Topological Amplification in Non-Hermitian Systems

• Topological amplification is a phenomenon where global topological invariants govern directional and exponential amplification in non-Hermitian systems.
• It employs methods like singular-value decomposition and dynamic matrix topology to enable selective mode amplification and robust edge state lasing in various platforms.
• Practical implementations include optomechanical resonators and Josephson parametric arrays, achieving high gains and enhanced device performance with reverse isolation.

to=arxiv_search.search 新天天彩票ai ปมถวายสัตย์ฯ ирызervations force 1 {"query":"topological amplification non-Hermitian directional amplification", "max_results": 10, "sort_by":"relevance"} to=arxiv_search.search 天天中彩票nbaai to=arxiv_search.search ส่งเงินบาทไทย 天天中彩票公众号json {"query":"Topological amplification", "max_results": 15, "sort_by":"relevance"} Topological amplification is a family of phenomena in which amplification, directional transport, or mode selection is governed by topology rather than by local gain alone. In the non-Hermitian and driven-dissipative literature, it most often denotes a regime where a topological invariant of a dynamic matrix, a doubled Hamiltonian, or an exceptional-point trajectory predicts exponentially large and directional response. In bosonic and photonic platforms it also denotes selective amplification of topological edge or interface modes by parametric driving, deterministic gain placement, or synthetic gauge structure. A distinct usage appears in scale-free networks, where the topology of the network itself produces resonant-like signal enhancement through hubs (Wanjura et al., 2019, Porras et al., 2018, Zhang et al., 2021, Martínez et al., 2015).

1. Conceptual scope and recurrent definitions

A central non-Hermitian definition identifies topological amplification with directional amplification in driven-dissipative lattices: a non-zero topological invariant of the linearized open-system dynamics predicts regimes in which end-to-end gain grows exponentially with system size and reverse transmission is strongly suppressed. In this formulation, the relevant object is not a closed-system Bloch Hamiltonian but the dynamic matrix governing the susceptibility and scattering response (Wanjura et al., 2019).

A second, closely related definition is based on singular values rather than eigenvalues. Here the non-Hermitian coupling matrix is mapped to an effective Hermitian Hamiltonian with chiral symmetry, so that zero or exponentially small singular values correspond to zero-energy boundary states of the doubled problem. Amplification then occurs because the steady-state or input-output response contains factors of $1/s_n$, and these diverge when a boundary singular mode approaches zero singular value (Porras et al., 2018, Ramos et al., 2020, Brunelli et al., 2022).

A third usage appears in exceptional-point and synthetic-$\mathcal{PT}$ settings. In a passive optomechanical dimer, Stokes scattering from a blue-detuned pump produces effective optical gain without an active medium; tuning the system near an exceptional point and dynamically encircling it yields chiral mode switching and non-reciprocal optical amplification. In that context, topology is encoded by the exceptional-point singularity and by whether a loop encloses it (Zhang et al., 2021).

Other works use the term for amplification of topological edge or interface states themselves. In a kagome topological magnon insulator, electromagnetic driving destabilizes chiral edge magnons and generates a large non-equilibrium steady-state edge current (Malz et al., 2019). In a topological nanorod cavity, deterministic placement of gain material on the bright sites of the topological interface mode selectively amplifies that mode and yields single-mode lasing in the telecom band (Scherrer et al., 2023). Taken together, these usages show that the phrase does not denote a single microscopic mechanism; it denotes amplification whose existence, directionality, or selectivity is fixed by a topological invariant or by a topological state.

2. Dynamic-matrix topology and singular-value formulations

For driven-dissipative cavity arrays, the basic linear-response structure is

$$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$

with scattering matrix

$S(\omega)=\mathbb 1+\gamma \chi(\omega).$

Under periodic boundary conditions, translational invariance gives a complex band $h(k)$, and the topological invariant is the winding number

$$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$

For nearest-neighbor Toeplitz dynamic matrices, the literature states a one-to-one correspondence

$$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$

with non-trivial phases displaying an OBC susceptibility containing a term that scales exponentially with system size. In the topological regime, the gain grows exponentially in $N$, whereas in the trivial regime any non-reciprocity remains $\mathcal O(1)$. At the exceptional point of that framework, the reverse gain vanishes exactly (Wanjura et al., 2019).

