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Dynamically encircling an exceptional point through phase-tracked closed-loop control

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Locking a resonator's drive frequency to its response phase lets it circle continuously around an exceptional point and swap its two vibration modes in either direction.

desk verdict A careful MEMS demonstration of phase-locked resonance tracking around an EP; the frequency curves are servo-enforced by a phase schedule from the same static fit, so the sheet-permutation claim outstrips the data. read the letter →

arxiv 2509.04940 v1 pith:KNVNOK5F submitted 2025-09-05 quant-ph physics.class-ph

Sen Zhang , Yangyu Huang , Lei Yu , Kaixuan He , Ning Zhou , Dingbang Xiao , Xuezhong Wu , Franco Nori
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Hui Jing Xin Zhou
This is my paper · ORCID
classification quant-phphysics.class-ph
keywords exceptionalpointphase-lockedloopnon-HermitianHamiltonianRiemannsurfacedynamicencirclingMEMSresonatoradiabaticcontrolmodeswitching
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Phase-tracked closed-loop control is claimed to solve a standing problem in non-Hermitian physics: dynamically encircling an exceptional point—the point where two vibration modes degenerate in frequency and line shape—without non-adiabatic jumps throwing the oscillation off its Riemann sheet. The authors lock a driven microelectromechanical (MEMS) disk resonator's excitation frequency to the response phase of one mode, forcing the oscillation to ride a preset path on the eigenfrequency surface while the pump that couples the two modes sweeps a loop around the exceptional point. They show that clockwise and counterclockwise encircling, from either Riemann sheet and from the parity-time (PT)-broken as well as PT-symmetric phases, always exchanges the two modes: the outcome is set by the starting sheet, not by the direction. If this holds, encircling an exceptional point ceases to be a fragile, asymmetric process and becomes a controllable, direction-agnostic switching operation in an integrated electrical device—and the same loop can also trigger non-adiabatic transitions deliberately.

What carries the argument

The load-bearing mechanism is the phase-locked loop working through the response-phase map θ(ω_d) = −Arg[χ₁(ω_d)] of the driven mode. In the weak-coupling regime (2g < max(γ₁,γ₂)) this map is injective, so a locked phase uniquely fixes the oscillation frequency. The phase detector compares the mode-1 response phase with a reference carrying a time-varying shift φ(t) = θ{Re[λ±(t)]} − θ{Re[λ±(0)]}, and a PID controller adjusts the drive frequency to hold the difference at a constant setpoint. Since φ(t) is precomputed from the best-fit susceptibility along the desired loop, the locked oscillation rides the target sheet of the eigenfrequency Riemann surface while the pump strength and detuning

What would settle it

Stabilize or measure the pump phase and record the mode-2 (idler) amplitude along with the mode-1 frequency during a phase-tracked encirclement, then check directly that the two-mode state swaps eigenstates when the loop closes: if the frequency ends on the opposite sheet without the mode-2 component changing correspondingly, the permutation claim fails. A second check: sweep the encircling loop progressively faster and watch when the trajectory leaves the Riemann surface—it should depart exactly when the phase-locked loop loses lock, not earlier; if the oscillation leaves the surface while th

Watch

Extended reading notes

Core claim

Central claim: a phase-locked loop can carry a driven non-Hermitian oscillator continuously around an exceptional point while staying on resonance—something open-loop encircling cannot do, because non-adiabatic transitions throw the state off its Riemann sheet. The authors demonstrate this in an on-chip silicon MEMS disk resonator whose two modes have engineered different damping and are coupled by an anti-Stokes parametric pump. As the pump strength and detuning trace rectangular and circular loops around the EP, the PLL forces the measured response phase to follow the precomputed eigenfrequency-mapped phase; the oscillation glides along the predicted Re[λ±] surface and ends on the opposite

Load-bearing premise

The static response-phase curve θ(ω_d) = −Arg[χ₁(ω_d)], characterized once from best-fit parameters, remains one-to-one and quantitatively accurate while the pump voltage and detuning are swept in real time, so locking the response phase to the precomputed target uniquely places the oscillation on the intended Riemann sheet; if the phase map drifts during the sweep, the loop either unlocks or rides the wrong sheet while the phase condition is still satisfied.

