Zero-Mode Transfer Probability

• Zero-mode transfer probability is a measure of transitions between paired zero modes, defined via occupation depletion in lattice models or amplitude squares in field settings.
• It quantifies edge-localized defects in the generalized Creutz ladder and nonlocal contributions in entanglement harvesting on the Einstein cylinder.
• Experimental readouts derive from boundary particle number deficits and interference oscillations, providing actionable insights into topological and quantum field dynamics.

Searching arXiv for recent and directly relevant uses of the term and associated contexts. Zero-mode transfer probability is a context-dependent quantity associated with dynamics in a zero-mode sector. In the generalized Creutz ladder, it is defined as the probability that an initially occupied edge zero mode has migrated to its particle–hole partner after a time-dependent quench,

$p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f),$

with $n_{1,\pm}(t_f)$ the final occupations of the two zero modes (Zhang et al., 11 Jun 2026). In relativistic quantum information on the $(1+1)$-dimensional Einstein cylinder, the same phrase denotes the absolute square of the zero-mode contribution to the two-detector transfer amplitude,

$T_0=\lvert X_0\rvert^2,$

within leading-order entanglement harvesting (Tjoa et al., 2020). This suggests that the term is not universal across subfields; rather, it labels different observables constructed from zero-mode degrees of freedom.

1. Definitions and scope

The two principal definitions appearing in the supplied literature are structurally different. One is an occupation-transfer probability between two topological edge zero modes in a lattice model; the other is a probability built from a nonlocal amplitude induced by the spatially constant mode of a quantum field.

Context	Quantity	Definition
Generalized Creutz ladder	Zero-mode transfer probability	$p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$
Einstein cylinder entanglement harvesting	Zero-mode transfer probability	$T_0=\lvert X_0\rvert^2$

In the Creutz-ladder setting, the occupations are

$$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$

and particle–hole symmetry at half filling gives

$n_{1,+}(t_f)+n_{1,-}(t_f)=1.$

The transfer probability therefore measures depletion of the initially occupied negative-energy zero mode into its partner (Zhang et al., 11 Jun 2026).

In the harvesting setting, the key object is the nonlocal amplitude $X\equiv\mathcal M$, decomposed as $X=X_0+X_{\rm osc}$, where $n_{1,\pm}(t_f)$0 is the zero-mode contribution and $n_{1,\pm}(t_f)$1 comes from the oscillatory sector. The zero-mode transfer probability is then the absolute square of the zero-mode part alone, $n_{1,\pm}(t_f)$2 (Tjoa et al., 2020).

2. Topological zero-mode transfer in the generalized Creutz ladder

The generalized Creutz ladder is described in real space by

$n_{1,\pm}(t_f)$3

The control parameters are the magnetic flux $n_{1,\pm}(t_f)$4 and the vertical hopping $n_{1,\pm}(t_f)$5, with $n_{1,\pm}(t_f)$6 (Zhang et al., 11 Jun 2026).

Under open-boundary conditions and in the topological regime $n_{1,\pm}(t_f)$7, two zero modes $n_{1,\pm}(t_f)$8 appear exponentially localized at the ends. At half filling the ground-state manifold is twofold degenerate, spanned by

$n_{1,\pm}(t_f)$9

The quantity $(1+1)$0 therefore measures a purely edge-localized defect: depletion of the initially occupied zero mode $(1+1)$1 into the unoccupied partner $(1+1)$2 (Zhang et al., 11 Jun 2026).

For comparison with bulk excitations, the same work defines

$(1+1)$3

which counts particle–hole excitations away from the zero-mode pair. The distinction is conceptually important: $(1+1)$4 is an edge observable tied to the zero-mode subspace, whereas $(1+1)$5 is a bulk excitation density.

3. Interference under closed-path quenches

When a quench path crosses two critical points successively, the zero-mode transfer probability decomposes into a smooth background and an oscillatory term,

$(1+1)$6

The corresponding bulk quantity obeys

$(1+1)$7

The oscillations are interpreted as interference of critical dynamics associated with zero modes, denoted ICDZM (Zhang et al., 11 Jun 2026).

For a closed path linking two topologically nontrivial phases, $(1+1)$8 at $(1+1)$9, the zero-mode signal shows a clear oscillation with

$T_0=\lvert X_0\rvert^2,$0

The period doubling follows from the relation

$T_0=\lvert X_0\rvert^2,$1

because the relevant edge-mode gap is one half of the bulk particle–hole gap in the nontrivial region (Zhang et al., 11 Jun 2026).

For a closed path that crosses the same critical point twice, $T_0=\lvert X_0\rvert^2,$2 at $T_0=\lvert X_0\rvert^2,$3, the bulk still shows ICD, but the zero-mode oscillation is strongly suppressed and vanishes within numerical resolution. For an open path through the topologically trivial phase, $T_0=\lvert X_0\rvert^2,$4 at $T_0=\lvert X_0\rvert^2,$5, the zero-mode pair ends up half occupied each, $T_0=\lvert X_0\rvert^2,$6, and $T_0=\lvert X_0\rvert^2,$7 (Zhang et al., 11 Jun 2026).

A compact two-passage description expresses the interference phase as

$T_0=\lvert X_0\rvert^2,$8

with $T_0=\lvert X_0\rvert^2,$9 the instantaneous branch energies and $p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$0 a nonuniversal offset. The same work derives this phase using WKB analysis: protocol 1 leads to a Whittaker–Hill-type equation after removing a trivial dynamical phase, while protocol 3 reduces by a $p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$1 rotation to a Weber equation. This framework accounts for the oscillatory structure, the suppression pattern, and the period doubling (Zhang et al., 11 Jun 2026).

