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Casimir-Polder interaction between atoms and hollow-core fibers

B. Beverungen1,*, D. Reiche1, K. Busch1,2, and F. Intravaia1

  • *Contact author: bettina.beverungen@physik.hu-berlin.de

Phys. Rev. A 114, 032802 – Published 2 September, 2026

DOI: https://doi.org/10.1103/q15y-p7nj

Abstract

The Casimir-Polder force acts on polarizable particles due to quantum fluctuations of the electromagnetic field that are modified by the presence of material bodies. We investigate the Casimir-Polder interaction for atoms near cylindrical fibers with hollow cores. This geometry represents one of the archetypal configurations encountered in numerous experimental setups designed to control and manipulate atoms in fundamental and quantum technological applications. Specifically, we analyze how the interplay of both geometrical and material-related length scales characterize the interaction, emphasizing the impact of the shell thickness. We develop a flexible and fast-converging numerical scheme for evaluating the interaction over a wide range of atom-cylinder separations at both zero and finite temperature. Furthermore, we provide a detailed analytical investigation of how various material properties modify the Casimir-Polder potential. Finally, we analyze and discuss a number of limiting cases and compare numerical computations with corresponding analytical asymptotic expressions. In particular, in this geometry the Casimir-Polder potential is able to distinguish between an Ohmic and non-Ohmic description of conductors. One of the most significant outcomes of our work is that the shell thickness emerges as a useful parameter for controlling the interaction, opening avenues for both fundamental physics and applications in quantum technologies.

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References (122)

  1. W. E. Lamb and R. C. Retherford, Fine structure of the hydrogen atom by a microwave method, Phys. Rev. 72, 241 (1947).
  2. H. B. G. Casimir and D. Polder, The influence of retardation on the London-van der Waals forces, Phys. Rev. 73, 360 (1948).
  3. J. Reichel, W. Hänsel, and T. W. Hänsch, Atomic micromanipulation with magnetic surface traps, Phys. Rev. Lett. 83, 3398 (1999).
  4. J. Fortágh and C. Zimmermann, Magnetic microtraps for ultracold atoms, Rev. Mod. Phys. 79, 235 (2007).
  5. C. Henkel, in Atom Chips, edited by J. Reichel and V. Vuletić (Wiley, Weinheim, 2011), Chap. 5.
  6. M. Keil, O. Amit, S. Zhou, D. Groswasser, Y. Japha, and R. Folman, Fifteen years of cold matter on the atom chip: Promise, realizations, and prospects, J. Mod. Opt. 63, 1840 (2016).
  7. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber, Phys. Rev. Lett. 104, 203603 (2010).
  8. P. Solano, P. Barberis-Blostein, F. K. Fatemi, L. A. Orozco, and S. L. Rolston, Super-radiance reveals infinite-range dipole interactions through a nanofiber, Nat. Commun. 8, 1857 (2017).
  9. F. Le Kien, S. S. Hejazi, V. G. Truong, S. Nic Chormaic, and T. Busch, Chiral force of guided light on an atom, Phys. Rev. A 97, 063849 (2018).
  10. R. Pennetta, A. Glicenstein, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, Hybrid trapping of cold atoms with surface forces and blue-detuned evanescent light on a nanophotonic waveguide, arXiv:2509.17767.
  11. M. Langbecker, M. Noaman, N. Kjærgaard, F. Benabid, and P. Windpassinger, Rydberg excitation of cold atoms inside a hollow-core fiber, Phys. Rev. A 96, 041402(R) (2017).
  12. F. Le Kien, D. F. Kornovan, S. Nic Chormaic, and T. Busch, Repulsive Casimir-Polder potentials of low-lying excited states of a multilevel alkali-metal atom near an optical nanofiber, Phys. Rev. A 105, 042817 (2022).
  13. R. Fermani, S. Scheel, and P. L. Knight, Trapping cold atoms near carbon nanotubes: Thermal spin flips and Casimir-Polder potential, Phys. Rev. A 75, 062905 (2007).
