Hierarchical equations of motion: Difference between revisions

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The hierarchical equation of motion for a system in a harmonic Markovian bath is<ref name=Tanimura>{{Citation | last = Tanimura| first = Yoshitaka | year = 1990 | authorlink = Yoshitaka Tanimura |title=Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath | journal = Phys. Rev. A | volume = 41| issue = 12 |pages= 6676–6687 | doi = 10.1103/PhysRevA.41.6676 | pmid = 9903081 | bibcode = 1990PhRvA..41.6676T }}</ref>

:<math> \frac{\partial}{\partial t}{\hat{\rho}}_n = - \left(\frac{i}{\hbar}\hat{H}^{\times}_A + n\gamma\right) \hat{\rho}_n - {i\over\hbar}\hat{V}^{\times}\hat{\rho}_{n+1} + {in\over\hbar}\hat{\Theta}\hat{\rho}_{n-1}</math>

where the superscript <math>^{\times}</math> denoting a commutator and the temperature-dependent super-operator <math>\hat{\Theta}</math> are defined below. The parameter <math>\gamma</math> is the frequency width of the Drude spectral function <math>J(\omega)</math> (see below).

== Hierarchical equations of motion ==

== Equations of motion for the density matrix ==

HEOMs are developed to describe the time evolution of the density matrix <math> \rho(t)</math> for an open quantum system. It is a non-perturbative, non-Markovian approach to propagating in time a quantum state. Motivated by the path integral formalism presented by Feynman and Vernon, Tanimura derive the HEOM from a combination of statistical and quantum dynamical techniques.<ref name="Tanimura"/><ref name=Tanimura06>{{Citation |last=Tanimura |first=Yoshitaka |year = 2006 |authorlink= Yoshitaka Tanimura |title=Stochastic Liouville, Langevin, Fokker-Planck, and Master Equation Approaches to Quantum Dissipative Systems| journal = J. Phys. Soc. Jpn. | volume = 75|issue=8 |pages= 082001 |doi=10.1143/JPSJ.75.082001 |bibcode=2006JPSJ...75h2001T }}</ref><ref name=Tanimura14>{{Citation |last=Tanimura |first=Yoshitaka |authorlink= Yoshitaka Tanimura | year = 2014 |title=Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities| journal = J. Chem. Phys. | volume = 141|issue=4 |pages= 044114 |doi=10.1063/1.4890441 |pmid=25084888 |arxiv=1407.1811 |bibcode=2014JChPh.141d4114T |s2cid=15745963 }}</ref>

Using a two level spin-boson system Hamiltonian

:<math> \hat{H} = \hat{H}_A(\hat{a}^{+},\hat{a}^{-}) + V(\hat{a}^{+},\hat{a}^{-})\sum_{j}c_j\hat{x}_j + \sum_{j}\bigleft[ {\ \hat{p}^2\over{2}} + \frac{1}{2}\hat{x}_{j}^{2} \bigright] </math>

By writing the density matrix in path integral notation and making use of Feynman–Vernon influence functional, all the bath coordinates <math>x_j</math> in the interaction terms can be grouped into this influence functional which in some specific cases can be calculated in closed form. Assuming a high temperature heat bath with the Drude spectral distribution <math> J(\omega) = \hbar\eta\gamma^2\omega/\pi(\gamma^2 + \omega^2) </math> and taking the time derivative of the path integral form density matrix the equation and writing it in hierarchal form yields

Characterising the bath phonons by the spectral density <math> J(\omega) = \sum\nolimits_j c_j^{2}\delta(\omega - \omega_j)</math>

Assuming a Drude spectral function

:<math> J(\omega)

By writing the density matrix in path integral notation and making use of Feynman–Vernon influence functional, all the bath coordinates in the interaction terms can be grouped into this influence functional which in some specific cases can be calculated in closed form. Assuming a high temperature heat bath with the Drude spectral distribution <math> J(\omega) = \hbar\eta\gamma^2\omega/\pi(\gamma^2 + \omega^2) </math> and taking the time derivative of the path integral form density matrix the equation and writing it in hierarchal form yields

= \sum\nolimits_j c_j^{2}\delta(\omega - \omega_j)

= \frac{ \hbar\eta\gamma^2\omega}{\pi(\gamma^2 + \omega^2)}</math>

and a high temperature heat bath, taking the time derivative of the system density matrix, and writing it in hierarchical form yields (<math>n = 0, 1, \ldots</math>)