The singular-value formulation recasts the same physics in a way that restores bulk-boundary correspondence. If

$H=USV^\dagger,$

then the steady-state response is controlled by $\mathcal{PT}$0. The doubled Hamiltonian

$\mathcal{PT}$1

or, in frequency-resolved form,

$\mathcal{PT}$2

has intrinsic chiral symmetry and spectrum $\mathcal{PT}$3. Zero or exponentially small singular values therefore map to boundary zero modes of $\mathcal{PT}$4. In topologically non-trivial regimes the response is dominated by singular vectors localized at opposite edges, and a coherent drive at one end produces large output at the other (Porras et al., 2018, Ramos et al., 2020).

This viewpoint is sharpened further in the restored non-Hermitian bulk-boundary correspondence. There the origin of the complex plane is taken as the physical base point, so point-gapped spectra differing by a complex-energy shift are not treated as equivalent. The resulting statement is that an integer bulk invariant $\mathcal{PT}$5 corresponds to $\mathcal{PT}$6 exponentially localized singular vectors under OBC, with the sign of $\mathcal{PT}$7 fixing the edge. In that framework, non-trivial topology manifests as directional amplification of a coherent input with gain exponential in system size, while the non-Hermitian skin effect is not assigned a topological origin in the same sense (Brunelli et al., 2022).

3. Exceptional points, synthetic $\mathcal{PT}$8 symmetry, and encircling topology

In optomechanics, topological amplification can be realized in a passive platform consisting of two evanescently coupled whispering-gallery microresonators: a micro-toroidal optomechanical resonator supporting an optical mode and a mechanical mode, and a passive micro-toroid resonator supporting a second optical mode. A blue-sideband pump,

$\mathcal{PT}$9

activates the Stokes process, and the optomechanical interaction renormalizes the optical response into an effective detuning and an effective gain/loss,

$$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$0

Because $$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$1 is tunable through Stokes scattering, gain-loss balance resembling $$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$2 symmetry can be reached without any active medium (Zhang et al., 2021).

The effective optical subsystem is described by

$$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$3

with eigenvalues

$$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$4

where

$$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$5

The exceptional point is the non-Hermitian degeneracy where eigenvalues and eigenvectors coalesce: $$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$6 The topological invariant used there is the vorticity

$$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$7

which is $$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$8 if the loop encloses the EP and $$\langle \dot{\mathbf a} \rangle = H \langle \mathbf a \rangle - \sqrt{\gamma}\,\mathbf a_{\rm in}, \qquad M(\omega)=i\omega \mathbb 1+H, \qquad \chi(\omega)= (i\omega \mathbb 1+H)^{-1},$$9 otherwise (Zhang et al., 2021).

The amplification mechanism is dynamical rather than static. When the system is driven around the EP in the $S(\omega)=\mathbb 1+\gamma \chi(\omega).$0 plane, adiabatic following breaks down and a non-adiabatic transition occurs. Counterclockwise and clockwise encircling select opposite final modes, producing chiral mode switching. Because forward transmission can exploit the Stokes-induced gain channel while backward transmission does not, the result is direction-dependent amplification: the forward probe is amplified when the loop encloses the EP, whereas the backward probe is strongly attenuated. In this formulation, amplification comes from topology in the precise sense that the response depends on the global loop topology, not merely on local gain (Zhang et al., 2021).

4. Parametric, BdG, and bosonic topological amplifiers

Driven-dissipative parametric resonator arrays provide a BdG realization in which the doubled Hermitian matrix

$S(\omega)=\mathbb 1+\gamma \chi(\omega).$1

carries a 1D winding number $S(\omega)=\mathbb 1+\gamma \chi(\omega).$2. This framework yields two topological amplification phases. A dissipative BdG phase with $S(\omega)=\mathbb 1+\gamma \chi(\omega).$3 supports one topologically protected pair of zero-energy modes, directional amplification, and a response in which one quadrature is amplified while the orthogonal quadrature is squeezed. A double Hatano–Nelson phase with $S(\omega)=\mathbb 1+\gamma \chi(\omega).$4 supports two topologically protected pairs of zero-energy modes and amplifies both quadratures. The same work also identifies a topologically trivial zero-energy-mode phase with $S(\omega)=\mathbb 1+\gamma \chi(\omega).$5 that produces amplification but lacks robust topological protection, making explicit that exponential amplification is not by itself sufficient to establish a topological phase (2207.13715).