  14. M. Gierling, P. Schneeweiss, G. Visanescu, P. Federsel, M. Häffner, D. P. Kern, T. E. Judd, A. Günther, and J. Fortágh, Cold-atom scanning probe microscopy, Nat. Nanotechnol. 6, 446 (2011).
  15. P. Schneeweiss, M. Gierling, G. Visanescu, D. P. Kern, T. E. Judd, A. Gunther, and J. Fortagh, Dispersion forces between ultracold atoms and a carbon nanotube, Nat. Nanotechnol. 7, 515 (2012).
  16. R. Folman, P. Krüger, J. Schmiedmayer, J. H. Denschlag, and C. Henkel, Microscopic atom optics: From wires to an atom chip, Adv. At Mol. Opt. Phys. 48, 263 (2002).
  17. D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator, Phys. Rev. Lett. 104, 143002 (2010).
  18. D. Hümmer, O. Romero-Isart, A. Rauschenbeutel, and P. Schneeweiss, Probing surface-bound atoms with quantum nanophotonics, Phys. Rev. Lett. 126, 163601 (2021).
  19. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Chiral quantum optics, Nature (London) 541, 473 (2017).
  20. Y. Meng, A. Dareau, P. Schneeweiss, and A. Rauschenbeutel, Near-ground-state cooling of atoms optically trapped 300 nm away from a hot surface, Phys. Rev. X 8, 031054 (2018).
  21. D. Lechner, R. Pennetta, M. Blaha, P. Schneeweiss, A. Rauschenbeutel, and J. Volz, Light-matter interaction at the transition between cavity and waveguide QED, Phys. Rev. Lett. 131, 103603 (2023).
  22. M. Boström and B. E. Sernelius, Van der Waals energy of an atom in the proximity of thin metal films, Phys. Rev. A 61, 052703 (2000).
  23. A. M. Contreras Reyes and C. Eberlein, Casimir-Polder interaction between an atom and a dielectric slab, Phys. Rev. A 80, 032901 (2009).
  24. G. Dufour, A. Gérardin, R. Guérout, A. Lambrecht, V. V. Nesvizhevsky, S. Reynaud, and A. Y. Voronin, Quantum reflection of antihydrogen from the Casimir potential above matter slabs, Phys. Rev. A 87, 012901 (2013).
  25. S. Y. Buhmann, Dispersion Forces I: Macroscopic Quantum Electrodynamics and Ground-State Casimir, Casimir–Polder and van der Waals Forces (Springer, Berlin, 2013), Vol. 247.
  26. S. Buhmann, Dispersion Forces II: Many-Body Effects, Excited Atoms, Finite Temperature and Quantum Friction (Springer, Berlin, 2013), Vol. 248.
  27. V. M. Nabutovskii, V. R. Belosludov, and A. M. Korotkikh, Interaction potential between small neutral particles and spherical or cylindrical surfaces, Zh. Eksp. Teor. Fiz. 77, 700 (1979) [Sov. Phys. JETP 50, 352 (1979)].
  28. A. M. Marvin and F. Toigo, Van der Waals interaction between a point particle and a metallic surface. I. Theory, Phys. Rev. A 25, 782 (1982).
  29. Y. S. Barash, Van der Waals interaction between thin conducting layers, Fiz. Tver. Tela 30, 2738 (1988) [Sov. Phys. Solid State 30, 1578 (1988)].
  30. Y. S. Barash and O. I. Notych, Long-range correlations in a liquid filling a thin slit or pore, Zh Eksp. Teor. Fiz. 98, 542 (1990) [Sov. Phys. JETP 71, 301 (1990)].
  31. Configurations where atoms are enclosed within a cylindrical structure are also of interest and have been previously considered in the literature. Despite some differences, a description of the system can rely on mathematical tools similar to those employed in this paper. See, for example, Refs. [121, 122].
  32. C. Eberlein, R. Zietal, Force on a neutral atom near conducting microstructures, Phys. Rev. A 75, 032516 (2007).
  33. C. Eberlein, R. Zietal, Retarded Casimir-Polder force on an atom near reflecting microstructures, Phys. Rev. A 80, 012504 (2009).
