Phase-Tracked Dynamical Encircling

• Phase-tracked dynamical encircling is a non-Hermitian control strategy where systems are driven along closed loops while continuously tracking evolving state phases.
• The method employs closed-loop modulation of gain-loss and detuning to induce nonadiabatic transitions and direction-dependent mode switching, as verified in waveguide, photonic, and mechanical experiments.
• Applications span robust state conversion, full 2π phase modulation, and real-time Riemann surface transport, though outcomes rely sensitively on loop parameters and loss profiles.

Searching arXiv for recent and foundational papers on phase-tracked dynamical encircling, exceptional-point encircling, and dynamic pole-zero phase control. arXiv શોધી રહ્યો છું for relevant papers. Searching arXiv. Phase-tracked dynamical encircling denotes a class of non-Hermitian control protocols in which a system is driven along a closed loop while the evolving state, resonance condition, or response phase is continuously followed during the loop itself. In exceptional-point (EP) physics, this means that transport is governed by the full non-Hermitian dynamics on the associated Riemann surfaces rather than by a purely quasi-static continuation of instantaneous eigenstates. In more recent photonic and mechanical settings, the same phrase also covers pole-zero-based phase control, where the operating point is tracked along an iso-amplitude contour in the complex frequency plane so that phase accumulation is observed continuously and constrained by topology (Doppler et al., 2016, Hassan et al., 2017, Krasnok, 22 May 2025, Zhang et al., 5 Sep 2025).

1. Conceptual basis and historical emergence

The foundational dynamical-encircling problem arose in two-mode non-Hermitian systems with loss or gain, where an EP occurs when two resonant modes coalesce both in eigenvalue and eigenvector. In the canonical waveguide realization, a smoothly deformed metallic waveguide with absorption is engineered so that propagation distance $x$ plays the role of time, and the two relevant propagating modes are coupled by a periodic boundary modulation. This mapping turns scattering through the device into a genuine dynamical evolution problem, making it possible to examine the breakdown of adiabaticity during transmission rather than only the static topology of the EP (Doppler et al., 2016).

The 2016 waveguide experiment established the central physical message that slow encircling does not simply realize an adiabatic state flip. Instead, dominant non-adiabatic effects produce a direction-dependent, asymmetric mode switch, so that left and right injection correspond to opposite encircling orientations and yield different outgoing modes. This result repositioned EP encircling from a purely topological eigenvalue-exchange problem to a dynamical transport problem in which topology and gain-loss imbalance act jointly (Doppler et al., 2016).

A complementary analytical milestone came from the exact solution of a two-level non-Hermitian Hamiltonian with constant off-diagonal exchange elements and a perfectly encircled EP. That work derived the full evolution in transfer-matrix form, showed explicitly why clockwise and counterclockwise loops select different terminal eigenstates, and clarified why the adiabatic limit becomes effectively rank-one in its input-output action: the final state is funneled into a preferred eigenvector irrespective of the initial condition (Hassan et al., 2017).

2. Mathematical formulations

The standard two-level EP-encircling model is a $2\times 2$ non-Hermitian Hamiltonian with detuning, coupling, and unequal loss or gain. Representative forms used in the literature include

$$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$

For the first model, the EP occurs at

$$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$

while in the scaled second model it occurs at $g=1$, $\delta=0$. In both cases the spectral topology is a branch point with self-intersecting Riemann sheets, and a closed loop around the EP permutes the eigensheets (Doppler et al., 2016, Hassan et al., 2017).

The dynamical loop is usually parameterized so that the system evolves continuously around the degeneracy. One widely used parametrization is

$$g(\tau)=1-\rho\cos(\gamma\tau),\qquad \delta(\tau)=\rho\sin(\gamma\tau),$$

with $\gamma>0$ corresponding to a clockwise loop and $\gamma<0$ to a counterclockwise loop. In waveguide mappings, the same logic is implemented spatially by making coupling and detuning vary smoothly along the propagation coordinate, for example through an envelope $o(x)$ and detuning profile $2\times 2$0, so that the traveling wave executes the loop during transmission (Doppler et al., 2016, Hassan et al., 2017).

