advantage for high performance sensing. On the other hand, to ensure the effective operation of the two electrostatically coupled resonators, their separation should be within 1 µm. Magnetically coupled elements allow some degree of freedom, and can function even at large separations, which provides a promising platform for magnetically coupled modes to be used in various sensing applications.
To avoid neighborhood restriction and nonconfigurable coupling of M/NEMS, one can use light as coupling media to enable low loss, long-distance controlled interactions, which be generalized for complex topologies and arbitrary network systems. [52] In 2012, contrary to coupling through a directly contact or electrostatic interaction, Zhang et al. demonstrated the coupling only through an optical cavity radiation field and managed to realize synchronization of two dissimilar silicon nitride (Si 3 N 4 ) self-sustaining optomechanical oscillators. [52d] In 2015, Shah et al. demonstrated a reconfigurable scheme to couple via light the two independent mechanical oscillators, with resonance frequency difference ≈80 kHz and located one from another at a spectacular distance of ≈3.2 km. [10b] It should be noted that coupling of nanomechanical resonators to an optical cavity eliminates the unavoidable problem of device nonuniformity and enables access to strong coupling regime being reproducible in various NEMS. Bagheri et al. [52e] experimentally demonstrated the first synchronization of two mechanically isolated nanoscale radio-frequency oscillators integrated in an optical cavity. Their technique of coupling mechanical oscillators via a single photonic bus creates an interesting future direction for long distances and large arrays coupling. [53] Recently, Xu et al. [54] investigated a robust, compact, stationary, and tunable scheme nonreciprocal coupling between two normal modes of a silicon nitride membrane inside a cryogenic Fabry-Perot optical cavity. They further developed this nonreciprocal approach to control thermal fluctuations, which represents a way to cool phononic resonators.
Moreover, the ability to arbitrarily tune the coupling strength [55] provides a versatile ground for realization of various regimes of nonlinear dynamics in resonator networks. In 2020, by interacting with a common intracavity field through dynamical backaction, Wu's group demonstrated the tunable coupling between the two long-distance dielectric membranes and self-organized synchronization (a ubiquitous collective phenomenon, in which each unit adjusts their rhythms to achieve synchrony through mutual interactions) of phonon lasers in a two-membrane-in-the-middle optomechanical system. [56] It is the first experimental proof of synchronization in an optomechanical system. Moreover, in contrast to previous experimental studies [52c-e,53] such a system enables monitoring of the real-time transient dynamics to facilitate the controllable synchronous states, and consequently a phononic memory can be realized by tuning the system parameters. This result is the landmark step in quantum information processing and complex networks. Moreover, recently and for the first time Sheng et al. [57] experimentally demonstrated a stochastic heat engine effect in a two-membrane-inthe-middle cavity optomechanical systems with controllability and scalability, as many theoretical models [58] have predicted for strongly coupled resonators.
The study of coupled resonators with long-distance optical interaction opens a path toward investigation of intriguing physics in large-scale networks of optomechanical oscillators arrays, [59] and is a good candidate among mechanical systems enabling long-distance quantum information networking over optical fiber networks.
In contrast to cavity-opto/electromechanical systems with electromagnetic cavity coupled to mechanical resonator, [5a,60] the phonon-cavity [61] approach has emerged as an indispensable tool to control and manipulate resonator coupling. Similar to photon cavity, [62] phonon cavity also is capable to host dynamical backaction onto the mechanical element. [36] In 2014, Mahboob et al. demonstrated the electromechanical systems with dynamically engineered mechanical two-mode squeezing via parametric downconversion in a phonon cavity. [63] The observation of correlations between the two massive phonon ensembles, opens the prospects of all-mechanical macroscopic entanglement at the single phonon level. In 2020, Hatanaka and Yamaguchi investigated the coupling conditions in a cavity waveguide system, where phononic crystal cavities built in suspended GaAs membranes. [64] The structure they designed is the landmark step in phonon manipulation topic enabling scalability and further device integration. Furthermore, the interaction through the phonon cavity offers the potential to realize strong coupling regime, [10c,36] where the rate of interaction between the two systems is larger than the dissipation of energy. Luo et al. realized tunable indirect coupling in electromechanical systems via phonon cavity for the first time. [65] These advances provide a practical pathway to explore the phonon-mediated long-distance electron interaction, phonon state transmission/transformation, and quantum memories with high tunability.
The previous experimental studies have shown that nanomechanical oscillators can be coupled strongly also to microwave field. [5a,66] Furthermore, the backaction effects of the microwave field in coupled cavity-resonator acts as a force to alter the amplification and/or damping of mechanic oscillations, [67] which have been used in various sensing applications. Strong coupling is also suitable for large-scale integration of many resonators within a single cavity. Faust et al. [66a] achieved an adjustment-free (dielectric coupling allows to maintain a large quality factor over a wide temperature range), all-integrated (good integration and scalability to large resonator arrays coupled to a single readout cavity) and self-driven (the cavity backaction allowing to omit the piezoactuator) nanoelectromechanical resonator array based on a dielectrically coupled microwave www.advelectronicmat.de cavity, demonstrating potential usefulness for various sensing and signal processing applications. In contrast to previous experimental studies of entanglement in microscopic-scale systems, [68] macroscopic-scale and long-distance entanglement studies based on microwave-optical [69] could bring more versatility to connect distinct quantum core units setting the basis for quantum network applications. Ockeloen-Korppi et al. [69a] achieved entanglement of two micromechanical oscillators coupled via an microwave-frequency electromagnetic cavity. Kotler et al. [69b] realized the quantum entangled state of two macroscopic objects by strongly correlating the motion of the two oscillating drumheads at the quantum level via driving the circuit with tailored microwave pulses. De Lépinay et al. [69c] demonstrated entanglement of two oscillators to be a further tool to reduce intrinsic noise in measurements enabling noisefree monitor of weak external forces. Such entangled macroscopic systems based on microwave coupling establishes the new regime for the quantum information processing, precision measurements, [5a] and fundamental tests of quantum mechanics. [70] Nowadays, in the study of the above three mode coupling mechanisms, researchers have achieved strong coupling and tunability of resonant mode through electrical, optical, and other means. Nevertheless, the strong coupling and tunability in the mode coupling mechanism are still the two major characteristics of the most concerned applications such as sensing ( Table 1).