4. Boundary-particle readout and edge defects

The zero-mode transfer probability in the Creutz ladder has a direct local readout through the particle number on the first rung,

$p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$2

In the initial half-filled ground state,

$p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$3

reflecting the half-integer boundary charge $p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$4. After the quench,

$p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$5

so that

$p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$6

The zero-mode transfer probability is therefore measurable through a boundary-particle-number deficit rather than by direct projection onto the zero-mode basis (Zhang et al., 11 Jun 2026).

The same analysis emphasizes that in a single shot one measures $p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$7, while the ensemble average reproduces $p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$8. This links a topological edge observable to an experimentally local quantity. As an edge defect, $p^{\rm Z}(t_f)=1-n_{1,-}(t_f)=n_{1,+}(t_f)$9 captures both the ICDZM oscillation in closed paths and the anomalous defect production in a one-way quench across a single critical point. For the latter, the edge defect scales as

$T_0=\lvert X_0\rvert^2$0

matching the quoted anomalous power law (Zhang et al., 11 Jun 2026).

A plausible implication is that $T_0=\lvert X_0\rvert^2$1 occupies an intermediate status between a microscopic occupation number and a macroscopic defect observable: it is defined in the instantaneous zero-mode basis, but it can be inferred from a boundary charge deviation.

5. Zero-mode transfer probability in vacuum entanglement harvesting

In the entanglement-harvesting problem on the $T_0=\lvert X_0\rvert^2$2-dimensional Einstein cylinder with periodic boundary conditions, the field operator is decomposed as

$T_0=\lvert X_0\rvert^2$3

and the pull-back Wightman function splits accordingly,

$T_0=\lvert X_0\rvert^2$4

In the Schrödinger picture, the zero-mode operator evolves as

$T_0=\lvert X_0\rvert^2$5

The initial field state is $T_0=\lvert X_0\rvert^2$6, with $T_0=\lvert X_0\rvert^2$7 a pure Gaussian satisfying

$T_0=\lvert X_0\rvert^2$8

(Tjoa et al., 2020).

At leading order in the Unruh–DeWitt coupling $T_0=\lvert X_0\rvert^2$9, the single-detector excitation probability is

$$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$0

with decomposition $$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$1. For Gaussian switching $$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$2, the zero-mode contribution is

$$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$3

The nonlocal amplitude governing harvesting is

$$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$4

with $$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$5. For two detectors at rest, the zero-mode part is

$$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$6

The zero-mode transfer probability is then

$$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$7

Its behavior is sharply constrained: $$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$8 as $$n_{1,\pm}(t_f)=\big\langle \psi(t_f)\big|\,\eta_{1,\pm}^\dagger(\lambda_f)\,\eta_{1,\pm}(\lambda_f)\,\big|\psi(t_f)\big\rangle,$$9, $n_{1,+}(t_f)+n_{1,-}(t_f)=1.$0 remains finite of order $n_{1,+}(t_f)+n_{1,-}(t_f)=1.$1 as $n_{1,+}(t_f)+n_{1,-}(t_f)=1.$2, and $n_{1,+}(t_f)+n_{1,-}(t_f)=1.$3 as $n_{1,+}(t_f)+n_{1,-}(t_f)=1.$4. The same analysis therefore concludes that the Minkowski limit does not commute with the presence of a zero mode (Tjoa et al., 2020).

6. Related terminologies and conceptual boundaries

The supplied literature also contains several adjacent usages that can be confused with zero-mode transfer probability but are not identical to the two definitions above. In nanophotonics, zero-mode waveguides modify Förster resonance energy transfer and transmission through subwavelength apertures. For immobilized single molecules in a 110 nm Al ZMW, the measured single-molecule FRET-rate constant changes from

$n_{1,+}(t_f)+n_{1,-}(t_f)=1.$5

with efficiencies

$n_{1,+}(t_f)+n_{1,-}(t_f)=1.$6

corresponding to a 50% enhancement of the FRET rate constant (Patra et al., 2022). In a related Alexa-dye study, the summary explicitly describes a $n_{1,+}(t_f)+n_{1,-}(t_f)=1.$7–$n_{1,+}(t_f)+n_{1,-}(t_f)=1.$8 fold increase in the “zero-mode” FRET transfer probability $n_{1,+}(t_f)+n_{1,-}(t_f)=1.$9 at separations beyond $X\equiv\mathcal M$0 nm (Baibakov et al., 2020). A distinct ZMW transmission theory derives

$X\equiv\mathcal M$1

for a single atom in a zero-mode waveguide, yielding a Fano-type transmission profile tunable by detuning, atom position, and geometry (Klimov, 29 Aug 2025).

In quantum walks and wave scattering, nearby terms again denote different observables. In chiral continuous-time quantum walks on multibranch graphs, exact zero transfer occurs for all $X\equiv\mathcal M$2 if and only if

$X\equiv\mathcal M$3

a destructive-interference condition on branch phases (Sett et al., 2019). On weighted spider and Cayley-tree graphs, the occupation probability at the root obeys

$X\equiv\mathcal M$4

for large central hopping, giving “almost zero transfer” rather than an exact zero-mode quantity (Vieira et al., 2024). In one-mode waveguides near a Fano resonance, exact zero transmission is the condition $X\equiv\mathcal M$5 at a unique real frequency, enforced by the pole-zero structure of the scattering matrix (Chesnel et al., 2019).

This suggests a sharp terminological boundary. In topological-quench dynamics and entanglement harvesting, zero-mode transfer probability is a zero-mode-sector observable defined from occupations or amplitudes. In ZMW nanophotonics, quantum walks, and Fano scattering, the same words or close variants may instead refer to waveguide geometry, destructive interference, or transmission suppression.