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Non-canonical distribution and non-equilibrium transport beyond weak system-bath coupling regime: A polaron transformation approach

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Abstract

The concept of polaron, emerged from condense matter physics, describes the dynamical interaction of moving particle with its surrounding bosonic modes. This concept has been developed into a useful method to treat open quantum systems with a complete range of system-bath coupling strength. Especially, the polaron transformation approach shows its validity in the intermediate coupling regime, in which the Redfield equation or Fermi’s golden rule will fail. In the polaron frame, the equilibrium distribution carried out by perturbative expansion presents a deviation from the canonical distribution, which is beyond the usual weak coupling assumption in thermodynamics. A polaron transformed Redfield equation (PTRE) not only reproduces the dissipative quantum dynamics but also provides an accurate and efficient way to calculate the non-equilibrium steady states. Applications of the PTRE approach to problems such as exciton diffusion, heat transport and light-harvesting energy transfer are presented.

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References

  1. A. G. Redfield, The theory of relaxation processes, Adv. Magn. Reson. 1, 1 (1965)

    Article  Google Scholar 

  2. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48(2), 119 (1976)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, New York: Oxford University Press, 2002

    MATH  Google Scholar 

  4. J. Cao, A phase-space study of Bloch–Redfield theory, J. Chem. Phys. 107(8), 3204 (1997)

    Article  ADS  Google Scholar 

  5. S. A. Crooker, J. A. Hollingsworth, S. Tretiak, and V. I. Klimov, Spectrally resolved dynamics of energy transfer in quantum-dot assemblies: Towards engineered energy flows in artificial materials, Phys. Rev. Lett. 89(18), 186802 (2002)

    Article  ADS  Google Scholar 

  6. D. Kim, S. Okahara, M. Nakayama, and Y. Shim, Experimental verification of Förster energy transfer between semiconductor quantum dots, Phys. Rev. B 78(15), 153301 (2008)

    Article  ADS  Google Scholar 

  7. S. I. E. Vulto, M. A. de Baat, R. J. W. Louwe, H. P. Permentier, T. Neef, M. Miller, H. van Amerongen, and T. J. Aartsma, Exciton simulations of optical spectra of the FMO complex from the green sulfur bacterium Chlorobium tepidum at 6 K, J. Phys. Chem. B 102(47), 9577 (1998)

    Article  Google Scholar 

  8. T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, Two-dimensional spectroscopy of electronic couplings in photosynthesis, Nature 434(7033), 625 (2005)

    Article  ADS  Google Scholar 

  9. G. S. Engel, T. R. Calhoun, E. L. Read, T. Ahn, T. Mancal, Y. C. Cheng, R. E. Blankenship, and G. R. Fleming, Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems, Nature 446(7137), 782 (2007)

    Article  ADS  Google Scholar 

  10. J. Wu, F. Liu, Y. Shen, J. Cao, and R. J. Silbey, Efficient energy transfer in light-harvesting systems (I): optimal temperature, reorganization energy and spatial–temporal correlations, New J. Phys. 12(10), 105012 (2010)

    Google Scholar 

  11. Y. Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems, J. Phys. Soc. Jpn. 75(8), 082001 (2006)

    Google Scholar 

  12. R. X. Xu and Y. J. Yan, Dynamics of quantum dissipation systems interacting with bosonic canonical bath: Hierarchical equations of motion approach, Phys. Rev. E 75(3), 031107 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  13. N. Makri and D. E. Makarov, Tensor propagator for iterative quantum time evolution of reduced density matrices (I): Theory, J. Chem. Phys. 102(11), 4600 (1995)

    Article  ADS  Google Scholar 

  14. J. Prior, A. W. Chin, S. F. Huelga, and M. B. Plenio, Efficient simulation of strong system-environment interactions, Phys. Rev. Lett. 105(5), 050404 (2010)

    Article  ADS  Google Scholar 

  15. S. Tornow, R. Bulla, F. B. Anders, and A. Nitzan, Dissipative two-electron transfer: A numerical renormalization group study, Phys. Rev. B 78(3), 035434 (2008)

    Article  ADS  Google Scholar 

  16. H. D. Meyer, U. Manthe, and L. Cederbaum, The multiconfigurational time-dependent Hartree approach, Chem. Phys. Lett. 165(1), 73 (1990)

