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Koopman wavefunctions and classical states in hybrid quantum–classical dynamics

  • *Corresponding author: Cesare Tronci

    *Corresponding author: Cesare Tronci

For Anthony Bloch, on the occasion of his 65th birthday

This work was made possible through the support of Grant 62210 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. CT also acknowledges partial support by the Institute of Mathematics and its Applications and by the Royal Society Grant IES\R3\203005.

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  • We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum–classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times, thereby addressing a series of stringent consistency requirements. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems.

    Mathematics Subject Classification: Primary: 53Dxx, 70-XX, 81-XX; Secondary: 92Exx.

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  • [1] A. Abedi, N. T. Maitra and E. K. U Gross, Correlated electron-nuclear dynamics: Exact factorization of the molecular wavefunction, J. Chem. Phys., 137 (2012). doi: 10.1063/1.4745836.
    [2] F. Agostini, S. Caprara and G. Ciccotti, Do we have a consistent non-adiabatic quantum-classical mechanics?, Europhys. Lett. EPL, 78 (2007), 6 pp. doi: 10.1209/0295-5075/78/30001.
    [3] I. V. Aleksandrov, The statistical dynamics of a system consisting of a classical and a quantum subsystem, Z. Naturforsch. A., 36 (1981), 902-908.  doi: 10.1515/zna-1981-0819.
    [4] A. V. Akimov, R. Long and O. V. Prezhdo, Coherence penalty functional: A simple method for adding decoherence in Ehrenfest dynamics, J. Chem. Phys., 140 (2014). doi: 10.1063/1.4875702.
    [5] C. Barceló, R. Carballo-Rubio, L. J. Garay, and R. Gómez-Escalante, Hybrid classical-quantum formulations ask for hybrid notions, Phys. Rev. A, 86 (2012). doi: 10.1103/PhysRevA.86.042120.
    [6] A. D. Bermúdez Manjarres, Projective representation of the Galilei group for classical and quantum-classical systems, J. Phys. A, 54 (2021), 9 pp. doi: 10.1088/1751-8121/ac28cc.
    [7] M. V. Berry, True quantum chaos? An instructive example, in New Trends in Nuclear Collective Dynamics, Springer Proceedings in Physics, 58, Springer, Berlin, Heidelberg, 1992,183-186. doi: 10.1007/978-3-642-76379-3_10.
    [8] G. Bhole, J. A. Jones, C. Marletto and V. Vedral, Witnesses of non-classicality for simulated hybrid quantum systems, J. Phys. Comm., 4 (2020). doi: 10.1088/2399-6528/ab772b.
    [9] I. Bialynicki-BirulaM. CieplakJ. Karminski and  and A. M. FurdynaTheory of Quanta, Oxford University Press, 1992. 
    [10] D. Bondar, R. Cabrera, R. R. Lompay, M. Y. Ivanov, and H. A. Rabitz, Operational dynamic modeling transcending quantum and classical mechanics, Phys. Rev. Lett., 109 (2012). doi: 10.1103/PhysRevLett.109.190403.
    [11] D. I. Bondar, F. Gay-Balmaz and C. Tronci, Koopman wavefunctions and classical-quantum correlation dynamics, Proc. A, 475 (2019), 18 pp. doi: 10.1098/rspa.2018.0879.
    [12] W. Boucher and J. Traschen, Semiclassical physics and quantum fluctuations, Phys. Rev. D, 37 (1988), 3522-3532.  doi: 10.1103/PhysRevD.37.3522.
    [13] M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos, 22 (2012), 33 pp. doi: 10.1063/1.4772195.
    [14] S. M. Carroll and J. Lodman, Energy non-conservation in quantum mechanics, Found. Phys., 51 (2021), 15 pp. doi: 10.1007/s10701-021-00490-5.
    [15] H. Cendra, J. E. Marsden, S. Pekarsky and T. S. Ratiu, Variational principles for Lie-Poisson and Hamilton-Poincaré equations, Mosc. Math. J., 3 (2003), 833-867, 1197-1198.
