---
title: Parameter-Sensitive Non-Hermitian Tunneling
url: https://www.emergentmind.com/topics/parameter-sensitive-non-hermitian-tunneling
type: topic
---

# Parameter-Sensitive Non-Hermitian Tunneling

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 [2207.08715], [2504.04082], [2003.02219], [2304.10757], [2102.06608], [2508.17377]. 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 [2207.08715], [1902.09479], [2504.04082]. 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 [2003.02219], [2507.17395], [2603.25451]. 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 [2508.17377].

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 [2208.05625], [2207.08715], [2402.00374], [2508.17377]. 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 [1902.09479]. Conversely, in PT-symmetric electromagnetic structures, fixed-point frequencies can support unit transmission, zero reflection, and nearly flat transmission phase over a broad bandwidth [2208.05625].

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 [2003.02219]; the non-Hermitian Hartman effect can occur “without any special symmetry in the system” when non-Hermiticity comes from non-reciprocal couplings [2207.08715]; 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 [1002.1221].

## 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 [1404.5972]. 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 [2507.17395].

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 \(\gamma/\lambda\) 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 \(T<1\), \(T>1\), or CPA-laser behavior despite the absence of bulk amplification or damping [2304.10757]. In the driven spin-orbit-coupled bosonic junction, balanced or unbalanced gain-loss strengths \(\beta_l,\beta_r\) compete against drive-renormalized tunneling amplitudes to determine whether spin-dependent tunneling is stable, decaying, or unstable [2005.04627]. In space-fractional quantum mechanics, the absorptive part \(V_i\) of a barrier \(V_r-iV_i\) modifies both the phase accumulation and the delay-time dependence on thickness [2504.04082].

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 \(\gamma\), synthetic magnetic flux \(\phi\), and synthetic electric fields \(E_\parallel\) and \(E_\perp\): changing them switches the system among compact localization, stable Bloch oscillations, Landau-Zener-induced transfer, amplified Bloch oscillations, and blow-up [2102.06608]. In the driven bosonic junction, the ratio \(\Omega/\omega\), the spin-orbit parameter \(\lambda\), and the Bessel-renormalized couplings \(J_0\) and \(J_{\pm\Omega/\omega}\) determine whether only spin-conserving, only spin-flipping, or no tunneling channels survive [2005.04627].

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 \(W\) competes with non-reciprocity \(\tilde t\), drives a transition from extended complex-energy states to localized states, closes the point gap, and destroys skin modes [2406.01984]. 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” [2003.02219]. 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 [2507.17395]. 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 \(\mathcal{PT}\)-symmetric Hamiltonian
\[
H_{\mathcal{PT}}(t)=i\gamma \sigma_z + J(t)\cos[\Phi(t)]\,\sigma_x + J(t)\sin[\Phi(t)]\,\sigma_y
\]
remains in the \(\mathcal{PT}\)-symmetric regime \(J(t)>\gamma\), but non-adiabatic transitions between instantaneous eigenstate branches become highly sensitive to \(\dot\Phi(t)=\Delta(t)\) because the non-Hermitian quantum metric diverges near the exceptional point \(J=\gamma\) [2508.17377]. 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 \(\gamma/\lambda\) 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 \(\gamma\) the interface can function as a simultaneous laser and coherent perfect absorber [2304.10757]. 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-\(\mathcal{PT}\) phase [2102.06608]. In the periodically driven non-Hermitian bosonic junction, high-frequency reduction yields effective couplings
\[
J_0=\nu \cos(\pi \lambda)\mathcal{J}_{0}\!\left(\frac{2\varepsilon}{\omega}\right), \qquad
J_{\pm\Omega/\omega}=\nu \sin(\pi \lambda)\mathcal{J}_{\pm\Omega/\omega}\!\left(\frac{2\varepsilon}{\omega}\right),
\]
so tunneling stability is controlled by Bessel-zero engineering, parity of \(\Omega/\omega\), and gain/loss thresholds [2005.04627].

## 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 \(R_L=R_R=0\), and some turning points support \(|T|=1\) together with nearly uniform transmission phase over a broad bandwidth [2208.05625]. In the optical double-EP heterostructure, the critical observables are the directional reflection coefficients \(r_f\) and \(r_b\), which vanish at \(\lambda_{\mathrm{EPf}}=822.4~\mathrm{nm}\) and \(\lambda_{\mathrm{EPb}}=790.4~\mathrm{nm}\), respectively [1902.09479]. In the Dirac interface, the natural observables are
\[
R=\sum_n |a_{nL}^-|^2,\qquad T=\sum_n |a_{nR}^+|^2,
\]
with \(R+T\neq 1\) allowed because the scattering matrix is non-unitary when the Hermitian reductions differ across the wall [2304.10757].

