Coherent Scattering Configurations

• Coherent scattering configurations are arrangements where phase-locked amplitudes add before squaring, resulting in enhanced (often N²-scaled) cross sections.
• They underpin processes in neutrino–nucleus interactions, optical backscattering, and diffraction, with coherence determined by phase preservation and channel indistinguishability.
• Recent advances focus on control techniques like channel matching and magnetic switching to optimize detection and isolate coherent contributions in practical setups.

Searching arXiv for recent and relevant papers on coherent scattering configurations across domains. A coherent scattering configuration is a scattering arrangement in which the relevant alternatives retain definite phase relations, so that amplitudes add before squaring rather than being reduced to an incoherent sum of probabilities. Across the cited literature, the operational meaning of coherence depends on the system. In a composite-target non-relativistic setting, coherence occurs when the incident particle “does not resolve the internal structure of the target,” and the cross section scales as $\left(\sum_i g_i\right)^2$ (Gasbarri et al., 2015). In neutrino–nucleus scattering, the coherent configuration is the elastic neutral-current process in which “the nucleus remains in the same quantum state after the scattering,” with the coherent term governed by $|F(q)|^2$ and the incoherent term by $1-|F(q)|^2$ (Bednyakov et al., 2019). In multiple-scattering optics, coherence is expressed through reciprocity between direct and reversed paths, most prominently in exact backscattering (Shatokhin et al., 2012, Rouabah et al., 2014). This suggests a common structure: coherence is determined by phase preservation, channel indistinguishability, and the absence of which-path information.

1. Definition and scope

In the non-relativistic $N$-body formulation, the basic coherent-scattering configuration consists of “one non-relativistic probe” interacting with a “bound $N$-particle target” through

$$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$

with coherence defined by the condition that “the projectile cannot resolve which constituent it scattered from” (Gasbarri et al., 2015). In that regime the target behaves as a pointlike object with total coupling $G=\sum_i g_i$, and only elastic internal transitions survive.

In neutrino and antineutrino scattering, the state-based definition is explicit: $$\nu A\to \nu A^{(*)}, \qquad \bar\nu A\to \bar\nu A^{(*)},$$ where the superscript $(*)$ indicates that the nucleus may remain in its original quantum state or be left in an excited state. The paper distinguishes “coherent” and “incoherent” by the nuclear final state: coherent scattering means that “the nucleus remains in the same quantum state after the scattering,” whereas incoherent scattering means that “the nucleus changes its quantum state” (Bednyakov et al., 2019).

In coherent backscattering of light, the defining configuration is different but structurally analogous. A photon can scatter along a direct path and along the reversed path; near exact backscattering these amplitudes “acquire the same phase after configuration averaging and therefore interfere constructively” (Shatokhin et al., 2012). In cold-atom and disordered-media treatments, this is the standard coherent-backscattering geometry (Rouabah et al., 2014, Estakhri et al., 2021).

Other literatures extend the term to channel engineering. In cylindrical scattering with multiple coherent illuminations, coherence is imposed extrinsically through the input channels, with effective channel excitation

$H_n^{inf}=\sum_{m=1}^{N}e^{-in\Phi_m}H_m,$

so that selected angular-momentum channels are suppressed or enhanced without changing the object itself (Lee et al., 2019). In multichannel photonics, coherent scattering control is formulated through the scattering matrix $|F(q)|^2$0, with outgoing amplitudes satisfying

$|F(q)|^2$1

so the scattering response depends on the coherent configuration of the incident channels (Krasnok et al., 2019).

2. Microscopic criteria and scaling laws

The most compact microscopic coherence criterion in the composite-target problem is the phase condition

$|F(q)|^2$2

or equivalently

$|F(q)|^2$3

Under this condition the scattering probability factorizes, the internal target state is preserved, and the cross section acquires the coherent enhancement

$|F(q)|^2$4

which becomes $|F(q)|^2$5 for equal couplings (Gasbarri et al., 2015).

For neutrino–nucleus scattering, the coherent and incoherent contributions are separated by their nucleon-number scaling and by complementary form-factor weights. For heavy unpolarized nuclei,

$|F(q)|^2$6

whereas

$|F(q)|^2$7

The coherent term is therefore proportional to $|F(q)|^2$8, while the incoherent term is proportional to $|F(q)|^2$9 (Bednyakov et al., 2019).

In optical coherent backscattering, the same amplitude-before-squaring logic appears in path space rather than constituent space. The “ladder” term is the incoherent sum of intensities, whereas the “crossed” term is the interference of direct and reversed multiple-scattering paths (Shatokhin et al., 2012). In the scalar cold-atom treatment of a Gaussian cloud, the enhancement function reaches a maximum of $1-|F(q)|^2$0 in exact backscattering, which is the standard coherent-backscattering factor of two (Rouabah et al., 2014).

