1 Introduction↩︎

Predicting when a quantum detector clicks is not fixed by the usual Born rule for position at a prescribed time. This arrival- and detection-time problem has led to several inequivalent proposals, including flux-based rules, detector-free trajectory arrival times, complex absorbing potentials, and absorbing boundary conditions [1]–[6]. A central point is that a detector-free arrival time and a detector-present click time are generally different physical objects: the detector is not a passive marker of a pre-existing crossing event, but part of the dynamical boundary problem that defines the click-time distribution.

Absorbing boundaries are often introduced to remove probability from an open quantum evolution. In detector models this loss is physical: the squared norm \(\|\Psi_t\|^2\) is the probability that no click has yet occurred, and its loss rate is the click-time density [2], [5], [6]. The boundary is therefore not merely a surface where probability disappears. Like an impedance in wave physics, it is a boundary law determining how incident components are absorbed and how the undetected components are reflected, mixed, or filtered within the no-click evolution. In numerical settings such interfaces are often introduced as absorbing or radiation devices [7], [8]; in the detector model considered here, the same response is part of the measurement, not merely a numerical device [2], [5], [6].

In this work we study a spin-coupled absorbing boundary condition, or spinor ABC, whose impedance is matrix-valued. For a nonrelativistic spin-\(1/2\) particle in a harmonic waveguide, the absorbing boundary couples the normal derivative of the spinor to its tangential derivatives. Tangential Fourier decomposition at the detecting surface turns the boundary condition into a \(2\times2\) matrix relation \(\partial_z\widehat{\Psi}=\mathcal{C}(\boldsymbol{\xi})\widehat{\Psi}\). The eigenvalues \(i\kappa\pm|\boldsymbol{\xi}|\) of \(\mathcal{C}(\boldsymbol{\xi})\) define two spin–momentum branches. For the harmonic transverse ground state, \(\ell_\perp=\omega^{-1/2}\) in dimensionless units; hence increasing \(\omega\) narrows the guide and raises the typical tangential scale sampled at the boundary, so that \(|\boldsymbol{\xi}|\sim\ell_\perp^{-1}\sim\sqrt{\omega}\). This local boundary scale underlies the delayed detector-present roof-flux response studied below.

At first sight this confinement dependence is surprising, since the bulk Hamiltonian used here is spin independent and separable, so changing \(\omega\) changes the normalized transverse profile but does not by itself change the plane-integrated first-pass longitudinal Pauli flux. Thus the square-root scale observed in the detector response is not generated by ordinary first-pass bulk propagation. It must enter when the wave interacts with the spinor ABC.

This response is detector-present. The observable is the roof flux, equivalently the norm loss, of a specified nonunitary absorbing-boundary evolution. It is not the no-detector Bohmian first-arrival distribution of a freely evolving Pauli wave [3], [4], [9], [10]. Nor is the spin-coupled absorbing boundary assumed to describe all possible detecting screens; it is a particular idealized hard-detector model, a restriction emphasized by recent scattering-theory comparisons of absorbing detectors [11].

The Das–Dürr waveguide geometry provides a sharp comparison between detector-free arrival and detector-present detection [4]. In the same separable guide, complex absorbing potentials and the spin-decoupled absorbing boundary behave as scalar absorbers: they may generate reflected tails through detector back-action, but they do not read transverse confinement in this way [12], [13]. By contrast, the spin-coupled absorbing boundary suppresses the prompt roof-flux peak, lowers the detected fraction in a fixed observation window, and converts the transverse confinement into a delayed oscillatory sector, while showing no appreciable dependence on the initial Bloch angles in the tested regimes [13]. Those simulations left the confinement dependence unexplained. Here we identify its local origin, derive the two-branch boundary symbol \(i\kappa\pm|\boldsymbol{\xi}|\), and show how this boundary scale organizes the observed \(\omega\)-dependent detector-response signature.