  34. V. B. Bezerra, E. R. Bezerra de Mello, G. L. Klimchitskaya, V. M. Mostepanenko, and A. A. Saharian, Exact Casimir–Polder potential between a particle and an ideal metal cylindrical shell and the proximity force approximation, Eur. Phys. J. C 71, 1614 (2011).
  35. H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
  36. Y. B. Zel'dovich, Theory of interaction between an atom and a metal, Zh. Eksp. Teor. Fiz. 5, 22 (1935), reprinted in Selected Works of Yakov Borisovich Zeldovich, Volume I: Chemical Physics and Hydrodynamics (Princeton University Press, Princeton, NJ, 1992), Chap. III.11.
  37. Y. S. Barash and A. A. Kyasov, Interaction potential for two filaments and for an atom interacting with a filament, Zh. Eksp. Teor. Fiz. 95, 69 (1989) [Sov. Phys. JETP 68, 39 (1989)].
  38. E. Stourm, M. Lepers, J. Robert, S. Nic Chormaic, K. Mølmer, and E. Brion, Spontaneous emission and energy shifts of a Rydberg rubidium atom close to an optical nanofiber, Phys. Rev. A 101, 052508 (2020).
  39. M. Boström and B. E. Sernelius, Fractional van der Waals interaction between thin metallic films, Phys. Rev. B 61, 2204 (2000).
  40. I. Pirozhenko and A. Lambrecht, Influence of slab thickness on the Casimir force, Phys. Rev. A 77, 013811 (2008).
  41. G. Schütz, A. Rembold, A. Pooch, S. Meier, P. Schneeweiss, A. Rauschenbeutel, A. Günther, W. T. Chang, I. S. Hwang, and A. Stibor, Biprism electron interferometry with a single atom tip source, Ultramicroscopy 141, 9 (2014).
  42. O. A. Schmidt, T. G. Euser, and P. S. Russell, Mode-based microparticle conveyor belt in air-filled hollow-core photonic crystal fiber, Opt. Express 21, 29383 (2013).
  43. G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. J. Russell, and R. Löw, Rydberg atoms in hollow-core photonic crystal fibres, Nat. Commun. 5, 4132 (2014).
  44. D. S. Bykov, O. A. Schmidt, T. G. Euser, and P. S. J. Russell, Flying particle sensors in hollow-core photonic crystal fibre, Nat. Photon. 9, 461 (2015).
  45. E. V. Blagov, G. L. Klimchitskaya, and V. M. Mostepanenko, Van der Waals interaction between microparticle and uniaxial crystal with application to hydrogen atoms and multiwall carbon nanotubes, Phys. Rev. B 71, 235401 (2005).
  46. Casimir Physics, Vol. 834, Lecture Notes in Physics, edited by D. Dalvit, P. Milonni, D. Roberts, and F. da Rosa (Springer, Berlin, 2011).
  47. F. Intravaia, C. Henkel, and M. Antezza, Fluctuation-Induced Forces Between Atoms and Surfaces: The Casimir-Polder Interaction in Casimir Physics, Vol. 834, Lecture Notes in Physics, edited by D. Dalvit, P. Milonni, D. Roberts, and F. da Rosa (Springer, Berlin, 2011), Chap. 11, pp. 345–391.
  48. G. L. Klimchitskaya and V. M. Mostepanenko, Current status of the problem of thermal Casimir force, Int. J. Mod. Phys. A 37, 2241002 (2022).
  49. C.-T. Tai, Dyadic Green functions in electromagnetic theory, IEEE Press Series on Electromagnetic Waves (IEEE Press, New York, 1994).
  50. J. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  51. L. Landau, E. Lifshitz, and L. Pitaevskii, Course of Theoretical Physics: Electrodynamics in Continuous Media, 2nd ed. (Butterworth-Heinmann, Oxford, 1980).
  52. H. J. H. Ma and J. F. Scott, Non-Ohmic variable-range hopping and resistive switching in SrTiO3 domain walls, Phys. Rev. Lett. 124, 146601 (2020).