Applications in Fundamental Physics Research
The unprecedented rich and highly engineerable linear and nonlinear features can be observed in these modal coupling M/ NEMS. Along with high quality factor and excellent scalability, these features are of advantage for sensing and communication applications, and can also present an ideal testbed to probe fundamental physical phenomena and gain deeper insight into quantum mechanics. A new field of M/NEMS has emerged to explore the intricate dynamics phenomena emerging from this coupling during the last decades. In the following, we will focus on the aspect of linear and nonlinear coupling, modal localization, synchronization, and phonon manipulation (see Figure 3).

Linear and Nonlinear Coupling
M/NEMS have demonstrated rich and complex linear (veering, crossover) and nonlinear (Hopf bifurcation, internal resonance, chaos, etc.) dynamics of mode coupling, which offers an excellent platform for fundamental physical research and various applications. Veering (avoided-crossing) occurs when frequencies of two linearly coupled modes are first approaching and then are deviating away from each other following the path that the other would take. [72] Such behavior is typically associated with mode localization phenomenon, which will be reviewed in-depth in the following section. This section mainly focuses on the nonlinear nature of mode coupling. [26a,f,32c,73] M/NEMS devices are easily driven into the nonlinear regime due to the scaling effect. [20,74] Graphene, [31a,b,75] carbon nanotubes, [76] and related 2D materials [27a] have been the ideal candidates for various studies of nonlinear dynamic phenomena related to their high-frequency tunability, large elastic modulus, and other unique mechanical properties.
If the coupling between a pair of modes which is generated by a strain, then the nonlinear effects originate from the motion of one resonator, which generates tension, that modifies the vibrational parameters of the other coupled resonator. Such coupling can be dispersive [32d] when the influenced parameter is the resonance frequency, or dissipative [29d] when the affected parameter is the damping coefficient. Periodic parametric modulation at the difference or sum of two mode frequencies results in frequency conversion, two-mode squeezing, phonon lasing operation, and so on. The parametric coupling strength can exceed other contributions enabling to overcome the limitations of poor frequency modulation efficiency, leading to quantum, strong-coupling regime. [5a,31c,34,77] In this regime, the rate of coupling becomes stronger than dissipation, which can provide high speed switching of mechanical resonators between different modes, enabling coherent manipulation leading toward practical applications. Recently, Fu et al. [78] demonstrated the control of energy transfer through geometric Stückelberg interferometry consisting of two parametrically coupled mechanical modes, which can achieve the fast and highly controllable method for controlling noise-resilient mechanical energy transfer. In addition to intermode coupling, near-neighbor and long-range coupling can also be induced by parametric modulation technique. [9,29d,78] Okamoto et al. [79] demonstrated a highly controllable parametric pump induced coupling between two mechanical modes, which can serve as a simple and compact approach to solve the trade-off problem between the Q-factor and operation speed. The coherent control of the coupling induced by parametric modulation avoids the problem related to long waiting time for initializing the high-Q characteristics due to their slow energy relaxation. The method also allows the quick repetitive operation within the intrinsic long ring-down time via rapid switching of the vibration amplitude, leading to on-demand energy transfer between mechanical modes of the resonator. Moreover, the parametric coupling by pumping on a sideband offers a far more flexible platform as it permits any pair of modes to be activated irrespectively of their frequency difference, [61b] which can be further exploited to achieve the mechanical-vibration register and paves the way toward rapid signal and information processing using N/ MEMS resonator. [25,80] In addition, parametric coupling can be realized by periodic modulation of certain parameters (e.g., spring or coupling constants) of the M/NEMS. The operational modes of parametric resonators can be used for parametric amplification, including degenerate and nondegenerate parametric amplification, and parametric oscillations. [81] Such approach has been demonstrated leading to various applications: from quantum physics to signal processing, such as frequency converters and high-accuracy timing device. [82] On the other hand, the nonlinear dynamics of mode coupling can be widely implemented in M/NEMS applications to expand functionality and improve performance such as internal resonance and mode synchronization. [83] The internal resonance, the nonlinearities coupling two or more vibrational modes with an integer ratio of their resonance frequencies, have been extensively investigated and exploited during the last few decades. The nonlinear coupling changes significantly the behavior of the system as the energy is transferred between the two vibration modes due to internal resonance. [28c] Several groups examined the nonlinear behavior of internal resonance in various structures, such as cantilevers, [27b,84] arch beams, [27a,85] and membranes from 2D material. [27a,31a] As one of the earliest experimental work, Antonio et al. [83c] demonstrated that the oscillation frequency of nonlinear self-sustaining micromechanical resonators can be stabilized through an internal resonance, which provides valuable opportunities for engineering low-frequency noise oscillators capitalizing on the nonlinear features of M/NEMS devices. An internal resonance that can compensate for energy losses to support a self-sustained oscillation (without some injection of energy (e.g., a stimulus), the oscillations would slow and stop, but with stimulation, the oscillations are able to continue indefinitely) has been demonstrated by same group. [20] The investigation of internal resonances ratios among various modes mostly have been 1:1, [27a,86] 87] In addition, the coupling ratio of noninteger index caused by internal resonance could also exhibits the possibility of entering the self-sustain oscillation and chaotic regime. [88] The group [28e] investigated the behavior of a two-mode cavity exhibiting a 1:3 internal resonance in an open-loop configuration and explored supercritical Hopf instabilities. And later, the same group [28d] showed the generation of simultaneous multi ple internal resonances in a microelectromechanical selfsustained oscillator, whereby multiple modes excited via nonlinear interactions. Given the large body of activity in this field, we refer the reader to another recent review. [16] To conclude, understanding the rich and complex nonlinear dynamical phenomena and energy transfer between different modes in M/NEMS can open new possibilities in study of coupled quantum systems and provide diverse pathways to a wide spectrum of revolutionary applications and technology.