    [16] D. ChruścińskiA. KossakowskiG. Marmo and E. C. G. Sudarshan, Dynamics of interacting classical and quantum systems, Open. Syst. Inf. Dyn., 18 (2011), 339-351.  doi: 10.1142/S1230161211000236.
    [17] G. Della Riccia and N. Wiener, Wave mechanics in classical phase space, Brownian motion, and quantum theory, J. Mathematical Phys., 7 (1966), 1372-1383.  doi: 10.1063/1.1705047.
    [18] L. Diósi and J. J. Halliwell, Coupling classical and quantum variables using continuous quantum measurement theory, Phys. Rev. Lett., 81 (1998), 2846-2849.  doi: 10.1103/PhysRevLett.81.2846.
    [19] F. Faure, Prequantum chaos: Resonances of the prequantum cat map, J. Mod. Dyn., 1 (2007), 255-285.  doi: 10.3934/jmd.2007.1.255.
    [20] R. P. Feynman, Negative probability, in Quantum Implications, Routledge & Kegan Paul, London, 1987,235–248.
    [21] M. S. FoskettD. D. Holm and C. Tronci, Geometry of nonadiabatic quantum hydrodynamics, Acta Appl. Math., 162 (2019), 63-103.  doi: 10.1007/s10440-019-00257-1.
    [22] M. S. Foskett and C. Tronci, Holonomy and vortex structures in quantum hydrodynamics, in Hamiltonian Systems: Dynamics, Analysis, Applications, Math. Sci. Res. Inst. Pub., 72, Cambridge University Press, 2022.
    [23] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv. in Appl. Math., 42 (2009), 176-275.  doi: 10.1016/j.aam.2008.06.002.
    [24] F. Gay-Balmaz and C. Tronci, Evolution of hybrid quantum-classical wavefunctions, Phys. D, 440 (2022). doi: 10.1016/j.physd.2022.133450.
    [25] F. Gay-Balmaz and C. Tronci, From quantum hydrodynamics to Koopman wavefunctions I, in Geometric Science of Information, Lecture Notes in Comput. Sci., 12829, Springer, Cham, 2021,302-310. doi: 10.1007/978-3-030-80209-7_34.
    [26] F. Gay-Balmaz and C. Tronci, Madelung transform and probability densities in hybrid quantum-classical dynamics, Nonlinearity, 33 (2020), 5383-5424.  doi: 10.1088/1361-6544/aba233.
    [27] F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys., 53 (2012), 36 pp. doi: 10.1063/1.4763467.
    [28] V. I. Gerasimenko, Dynamical equations of quantum-classical systems, Teoret. Mat. Fiz., 50 (1982), 77-87. 
    [29] D. Giannakis, A. Ourmazd, P. Pfeffer, J. Schumacher, and J. Slawinska, Embedding classical dynamics in a quantum computer, Phys. Rev. A, 105 (2022), 47 pp. doi: 10.1103/physreva.105.052404.
    [30] V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. Physics, 127 (1980), 220-253.  doi: 10.1016/0003-4916(80)90155-4.
    [31] M. J. W. Hall and M. Reginatto, Ensembles on Configuration Space. Classical, Quantum, and Beyond, Fundamental Theories of Physics, 184, Springer, Cham, 2016. doi: 10.1007/978-3-319-34166-8.
    [32] D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, in Geometry, Mechanics, and Dynamics, Springer, New York, 2002,113-167. doi: 10.1007/b97525.
    [33] D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.
    [34] D. D. Holm, J. I. Rawlinson and C. Tronci, The bohmion method in nonadiabatic quantum hydrodynamics, J. Phys. A, 54 (2021), 24 pp. doi: 10.1088/1751-8121/ac2ae8.
    [35] J. Hurst, P.-A. Hervieux and G. Manfredi, Phase-space methods for the spin dynamics in condensed matter systems, Phil. Trans. Roy. Soc. A, 375 (2017), 14 pp. doi: 10.1098/rsta.2016.0199.
    [36] R. S. Ismagilov, M. Losik and P. Michor, A 2-cocycle on a symplectomorphism group, Mosc. Math. J., 6 (2006), 307-315,407.
    [37] H.-R. Jauslin and D. Sugny, Dynamics of mixed quantum-classical systems, geometric quantization and coherent states, in Mathematical Horizons for Quantum Physics, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 20, World Sci. Publ., Hackensack, NJ, 2010, 65-96. doi: 10.1142/9789814313322_0003.
    [38] I. Joseph, Koopman-von Neumann approach to quantum simulation of nonlinear classical dynamics, Phys. Rev. Res., 2 (2020). doi: 10.1103/PhysRevResearch.2.043102.