2 Model and detector observable↩︎

We work in units \(\hbar=m=1\). The bulk Hamiltonian is spin independent, \[H=-\frac{1}{2}\Delta+\frac{1}{2}\omega^2\bigl[x^2+y^2\bigr],\] and the initial state is factorized as \[\Psi_0(x,y,z)=\chi_\omega(x,y)\,\phi_0(z)\,\eta ,\] where \(\eta\in\mathbb{C}^2\) is a constant spinor and \(\chi_\omega\) is the transverse harmonic ground state. For the finite-window confinement study below, \(\phi_0\) is a right-moving Gaussian longitudinal packet centered midway along the guide. The finite guide has a reflecting lower end at \(z=0\), and the detecting surface is the roof \[\Sigma_L=\{z=L\}.\] On this surface we impose the spin-coupled absorbing boundary condition \[(\boldsymbol{\sigma}\cdot\boldsymbol{\nabla})\Psi=i\kappa\sigma_z\Psi,\qquad \kappa>0. \label{eqn:spinorABC}\tag{1}\] This is the nonrelativistic limit of the semi-ideal Dirac absorbing boundary [14]; well-posedness and contraction-semigroup formulations are discussed in Refs. [15], [16]. For the Pauli current \[\boldsymbol{j}^P=\operatorname{Im}(\Psi^\dagger\boldsymbol{\nabla}\Psi) +\frac{1}{2}\boldsymbol{\nabla}\times(\Psi^\dagger\boldsymbol{\sigma}\Psi),\] the spinor ABC gives the outward roof flux \[j_z^P(x,y,L,t)=\kappa\rho(x,y,L,t),\qquad \rho=\Psi^\dagger\Psi .\] By the continuity equation, with no flux through the remaining faces, the detector-present click-time density is \[g(t;\omega) = \kappa\int_{\Sigma_L}\rho(x,y,L,t)\,dxdy = -\frac{d}{dt}\|\Psi_t\|^2 .\] For a finite observation horizon \(T\), define the detected fraction \[D_{T}(\omega)=\int_0^{T}g(t;\omega)\,dt,\] and the finite-window restricted mean detection time, \[\mu^*( T;\omega) = \int_0^{ T}S(t;\omega)\,dt, \qquad S(t;\omega)=\|\Psi_t\|^2 ,\] or, \[\mu^*( T;\omega)=\mathbb{E}[\min(\tau, T)].\] Here \(\tau\) is the random detector click-time. We use this restricted mean detection time because, at strong confinement, a substantial part of the probability remains undetected at the end of the finite observation window. Bohmian histograms shown in the plots are only Monte Carlo samples of the same detector-present flux law. These trajectories obey \[\boldsymbol{\dot{Q}}(t)= \frac{\boldsymbol{j}^P[\Psi_t^{\rm ABC}](\boldsymbol{Q}(t))}{\rho[\Psi_t^{\rm ABC}](\boldsymbol{Q}(t))}.\] Here \(\boldsymbol{Q}(t)\) denotes a sampled Pauli-current trajectory and \(\Psi_t^{\mathrm{ABC}}\) denotes the nonunitary spinor-ABC evolution.

We now present the confinement-dependent roof flux and then derive the boundary-symbol mechanism responsible for the \(\sqrt{\omega}\) scale.

3 Confinement-induced spin–momentum filtering by the spin-coupled ABC↩︎

Figure 1 shows the central numerical observation. The red curve in Fig. 1 is the detector-free one-dimensional Gaussian current through \(z=L\) for the same longitudinal packet, with the transverse factor integrated out. It is not a detector model or fit input; it only marks the free crossing scale defined in detail in Ref. [13]. Only the transverse confinement parameter \(\omega\) is varied; the longitudinal packet, detector parameter, initial spinor, and observation window are fixed. Increasing \(\omega\) narrows the transverse ground state and strongly suppresses the prompt roof-flux peak. This prompt suppression is the most direct finite-window signature of the boundary response. The detected fraction drops from \(D_{20}\simeq0.93\) at \(\omega=1\) to \(D_{20}\simeq0.42\) at \(\omega=300\). Figure 1 (c) summarizes the same confinement sweep: over the sampled \(\omega\)-values, the restricted mean detection time is fitted by \[\mu^*(20;\omega)\simeq 4.084+0.638\sqrt{\omega}. \label{eqn:mu}\tag{2}\] The fit in Eq. 2 is a finite-window observation, not an asymptotic law. Its coefficients would depend on the prepared wave packet, detector parameter, guide geometry, and observation time. The robust feature is the square-root scale. We now identify its boundary origin and the finite-guide mechanism that makes this scale visible in the roof-flux detection time statistic.

