Quantum stick-slip motion in nanoscaled friction
Summarize this article with:
AbstractFriction in atomistic systems is usually described by the classical Prandtl-Tomlinson model suitable for capturing the dragging force of a nanoparticle in a periodic potential. Here we consider the quantum mechanical version of this model in which the dissipation is facilitated by releasing heat to an external bath reservoir. The time evolution of the system is captured with the Liouville-von Neumann equation through the density matrix of the system in the Markov approximation. We examine several kinetic and dissipative properties of the nanoparticle by delineating classical vs quantum mechanical effects. We find that the Landau-Zener tunneling is a key factor in the overall reduction of the frictional dissipation when compared to the classical motion in which such tunneling is absent. Other regimes of motion, controlled by the corrugation parameter and other properties, are also found. This in-depth study analyzes the interplay between velocity, strength of interaction, and temperature to control the frictional force and provide useful guidelines for experimental data interpretation.Featured image: Dissipative friction of a nanoparticle moving above a periodic chain of atomsPopular summaryThis study examines quantum mechanical and classical aspects of contact nanoscale friction of a nanoparticle above a periodic chain of atoms. The frictional process is interpretated as a dissipative mechanism as a result of heat exchange with the environment. Our theory and subsequent analysis examines several dynamic properties, including the lateral frictional force which is accessible experimentally. Particular emphasis is given on the emerging Landau-Zener tunneling, a quantum process that allows particles to cross energy gaps in their eigenstates and reduces frictional dissipation relative to classical predictions. Our study puts forward a comprehensive theory suitable for providing theoretical guidance for guiding and interpreting experimental results in nanomechanics and related disciplines.► BibTeX data@article{Le2026quantumstickslip, doi = {10.22331/q-2026-04-16-2070}, url = {https://doi.org/10.22331/q-2026-04-16-2070}, title = {Quantum stick-slip motion in nanoscaled friction}, author = {Le, Dai-Nam and Rodriguez-Lopez, Pablo and Woods, Lilia M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2070}, month = apr, year = {2026} }► References [1] See Supplementary Information for detailed derivations of coupling potential between the nanoparticle and the atomic chain, matrix elements of quantum Prandtl-Tomlison Hamiltonian in moving harmonic oscillator basis set, numerical calculation for quasi-static eigenenergies and eigenstates, quantum evolution with and without external bath reservoir, classical evolution with and without external bath reservoir; detail explainations for tunneling and slipping of quantum and classical trajectories and an animation (GIF) of quantum and classical motions; see also Refs. gradshteyn2014table, Wilkie2005 therein. [2] Clara M. Almeida, Rodrigo Prioli, Benjamin Fragneaud, Luiz Gustavo Cançado, Ricardo Paupitz, Douglas S. Galvão, Marcelo De Cicco, Marcos G. Menezes, Carlos A. Achete, and Rodrigo B. Capaz. Giant and Tunable Anisotropy of Nanoscale Friction in Graphene. Sci. Rep., 6 (1): 31569, aug 2016. ISSN 2045-2322. 10.1038/srep31569. URL http://dx.doi.org/10.1038/srep31569. https://doi.org/10.1038/srep31569 [3] Luca Arceci, Simone Barbarino, Rosario Fazio, and Giuseppe E. Santoro. Dissipative Landau-Zener problem and thermally assisted Quantum Annealing. Phys. Rev. B, 96: 054301, Aug 2017. 10.1103/PhysRevB.96.054301. URL https://doi.org/10.1103/PhysRevB.96.054301. https://doi.org/10.1103/PhysRevB.96.054301 [4] Ranjan Kumar Barik and Lilia M Woods. Frictional Properties of Two-Dimensional Materials: Data-Driven Machine Learning Predictive Modeling. ACS Appl. Mater. Interfaces, 16 (30): 40149–40159, 2024. 10.1021/acsami.4c05532. URL https://doi.org/10.1021/acsami.4c05532. https://doi.org/10.1021/acsami.4c05532 [5] L. Bartels, G. Meyer, and K.-H. Rieder. Basic Steps of Lateral Manipulation of Single Atoms and Diatomic Clusters with a Scanning Tunneling Microscope Tip. Phys. Rev. Lett., 79: 697–700, Jul 1997. 10.1103/PhysRevLett.79.697. URL https://doi.org/10.1103/PhysRevLett.79.697. https://doi.org/10.1103/PhysRevLett.79.697 [6] Eric Bonvin, Louisiane Devaud, Massimiliano Rossi, Andrei Militaru, Lorenzo Dania, Dmitry S. Bykov, Markus Teller, Tracy E. Northup, Lukas Novotny, and Martin Frimmer. Hybrid paul-optical trap with large optical access for levitated optomechanics. Phys. Rev. Res., 6: 043129, Nov 2024. 