    Article  ADS  Google Scholar 

  17. M. Thoss, H. Wang, and W. H. Miller, Self-consistent hybrid approach for complex systems: Application to the spinboson model with Debye spectral density, J. Chem. Phys. 115(7), 2991 (2001)

    Article  ADS  Google Scholar 

  18. 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(11), 115412 (2012)

    Google Scholar 

  19. J. Moix, J. Ma, and J. Cao, Förster resonance energy transfer, absorption and emission spectra in multichromophoric systems (III): Exact stochastic path integral evaluation, J. Chem. Phys. 142(9), 094108 (2015)

    Google Scholar 

  20. H. Fröhlich, Electrons in lattice fields, Adv. Phys. 3, 325 (1954)

    Article  MATH  ADS  Google Scholar 

  21. T. Holstein, Studies of polaron motion, Ann. Phys. 8(3), 325 (1959)

    Article  MATH  ADS  Google Scholar 

  22. R. Silbey and R. A. Harris, Variational calculation of the dynamics of a two level system interacting with a bath, J. Chem. Phys. 80(6), 2615 (1984)

    Article  ADS  Google Scholar 

  23. R. A. Harris and R. Silbey, Variational calculation of the tunneling system interacting with a heat bath (II): Dynamics of an asymmetric tunneling system, J. Chem. Phys. 83(3), 1069 (1985)

    Article  ADS  Google Scholar 

  24. M. Grover and R. Silbey, Exciton migration in molecular crystals, J. Chem. Phys. 54(11), 4843 (1971)

    Article  ADS  Google Scholar 

  25. S. Jang, Y. C. Cheng, D. R. Reichman, and J. D. Eaves, Theory of coherent resonance energy transfer, J. Chem. Phys. 129(10), 101104 (2008)

    Article  ADS  Google Scholar 

  26. A. Nazir, Correlation-dependent coherent to incoherent transitions in resonant energy transfer dynamics, Phys. Rev. Lett. 103(14), 146404 (2009)

    Article  ADS  Google Scholar 

  27. D. P. S. McCutcheon and A. Nazir, Quantum dot Rabi rotations beyond the weak exciton–phonon coupling regime, New J. Phys. 12(11), 113042 (2010)

    Article  ADS  Google Scholar 

  28. C. K. Lee, J. M. Moix, and J. Cao, Coherent quantum transport in disordered systems: A unified polaron treatment of hopping and band-like transport, J. Chem. Phys. 142(16), 164103 (2015)

    Article  ADS  Google Scholar 

  29. C. K. Lee, J. Moix, and J. Cao, Accuracy of second order perturbation theory in the polaron and variational polaron frames, J. Chem. Phys. 136(20), 204120 (2012)

    Article  ADS  Google Scholar 

  30. C. K. Lee, J. Cao, and J. Gong, Noncanonical statistics of a spin-boson model: Theory and exact Monte Carlo simulations, Phys. Rev. E 86(2), 021109 (2012)

    Article  ADS  Google Scholar 

  31. H. Dong, S. Yang, X. F. Liu, and C. P. Sun, Quantum thermalization with couplings, Phys. Rev. A 76(4), 044104 (2007)

    Article  ADS  Google Scholar 

  32. D. Z. Xu, S. W. Li, X. F. Liu, and C. P. Sun, Noncanonical statistics of a finite quantum system with non-negligible system-bath coupling, Phys. Rev. E 90(6), 062125 (2014)

    Article  ADS  Google Scholar 

  33. C. Wang, J. Ren and J. Cao, Nonequilibrium energy transfer at nanoscale: A unified theory from weak to strong coupling, Scientific Reports 5, 11787 (2015)

    Article  ADS  Google Scholar 

  34. D. Z. Xu, C. Wang, Y. Zhao, and J. Cao, Polaron effects on the performance of light-harvesting systems: A quantum heat engine perspective, New J. Phys. 18(2), 023003 (2016)

    Article  ADS  Google Scholar 

  35. A. Ishizaki and G. R. Fleming, On the adequacy of the Redfield equation and related approaches to the study of quantum dynamics in electronic energy transfer, J. Chem. Phys. 130(23), 234110 (2009)