A distinct but related formalism appears in topological phase control via complex poles and zeros. For a side-coupled one-port resonator, the reflection coefficient

$2\times 2$1

has a zero at $2\times 2$2 and a pole at $2\times 2$3. Because the reflection phase is the argument of a meromorphic response, the total phase accumulation along a closed contour is governed by Cauchy’s Argument Principle,

$2\times 2$4

When the operating point is constrained to an iso-amplitude Apollonian circle that encloses one zero and no pole, a full $2\times 2$5 phase shift is accumulated while $2\times 2$6 remains fixed (Krasnok, 22 May 2025).

A further mathematical variant appears in closed-loop mechanical control. In the MEMS implementation, the relevant quantity is the response phase

$2\times 2$7

with $2\times 2$8 the driven-mode susceptibility. A phase-locked loop enforces $2\times 2$9, with the programmable reference phase $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$0 updated so that the drive frequency continuously follows the real-eigenvalue sheet of the instantaneous Hamiltonian. In this formulation, phase tracking is not only a diagnostic but also the feedback variable that keeps the system on resonance while the EP loop is traversed (Zhang et al., 5 Sep 2025).

3. Dynamical mechanisms: adiabatic breakdown, NAT, and phase selection

The defining phenomenon of dynamical EP encircling is the failure of ordinary adiabatic intuition. In non-Hermitian evolution, complex eigenvalues, nonorthogonal eigenvectors, and gain-loss competition prevent a simple statement that the state follows an instantaneous eigenmode around the loop. The exact two-level analysis isolates an accumulated gain-loss factor

$$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$1

and shows that the eigenstate spending longer in the amplifying sector dominates. This is the origin of the direction-dependent selection rule: clockwise encirclement asymptotically yields one eigenstate, counterclockwise encirclement the complementary one (Hassan et al., 2017).

In many platforms this selection occurs through a delayed non-adiabatic transition (NAT). The evolving state may remain on a higher-loss sheet for some interval, become unstable, and then jump to the lower-loss sheet. The jump is governed by the gain-loss landscape and the branch-cut structure of the Riemann surface rather than by a Hermitian Berry-phase picture. This is why the output can depend only on encircling direction and not on the initial branch or mode preparation (Doppler et al., 2016, Long et al., 2022).

The role of the starting point is equally decisive. In coupled ferromagnetic-waveguide systems with two EPs, starting in the $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$2-symmetric phase yields chiral behavior, while starting in the $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$3-broken phase yields non-chiral behavior: the final state becomes the same gain-like mode regardless of direction. Anti-$$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$4-symmetric systems reverse this rule. There, chiral dynamics appears when the loop starts and ends in the broken phase, because that is where the two eigenstates have the same imaginary part (Zhang et al., 2018, Zhang et al., 2018).

An important correction to early intuition is that chirality is not strictly equivalent to literal EP enclosure. A fiber-based polarization emulator directly observed chiral state transfer on an EP-excluding loop that remained sufficiently close to the singularity. The decisive ingredients were the local Riemann-surface geometry, eigenvector nonorthogonality, and finite-rate evolution. Likewise, in largely detuned multimode optomechanics, the dominant explanation is NAT and branch-loss structure rather than phase accumulation itself. This suggests that phase-tracked dynamical encircling should be understood as a dynamical non-Hermitian transport phenomenon whose topological content can remain operative even when the loop does not geometrically wind an EP in the strict sense (Nasari et al., 2022, Long et al., 2022).

4. Experimental realizations and observables

Phase-tracked dynamical encircling has been realized across wave, optical, mechanical, and solid-state quantum platforms. What varies from platform to platform is the tracked quantity: output phase difference, polarization trajectory, reconstructed state overlaps, resonance phase in a feedback loop, or the instantaneous phase of a pole-zero response. The common structure is a closed loop in a non-Hermitian control space and a measurement that resolves the dynamical path rather than only its endpoints.

