Backward Coherence in Physics & Modeling

• Backward coherence is a multifaceted concept describing future-state constraints manifested in QCD radiation, neural sequence regularization, and quantum thermodynamic analyses.
• It is applied to enhance long-range consistency in translation models, stabilize feature embeddings in continual visual search, and enforce Bayesian reflection in predictive inference.
• Backward coherence findings enable actionable insights into collider color flow, improved hidden-state stability, and directional optical interference in advanced experimental setups.

Backward coherence is a technical expression used in multiple, non-equivalent senses across the literature. In collider phenomenology it denotes a QCD-coherence effect in which backward top production is associated with enhanced soft radiation and larger pair recoil, thereby imprinting color structure on the top-quark forward–backward asymmetry (Gripaios et al., 2013). In sequence modeling it denotes the use of future-aware right-to-left information to regularize left-to-right generation and improve long-range consistency (Zhang et al., 2022). In process theories and quantum thermodynamics it appears in analyses of time reversal, where backward-time states, backward work statistics, and backward initial coherence are tightly constrained (Coecke et al., 2017, Francica et al., 2023). In Bayesian and predictive modeling it names reflection-style or sequential consistency conditions that propagate constraints from future credences or terminal predictive laws back to present assignments (Fuchs et al., 2011, Hahn, 2015). In quantum optics, photonics, and molecular dynamics it refers to coherence phenomena realized in counter-propagating or backward-emission geometries, often with distinctive symmetry or interference signatures (Lai et al., 2024, Swain et al., 2024, Mridha et al., 2018).

1. QCD coherence, color flow, and the top forward–backward asymmetry

In the top-quark literature, backward coherence is a consequence of soft-gluon coherence in $q\bar q \to t\bar t$. Soft radiation is organized by color dipoles, with emission written schematically as

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$

where the color-charge operators encode the color connections and $W_{ij}(k)$ is the eikonal kernel. In an $s$-channel color-octet exchange, color flows from the incoming quark to the outgoing top and from the incoming antiquark to the outgoing antitop. When the top is produced forward, the color line is gently deflected and radiates less; when the top is produced backward, the deflection is larger, coherent radiation is stronger, and the $t\bar t$ system receives larger recoil. Backward events therefore preferentially populate larger $p_T^{t\bar t}$, so $A_{\mathrm{FB}}(p_T^{t\bar t})$ falls with increasing pair transverse momentum (Gripaios et al., 2013).

The same work contrasts this with an $s$-channel color-singlet exchange. In that case the outgoing $t$ and $\bar t$ are not color-connected to the incoming $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$0 and $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$1 in a way that correlates soft radiation with the production angle, so the asymmetry remains roughly constant, or mildly increases, as a function of $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$2. The distinction is explicit in the antenna decomposition

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$3

for which singlet exchange receives only $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$4 and $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$5 antennae, while octet exchange is dominated by $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$6 and $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$7. Fixed-order EFT fits and coherent-shower studies with HERWIG++ both favor the $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$8-channel octet pattern, particularly because the observed $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$9 decreases with $W_{ij}(k)$0 (Gripaios et al., 2013).

A closely related Monte Carlo literature shows that coherent parton showers generate nonzero asymmetries even when the hard process is only LO accurate. In this setting the asymmetry is defined from $W_{ij}(k)$1, and coherent ISR–FSR interference in the initial–final dipoles $W_{ij}(k)$2–$W_{ij}(k)$3 and $W_{ij}(k)$4–$W_{ij}(k)$5 produces a characteristic pattern: positive $W_{ij}(k)$6 at low $W_{ij}(k)$7, a sign flip, and negative asymmetry at higher $W_{ij}(k)$8. The effect is amplified by Sudakov suppression at very small $W_{ij}(k)$9 and by recoil-induced migration across $s$0, summarized by $s$1 with $s$2 in coherent recoil schemes (Winter et al., 2013).

Within this usage, “backward coherence” does not mean retrocausal dynamics. It means that QCD coherence ties radiation to the backward hemisphere of the event in a color-structure-dependent way. The observable role of $s$3 is therefore diagnostic rather than merely kinematic.

2. Future-aware sequence modeling and hidden-state stability

In neural machine translation, backward coherence refers to improving a left-to-right translation by distilling information from a right-to-left decoder into the forward decoder. The architecture uses a shared Transformer encoder, a standard left-to-right decoder, and a separate right-to-left decoder with reversed causal masking. The forward model factorizes as

$s$4

while the backward model uses

$s$5

Training combines forward and backward cross-entropy with logit-level KL distillation and hidden-state MSE alignment, together with a teacher-annealing schedule $s$6. The stated motivation is that ordinary one-step-ahead training biases the model toward local structure, whereas a backward decoder “knows” the suffix and can regularize the forward decoder toward globally coherent plans (Zhang et al., 2022).