4 Boundary-symbol mechanism and finite-guide memory↩︎

We discuss the mechanism through three logical steps: (i) ordinary first-pass bulk propagation is excluded as the source of the \(\omega\)-dependence; (ii) the spinor ABC is shown to introduce the local roof scale \(R=|\boldsymbol{\xi}|\); and (iii) finite-guide memory provides a natural route by which undetected, branch-reweighted components can return to the roof and contribute to the delayed detection-time sector in Fig. 1.

(i) The square-root scale is not generated by ordinary first-pass bulk propagation. Before significant feedback from the detecting roof, the spin-independent separable bulk evolution has the first-pass form \[\Psi_{\rm fp}(x,y,z,t)=\chi_\omega(x,y,t)\phi_{\rm fp}(z,t)\eta,\] where \(\phi_{\rm fp}(z,0)=\phi_0(z)\). The transverse harmonic ground state changes only by an overall phase, so its density is unchanged, and the spinor \(\eta\) remains constant because the bulk Hamiltonian is spin independent. For this first-pass field, the Pauli current can then be written \[\boldsymbol{j}_{\rm fp}^{P} = \boldsymbol{j}_{\rm fp}^{\rm conv} + \frac{1}{2}\boldsymbol{\nabla}\times(\rho_{\rm fp}\boldsymbol{s}_\eta), \quad \boldsymbol{s}_\eta:=\eta^\dagger\boldsymbol{\sigma}\eta, \quad \rho_{\rm fp}:=|\Psi_{\rm fp}|^2 .\] Since \(\boldsymbol{s}_\eta\) is constant in the first-pass separable approximation, the \(z\)-component of the spin-curl term is a transverse divergence. By Stokes’ theorem on the transverse cross-section, together with the lateral Dirichlet conditions, its cross-section integral vanishes. Hence, for an interior plane \(\Sigma_b\) below the roof, \[\int_{\Sigma_b}j^{P}_{{\rm fp},z}\,dA = \int_{\Sigma_b} \operatorname{Im}(\Psi_{\rm fp}^\dagger\partial_z\Psi_{\rm fp})\,dA = \operatorname{Im} (\phi_{\rm fp}^*\partial_z\phi_{\rm fp})(z_b,t).\] The leading plane-integrated first-pass flux is therefore independent of the transverse confinement parameter \(\omega\). More details on the first-pass bulk calculation are given in the Supplementary Information [17], Sec. S5.