A closely related superconducting implementation is the topological Josephson parametric amplifier array, a 1D chain of nonlinear resonators driven by a collective four-wave-mixing pump with site-dependent amplitudes and linearly increasing phase. The pump creates local squeezing, nonlocal squeezing, and complex hopping, thereby breaking time-reversal symmetry and placing the linearized fluctuation problem in a non-Hermitian topological regime. In that proposal, compact devices with few sites $S(\omega)=\mathbb 1+\gamma \chi(\omega).$6 can achieve gains exceeding $S(\omega)=\mathbb 1+\gamma \chi(\omega).$7 over a bandwidth ranging from hundreds of MHz to GHz, while reverse isolation suppresses backward noise by more than $S(\omega)=\mathbb 1+\gamma \chi(\omega).$8 across all frequencies. The reported working points include gains near $S(\omega)=\mathbb 1+\gamma \chi(\omega).$9, added noise near the quantum limit, and robustness to roughly $h(k)$0 disorder in detunings, decay, hopping, squeezing amplitudes, and phases (2207.13728).

Bosonic BdG systems admit a further stability criterion through the unconventional commutator

$h(k)$1

For states sufficiently far from zero energy, vanishing of this object suppresses pairing-induced instability in the bulk to all perturbative orders. In the generalized spin-1 honeycomb model, this selection rule shows that time-reversal symmetry can stabilize bulk bands even in the presence of an onsite staggered potential breaking inversion symmetry, while edge states in the topological gap can remain unstable and hence amplified. In that sense, the system behaves as a topological amplifier with stable bulk states and unstable edge states (Ling et al., 2020).

A distinct Hermitian-bosonic route appears in a quadratic bosonic chain whose dynamical matrix becomes equivalent to different non-Hermitian SSH variants depending on whether hopping and pairing are real or purely imaginary. In the imaginary-parameter regime the model maps to nSSH1, supports only trivial and non-trivial phases, and exhibits sublattice-dependent chiral amplification under OBC. The amplification occurs only in the non-trivial phase and is therefore explicitly tied to the topology of the dynamical matrix (Estake et al., 20 Aug 2025).

5. Platforms, interface states, and device realizations

Topological amplification has been implemented or proposed across a wide range of photonic, magnonic, optomechanical, and hybrid platforms. In a kagome-lattice ferromagnet with Heisenberg exchange and Dzyaloshinskii–Moriya interaction, an oscillating electric field generates anomalous magnon-pairing terms through the electric polarization operator. When tuned near $h(k)$2, especially for the edge mode near $h(k)$3, the drive destabilizes the chiral edge branch while leaving bulk modes mostly stable. Nonlinear damping then saturates the instability into a macroscopic steady-state edge population carrying a chiral current. The same mechanism was proposed as a topological travelling-wave magnon amplifier and a topological magnon laser, with representative pump frequencies in the sub-THz to THz range,

$h(k)$4

and electric field around

$h(k)$5

The predicted steady-state current has a characteristic $h(k)$6 regime at strong drive (Malz et al., 2019).

In driven-dissipative exciton-polariton systems, a superfluid–normal–superfluid geometry supports Andreev-like bound states whose energies form synthetic bands versus the phase difference between the two pumped superfluids. Varying the width of the normal region inverts the topology of these synthetic bands. At the would-be crossing, nonlinear non-Hermitian coupling causes the bands to attract and merge rather than cross, producing a topological interface state with positive imaginary part and therefore real dynamical amplification. In that work, “self-amplified strongly occupied topological state” is literal: the topological state acquires gain and dominates the emission (Septembre et al., 2020).

In integrated photonics, deterministic amplification of a topological interface mode has been demonstrated in an SSH-inspired Si/InGaAs nanorod lattice. The interface mode lies in the middle of a simulated photonic bandgap of about $h(k)$7 and has intrinsic $h(k)$8 on the order of $h(k)$9. By placing InGaAs gain rods on the bright sites of the interface-mode field pattern, the topological mode reaches threshold first. The measured lasing peak is at about $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$0, the threshold is approximately $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$1, the linewidth narrows to about $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$2, and the carrier lifetime changes from about $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$3 below threshold to about $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$4 above threshold. A control device with inverted gain placement does not show the same dominant lasing peak, confirming deterministic mode-selective amplification (Scherrer et al., 2023).