  53. F. London and H. London, in Proceedings of the Royal Society of London, No 866 in Series A, Mathematical and Physical Sciences (1934-1990) (JSTOR, London, 1935), pp. 71–88.
  54. G. Bimonte, H. Haakh, C. Henkel, and F. Intravaia, Optical BCS conductivity at imaginary frequencies and dispersion energies of superconductors, J. Phys. A: Math. Theor. 43, 145304 (2010).
  55. M. S. Tomaš, Green function for multilayers: Light scattering in planar cavities, Phys. Rev. A 51, 2545 (1995).
  56. L. Li, P. Kooi, M. Leong, and T. Yeo, Analytic representation of scattering dyadic Green's functions' coefficients for cylindrically multilayered chiral media, J. Electromagn. Waves Appl. 9, 1207 (1995).
  57. L.-W. Li, M.-S. Leong, T.-S. Yeo, and P.-S. Kooi, Electromagnetic dyadic Green's functions in spectral domain for multilayered cylinders, J. Electromagn. Waves Appl. 14, 961 (2000).
  58. L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yee, Electromagnetic dyadic Green's function in spherically multilayered media, IEEE Trans. Microwave Theory Tech. 42, 2302 (1994).
  59. G. Mie, Beiträge zur optik trüber medien, speziell kolloidaler metallösungen, Ann. Phys. 330, 377 (1908).
  60. D. Tzarouchis and A. Sihvola, Light scattering by a dielectric sphere: Perspectives on the Mie resonances, Appl. Sci. 8, 184 (2018).
  61. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  62. M. Abramowitz and I. Stengun, Handbook of Mathematical Functions (Dover, New York, 1971).
  63. Our notation slightly differs from the literature [49], but is more suitable for our purposes.
  64. P. K. Rekdal, S. Scheel, P. L. Knight, and E. A. Hinds, Thermal spin flips in atom chips, Phys. Rev. A 70, 013811 (2004).
  65. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, 2007).
  66. G. Xu and S. G. Johnson, Modified discrete Laguerre polynomials for efficient computation of exponentially bounded Matsubara sums, arXiv:2101.01655.
  67. S. Engblom, Gaussian quadratures with respect to discrete measures, Technical Report No. 2006-007, Department of Information Technology, Uppsala University.
  68. H. Monien, Gaussian quadrature for sums: A rapidly convergent summation scheme, Math. Comput. 79, 857 (2010).
  69. D. E. Amos, Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order, ACM Trans. Math. Softw. 12, 265 (1986).
  70. D. A. Steck, Rubidium 87 D line data, available online at http://steck.us/alkalidata (revision 2.1.5, 13) (2015).
  71. T. Pflug, M. Olbrich, and A. Horn, Electron dynamics in fused silica after strong field laser excitation detected by spectroscopic imaging pump-probe ellipsometry, Phys. Rev. B 106, 014307 (2022).
  72. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2: Theory of the Condensed State (Pergamon, Oxford, 1980).
  73. I. Brevik, J. B. Aarseth, J. S. Høye, and K. A. Milton, Temperature dependence of the Casimir effect, Phys. Rev. E 71, 056101 (2005).
  74. A. W. Rodriguez, F. Capasso, and S. G. Johnson, The Casimir effect in microstructured geometries, Nat. Photon. 5, 211 (2011).
  75. J. Sipe, The dipole antenna problem in surface physics: A new approach, Surf. Sci. 105, 489 (1981).
  76. M. Oelschläger, K. Busch, and F. Intravaia, Nonequilibrium atom-surface interaction with lossy multilayer structures, Phys. Rev. A 97, 062507 (2018).
  77. R. O. Behunin, D. A. R. Dalvit, R. S. Decca, C. Genet, I. W. Jung, A. Lambrecht, A. Liscio, D. López, S. Reynaud, G. Schnoering, et al., Kelvin probe force microscopy of metallic surfaces used in Casimir force measurements, Phys. Rev. A 90, 062115 (2014).