(ii) The observed confinement dependence therefore must enter when the wave interacts with the detecting boundary. The spinor ABC introduces this scale through the tangential symbol \(\mathcal{C}(\boldsymbol{\xi})\) of the boundary operator. The same roof condition, Eq. 1 , can be written componentwise as \[\begin{align} \partial_z \psi_\uparrow &= i\kappa\,\psi_\uparrow - (\partial_x-i\partial_y)\psi_\downarrow, \\ \partial_z \psi_\downarrow &= i\kappa\,\psi_\downarrow + (\partial_x+i\partial_y)\psi_\uparrow. \end{align} \label{eq:abc-comp}\tag{3}\] Equivalently, \[\partial_z\Psi=\mathcal{C}\Psi, \quad \mathcal{C}= \begin{pmatrix} i\kappa & -D_-\\ D_+ & i\kappa \end{pmatrix}, \quad D_\pm=\partial_x\pm i\partial_y . \label{eqn:boundary}\tag{4}\] Therefore the detector couples normal absorption to tangential derivatives. For a tangential Fourier mode \(\boldsymbol{\xi}=(\xi_x,\xi_y)\), Eq. 4 gives \[\mathcal{C}(\boldsymbol{\xi})=i\kappa I+(\hat{z}\times \boldsymbol{\xi})\cdot\boldsymbol{\sigma} .\] Writing \[R:=|\boldsymbol{\xi}|,\qquad \Gamma(\boldsymbol{\xi})=\frac{(\hat{z}\times \boldsymbol{\xi})\cdot\sigma}{R}, \qquad \Gamma^2=I.\] Thus \[\mathcal{C}(\boldsymbol{\xi})=i\kappa I+R\Gamma(\boldsymbol{\xi}).\] The two tangential spin–momentum projectors are then \[\Pi_\pm(\boldsymbol{\xi})=\frac{1}{2}\bigl(I\pm\Gamma(\boldsymbol{\xi})\bigr),\] and the corresponding boundary eigenbranches are \[\lambda_\pm(\boldsymbol{\xi})=i\kappa\pm R.\] This is the key point: the detector imposes not only the impedance scale \(\kappa\), but also the tangential spin–momentum scale \(R=|\boldsymbol{\xi}|\). The same branch structure gives the local normal response. Let \(W(\boldsymbol{\xi},t)=\widehat{\Psi}(\boldsymbol{\xi},L,t)\) be the roof trace. For a small auxiliary inward depth \(\epsilon=L-z\), the homogeneous normal continuation generated by the boundary relation is \[\widehat{\Psi}(\boldsymbol{\xi},L-\epsilon,t) = e^{-i\kappa\epsilon}B_{\rm br}(\epsilon,\boldsymbol{\xi})W(\boldsymbol{\xi},t) +\widehat{\cal R}_{\rm Duh},\] where \[B_{\rm br}(\epsilon,\boldsymbol{\xi})W = e^{-R\epsilon}\Pi_+ W + e^{R\epsilon}\Pi_- W. \label{eqn:filter}\tag{5}\] This is a matrix-valued branch filter acting on the spinor amplitude \(W(\boldsymbol{\xi},t)\): it decomposes the roof trace into the two tangential spin–momentum components \(\Pi_\pm W\) and assigns them the auxiliary normal weights \(e^{-R\epsilon}\) and \(e^{R\epsilon}\). The term \(\widehat{\mathcal{R}}_{\rm Duh}\) is a Duhamel remainder measuring the deviation of the true time dependent Schrödinger equation (TDSE) solution from this homogeneous boundary continuation away from the roof; it has no zeroth- or first-order contribution at \(\epsilon=0\), as shown in the Supplementary Information [17], Sec. S7. All physical detection statistics remain defined at the roof, \(\epsilon=0\). Therefore \(B_{\rm br}\) is not a physical propagation law through a detector layer, but a local boundary-symbol probe \[B_{\rm br}(0,\boldsymbol{\xi})=I,\qquad \left. \partial_\epsilon B_{\rm br}(\epsilon,\boldsymbol{\xi}) \right|_{\epsilon=0} = -R\,\Gamma(\boldsymbol{\xi}). \label{eqn:FirstVariation}\tag{6}\] The branch structure is not a direct multiplicative suppression of the roof density; it enters the first inward normal response. Therefore the same tangential radius \(R=|\boldsymbol{\xi}|\) that labels the roof mode controls the local normal boundary response. For the harmonic transverse ground state, \(\ell_\perp=\omega^{-1/2}\), so the typical tangential Fourier scale is set by \[R=\lvert\boldsymbol{\xi}\rvert \sim \ell_\perp^{-1} \sim \sqrt{\omega}. \label{eqn:HO95scale}\tag{7}\] This is the local boundary scale. To express how it enters a normalized first time moment, define an auxiliary boundary layer response measure \[d\mu^{\rm bl}_{\omega,\epsilon}(\boldsymbol{\xi},t) := |B_{\rm br}(\epsilon,\boldsymbol{\xi})W(\boldsymbol{\xi},t)|^2\,d^2\xi\,dt .\] This is not a click-time distribution at \(z=L-\epsilon\): the variable \(t\) is only the time label of the roof trace \(W(\boldsymbol{\xi},t)\), and the physical detector observable remains the roof flux at \(z=L\). Since the scalar phase drops out of densities and the Duhamel term has no first-order contribution at the roof, the first variation is computed from the homogeneous branch probe. With \(B_{\rm br}=e^{-\epsilon R\Gamma}\), \[|B_{\rm br}W|^2=W^\dagger e^{-2\epsilon R\Gamma}W=|W|^2b_{\omega,\epsilon}.\] where \[b_{\omega,0}=1,\qquad \partial_\epsilon b_{\omega,\epsilon}|_{\epsilon=0}=Ra_\omega,\] and \[a_\omega(\boldsymbol{\xi},t) := -2\frac{W^\dagger\Gamma(\boldsymbol{\xi})W}{W^\dagger W}, \qquad |a_\omega|\le2 .\]

