Hierarchical equations of motion: Difference between revisions

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authorlink = Akihito Ishizaki | author2link = Yoshitaka Tanimura |year = 2005 | title=Quantum Dynamics of System Strongly Coupled to Low-Temperature Colored Noise Bath: Reduced Hierarchy Equations Approach | journal = J. Phys. Soc. Jpn. | volume = 74| issue = 12 |pages= 3131–3134 | doi = 10.1143/JPSJ.74.3131 }}</ref> Brownian,<ref name=Tanaka>{{Citation | last = Tanaka | first = Midori |author2= Tanimura, Yoshitaka | year = 2009 |

authorlink = Midori Tanaka | author2link = Yoshitaka Tanimura |title=Quantum Dissipative Dynamics of Electron Transfer Reaction System: Nonperturbative Hierarchy Equations Approach | journal = J. Phys. Soc. Jpn. | volume = 78| issue = 7 |pages= 073802 (2009) | doi = 10.1143/JPSJ.78.073802 }}</ref> Lorentzian,<ref name=Nori>{{Citation | last = Ma | first = Jian |author2= Sun, Zhe |author3= Wang, Xiaoguanag |author4= Nori, Franco | year = 2012 |

authorlink = Midori Tanaka | author2link = Yoshitaka Tanimura |title=Entanglement dynamics of two qubits in a common bath | journal = Phys. Rev. A | volume = 85|pages= 062323 (2012) | url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.85.0623232| doi = 10.1103/PhysRevAphysreva.85.0623232 | doi-broken-date = 2019-08-20 062323}}</ref> and Sub-Ohmic, <ref name=Cao>{{Citation | last = Duan | first = Chenru | year = 2017 | authorlink = Chenru Duan |author2= Zhoufei, Tang |author3= Jianshu, Cao |author4= Jianlan, Wu|title=Zero-temperature localization in a sub-Ohmic spin-boson model investigated by an extended hierarchy equation of motion | journal = Phys. Rev. B | volume = 95| issue = 21 |pages= 214308 | doi = 10.1103/PhysRevB.95.214308 }}</ref> or even arbitrary bath response functions at any temperature.<ref>{{Cite journal|last=Tanimura|first=Yoshitaka|date=1990-06-01|title=Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath|url=https://link.aps.org/doi/10.1103/PhysRevA.41.6676|journal=Physical Review A|language=en|volume=41|issue=12|pages=6676–6687|doi=10.1103/PhysRevA.41.6676|issn=1050-2947}}</ref>

In the Drude case, by modifying the correlation function that describes the noise correlation function strongly non-Markovian and non-perturbative system-bath interactions can be dealt with.<ref name="Tanimura"/><ref name="IshizakiTanimura"/> The equations of motion in this case can be written in the form