Editorial extensions

If this is right

  • Continuous, resonance-maintained encircling becomes an ordinary control operation: the phase-tracking loop holds the oscillation on the eigenfrequency surface at every point of the loop, so the non-adiabatic jumps that interrupt open-loop encirclements do not occur.
  • The outcome is fixed by the starting sheet, not the direction: clockwise and counterclockwise encirclements both permute the two sheets, from starting points in both the PT-symmetric and PT-broken phases, so the chiral mode-switching asymmetry of conventional dynamical encircling is absent.
  • The method is path-general: it reproduces the predicted Re[λ±] trajectories for rectangular loops (one parameter varied at a time) and circular loops (both parameters varied simultaneously), meaning arbitrary enclosing paths can be programmed.
  • The same control can force the opposite behavior: deliberately switching the tracked phase between the high- and low-frequency sheets during an encirclement induces controlled non-adiabatic transitions in the low-loss region, giving an in-situ knob for Stokes-phenomenon physics.
  • Reading out the instantaneous hybrid state |ψ⟩ ∝ (ω_d − Ω₂ + iγ₂/2, g)ᵀ along the trajectory would recover the instantaneous Hamiltonian, the imaginary parts Im[λ±], and the eigenstates—extending the method from frequency tracking to full state tracking.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The loop effectively substitutes a bandwidth condition for the usual adiabaticity condition of slow driving: a testable extension is that the trajectory should stay on the Riemann surface up to sweep rates set by the phase-lock bandwidth rather than by the eigenvalue gap, so encircling fast enough to trigger Stokes jumps in open loop should remain smooth until the loop unlocks.
  • Because the protocol needs only one measurable phase, it should transfer to higher-order exceptional points; a natural test is a third-order EP, where the prediction would be a three-cycle permutation of sheets again fixed by the starting sheet.
  • The sheet exchange is inferred from the mode-1 frequency alone, since the paper states the mode-2 phase is unreadable under the uncontrolled pump phase; a direct confirmation of genuine state permutation awaits a coherently coupled (non-parametric) device where both mode amplitudes can be measured—the paper names this as its own future direction.
  • Read as an engineering primitive, the effect is a deterministic topological switch: start on either sheet, close any loop around the exceptional point, and the oscillation ends on the opposite sheet regardless of the loop's orientation—a property that could be exploited in MEMS oscillators and gyroscopes as a state-switching element.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper reports a closed-loop control scheme, based on a phase-locked loop (PLL), for dynamically encircling an exceptional point (EP) in a non-Hermitian MEMS disk resonator. Two mechanical modes with different damping rates are coupled by a parametric pump, yielding an effective non-Hermitian Hamiltonian with eigenvectors forming a Riemann surface in the (Vp, δp) parameter plane. Static frequency-response spectra are fit to a susceptibility model (Eqs. 8–10) to extract the complex eigenvalues and to construct the Riemann surfaces. The authors then impose a time-varying phase schedule φ(t) = θ{Re[λ±(t)]} − θ{Re[λ±(0)]}, computed from the same best-fit parameters, and use a PID-controlled PLL to force the response phase θ to track this schedule while Vp(t) and δp(t) execute rectangular or circular loops around the EP. They report that the measured oscillation frequency follows the theoretical Re[λ±] surface, and they interpret this as a demonstration that excitations are permuted between the two Riemann sheets, independent of encircling direction and of the starting sheet or phase (PT-symmetric or PT-broken). They also demonstrate induced non-adiabatic transitions by deliberately switching the phase setpoint between the high- and low-frequency branches.

Significance. The static spectral characterization and the Floquet reduction are carefully executed: the fits populate the expected eigenvalue surfaces and the EP location is consistent with the parity-time transition. The PLL-based phase tracking is an interesting engineering contribution that may enable real-time resonance maintenance in tunable non-Hermitian systems. However, the central dynamical claim — that the protocol genuinely transports excitations on the Riemann surface and permutes sheets — is not independently established because the phase schedule is derived from the same static model used to draw the theoretical surfaces, and the second mode's phase is never measured. If the claims were backed by an independent observable, the work would constitute a notable advance in dynamic EP encirclement; in its current form, the experimental evidence primarily demonstrates that a PLL can track a precomputed phase schedule within the validity of the static phase map.

