Mean-force Hamiltonians from influence functionals
DOI (Low Temperature Physics):
https://doi.org/10.1063/10.0044810Ключові слова:
Hamiltonian of mean force, strong-coupling thermodynamics, Hubbard–Stratonovich transformation, influence functional, open quantum systemsАнотація
Гамільтоніан середньої сили (HMF) є стандартною від¬правною точкою для термодинаміки сильного зв’язку, проте точні операторні форми відомі лише для обмежених випадків. Ми пропонуємо концепцію «замороженої густини» (quenched density), яка використовує перетворення Габбарда–Стратоно¬вича для перезапису зведеного рівноважного стану як середнього значення локальних пропагаторів в уявному часі. Цей підхід строго розділяє статистичне визначення середовища від алгебраїчної структури відгуку системи. Ми застосовуємо цей метод до мінімального випадку гармонічного середовища зі зв’язком, що комутує з гамільтоніаном системи. За цим сценарієм поправка до HMF має точний вираз у замкненій формі. Ми перевірили цей результат за допомогою обчислень через взяття сліду для скінченного термостата та стохастичної вибірки в уявному часі на п’ятирівневій моделі з проєкторним зв’язком.
Посилання
1. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
2. P. Hänggi, G.-L. Ingold, and P. Talkner, “Finite quantum dissipation: The challenge of obtaining specific heat,” New J. Phys. 10, 115008 (2008). https://doi.org/10.1088/1367-2630/10/11/115008
3. G.-L. Ingold, P. Hänggi, and P. Talkner, “Specific heat anomalies of open quantum systems,” Phys. Rev. E 79, 061105 (2009). https://doi.org/10.1103/PhysRevE.79.061105
4. M. Campisi, P. Talkner, and P. Hänggi, “Fluctuation theorem for arbitrary open quantum systems,” Phys. Rev. Lett. 102, 210401 (2009). https://doi.org/10.1103/PhysRevLett.102.210401
5. C. Jarzynski, “Nonequilibrium work theorem for a system strongly coupled to a thermal environment,” J. Stat. Mech.: Theory Exp. 2004, P09005. https://doi.org/10.1088/1742-5468/2004/09/P09005
6. F. C. Binder, J. Goold, and K. Modi, “Energetic fluctuations in an open quantum process,” Phys. Rev. E 100, 062120 (2019).
7. M. Esposito, M. A. Ochoa, and M. Galperin, “Nature of heat in strongly coupled open quantum systems,” Phys. Rev. B 92, 235440 (2015). https://doi.org/10.1103/PhysRevB.92.235440
8. Á Rivas, “Strong coupling thermodynamics of open quantum systems,” Phys. Rev. Lett. 124, 160601 (2020). https://doi.org/10.1103/PhysRevLett.124.160601
9. A. M. Lacerda, A. Purkayastha, M. Kewming, G. T. Landi, and J. Goold, “Quantum thermodynamics with fast driving and strong coupling via the mesoscopic leads approach,” Phys. Rev. B 107, 195117 (2023). https://doi.org/10.1103/PhysRevB.107.195117
10. P. Strasberg and M. Esposito, “Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force,” Phys. Rev. E 101, 050101 (2020). https://doi.org/10.1103/PhysRevE.101.050101
11. H. J. D. Miller and J. Anders, “Entropy production and time asymmetry in the presence of strong interactions,” Phys. Rev. E 95, 032123 (2017). https://doi.org/10.1103/PhysRevE.95.032123
12. F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, “Operational thermodynamics of open quantum systems,” arXiv:1406.2801v1 (2014).