10.1103/PhysRevResearch.6.043129. URL https://doi.org/10.1103/PhysRevResearch.6.043129. https://doi.org/10.1103/PhysRevResearch.6.043129 [7] D. Bouwmeester, G. P. Karman, C. A. Schrama, and J. P. Woerdman. Observation of interference in transitions due to local geometric phases. Phys. Rev. A, 53: 985–989, Feb 1996. 10.1103/PhysRevA.53.985. URL https://doi.org/10.1103/PhysRevA.53.985. https://doi.org/10.1103/PhysRevA.53.985 [8] S. Cahangirov, C. Ataca, M. Topsakal, H. Sahin, and S. Ciraci. Frictional Figures of Merit for Single Layered Nanostructures. Phys. Rev. Lett., 108: 126103, Mar 2012. 10.1103/PhysRevLett.108.126103. URL https://doi.org/10.1103/PhysRevLett.108.126103. https://doi.org/10.1103/PhysRevLett.108.126103 [9] A. O. Caldeira and A. J. Leggett. Influence of Dissipation on Quantum Tunneling in Macroscopic Systems. Phys. Rev. Lett., 46: 211–214, Jan 1981. 10.1103/PhysRevLett.46.211. URL https://doi.org/10.1103/PhysRevLett.46.211. https://doi.org/10.1103/PhysRevLett.46.211 [10] Ian Counts, Dorian Gangloff, Alexei Bylinskii, Joonseok Hur, Rajibul Islam, and Vladan Vuletić. Multislip Friction with a Single Ion. Phys. Rev. Lett., 119: 043601, Jul 2017. 10.1103/PhysRevLett.119.043601. URL https://doi.org/10.1103/PhysRevLett.119.043601. https://doi.org/10.1103/PhysRevLett.119.043601 [11] Diem Thi-Xuan Dang, Dai-Nam Le, and Lilia M. Woods. Twisting in $h$-BN bilayers and their angle-dependent properties. Phys. Rev. B, 112: 085102, Aug 2025a. 10.1103/wzz2-pszx. URL https://doi.org/10.1103/wzz2-pszx. https://doi.org/10.1103/wzz2-pszx [12] Diem Thi-Xuan Dang, Dai-Nam Le, and Lilia M. Woods. Dissecting van der Waals interactions with density functional theory – Wannier-basis approach. Comput. Phys. Commun., 310: 109525, 2025b. ISSN 0010-4655. 10.1016/j.cpc.2025.109525. URL https://doi.org/10.1016/j.cpc.2025.109525. https://doi.org/10.1016/j.cpc.2025.109525 [13] Thomas. Dittrich, Peter Hänggi, Gert-Ludwig Ingold, Bernhard Krammer, Gerd Schön, and Wilhelm Zwerger. Quantum transport and dissipation. Wiley-VCH, Weinheim, Germany, 1998. ISBN 3527292616. [14] M. Belén Farías, Fernando C. Lombardo, Alejandro Soba, Paula I. Villar, and Ricardo S. Decca. Towards detecting traces of non-contact quantum friction in the corrections of the accumulated geometric phase. npj Quantum. Inf., 6 (1), February 2020. ISSN 2056-6387. URL http://dx.doi.org/10.1038/s41534-020-0252-x. https://doi.org/10.1038/s41534-020-0252-x [15] Dorian Gangloff, Alexei Bylinskii, Ian Counts, Wonho Jhe, and Vladan Vuletić. Velocity tuning of friction with two trapped atoms. Nat. Phys., 11: 915–919, 2015. 10.1038/NPHYS3459. URL https://doi.org/10.1038/nphys3459. https://doi.org/10.1038/NPHYS3459 [16] S. Gasparinetti, P. Solinas, and J. P. Pekola. Geometric Landau-Zener Interferometry. Phys. Rev. Lett., 107: 207002, Nov 2011. 10.1103/PhysRevLett.107.207002. URL https://doi.org/10.1103/PhysRevLett.107.207002. https://doi.org/10.1103/PhysRevLett.107.207002 [17] Enrico Gnecco, Raphael Roth, and Alexis Baratoff. Analytical expressions for the kinetic friction in the Prandtl-Tomlinson model. Phys. Rev. B, 86: 035443, Jul 2012. 10.1103/PhysRevB.86.035443. URL https://doi.org/10.1103/PhysRevB.86.035443. https://doi.org/10.1103/PhysRevB.86.035443 [18] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of integrals, series, and products. Academic press, 2014. 10.1016/C2010-0-64839-5. URL https://doi.org/10.1016/C2010-0-64839-5. https://doi.org/10.1016/C2010-0-64839-5 [19] Zhongkai Huang and Yang Zhao. Dynamics of dissipative Landau-Zener transitions. Phys. Rev. A, 97: 013803, Jan 2018. 10.1103/PhysRevA.97.013803. URL https://doi.org/10.1103/PhysRevA.97.013803. https://doi.org/10.1103/PhysRevA.97.013803 [20] Modi Ke, Dai-Nam Le, and Lilia M. Woods. Casimir interactions between two parallel graphene sheets carrying steady-state drift currents. Phys. Rev. B, 113: 075414, Feb 2026. 10.1103/84n7-1r3b. URL https://doi.org/10.1103/84n7-1r3b. https://doi.org/10.1103/84n7-1r3b [21] Florian R. Krajewski and Martin H. Müser. Quantum Creep and Quantum-Creep Transitions in 1D Sine-Gordon Chains. Phys. Rev. Lett., 92: 030601, Jan 2004. 10.1103/PhysRevLett.92.030601. URL https://doi.org/10.1103/PhysRevLett.92.030601. https://doi.org/10.1103/PhysRevLett.92.030601 [22] Florian R. Krajewski and Martin H. Müser. Quantum dynamics in the highly discrete, commensurate Frenkel Kontorova model: A path-integral molecular dynamics study. J. Chem. Phys., 122 (12): 124711, 03 2005. 10.1063/1.1869392. URL https://doi.org/10.1063/1.1869392. https://doi.org/10.1063/1.1869392 [23] L. Landau. Zur Theorie der Energieubertragung. II. Physikalische Zeitschrift der Sowjetunion, 2: 46–51, 1932. [24] Dai-Nam Le, Ngoc-Tram D. Hoang, and Van-Hoang Le. Exact analytical solutions of the Schrödinger equation for a two dimensional purely sextic double-well potential. J. Math. Phys., 59 (3): 032101, 2018. 