Mode Localization
In a periodic mechanical system, e.g., two nearly identical resonators weakly coupled to each other, the propagation of vibration can be localized as a result of small periodicity breaking structural irregularities. [37a] This phenomenon is referred to as mode localization, which will lead to abrupt divergence within the loci of the eigenvalues, i.e., "curve veering," and drastic energy confinement, being the different manifestations of the same phenomenon. The strength of mode localization depends on the magnitude of the perturbation and the strength of the coupling, with weaker coupling resulting in stronger mode localization.
The theory of mode localization was first introduced by Anderson's Nobel Prize winning work, [89] and later research demonstrated that it can be applied to M/NEMS sensing applications. As a promising sensing mechanism, these weakly coupled resonators based on the mode localization phenomenon have been implemented in ultrahigh sensitive mass sensors, [21,37a] force sensors, [47d,90] displacement sensors, [91] accelerometers, [92] and electrometers. [93] The ringdown time and sensitivity can be adjusted also by changing energy exchange rate between coupled resonators. Over the last decades, various groups theoretically and experimentally demonstrated that the sensors based on the output metrics of eigenstate or amplitude ratio have the potential to improve the sensitivity by more than 2 orders of magnitude [21,37a,b,90a] compared to their state-of-the-art one degree of freedom counterpart devices based on frequency ratio. This type of sensors can be also relatively immune to common mode noise, originating from finite temperature or pressure variations. [94] Moreover, tracking the localization in the frequency domain and detection the position where the analyte is adhered to, [95] can extend the linear sensing range. Recently, Zhao et al. [4b] demonstrated parametric modulation technique with enhanced performance of mode-localized M/NEMS sensors. The parameters such as resolution, dynamic-tunable scale factor, high tunability of the operational bandwidth and inherent noise filtering are much better compared to the conventional mode-localized approach, which exhibits the low tunability of mechanical coupling and dynamic pull-in (e.g., mechanical loading) utilizing electrostatic coupling. The advantages of mode localization sensors will also be discussed in-depth in the next part of this section.

