Parameter-Sensitive Non-Hermitian Tunneling

• Parameter-sensitive non-Hermitian tunneling refers to transport phenomena where slight variations in non-Hermitian parameters induce abrupt changes in transmission, reflection, and delay.
• It encompasses mechanisms such as skin effect, exceptional-point branch transfer, and non-unitary interface matching to control wave and state transfer.
• Applications in photonics, acoustics, and sensing highlight the tunable interplay between gain/loss, non-reciprocity, and disorder in practical systems.

Parameter-sensitive non-Hermitian tunneling denotes a family of transport, scattering, and state-transfer phenomena in which tunneling-like response changes sharply under variations of non-Hermitian control parameters such as non-reciprocity, gain/loss strength, onsite dissipation, driving, disorder, exceptional-point proximity, or imaginary potential. In current usage, the phrase spans several distinct settings: barrier penetration and tunneling time in complex potentials, directional tunneling induced by the non-Hermitian skin effect, non-unitary interface transmission, Landau-Zener-type interband transfer in driven lattices, and non-adiabatic branch transfer in parameter space (Longhi, 2022, Umar et al., 5 Apr 2025, Yi et al., 2020, Terh et al., 2023, Parkavi et al., 2021, Liu et al., 24 Aug 2025). What unifies these cases is not a single Hamiltonian form, but the fact that small parameter changes can switch the system among reflectionless transmission, suppressed transport, chiral tunneling, amplified oscillations, width-independent delay, or instability.

1. Conceptual domain

The literature treats non-Hermitian tunneling in several technically different senses. In the narrowest sense, it refers to wave transmission through a spatially localized non-Hermitian barrier, as in tight-binding heterojunctions, optical multilayers, or complex rectangular potentials (Longhi, 2022, Yim et al., 2019, Umar et al., 5 Apr 2025). In a broader lattice sense, it includes directional transport across interfaces or boundaries where asymmetric couplings, skin localization, or non-Bloch structure act as an effective barrier (Yi et al., 2020, Langfeldt et al., 23 Jul 2025, Wang et al., 26 Mar 2026). A further extension treats “tunneling” as transfer between instantaneous eigenstate branches rather than motion through real space; this is the meaning adopted for non-adiabatic sensing near exceptional points (Liu et al., 24 Aug 2025).

This diversity matters because the same word can otherwise obscure distinct observables. Some papers study transmission and reflection coefficients, some study tunneling delay, some analyze boundary accumulation and skin-direction transport, and some use output populations or Fisher information as the tunneling readout (Jazayeri, 2022, Longhi, 2022, Lu et al., 2024, Liu et al., 24 Aug 2025). A common misconception is that non-Hermitian tunneling necessarily implies high-transmission barrier penetration. That is not generally so: in the double-exceptional-point optical heterostructure, the decisive effect is unidirectional reflectionlessness while transmission is “negligibly small” because of strong absorption (Yim et al., 2019). Conversely, in PT-symmetric electromagnetic structures, fixed-point frequencies can support unit transmission, zero reflection, and nearly flat transmission phase over a broad bandwidth (Jazayeri, 2022).

A second misconception is that non-Hermitian tunneling must be tied to balanced gain and loss or to $\mathcal{PT}$ symmetry. Several studies explicitly show otherwise. On-site dissipation can induce skin modes only under symmetry-dependent conditions rather than generically (Yi et al., 2020); the non-Hermitian Hartman effect can occur “without any special symmetry in the system” when non-Hermiticity comes from non-reciprocal couplings (Longhi, 2022); and the two-center complex delta potential demonstrates that physically relevant quasi-Hermitian scattering can depend more directly on balanced opposite imaginary parts than on exact $\mathcal{PT}$ symmetry (Mehri-Dehnavi et al., 2010).

2. Control parameters and response variables

The most elementary control parameter is non-reciprocity. In the two-site asymmetric-tunneling model, the single parameter $\alpha$ enters the effective Hamiltonian

$$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$

with $\beta=(1+\alpha)/(1-\alpha)$ directly controlling the difference between forward and backward normalized tunneling probabilities (Santos et al., 2014). In non-reciprocal active acoustic metamaterials, the same role is played by $\eta$, which sets asymmetric couplings $1\pm\eta$, discrete attenuation $q_d=\sqrt{(1-\eta)/(1+\eta)}$, and continuous attenuation $q_c=\eta\omega_0/c$; increasing $\eta$ darkens the interface but also increases reflection (Langfeldt et al., 23 Jul 2025).