10.1063/1.4997532. URL https://doi.org/10.1063/1.4997532. https://doi.org/10.1063/1.4997532 [25] Dai-Nam Le, Pablo Rodriguez-Lopez, and Lilia M. Woods. Nonlinear effects in manybody van der waals interactions. Phys. Rev. Res., 6: 013289, Mar 2024. 10.1103/PhysRevResearch.6.013289. URL https://doi.org/10.1103/PhysRevResearch.6.013289. https://doi.org/10.1103/PhysRevResearch.6.013289 [26] Xin-Z. Liu, Zhijiang Ye, Yalin Dong, Philip Egberts, Robert W. Carpick, and Ashlie Martini. Dynamics of Atomic Stick-Slip Friction Examined with Atomic Force Microscopy and Atomistic Simulations at Overlapping Speeds. Phys. Rev. Lett., 114: 146102, Apr 2015. 10.1103/PhysRevLett.114.146102. URL https://doi.org/10.1103/PhysRevLett.114.146102. https://doi.org/10.1103/PhysRevLett.114.146102 [27] Fernando C. Lombardo and Paula I. Villar. Geometric phases in open systems: A model to study how they are corrected by decoherence. Phys. Rev. A, 74: 042311, Oct 2006. 10.1103/PhysRevA.74.042311. URL https://doi.org/10.1103/PhysRevA.74.042311. https://doi.org/10.1103/PhysRevA.74.042311 [28] Fernando C. Lombardo, Ricardo S. Decca, Ludmila Viotti, and Paula I. Villar. Detectable Signature of Quantum Friction on a Sliding Particle in Vacuum. Adv. Quantum. Technol., 4 (5): 2000155, 2021. URL https://doi.org/10.1002/qute.202000155. https://doi.org/10.1002/qute.202000155 [29] Gabriele Losi, Omar Chehaimi, and M. Clelia Righi. Tribchem: A software for the first-principles, high-throughput study of solid interfaces and their tribological properties. J. Chem. Theory Comput., 19 (15): 5231–5241, 2023. 10.1021/acs.jctc.3c00459. URL https://doi.org/10.1021/acs.jctc.3c00459. https://doi.org/10.1021/acs.jctc.3c00459 [30] E Meyer, T Gyalog, R M Overney, and K Dransfeld. Nanoscience: Friction and Rheology on the Nanometer Scale. World Scientific, 1998. 10.1142/3026. URL https://doi.org/10.1142/3026. https://doi.org/10.1142/3026 [31] Martin H. Müser. Velocity dependence of kinetic friction in the Prandtl-Tomlinson model. Phys. Rev. B, 84: 125419, Sep 2011. 10.1103/PhysRevB.84.125419. URL https://doi.org/10.1103/PhysRevB.84.125419. https://doi.org/10.1103/PhysRevB.84.125419 [32] Nicholas A. Peters, Tzu-Chieh Wei, and Paul G. Kwiat. Mixed-state sensitivity of several quantum-information benchmarks. Phys. Rev. A, 70: 052309, Nov 2004. 10.1103/PhysRevA.70.052309. URL https://doi.org/10.1103/PhysRevA.70.052309. https://doi.org/10.1103/PhysRevA.70.052309 [33] V. L. Pokrovsky and N. A. Sinitsyn. Landau-Zener transitions in a linear chain. Phys. Rev. B, 65: 153105, Apr 2002. 10.1103/PhysRevB.65.153105. URL https://doi.org/10.1103/PhysRevB.65.153105. https://doi.org/10.1103/PhysRevB.65.153105 [34] L. Prandtl. Ein Gedankenmodell zur kinetischen Theorie der festen Körper. J. Appl. Math. Mech. 8,, 8 (2): 85–106, 1928. 10.1002/zamm.19280080202. URL https://doi.org/10.1002/zamm.19280080202. https://doi.org/10.1002/zamm.19280080202 [35] Paolo Restuccia, Gabriele Losi, Omar Chehaimi, Margherita Marsili, and M. Clelia Righi. High-Throughput First-Principles Prediction of Interfacial Adhesion Energies in Metal-on-Metal Contacts. ACS Appl. Mater. Interfaces, 15 (15): 19624–19633, 2023. 10.1021/acsami.3c00662. URL https://doi.org/10.1021/acsami.3c00662. https://doi.org/10.1021/acsami.3c00662 [36] Sergei M Rytov, Yurii A Kravtsov, and Valeryan I Tatarskii. Principles of Statistical Radiophysics 3. Springer, Berlin, Germany, 1989. ISBN 978-3-642-72687-3. URL https://link.springer.com/book/9783642726873. https://link.springer.com/book/9783642726873 [37] J. J. Sakurai and Jim Napolitano.
Modern Quantum Mechanics.
Cambridge University Press, 3 edition, 2020. 10.1017/9781108587280. URL https://doi.org/10.1017/9781108587280. https://doi.org/10.1017/9781108587280 [38] Erik Schäffer, Simon F. Nørrelykke, and Jonathon Howard. Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers. Langmuir, 23 (7): 3654–3665, 2007. 10.1021/la0622368. URL https://doi.org/10.1021/la0622368. https://doi.org/10.1021/la0622368 [39] S.N. Shevchenko, S. Ashhab, and Franco Nori. Landau–Zener–Stückelberg interferometry. Phys. Rep., 492 (1): 1–30, 2010. ISSN 0370-1573. 10.1016/j.physrep.2010.03.002. URL https://doi.org/10.1016/j.physrep.2010.03.002. https://doi.org/10.1016/j.physrep.2010.03.002 [40] Pier Luigi Silvestrelli, S. Subashchandrabose, Alberto Ambrosetti, and Maria Clelia Righi. Sliding properties of transition metal dichalcogenide bilayers. J. Chem. Phys., 163 (8): 084709, 08 2025. ISSN 0021-9606. 10.1063/5.0283532. URL https://doi.org/10.1063/5.0283532. https://doi.org/10.1063/5.0283532 [41] A. Socoliuc, R. Bennewitz, E. Gnecco, and E. Meyer. Transition from Stick-Slip to Continuous Sliding in Atomic Friction: Entering a New Regime of Ultralow Friction. Phys. Rev. Lett., 92: 134301, Apr 2004. 10.1103/PhysRevLett.92.134301. URL https://doi.org/10.1103/PhysRevLett.92.134301. https://doi.org/10.1103/PhysRevLett.92.134301 [42] Anthony Stone. The Theory of Intermolecular Forces.