  78. S. Kalusniak, S. Sadofev, and F. Henneberger, ZnO as a tunable metal: New types of surface plasmon polaritons, Phys. Rev. Lett. 112, 137401 (2014).
  79. B. Das, Obtaining Wien's displacement law from Planck's law of radiation, The Physics Teacher 40, 148 (2002).
  80. B. W. Williams, A specific mathematical form for Wien's displacement law as vmax/T = constant, J. Chem. Educ. 91, 623 (2014).
  81. K. Asheichyk and M. Krüger, Radiative heat transfer with a cylindrical waveguide decays logarithmically slow, Phys. Rev. Lett. 129, 170605 (2022).
  82. J.-Y. Wang, Y. Zhang, C.-L. Zhou, Y. Shuai, and H.-L. Yi, Enhanced radiative heat transfer via propagating surface modes in a dielectric nanowire, Phys. Rev. B 109, 075423 (2024).
  83. J.-Y. Wang, Y. Zhang, X.-P. Luo, M. Antezza, and H.-L. Yi, Twist-induced near-field radiative thermal regulator assisted by cylindrical surface modes, Int. J. Heat Mass Transfer 240, 126645 (2025).
  84. W. Zhao, Á. R. Echarri, A. Eljarrat, H. C. Nerl, T. Kiel, B. Haas, H. Halim, Y. Lu, C. T. Koch, and K. Busch, Time-domain study of surface plasmon polariton propagation in silver nanowires, Phys. Rev. B 113, 085425 (2026).
  85. Á. R. Echarri, W. Zhao, K. Busch, and F. J. García de Abajo, Relativistic electron energy-loss spectroscopy in cylindrical waveguides and holes, Phys. Rev. B 111, 205436 (2025).
  86. R. F. Egerton, Electron energy-loss spectroscopy in the TEM, Rev. Mod. Phys. 72, 016502 (2009).
  87. R. Ruppin, Extinction properties of thin metallic nanowires, Opt. Commun. 190, 205 (2001).
  88. G. V. Dedkov, Van der Waals interactions of moving particles with surfaces of cylindrical geometry, Universe 7, 106 (2021).
  89. D. Reiche, F. Intravaia, and K. Busch, Wading through the void: Exploring quantum friction and nonequilibrium fluctuations, APL Photonics 7, 030902 (2022).
  90. G. Wegner, D.-N. Huynh, N. A. Mortensen, F. Intravaia, and K. Busch, Halevi's extension of the Euler-Drude model for plasmonic systems, Phys. Rev. B 107, 115425 (2023).
  91. J. Fiedler, C. Persson, and S. Y. Buhmann, Spectroscopy of nanoparticles without light, Phys. Rev. Appl. 13, 014025 (2020).
  92. W. Yan, M. Wubs, and N. A. Mortensen, Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity, Phys. Rev. B 86, 205429 (2012).
  93. F. Monticone, N. A. Mortensen, A. I. Fernández-Domínguez, Y. Luo, X. Zheng, C. Tserkezis, J. B. Khurgin, T. V. Shahbazyan, A. J. Chaves, N. M. R. Peres, et al., Nonlocality in photonic materials and metamaterials: Roadmap, Opt. Mater. Express 15, 1544 (2025).
  94. R. Esquivel, C. Villarreal, and W. L. Mochan, Exact surface impedance formulation of the Casimir force: Application to spatially dispersive metals, Phys. Rev. A 68, 052103 (2003).
  95. A. M. Contreras-Reyes and W. L. Mochán, Surface screening in the Casimir force, Phys. Rev. A 72, 034102 (2005).
  96. V. B. Svetovoy, Application of the Lifshitz theory to poor conductors, Phys. Rev. Lett. 101, 163603 (2008).
  97. D. Reiche, K. Busch, and F. Intravaia, Quantum thermodynamics of overdamped modes in local and spatially dispersive materials, Phys. Rev. A 101, 012506 (2020).