Empirically, the reported gains are concentrated on longer sentences and on metrics associated with long-range consistency. On IWSLT 2014 De-En, the full model improves the Transformer-base TEST score from $s$7 to $s$8, while removing teacher annealing, hidden-state KD, or logit KD reduces performance. On WMT 2014 and WMT 2017 En-De, both base and big configurations outperform the corresponding Transformer baselines. The paper interprets these improvements as better topic continuity, lexical choice consistency, tense agreement, and long-distance reordering, while leaving inference unchanged because only the left-to-right decoder is used at test time (Zhang et al., 2022).

A different sequence-oriented usage appears in a study of naturalistic language comprehension. There, backward coherence is the integration of what has just been heard into an evolving situation model. The study formalizes a slow drift signal from a leaky-integrated LLM hidden state,

$s$9

and a fast shift signal given by the log-probability of an event-boundary token. Drift predictions were prevalent in default-mode network hubs, whereas shift predictions were evident bilaterally in primary auditory cortex and language association cortex. A timescale sweep found that drift generalization in DMN–parietal regions peaks at intermediate $t\bar t$0–$t\bar t$1, suggesting a finite preferred integration window for backward-looking coherence during narrative comprehension (Stauba et al., 23 Dec 2025).

An explicitly dynamical formulation is developed for recurrent networks. Here backward coherence means that the current hidden state $t\bar t$2 should be reconstructible from $t\bar t$3 by a learned backward projector $t\bar t$4, with regularizer

$t\bar t$5

Under contraction and summable backward drift, the hidden-state sequence becomes a quasi-reverse-martingale, yielding almost-sure convergence, geometric rates under stronger conditions, finite pathwise stopping times, and time-uniform confidence sequences. Simulations report $t\bar t$6–$t\bar t$7 reductions in the empirical quasi-martingale total $t\bar t$8 and stability reached $t\bar t$9–$p_T^{t\bar t}$0 earlier than in an unregularized RNN; real-data studies report stable representations $p_T^{t\bar t}$1 hours earlier on PhysioNet 2012 ICU mortality prediction and about fourfold lower one-month-ahead forecast error under concept drift on FRED-MD (Chang, 8 Jun 2026).

Taken together, these sequence-model uses suggest a shared computational motif: backward coherence acts as a constraint from future or later states onto current representations. The exact mechanism differs—bidirectional distillation, leaky contextual integration, or learned backward projection—but the formal role is consistently to regularize local updates by longer-range structure.

3. Backward-consistent embeddings in continual visual search

In continual visual search, backward coherence is formulated as backward-consistent feature embedding. The practical problem is that a gallery grows session by session, while the embedding model is updated. If the new model is not compatible with previously stored gallery embeddings, one must re-extract the entire gallery after each update. The proposed solution keeps the old gallery features frozen and trains the new model so that similarities remain comparable across sessions (Wan et al., 2022).

The method introduces three losses. The first is inter-session data coherence, which anchors current or replayed samples to cross-session class centers

$p_T^{t\bar t}$2

using separate terms for classes shared with the current session and for replay exemplars from all previous classes. The second is neighbor-session model coherence, a two-sample three-embedding triplet distillation objective

$p_T^{t\bar t}$3

with margin $p_T^{t\bar t}$4. The third is intra-session discrimination through normalized softmax with temperature $p_T^{t\bar t}$5. The total objective is

$p_T^{t\bar t}$6

with defaults $p_T^{t\bar t}$7 and $p_T^{t\bar t}$8 (Wan et al., 2022).

The reported advantage is that only new gallery images require feature extraction, while old session features $p_T^{t\bar t}$9 remain untouched. The method also handles novel classes in later sessions and blurry boundaries between old and new categories. Empirically, it improves AR@1 over continual-learning baselines across disjoint, blurry, and general-incremental settings. On CIFAR100 under the disjoint setup, the reported AR@1 is $A_{\mathrm{FB}}(p_T^{t\bar t})$0 for CVS versus $A_{\mathrm{FB}}(p_T^{t\bar t})$1 for RWalk and $A_{\mathrm{FB}}(p_T^{t\bar t})$2 for BCT; on Tiny ImageNet it reports $A_{\mathrm{FB}}(p_T^{t\bar t})$3 versus $A_{\mathrm{FB}}(p_T^{t\bar t})$4 for RWalk; on fine-grained general-incremental benchmarks it reports $A_{\mathrm{FB}}(p_T^{t\bar t})$5 on Stanford Dog, $A_{\mathrm{FB}}(p_T^{t\bar t})$6 on iNat-M, and $A_{\mathrm{FB}}(p_T^{t\bar t})$7 on Product-M (Wan et al., 2022).