Oxford University Press, 01 2013. ISBN 9780199672394. 10.1093/acprof:oso/9780199672394.001.0001. URL https://doi.org/10.1093/acprof:oso/9780199672394.001.0001. https://doi.org/10.1093/acprof:oso/9780199672394.001.0001 [43] Izabela Szlufarska, Michael Chandross, and Robert W Carpick. Recent advances in single-asperity nanotribology. J. Phys. D: Appl. Phys., 41 (12): 123001, may 2008. 10.1088/0022-3727/41/12/123001. URL https://dx.doi.org/10.1088/0022-3727/41/12/123001. https://doi.org/10.1088/0022-3727/41/12/123001 [44] G.A. Tomlinson. CVI. A molecular theory of friction . London Edinb. Dublin Philos. Mag. J. Sci., 7 (46): 905–939, 1929. 10.1080/14786440608564819. URL https://doi.org/10.1080/14786440608564819. https://doi.org/10.1080/14786440608564819 [45] D. M. Tong, E. Sjöqvist, L. C. Kwek, and C. H. Oh. Kinematic Approach to the Mixed State Geometric Phase in Nonunitary Evolution. Phys. Rev. Lett., 93: 080405, Aug 2004. 10.1103/PhysRevLett.93.080405. URL https://doi.org/10.1103/PhysRevLett.93.080405. https://doi.org/10.1103/PhysRevLett.93.080405 [46] Paola C. Torche, Andrea Silva, Denis Kramer, Tomas Polcar, and Ondrej Hovorka. Multi-scale model predicting friction of crystalline materials. Adv. Mater. Interfaces, 9 (4): 2100914, 2022. 10.1002/admi.202100914. URL https://doi.org/10.1002/admi.202100914. https://doi.org/10.1002/admi.202100914 [47] A.I. Vakis, V.A. Yastrebov, J. Scheibert, L. Nicola, D. Dini, C. Minfray, A. Almqvist, M. Paggi, S. Lee, G. Limbert, J.F. Molinari, G. Anciaux, R. Aghababaei, S. Echeverri Restrepo, A. Papangelo, A. Cammarata, P. Nicolini, C. Putignano, G. Carbone, S. Stupkiewicz, J. Lengiewicz, G. Costagliola, F. Bosia, R. Guarino, N.M. Pugno, M.H. Müser, and M. Ciavarella. Modeling and simulation in tribology across scales: An overview. Tribol. Int., 125: 169–199, 2018. 10.1016/j.triboint.2018.02.005. URL https://doi.org/10.1016/j.triboint.2018.02.005. https://doi.org/10.1016/j.triboint.2018.02.005 [48] Ludmila Viotti, Fernando C. Lombardo, and Paula I. Villar. Geometric phase in a dissipative Jaynes-Cummings model: Theoretical explanation for resonance robustness. Phys. Rev. A, 105: 022218, Feb 2022. 10.1103/PhysRevA.105.022218. URL https://doi.org/10.1103/PhysRevA.105.022218. https://doi.org/10.1103/PhysRevA.105.022218 [49] Ludmila Viotti, Ana Laura Gramajo, Paula I. Villar, Fernando C. Lombardo, and Rosario Fazio. Geometric phases along quantum trajectories. Quantum, 7: 1029, June 2023. ISSN 2521-327X. 10.22331/q-2023-06-02-1029. URL https://doi.org/10.22331/q-2023-06-02-1029. https://doi.org/10.22331/q-2023-06-02-1029 [50] A. I. Volokitin and B. N. J. Persson. Near-field radiative heat transfer and noncontact friction. Rev. Mod. Phys., 79: 1291–1329, Oct 2007. 10.1103/RevModPhys.79.1291. URL https://doi.org/10.1103/RevModPhys.79.1291. https://doi.org/10.1103/RevModPhys.79.1291 [51] Ulrich Weiss.
Quantum Dissipative Systems. World Scientific, Singapore, 4th edition, 2012. 10.1142/8334. URL https://doi.org/10.1142/8334. https://doi.org/10.1142/8334 [52] Joshua Wilkie and Murat Çetinbaş. Variable-stepsize Runge–Kutta methods for stochastic Schrödinger equations. Phys. Lett. A, 337 (3): 166–182, 2005. ISSN 0375-9601. 10.1016/j.physleta.2005.01.064. URL https://doi.org/10.1016/j.physleta.2005.01.064. https://doi.org/10.1016/j.physleta.2005.01.064 [53] M. Wolloch, G. Levita, P. Restuccia, and M. C. Righi.
Interfacial Charge Density and Its Connection to Adhesion and Frictional Forces. Phys. Rev. Lett., 121: 026804, Jul 2018. 10.1103/PhysRevLett.121.026804. URL https://doi.org/10.1103/PhysRevLett.121.026804. https://doi.org/10.1103/PhysRevLett.121.026804 [54] M. Wolloch, G. Losi, M. Ferrario, and M. C. Righi. High-throughput screening of the static friction and ideal cleavage strength of solid interfaces. Sci. Rep., 9: 17062, 2019. 10.1038/s41598-019-49907-2. URL https://doi.org/10.1038/s41598-019-49907-2. https://doi.org/10.1038/s41598-019-49907-2 [55] Huan Xu, Weizhong Chen, and Yifei Zhu. Influence of the bond defect in driven Frenkel-Kontorova chains. Phys. Rev. B, 75: 224303, Jun 2007. 10.1103/PhysRevB.75.224303. URL https://doi.org/10.1103/PhysRevB.75.224303. https://doi.org/10.1103/PhysRevB.75.224303 [56] Makoto Yamaguchi, Tatsuro Yuge, and Tetsuo Ogawa. Markovian quantum master equation beyond adiabatic regime. Phys. Rev. E, 95: 012136, Jan 2017. 10.1103/PhysRevE.95.012136. URL https://doi.org/10.1103/PhysRevE.95.012136. https://doi.org/10.1103/PhysRevE.95.012136 [57] Chih-Wen Yang, Kwan-tai Leung, Ren-Feng Ding, Hsien-Chen Ko, Yi-Hsien Lu, Chung-Kai Fang, and Ing-Shouh Hwang. Lateral force microscopy of interfacial nanobubbles: Friction reduction and novel frictional behavior. Sci. Rep., 8 (21): 3125, february 2018. 10.1038/s41598-018-21264-6. URL https://doi.org/10.1038/s41598-018-21264-6. https://doi.org/10.1038/s41598-018-21264-6 [58] Tommaso Zanca, Franco Pellegrini, Giuseppe E. Santoro, and Erio Tosatti. Frictional lubricity enhanced by quantum mechanics. Proc. Natl. Acad. Sci. U.S.A., 115: 3547–3550, April 2018. 