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Kaltenmeyer (talk | contribs) clean up, typo(s) fixed: , → , , hierachy → hierarchy |
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<math> \hat{A}^{\circ} \hat{\rho} = \hat{A}\hat{\rho} + \hat{\rho} \hat{A}</math>
As with the Kubo's Stochastic Liouville Equation in hierarchal form, the counter <math> n </math> can go up to infinity which is a problem numerically, however Tanimura and Kubo provide a method by which the infinite hierarchy can be truncated to a finite set of <math> N </math> differential equations where <math> N </math> is determined by some constraint sensitive to the characteristics of the system i.e. frequency, amplitude of fluctuations, bath coupling etc. The "Terminator" defines the depth of the hierarchy. A simple relation to eliminate the <math> \hat{\rho}_{n+1}</math> term is found. <math>\ \hat{\rho}_{N+1} = - \hat{\Theta} \hat{\rho}_N/ \hbar\gamma</math>.<ref name=Tanimura91>{{Citation |last=Tanimura |first=Yoshitaka |authorlink= Yoshitaka Tanimura| author2= Wolynes, Peter | year = 1991 | title=Quantum and classical Fokker-Planck equations for a Gaussian-Markovian noise bath| journal = Phys. Rev. A | volume = 43 |issue=8 |pages=4131–4142 |doi=10.1103/PhysRevA.43.4131 }}</ref> With this terminator the hierarchy is closed at the depth <math> N </math> of the
<math> \frac{\partial}{\partial t}{\hat{\rho}}_N = -i (\hat{H}_A + N\gamma) \hat{\rho}_N - {1\over \gamma\hbar^2}\hat{V}^{\times}\hat{\Theta}\hat{\rho}_{N} + {iN\over\hbar}\hat{\Theta}\hat{\rho}_{N-1}</math>.
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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/PhysRevA.85.0623232 | doi-broken-date = 2019-08-20 }}</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"/>
<math>
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</math>
Performing a Wigner transformation on this HEOM, the quantum Fokker-Planck equation with low temperature correction terms emerges.<ref name="Tanimura152">{{Citation|last=Tanimura|first=Yoshitaka|title=Real-time and imaginary-time quantum hierarchical Fokker-Planck equations|journal=J. Chem. Phys.|volume=141|issue=14|pages=044114|year=2015|arxiv=1502.04077|doi=10.1063/1.4916647|authorlink=Yoshitaka Tanimura}}</ref><ref>{{Cite journal|last=Tanimura|first=Yoshitaka|last2=Wolynes|first2=Peter G.|date=1991-04-01|title=Quantum and classical Fokker-Planck equations for a Gaussian-Markovian noise bath|url=https://link.aps.org/doi/10.1103/PhysRevA.43.4131|journal=Physical Review A|language=en|volume=43|issue=8|pages=4131–4142|doi=10.1103/PhysRevA.43.4131|issn=1050-2947}}</ref
===Computational Cost===
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===Implementations===
The HEOM method is implemented in a number of freely available codes. A number of these are at the website of [[Yoshitaka Tanimura]] <ref>url=http://theochem.kuchem.kyoto-u.ac.jp/resarch/resarch08.htm</ref> including a version for GPU <ref name=Tsuchimoto>{{Citation | last = Tsuchimoto | first = Masashi |author2= Tanimura, Yoshitaka| year = 2015 | authorlink = Masashi Tsuchimoto | author2link = Yoshitaka Tanimura |title=Spins Dynamics in a Dissipative Environment: Hierarchal Equations of Motion Approach Using a Graphics Processing Unit (GPU) | journal = Journal of Chemical Theory and Computation | volume = 11 | issue = 7 | pages = 3859–3865| doi = 10.1021/acs.jctc.5b00488 | pmid = 26574467 }}</ref> which used improvements introduced in the thesis of David Wilkins
==See also==
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