This usage is narrower than the sequence-model definitions, but the logic is parallel. Backward coherence here is not about future tokens or backward propagation in time; it is the requirement that a later embedding function remain geometrically compatible with earlier, frozen representations.

4. Time reversal, eternal noise, and backward work statistics

In categorical process theories, backward coherence is largely absent rather than enforced. A causal process theory is terminal if discarding after a process equals discarding before it, $A_{\mathrm{FB}}(p_T^{t\bar t})$8, which is equivalent to uniqueness of deterministic effects. Under any diagram-respecting time-reversal functor $A_{\mathrm{FB}}(p_T^{t\bar t})$9, the unique normalized effect $s$0 becomes a state $s$1. The main theorem states that in the time-reverse of any causal process theory, every system has exactly one normalized state, namely $s$2; the paper describes this as “eternal noise” (Coecke et al., 2017).

For quantum theory the time-reversed state is

$s$3

the maximally mixed state. Since this state has zero off-diagonal elements in every basis, basis-dependent coherence measures such as the $s$4-norm of coherence vanish. The argument therefore excludes nontrivial backward-time coherence in any causal process theory, unless one gives up terminality or introduces extra controllable states, in which case signaling pathologies reappear. The paper also shows that time-reversing measurements yields classically controlled preparations or decohering maps, and that unitaries do not restore backward coherence because $s$5 (Coecke et al., 2017).

A distinct but related use occurs in work fluctuation theorems with initial quantum coherence. There the forward process begins from a state $s$6 that may be coherent in the initial energy basis, while the backward process begins from a state $s$7 that may be coherent in the final energy basis. The formalism introduces a forward coherence variable $s$8 and a backward coherence variable $s$9, leading to the detailed fluctuation relation

$t$0

When both initial states are incoherent, $t$1, and the relation reduces to the standard Tasaki–Crooks formula. In this setting backward coherence is therefore explicit rather than forbidden: it appears as a correction to the usual thermodynamic arrow through the factor $t$2 (Francica et al., 2023).

The paper further derives integral fluctuation theorems such as

$t$3

and bounds

$t$4

It also shows that forward and backward nonclassicality need not be symmetric: a forward work quasiprobability can be nonnegative while the corresponding backward quasiprobability remains contextual or negative. In this literature, backward coherence is the coherent content of the backward initial state required to formulate a detailed fluctuation theorem compatible with coherent forward preparation (Francica et al., 2023).

These two physical uses point in opposite directions. In process-theoretic time reversal, backward coherence is eliminated by causal compatibility. In coherent fluctuation theorems, it survives as a controlled quasiprobabilistic correction. The contrast is structural rather than terminological.

5. Reflection, predictive martingales, and backward induction

In personalist Bayesianism, backward coherence denotes a diachronic constraint from anticipated future credences onto present credences. The central statement is van Fraassen’s reflection principle: $t$5 whenever $t$6. A Dutch-book argument shows that if this equality fails, an agent can be made to incur a sure loss using fair bets placed at times $t$7 and $t$8. Combined with synchronic coherence, reflection implies Goldstein’s constraint

$t$9

The same framework argues that Bayesian conditioning is not forced by coherence alone; it requires an additional strategy assumption, $\bar t$0, from which $\bar t$1 follows (Fuchs et al., 2011).

The paper also applies this logic to quantum measurement. If a first measurement is described by trace-decreasing completely positive maps $\bar t$2, and a later measurement by a POVM $\bar t$3, then backward coherence gives

$\bar t$4

For projective Lüders measurements, $\bar t$5. The interpretation is explicitly non-dynamical: the “decohered” state is not produced by a physical decoherence process but is the state the agent should already use when pricing bets about a later measurement given that an intermediate measurement will occur. The paper concludes that the usual decoherence narrative has the plot “turned exactly backward” (Fuchs et al., 2011).