10.1073/pnas.1801144115. URL https://doi.org/10.1073/pnas.1801144115. https://doi.org/10.1073/pnas.1801144115 [59] Clarence Zener. Non-Adiabatic Crossing of Energy Levels. Proc. R. Soc. London Ser. A, 137: 696–702, 1932. 10.1098/rspa.1932.0165. URL https://doi.org/10.1098/rspa.1932.0165. https://doi.org/10.1098/rspa.1932.0165 [60] A. Zenesini, H. Lignier, G. Tayebirad, J. Radogostowicz, D. Ciampini, R. Mannella, S. Wimberger, O. Morsch, and E. Arimondo. Time-Resolved Measurement of Landau-Zener Tunneling in Periodic Potentials. Phys. Rev. Lett., 103: 090403, Aug 2009. 10.1103/PhysRevLett.103.090403. URL https://doi.org/10.1103/PhysRevLett.103.090403. https://doi.org/10.1103/PhysRevLett.103.090403Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-16 07:42:12: Could not fetch cited-by data for 10.22331/q-2026-04-16-2070 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-16 07:42:13: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractFriction in atomistic systems is usually described by the classical Prandtl-Tomlinson model suitable for capturing the dragging force of a nanoparticle in a periodic potential. Here we consider the quantum mechanical version of this model in which the dissipation is facilitated by releasing heat to an external bath reservoir. The time evolution of the system is captured with the Liouville-von Neumann equation through the density matrix of the system in the Markov approximation. We examine several kinetic and dissipative properties of the nanoparticle by delineating classical vs quantum mechanical effects. We find that the Landau-Zener tunneling is a key factor in the overall reduction of the frictional dissipation when compared to the classical motion in which such tunneling is absent. Other regimes of motion, controlled by the corrugation parameter and other properties, are also found. This in-depth study analyzes the interplay between velocity, strength of interaction, and temperature to control the frictional force and provide useful guidelines for experimental data interpretation.Featured image: Dissipative friction of a nanoparticle moving above a periodic chain of atomsPopular summaryThis study examines quantum mechanical and classical aspects of contact nanoscale friction of a nanoparticle above a periodic chain of atoms. The frictional process is interpretated as a dissipative mechanism as a result of heat exchange with the environment. Our theory and subsequent analysis examines several dynamic properties, including the lateral frictional force which is accessible experimentally. Particular emphasis is given on the emerging Landau-Zener tunneling, a quantum process that allows particles to cross energy gaps in their eigenstates and reduces frictional dissipation relative to classical predictions. Our study puts forward a comprehensive theory suitable for providing theoretical guidance for guiding and interpreting experimental results in nanomechanics and related disciplines.► BibTeX data@article{Le2026quantumstickslip, doi = {10.22331/q-2026-04-16-2070}, url = {https://doi.org/10.22331/q-2026-04-16-2070}, title = {Quantum stick-slip motion in nanoscaled friction}, author = {Le, Dai-Nam and Rodriguez-Lopez, Pablo and Woods, Lilia M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2070}, month = apr, year = {2026} }► References [1] See Supplementary Information for detailed derivations of coupling potential between the nanoparticle and the atomic chain, matrix elements of quantum Prandtl-Tomlison Hamiltonian in moving harmonic oscillator basis set, numerical calculation for quasi-static eigenenergies and eigenstates, quantum evolution with and without external bath reservoir, classical evolution with and without external bath reservoir; detail explainations for tunneling and slipping of quantum and classical trajectories and an animation (GIF) of quantum and classical motions; see also Refs. gradshteyn2014table, Wilkie2005 therein. [2] Clara M. Almeida, Rodrigo Prioli, Benjamin Fragneaud, Luiz Gustavo Cançado, Ricardo Paupitz, Douglas S. Galvão, Marcelo De Cicco, Marcos G. Menezes, Carlos A. Achete, and Rodrigo B. Capaz. Giant and Tunable Anisotropy of Nanoscale Friction in Graphene. Sci. Rep., 6 (1): 31569, aug 2016. ISSN 2045-2322. 10.1038/srep31569. URL http://dx.doi.org/10.1038/srep31569. https://doi.org/10.1038/srep31569 [3] Luca Arceci, Simone Barbarino, Rosario Fazio, and Giuseppe E. Santoro. Dissipative Landau-Zener problem and thermally assisted Quantum Annealing. Phys. Rev. B, 96: 054301, Aug 2017. 10.1103/PhysRevB.96.054301. URL https://doi.org/10.1103/PhysRevB.96.054301. https://doi.org/10.1103/PhysRevB.96.054301 [4] Ranjan Kumar Barik and Lilia M Woods. Frictional Properties of Two-Dimensional Materials: Data-Driven Machine Learning Predictive Modeling. ACS Appl. Mater. Interfaces, 16 (30): 40149–40159, 2024. 10.1021/acsami.4c05532. URL https://doi.org/10.1021/acsami.4c05532. https://doi.org/10.1021/acsami.4c05532 [5] L. Bartels, G. Meyer, and K.-H. Rieder. Basic Steps of Lateral Manipulation of Single Atoms and Diatomic Clusters with a Scanning Tunneling Microscope Tip. Phys. Rev. Lett., 79: 697–700, Jul 1997. 