13. F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, “Quantum thermodynamics of general quantum processes,” Phys. Rev. E 91, 032119 (2015). https://doi.org/10.1103/PhysRevE.91.032119
14. C. Jarzynski, “Stochastic and macroscopic thermodynamics of strongly coupled systems,” Phys. Rev. X 7, 011008 (2017). https://doi.org/10.1103/PhysRevX.7.011008
15. P. Talkner and P. Hänggi, “Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical,” Rev. Mod. Phys. 92, 041002 (2020). https://doi.org/10.1103/RevModPhys.92.041002
16. U. Seifert, “First and second law of thermodynamics at strong coupling,” Phys. Rev. Lett. 116, 020601 (2016). https://doi.org/10.1103/PhysRevLett.116.020601
17. L. A. Correa and J. Glatthard, “Potential renormalisation, Lamb shift and mean-force Gibbs state – to shift or not to shift?,” Quantum 9, 1868 (2025). https://doi.org/10.22331/q-2025-09-30-1868
18. J. D. Cresser and J. Anders, “Weak and ultrastrong coupling limits of the quantum mean force Gibbs state,” Phys. Rev. Lett. 127, 250601 (2021). https://doi.org/10.1103/PhysRevLett.127.250601
19. A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, “Open quantum system dynamics and the mean force Gibbs state,” AVS Quantum Sci. 4, 012301 (2022). https://doi.org/10.1116/5.0073853
20. C. R. Hogg, J. Glatthard, F. Cerisola, and J. Anders, “Tutorial on the stochastic simulation of dissipative quantum oscillators,” J. Chem. Phys. 161, 071501 (2024). https://doi.org/10.1063/5.0222528
21. C. R. Hogg, F. Cerisola, J. D. Cresser, S. A. R. Horsley, and J. Anders, “Enhanced entanglement in multi-bath spin-boson models,” Quantum 8, 1357 (2024). https://doi.org/10.22331/q-2024-05-23-1357
22. S. Jang, Y.-C. Cheng, D. R. Reichman, and R. J. Silbey, “Nonequilibrium theory of electronic excitation energy transfer: The small polaron quantum master equation,” J. Chem. Phys. 129, 101104 (2008). https://doi.org/10.1063/1.2977974
23. M. Shubrook, J. Iles-Smith, and A. Nazir, “Non-Markovian quantum heat statistics with the reaction coordinate mapping,” Quantum Sci. Technol. 10, 025063 (2025). https://doi.org/10.1088/2058-9565/adc6b7
24. X. Du, J. Wang, and Y. Ma, “Generalized Hamiltonian of mean-force approach to open quantum systems coupled to finite baths in thermoequilibrium,” Phys. Rev. E 111, 024139 (2025). https://doi.org/10.1103/PhysRevE.111.024139
25. A. Nazir and G. Schaller, “The reaction coordinate mapping in quantum thermodynamics,” in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Fundamental Theories of Physics, Vol. 195, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer, Cham, 2018), pp. 551–577.
26. N. Anto-Sztrikacs, A. Nazir, and D. Segal, “Effective-Hamiltonian theory of open quantum systems at strong coupling,” PRX Quantum 4, 020307 (2023). https://doi.org/10.1103/PRXQuantum.4.020307
27. J. Iles-Smith, A. G. Dijkstra, N. Lambert, and A. Nazir, “Energy transfer in structured and unstructured environments: Master equations beyond the Born–Markov approximations,” J. Chem. Phys. 144, 044110 (2016). https://doi.org/10.1063/1.4940218
28. J. M. Moix, Y. Zhao, and J. Cao, “Equilibrium-reduced density matrix formulation: Influence of noise, disorder, and temperature on localization in excitonic systems,” Phys. Rev. B 85, 115412 (2012). https://doi.org/10.1103/PhysRevB.85.115412
29. X. Chen, “The rigorous stochastic matrix multiplication scheme for the calculations of reduced equilibrium density matrices of open multilevel quantum systems,” J. Chem. Phys. 140, 154101 (2014). https://doi.org/10.1063/1.4870496
30. N. Makri, “Exploiting classical decoherence in dissipative quantum dynamics: Memory, phonon emission, and the blip sum,” Chem. Phys. Lett. 593, 93 (2014). https://doi.org/10.1016/j.cplett.2013.11.064
31. Y. Tanimura, “Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities,” J. Chem. Phys. 141, 044114 (2014). https://doi.org/10.1063/1.4890441
32. L. Song and Q. Shi, “Calculation of correlated initial state in the hierarchical equations of motion method using an imaginary time path integral approach,” J. Chem. Phys. 143, 194106 (2015). https://doi.org/10.1063/1.4935799
33. P. L. Walters and F. Wang, “Path integral quantum algorithm for simulating non-Markovian quantum dynamics in open quantum systems,” Phys. Rev. Res. 6, 013135 (2024). https://doi.org/10.1103/PhysRevResearch.6.013135
34. R. P. Feynman and F. L. Vernon, “The theory of a general quantum system interacting with a linear dissipative system,” Ann. Phys. (N.Y.) 24, 118 (1963). https://doi.org/10.1016/0003-4916(63)90068-X
35. A. O. Caldeira and A. J. Leggett, “Quantum tunnelling in a dissipative system,” Ann. Phys. (N.Y.) 149, 374 (1983). https://doi.org/10.1016/0003-4916(83)90202-6
36. H. Grabert, P. Schramm, and G.-L. Ingold, “Quantum Brownian motion: The functional integral approach,” Phys. Rep. 168, 115 (1988). https://doi.org/10.1016/0370-1573(88)90023-3
37. M. Xu, V. Vadimov, J. T. Stockburger, and J. Ankerhold, “Colloquium: Simulating non-Markovian dynamics in open quantum systems,” Rev. Mod. Phys. 98, 021002 (2026). https://doi.org/10.1103/w3nw-hbjc
38. J. Hubbard, “Calculation of partition functions,” Phys. Rev. Lett. 3, 77 (1959). https://doi.org/10.1103/PhysRevLett.3.77