Synchronization
Synchronization, the emergence of self-tuning rhythms of the coupled systems, is a ubiquitous phenomenon both in physical and biological sciences. [96] A large number of research has shown that, synchronized micro/nanoelectromechanical resonators have the potential for applications in signal processing, microwave communication, neural computing, timing, sensing, and other fields. In biological systems, the first systematic studies of synchronization started with Peskin's attempt to understand the generation of a heartbeat by modeling self-synchronization of cardiac pacemaker cells. [97] Furthermore, rhythmic behavior showed in biological neural networks opens up a new perspective: neurocomputer utilizing nanomechanical oscillators arrays. [98] In physical systems synchronization can be used to enhance detector sensitivity by reduction of phase noise [53,99] and improvement of frequency stability through the inherent phase locking characteristic. [39c] To our knowledge, Agrawal et al., [39c] for the first time reported a significant improvement of the frequency stability of electrostatically driven synchronized micromechanical oscillators. Later, Pu et al. [100] demonstrated and theoretical analyzed that synchronization can improve the frequency stability of a piezoresistive micromechanical oscillator. Avoiding degradation of frequency stability by scaling effect [101] and nonlinearity, [20,74] the frequency stability improvement due to synchronization can lead to a large number of applications, especially in timing and frequency-control domains. [83c,99] Moreover, Cross [102] demonstrated that the effective noise intensity can be reduced by a factor 1/N (here N is the number of coupled resonators), and the frequency precision can be improved by a factor of N via synchronizing a lattice of resonators with unequal frequencies.
Shoshani et al. [103] demonstrated that oscillator phase noise can be remarkable reduced by three order of magnitude by www.advelectronicmat.de synchronizing the oscillator with a weak harmonic drive. Specifically, one can use synchronization to achieve clean signal amplification.
The complex dynamics displayed by synchronization is essential for understanding the performance and fine tuning of ultimate sensitive devices. [104] Researches show that synchronization can be achieved through both the mode-dependent dissipation mechanism, [102] such as the famous example of Huygens' clocks, and reactive coupling [105] through nonlinear frequency pulling. [106] Synchronized states are often represented by Arnold's tongues or frequency entrainment. [37c] In addition, the form of synchronization can be of unidirectional coupling (injection locking [107] ) and bidirectional coupling (subharmonic synchronization [99,108] ) types. The range of frequency locking is called the synchronization range, and recent studies showed that synchronization domain can be significantly expanded by nonlinearity. [20,39c] Taheri-Tehrani et al. [109] studied synchronization of a micromachined oscillator subject to different electromechanical nonlinearities and presented an analytical model that achieves good quantitative agreement in each the nonlinear regime. Later, the same group [110] demonstrated that the synchronization range can be also increased by strong drive of the lower frequency oscillator or weak drive of the higher frequency resonator. This result has the strong potential for applications related to detection and processing of weak signals. Pu et al. [111] showed that a wider synchronization region can be obtained by larger coupling strength or driving force. These studies provide the new strategy to enhance the synchronization of resonators.
The resonators based on M/NEMS give many unique advantages for experimental studies of synchronization due to their highly engineerable nonlinearities, [105a] high quality factor, and excellent scalability. [112] Such systems are also useful for exploring quantum synchronization: [83d,113] synchronization of a quantum tunneling system to an external driving, [114] clock synchronization by means of quantum and classical communication protocols, [115] quantum behavior of classically synchronized systems, [116] etc. Moreover, it was suggested that optomechanical systems are also the unique platform for investigation the problem of mechanical resonators synchronization. [52d,53,117] Due to their inherent nonlinearity associated with the radiation pressure interaction, optomechanical systems can exhibit self-sustained oscillations, [118] which can enable access to synchronize different mechanical resonators. [119] Using resonators linked through cavity field can form an all-to-all coupling [120] and overcome the restrictions originating from the neighborhood and nonconfigurable coupling, which are the major challenges in the field of nanoscale resonator synchronization. [112a,118,120a] It is worth noting that the coupled microdisks and nanomechanical resonators with an optical racetrack cavity can also be used for   [56] showed the synchronization effect in an optomechanical multimembrane system. [119] It should be noted that the synchronized frequency can be controlled by tuning the system parameters: the effect that could be exploited for realization of a phononic memory. Mechanical synchronization in multiple coupled optomechanical systems and arrays have also been realized, [52d ,53,117b,120a,121] which could open the practical route toward high power and low noise integrated frequency sources and synchronized resonator networks.

Phonon Manipulation
Traveling at significantly lower velocity (about 5 orders of magnitude) than photons, phonons can be better manipulated to store and transmit information with high fidelity. The ability to dynamically manipulate phonon transport in M/NEMS has attracted significant interest. The mechanical mode manipulation is crucial for practical applications, such as phonon-based classical [32b,122] and quantum [7b,123] information processing, low-power mechanical filters, logic-circuits, [124] switches, [24] memories, [22] and shift registers. [25] So far the efforts toward phonon manipulation in NEMS/MEMS have been made in different systems, such as single mechanical resonator, [26d,32b,125] neighboring mechanical resonators [9] or spatially separated mechanical systems with indirect coupling. [10c,65] Studies have shown that information encoded vibrational modes can propagate through mode coupling, resulting in phonons to be stored, transferred, and manipulated. Hence, coupled resonators as architected unit cells can be assembled into periodic arrays structures to create phononic crystal structures and consequential phononic bands, [126] which are similar to the atomic lattices in crystalline solids and photonic crystal lattices. The low propagation losses and design flexibility allows the phonon propagation to be dynamically controlled, [61a,127] which opens the new avenue toward possible applications in the fields of signal processing and electromechanics. [128] Hatanaka et al. [129] constructed a phonon waveguide via a 1D array of mechanical resonators and further realized mechanical random access memory, which confirms the viability of phonon waveguide interconnects as building blocks of mechanic logic circuits. Wang et al. [130] designed and systematically investigated the first hexagonal boron nitride (hBN) phononic crystal waveguides. Combining the unique piezoelectric properties of hBN, such are additively integratable in hBN platform, may open the way toward building future integrated phononic and hybrid quantum circuits. Fu et al. realized a Stuckelberg interferometer with two mechanical coupled cantilevers based on optomechanical system and determined the nonadiabatic Stokes phase accumulated in the Landau-Zener transitions of a classical system for the first time. Their experiment showed the classical analog of the Stuckelberg interferometer in a quantum two-level system, and demonstrated that the spatially separated mechanical oscillation transfer efficiency can be as high as 97% with controllable dissipation, which would pave the way to the further control of information and energy in coupled resonator based on phononic quantum networks.
Moreover, the phonon-mediated coupling opens the door of realizing strong direct interaction between any distant mechanical resonators by either statically [26a] or dynamically [131] indirect coupling to an intermediate mechanical mode. Recently, Yuan et al. [132] demonstrated the geometric motion transfer by controlling the phonon mediated coupling between the two mechanical resonators, where they are indirectly and dynamically coupled to the intermediate mode by two parametric pumps in a three mode mechanical system. They also experimentally investigated the real-time dynamics of the phonon mediated motion transfer in the three-mode system, which opens the prospect of noise-resilient information processing in scalable phonon-based devices. Gong et al. [133] presented coherent control of the phonon-mediated dynamics for addressable motion in a coupled-mechanical-resonator transducer consisting of three nearest-neighboring coupled cantilevers, where long-range transfer of both iterating and intramode motions can be controlled by phonon mediated coupling avoiding undesired excitation of an intermediate mode.