10.1103/PhysRevLett.79.697. URL https://doi.org/10.1103/PhysRevLett.79.697. https://doi.org/10.1103/PhysRevLett.79.697 [6] Eric Bonvin, Louisiane Devaud, Massimiliano Rossi, Andrei Militaru, Lorenzo Dania, Dmitry S. Bykov, Markus Teller, Tracy E. Northup, Lukas Novotny, and Martin Frimmer. Hybrid paul-optical trap with large optical access for levitated optomechanics. Phys. Rev. Res., 6: 043129, Nov 2024. 10.1103/PhysRevResearch.6.043129. URL https://doi.org/10.1103/PhysRevResearch.6.043129. https://doi.org/10.1103/PhysRevResearch.6.043129 [7] D. Bouwmeester, G. P. Karman, C. A. Schrama, and J. P. Woerdman. Observation of interference in transitions due to local geometric phases. Phys. Rev. A, 53: 985–989, Feb 1996. 10.1103/PhysRevA.53.985. URL https://doi.org/10.1103/PhysRevA.53.985. https://doi.org/10.1103/PhysRevA.53.985 [8] S. Cahangirov, C. Ataca, M. Topsakal, H. Sahin, and S. Ciraci. Frictional Figures of Merit for Single Layered Nanostructures. Phys. Rev. Lett., 108: 126103, Mar 2012. 10.1103/PhysRevLett.108.126103. URL https://doi.org/10.1103/PhysRevLett.108.126103. https://doi.org/10.1103/PhysRevLett.108.126103 [9] A. O. Caldeira and A. J. Leggett. Influence of Dissipation on Quantum Tunneling in Macroscopic Systems. Phys. Rev. Lett., 46: 211–214, Jan 1981. 10.1103/PhysRevLett.46.211. URL https://doi.org/10.1103/PhysRevLett.46.211. https://doi.org/10.1103/PhysRevLett.46.211 [10] Ian Counts, Dorian Gangloff, Alexei Bylinskii, Joonseok Hur, Rajibul Islam, and Vladan Vuletić. Multislip Friction with a Single Ion. Phys. Rev. Lett., 119: 043601, Jul 2017. 10.1103/PhysRevLett.119.043601. URL https://doi.org/10.1103/PhysRevLett.119.043601. https://doi.org/10.1103/PhysRevLett.119.043601 [11] Diem Thi-Xuan Dang, Dai-Nam Le, and Lilia M. Woods. Twisting in $h$-BN bilayers and their angle-dependent properties. Phys. Rev. B, 112: 085102, Aug 2025a. 10.1103/wzz2-pszx. URL https://doi.org/10.1103/wzz2-pszx. https://doi.org/10.1103/wzz2-pszx [12] Diem Thi-Xuan Dang, Dai-Nam Le, and Lilia M. Woods. Dissecting van der Waals interactions with density functional theory – Wannier-basis approach. Comput. Phys. Commun., 310: 109525, 2025b. ISSN 0010-4655. 10.1016/j.cpc.2025.109525. URL https://doi.org/10.1016/j.cpc.2025.109525. https://doi.org/10.1016/j.cpc.2025.109525 [13] Thomas. Dittrich, Peter Hänggi, Gert-Ludwig Ingold, Bernhard Krammer, Gerd Schön, and Wilhelm Zwerger. Quantum transport and dissipation. Wiley-VCH, Weinheim, Germany, 1998. ISBN 3527292616. [14] M. Belén Farías, Fernando C. Lombardo, Alejandro Soba, Paula I. Villar, and Ricardo S. Decca. Towards detecting traces of non-contact quantum friction in the corrections of the accumulated geometric phase. npj Quantum. Inf., 6 (1), February 2020. ISSN 2056-6387. URL http://dx.doi.org/10.1038/s41534-020-0252-x. https://doi.org/10.1038/s41534-020-0252-x [15] Dorian Gangloff, Alexei Bylinskii, Ian Counts, Wonho Jhe, and Vladan Vuletić. Velocity tuning of friction with two trapped atoms. Nat. Phys., 11: 915–919, 2015. 10.1038/NPHYS3459. URL https://doi.org/10.1038/nphys3459. https://doi.org/10.1038/NPHYS3459 [16] S. Gasparinetti, P. Solinas, and J. P. Pekola. Geometric Landau-Zener Interferometry. Phys. Rev. Lett., 107: 207002, Nov 2011. 10.1103/PhysRevLett.107.207002. URL https://doi.org/10.1103/PhysRevLett.107.207002. https://doi.org/10.1103/PhysRevLett.107.207002 [17] Enrico Gnecco, Raphael Roth, and Alexis Baratoff. Analytical expressions for the kinetic friction in the Prandtl-Tomlinson model. Phys. Rev. B, 86: 035443, Jul 2012. 10.1103/PhysRevB.86.035443. URL https://doi.org/10.1103/PhysRevB.86.035443. https://doi.org/10.1103/PhysRevB.86.035443 [18] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of integrals, series, and products. Academic press, 2014. 10.1016/C2010-0-64839-5. URL https://doi.org/10.1016/C2010-0-64839-5. https://doi.org/10.1016/C2010-0-64839-5 [19] Zhongkai Huang and Yang Zhao. Dynamics of dissipative Landau-Zener transitions. Phys. Rev. A, 97: 013803, Jan 2018. 10.1103/PhysRevA.97.013803. URL https://doi.org/10.1103/PhysRevA.97.013803. https://doi.org/10.1103/PhysRevA.97.013803 [20] Modi Ke, Dai-Nam Le, and Lilia M. Woods. Casimir interactions between two parallel graphene sheets carrying steady-state drift currents. Phys. Rev. B, 113: 075414, Feb 2026. 10.1103/84n7-1r3b. URL https://doi.org/10.1103/84n7-1r3b. https://doi.org/10.1103/84n7-1r3b [21] Florian R. Krajewski and Martin H. Müser. Quantum Creep and Quantum-Creep Transitions in 1D Sine-Gordon Chains. Phys. Rev. Lett., 92: 030601, Jan 2004. 10.1103/PhysRevLett.92.030601. URL https://doi.org/10.1103/PhysRevLett.92.030601. https://doi.org/10.1103/PhysRevLett.92.030601 [22] Florian R. Krajewski and Martin H. Müser. Quantum dynamics in the highly discrete, commensurate Frenkel Kontorova model: A path-integral molecular dynamics study. J. Chem. Phys., 122 (12): 124711, 03 2005. 10.1063/1.1869392. URL https://doi.org/10.1063/1.1869392. https://doi.org/10.1063/1.1869392 [23] L. Landau. Zur Theorie der Energieubertragung. II. Physikalische Zeitschrift der Sowjetunion, 2: 46–51, 1932. [24] Dai-Nam Le, Ngoc-Tram D. Hoang, and Van-Hoang Le. Exact analytical solutions of the Schrödinger equation for a two dimensional purely sextic double-well potential. J. Math. Phys., 59 (3): 032101, 2018. 