To normalize the auxiliary boundary-response measure, let \(d\nu^{(0)}_\omega=d\mu^{\rm bl}_{\omega,0}/\int d\mu^{\rm bl}_{\omega,0}\). After normalization, \[\mathbb{E}_{\nu^{\rm bl}_{\omega,\epsilon}}[F] = \frac{\mathbb{E}_{\nu^{(0)}_\omega}[F b_{\omega,\epsilon}]}{ \mathbb{E}_{\nu^{(0)}_\omega}[b_{\omega,\epsilon}]} .\] Differentiating at \(\epsilon=0\) gives \[\partial_\epsilon \mathbb{E}_{\nu^{\rm bl}_{\omega,\epsilon}}[F]\big|_{\epsilon=0} = \operatorname{Cov}_{\nu^{(0)}_\omega}(F,Ra_\omega).\] Take \(F=t\), the local first-order boundary sensitivity is \[\Lambda_\omega = \operatorname{Cov}_{\nu^{(0)}_\omega}(t,Ra_\omega).\] For the harmonic transverse family, \(R=\sqrt{\omega}s\), with \(s\) dimensionless and order one. Hence \[\Lambda_\omega = \sqrt{\omega}\, \operatorname{Cov}_{\nu^{(0)}_\omega}(t,sa_\omega) =:\sqrt{\omega}\,\beta_\omega . \label{eqn:source}\tag{8}\] Thus \(\Lambda_\omega\) is a local first-order boundary sensitivity, not the observed finite-window delay itself. Equation 8 shows that the first boundary-response contribution factorizes into the explicit transverse scale \(\sqrt{\omega}\) and a dimensionless roof-trace covariance \(\beta_\omega\). The latter is not universal: it depends on the detector-present roof trace and may change with the prepared wave packet, detector parameter, guide geometry, and observation window. The finite-grid diagnostics in the Supplementary Information [17], Sec. S7, show \(\beta_\omega=O(1)\) over the sweep, with no competing power-law growth. Thus the local calculation identifies the \(\sqrt{\omega}\) scale, not the setup-dependent value or sign of \(B\) in the finite-window fit \(A+B\sqrt{\omega}\).

(iii) In the finite guide, the local boundary response becomes visible in the roof-flux statistic through finite-guide memory. During the first near-roof encounter, the spinor ABC does not remove a scalar fraction of the incident packet. The amplitude that is not absorbed has already sampled the matrix boundary symbol, has been attenuated by detector loss, and has been reweighted between the two tangential spin–momentum branches. The surviving no-click amplitude that subsequently propagates back into the guide is therefore not a scalar reflected copy of the incident longitudinal packet.

This distinction is important for interpreting the delayed oscillatory sector. No detected probability is returning to the detector. The returning object is the undetected amplitude after it has been filtered by the spinor ABC. Since this component inherits the local boundary scale \(R=|\boldsymbol{\xi}|\), and since the harmonic transverse family samples \(R\sim \ell_\perp^{-1}\sim\sqrt{\omega}\), finite-guide memory converts the local branch reweighting into a delayed roof-flux contribution. The numerical coefficients of a finite-window fit such as \(A+B\sqrt{\omega}\) remain dependent on the source packet, detector parameter, guide length, and observation window. The robust point is that the spinor boundary makes the detector-present roof flux sensitive to the tangential momentum scale. A reduced flux-level bookkeeping formulation of this memory effect is given in the Supplementary Information, [17] Sec. S8; it is not an exact mode-resolved TDSE decomposition and is not used as a fit model.