A second predictive use appears in model specification by sequential coherence and backward induction. For predictive densities $\bar t$6, sequential coherence is the integral condition

$\bar t$7

which the paper interprets as a predictive martingale: the expected predictive density tomorrow equals the predictive density today. Rather than starting from a prior and likelihood, the method stipulates a large-sample predictive $\bar t$8 and works backward to earlier $\bar t$9, thereby constructing a coherent, prior-free uncertainty assessment (Hahn, 2015).

The kernel-density example makes this concrete. Taking

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$00

as the terminal predictive form, backward induction yields

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$01

with $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$02, $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$03. Multiple backward steps produce a product mixture with $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$04, and the posterior mean density becomes $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$05. The same martingale structure yields Azuma–Hoeffding concentration bounds for $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$06 (Hahn, 2015).

In these inferential settings, backward coherence is an intertemporal consistency requirement. It does not refer to physical backward propagation, but to the obligation that current predictive or betting commitments already encode structured expectations about future states of belief.

6. Counter-propagating optics, molecular interference, and directional gain

In backward spontaneous four-wave mixing in laser-cooled $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$07, backward coherence denotes the robustness of two-photon temporal coherence in a counter-propagating geometry. For nondegenerate biphotons, one photon experiences slow-light EIT and absorption while the other is essentially lossless, giving

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$08

and an e-fold coherence time

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$09

For degenerate biphotons with $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$10, $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$11, and $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$12, the total attenuation becomes position-independent, the phase-mismatch imaginary parts cancel, and

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$13

The experiments report a loss-insensitive coherence time of about $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$14 ns in the degenerate case and scaling up to $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$15 by reducing the coupling power, together with $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$16 visibility in a two-photon beating experiment (Lai et al., 2024).

A subsequent theory paper re-derives the same protection using a Heisenberg–Langevin treatment with explicit quantum noise. The coupled equations are PT symmetric, and the Langevin contribution exactly cancels the loss-induced exponential tilt in the joint temporal amplitude, leaving

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$17

so that the coherence time remains $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$18 and depends only on geometry and group velocity, not on absorption or dephasing, provided the backward degenerate symmetry is preserved (Lai et al., 7 Apr 2025).

In dissociative electron attachment, backward coherence has a different content: a forward–backward angular asymmetry in fragment-ion emission generated by interference between opposite-parity dissociation pathways. At the $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$19 eV peak in $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$20, coherently excited $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$21 and $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$22 resonances, accessed predominantly by $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$23- and $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$24-wave capture, produce an odd-in-$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$25 cross-term in

$$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$26

For HD, both the $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$27 and $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$28 channels show identical forward–backward asymmetry within uncertainties, implying the same sign of the interference term in both channels. The paper emphasizes that the permanent dipole moment of HD plays no role in this process (Swain et al., 2024).

Directional asymmetry also appears in coherence-enhanced transient lasing. In a three-level $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$29 or $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$30 system, a resonant coherent drive on the $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$31 transition builds $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$32 and can yield an order-of-magnitude enhancement of forward XUV or X-ray output on the $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$33 transition. The paper states, however, that the same coherent drive always suppresses the backward gain. In the forward equations the lasing polarization $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$34 is fed by the coherence-assisted term involving $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$35, whereas the backward polarization $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$36 lacks the corresponding drive-assisted contribution. The result is a one-sided coherence injection that enhances forward amplification but not backward amplification (Jha et al., 2011).

A further backward-propagating realization occurs in gas-filled hollow-core photonic crystal fibers. There, backward stimulated Raman scattering in hydrogen is driven by a pump–backward-Stokes beat that writes a moving fine-period Bragg grating with spatial phase $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$37, $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$38, and period $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$39. Even though the backward Raman gain is lower than the forward gain at $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$40 bar, simulations and experiments show that the backward Raman coherence grows strongly near the input end of the fiber, depletes the pump early, and can make the backward Stokes dominate the forward Stokes. The paper reports quantum conversion efficiencies exceeding $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$41 for noise-seeded backward Stokes at $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$42 nm from a $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$43-nm pump, increasing to $$dP \propto \frac{\alpha_s}{\pi}\sum_{i<j}(\mathbf{T}_i\!\cdot\!\mathbf{T}_j)\,W_{ij}(k),$$44 under weak self-seeding by back-reflected forward Stokes (Mridha et al., 2018).

Across these optical and molecular uses, the term consistently identifies coherence that is specific to a backward geometry or backward hemisphere. The operative mechanisms differ—space-time symmetry, parity-interference asymmetry, directional phase matching, or moving Raman gratings—but all make backward propagation or backward emission a physically distinguished coherent channel.