10.1063/1.4997532. URL https://doi.org/10.1063/1.4997532. https://doi.org/10.1063/1.4997532 [25] Dai-Nam Le, Pablo Rodriguez-Lopez, and Lilia M. Woods. Nonlinear effects in manybody van der waals interactions. Phys. Rev. Res., 6: 013289, Mar 2024. 10.1103/PhysRevResearch.6.013289. URL https://doi.org/10.1103/PhysRevResearch.6.013289. https://doi.org/10.1103/PhysRevResearch.6.013289 [26] Xin-Z. Liu, Zhijiang Ye, Yalin Dong, Philip Egberts, Robert W. Carpick, and Ashlie Martini. Dynamics of Atomic Stick-Slip Friction Examined with Atomic Force Microscopy and Atomistic Simulations at Overlapping Speeds. Phys. Rev. Lett., 114: 146102, Apr 2015. 10.1103/PhysRevLett.114.146102. URL https://doi.org/10.1103/PhysRevLett.114.146102. https://doi.org/10.1103/PhysRevLett.114.146102 [27] Fernando C. Lombardo and Paula I. Villar. Geometric phases in open systems: A model to study how they are corrected by decoherence. Phys. Rev. A, 74: 042311, Oct 2006. 10.1103/PhysRevA.74.042311. URL https://doi.org/10.1103/PhysRevA.74.042311. https://doi.org/10.1103/PhysRevA.74.042311 [28] Fernando C. Lombardo, Ricardo S. Decca, Ludmila Viotti, and Paula I. Villar. Detectable Signature of Quantum Friction on a Sliding Particle in Vacuum. Adv. Quantum. Technol., 4 (5): 2000155, 2021. URL https://doi.org/10.1002/qute.202000155. https://doi.org/10.1002/qute.202000155 [29] Gabriele Losi, Omar Chehaimi, and M. Clelia Righi. Tribchem: A software for the first-principles, high-throughput study of solid interfaces and their tribological properties. J. Chem. Theory Comput., 19 (15): 5231–5241, 2023. 10.1021/acs.jctc.3c00459. URL https://doi.org/10.1021/acs.jctc.3c00459. https://doi.org/10.1021/acs.jctc.3c00459 [30] E Meyer, T Gyalog, R M Overney, and K Dransfeld. Nanoscience: Friction and Rheology on the Nanometer Scale. World Scientific, 1998. 10.1142/3026. URL https://doi.org/10.1142/3026. https://doi.org/10.1142/3026 [31] Martin H. Müser. Velocity dependence of kinetic friction in the Prandtl-Tomlinson model. Phys. Rev. B, 84: 125419, Sep 2011. 10.1103/PhysRevB.84.125419. URL https://doi.org/10.1103/PhysRevB.84.125419. https://doi.org/10.1103/PhysRevB.84.125419 [32] Nicholas A. Peters, Tzu-Chieh Wei, and Paul G. Kwiat. Mixed-state sensitivity of several quantum-information benchmarks. Phys. Rev. A, 70: 052309, Nov 2004. 10.1103/PhysRevA.70.052309. URL https://doi.org/10.1103/PhysRevA.70.052309. https://doi.org/10.1103/PhysRevA.70.052309 [33] V. L. Pokrovsky and N. A. Sinitsyn. Landau-Zener transitions in a linear chain. Phys. Rev. B, 65: 153105, Apr 2002. 10.1103/PhysRevB.65.153105. URL https://doi.org/10.1103/PhysRevB.65.153105. https://doi.org/10.1103/PhysRevB.65.153105 [34] L. Prandtl. Ein Gedankenmodell zur kinetischen Theorie der festen Körper. J. Appl. Math. Mech. 8,, 8 (2): 85–106, 1928. 10.1002/zamm.19280080202. URL https://doi.org/10.1002/zamm.19280080202. https://doi.org/10.1002/zamm.19280080202 [35] Paolo Restuccia, Gabriele Losi, Omar Chehaimi, Margherita Marsili, and M. Clelia Righi. High-Throughput First-Principles Prediction of Interfacial Adhesion Energies in Metal-on-Metal Contacts. ACS Appl. Mater. Interfaces, 15 (15): 19624–19633, 2023. 10.1021/acsami.3c00662. URL https://doi.org/10.1021/acsami.3c00662. https://doi.org/10.1021/acsami.3c00662 [36] Sergei M Rytov, Yurii A Kravtsov, and Valeryan I Tatarskii. Principles of Statistical Radiophysics 3. Springer, Berlin, Germany, 1989. ISBN 978-3-642-72687-3. URL https://link.springer.com/book/9783642726873. https://link.springer.com/book/9783642726873 [37] J. J. Sakurai and Jim Napolitano.
Modern Quantum Mechanics.
Cambridge University Press, 3 edition, 2020. 10.1017/9781108587280. URL https://doi.org/10.1017/9781108587280. https://doi.org/10.1017/9781108587280 [38] Erik Schäffer, Simon F. Nørrelykke, and Jonathon Howard. Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers. Langmuir, 23 (7): 3654–3665, 2007. 10.1021/la0622368. URL https://doi.org/10.1021/la0622368. https://doi.org/10.1021/la0622368 [39] S.N. Shevchenko, S. Ashhab, and Franco Nori. Landau–Zener–Stückelberg interferometry. Phys. Rep., 492 (1): 1–30, 2010. ISSN 0370-1573. 10.1016/j.physrep.2010.03.002. URL https://doi.org/10.1016/j.physrep.2010.03.002. https://doi.org/10.1016/j.physrep.2010.03.002 [40] Pier Luigi Silvestrelli, S. Subashchandrabose, Alberto Ambrosetti, and Maria Clelia Righi. Sliding properties of transition metal dichalcogenide bilayers. J. Chem. Phys., 163 (8): 084709, 08 2025. ISSN 0021-9606. 10.1063/5.0283532. URL https://doi.org/10.1063/5.0283532. https://doi.org/10.1063/5.0283532 [41] A. Socoliuc, R. Bennewitz, E. Gnecco, and E. Meyer. Transition from Stick-Slip to Continuous Sliding in Atomic Friction: Entering a New Regime of Ultralow Friction. Phys. Rev. Lett., 92: 134301, Apr 2004. 10.1103/PhysRevLett.92.134301. URL https://doi.org/10.1103/PhysRevLett.92.134301. https://doi.org/10.1103/PhysRevLett.92.134301 [42] Anthony Stone. The Theory of Intermolecular Forces.