Delay-time diagnostics reveal a more delicate parameter dependence. In non-Hermitian lattice barriers, the large-\(N\) asymptotic transmission takes the form
\[
T(q)=\overline{T}(q)\,\beta_s^N,
\]
and the tunneling phase time becomes
\[
\tau= -\frac{d\overline{\phi}/dq}{2\kappa \sin q} - N\,\frac{d\phi_s/dq}{2\kappa \sin q}.
\]
The Hartman effect therefore persists if and only if \(d\phi_s/dq=0\) [2207.08715]. 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 \(d\), implying absence of Hartman saturation, although a “potential manifestation” can occur for specific combinations of the absorption component \(V_i\) and the Lévy index \(\alpha\) [2504.04082]. 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 [2208.05625]. 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 [2504.04082].

Population-based diagnostics are central when tunneling is understood as state transfer. In the two-site asymmetric model, the normalized probabilities
\[
\mathscr{P}^{\mathrm{norm}}_{A\to B}(t)=
\frac{\beta\sin^2(\omega t)}{\cos^2(\omega t)+\beta\sin^2(\omega t)},\qquad
\mathscr{P}^{\mathrm{norm}}_{B\to A}(t)=
\frac{\beta^{-1}\sin^2(\omega t)}{\cos^2(\omega t)+\beta^{-1}\sin^2(\omega t)}
\]
show directly that forward and backward tunneling are unequal for \(\beta\neq 1\) [1404.5972]. 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 [2508.17377].

More geometric diagnostics also appear. The Fisher-Rao metric in the biorthogonal setting,
\[
g_{ij}= \langle \partial_i \tilde{\psi}(t) | \partial_j \psi(t) \rangle
- \langle \partial_i \tilde{\psi}(t) | \psi(t) \rangle
\langle \tilde{\psi}(t) | \partial_j \psi(t) \rangle,
\]
provides an intrinsic measure of parameter sensitivity for non-Hermitian state manifolds [2402.00374]. In disordered non-reciprocal systems, the biorthogonal participation ratio \(p_2(L,W)\) plays a similar role for transport: its finite-size scaling locates disorder-tuned transitions between extended and localized phases [2406.01984].

## 5. Symmetry, topology, and pseudo-Hermitian structure

\(\mathcal{PT}\) 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 [2208.05625]. However, the non-Hermitian Hartman effect can arise without any special symmetry when non-Hermiticity comes from non-reciprocal couplings [2207.08715], and the two-delta scattering problem shows that the leading nonlocal effects of non-Hermiticity are tied more generally to \(\Im(\zeta_+)=-\Im(\zeta_-)\) than to strict \(\zeta_+=\zeta_-^*\) [1002.1221].

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 [2003.02219]. 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 \([\mathcal P,\Gamma_0]=0\), \(\{\mathcal P,\mathcal T\}=0\), and \([\Gamma_0,\mathcal T]=0\) [2003.02219]. 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 \(t_1\neq t_2\) controls the generalized Brillouin zone and the skin exponent
\[
\kappa=\frac{1}{2}\ln\left|\frac{t_1}{t_2}\right|
\]
in the analytically tractable limit \(\delta=t_3\) [2603.25451]. 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 [2603.25451]. 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 \(\eta\) and an equivalent Hermitian Hamiltonian in parameter regions without spectral singularities or complex bound states [1002.1221]. 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-\(\mathcal{PT}\)-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 [2108.11648]. 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 [1902.09479]. PT-symmetric electromagnetic periodic structures realize fixed-point transport, extended states in the bandgap, and ideal superluminal tunneling in the phase-delay sense [2208.05625]. Trapped-ion experiments measure Fisher information directly for non-adiabatic exceptional-point sensing based on branch transfer [2508.17377]. 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 [2507.17395].

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 [2508.17377]. In transport control, periodically driven bosonic junctions and broken-\(\mathcal{PT}\) 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 [2005.04627], [2102.06608]. 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 [2603.25451].

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 \(\eta\) produces a darker interface but less transmitted energy and a smaller experimental stability margin [2507.17395]. 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 [1609.04276]. 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 [2207.08715], [2504.04082]. Fourth, the relation between symmetry and transport remains subtle: \(\mathcal{PT}\) symmetry can organize the spectrum, but it is neither necessary for directional tunneling nor sufficient by itself to determine physical response [1002.1221], [2208.05625].

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.

Source: https://www.emergentmind.com/topics/parameter-sensitive-non-hermitian-tunneling