    Article  ADS  Google Scholar 

  36. L. A. Pachón and P. Brumer, Computational methodologies and physical insights into electronic energy transfer in photosynthetic light-harvesting complexes, J. Phys. Chem. Lett. 2, 2728 (2011), arXiv: 1203.3978

    Article  Google Scholar 

  37. H. J. Carmichael, Statistical Methods in Quantum Optics, Springer, 1999

    Book  MATH  Google Scholar 

  38. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative twostate system, Rev. Mod. Phys. 59(1), 1 (1987)

    Article  ADS  Google Scholar 

  39. U. Weiss, Quantum Dissipative Systems, Singapore: World Scientific, 2008

    Book  MATH  Google Scholar 

  40. J. Cao, Effects of bath relaxation on dissipative two-state dynamics, J. Chem. Phys. 112(15), 6719 (2000)

    Article  ADS  Google Scholar 

  41. A. W. Chin, J. Prior, S. F. Huelga, and M. B. Plenio, Generalized polaron ansatz for the ground state of the sub-ohmic spin-boson model: An analytic theory of the localization transition, Phys. Rev. Lett. 107(16), 160601 (2011)

    Article  ADS  Google Scholar 

  42. Q. J. Tong, J. H. An, H. G. Luo, and C. H. Oh, Quantum phase transition in the delocalized regime of the spin-boson model, Phys. Rev. B 84(17), 174301 (2011)

    Article  ADS  Google Scholar 

  43. D. P. S. McCutcheon, N. S. Dattani, E. M. Gauger, B. W. Lovett, and A. Nazir, A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots, Phys. Rev. B 84(8), 081305 (2011)

    Article  ADS  Google Scholar 

  44. D. Ruelle, Statistical Mechanics: Rigorous Results, New York: Benjamin, 1969

    MATH  Google Scholar 

  45. R. P. Feynman, Statistical Mechanics. A set of lectures, Longman: Addison Wesley, 1998

    MATH  Google Scholar 

  46. M. D. Girardeau and R. M. Mazo, Advances in Chemical Physics, Vol. 24, New York: Wiley, 1973

  47. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics (II): Nonequilibrium Statistical Mechanics, Berlin: Springer-Verlag, 1983

    MATH  Google Scholar 

  48. A. Nitzan, Chemical Dynamics in Condensed Phases: Relaxation, Transfer and Reactions in Condensed Molecular Systems, Oxford: Oxford University Press, 2006

    Google Scholar 

  49. R. P. Feynman and F. L. Jr Vernon, The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. 24, 118 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  50. W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75(3), 715 (2003)

    MathSciNet  MATH  Google Scholar 

  51. J. P. Paz and W. H. Zurek, Quantum limit of decoherence: Environment induced superselection of energy eigenstates, Phys. Rev. Lett. 82(26), 5181 (1999)

    Article  ADS  Google Scholar 

  52. D. Braun, F. Haake, and W. T. Strunz, Universality of decoherence, Phys. Rev. Lett. 86(14), 2913 (2001)

    Article  ADS  Google Scholar 

  53. W. G. Wang, J. B. Gong, G. Casati, and B. Li, Entanglement-induced decoherence and energy eigenstates, Phys. Rev. A 77(1), 012108 (2008)

    Article  ADS  Google Scholar 

  54. C. Gogolin, Environment-induced super selection without pointer states, Phys. Rev. E 81(5), 051127 (2010)

    Article  ADS  Google Scholar 

  55. E. N. Zimanyi and R. J. Silbey, Theoretical description of quantum effects in multi-chromophoric aggregates, Philos. Trans. R. Soc. A 370(1972), 3620 (2012)

  56. A. Troisi and G. Orlandi, Charge-transport regime of crystalline organic semiconductors: Diffusion limited by thermal off-diagonal electronic disorder, Phys. Rev. Lett. 96(8), 086601 (2006)

    Article  ADS  Google Scholar 

  57. T. Sakanoue and H. Sirringhaus, Band-like temperature dependence of mobility in a solution-processed organic semiconductor, Nat. Mater. 9(9), 736 (2010)

    Article  ADS  Google Scholar 

  58. J. Singh, E. R. Bittner, D. Beljonne, and G. D. Scholes, Fluorescence depolarization in poly[2-methoxy-5-((2-ethylhexyl)oxy)-1,4-phenylenevinylene]: Sites versus eigenstates hopping, J. Chem. Phys. 131(19), 194905 (2009)