Application in Sensing: Near-Neighbor Coupling
The considerable research efforts on exploring the resonators mode coupling has already led to exciting breakthrough results across a variety of disciplines from fundamental science to device engineering. Appropriate engineering of the mode coupling mechanisms into the design of M/NEMS leads to unprecedented rich linear and nonlinear phenomena, which have been used in band-pass filters, [19,134] amplifiers, [23,135] logics, [22] various sensing applications such as mass sensing, [21,37a] force sensors, [47d,90a] gyrooperation, [136] etc. [137] In this part we will focus on utilization of mode coupling in various of sensing applications (see Figure 4).
Over the past few decades, emerging M/NEMS technology enabled the field of precision measurement to be developed rapidly. Due to the tiny size, high quality factor, and other virtues of micro/nanomechanical resonators, their resonance frequency appears to be extremely sensitive to small perturbations and adsorption phenomena. This feature has been extensively used for many sensing applications including mass and stiffness change detection, and plays important roles in many different areas, such as inertial sensing, [141] surface chemical material characterization, [142] study of biomechanics [143] and cell mechanobiology, [144] and so on.
The conventional approach is based on utilization of a single degree of freedom (1-DoF) resonator, where quasi-digital output signal could be achieved by frequency shift leading to ultrahigh sensitivity at cryogenic temperatures. These 1-DoF resonators are attractive to researchers compared to other alternative sensing methods due to their fast response and high sensitivity. However, multidegree of freedom (m-DoF) coupled resonators utilizing mode localization, have emerged as a new sensing paradigm, which has been theoretically and experimentally verified demonstrating significant enhancements (more than 2 orders of magnitude compared to 1-Dof) in the sensitivity to induced perturbation. [18,21,37a,41,136] In contrast to the eigenvalue method, which utilizes the frequency shift caused by change of mass/ stiffness, the eigenmode-shift approach offers the advantage of www.advelectronicmat.de intrinsic common mode rejection [37a,94] that makes it less susceptible to ambient pressure and temperature drift. [145] Various activities have been focused on enhancing the performance of sensors based on mode localization, especially aiming at high sensitivity. For example, owing to the sensitivity depends on the degree of the initial delocalization, the requirement for fabrication technology is very high. Alternatively, with the rapid development in micro/nanofabrication, it is now possible to extend the number of degrees of freedom of the resonators, which would help to further increase in sensitivity. 3-DoF mode localized sensors have been already reported [90a,136,139a] with the sensitivity improved compared to 2-DoF. With the absorption of energy by the third resonator located in between two identical resonators, the energy propagation can be reduced and the mode localization can be enhanced. In contrast to the approaches using three identical resonators, a novel structure was described by Zhao et al. [136] the resonator in the middle with higher stiffness and the two identical resonators on either side, which could improve the sensitivity even further. Spletzer et al. [37b] proposed an array of 15 weakly coupled microcantilevers. Such array-based sensors provide multifunction or multidegree of freedom sensing, which can not only enable mass sensing, but also is capable to identify on which resonator the extra mass specimen has been docked. [146] In addition, the quality factor of the coupled resonators can affect the sensitivity. It can be improved by encapsulating such sensors in vacuum. These are important guidance for the mode coupling based sensors design. Recently, Peng et al. [147] investigated the influence of eigenfrequency shift in coupled resonator array sensors based on mode localization approach, which can be introduced by residual stress during the packaging process and can significantly affect the functionality Figure 4. The applications of mode coupling in the field of sensing, which include inertial sensor. Reproduced with permission. [4b] Copyright 2023, Physical Review Applied strain sensor. Reproduced with permission. [138] Copyright 2007, IEEE force sensor. Reproduced with permission. [90a] Copyright 2015, Elsevier stiffness sensor. Reproduced with permission. [136] Copyright 2016, IEEE acceleration sensor. Reproduced with permission. [139] Copyright 2018, IEEE mass sensor. Reproduced with permission. [21] Copyright 2010, AIP Publishing displacement sensor. Reproduced with permission. [91b] Copyright 2012, IEEE electrometer. Reproduced with permission. [93] Copyright 2010, IEEE, Reproduced with permission. [140] Copyright 2016, IEEE and so on. The schematic of mode localization is shown in the center of the picture. The left image shows unperturbed cases with identical vibration amplitudes of resonator 1 and 3. The right image shows the perturbed cases with mode localization, where vibration energy is locally confined on one of the resonators. The simple spring-damper-mass model of 1, 2, 3 and more coupled-resonator systems are depicted in the second outer circle of the figure. www.advelectronicmat.de of the device. Hence, in order to obtain highly sensitive mode coupling sensors, the device geometry should be well optimized and the measurement setup along with fabrication process should be improved.