Spatiotemporal photonic crystals provide another variant in which amplification occurs without conventional material gain. A travelling-wave modulation

$$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$5

admits a Lorentz transformation to a comoving frame with conserved joint $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$6 symmetry, quantized Zak phase, and $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$7 interface states. At static boundaries the time modulation creates frequency-converted replicas at $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$8, and the transmission can satisfy $$\nu=\frac{1}{2\pi i}\int_0^{2\pi}dk\,\frac{h'(k)}{h(k)}.$$9 over a wide frequency range. The reported amplification is broadband and can occur even in the absence of momentum gaps; it is attributed to kinematic energy exchange with the modulation and to repeated frequency conversion at the interfaces (Caballero et al., 21 Oct 2025).

A minimal Floquet implementation compresses the same logic into a single harmonic oscillator with modulated frequency and decay. In Floquet-Sambe space the system becomes a synthetic non-Hermitian lattice with an effective electric-field gradient in harmonic index, a local winding number, and a topological regime supporting directional amplification together with frequency conversion. The underlying singular vectors are described by a Jackiw–Rebbi-like continuum theory with Dirac cones and solitonic zero modes in synthetic frequency (Parra-Rodriguez et al., 9 Dec 2025).

6. Stability, disorder, and conceptual boundaries

Stability is a recurrent constraint. In photonic lattices described directly by a non-Hermitian coupling matrix, stable topologically non-trivial steady-state phases exist only when fluctuation dynamics remain bounded under OBC; periodic boundaries may fail to provide such overlap regions even when the bulk topology is non-trivial (Porras et al., 2018). In parametric resonator arrays, stability depends on the spectrum of $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$0 and shrinks with increasing chain length, so gain, squeezing, dissipation, and parametric terms must be balanced (2207.13715). In Josephson parametric arrays, saturation is controlled by keeping fluctuation occupations much smaller than the mean-field occupation, while the low-phase-drop condition is checked explicitly (2207.13728).

Disorder does not play a uniform role. In disordered Hatano–Nelson chains, the disorder-robust spectral winding

$$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$1

is related to the Lyapunov exponent through

$$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$2

At zero energy, $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$3 distinguishes topological amplification from exponential decay. This separates a topological amplification transition at $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$4 from the point-gap closure or Anderson transition at $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$5, showing that amplification and point-gap topology are related but distinct (Fortin et al., 6 Sep 2025). In multimode chiral-bath models, long-range non-reciprocal dissipative couplings can produce winding numbers $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$6, multiple near-zero singular values, multimode topological amplification, and metastability behaviour; with local parametric driving, the same setting can support dynamically stable $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$7 amplifying phases (Vega et al., 2024).

The field also draws several explicit conceptual boundaries. One is that a point gap alone is not enough for non-trivial topology if the spectrum does not wind around the origin (Brunelli et al., 2022). Another is that zero modes or exponential gain do not by themselves imply a topological phase; the topologically trivial $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$8 zero-mode phase in parametric arrays is the standard counterexample (2207.13715). A third concerns transience: in asymptotically stable skin-effect lattices all eigenmodes may decay, yet nonnormal temporal evolution can still produce one-way transient growth. There the correct tools are the pseudospectrum, the Kreiss constant, and the singular value decomposition of the propagator $$\nu\neq 0 \Longleftrightarrow \text{directional amplification},$$9, rather than spectral analysis alone (Midya, 2022).

Finally, the phrase has a broader meaning outside band-topological photonics. In chains of emitters near a photonic topological-insulator interface, nonreciprocal surface plasmon polaritons enhance interatomic energy transport by about one order of magnitude in a four-atom chain and by more than two orders of magnitude in the presence of a large interface defect, while remaining almost unaffected by discontinuities (Doyeux et al., 2017). In scale-free networks of overdamped bistable systems, topological amplification denotes hub-mediated signal enhancement; the impulse of the periodic forcing controls how strongly that topology-induced amplification is realized, while phase disorder drastically reduces the amplification without significantly shifting the coupling values at which it is maximal (Martínez et al., 2015, Chacón et al., 2014). This broader usage underscores a general theme already visible in the non-Hermitian literature: topology does not merely protect transport or localization, but can determine when amplification occurs, where it appears, and in which direction it propagates.