Lithuanian Journal of Physics article / Lietuvos fizikos žurnalo straipsnis

APPLICATION OF THE FROZEN-MODES APPROXIMATION TO CLASSICAL HARMONIC OSCILLATOR SYSTEMS
Justina Vaičaitytė^a,b, Leonas Valkunas^a, and Andrius Gelžinis^a,b
^aDepartment of Molecular Compound Physics, Center for Physical Sciences and Technology, Saulėtekio 3, 10257 Vilnius, Lithuania
^bInstitute of Chemical Physics, Faculty of Physics, Vilnius University, Saulėtekio 9, 10222 Vilnius, Lithuania
Email: andrius.gelzinis@ftmc.lt

Received 2 April 2024; revised 6 May 2024; accepted 10 May 2024

The problems of open classical systems usually correspond to a motion of a test particle that interacts with a large number of bath oscillators. Often, the test particle itself can be considered a harmonic oscillator. For such composite systems, exact numerical solutions are available, but they can become increasingly costly for a large number of bath oscillators. Here we take inspiration from the recent work on open quantum systems and investigate the applicability of the frozen-modes approximation to such classical systems. This approach assumes that some part of the low-frequency bath modes are frozen, thus only their initial values need to be considered. We show that by applying the frozen-modes approximation one can significantly increase the accuracy of the perturbative multiple-scales solution, especially for slow baths. This approach provides a good accuracy even for strong system–bath couplings, a regime that is not accessible to straightforward applications of the perturbation theory. We also suggest a rule for the splitting of spectral density to the fast and slow bath modes. We find that our approach gives excellent results for the ohmic spectral density, but it could be applied for other similar spectral densities as well.
Keywords: classical oscillators, multiple-scales method, frozen modes

UŽŠALDYTŲ MODŲ ARTINIO TAIKYMAS KLASIKINIŲ HARMONINIŲ OSCILIATORIŲ SISTEMOMS
Justina Vaičaitytė^a,b, Leonas Valkūnas^a, Andrius Gelžinis^a,b

^aFizinių ir technologijos mokslų centro Molekulinių darinių fizikos skyrius, Vilnius, Lietuva
^bVilniaus universiteto Fizikos fakulteto Cheminės fizikos institutas, Vilnius, Lietuva

Atvirųjų klasikinių sistemų uždaviniai dažniausiai yra susiję su dalelės, sąveikaujančios su dideliu skaičiumi aplinkos osciliatorių, judėjimu. Gana dažnai nagrinėjama dalelė irgi gali būti laikoma harmoniniu osciliatoriumi. Tokių sistemų uždavinį galima skaitiškai išspręsti tiksliai, bet reikalingi skaičiavimo pajėgumai smarkiai auga didėjant aplinkos osciliatorių skaičiui. Pastaraisiais metais pasirodė darbų, kuriuose užšaldytų modų artinys buvo pritaikytas atvirosioms kvantinėms sistemoms. Šiame darbe mes pritaikome šį metodą atvirosioms klasikinėms sistemoms. Taikant šį artinį laikoma, kad dalis žemo dažnio aplinkos modų yra užšaldytos, todėl reikia įskaityti tik jų koordinačių ir judesio kiekių pradines vertes. Parodome, kad užšaldytų modų artinio taikymas gali gerokai praplėsti trikdžių teorija paremto daugelio skalių metodo taikymo ribas. Gaunamas geras tikslumas net esant stipriai sistemos osciliatoriaus sąveikai su aplinkos osciliatoriais, ką yra sunku pasiekti taikant įprastus artutinius metodus. Taip pat pasiūlome taisyklę, kaip padalinti aplinkos spektrinį tankį į greitas ir lėtas modas. Pasiūlytas metodas veikia itin gerai, kai sistema aprašoma ominio tipo spektriniu tankiu, bet jis yra tinkamas ir kitiems aplinkos modeliams.