Moreover, it is well known that the smaller is the coupling strength, the higher is the sensitivity. However, it was theoretically analyzed and experimentally verified that the coupling strength cannot be decreased infinitely, being limited by 3 dB bandwidth, [92,148] the amplitude measurement error and the frequency misalignment error. [140] Glean et al. [149] explored the strength of the coupling between elements in an indirectly coupled sensor array, which can be affected by system frequency and mass ratio. With the decreasing of the frequency separation, the coupling strength between elements increases linearly and the maximum value of it depends on the ratio of masses of the sensing element to primary resonator.
Another important consideration for mode localized sensors appeared to be the unavoidable process induced by the variability, [150] which significantly affects the performance of sensors' sensitivity. Therefore, it is critical to compensate for these variations, such as tuning the stiffness of the resonators via external direct current or electrothermal loading.
Moreover, as one of the most essential considerations for optimization of sensing performance is the resolution limit, the mode-localized sensors have been widely investigated. The results elaborated in prior work [148] clearly demonstrated, that the fundamental physical noise (including intrinsic and extrinsic noise, e.g., the electronic interfacial circuits) has the significant impact of ultimate resolution and the dynamic range of such mode-localized sensors. Hence it is critical to theoretically and experimentally investigate the strategies of noise optimization for mode-localized sensors, [151] including the device topology and the low noise interface circuitry. Besides, the resolution also depends on the strength of mode coupling. It was observed [148] that a weaker effective coupling between the resonators can improve the resolution by 3 to 4 orders. The other correlational studies proposed that reduction of the amplitude fluctuations, [140] nonlinearity cancellation, [152] and operation near the veering zone [153] can also contribute to substantial improvements in the resolution of mode localization sensing.
Dynamic range and linearity are the other two key parameters of the mode localization sensors. The detailed theoretical analyzes and experimental studies revealed that a mode aliasing effect, [136] high sensitivity, [154] and noise can lead to the decrease of the dynamic range. However, Holzinger et al. [155] showed that when the quality factor is sufficient, the dynamic range can reach at least an order of magnitude higher values, than the existing state-of-the-art solutions. Zhang et al. [156] demonstrated that driving both resonators instead of a single one, can extend the linear sensing range at the cost of an order lower sensitivity. Moreover, the dynamic range and linearity can also be improved by an intentional stiffness perturbation (a bias operation point) proposed by Zhao et al., [90a] and some readout schemes were demonstrated by Zhang et al. [157] Real-time sensing, as one of the central techniques for the further development of the mode-localized sensors, has been explored through closed-loop readout scheme [92] based on self-oscillation configurations of mode-localized sensors. However, due to the mode-aliasing phenomenon, [90a] it is hard to obtain directly or accurately measure the sensors' transient responses. Yang et al. [139b] for the first time applied closed-loop readout technique for mode localized sensors, and confirmed the significant improvement of the accelerometer performance compared to previous reports. [91a] Several different output metrics and output signal options have been used in previous research. In addition to eigenvalue and eigenvector method, the sensitivity of mode-shape-based mass identification have been explored by Ryan et al. [158] Later, Glean et al. [159] investigated vibration localization in arrays of resonators used for ultrasensitive mass detection. They explored the effect of mass ratio and system frequency spacing on system frequencies and mode shapes, and described an algorithm for determining the amount(s) and location(s) of added mass. Recently, to fully utilize the eigen-information provided by the coupled resonators, Tao et al. [160] presented an effective and promising inverse eigenvalue sensing approach, demonstrating simplicity and high accuracy. Experimental results demonstrated the minimum stiffness error as low as 2 × 10 −5 , coupling strength error of 4.6 × 10 −4 and a dynamic range of 66 dB. To avoid the instability and errors induced by measuring the eigenvectors, this technique provides full parameter extraction capability to sense mass/stiffness and coupling strength, and has the potential of enhanced resolution, dynamic range, and linearity. The other emerging coupled sensing methods have been also widely exploited: [161] amplitude difference [18] and phase difference. [162] However, only very few systematic studies of advantages and disadvantages of particular methods have been reported so far. In order to optimize the performance of sensing, it is imperative for this emerging technology of mode localization sensors to find a suitable output metrics.
From gravitational astronomy to atomic force microscopy domain, high precision measurement of small forces has been the significant goal. Except for the reduction in bandwidth and contact counts, these ultrahigh sensitivity mode-localized coupled resonators have been the dominant choice for detecting mass and stiffness change. The constructive utilization of mode coupling in M/ NEMS can provide diverse pathways to develop new technologies and applications related to sensors, as reviewed in this paper.