Platform	Tracked quantity or control variable	Reported outcome
Two-mode waveguide (Doppler et al., 2016)	Spatially varying $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$5, $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$6; modal transmission	Robust asymmetric mode switch
NV center in diamond (Liu et al., 2020)	Time-dependent dilated Hamiltonian; state tomography	Asymmetric and symmetric mode switches
Fiber polarization emulator (Nasari et al., 2022)	Round-trip Stokes parameters on the Poincaré sphere	Direct observation of delayed jumps and chirality
MEMS disk resonator (Zhang et al., 5 Sep 2025)	PLL-tracked response phase $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$7	Continuous transport on real-eigenvalue Riemann surfaces
Side-coupled resonator (Krasnok, 22 May 2025)	Iso-amplitude pole-zero encirclement in complex frequency	Full $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$8 phase shift at constant amplitude

The microwave ferromagnetic-waveguide experiments are especially important because they introduced in situ control of loop size via an external magnetic field and enabled encircling of zero, one, or two EPs in a single device. The same family of systems also made phase tracking explicit through the measured output phase difference $$H= \begin{pmatrix} \delta-i\gamma_1/2 & g\ g & -i\gamma_2/2 \end{pmatrix}, \qquad H(t)= \begin{pmatrix} -i\tilde g(t)-\tilde\delta(t) & \kappa\ \kappa & i\tilde g(t)+\tilde\delta(t) \end{pmatrix}.$$9, with $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$0 identifying a symmetric mode and $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$1 an anti-symmetric mode (Zhang et al., 2017, Zhang et al., 2018).

In the quantum regime, a time-dependent non-Hermitian Hamiltonian was implemented with a single nitrogen-vacancy center by dilating the non-unitary two-level dynamics into a larger Hermitian two-qubit system. The experimentally accessible state $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$2 was reconstructed by quantum state tomography and mapped back to the target non-Hermitian state, allowing the direction dependence and start-point dependence of dynamical encircling to be verified in a real quantum system (Liu et al., 2020).

The MEMS realization introduced a different operational meaning of phase tracking: a closed-loop controller uses the system’s measured phase response to keep the drive continuously on the chosen eigenfrequency branch. This turns phase from a passive observable into the variable that enables branch following itself, and it suppresses the open-loop drift that ordinarily triggers unintended non-adiabatic switching (Zhang et al., 5 Sep 2025).

5. Multiple EPs, higher-order singularities, and extended geometry

The simplest EP encircling problem involves a single second-order branch point, but the literature has progressively generalized this to pairs of EPs, higher-order EPs, exceptional curves, and compactified parameter spaces. In ferromagnetic-waveguide systems, a lossless diabolic point can split into two EPs once asymmetry is introduced. This produces symmetry recovery: as the control parameter varies, the system can pass from a $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$3-symmetric phase to a broken phase, then recover symmetry before entering another broken phase. Encircling zero or two EPs yields no net state flip, whereas encircling one EP yields chiral switching (Zhang et al., 2017).

A geometric extension maps the full two-parameter Hamiltonian space onto a Riemann sphere. Under stereographic projection, the infinite boundary of the conventional $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$4 plane collapses to the north vertex. Continuous loops passing through that vertex can preserve the chiral EP response while avoiding the distributed loss accumulated on ordinary finite loops. In a silicon implementation, this geometry enabled near 100% asymmetrical polarization conversion efficiency between TE and TM modes with mode crosstalk below $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$5 dB at 1550 nm (Li et al., 2021).