Periodic coherent diffraction yields the same scaling principle in lattice form. For a finite 1D array,

$1-|F(q)|^2$1

so at a diffraction maximum the peak height scales as $1-|F(q)|^2$2, while the peak width scales as $1-|F(q)|^2$3 (Weitenberg et al., 2011). In linear ion crystals the far-field model is written as

$1-|F(q)|^2$4

with the $1-|F(q)|^2$5 term carrying the coherent interference structure and $1-|F(q)|^2$6 representing incoherent scattered light (Verde et al., 2024).

3. Kinematics, geometry, and channel matching

A recurrent kinematic condition is that the probe should not resolve the target’s internal structure. In neutrino–nucleus scattering the paper states explicitly that “the coherency requirement reads as $1-|F(q)|^2$7,” equivalently the exchanged-momentum wavelength must be large compared with nuclear size (Bednyakov et al., 2019). In the non-relativistic composite-target problem the same content appears as $1-|F(q)|^2$8 (Gasbarri et al., 2015).

For coherent diffraction from a single atomic plane, the condition is not full 3D Bragg matching but only in-plane matching: $1-|F(q)|^2$9 Because the sample is only one lattice plane thick, “scattering from a single plane yields diffraction peaks for any incidence angle” (Weitenberg et al., 2011). In coherent Fourier scatterometry on periodic gratings, a focused beam covering several periods is decomposed into Bloch/Floquet components over the first Brillouin zone, and the observable in the back focal plane is obtained by coherent superposition of the Fourier-plane fields (Hammerschmidt et al., 2023).

In coherent backscattering, the geometric condition is exact backscattering. The reversed paths are phase matched when the observation direction is near the backscattering direction, and the coherent part survives disorder averaging only in that configuration (Shatokhin et al., 2012, Estakhri et al., 2021). In the $N$0-dimensional vector theory of disordered transverse media, the coherent peak is centered at

$N$1

with width set by the diffusive factor

$N$2

and the polarization channel selected by reciprocity depends on whether the field is in the scalar or vector regime (Cherroret, 2018).

Channel matching can also be imposed externally. In multiwave cylindrical scattering, one prescribes the effective excitation $N$3 of chosen cylindrical harmonics by selecting illumination directions $N$4, amplitudes, and phases $N$5 (Lee et al., 2019). In multichannel coherent perfect absorption, the required coherent configuration is the eigenvector of $N$6 whose eigenvalue vanishes: $N$7 In a mirror-symmetric two-port system this reduces to the symmetric or antisymmetric combinations $N$8 with eigenvalues $N$9 (Krasnok et al., 2019).

4. Representative physical realizations

In neutrino physics, coherent scattering is discussed for both accelerator and reactor environments. The coherent elastic neutrino–nucleus scattering signal is maximal at low recoil, while the incoherent admixture grows as momentum transfer increases. For $N$0 and neutrino energies of $N$1–$N$2 MeV, the incoherent cross section is “about 15–20% of the coherent one,” so recoil-only measurements can contain a substantial inelastic admixture if deexcitation $N$3 rays are not detected (Bednyakov et al., 2019). Reactor CEvNS experiments such as NEON are therefore driven by the need for very high light yield and sub-keVee thresholds (Choi et al., 2022).

In cold-atom optics, the minimal coherent-backscattering configuration is “two laser-driven two-level atoms exchanging one scattered photon and observed in the backscattering direction” (Shatokhin et al., 2012). For larger cold atomic clouds, coherent backscattering emerges from reciprocal multiple-scattering paths, while coherent forward scattering appears through the structure factor and its double-scattering correction (Rouabah et al., 2014). In bounded sparse disordered media, the coherent-backscattering cone survives even for a few hundred particles, but its width and enhancement depend strongly on density, particle number, volume size, and boundary geometry (Estakhri et al., 2021).

In periodic atomic matter, coherent scattering can be realized as diffraction from a single two-dimensional Mott-insulator plane, where the field amplitudes radiated from different lattice sites add with definite phase relations and produce far-field diffraction peaks (Weitenberg et al., 2011). In trapped-ion crystals, each ion acts as a coherent single-photon emitter, and the far-field pattern is the Fourier signature of the ion coordinates. In the spin-selective mode, the “set of active emitters is spin dependent,” so the fringe pattern becomes a direct probe of spin texture (Verde et al., 2024).

In nuclear forward scattering, the coherent configuration is a collective, spatially phase-coherent nuclear exciton excited in a $N$4 crystal by a short synchrotron pulse. Its coherent forward decay can be suppressed or released by abrupt rotations of the hyperfine magnetic field, because the transformed nuclear transition currents interfere constructively or destructively depending on the switching time (Pálffy et al., 2010).