39. R. L. Stratonovich, “On a method of calculating quantum distribution functions,” Sov. Phys. Doklady 2, 416 (1957).
40. J. T. Stockburger and H. Grabert, “Exact c-number representation of non-markovian quantum dissipation,” Phys. Rev. Lett. 88, 170407 (2002). https://doi.org/10.1103/PhysRevLett.88.170407
41. J. T. Stockburger, “Simulating spin-boson dynamics with stochastic Liouville-von Neumann equations,” Chem. Phys. 296, 159 (2004). https://doi.org/10.1016/j.chemphys.2003.09.014
42. G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, “Partition-free approach to open quantum systems in harmonic environments: An exact stochastic Liouville equation,” Phys. Rev. B 95, 125124 (2017). https://doi.org/10.1103/PhysRevB.95.125124
43. M. Wiedmann, J. T. Stockburger, and J. Ankerhold, “Non-Markovian dynamics of a quantum heat engine: Out-of-equilibrium operation and thermal coupling control,” New J. Phys. 22, 033007 (2020). https://doi.org/10.1088/1367-2630/ab725a
44. M. A. Lane, D. Matos, I. J. Ford, and L. Kantorovich, “Exactly thermalized quantum dynamics of the spin-boson model coupled to a dissipative environment,” Phys. Rev. B 101, 224306 (2020). https://doi.org/10.1103/PhysRevB.101.224306
45. D. Matos, M. A. Lane, I. J. Ford, and L. Kantorovich, “Efficient choice of colored noise in the stochastic dynamics of open quantum systems,” Phys. Rev. E 102, 062134 (2020).https://doi.org/10.1103/PhysRevE.102.062134
46. D. Matos, L. Kantorovich, and I. J. Ford, “Stochastic entropy production for continuous measurements of an open quantum system,” J. Phys. Commun. 6, 125003 (2022). https://doi.org/10.1088/2399-6528/aca742
47. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010). https://doi.org/10.1103/RevModPhys.82.1155
48. Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357 (2001). https://doi.org/10.1103/RevModPhys.73.357
49. G. M. Palma, K.-A. Suominen, and A. K. Ekert, “Quantum computers and dissipation,” Proc. Roy. Soc. Lond. A 452, 567 (1996). https://doi.org/10.1098/rspa.1996.0029
50. M. Campisi, P. Talkner, and P. Hänggi, “Thermodynamics and fluctuation theorems for a strongly coupled open quantum system: An exactly solvable case,” J. Phys. A: Math. Theor. 42, 392002 (2009). https://doi.org/10.1088/1751-8113/42/39/392002
51. U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 2012).
52. G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, “Driving spin-boson models from equilibrium using exact quantum dynamics,” Phys. Rev. B 97, 224310 (2018). https://doi.org/10.1103/PhysRevB.97.224310
53. G. McCaul and D. I. Bondar, “How to win friends and influence functionals: Deducing stochasticity from deterministic dynamics,” Eur. Phys. J. Spec. Top. 230, 733 (2021). https://doi.org/10.1140/epjs/s11734-021-00068-2
54. A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson's Challenge, Lecture Notes in Chemistry, Vol. 72 (Springer, Berlin, 2000).
55. “Reduced-density-matrix mechanics: With application to many-electron atoms and molecules,” in Adv. Chem. Phys., edited by D. A. Mazziotti (John Wiley & Sons, 2007), Vol. 134. https://doi.org/10.1002/0470106603
56. Y.-K. Liu, M. Christandl, and F. Verstraete, “Quantum computational complexity of the N-representability problem: QMA complete,” Phys. Rev. Lett. 98, 110503 (2007). https://doi.org/10.1103/PhysRevLett.98.110503
57. A. A. Klyachko, “Quantum marginal problem and N-representability,” J. Phys. Conf. Ser. 36, 72 (2006). https://doi.org/10.1088/1742-6596/36/1/014
58. K. V. Hovhannisyan, S. Nemati, C. Henkel, and J. Anders, “Hamiltonian of mean force for strongly-coupled systems,” Phys. Rev. E 102, 012104 (2020). https://doi.org/10.1103/PhysRevE.102.012104
59. In the sequel, we shall revisit and resolve this issue.
60. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
61. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed. (World Scientific, Singapore, 2009).
62. L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
63. C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Springer Tracts in Modern Physics (Springer, Berlin, 1998), Vol. 145.
64. A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, 2010).
65. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Rev. Mod. Phys. 59, 1 (1987). https://doi.org/10.1103/RevModPhys.59.1
66. M. Xu and J. Ankerhold, “Quantum Brownian motion for embedded systems,” Eur. Phys. J. Spec. Top. 235, 2721 (2026). https://doi.org/10.1140/epjs/s11734-025-01925-0
67. F. Cerisola, M. Berritta, S. Scali, S. A. R. Horsley, J. D. Cresser, and J. Anders, “Quantum-classical correspondence in spin-boson equilibrium states at arbitrary coupling,” New J. Phys. 26, 053032 (2024). https://doi.org/10.1088/1367-2630/ad4818
68. G. McCaul, D. V. Zhdanov, and D. I. Bondar, “Wave operator representation of quantum and classical dynamics,” Phys. Rev. A 108, 052208 (2023). https://doi.org/10.1103/PhysRevA.108.052208