Higher-order structures modify the usual chiral picture even more strongly. In a few-mode planar waveguide with a customized multilayer gain-loss profile, two embedded EP2s generate a third-order branch-point topology among three modes. Adiabatic encirclement yields the cube-root permutation

$$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$6

but dynamical encirclement breaks chirality: both clockwise and counterclockwise propagation select the same output mode $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$7, with reported conversion efficiencies about 78% for CW and about 73% for CCW (Dey et al., 2020).

Multimode optomechanical systems exhibit similarly rich behavior. Depending on parameters, they can host no EP, a pair of second-order EPs, or a third-order EP. The outcome then depends on loop radius, orientation, speed, starting point, and the number or order of enclosed EPs. A single second-order EP can show either non-reciprocal or chiral transfer depending on radius; two EP2s yield chiral transfer with strong starting-point dependence; a third-order EP can display both chiral and non-reciprocal state transfer within a single loop family (Fan et al., 2023). In four-mode $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$8-symmetric bosonic systems, dynamical crossing of a diabolic curve while encircling exceptional curves connects otherwise disconnected sheet pairs and produces a programmable symmetric-asymmetric multimode switch (Arkhipov et al., 2022).

6. Applications, interpretation, and limitations

The principal application of phase-tracked dynamical encircling is state conversion that is robust against input-state variation because the dynamics selects a preferred terminal branch. This underlies the asymmetric waveguide mode switch, the optical omni-polarizer that converts arbitrary forward input to one polarization eigenstate and reverse input to its complementary biorthogonal state, and the asymmetric polarization-locked silicon device in which output is locked to TE or TM according to propagation direction (Doppler et al., 2016, Hassan et al., 2017, Li et al., 2021).

A different application is pure phase modulation. In pole-zero engineering, two distinct approaches realize a full $$\delta_{\mathrm{EP}}=0,\qquad g_{\mathrm{EP}}=\frac{|\gamma_1-\gamma_2|}{4},$$9 phase shift while maintaining fixed reflected amplitude: either the excitation itself follows a complex-frequency iso-amplitude circle around a static reflection zero, or the resonator parameters are modulated so that the zero encircles a fixed monochromatic excitation. Because the phase change depends on $g=1$0, the effect is topological and remains stable against perturbations that do not change which singularities are enclosed (Krasnok, 22 May 2025).

Closed-loop phase tracking adds an instrumentation-level application: continuous in situ transport on Riemann surfaces. The MEMS phase-locked-loop architecture is designed specifically to maintain steady-state resonance while the EP loop is traversed, and it can also be used to induce branch switching intentionally by changing the tracked sheet. This points toward electrically controlled studies of branch cuts, EP braiding, and real-time reconstruction of non-Hermitian eigenstate transport (Zhang et al., 5 Sep 2025).

Several limitations recur across the literature. The outcome depends sensitively on start point, loop speed, and loss landscape; fast loops can suppress transfer, and in multimode settings higher-order topology may destroy chirality rather than enhance it (Zhang et al., 2018, Dey et al., 2020, Fan et al., 2023). In pole-zero phase control, the loop must remain on the constant-$g=1$1 contour and must enclose the zero without crossing the pole; otherwise amplitude-phase coupling reappears (Krasnok, 22 May 2025). In the closed-loop MEMS method, the phase-to-frequency map is one-to-one only in the weak-coupling regime $g=1$2 (Zhang et al., 5 Sep 2025).

A persistent misconception is that the phenomenon is simply Berry-phase accumulation around an EP. The current literature does not support that reduction. Some platforms do explicitly track phase and exploit topological phase winding, but many of the most characteristic outputs—direction-dependent state selection, NAT-triggered jumps, and initial-state-independent terminal modes—arise from the interplay of branch topology with gain-loss filtering and non-Hermitian adiabatic breakdown. A plausible implication is that “phase-tracked dynamical encircling” is best regarded not as a single standardized protocol, but as a family of control strategies unified by continuous branch-sensitive evolution on a non-Hermitian landscape (Hassan et al., 2017, Nasari et al., 2022, Long et al., 2022, Krasnok, 22 May 2025).