major comments (3)
  1. [Results, 'Smooth encircling of EP'; Eqs. (8)–(10)] The phase schedule φ(t) = θ{Re[λ±(t)]} − θ{Re[λ±(0)]} is computed as −Arg{χ1[Re(λ±)]} using the best-fit parameters obtained from Fig. 2d–e. The PLL then forces the response phase θ to follow this schedule. Because θ(ωd; Vp, δp) is the same fitted static map, the resulting ωd is essentially the inverse image of the prescribed phase under that map. Agreement between the moving-averaged frequency and the gray theoretical curve in Fig. 3d therefore demonstrates self-consistency between the schedule and the static model, not an independent verification that an excitation traverses the Riemann surface or that the two sheets are permuted. This is the central claim of the paper, and it needs an independent probe (e.g., mode-2 phase or amplitude, or a perturbation test) or a reframing of the claim as 'phase-scheduled resonance tracking' rather than 'sheet permutation.'
  2. [Conclusions and Supplementary Note 2C] The paper explicitly states that the phase of the mode-2 idler wave is undetectable because of the uncontrolled pump phase, and in the Conclusions that 'we currently face limitations in measuring the phase information of mode 2.' Thus the state on the second Riemann sheet is inferred solely from the mode-1 oscillation frequency. The direction-independent permutation conclusion therefore rests on the phase schedule, which already contains the branch-switch information, rather than on a measured two-state trajectory. This is a load-bearing gap for the topological claim; without a second observable or a model-independent signature, the experimental data cannot distinguish a genuine state flip from a servo-enforced frequency sweep that merely reproduces the input schedule.
  3. [Results, 'Concept' and 'Smooth encircling of EP'] The injectivity of the phase–frequency map is invoked via the weak-coupling condition 2g < max(γ1, γ2), and the entire loop assumes that this static map remains quantitatively valid while Vp(t) and δp(t) are swept dynamically. This assumption is not tested. If the static phase map drifts during the sweep (for example, because of the Vp-dependent frequency shift κVp²/(8ω) or slow thermal drift), the loop could remain locked while riding a different sheet or a slightly off-resonance point, still satisfying the phase condition. The error band in Fig. 3d reports frequency jitter but does not quantify the accuracy of the phase map. A control experiment with an intentionally detuned phase schedule, or simultaneous open-loop verification at selected points along the loop, would be needed to support the claim that the trajectory actually follows the intended Riemann sheet.
minor comments (5)
  1. [Abstract and Introduction] The phrase 'smooth, dynamic encircling' is used, but the encircling is predetermined by a phase schedule computed from the static model. Suggest using 'phase-scheduled' or 'emulated' where appropriate to avoid overclaiming.
  2. [Fig. 3d] Please define explicitly the moving-average window and the rolling standard deviation used for the error band. Currently 'moving averaged value' and 'error band' are not quantified, making it difficult to assess the scatter of the measured frequency.
  3. [Methods, 'Spectral fitting'] The phase compensation φ = −65° is introduced in Methods but the reader must infer how it is incorporated into Eq. (10). Clarify whether the compensated phase is θ_comp = θ − φ and how this enters the fitting.
  4. [Methods, 'Smoothly encircling of EP with a circular trajectory'] The circular-path parameters are given only in the Methods. Since the circular path is an important demonstration, consider showing the trajectory in the main text or in a main-text figure.
  5. [Conclusions] The 'hybrid state' |ψ⟩ expression is introduced without derivation. Either derive it from the steady-state solution (Eq. S.18–S.20) or cite the relevant supplementary section explicitly.

Circularity Check

2 steps flagged · score 6.0 of 10

The closed-loop frequency trajectories are servo-enforced by a phase schedule computed from the same fitted static model used to draw the Riemann surfaces, so the phase-tracked 'sheet permutation' claim reduces to the input schedule and unmeasured mode-2 information.

  1. fitted input called prediction [Results, 'Smooth encircling of EP' (around Fig. 3a-d); Methods, 'Spectral fitting', Eqs. (8)-(10)]
    "The phase θ{Re[λ±(t)]} for each loop is predetermined by calculating −Arg{χ1[Re(λ±)]} using the best-fit parameters of the non-Hermitian system derived from Fig. 2d-e. ... Throughout the smooth encircling process, the time-varying phase shift is defined as φ(t)=θ{Re[λ±(t)]}−θ{Re[λ±(0)]}. ... A PID controller is then employed to adjust the reference frequency ωd, ensuring that the relative phase θ−φ remains locked at a constant setpoint value θ0."

    The control loop is commanded to enforce θ(t)=θ0+φ(t), where φ(t) is evaluated from the same best-fit parameters that generate the Re[λ±] surfaces in Fig. 2d-e. Since the phase-frequency relation θ(ωd)=−Arg[χ1(ωd)] (Methods, Eq. 8) is assumed injective, the locked frequency is the unique solution of θ(ωd)=θ{Re[λ±(t)]}, i.e. ωd=Re[λ±(t)]. Thus the moving-averaged frequency trajectory in Fig. 3d and Fig. 4 is not an independent prediction of the model; it is the inverse image of the prescribed phase under the fitted static map. The agreement with the gray theoretical curve confirms that the PLL tracks its own schedule and that the static map remains approximately valid during the sweep, but it does not independently demonstrate transport on the Riemann surface.