Gain/loss and dissipation provide a second major control axis. In the non-Hermitian Dirac interface problem, anomalous Klein tunneling is governed by whether the ratio $\mathcal{PT}$0 is matched across the interface: matched ratios reduce the problem to Hermitian Dirac scattering, whereas mismatched ratios make the interface itself non-unitary and permit $\mathcal{PT}$1, $\mathcal{PT}$2, or CPA-laser behavior despite the absence of bulk amplification or damping (Terh et al., 2023). In the driven spin-orbit-coupled bosonic junction, balanced or unbalanced gain-loss strengths $\mathcal{PT}$3 compete against drive-renormalized tunneling amplitudes to determine whether spin-dependent tunneling is stable, decaying, or unstable (Luo et al., 2020). In space-fractional quantum mechanics, the absorptive part $\mathcal{PT}$4 of a barrier $\mathcal{PT}$5 modifies both the phase accumulation and the delay-time dependence on thickness (Umar et al., 5 Apr 2025).

Driving and externally imposed fields often convert static non-Hermitian structure into parameter-sensitive tunneling dynamics. In the non-Hermitian diamond chain, the decisive parameters are the gain-loss strength $\mathcal{PT}$6, synthetic magnetic flux $\mathcal{PT}$7, and synthetic electric fields $\mathcal{PT}$8 and $\mathcal{PT}$9: changing them switches the system among compact localization, stable Bloch oscillations, Landau-Zener-induced transfer, amplified Bloch oscillations, and blow-up (Parkavi et al., 2021). In the driven bosonic junction, the ratio $\alpha$0, the spin-orbit parameter $\alpha$1, and the Bessel-renormalized couplings $\alpha$2 and $\alpha$3 determine whether only spin-conserving, only spin-flipping, or no tunneling channels survive (Luo et al., 2020).

Disorder is a further control parameter when tunneling is understood as wave penetration through a non-Hermitian medium. In the generalized Hatano-Nelson class, increasing disorder $\alpha$4 competes with non-reciprocity $\alpha$5, drives a transition from extended complex-energy states to localized states, closes the point gap, and destroys skin modes (Wang et al., 2024). This is not a barrier-transmission problem in the narrow sense, but it is directly relevant to parameter-sensitive transport because it identifies sharp disorder-tuned transitions in bulk penetration and boundary accumulation.

3. Mechanisms of non-Hermitian tunneling

One major mechanism is skin-induced effective barrier formation. In the Rice-Mele-based analysis of on-site-dissipation-induced skin modes, tunneling becomes chiral: it “favors the direction where the skin modes are localized” (Yi et al., 2020). In the active acoustic metamaterial, a reciprocal region coupled to two mirrored non-reciprocal regions develops a quiet interior zone; the wave amplitude decays strongly inside the interface and reappears on the far side, creating a tunneling analogue generated by the non-Hermitian skin effect rather than by a scalar potential barrier (Langfeldt et al., 23 Jul 2025). In both settings, the effective barrier is a consequence of asymmetric coupling and non-Bloch accumulation.

A second mechanism is exceptional-point or criticality-enhanced branch transfer. In the non-adiabatic sensing paradigm, the two-level $\alpha$6-symmetric Hamiltonian

$\alpha$7

remains in the $\alpha$8-symmetric regime $\alpha$9, but non-adiabatic transitions between instantaneous eigenstate branches become highly sensitive to $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$0 because the non-Hermitian quantum metric diverges near the exceptional point $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$1 (Liu et al., 24 Aug 2025). Here tunneling is not spatial; it is branch switching in parameter space, amplified by critical geometry and modulated by the imaginary intraband Berry connection.

A third mechanism is non-unitary interface matching. In the non-Hermitian Dirac equation, the bulk spectrum can remain real and pairwise orthogonal, yet an interface between two spatially uniform domains can still show anomalous Klein tunneling if $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$2 changes across the wall. Reflection can be suppressed while transmitted flux becomes substantially higher or lower than the incident flux, and under a sign flip of $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$3 the interface can function as a simultaneous laser and coherent perfect absorber (Terh et al., 2023). The non-conservation is localized at the interface, not distributed through the bulk.

A fourth mechanism is driven interband transfer. In the non-Hermitian diamond chain, isolated complex bands permit electric-field-induced Landau-Zener tunneling that supports stable Bloch oscillations and large-amplitude super Bloch oscillations even in a broken-$$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$4 phase (Parkavi et al., 2021). In the periodically driven non-Hermitian bosonic junction, high-frequency reduction yields effective couplings

$$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$5

so tunneling stability is controlled by Bessel-zero engineering, parity of $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$6, and gain/loss thresholds (Luo et al., 2020).