    Article  ADS  Google Scholar 

  59. M. Bednarz, V. A. Malyshev, and J. Knoester, Temperature dependent fluorescence in disordered Frenkel chains: Interplay of equilibration and local band-edge level structure, Phys. Rev. Lett. 91(21), 217401 (2003)

    Article  ADS  Google Scholar 

  60. J. Moix, J. Wu, P. Huo, D. Coker, and J. Cao, Efficient energy transfer in light-harvesting systems (III): The influence of the eighth bacteriochlorophyll on the dynamics and efficiency in FMO, J. Phys. Chem. Lett. 2(24), 3045 (2011)

    Article  Google Scholar 

  61. G. T. de Laissardière, J. P. Julien, and D. Mayou, Quantum transport of slow charge carriers in quasicrystals and correlated systems, Phys. Rev. Lett. 97, 026601 (2006)

    Article  ADS  Google Scholar 

  62. V. Coropceanu, J. Cornil, D. A. da Silva Filho, Y. Olivier, R. Silbey, and J. L. Bredas, Charge transport in organic semiconductors, Chem. Rev. 107(4), 926 (2007)

    Article  Google Scholar 

  63. F. Ortmann, F. Bechstedt, and K. Hannewald, Theory of charge transport in organic crystals: Beyond Holstein’s small-polaron model, Phys. Rev. B 79(23), 235206 (2009)

    Article  ADS  Google Scholar 

  64. S. Ciuchi, S. Fratini, and D. Mayou, Transient localization in crystalline organic semiconductors, Phys. Rev. B 83(8), 081202 (2011)

    Article  ADS  Google Scholar 

  65. Y. Cheng and R. J. Silbey, A unified theory for charge-carrier transport in organic crystals, J. Chem. Phys. 128(11), 114713 (2008)

    Article  ADS  Google Scholar 

  66. J. M. Moix, M. Khasin, and J. Cao, Coherent quantum transport in disordered systems (I): The influence of dephasing on the transport properties and absorption spectra on one-dimensional systems, New J. Phys. 15(8), 085010 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  67. C. Chuang, C. K. Lee, J. M.Moix, J. Knoester, and J. Cao, Quantum diffusion on molecular tubes: Universal scaling of the 1D to 2D transition, arXiv: 1511.01198 (2015)

    Google Scholar 

  68. A. O. Niskanen, Y. Nakamura, and J. P. Pekola, Information entropic superconducting microcooler, Phys. Rev. B 76(17), 174523 (2007)

    Article  ADS  Google Scholar 

  69. K. Le Hur, Kondo resonance of a microwave photon, Phys. Rev. B 85(14), 140506 (2012)

    Article  ADS  Google Scholar 

  70. M. Galperin, M. A. Ratner, and A. Nitzan, Molecular transport junctions: Vibrational effects, J. Phys.: Condens. Matter 19(10), 103201 (2007)

    ADS  Google Scholar 

  71. J. C. Cuevas and E. Scheer, Molecular Electronics: An Introduction to Theory and Experiment, Singapore: World Scientific, 2010

    Book  Google Scholar 

  72. D. Segal and A. Nitzan, Spin-boson thermal rectifier, Phys. Rev. Lett. 94(3), 034301 (2005)

    Article  ADS  Google Scholar 

  73. D. Segal, Stochastic pumping of heat: Approaching the Carnot efficiency, Phys. Rev. Lett. 101(26), 260601 (2008)

    Article  ADS  Google Scholar 

  74. J. Ren, P. Hänggi, and B. Li, Berry-phase-induced heat pumping and its impact on the fluctuation theorem, Phys. Rev. Lett. 104(17), 170601 (2010)

    Article  ADS  Google Scholar 

  75. A. Caldeira and A. J. Leggett, Influence of dissipation on quantum tunneling in macroscopic systems, Phys. Rev. Lett. 46(4), 211 (1981)

    Article  ADS  Google Scholar 

  76. N. B. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84(3), 1045 (2012)

    Article  ADS  Google Scholar 

  77. L. Zhu, S. Kirchner, Q. M. Si, and A. Georges, Quantum critical properties of the Bose–Fermi Kondo model in a large-Nlimit, Phys. Rev. Lett. 93(26), 267201 (2004)

