Hierarchical Equations of Motion Approach to Quantum Thermodynamics

  • Akihito Kato
  • Yoshitaka TanimuraEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 195)


We present a theoretical framework to investigate quantum thermodynamic processes under non-Markovian system-bath interactions on the basis of the hierarchical equations of motion (HEOM) approach, which is convenient to carry out numerically “exact” calculations. This formalism is valuable because it can be used to treat not only strong system-bath coupling but also system-bath correlation or entanglement, which will be essential to characterize the heat transport between the system and quantum heat baths. Using this formalism, we demonstrated an importance of the thermodynamic effect from the bath-system-bath tri-partite correlations (TPC) for a two-level heat transfer model and a three-level autonomous heat engine model under the conditions that the conventional quantum master equation approaches are failed. Our numerical calculations show that TPC contributions, which distinguish the heat current from the energy current, have to be take into account to satisfy the thermodynamic laws.



The authors are grateful for motivating us to write this article with Yoshi Oono. A. K. is supported by JSPS KAKENHI Grant Number 17H02946. Y. T. is supported by JSPS KAKENHI Grant Number A26248005.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute for Molecular ScienceNational Institutes of Natural SciencesOkazakiJapan
  2. 2.Department of ChemistryGraduate School of Science, Kyoto UniversityKyotoJapan

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