4. Observables, diagnostics, and asymptotics

The most direct diagnostics are transmission and reflection. In PT-symmetric electromagnetic models, fixed points of the band structure correspond to frequencies at which the finite structure shows $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$7, and some turning points support $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$8 together with nearly uniform transmission phase over a broad bandwidth (Jazayeri, 2022). In the optical double-EP heterostructure, the critical observables are the directional reflection coefficients $$H=-\hbar g \begin{bmatrix} 0 & 1-\alpha\ 1+\alpha & 0 \end{bmatrix},$$9 and $\beta=(1+\alpha)/(1-\alpha)$0, which vanish at $\beta=(1+\alpha)/(1-\alpha)$1 and $\beta=(1+\alpha)/(1-\alpha)$2, respectively (Yim et al., 2019). In the Dirac interface, the natural observables are

$\beta=(1+\alpha)/(1-\alpha)$3

with $\beta=(1+\alpha)/(1-\alpha)$4 allowed because the scattering matrix is non-unitary when the Hermitian reductions differ across the wall (Terh et al., 2023).

Delay-time diagnostics reveal a more delicate parameter dependence. In non-Hermitian lattice barriers, the large-$\beta=(1+\alpha)/(1-\alpha)$5 asymptotic transmission takes the form

$\beta=(1+\alpha)/(1-\alpha)$6

and the tunneling phase time becomes

$\beta=(1+\alpha)/(1-\alpha)$7

The Hartman effect therefore persists if and only if $\beta=(1+\alpha)/(1-\alpha)$8 (Longhi, 2022). This establishes a model-independent criterion within the tight-binding heterojunction framework. By contrast, in non-Hermitian space-fractional quantum mechanics the thick-barrier asymptotics are generically linear in barrier width $\beta=(1+\alpha)/(1-\alpha)$9, implying absence of Hartman saturation, although a “potential manifestation” can occur for specific combinations of the absorption component $\eta$0 and the Lévy index $\eta$1 (Umar et al., 5 Apr 2025). The literature thus does not support a universal verdict on non-Hermitian Hartman behavior; it is strongly model dependent.

Several works stress that group-delay anomalies should not be interpreted as superluminal signal transport. The electromagnetic study that introduces “ideal superluminal tunneling” explicitly defines it through nearly frequency-independent transmission phase and notes that the effect is a phase/group-delay phenomenon rather than a causality violation (Jazayeri, 2022). The space-fractional study likewise frames tunneling time through a stationary-phase construction and treats the result as a phase-time quantity rather than a literal traversal clock (Umar et al., 5 Apr 2025).

Population-based diagnostics are central when tunneling is understood as state transfer. In the two-site asymmetric model, the normalized probabilities

$\eta$2

show directly that forward and backward tunneling are unequal for $\eta$3 (Santos et al., 2014). In the non-adiabatic sensing problem, output population and Fisher information replace transmission as the operative measures, because the tunneling event is branch transfer rather than spatial penetration (Liu et al., 24 Aug 2025).

More geometric diagnostics also appear. The Fisher-Rao metric in the biorthogonal setting,

$\eta$4

provides an intrinsic measure of parameter sensitivity for non-Hermitian state manifolds (Lu et al., 2024). In disordered non-reciprocal systems, the biorthogonal participation ratio $\eta$5 plays a similar role for transport: its finite-size scaling locates disorder-tuned transitions between extended and localized phases (Wang et al., 2024).

5. Symmetry, topology, and pseudo-Hermitian structure

$\eta$6 symmetry is important but not exclusive. PT-symmetric electromagnetic models are notable because the periodic-boundary and open-boundary band structures coincide, which allows fixed points, extended states in the bandgap, and turning points of the band structure to be read directly in finite-sample scattering (Jazayeri, 2022). However, the non-Hermitian Hartman effect can arise without any special symmetry when non-Hermiticity comes from non-reciprocal couplings (Longhi, 2022), and the two-delta scattering problem shows that the leading nonlocal effects of non-Hermiticity are tied more generally to $\eta$7 than to strict $\eta$8 (Mehri-Dehnavi et al., 2010).

Time-reversal and inversion constraints can forbid or enable skin-induced tunneling asymmetry. For on-site dissipation, the paper on chiral tunneling establishes a no-go theorem: if the parent Hermitian Hamiltonian has spinless time-reversal symmetry, on-site dissipation cannot induce skin modes (Yi et al., 2020). In the spinful case, skin modes become possible when inversion is broken, when inversion anticommutes with the dissipation matrix, or under a specific representation satisfying $\eta$9, $1\pm\eta$0, and $1\pm\eta$1 (Yi et al., 2020). The resulting tunneling asymmetry is therefore symmetry-selective rather than generic.