In electron scattering, a coherent configuration arises when an optically modulated free-electron wavepacket with discrete photon sidebands is allowed to propagate before colliding with a localized atom. Different incident momentum components can then scatter into the same outgoing momentum, and the cross section contains interference terms proportional to

$N$5

yielding modulation of elastic and inelastic scattering with Å-scale spatial sensitivity (Morimoto et al., 2021).

5. Control, detection, and readout

A coherent scattering configuration is often inseparable from the way coherence is detected or controlled. In neutrino–nucleus scattering, elastic and inelastic neutral-current events are “practically indistinguishable” if only the recoil energy is observed. The paper identifies two practical discriminants: detection of deexcitation $N$6 rays from $N$7, and operation at very low recoil threshold where the incoherent contribution is strongly suppressed (Bednyakov et al., 2019).

In nonlinear coherent backscattering from cold atoms, the pump–probe formalism replaces a full two-atom master equation by single-atom responses to a bichromatic field, which are then recombined into ladder and crossed diagrams. The coherent configuration is therefore encoded not only in geometry but also in the spectral response functions used to build the multiple-scattering signal (Shatokhin et al., 2012).

In nuclear forward scattering, coherent control is achieved by magnetic switching. A rotation of the hyperfine magnetic field into the beam direction at

$N$8

suppresses the dominant first-order coherent forward decay, while later switchings restore selected polarization channels and produce delayed pulses (Pálffy et al., 2010). In multichannel photonics, coherent perfect absorption and virtual perfect absorption are controlled by matching the incident waveform to a zero of an $N$9-matrix eigenvalue, either at real frequency or at complex frequency (Krasnok et al., 2019).

External channel engineering provides another control paradigm. For passive cylindrical scatterers, one can suppress a selected multipole channel by imposing

$$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$0

or isolate one channel by choosing $$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$1 and $$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$2 for $$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$3 (Lee et al., 2019). A different but related target is coherent orthogonal scattering, defined by

$$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$4

so that the actual output is orthogonal to the output that would have been obtained from the same input without the scatterer (Guo et al., 2024).

In multiply scattering optical media, non-invasive control is achieved by learning the coherent forward operator $$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$5 and the return operator $$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$6 from paired input/output measurements. For fluorescence,

$$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$7

whereas for SHG the coherent model becomes

$$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$8

This architecture makes the coherent scattering configuration explicit as propagation, local contrast generation, and return-path transport (d'Arco et al., 2022).

6. Limits, crossover behavior, and terminological issues

A central misconception corrected in several papers is that coherence is not an all-or-nothing property. In neutrino–nucleus scattering, the transition between coherent and incoherent regimes is “smooth,” because the same nuclear form factor redistributes strength continuously from $$\hat H_{\text{int}}=\sum_{i=1}^N g_i\,V(\hat{\mathbf Y}-\hat{\mathbf X}-\hat{\mathbf r}_i),$$9 to $G=\sum_i g_i$0 as momentum transfer increases (Bednyakov et al., 2019). In the composite-target problem, partial coherence corresponds to intermediate scaling $G=\sum_i g_i$1 with $G=\sum_i g_i$2 (Gasbarri et al., 2015).

In optical multiple scattering, saturation and inelastic processes modify coherence without eliminating it. In strongly driven cold atoms, inelastic resonance-fluorescence processes reduce reciprocity at the spectral level and redistribute the CBS signal over several frequency components (Shatokhin et al., 2012). In finite sparse random media, boundaries introduce a coherent specular reflection distinct from the CBS cone, and finite size “severely reduces the probability of long photon trajectories, so the CBS cone broadens” (Estakhri et al., 2021).

The $G=\sum_i g_i$3-dimensional vector theory shows that coherence can undergo a genuine regime change. For a beam injected at transverse momentum $G=\sum_i g_i$4, the polarization-randomization scale is

$G=\sum_i g_i$5

so a long scalar-like regime can be followed by a vector regime in which the CBS selection rules change qualitatively (Cherroret, 2018).

Oriented crystals furnish another cautionary example. In ultrathin crystals, both coherent and incoherent scattering depend strongly on whether the beam is aligned with a crystallographic axis or plane. For scattering by crystal planes of atoms, the paper states explicitly that “there is no Debye-Waller factor in the term that determines incoherent effects in scattering,” whereas the coherent term retains the orientation-dependent interference structure (Shul'ga et al., 2020).

Terminology is also system dependent. The neutrino–nucleus paper notes that, for tens-of-MeV neutrinos, “elastic” and “inelastic” are the more precise labels, even though “coherent” and “incoherent” are retained by tradition (Bednyakov et al., 2019). A plausible implication is that the most stable cross-domain definition of a coherent scattering configuration is not tied to a single wavelength criterion but to a combination of three features: preservation of the relevant final state, phase-matched addition of alternatives, and measurement conditions that do not reveal which constituent, path, or channel carried the interaction.