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Detection time distribution predicted using absorbing boundary conditions and imaginary potentials

Alireza Jozani* and Roderich Tumulka†

  • *Contact author: alireza.jozani@uni-tuebingen.de; jozani.alireza@gmail.com
  • †Contact author: roderich.tumulka@uni-tuebingen.de
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Phys. Rev. Research 8, 033072 – Published 17 July, 2026

DOI: https://doi.org/10.1103/g4wn-ctbl

Abstract

There are several inequivalent proposals in the literature for how to compute the probability distribution of the time that a detector registers for the arrival of a quantum particle. For three of these proposals, based on two kinds of absorbing boundary conditions and imaginary potentials, we compute the predicted distribution for an experimental setup involving a single nonrelativistic quantum particle with spin 0 or 1/2 in a waveguide along the z axis with the detector waiting downstream. We find that the distribution shows signs of partial reflection of the wave function off of the detector; for a spin-1/2 wave function, it is independent of the initial spin orientation for the parameters tested but does depend, for boundary conditions coupling to the spin, on the width of the waveguide. We also compare our predictions with the competing ones of Das and Dürr [Sci. Rep. 9, 2242 (2019)].

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References (56)

  1. G. R. Allcock, The time of arrival in quantum mechanics. II. The individual measurement, Ann. Phys. (NY) 53, 286 (1969).
  2. J. G. Muga and C. R. Leavens, Arrival time in quantum mechanics, Phys. Rep. 338, 353 (2000).
  3. N. Vona, G. Hinrichs, and D. Dürr, What does one measure when one measures the arrival time of a quantum particle? Phys. Rev. Lett. 111, 220404 (2013).
  4. Y. Aharonov and D. Bohm, Time in the quantum theory and the uncertainty relation for time and energy, Phys. Rev. 122, 1649 (1961).
  5. J. Kijowski, On the time operator in quantum mechanics and the Heisenberg uncertainty relation for energy and time, Rep. Math. Phys. 6, 361 (1974).
  6. R. Tumulka, Distribution of the time at which an ideal detector clicks, Ann. Phys. (NY) 442, 168910 (2022).
  7. S. Das and D. Dürr, Arrival time distributions of spin-1/2 particles, Sci. Rep. 9, 2242 (2019).
  8. M. Daumer, D. Dürr, S. Goldstein, and N. Zanghì, On the quantum probability flux through surfaces, J. Stat. Phys. 88, 967 (1997); reprinted in Ref. [32].
  9. R. Kaimal and R. Tumulka, Scattering cross section formula derived from macroscopic model of detectors, arXiv:2601.01625.
  10. R. Tumulka, Foundations of Quantum Mechanics (Springer, Heidelberg, 2022).
  11. S. Goldstein, R. Tumulka, and N. Zanghì, On the spin dependence of detection times and the nonmeasurability of arrival times, Sci. Rep. 14, 3775 (2024).
  12. S. Goldstein, R. Tumulka, and N. Zanghì, Arrival times versus detection times, Found. Phys. 54, 63 (2024).
  13. D. Dürr, S. Goldstein, and N. Zanghì, Quantum equilibrium and the role of operators as observables in quantum theory, J. Stat. Phys. 116, 959 (2004); reprinted in Ref. [32].
  14. C. Beck and D. Lazarovici, The POVM theorem in Bohmian mechanics, Entropy 27, 391 (2025).
  15. N. Gisin, Stochastic quantum dynamics and relativity, Helv. Phys. Acta 62, 363 (1989).
  16. P. H. Eberhard, Bell's theorem and the different concepts of locality, Nuovo Cimento B 46, 392 (1978).
  17. S. Tahvildar-Zadeh and S. Zhou, Detection time of Dirac particles in one space dimension, in Physics and the Nature of Reality: Essays in Memory of Detlef Dürr, edited by A. Bassi, S. Goldstein, R. Tumulka, and N. Zanghì (Springer, Heidelberg, 2024), pp. 187–201.
  18. C. Peters, S. Tahvildar-Zadeh, and R. Tumulka, No-signaling theorem covering also absorbing boundary conditions and imaginary potentials (unpublished).
  19. R. Werner, Arrival time observables in quantum mechanics, Ann. Inst. Henri Poincaré Phys. Théor. 47, 429 (1987).
  20. R. Tumulka, Detection time distribution for the Dirac equation, arXiv:1601.04571.
  21. R. Tumulka, Absorbing boundary condition as limiting case of imaginary potentials, Commun. Theor. Phys. 75, 015103 (2023).
  22. V. Dubey, C. Bernardin, and A. Dhar, Quantum dynamics under continuous projective measurements: Non-Hermitian description and the continuum space limit, Phys. Rev. A 103, 032221 (2021).
  23. L. Frolov, S. Teufel, and R. Tumulka, Existence of Schrödinger evolution with absorbing boundary condition, Math. Phys. Anal. Geom. 28, 27 (2025).
  24. L. Frolov, Detecting screens modeled by Schrödinger operators that generate C0 contraction semigroups, Rev. Math. Phys. 2550036 (2025), doi: 10.1142/S0129055X25500369; arXiv:2506.00231.
  25. H. A. Bethe, A continuum theory of the compound nucleus, Phys. Rev. 57, 1125 (1940).
  26. R. Tumulka, Detection time distribution for several quantum particles, Phys. Rev. A 106, 042220 (2022).