5 Conclusion↩︎

We have shown that the spinor ABC is not a scalar absorbing sink but a local spin–momentum impedance. At the detecting roof, each tangential mode decomposes into two spin–momentum components \(\Pi_\pm W\), with boundary eigenbranches \(i\kappa\pm|\boldsymbol{\xi}|\). In a harmonic guide the roof trace samples \(R=|\boldsymbol{\xi}|\sim\ell_\perp^{-1}\sim\sqrt{\omega}\), so increasing transverse confinement directly changes the local detector-present boundary response. Equation 5 gives the local meaning of this filtering: after the scalar phase is removed, the auxiliary inward normal response weights the two spin–momentum components by \(e^{\mp R\epsilon}\). The \(e^{-R\epsilon}\Pi_+\) factor is an evanescent normal factor generated by the boundary symbol itself, not by propagation through a classically forbidden bulk region or a potential barrier. The complementary \(e^{R\epsilon}\Pi_-\) factor is the opposite branch of the same auxiliary continuation, not a gain channel. The physical detection statistic remains the roof flux at \(z=L\).

Since the spin-independent separable bulk first-pass flux is independent of \(\omega\), the observed confinement dependence is generated at the detecting boundary rather than by ordinary bulk propagation. Together with finite-guide memory of the undetected, branch-reweighted amplitude, the local scale \(R\sim\sqrt{\omega}\) organizes the numerical signatures in Fig. 1: suppression of the prompt roof flux, reduction of \(D_{20}\), and growth of the finite-window restricted mean.

Accordingly, the fit \[\mu^*(20;\omega)\simeq 4.084+0.638\sqrt{\omega}\] should be read only as a finite-window diagnostic for the specified prepared wave packet, detector parameter, guide geometry, and observation time. The constants are effective and setup dependent. The robust observable signature is the confinement-controlled deformation of the detector-present roof-flux distribution by a matrix-valued absorbing boundary, not a universal arrival-time law.

A direct physical realization of this response would require engineering an absorbing interface whose effective entrance impedance realizes the tangential branches \(i\kappa\pm|\boldsymbol{\xi}|\). The controls in the Supplemental Material support this specificity in two complementary ways. First, when the spinor ABC is retained, representative bulk spin perturbations, including Zeeman-type and spin–orbit terms, do not substantially remove the delayed sector. Second, when the spinor boundary is replaced by generic spin-dependent absorbing Hamiltonian layers, the spinor-ABC confinement trend is not robustly reproduced. The inverse-engineered layer approaches the response only when the branch impedances \(i\kappa\pm|\boldsymbol{\boldsymbol{\xi}}|\) are built in by construction; detailed controls, parameter scans, and physical scale estimates are given in the Supplementary Information [17], Secs. S9–S11.

The broader implication is that absorbing interfaces with internal degrees of freedom need not behave as passive scalar probability sinks. Even when the bulk Hamiltonian is spin independent and separable, the detector boundary can read tangential momentum through its local symbol and imprint that scale on click-time statistics. Thus detector-present click-time statistics can contain information about the boundary impedance, not only about the freely propagated incident packet.

Declarations and acknowledgments↩︎

5.0.0.1 Data and code availability.

Processed data underlying Fig. 1 and the Supplementary figures, representative density animations, and the Python scripts for the Crank–Nicolson/GMRES simulations, diagnostics, post-processing, and figure generation are available at https://github.com/jloOop/Boundary-Spin-Momentum-Filtering-Data. The repository includes parameters, reduced data, and reproducibility notes needed to reproduce the reported figures. Larger raw HPC outputs are omitted because of size; selected additional outputs are available from the author upon reasonable request. The solver follows the implementation described in Appendix A of Ref. [13].

5.0.0.2 Competing interests.

The author declares no competing interests.

5.0.0.3 Author contributions.

A.J. is the sole author of this work and conceived the study, developed the theoretical analysis, implemented and validated the simulations, analyzed the data, prepared the figures and repository, and wrote the manuscript and Supplementary Information.

5.0.0.4 Declaration of AI-assisted tools.

ChatGPT (OpenAI) was used for language polishing and for assistance with drafting, debugging, and checking Python scripts for numerical post-processing and figure preparation. The author independently reviewed and verified the code, calculations, figures, references, and manuscript text, and takes full responsibility for the analysis, results, and conclusions.

5.0.0.5 Acknowledgments.

The author thanks Roderich Tumulka for discussions, critical questions, and invaluable input. The numerical calculations used computing resources provided by PC2, NHR@ZIB, and bwForCluster Helix. The author acknowledges support by the state of Baden-Württemberg through bwHPC Cluster and the German Research Foundation (DFG) through grant INST 35/1597-1 FUGG.

References↩︎