Koopman wavefunctions and classical states in hybrid quantum-classical dynamics. (English) Zbl 1506.37113
The authors study the interaction dynamics for systems with classical and quantum degrees of freedom. The second section is a large and excellent review of the Koopman-van Hove (KvH) approach of classical mechanics. This classical KvH theory is extended in Section 3 in order to include the coupling to a quantum system. The variational structure of the quantum-classical wave equation is studied in Section 4 upon using the exact factorization of the hybrid wavefunction. Section 5 provides the formulation of the closure model ensuring that the classical density retains its initial sign. A conclusive discussion is the content of the last section which ends with a list of open problems.
Reviewer: Mircea Crâşmăreanu (Iaşi)
MSC:
| 37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |
| 81Q80 | Special quantum systems, such as solvable systems |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 81S08 | Canonical quantization |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
Keywords:
Koopman wavefunction; quantum dynamics; prequantization; mixed quantum-classical dynamics; momentum mapReferences:
| [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). |
| [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. · Zbl 1244.81002 |
| [3] | I. V. Aleksandrov, The statistical dynamics of a system consisting of a classical and a quantum subsystem, Z. Naturforsch. A., 36, 902-908 (1981) · 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). |
| [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). |
| [6] | A. D. Bermúdez Manjarres, Projective representation of the Galilei group for classical and quantum-classical systems, J. Phys. A, 54 (2021), 9 pp. · Zbl 1473.83085 |
| [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. |
| [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). |
| [9] | 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). |
| [10] | D. I. Bondar, F. Gay-Balmaz and C. Tronci, Koopman wavefunctions and classical-quantum correlation dynamics, Proc. A, 475 (2019), 18 pp. · Zbl 1472.81012 |
| [11] | W. Boucher; J. Traschen, Semiclassical physics and quantum fluctuations, Phys. Rev. D, 37, 3522-3532 (1988) · doi:10.1103/PhysRevD.37.3522 |
| [12] | M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos, 22 (2012), 33 pp. · Zbl 1319.37013 |
| [13] | S. M. Carroll and J. Lodman, Energy non-conservation in quantum mechanics, Found. Phys., 51 (2021), 15 pp. · Zbl 07423818 |
| [14] | 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. · Zbl 1156.37314 |
| [15] | D. Chruściński; A. Kossakowski; G. Marmo; E. C. G. Sudarshan, Dynamics of interacting classical and quantum systems, Open. Syst. Inf. Dyn., 18, 339-351 (2011) · Zbl 1243.82043 · doi:10.1142/S1230161211000236 |
| [16] | G. Della Riccia; N. Wiener, Wave mechanics in classical phase space, Brownian motion, and quantum theory, J. Mathematical Phys., 7, 1372-1383 (1966) · doi:10.1063/1.1705047 |
| [17] | L. Diósi; J. J. Halliwell, Coupling classical and quantum variables using continuous quantum measurement theory, Phys. Rev. Lett., 81, 2846-2849 (1998) · Zbl 0947.81012 · doi:10.1103/PhysRevLett.81.2846 |
| [18] | F. Faure, Prequantum chaos: Resonances of the prequantum cat map, J. Mod. Dyn., 1, 255-285 (2007) · Zbl 1145.81034 · doi:10.3934/jmd.2007.1.255 |
| [19] | R. P. Feynman, Negative probability, in Quantum Implications, Routledge & Kegan Paul, London, 1987,235-248. |
| [20] | M. S. Foskett; D. D. Holm; C. Tronci, Geometry of nonadiabatic quantum hydrodynamics, Acta Appl. Math., 162, 63-103 (2019) · Zbl 1421.35274 · doi:10.1007/s10440-019-00257-1 |
| [21] | 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. |
| [22] | F. Gay-Balmaz; T. S. Ratiu, The geometric structure of complex fluids, Adv. in Appl. Math., 42, 176-275 (2009) · Zbl 1161.37052 · doi:10.1016/j.aam.2008.06.002 |
| [23] | F. Gay-Balmaz and C. Tronci, Evolution of hybrid quantum-classical wavefunctions, Phys. D, 440 (2022). · Zbl 1485.76088 |
| [24] | 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. · Zbl 1485.76088 |
| [25] | F. Gay-Balmaz; C. Tronci, Madelung transform and probability densities in hybrid quantum-classical dynamics, Nonlinearity, 33, 5383-5424 (2020) · Zbl 1454.35310 · doi:10.1088/1361-6544/aba233 |
| [26] | F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys., 53 (2012), 36 pp. · Zbl 1287.37045 |
| [27] | V. I. Gerasimenko, Dynamical equations of quantum-classical systems, Teoret. Mat. Fiz., 50, 77-87 (1982) |