  27. J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Math. Proc. Cambridge Philos. Soc. 43, 50 (1947).
  28. A. Jozani, Spin-momentum impedance and filtering by a spin-coupled absorbing boundary condition, arXiv:2606.25650v1.
  29. Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, and W. G. Unruh, Measurement of time of arrival in quantum mechanics, Phys. Rev. A 57, 4130 (1998).
  30. J. J. Halliwell, Arrival times in quantum theory from an irreversible detector model, Prog. Theor. Phys. 102, 707 (1999).
  31. D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables I, Phys. Rev. 85, 166 (1952).
  32. D. Dürr, S. Goldstein, and N. Zanghì, Quantum Physics Without Quantum Philosophy (Springer, Berlin, 2013).
  33. It might seem that Eq. (32) implies that when R=XD and Q(τ)=XWID are integrated out and only T=TD and τ=TWID are considered, their distributions coincide; that is not the case because, in the presence of the spinless ABC (3), jzP at z=L can be negative, so that the f factor in Eq.  (38) is not always 1, so Eq.  (38) integrated over r is not the same expression as Eq.  (32), and Eq.  (33) not the same as the density of τ=TWID.
  34. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2004).
  35. E. Hairer and G. Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, 2nd ed. (Springer, Berlin, 1996), Vol. 14.
  36. W. van Dijk and F. M. Toyama, Accurate numerical solutions of the time-dependent Schrödinger equation, Phys. Rev. E 75, 036707 (2007).
  37. M. D. Feit, J. A. Fleck, Jr., and A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47, 412 (1982).
  38. D. Kosloff and R. Kosloff, A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics, J. Comput. Phys. 52, 35 (1983).
  39. A. Jozani and R. Tumulka, Numerical–SchrodingerEquation-and-BohmianTrajectories, GitHub repository, 2026, https://github.com/jloOop/Numerical–SchrodingerEquation-and-BohmianTrajectories.
  40. W. Brenig and R. Haag, Allgemeine Quantentheorie der Stoßprozesse, Fortschr. Phys. 7, 183 (1959).
  41. R. G. Newton, Scattering Theory of Waves and Particles (Springer, New York, NY, 1982).
  42. W. Cavendish and S. Das, Absorbing detectors meet scattering theory, Phys. Rev. A 112, 062217 (2025).
  43. J. P. Palao, J. G. Muga, and R. Sala, Composite absorbing potentials, Phys. Rev. Lett. 80, 5469 (1998).
  44. D. E. Manolopoulos, Derivation and reflection properties of a transmission-free absorbing potential, J. Chem. Phys. 117, 9552 (2002)
  45. M. Visser, Some general bounds for 1-D scattering, Phys. Rev. A 59, 427 (1999).
  46. P. Garrido, S. Goldstein, J. Lukkarinen, and R. Tumulka, Paradoxical reflection in quantum mechanics, Am. J. Phys. 79, 1218 (2011).
  47. For the reasons pointed out in [33], Eq. (32) does not imply that the two distributions coincide.
  48. D. Heine, W. Rohringer, D. Fischer, M. Wilzbach, T. Raub, S. Loziczky, X. Liu, S. Groth, B. Hessmo, and J. Schmiedmayer, A single atom detector integrated on an atom chip: Fabrication, characterization and application, New J. Phys. 12, 095005 (2010).
  49. A. Ruschhaupt, J. G. Muga, and G. C. Hegerfeldt, Detector models for the quantum time of arrival, in Time in Quantum Mechanics: Lecture Notes in Physics 789, edited by J. G. Muga, A. Ruschhaupt, and A. del Campo, (Springer, Heidelberg, 2009), Vol. 2, pp. 65–96.
  50. J. G. Muga, J. P. Palao, B. Navarro, and I. L. Egusquiza, Complex absorbing potentials, Phys. Rep. 395, 357 (2004).
  51. B. Navarro, I. L. Egusquiza, J. G. Muga, and G. C. Hegerfeldt, Optimal atomic detection of ultracold atoms by control of detuning and spatial dependence of laser intensity, J. Phys. B: At. Mol. Opt. Phys. 36, 3899 (2003).
  52. J. A. Damborenea, I. L. Egusquiza, G. C. Hegerfeldt, and J. G. Muga, On atomic time-of-arrival measurements with a laser of finite beam width, J. Phys. B: At. Mol. Opt. Phys. 36, 2657 (2003).
  53. A. Haase, B. Hessmo, and J. Schmiedmayer, Detecting magnetically guided atoms with an optical cavity, Opt. Lett. 31, 268 (2006).
  54. Here, “lexicographic” means we sweep z fastest, then y, and then x.
  55. Here, the predictor displacement is a first guess of where the particle will go: It is the first RK2 substep ΔQpred:=Qn+12−Qn=Δt2k̃1. The corrector displacement is a refined version of that guess and is given by the second substep ΔQcorr:=Qn+1−Qn=Δtk̃2. Because |k̃|≤vmax by construction, we have |ΔQpred|≤12cflhmin and |ΔQcorr|≤cflhmin.
  56. Equivalently, one can view the capping as introducing an adaptive effective stage time Δteff=min{Δtstage,cflhmin/|k|}, which preserves the guidance direction and only shortens the stage when the proposed displacement would exceed a fraction of a cell. Since v(·,t) is frozen over [tn,tn+1], reducing Δteff changes only the numerical step length, not the underlying physics.

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Phys. Rev. Research 8, 033072– Published 17 July, 2026
  • Received 10 April 2026
  • Accepted 25 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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