    Article  ADS  Google Scholar 

  78. K. Saito and T. Kato, Kondo signature in heat transfer via a local two-state system, Phys. Rev. Lett. 111(21), 214301 (2013)

    Article  ADS  Google Scholar 

  79. A. Ishizaki and G. R. Fleming, Quantum coherence in photosynthetic light harvesting, Annu. Rev. Condens. Matter Phys. 3(1), 333 (2012)

    Article  Google Scholar 

  80. J. L. Wu, R. J. Silbey, and J. Cao, Generic mechanism of optimal energy transfer efficiency: A scaling theory of the mean first-passage time in exciton systems, Phys. Rev. Lett. 110(20), 200402 (2013)

    Article  ADS  Google Scholar 

  81. S. F. Huelga and M. B. Plenio, Vibrations, quanta and biology, Contemp. Phys. 54(4), 181 (2013)

    Article  Google Scholar 

  82. M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81(4), 1665 (2009)

    MathSciNet  MATH  Google Scholar 

  83. T. V. Tscherbul and P. Brumer, Long-lived quasistationary coherences in a V-type system driven by incoherent light, Phys. Rev. Lett. 113(11), 113601 (2014)

    Article  ADS  Google Scholar 

  84. J. Olšina, A. G. Dijkstra, C. Wang, and J. Cao, Can natural sunlight induce coherent exciton dynamics? arXiv: 1408.5385 (2014)

    Google Scholar 

  85. H. T. Quan, Y. X. Liu, C. P. Sun, and F. Nori, Quantum thermodynamic cycles and quantum heat engines, Phys. Rev. E 76(3), 031105 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  86. M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim, and A. Svidzinsky, Quantum heat engine power can be increased by noise-induced coherence, Proc. Natl. Acad. Sci. USA 108(37), 15097 (2011)

    Article  ADS  Google Scholar 

  87. K. E. Dorfman, D. V. Voronine, S. Mukamel, and M. O. Scully, Photosynthetic reaction center as a quantum heat engine, Proc. Natl. Acad. Sci. USA 110(8), 2746 (2013)

    Article  ADS  Google Scholar 

  88. H. E. D. Scovil and E. O. Schulz-DuBois, Three-level masers as heat engines, Phys. Rev. Lett. 2(6), 262 (1959)

    Article  ADS  Google Scholar 

  89. J. E. Geusic, E. O. Schulz-DuBios, and H. E. D. Scovil, Quantum equivalent of the carnot cycle, Phys. Rev. 156(2), 343 (1967)

    Article  ADS  Google Scholar 

  90. E. Geva and R. Kosloff, The quantum heat engine and heat pump: An irreversible thermodynamic analysis of the threelevel amplifier, J. Chem. Phys. 104(19), 7681 (1996)

    Article  ADS  Google Scholar 

  91. E. Boukobza and D. J. Tannor, Three-level systems as amplifiers and attenuators: A thermodynamic analysis, Phys. Rev. Lett. 98(24), 240601 (2007)

    Article  ADS  Google Scholar 

  92. D. Gelbwaser-Klimovsky and A. Aspuru-Guzik, Strongly coupled quantum heat machines, J. Phys. Chem. Lett. 6(17), 3477 (2015)

    Article  Google Scholar 

  93. J. Cao and R. J. Silbey, Optimization of exciton trapping in energy transfer processes, J. Phys. Chem. A 113(50), 13825 (2009)

    Article  Google Scholar 

  94. C. Wang, J. Ren, and J. Cao, Optimal tunneling enhances the quantum photovoltaic effect in double quantum dots, New J. Phys. 16(4), 045019 (2014)

    Article  ADS  Google Scholar 

  95. A. G. Dijkstra, C. Wang, J. Cao, and G. R. Fleming, Coherent exciton dynamics in the presence of underdamped vibrations, J. Phys. Chem. Lett. 6(4), 627 (2015)

    Article  Google Scholar 

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Xu, D., Cao, J. Non-canonical distribution and non-equilibrium transport beyond weak system-bath coupling regime: A polaron transformation approach. Front. Phys. 11, 110308 (2016). https://doi.org/10.1007/s11467-016-0540-2

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