    [39] R. Kapral, Progress in the theory of mixed quantum-classical dynamics, Ann. Rev. Phys. Chem., 57 (2006), 129-157.  doi: 10.1146/annurev.physchem.57.032905.104702.
    [40] B. KhesinG. Misiołek and K. Modin, Geometry of the Madelung transform, Arch. Ration. Mech. Anal., 234 (2019), 549-573.  doi: 10.1007/s00205-019-01397-2.
    [41] B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Nat. Acad. Sci., 17 (1931), 315-318.  doi: 10.1073/pnas.17.5.31.
    [42] B. Kostant, Line bundles and the prequantized Schrödinger equation, in Colloquium on Group Theoretical Methods in Physics, Centre de Physique Théorique, Marseille, 1972.
    [43] B. Kostant, Quantization and unitary representations. I. Prequantization, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., 170, Springer, Berlin, 1970, 87–208.
    [44] E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1927), 322-326.  doi: 10.1007/BF01400372.
    [45] J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7.
    [46] A. Peres and D. R. Terno, Hybrid quantum-classical dynamics, Phys. Rev. A, 63 (2001). doi: 10.1103/PhysRevA.63.022101.
    [47] O. V. Prezhdo and V. V. Kisil, Mixing quantum and classical mechanics, Phys. Rev. A (3), 56 (1997), 162-175.  doi: 10.1103/PhysRevA.56.162.
    [48] C. Rangan and A. M. Bloch, Control of finite-dimensional quantum systems: Application to a spin-$\frac12$ particle coupled with a finite quantum harmonic oscillator, J. Math. Phys., 46 (2005), 9 pp. doi: 10.1063/1.1852701.
    [49] L. L. Salcedo, Absence of classical and quantum mixing, Phys. Rev. A, 54 (1996), 3657-3660.  doi: 10.1103/PhysRevA.54.3657.
    [50] J.-M. Souriau, Quantification géométrique, Comm. Math. Phys., 1 (1966), 374-398. 
    [51] E. C. G. Sudarshan, Interaction between classical and quantum systems and the measurement of quantum observables, Pramana, 6 (1976), 117-126.  doi: 10.1007/BF02847120.
    [52] I. Tavernelli, B. F. E. Curchod and U. Rothlisberger, Mixed quantum-classical dynamics with time-dependent external fields: A time-dependent density-functional-theory approach, Phys. Rev A, 81 (2010). doi: 10.1103/PhysRevA.81.052508.
    [53] G. 't Hooft, Quantum-mechanical behaviour in a deterministic model, Found. Phys. Lett., 10 (1997), 105-111.  doi: 10.1007/BF02764232.
    [54] D. R. Terno, Inconsistency of quantum-classical dynamics, and what it implies, Found. Phys., 36 (2006), 102-111.  doi: 10.1007/s10701-005-9007-y.
    [55] C. Tronci, Hybrid models for perfect complex fluids with multipolar interactions, J. Geom. Mech., 4 (2012), 333-363.  doi: 10.3934/jgm.2012.4.333.
    [56] C. Tronci, Momentum maps for mixed states in quantum and classical mechanics, J. Geom. Mech., 11 (2019), 639-656.  doi: 10.3934/jgm.2019032.
    [57] C. Tronci and F. Gay-Balmaz, From quantum hydrodynamics to Koopman wavefunctions II, in Geometric Science of Information, Lecture Notes in Comput. Sci., 12829, Springer, Cham, 2021,311-319. doi: 10.1007/978-3-030-80209-7_35.
    [58] C. Tronci and I. Joseph, Koopman wavefunctions and Clebsch variables in Vlasov-Maxwell kinetic theory, J. Plasma Phys., 87 (2021). doi: 10.1017/S0022377821000805.
    [59] J. C. Tully, Mixed quantum-classical dynamics, Faraday Discuss., 110 (1998), 407-419.  doi: 10.1039/A801824C.
    [60] L. van Hove, On certain unitary representations of an infinite group of transformations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812838988.
    [61] A. Widom and Y. Srivastava, Lagrangian formulation of Bohr's measurement theory, Nuovo Cimento B (11), 107 (1992), 71-75.  doi: 10.1007/BF02726886.
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