  2. self definitional [Conclusions (p. 10); Supplementary Note 2C (p. 7)]
    "These findings robustly demonstrate that a phase-tracked dynamic encircling of the EP can permute excitations between the two sheets, regardless of the direction of encircling. ... we currently face limitations in measuring the phase information of mode 2 due to its strong dependence on the uncontrolled pump phase. ... Since the pump phase is arbitrary and uncontrollable in this study, the overall phase of the mode-2 idler wave is undetectable."

    The claimed 'permutation of excitations between the two sheets' is the central topological conclusion. It is read off from the mode-1 frequency trajectory, but that trajectory is exactly the branch λ± written into the predetermined phase schedule: the sign of ± is switched when the scheduled phase crosses the branch cut, and the PLL then forces the oscillator frequency to follow that branch. With mode-2 phase undetectable, the data cannot distinguish a genuine state flip on the second sheet from the oscillator simply tracking the prescribed frequency branch. The sheet label used to conclude permutation is therefore supplied by the input schedule rather than independently observed.

full rationale

The paper presents a real experimental implementation of a phase-locked loop around an exceptional point in a MEMS system, and the static characterization, fitting, and PLL operation are self-consistent. The circularity is not in the control electronics but in the epistemic framing of the result. The phase schedule is computed from the same best-fit static susceptibility used to draw the theoretical eigenvalue surfaces, and the injective phase-frequency map forces the locked frequency to reproduce the input eigenvalue trajectory. Consequently, the 'alignment' of the measured frequency with the theoretical Re[λ±] curve is a consistency check of the static model and the PLL's tracking ability, not an independent confirmation of the topological sheet permutation. The paper itself admits that the mode-2 phase information—which would be needed to verify on which sheet the state actually resides—is undetectable. Thus the central topological claim reduces, by the paper's own equations, to the branch choice embedded in the predetermined phase schedule. I do not count the self-citation to [31] as load-bearing circularity: the injectivity assumption is checkable from the paper's own fits, and the main reduction is the phase-schedule/frequency-trajectory tautology. The work still demonstrates a working phase-tracked closed-loop encircling protocol, which has independent experimental value, but the 'permutation' claim is only partially supported. Hence a score of 6, not higher: the reported frequency tracking is real, but the topological assertion is largely defined into the input rather than measured from the output.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on five fitted quantities (damping rates, electrostatic tuning coefficient, circuit phase, spectral fitting scale/feedthrough, and implicit PID gains) plus the weak-coupling injectivity assumption, the validity of the static phase map under dynamic sweeps, the rotating-wave reduction of the parametric pump, and the neglect of the Vp^2 detuning shift. No new physical entities, particles, or forces are introduced.

free parameters (5)
  • Damping rates gamma1, gamma2 = 2π x 676 mHz, 2π x 904 mHz
    Measured from decay signals of the two modes (Q1=74,658, Q2=56,424); they set the non-Hermiticity gamma = 2π x 228 mHz and enter the phase map used to program the encircling loop.
  • Electrostatic tuning coefficient kappa = 70,186 N m^-1 kg^-1 V^-2
    Measured; determines g = kappa V0 Vp / (4 omega1) and the tuning of Omega1, Omega2, i.e., the whole Riemann surface used for the phase setpoints.
  • Circuitry phase shift phi = -65 degrees
    Found by fitting open-loop phase data; compensated before the response signal enters the phase detector, and therefore enters the effective setpoint.
  • Feedthrough b and drive scale f = complex, fitted per spectrum
    Parameters of the spectral fitting function F(omega_d) = f chi1 e^{i phi} + b used to extract complex eigenvalues from open-loop spectra; these fits produce the Riemann surfaces and the EP location that define the encircling loops.
  • PID gains of the adaptive PLL = not stated
    Chosen implicitly by hand; closed-loop stability and tracking fidelity depend on them, but they are not reported.
assumptions (5)
  • domain assumption Rotating-wave approximation: only the 0th idler of mode 1 and the 1st idler of mode 2 are kept; higher idlers and second-order time derivatives are neglected
    Supplementary Note 2, Eqs. (S.3)-(S.7); standard for this pump-drive geometry but limits validity to weak modulation and near-resonant drive.
  • domain assumption Weak-coupling injectivity: 2g < max(gamma1, gamma2), so the phase response theta(omega_d) is injective
    Results, 'Concept' paragraph: the system is operated in the weak-coupling region to ensure the mapping remains injective, otherwise bifurcation takes place. This injectivity is what makes phase-locking uniquely determine the frequency.
  • ad hoc to paper Static phase map remains valid during dynamic parameter sweeps
    The preset phi(t) is computed from static best-fit parameters, and the paper does not independently verify that the open-loop phase response at the swept (Vp(t), delta_p(t)) equals the static one; this is the load-bearing premise of the dynamic demonstration.
  • domain assumption The Vp^2-dependent detuning shift of Omega1 and Omega2 is negligible in the measurement range
    Results, 'On-chip non-Hermitian system': the authors state this effect is negligible within the measurement range; it is dropped from the effective detuning control.
  • domain assumption Steady-state linear response: constant drive amplitude F and linear susceptibility chi1 describe the driven mode
    Methods, 'Spectral fitting'; no nonlinearity check is reported for the closed-loop operating amplitudes, and the drive amplitude is kept constant.

