---
title: Zero-Mode Transfer Probability
url: https://www.emergentmind.com/topics/zero-mode-transfer-probability
type: topic
---

# Zero-Mode Transfer Probability

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 [2606.13200]. 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 [2002.11790]. 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 [2606.13200].

In the harvesting setting, the key object is the nonlocal amplitude \(X\equiv\mathcal M\), decomposed as \(X=X_0+X_{\rm osc}\), where \(X_0\) is the zero-mode contribution and \(X_{\rm osc}\) comes from the oscillatory sector. The zero-mode transfer probability is then the absolute square of the zero-mode part alone, \(T_0=\lvert X_0\rvert^2\) [2002.11790].

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

The generalized Creutz ladder is described in real space by
\[
H=-\sum_j\Big[
K e^{i\theta}a_{j+1}^\dagger a_j
+K e^{-i\theta}b_{j+1}^\dagger b_j
+J_Y\,a_j^\dagger b_j
+K\,a_j^\dagger b_{j+1}
+K\,b_j^\dagger a_{j+1}
+\mathrm{H.c.}
\Big].
\]
The control parameters are the magnetic flux \(\theta\) and the vertical hopping \(J_Y\), with \(\mu=J_Y/(2K)\) [2606.13200].

Under open-boundary conditions and in the topological regime \(|\mu|<1\), two zero modes \(\eta_{1,\pm}\) appear exponentially localized at the ends. At half filling the ground-state manifold is twofold degenerate, spanned by
\[
\prod_j \eta_{j,-}^\dagger|0\rangle
\quad\text{and}\quad
\eta_{1,+}^\dagger\,\eta_{1,-}\,\prod_j\eta_{j,-}^\dagger|0\rangle.
\]
The quantity \(p^{\rm Z}\) therefore measures a purely edge-localized defect: depletion of the initially occupied zero mode \(\eta_{1,-}\) into the unoccupied partner \(\eta_{1,+}\) [2606.13200].

For comparison with bulk excitations, the same work defines
\[
p^{\rm B}(t_f)=\frac1L\sum_{j=2}^L
\big\langle\psi(t_f)\big|\,
\eta_{j,+}^\dagger(\lambda_f)\eta_{j,+}(\lambda_f)\,
\big|\psi(t_f)\big\rangle,
\]
which counts particle–hole excitations away from the zero-mode pair. The distinction is conceptually important: \(p^{\rm Z}\) is an edge observable tied to the zero-mode subspace, whereas \(p^{\rm B}\) 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,
\[
p^{\mathrm Z}(\tau_Q)=p_0^{\mathrm Z}(\tau_Q)+p_{\rm osc}^{\mathrm Z}(\tau_Q),
\qquad
p_{\rm osc}^{\mathrm Z}(\tau_Q)\sim \cos\bigl[\Delta\varphi_{\mathrm Z}(\tau_Q)\bigr].
\]
The corresponding bulk quantity obeys
\[
p^{\mathrm B}(\tau_Q)=p_0^{\mathrm B}(\tau_Q)+p_{\rm osc}^{\mathrm B}(\tau_Q),
\qquad
p_{\rm osc}^{\mathrm B}(\tau_Q)\sim \cos\bigl[\Delta\varphi_{\mathrm B}(\tau_Q)\bigr].
\]
The oscillations are interpreted as interference of critical dynamics associated with zero modes, denoted ICDZM [2606.13200].

For a closed path linking two topologically nontrivial phases, \(\theta:\pi/2\to5\pi/2\) at \(\mu=0\), the zero-mode signal shows a clear oscillation with
\[
p_0^{\mathrm Z}\sim\tau_Q^{-1.332},
\qquad
T_{\mathrm Z}\simeq 2\,T_{\mathrm B},
\qquad
T_{\mathrm B}=\pi/(4K).
\]
The period doubling follows from the relation
\[
\Delta_{\mathrm Z}=\tfrac12\Delta_{\mathrm B}\Rightarrow T_{\mathrm Z}=2T_{\mathrm B},
\]
because the relevant edge-mode gap is one half of the bulk particle–hole gap in the nontrivial region [2606.13200].

For a closed path that crosses the same critical point twice, \(\theta:\pi/2\to3\pi/2\to\pi/2\) at \(\mu=0\), 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, \(\mu:0\to2.5\to0\) at \(\theta=\pi/2\), the zero-mode pair ends up half occupied each, \(p_0^{\mathrm Z}\approx\tfrac12\), and \(T_{\mathrm Z}\simeq T_{\mathrm B}\) [2606.13200].

A compact two-passage description expresses the interference phase as
\[
\Delta\varphi
=\int_{t_1}^{t_2}\!\bigl[E_+(t)-E_-(t)\bigr]\,dt+\delta,
\]
with \(E_\pm(t)\) the instantaneous branch energies and \(\delta\) 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 \(\sigma_y\) rotation to a Weber equation. This framework accounts for the oscillatory structure, the suppression pattern, and the period doubling [2606.13200].

## 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,
\[
\hat N_1=a_1^\dagger a_1+b_1^\dagger b_1,
\qquad
\Delta N_1=\langle\hat N_1\rangle_{t_i}-\langle\hat N_1\rangle_{t_f}\equiv d_{\rm left}.
\]
In the initial half-filled ground state,
\[
\langle N_1\rangle_{t_i}=3/2,
\]
reflecting the half-integer boundary charge \(+1/2\). After the quench,
\[
\langle N_1\rangle_{t_f}=3/2-p^{\rm Z},
\]
so that
\[
d_{\rm left}=p^{\rm Z}.
\]
The zero-mode transfer probability is therefore measurable through a boundary-particle-number deficit rather than by direct projection onto the zero-mode basis [2606.13200].