Mass Sensor
Coupled MEMS resonators have found particular attention for mass sensing. [21,37a,163] The concept of mode localization sensors was first theoretically and experimentally demonstrated in weakly coupled resonators to detect the induced mass of a target by Spletzer et al. in 2006. [37a] This new technology, without doubt, is the excellent candidate for ultrahigh mass sensing. Later, using a pair of mechanically coupled nanocantilevers [4a] demonstrated the promising paradigm for ultrahigh sensitive mass detection based on stochastic and deterministic resonant responses, when initial differences in the active masses of the two cantilevers are no longer required to be near zero. In contrast to mechanically coupled structures, utilizing electrical coupling enables significantly weaker spring constants, and consequently stronger mode localization for mass addition. Thiruvenkatanathan et al. [21] demonstrated that the parametric sensitivities of mode localized sensors can be enhanced by 400% with an electrical tuning method, which could be helpful for electrically mechanical sensing. Moreover, Lin et al. [164] presented a novel dual resonator sensing platform, www.advelectronicmat.de where two spatially separated silicon microresonators mechanically were coupled together via a thin beam. The approach demonstrated its application potential as a mass sensing platform for biochemical sensing in liquid.

Stiffness Sensor
The concept [90a,136,165] has also been employed for sensing small changes in the stiffness of micro/nanomechanical resonators, thereby extending the applicability of this new paradigm for wide range of applications, including strain, [138] pressure, [166] acceleration, [141] charge detection, [167] and magnetic field [168] sensing.
Zhang et al. [169] experimentally demonstrated the 2-DoF MEMS resonant accelerometer based on mode localization phenomenon, which has the potential to improve the sensitivity of existing resonant accelerometers by several orders of magnitude. Later, the same group [91a] extended previous work by presenting a new theoretical model of an accelerometer with the capability to predict in details certain parameters such as sensitivity, linearity, and bandwidth. They proposed certain approaches to achieve an optimized performance by improving the device topology, adding electrostatic electrodes, increasing the resonant frequency and developing a closed-loop circuit based on phaselocked loop detection and variable gain control. [92,170] In order to further increase the sensitivity of the mode localized accelerometer, the number of DoF of the resonators was increased from 2 to 3. [171] Recently, Peng et al. [172] reported a four degree of freedom series-parallel resonator array with improved adjustability and sensitivity. Furthermore, Kang et al. [139a] described a closed loop accelerometer with complex three degree of freedom system using a self-elimination method to facilitate the device matching. The experimental results showed that the sensitivity based on closed-loop measurement method can be further improved compared to open-loop measurement. This transduction scheme shows the great potential for developing highperformance accelerometers. And, the real-time measurement of acceleration was successfully realized through closed-loop readout scheme in the mode-localized devices. [139b] Unlike previous attempts, recently there is the trend to incorporate the mechanism of mode localization sensing into the electrical domain and monitor minute charge fluctuations across an input capacitor. [140,173] In 2010, Seshia et al. [93] reported a highly sensitive prototype micromechanical electrometer that employs the phenomena of mode localization: an addition of charge on one of the coupled resonators through input capacitor induces a perturbation of the nearly periodic system, enabling monitoring minute charge fluctuations across the capacitor. The experiment showed ≈300 times enhancement of the sensitivity compared to the method utilizing frequency shift. In 2011 it was demonstrated [174] the possibility of high-sensitivity charge detection by using antisymmetric vibration in two coupled GaAs oscillators via increasing the actuation power into the nonlinear response regime. In 2016, Zhang et al. [140] improved mode-localized electrometer performance through optimizing the topology of coupled resonators. By theoretically analyzing and experimentally verifying the frequency misalignment and amplitude ratio measurement errors, they greatly improved the limitations of the coupling factor, demonstrating an electrometer with the resolution of ≈8000 electrons.

Displacement Sensor
Another interesting application of mode-localized sensors is the displacement sensing, [91b] which is essential in many applications such as atomic force microscopy [175] and inertial sensing. [176] In 2012, Seshia et al. [91b] proposed the construction of a new operating concept in micromechanical displacement transduction that employs the phenomenon of mode localization for monitoring minute inertial displacements in electroelastically coupled resonators. It was demonstrated that the sensitivity of eigenstate output method may be 1000 times greater compared to resonant frequency method. In 2016, Villiers et al. [177] implemented an allmechanical heterodyne detection scheme, where the motion of the fundamental mode is resolved via the second mode through strong coupling, which has the potential be developed into an ultraprecise displacement measurement protocol.

Open Questions and Challenges
Promising results related to mode coupling in M/NEMS have been obtained so far, however, there are still many areas requiring further investigation. For example, future applications may need better coupling tunability and integration of coupled resonators at larger scale. Moreover, in the field of mechanical sensing, there are still a large number of challenges waiting to be solved. A more interesting research field is in quantum, in particular in the study of quantum networks by using phonon modes in coupled mechanical resonators.