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Pith. "Pith review of Dynamically encircling an exceptional point through phase-tracked closed-loop control." pith.science (2026). https://pith.science/paper/KNVNOK5F

@misc{pith2026250904940,
  author       = {Pith},
  title        = {Pith review of: Dynamically encircling an exceptional point through phase-tracked closed-loop control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KNVNOK5F}},
  note         = {Machine review of arXiv:2509.04940}
}
read the original abstract

The intricate complex eigenvalues of non-Hermitian Hamiltonians manifest as Riemann surfaces in control parameter spaces. At the exceptional points (EPs), the degeneracy of both eigenvalues and eigenvectors introduces noteworthy topological features, particularly during the encirclement of the EPs. Traditional methods for probing the state information on the Riemann surfaces involve static measurements; however, realizing continuous encircling remains a formidable challenge due to non-adiabatic transitions that disrupt the transport paths. Here we propose an approach leveraging the phase-locked loop (PLL) technique to facilitate smooth, dynamic encircling of EPs while maintaining resonance. Our methodology strategically ties the excitation frequencies of steady states to their response phases, enabling controlled traversal along the Riemann surfaces of real eigenvalues. This study advances the concept of phase-tracked dynamical encircling and explores its practical implementation within a fully electrically controlled non-Hermitian microelectromechanical system, highlighting robust in-situ tunability and providing methods for exploring non-Hermitian topologies.

Figures

Figures reproduced from arXiv: 2509.04940 by the authors.

Figure 5
Figure 5. The Newtonian equatio [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 1
Figure 1. Concept of phase-tracked smooth encircling. a, The non-Hermitian system comprises two coupled modes (labeled as Ωଵ and Ωଶ for their natural frequencies) with distinct damping rates. An adaptive phase-tracked closed-loop oscillation is established to trace the eigenfrequencies. PLL, phase-locked loop. b, The correspondence between the eigenfrequencies Re(𝜆±) and the response phase 𝜃[Re(𝜆±)] as illustrated in a typica… view at source ↗
Figure 2
Figure 2. On-chip non-Hermitian microelectromechanical system. a, The layout of this non￾Hermitian resonator, where the left one shows the 3D model diagram, the right one shows a wafer containing 333 devices, and the middle one displays the microscope image of a single device (scale bar: 1mm). b, A pair of non-degenerate six-node standing-wave modes (labeled as 1 and 2) of a MEMS disk resonator are engineered to exhibit a con… view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: Setup for adaptive phase-tracking. a, Mode 1 of the non-Hermitian resonator is actuated by a sinusoidal wave generated by a reference oscillator. The displacement of mode 1, which includes an additional response phase 𝜃, is demodulated using a parallel reference signal…
Figure 4
Figure 4. Figure 4: Results of smooth encircling starting from PT-symmetric phase obtained using the adaptive phase-tracking technique. a, The phase-tracked closed-loop oscillation frequencies for the CW encircling process starting from the high-frequency sheet (red curve), the CW encircl…
Figure 5
Figure 5. Figure 5: Results of smooth encircling starting from PT-broken phase obtained using the adaptive phase-tracking technique. a, The phase-tracked closed-loop oscillation frequencies for the CW encircling process starting from the high-frequency sheet (red curve), the CW encircling…

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    Chemical mechanical polishing (CMP): The handle and oxide layers of the structure SOI are subsequently removed using a CMP process, leaving a 100 m-thick structural layer firmly bonded to the substrate SOI

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    Structure release Supplementary Figure 7. Fabrication of the non-Hermitian MEMS resonator . (a) MEMS fabrication process of the devices. ( b) The fabricated 6-inch wafer with 333 devices. ( c) Details of the dicing lanes. ( d) The microscopic picture of a non-Hermitian MEMS di...

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.