Topology becomes decisive in lattice settings. In the extended non-Hermitian SSH chain, asymmetric intracell tunneling $1\pm\eta$2 controls the generalized Brillouin zone and the skin exponent

$1\pm\eta$3

in the analytically tractable limit $1\pm\eta$4 (Wang et al., 26 Mar 2026). Along exceptional-point-constrained manifolds, periodic-boundary point-gap transitions and open-boundary line-gap transitions become locked, so a single parameter sweep can simultaneously reverse skin-direction transport and create or destroy zero-energy boundary modes (Wang et al., 26 Mar 2026). This identifies a precise condition under which bulk spectral evolution reliably diagnoses boundary-sensitive transport.

Pseudo-Hermitian and quasi-Hermitian strands also belong to the subject, although the available documentation is uneven. The two-center complex delta potential admits a positive-definite metric $1\pm\eta$5 and an equivalent Hermitian Hamiltonian in parameter regions without spectral singularities or complex bound states (Mehri-Dehnavi et al., 2010). A distinct line of work is represented by “A weak pseudo-Hermitian two band model, artificial Hawking radiation and tunneling,” whose abstract states that it examines artificial Hawking radiation in a non-$1\pm\eta$6-symmetric weakly pseudo-Hermitian two-band model containing a tilting parameter and determines tunneling probability through an event horizon acting as a classically forbidden barrier (Bagchi et al., 2021). The supplied arXiv record, however, does not provide the Hamiltonian, pseudo-Hermiticity relation, or tunneling derivation, so only that broad characterization can be established from the record.

6. Implementations, applications, and unresolved issues

Parameter-sensitive non-Hermitian tunneling now spans several experimental platforms. Optical multilayer heterostructures realize direction-dependent reflectionless scattering at double exceptional points and provide a route to two-parameter sensing of temperature and stress (Yim et al., 2019). PT-symmetric electromagnetic periodic structures realize fixed-point transport, extended states in the bandgap, and ideal superluminal tunneling in the phase-delay sense (Jazayeri, 2022). Trapped-ion experiments measure Fisher information directly for non-adiabatic exceptional-point sensing based on branch transfer (Liu et al., 24 Aug 2025). Active acoustic metamaterials implement a mirrored non-reciprocal interface with microphones, loudspeakers, and distributed feedback control, and both finite-element simulations and experiment observe the predicted tunneling phenomenon (Langfeldt et al., 23 Jul 2025).

The application space is correspondingly broad. In sensing, the main promise is that tunneling response can be more useful than static eigenvalue splitting. The non-adiabatic sensing work explicitly proposes a route that avoids the quasi-static and eigenstate-collapse limitations of conventional exceptional-point sensing, using phase-change-rate-modulated tunneling with geometric amplification validated by Fisher-information measurements (Liu et al., 24 Aug 2025). In transport control, periodically driven bosonic junctions and broken-$1\pm\eta$7 flat-band lattices show that gain/loss, drive frequency, and synthetic fields can be used to select tunneling channels, stabilize oscillatory transport, or suppress it (Luo et al., 2020, Parkavi et al., 2021). In topology, EP-constrained parameter sweeps provide a way to infer open-boundary transport reorganizations from periodic-boundary spectral data in photonic, circuit, and cold-atom settings (Wang et al., 26 Mar 2026).

Several unresolved issues remain. First, the same parameter that enhances sensitivity often also amplifies reflection, loss, or instability. The acoustic metamaterial makes this trade-off explicit: larger $1\pm\eta$8 produces a darker interface but less transmitted energy and a smaller experimental stability margin (Langfeldt et al., 23 Jul 2025). Second, metrological enhancement in non-Hermitian protocols is often conditional. In the single-qubit non-Hermitian-operator schemes, large QFI enhancement is inseparable from postselection and finite success probability (Guo et al., 2016). Third, there is no universal rule for delay-time anomalies: some non-Hermitian barriers exhibit Hartman saturation under a precise non-Bloch phase condition, while others do not (Longhi, 2022, Umar et al., 5 Apr 2025). Fourth, the relation between symmetry and transport remains subtle: $1\pm\eta$9 symmetry can organize the spectrum, but it is neither necessary for directional tunneling nor sufficient by itself to determine physical response (Mehri-Dehnavi et al., 2010, Jazayeri, 2022).

Taken together, these works suggest that parameter-sensitive non-Hermitian tunneling is best understood not as a single effect, but as a technical umbrella for transport phenomena in which non-Hermitian structure makes tunneling unusually reconfigurable. Depending on the model, the decisive parameter may be a hopping asymmetry, a gain/loss ratio, a driving amplitude, a disorder scale, a fractional index, or an exceptional-point distance; the observable may be transmission, reflection, delay, boundary accumulation, population transfer, or Fisher information. The common feature is a sharply tunable transition between distinct transport regimes that have no generic Hermitian counterpart.