The same analysis emphasizes that in a single shot one measures \(N_1^{\rm(exp)}=0,1,2\), while the ensemble average reproduces \(\langle N_1\rangle\). This links a topological edge observable to an experimentally local quantity. As an edge defect, \(p^{\rm Z}\) 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
\[
d_{\rm left}\sim\tau_Q^{-1.332},
\]
matching the quoted anomalous power law [2606.13200].

A plausible implication is that \(p^{\rm Z}\) 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 \((1+1)\)-dimensional Einstein cylinder with periodic boundary conditions, the field operator is decomposed as
\[
\hat\phi(t,x)=\hat\phi_{zm}(t)+\hat\phi_{osc}(t,x),
\]
and the pull-back Wightman function splits accordingly,
\[
W(x,x')=W_{zm}(t,t')+W_{osc}(x,x').
\]
In the Schrödinger picture, the zero-mode operator evolves as
\[
\hat\phi_{zm}(t)=\hat Q_S+\frac{t}{L}\hat P_S,
\qquad [\hat Q_S,\hat P_S]=i.
\]
The initial field state is \(\rho_\phi=\rho_{zm}\otimes|0\rangle\langle0|_{osc}\), with \(\rho_{zm}\) a pure Gaussian satisfying
\[
\langle Q_S\rangle=\langle P_S\rangle=0,\quad
\langle Q_S^2\rangle=\tfrac1{2\gamma},\quad
\langle P_S^2\rangle=\tfrac\gamma2,\quad
\langle\{Q_S,P_S\}\rangle=0
\]
[2002.11790].

At leading order in the Unruh–DeWitt coupling \(\lambda\), the single-detector excitation probability is
\[
P_A=\mathcal L_{AA}
=\lambda^2\!\int\!dt\,dt'\,\chi(t)\chi(t')e^{-i\Omega(t-t')}
\,W\bigl(t,x_A;t',x_A\bigr),
\]
with decomposition \(P_A=P_{A,0}+P_{A,osc}\). For Gaussian switching \(\chi(t)=e^{-t^2/(2T^2)}\), the zero-mode contribution is
\[
P_{A,0}
=\lambda^2\,\frac{\pi\,e^{-T^2\Omega^2}\,\bigl(LT-\gamma T^3\Omega\bigr)^2}{\gamma\,L^2}.
\]

The nonlocal amplitude governing harvesting is
\[
X\equiv\mathcal M
=-\lambda^2\!\int\!dt\!\int\!dt'\,
\chi_A(t)\chi_B(t')e^{i(\Omega t+\Omega t')}
\Bigl[\Theta(t-t')W\bigl(x_A(t),x_B(t')\bigr)
+\Theta(t'-t)W\bigl(x_B(t'),x_A(t)\bigr)\Bigr],
\]
with \(X=X_0+X_{osc}\). For two detectors at rest, the zero-mode part is
\[
X_{0}
=-\,\lambda^2\,
\frac{e^{-T^2\Omega^2}}{\gamma\,L^2}
\Bigl[\pi L^2T^2
-2i\sqrt{\pi}\,\gamma LT^3
-\pi\gamma^2T^6\Omega^2\Bigr].
\]
The zero-mode transfer probability is then
\[
T_0=\lvert X_0\rvert^2
=\lambda^4\,
\frac{e^{-2T^2\Omega^2}}{\gamma^2L^4}
\Bigl[
\pi^2L^4T^4
+4\pi\gamma^2L^2T^6
+\pi^2\gamma^4T^{12}\Omega^4
\Bigr].
\]
Its behavior is sharply constrained: \(T_0\to0\) as \(T\to\infty\), \(T_0\) remains finite of order \(O(\lambda^4)\) as \(L\to\infty\), and \(T_0\to0\) as \(\gamma\to\infty\). The same analysis therefore concludes that the Minkowski limit does not commute with the presence of a zero mode [2002.11790].

## 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
\[
k_T^{\rm glass}=2.8\pm1.0\;\mathrm{ns}^{-1}
\quad\text{to}\quad
k_T^{\rm ZMW}=4.1\pm1.5\;\mathrm{ns}^{-1},
\]
with efficiencies
\[
E^{\rm glass}=0.85\pm0.06,
\qquad
E^{\rm ZMW}=0.76\pm0.10,
\]
corresponding to a 50% enhancement of the FRET rate constant [2205.01190]. In a related Alexa-dye study, the summary explicitly describes a \(1.5\)–\(2.2\) fold increase in the “zero-mode” FRET transfer probability \(k'_{\rm FRET}/(k'_{\rm FRET}+k'_D+k'_{NR})\) at separations beyond \(10\) nm [2004.04513]. A distinct ZMW transmission theory derives
\[
T(\omega)\equiv |t(\omega)|^2
=\left|t_0+\frac{i g^2}{\Delta+i\Gamma/2}\right|^2
\]
for a single atom in a zero-mode waveguide, yielding a Fano-type transmission profile tunable by detuning, atom position, and geometry [2508.21514].

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 \(t\) if and only if
\[
\sum_{p=1}^b e^{-i\Theta_p}=0,
\]
a destructive-interference condition on branch phases [1902.11115]. On weighted spider and Cayley-tree graphs, the occupation probability at the root obeys
\[
P_{\rm root}(t)\sim O(J^{-2})
\]
for large central hopping, giving “almost zero transfer” rather than an exact zero-mode quantity [2404.04094]. In one-mode waveguides near a Fano resonance, exact zero transmission is the condition \(T(\omega_*)=0\) at a unique real frequency, enforced by the pole-zero structure of the scattering matrix [1907.08439].

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.

Source: https://www.emergentmind.com/topics/zero-mode-transfer-probability