Arbitrary-order exceptional points in a nanomechanical cavity

Higher-order exceptional points (EPs) govern non-Hermitian system dynamics through their enriched and sharpened spectral topology, yet the intrinsic topological fragility hinders robust experimental realization. Here, we present a scalable architecture that implements arbitrary-order EPs via a recurrent network comprising a single nanomechanical resonator and unlimited virtual resonators. We experimentally realize mechanical EPs up to the seventh order and confirm this architecture’s scalability. Moreover, we reveal that the fundamental noise component and the measured signal share the same system coupling channel and thus undergo identical root-response amplification near EPs of arbitrary order, consistent with our signal-to-noise ratio measurements. Our work establishes a general platform for exploring higher-order EP-based phenomena while clarifying the fundamental boundary of non-Hermitian sensitivity enhancement across diverse physical systems.

💡 Research Summary

In this work the authors address two central challenges in non‑Hermitian physics: (i) the scalable realization of arbitrary‑order exceptional points (EPs) and (ii) the long‑standing debate over whether higher‑order EPs can fundamentally improve sensor sensitivity. Conventional approaches to high‑order EPs rely on physically coupling many resonators, which quickly becomes impractical because each added mode introduces extra loss channels, parameter‑tuning complexity, and additional sources of thermal and quantum noise. To overcome these obstacles the authors propose a hybrid physical‑digital architecture based on a recurrent network. A single nanomechanical resonator (mode 0) is coupled to an unlimited number of “virtual” resonators (modes 1…n) that are simulated in real time on an FPGA. By pre‑defining the complex eigenfrequencies and coupling coefficients of the virtual modes, the full N‑dimensional non‑Hermitian Hamiltonian can be engineered while keeping the physical system one‑dimensional.

The experimental platform consists of a silicon “fishbone” optomechanical cavity. Two optical modes (≈1563 nm for readout and ≈1608 nm for feedback) are used to transduce the mechanical displacement and to apply the digitally generated feedback force, respectively. The FPGA computes the response of the virtual resonators and feeds the resulting signal back to the pump laser, thereby realizing the desired complex coupling J₀j e^{iφ_j}. By adjusting the feedback gain and the delay phase φ_d, the authors demonstrate both parity‑time (PT) and anti‑PT symmetric configurations, achieving second‑ and third‑order linear EPs (LEPs).

To explore nonlinear EPs (NEPs) they exploit feedback saturation in the phonon‑lasing regime. When the gain is high enough, the system settles into a single lasing mode whose frequency response to a perturbation ε follows εⁿ, where n is the EP order. The authors experimentally verify this scaling for n = 3 and n = 7, confirming the existence of seventh‑order EPs—an unprecedented achievement in any platform.

A key contribution of the paper is the systematic analysis of noise. Because the measured signal and the fundamental noise (thermal and vacuum fluctuations) both enter the system through the same coupling channel, they experience identical root‑response amplification near the EP. Allan deviation and linewidth measurements show that the noise grows as εⁿ, exactly canceling the signal’s enhanced response. Consequently, the signal‑to‑noise ratio (SNR) does not improve with EP order; no SNR gain is observed near the third‑ or seventh‑order NEPs. This experimental result resolves the controversy over “exceptional‑point‑enhanced sensitivity” by demonstrating that, for fundamental noise, higher‑order EPs offer no intrinsic advantage.

However, the authors point out that when the dominant noise source is technical (e.g., laser intensity noise, electronic readout noise) and is not coupled through the EP dynamics, the large response factor of a high‑order EP can still translate into a practical SNR improvement. Thus, EP‑based sensing remains valuable in carefully engineered noise environments.

Finally, the authors emphasize the universality of their approach. Since the virtual resonators are purely digital, the same scheme can be ported to spin ensembles, superconducting circuits, or thermal systems by merely reprogramming the FPGA and adjusting the physical coupling interface. This opens a pathway for systematic exploration of non‑Hermitian topology, high‑order EP physics, and their applications in precision metrology across a broad range of platforms.