Oxford University Press, 01 2013. ISBN 9780199672394. 10.1093/acprof:oso/9780199672394.001.0001. URL https://doi.org/10.1093/acprof:oso/9780199672394.001.0001. https://doi.org/10.1093/acprof:oso/9780199672394.001.0001 [43] Izabela Szlufarska, Michael Chandross, and Robert W Carpick. Recent advances in single-asperity nanotribology. J. Phys. D: Appl. Phys., 41 (12): 123001, may 2008. 10.1088/0022-3727/41/12/123001. URL https://dx.doi.org/10.1088/0022-3727/41/12/123001. https://doi.org/10.1088/0022-3727/41/12/123001 [44] G.A. Tomlinson. CVI. A molecular theory of friction . London Edinb. Dublin Philos. Mag. J. Sci., 7 (46): 905–939, 1929. 10.1080/14786440608564819. URL https://doi.org/10.1080/14786440608564819. https://doi.org/10.1080/14786440608564819 [45] D. M. Tong, E. Sjöqvist, L. C. Kwek, and C. H. Oh. Kinematic Approach to the Mixed State Geometric Phase in Nonunitary Evolution. Phys. Rev. Lett., 93: 080405, Aug 2004. 10.1103/PhysRevLett.93.080405. URL https://doi.org/10.1103/PhysRevLett.93.080405. https://doi.org/10.1103/PhysRevLett.93.080405 [46] Paola C. Torche, Andrea Silva, Denis Kramer, Tomas Polcar, and Ondrej Hovorka. Multi-scale model predicting friction of crystalline materials. Adv. Mater. Interfaces, 9 (4): 2100914, 2022. 10.1002/admi.202100914. URL https://doi.org/10.1002/admi.202100914. https://doi.org/10.1002/admi.202100914 [47] A.I. Vakis, V.A. Yastrebov, J. Scheibert, L. Nicola, D. Dini, C. Minfray, A. Almqvist, M. Paggi, S. Lee, G. Limbert, J.F. Molinari, G. Anciaux, R. Aghababaei, S. Echeverri Restrepo, A. Papangelo, A. Cammarata, P. Nicolini, C. Putignano, G. Carbone, S. Stupkiewicz, J. Lengiewicz, G. Costagliola, F. Bosia, R. Guarino, N.M. Pugno, M.H. Müser, and M. Ciavarella. Modeling and simulation in tribology across scales: An overview. Tribol. Int., 125: 169–199, 2018. 10.1016/j.triboint.2018.02.005. URL https://doi.org/10.1016/j.triboint.2018.02.005. https://doi.org/10.1016/j.triboint.2018.02.005 [48] Ludmila Viotti, Fernando C. Lombardo, and Paula I. Villar. Geometric phase in a dissipative Jaynes-Cummings model: Theoretical explanation for resonance robustness. Phys. Rev. A, 105: 022218, Feb 2022. 10.1103/PhysRevA.105.022218. URL https://doi.org/10.1103/PhysRevA.105.022218. https://doi.org/10.1103/PhysRevA.105.022218 [49] Ludmila Viotti, Ana Laura Gramajo, Paula I. Villar, Fernando C. Lombardo, and Rosario Fazio. Geometric phases along quantum trajectories. Quantum, 7: 1029, June 2023. ISSN 2521-327X. 10.22331/q-2023-06-02-1029. URL https://doi.org/10.22331/q-2023-06-02-1029. https://doi.org/10.22331/q-2023-06-02-1029 [50] A. I. Volokitin and B. N. J. Persson. Near-field radiative heat transfer and noncontact friction. Rev. Mod. Phys., 79: 1291–1329, Oct 2007. 10.1103/RevModPhys.79.1291. URL https://doi.org/10.1103/RevModPhys.79.1291. https://doi.org/10.1103/RevModPhys.79.1291 [51] Ulrich Weiss.
Quantum Dissipative Systems. World Scientific, Singapore, 4th edition, 2012. 10.1142/8334. URL https://doi.org/10.1142/8334. https://doi.org/10.1142/8334 [52] Joshua Wilkie and Murat Çetinbaş. Variable-stepsize Runge–Kutta methods for stochastic Schrödinger equations. Phys. Lett. A, 337 (3): 166–182, 2005. ISSN 0375-9601. 10.1016/j.physleta.2005.01.064. URL https://doi.org/10.1016/j.physleta.2005.01.064. https://doi.org/10.1016/j.physleta.2005.01.064 [53] M. Wolloch, G. Levita, P. Restuccia, and M. C. Righi.
Interfacial Charge Density and Its Connection to Adhesion and Frictional Forces. Phys. Rev. Lett., 121: 026804, Jul 2018. 10.1103/PhysRevLett.121.026804. URL https://doi.org/10.1103/PhysRevLett.121.026804. https://doi.org/10.1103/PhysRevLett.121.026804 [54] M. Wolloch, G. Losi, M. Ferrario, and M. C. Righi. High-throughput screening of the static friction and ideal cleavage strength of solid interfaces. Sci. Rep., 9: 17062, 2019. 10.1038/s41598-019-49907-2. URL https://doi.org/10.1038/s41598-019-49907-2. https://doi.org/10.1038/s41598-019-49907-2 [55] Huan Xu, Weizhong Chen, and Yifei Zhu. Influence of the bond defect in driven Frenkel-Kontorova chains. Phys. Rev. B, 75: 224303, Jun 2007. 10.1103/PhysRevB.75.224303. URL https://doi.org/10.1103/PhysRevB.75.224303. https://doi.org/10.1103/PhysRevB.75.224303 [56] Makoto Yamaguchi, Tatsuro Yuge, and Tetsuo Ogawa. Markovian quantum master equation beyond adiabatic regime. Phys. Rev. E, 95: 012136, Jan 2017. 10.1103/PhysRevE.95.012136. URL https://doi.org/10.1103/PhysRevE.95.012136. https://doi.org/10.1103/PhysRevE.95.012136 [57] Chih-Wen Yang, Kwan-tai Leung, Ren-Feng Ding, Hsien-Chen Ko, Yi-Hsien Lu, Chung-Kai Fang, and Ing-Shouh Hwang. Lateral force microscopy of interfacial nanobubbles: Friction reduction and novel frictional behavior. Sci. Rep., 8 (21): 3125, february 2018. 10.1038/s41598-018-21264-6. URL https://doi.org/10.1038/s41598-018-21264-6. https://doi.org/10.1038/s41598-018-21264-6 [58] Tommaso Zanca, Franco Pellegrini, Giuseppe E. Santoro, and Erio Tosatti. Frictional lubricity enhanced by quantum mechanics. Proc. Natl. Acad. Sci. U.S.A., 115: 3547–3550, April 2018. 10.1073/pnas.1801144115. URL https://doi.org/10.1073/pnas.1801144115. https://doi.org/10.1073/pnas.1801144115 [59] Clarence Zener. Non-Adiabatic Crossing of Energy Levels. Proc. R. Soc. London Ser. A, 137: 696–702, 1932. 10.1098/rspa.1932.0165. URL https://doi.org/10.1098/rspa.1932.0165. https://doi.org/10.1098/rspa.1932.0165 [60] A. Zenesini, H. Lignier, G. Tayebirad, J. Radogostowicz, D. Ciampini, R. Mannella, S. Wimberger, O. Morsch, and E. Arimondo. Time-Resolved Measurement of Landau-Zener Tunneling in Periodic Potentials. Phys. Rev. Lett., 103: 090403, Aug 2009. 10.1103/PhysRevLett.103.090403. URL https://doi.org/10.1103/PhysRevLett.103.090403. https://doi.org/10.1103/PhysRevLett.103.090403Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-16 07:42:12: Could not fetch cited-by data for 10.22331/q-2026-04-16-2070 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-16 07:42:13: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.