Nonlocal Quantum Backscattering Channels

• Nonlocal quantum backscattering channels are mechanisms enabling quantum states at edge or interface modes in topological systems to reflect or transmit via spatially separated processes.
• They employ semiclassical tunneling and resonance-assisted mini-gap formation, with models showing exponential suppression of backscattering with smooth geometries.
• Experimental signatures such as 4π-periodic Andreev bound states and spatially oscillatory conductance in mesoscopic structures validate these nonlocal interactions.

Nonlocal quantum backscattering channels are mechanisms by which quantum states, particularly those associated with edge or interface modes in topological systems, experience reflection or transmission events mediated through spatially or degree-of-freedom-separated processes, rather than via conventional, purely local scattering. These channels underlie key phenomena in topological condensed matter, quantum transport, and quantum information theory, and can arise from physical processes such as impurity-mediated interactions, domain-wall smoothness effects, interface-induced hybridizations, and nonlocal correlations in the hidden-variable interpretations of quantum mechanics.

1. Model Frameworks and Fundamental Mechanisms

Nonlocal quantum backscattering arises in systems where symmetry protection or geometric constraints would, in the ideal case, forbid backscattering, but secondary mechanisms reintroduce it through nonlocality. In the context of valley Hall edge channels, the low-energy effective Hamiltonian takes the form

$H = m(x)\sigma_z + v[\sigma_x k_x + \sigma_y k_y],$

where $m(x)$ is a mass profile varying across a domain wall. In the sharp-wall limit, a spectrally isolated chiral mode emerges, and backscattering requiring large momentum transfer $K\leftrightarrow K'$ is forbidden in the Dirac approximation but reappears when the full tight-binding lattice dispersion is included (Shah et al., 2021).

In quantum spin Hall (QSH) and Josephson SNSN' geometries, backscattering mechanisms involve both phase-dependent and independent Andreev bound states (ABS) localized at spatially distinct regions. Interedge coupling mediated by TR-symmetric interaction terms introduces nonlocal channels, producing resonance-assisted mini-gaps and 4$\pi$-periodic Josephson modes (Heinz et al., 16 Oct 2024).

In helical edge transport, dynamical impurities modeled as two-level systems (TLS) can mediate nonlocal elastic backscattering through purely electrostatic (Coulomb) interactions, whereby elastic backscattering is only allowed when the impurity flips state, ensuring global time-reversal (TR) invariance (McGinley et al., 2020).

Further, in the pilot-wave formulation of quantum mechanics, quantum nonequilibrium in one spatial region can be transferred nonlocally via entanglement, constituting a generalization of the nonlocal channel concept to information-theoretic flows (Kandhadai et al., 2019).

2. Microscopic Theories and Semiclassical Tunneling

A prototypical semiclassical framework for nonlocal backscattering in valley Hall edge channels is formulated as a tunneling problem in quasimomentum space. Upon Fourier transformation, the problem maps to

$$H = m(i\lambda \partial_{k_x})\sigma_z + \mathbf{h}(k_x, k_y)\cdot\boldsymbol{\sigma},$$

with a small parameter $\lambda \ll 1$ akin to an effective $\hbar$ in reciprocal-space WKB theory. The leading-order solution in the WKB approximation yields a Hamilton–Jacobi equation, restricting classical propagation to regions where $|\mathbf{h}(k)|\le|E|$. The backscattering amplitude is expressed in Landau–Zener form:

$$R(E, w) \approx \exp\left[-2\,\mathrm{Im} \int_{k^t_1}^{k^t_2} Q(k_x) \, dk_x\right],$$

where $Q(k_x)$ is determined via the dispersion and the spatial profile of $m(x)$. For a smooth wall ($w\gg a$), the tunneling action scales as $S(0,w) = (2\pi^2/3\sqrt{3})(w/a)$, leading to exponential suppression of backscattering with increasing wall smoothness—highlighting the nonlocality, as effective scatterers develop discretely where the energetic and geometric conditions align, often at locations where the local propagation axis aligns through high-symmetry points in the Brillouin zone (Shah et al., 2021).

3. Interaction-Mediated and Impurity-Induced Channels

Quantum spin Hall edge states, protected against local TR-symmetric elastic backscattering, become susceptible to nonlocal channels when coupled to dynamical nonmagnetic impurities. In the model of a helical Luttinger liquid coupled to a TLS by charge–charge interaction,

$$H_\text{tot} = H_\text{edge} + H_\text{imp} + H_\text{int},$$

with $H_\text{int}$ involving spatially extended electrostatic form-factors $V_{x,z}(r)$. Crucially, commuting and noncommuting TLS operators in combination allow the process:

• Forward scattering (via $J_z\sigma^z$), then backscatter (via $J_x\sigma^x$)
• Backscatter first, then forward-scatter,

which, due to noncommutation, yields an effective symmetry-allowed term $\propto \cos(2\varphi(0))\sigma^y$ that mediates backscattering concurrent with a flip of the TLS. This process does not require electron tunneling into the impurity and demonstrates a genuinely nonlocal character, with the coupling strength algebraically decaying ($\propto d^{-2}$ for distance $d$), contrasting with exponential decay for tunneling-mediated processes (McGinley et al., 2020).