| [28] | 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. |
| [29] | V. Guillemin; S. Sternberg, The moment map and collective motion, Ann. Physics, 127, 220-253 (1980) · Zbl 0453.58015 · doi:10.1016/0003-4916(80)90155-4 |
| [30] | M. J. W. Hall and M. Reginatto, Ensembles on Configuration Space. Classical, Quantum, and Beyond, Fundamental Theories of Physics, 184, Springer, Cham, 2016. · Zbl 1341.81006 |
| [31] | D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, in Geometry, Mechanics, and Dynamics, Springer, New York, 2002,113-167. · Zbl 1114.37308 |
| [32] | D. D. Holm; J. E. Marsden; T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137, 1-81 (1998) · Zbl 0951.37020 · doi:10.1006/aima.1998.1721 |
| [33] | D. D. Holm, J. I. Rawlinson and C. Tronci, The bohmion method in nonadiabatic quantum hydrodynamics, J. Phys. A, 54 (2021), 24 pp. · Zbl 1507.81042 |
| [34] | 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. · Zbl 1404.81159 |
| [35] | R. S. Ismagilov, M. Losik and P. Michor, A 2-cocycle on a symplectomorphism group, Mosc. Math. J., 6 (2006), 307-315,407. · Zbl 1132.53043 |
| [36] | 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. |
| [37] | I. Joseph, Koopman-von Neumann approach to quantum simulation of nonlinear classical dynamics, Phys. Rev. Res., 2 (2020). |
| [38] | R. Kapral, Progress in the theory of mixed quantum-classical dynamics, Ann. Rev. Phys. Chem., 57, 129-157 (2006) · doi:10.1146/annurev.physchem.57.032905.104702 |
| [39] | B. Khesin; G. Misiołek; K. Modin, Geometry of the Madelung transform, Arch. Ration. Mech. Anal., 234, 549-573 (2019) · Zbl 1425.53107 · doi:10.1007/s00205-019-01397-2 |
| [40] | B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Nat. Acad. Sci., 17, 315-318 (1931) · JFM 57.1010.02 · doi:10.1073/pnas.17.5.31 |
| [41] | B. Kostant, Line bundles and the prequantized Schrödinger equation, in Colloquium on Group Theoretical Methods in Physics, Centre de Physique Théorique, Marseille, 1972. |
| [42] | 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. · Zbl 0223.53028 |
| [43] | E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40, 322-326 (1927) · JFM 52.0969.06 · doi:10.1007/BF01400372 |
| [44] | J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. · Zbl 1241.53069 |
| [45] | A. Peres and D. R. Terno, Hybrid quantum-classical dynamics, Phys. Rev. A, 63 (2001). |
| [46] | O. V. Prezhdo; V. V. Kisil, Mixing quantum and classical mechanics, Phys. Rev. A (3), 56, 162-175 (1997) · doi:10.1103/PhysRevA.56.162 |
| [47] | 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. · Zbl 1076.81010 |
| [48] | L. L. Salcedo, Absence of classical and quantum mixing, Phys. Rev. A, 54, 3657-3660 (1996) · doi:10.1103/PhysRevA.54.3657 |
| [49] | J.-M. Souriau, Quantification géométrique, Comm. Math. Phys., 1, 374-398 (1966) · Zbl 1148.81307 |
| [50] | E. C. G. Sudarshan, Interaction between classical and quantum systems and the measurement of quantum observables, Pramana, 6, 117-126 (1976) · doi:10.1007/BF02847120 |
| [51] | 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). |
| [52] | G. ’t Hooft, Quantum-mechanical behaviour in a deterministic model, Found. Phys. Lett., 10, 105-111 (1997) · doi:10.1007/BF02764232 |
| [53] | D. R. Terno, Inconsistency of quantum-classical dynamics, and what it implies, Found. Phys., 36, 102-111 (2006) · Zbl 1105.81004 · doi:10.1007/s10701-005-9007-y |
| [54] | C. Tronci, Hybrid models for perfect complex fluids with multipolar interactions, J. Geom. Mech., 4, 333-363 (2012) · Zbl 1295.76005 · doi:10.3934/jgm.2012.4.333 |
| [55] | C. Tronci, Momentum maps for mixed states in quantum and classical mechanics, J. Geom. Mech., 11, 639-656 (2019) · Zbl 1448.81070 · doi:10.3934/jgm.2019032 |
| [56] | 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. · Zbl 1485.76089 |
| [57] | C. Tronci and I. Joseph, Koopman wavefunctions and Clebsch variables in Vlasov-Maxwell kinetic theory, J. Plasma Phys., 87 (2021). |
| [58] | J. C. Tully, Mixed quantum-classical dynamics, Faraday Discuss., 110, 407-419 (1998) · doi:10.1039/A801824C |
| [59] | L. van Hove, On certain unitary representations of an infinite group of transformations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. · Zbl 0989.81051 |
| [60] | A. Widom; Y. Srivastava, Lagrangian formulation of Bohr’s measurement theory, Nuovo Cimento B (11), 107, 71-75 (1992) · doi:10.1007/BF02726886 |
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