Tunability
The completely controllable coupling between multiple resonators in M/NEMS, has attracted considerable attention in the fields of quantum simulation, [178] mechanical functional devices, [179] topological physics, [180] metamaterial, [181] and information processes. [65,182] The parametrically and strongly coupling multiple vibration modes enable the resonators to be dynamically manipulated, and phonons to be coherently exchanged between the modes. The approach has the potential to be developed into a novel and highly functional electromechanical computer. However, the ability to define and precisely control the modal coupling in M/NEMS systems (especially in nonlinear domain) remains the key challenge, so far limiting practical applications. It may be possible to achieve a mechanical coupling strength regime that goes from a strong coupling regime to an ultrastrong coupling regime (coupling strength reaches 10% of the mechanical frequencies). [183]

Scalability
Realization of a controllable network based on M/NEMS is expected to provide the huge potential in investigating various interesting phenomena, such as exotic states, [184] symmetry breaking, [185] topological transport at high dimensions, [186] synchronization of networks [120b] and multifunctional and integrative metamaterials. [187] Furthermore, studies showed that arrays of coupled resonators displayed excellent performance, [200] such as low motional impedance, high power handling capability, and significantly improvement of the phase noise. [188] As an ideal network, one of the most desired objectives would be the realization perfect reconfigurability: the capability to increase the number of individual nodes freely. Although there are many studies that investigate the unit cells based on M/NEMS, [61a,126a,181] once manufactured, no cell of these arrays can be tuned individually, thus limiting the implementation of arbitrary configuration and further integration. Moreover, with the increasing number of coupled resonators, the individual tunability of manipulation will decrease, thus it is necessary and challenging for physics and engineering to achieve a network with flexible scalability while maintaining individual tunability based on M/NEMS.

Sensing
As discussed above, considerable effort has been made to develop the performance of sensors based on the mode localization phenomenon. However, there are several fundamental issues that still lack a clear understanding. In this respect one can mention the following issues: dynamic behavior of stochastic response involving nonuniform geometries and complicated transduction methods (e.g., investigation localization at the higher order modes or in the linear regime); the ultimate limits of sensing (e.g., due to damping effect); lownoise readout of mode shapes for rapid fluctuations in system parameters and closed loop control for automated parameter identification. These important issues motivate research and once being solved can open new opportunities and emerging applications in this field. In the future, it may be possible to beyond the limits of classical sensing by using multimode coupled mechanical resonators. For example, multimode entanglement can open up possibility of constructing novel precision measurement systems beyond the standard quantum limits.

Quantum
The M/NEMS supporting multimode coupling creates a rich platform for explorations of fundamental quantum physics, [189] with promising applications toward quantum read out, [123,190] quantum sensing, [191] quantum computing, [192] quantum network, and so on. Over a small footprint, multimode system gives rise to even richer physics and new measurement strategies, including store long-lived quantum memories with minimal crosstalk, [193] realize quantum squeezing of the mechanical motion [194] and measurement-based quantum control of Figure 5. Promising research directions, including tunability. Reproduced with permission. [199] Copyright 2021, American Chemical Society scalability. Reproduced with permission. [200] Copyright 2023, American Physical Society sensing. Reproduced with permission. [201] Copyright 2023, American Physical Society and quantum. Reproduced with permission. [190a] Copyright 2022, Springer Nature. www.advelectronicmat.de mechanical motion, [191a,195] generate nonclassica, [123,190b] and entangled states of motion, [14,95,196] is a compelling system for processing quantum information.
The quantum networks, being one of the highest priority objectives of research and challenges in quantum physics and engineering, are the excellent candidates for encrypted longdistance communication, distributed quantum computing, and nonlocal sensing and metrology. [197] The key obstacle to the development of this field is the ability to efficiently interface multiqubit registers within the reasonable coherence time, enabling processing and storing of quantum information. Although almost all of individual components needed for quantum networks have been demonstrated in various experiments, [198] while realization of a full-featured quantum network node has not been realized yet. Mechanical modes can be coupled with a variety of quantum systems, such as superconducting qubits, semiconductor qubits, color center qubits, as well as a variety of physical fields such as light fields, electric fields, microwave fields. Therefore, mechanical modes, especially coupled mechanical resonators systems with multiple acoustic degrees of freedom, will play an important role in quantum networks, such as coupling multiple qubits, quantum transduction between different quantum systems, and so on ( Figure 5).

Summary
The mode coupling in M/NEMS with various methods have attracted a wide range of interest from researchers in different fields, and have been widely used for fundamental physics studies, sensing applications, and classical/quantum information processing. In this review, we start from our own perspective on mechanical mode coupling, and divide such coupling into three types: intermode, near-neighbor, and long-range coupling. The coupling mechanism, application prospects, and important progress of these coupling types are reviewed. We then summarize several fundamental dynamical phenomena emerging from mode coupling, such as linear and nonlinear coupling, modal localization, synchronization, and phonon manipulation based on M/NEMS. In addition to the fundamental research, we also reviewed the applications of coupled resonators in sensing, summarizing the latest advances by using coupled mechanical systems, especially we focus on several proposals that can improve sensing accuracies. We ended up the review by listing some of the open questions and major challenges faced in the field of coupled mechanical resonators. We hope that this review will be helpful to researchers/students in the fields of M/NEMS, optomechanics, phononic science and technology, and hybrid quantum systems with nanomechanics.