  98. P. T. Kristensen, B. Beverungen, F. Intravaia, and K. Busch, High-accuracy Casimir-Polder force calculations using the discontinuous Galerkin time-domain method, Phys. Rev. B 108, 205424 (2023).
  99. H. Zhang and K. Ding, Nanoscale Casimir force softening originated from quantum surface responses, Phys. Rev. Lett. 134, 146901 (2025).
  100. P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces, Phys. Rev. B 77, 035431 (2008).
  101. H. R. Haakh and C. Henkel, Magnetic near fields as a probe of charge transport in spatially dispersive conductors, Eur. Phys. J. B 85, 46 (2012).
  102. F. Intravaia and K. Busch, Fluorescence in nonlocal dissipative periodic structures, Phys. Rev. A 91, 053836 (2015).
  103. R. Schmidt and S. Scheel, Local density of states near spatially dispersive nanospheres, Phys. Rev. A 93, 033804 (2016).
  104. D. Reiche, D. A. R. Dalvit, K. Busch, and F. Intravaia, Spatial dispersion in atom-surface quantum friction, Phys. Rev. B 95, 155448 (2017).
  105. D. Reiche, M. Oelschläger, K. Busch, and F. Intravaia, Extended hydrodynamic description for nonequilibrium atom-surface interactions, J. Opt. Soc. Am. B 36, C52 (2019).
  106. K. A. Milton, P. Parashar, N. Pourtolami, and I. Brevik, Casimir-Polder repulsion: Polarizable atoms, cylinders, spheres, and ellipsoids, Phys. Rev. D 85, 025008 (2012).
  107. D. T. Alves, L. Queiroz, E. C. M. Nogueira, and N. M. R. Peres, Curvature-induced repulsive effect on the lateral Casimir-Polder–van der Waals force, Phys. Rev. A 107, 062821 (2023).
  108. M. T. H. Reid, A. W. Rodriguez, J. White, and S. G. Johnson, Efficient computation of Casimir interactions between arbitrary 3D objects, Phys. Rev. Lett. 103, 040401 (2009).
  109. T. Emig and G. Bimonte, Multiple scattering expansion for dielectric media: Casimir effect, Phys. Rev. Lett. 130, 200401 (2023).
  110. G. Bimonte and T. Emig, Casimir and Casimir-Polder interactions for magneto-dielectric materials: Surface scattering expansion, Phys. Rev. A 108, 052807 (2023).
  111. G. Bimonte and T. Emig, Complex orientation dependence of Casimir-Polder interaction induced by curvature and optical properties of the surface and the surrounding medium, Phys. Rev. B 112, 205405 (2025).
  112. J. Segura, Monotonicity properties for ratios and products of modified Bessel functions and sharp trigonometric bounds, Results Math. 76, 221 (2021).
  113. Á. Baricz, Bounds for modified Bessel functions of the first and second kinds, Proc. Edinburgh Math. Soc. 53, 575 (2010).
  114. D. E. Amos, Computation of modified Bessel functions and their ratios, Math. Comput. 28, 239 (1974).
  115. Z.-H. Yang and S.-Z. Zheng, The monotonicity and convexity for the ratios of modified Bessel functions of the second kind and applications, Proc. Am. Math. Soc. 145, 2943 (2017).
  116. J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev. 59, 65 (2017).
  117. D. Huybrechs, On the Fourier extension of nonperiodic functions, SIAM J. Numer. Anal. 47, 4326 (2010).
  118. G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature rules, Math. Comput. 23, 221 (1969).
  119. A. C. Genz and A. A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, J. Comput. Appl. Math. 6, 295 (1980).
  120. S. G. Johnson, The HCubature.jl package for multi-dimensional adaptive integration in Julia, https://github.com/JuliaMath/HCubature.jl.
  121. H. R. Haakh and S. Scheel, Modified and controllable dispersion interaction in a one-dimensional waveguide geometry, Phys. Rev. A 91, 052707 (2015).
  122. E. K. Osestad, P. Parviainen, and J. Fiedler, A novel gas sensing principle based on quantum fluctuations, EPJ Quantum Technol. 12, 37 (2025).

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