References / Nuorodos

[1] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002),
https://doi.org/10.1007/3-540-44874-8_4
[2] U. Weiss, Quantum Dissipative Systems (World Scientific, 2008),
https://doi.org/10.1142/9789812791795
[3] I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017),
https://doi.org/10.1103/RevModPhys.89.015001
[4] M. Razavy, Classical and Quantum Dissipative Systems (Imperial College Press, 2005),
https://doi.org/10.1142/9781860949180
[5] K. Zhou and B. Liu, Molecular Dynamics Simulation: Fundamentals and Applications (Elsevier, 2022),
https://doi.org/10.1016/C2017-0-04711-0
[6] J.S. Bader and B.J. Berne, Quantum and classical relaxation rates from classical simulations, J. Chem. Phys. 100, 8359 (1994),
https://doi.org/10.1063/1.466780
[7] H. Hasegawa, Classical small systems coupled to finite baths, Phys. Rev. E 83, 021104 (2011),
https://doi.org/10.1103/PhysRevE.83.021104
[8] A.O. Caldeira and A.J. Leggett, Quantum tunnelling in a dissipative system, Ann. Phys. 149, 374 (1983),
https://doi.org/10.1016/0003-4916(83)90202-6
[9] G.W. Ford and M. Kac, On the quantum Langevin equation, J. Stat. Phys. 46, 803 (1987),
https://doi.org/10.1007/BF01011142
[10] H. Hasegawa, Responses to applied forces and the Jarzynski equality in classical oscillator systems coupled to finite baths: An exactly solvable nondissipative nonergodic model, Phys. Rev. E 84, 011145 (2011),
https://doi.org/10.1103/PhysRevE.84.011145
[11] J.S. Briggs and A. Eisfeld, Coherent quantum states from classical oscillator amplitudes, Phys. Rev. A 85, 052111 (2012),
https://doi.org/10.1103/PhysRevA.85.052111
[12] T.E. Skinner, Exact mapping of the quantum states in arbitrary N-level systems to the positions of classical coupled oscillators, Phys. Rev. A 88, 012110 (2013),
https://doi.org/10.1103/PhysRevA.88.012110
[13] T. Mančal, Excitation energy transfer in a classical analogue of photosynthetic antennae, J. Phys. Chem. B 117, 11282 (2013),
https://doi.org/10.1021/jp402101z
[14] A. Montoya-Castillo, T.C. Berkelbach, and D.R. Reichman, Extending the applicability of Redfield theories into highly non-Markovian regimes, J. Chem. Phys. 143, 194108 (2015),
https://doi.org/10.1063/1.4935443
[15] J.H. Fetherolf and T.C. Berkelbach, Linear and nonlinear spectroscopy from quantum master equations, J. Chem. Phys. 147, 244109 (2017),
https://doi.org/10.1063/1.5006824
[16] H.-H. Teh, B.-Y. Jin, and Y.-C. Cheng, Frozen-mode small polaron quantum master equation with variational bound for excitation energy transfer in molecular aggregates, J. Chem. Phys. 150, 224110 (2019),
https://doi.org/10.1063/1.5096287
[17] A.A. Kananenka, C.-Y. Hsieh, J. Cao, and E. Geva, Accurate long-time mixed quantum-classical Liouville dynamics via the transfer tensor method, J. Phys. Chem. Lett. 7, 4809 (2016),
https://doi.org/10.1021/acs.jpclett.6b02389
[18] C. Chuang, D.I. Bennett, J.R. Caram, A. Aspuru-Guzik, M.G. Bawendi, and J. Cao, Generalized Kasha's model: T-dependent spectroscopy reveals short-range structures of 2D excitonic systems, Chem 5, 3135 (2019),
https://doi.org/10.1016/j.chempr.2019.08.013
[19] A. Gelzinis and L. Valkunas, Analytical derivation of equilibrium state for open quantum system, J. Chem. Phys. 152, 051103 (2020),
https://doi.org/10.1063/1.5141519
[20] Y. Lai and E. Geva, On simulating the dynamics of electronic populations and coherences via quantum master equations based on treating off-diagonal electronic coupling terms as a small perturbation, J. Chem. Phys. 155, 204101 (2021),
https://doi.org/10.1063/5.0069313
[21] J.G. Simmonds and J.E. Mann, A First Look at Perturbation Theory (Dover Publications, 1997)
[22] A.H. Nayfeh, Introduction to Perturbation Techniques (Wiley-VCH, 1981)
[23] M.H. Holmes, Introduction to Perturbation Methods (Springer, 1995),
https://doi.org/10.1007/978-1-4612-5347-1
[24] T.C. Berkelbach, D.R. Reichman, and T.E. Markland, Reduced density matrix hybrid approach: An eﬀicient and accurate method for adiabatic and non-adiabatic quantum dynamics, J. Chem. Phys. 136, 034113 (2012),
https://doi.org/10.1063/1.3671372