4. Nonlocal Channels in Mesoscopic and Superconducting Structures

In QSH Josephson junctions in N'SNSN' geometries, nonlocal backscattering is realized through the coupling of ABS at spatially separated locations. The system supports ABS in "stub" (N'S) regions at discrete, flux-tunable energies $E_n$, and in the central superconducting segment at phase-dependent energies $\epsilon_m(\varphi)$. When these energies are resonant, interedge backscattering via the coupling term $M_s$ induces mini-gap openings and promotes a 4$\pi$-periodic ABS branch:

$$E_\pm(\varphi) = \pm\sqrt{\epsilon_0(\varphi)^2 + |V_{0,0}|^2},$$

where $V_{0,0}$ is a weakly phase-dependent wavefunction overlap. The resulting supercurrent contains an anomalous 4$\pi$ component, whose ratio to the conventional 2$\pi$ component is determined by a new backscattering length scale $\ell_{bs}^{-1} \sim \int dx\, |M_s(x)|/(\hbar v_F)$. Distinct modification of superconducting quantum interference (SQI) patterns and direct tunability of the backscattering channel by magnetic flux constitute key signatures (Heinz et al., 16 Oct 2024).

5. Disorder- and Geometry-Driven Nonlocality in Edge Transport

Backscattering via impurity-pinned antidots represents a prominent nonlocal channel in quantum Hall constrictions, as demonstrated by scanning gate microscopy (SGM). When two otherwise spatially separated edge trajectories are brought within $\sim 200$ nm in the presence of a localized antidot (AD), electrons tunnel nonlocally from one edge to the other via the AD orbital. The conductance $G_T$ then exhibits resonant oscillations, described by Aharonov–Bohm phase accumulation and mediated by two-step tunneling, $t_1$ and $t_2$, through the AD. Such channels are active only if the geometric and energetic alignment is met ($\Omega$, the AD's area, sets the periodicity). SGM allows spatial imaging of these hotspots, and the nonlocal coupling can be tuned by adjusting tip-induced potentials, confirming the long-range and geometric origin of nonlocal backscattering (Paradiso et al., 2012).

In QSH systems, spatially resolved SGM has revealed that well-localized backscattering "puddles" (potential/dephasing centers) suppress conductance in discrete micron-separated regions, but charge transport between these is ballistic. Weak fluctuations between puddles, when coupled by longer-range paths, enable multi-scatterer nonlocal backscattering channels. The corresponding interference can be detected as complex oscillatory features in the transmission, with a backscattering length $\ell_{BS}$ that can greatly exceed the device size, confirming ballisticity except at sparse nonlocal defect sites (König et al., 2012).

6. Quantum Information and Nonlocality in Hidden-Variable Theories

In the de Broglie–Bohm pilot-wave framework, the notion of nonlocal quantum backscattering extends to the transfer of quantum nonequilibrium between spatially separated subsystems via entanglement. The key mathematical structure involves the guidance equation for each particle or mode, and a continuity equation for the probability density $\rho\neq |\Psi|^2$. If one subsystem (A) is prepared in nonequilibrium, its statistics can instantaneously affect the marginal distribution of another (B) through a linear kernel $K(x_B,x_A;t)$, resulting in a nonlocal backscattering of information:

$$\delta\rho_B(x_B, t) \approx \int dx_A\, K(x_B, x_A; t)\, \delta\rho_A(x_A, 0).$$

This mechanism has been applied to the black hole information problem: nonequilibrium behind an event horizon in one quantum field region can transduce, via entangled Unruh vacuum pairs, to information in the observable exterior, without any classical transfer of energy or signaling, but via nonlocal reconfiguration of the "hidden-variable" distribution (Kandhadai et al., 2019).

7. Experimental Signatures, Suppression, and Control

Nonlocal quantum backscattering channels manifest in a range of transport and interference signatures:

• Exponential suppression of backscattering with increasing domain wall smoothness, with characteristic action scaling as $S\propto w/a$ (where $w$ is wall width, $a$ the lattice constant), conferring robust valley protection except at well-defined geometric loci (Shah et al., 2021).
• Power-law or nearly temperature-independent resistance contributions in QSH systems exposed to dynamical impurities, with signatures such as nonexponential spatial decay of scattering and visible conductance reduction even in ultraclean samples (McGinley et al., 2020).
• Resonant modifications of SQI patterns and direct control of 4$\pi$-periodic modes by external flux in Josephson junctions, indicating the participation of nonlocal ABS-mediated channels (Heinz et al., 16 Oct 2024, König et al., 2012).
• SGM-resolved spatial mapping of conductance fluctuations tied to antidot-mediated tunneling paths, enabling identification and in situ tuning of nonlocal scattering sites (Paradiso et al., 2012).

A plausible implication is that in topological and mesoscopic systems, manipulation of geometry, impurity configuration, and coupling strengths enables selective activation or suppression of nonlocal backscattering, offering routes both to enhance coherence and to probe fundamental quantum nonlocality.