:<math> \frac{\partial}{\partial t}{\hat{\rho}}_n = - (\frac{i}{\hbar}\hat{H}^{\times}_A + n\gamma) \hat{\rho}_n - {i\over\hbar}\hat{V}^{\times}\hat{\rho}_{n+1} + {in\over\hbar}\hat{\Theta}\hat{\rho}_{n-1}</math>

:<math> \frac{\partial}{\partial t}{\hat{\rho}}_n = - \left(\frac{i}{\hbar}\hat{H}^{\times}_A + n\gamma \right) \hat{\rho}_n - {i\over\hbar}\hat{V}^{\times}\hat{\rho}_{n+1} + {in\over\hbar}\hat{\Theta}\hat{\rho}_{n-1}</math>

where <math> \Theta </math> destroys system excitation and hence can be referred to as the relaxation operator.

whereHere <math> \Theta </math> destroysreduces the system excitation and hence can beis referred to as the relaxation operator.:

:<math> \hat{\Theta} = -\frac{\eta\gamma}{\beta} \big( \hat{V}^{\times} - i \frac{\beta\hbar\gamma}{2} \hat{V}^{\circ }\big) </math>

:<math> \hat{\Theta} = -\frac{\eta\gamma}{\beta} \bigleft( \hat{V}^{\times} - i \frac{\beta\hbar\gamma}{2} \hat{V}^{\circ } \bigright) </math>

The second term in <math>\hat{\Theta} </math> is the temperature correction term with the inverse temperature <math> \beta = 1/k_B T</math> and the "Hyper-operator" notation is introduced.

The second term in <math>\hat{\Theta} </math> is the temperature correction term with the inverse temperature <math> \beta = 1/k_B T</math> and the following "Hypersuper-operator" notation is introduced.:

:<math> \hat{A}^{\times} \hat{\rho} = \hat{A}\hat{\rho} - \hat{\rho} \hat{A}</math>

:<math>

:<math> \hat{A}^{\circ} \hat{\rho} = \hat{A}\hat{\rho} + \hat{\rho} \hat{A}</math>

\begin{align}

:<math> \hat{A}^{\times} \hat{\rho} &= \hat{A}\hat{\rho} - \hat{\rho} \hat{A}</math>

\\

:<math> \hat{A}^{\circ} \hat{\rho} &= \hat{A}\hat{\rho} + \hat{\rho} \hat{A}</math>

\end{align}

</math>

The counter <math> n </math> provides for <math>n = 0</math> the system density matrix.

As with the Kubo's stochastic Liouville equation in hierarchalhierarchical form, theit countergoes <math>up nto </math>infinity canin gothe up to infinityhierarchy which is a problem numerically, however. Tanimura and Kubo, however, provide a method by which the infinite hierarchy can be truncated to a finite set of <math> N </math> differential equations. whereThis "terminator" <math> N </math> defines the depth of the hierarchy and 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 |pmid=9905511 |bibcode=1991PhRvA..43.4131T }}</ref> With this terminator the hierarchy is closed at the depth <math> N </math> of the hierarchy by the final term:

:<math>

:<math> \frac{\partial}{\partial t}{\hat{\rho}}_N = -(\frac{i}{\hbar}\hat{H}^{\times}_A + N\gamma) \hat{\rho}_N - {i\over \gamma\hbar^2}\hat{V}^{\times}\hat{\Theta}\hat{\rho}_{N} + {iN\over\hbar}\hat{\Theta}\hat{\rho}_{N-1}</math>.

\hat{\rho}_{N+1} = - \hat{\Theta} \hat{\rho}_N/ \hbar\gamma.

</math>

The closing line of the hierarchy is thus:

The statistical nature of the HEOM approach allows information about the bath noise and system response to be encoded into the equation of motion doctoring the infinite energy problem of Kubo's SLE by introducing the relaxation operator ensuring a return to equilibrium.

:<math> \frac{\partial}{\partial t}{\hat{\rho}}_N = -\left( \frac{i}{\hbar}\hat{H}^{\times}_A + N\gamma \right) \hat{\rho}_N - {i\over \gamma\hbar^2}\hat{V}^{\times}\hat{\Theta}\hat{\rho}_{N} + {iN\over\hbar}\hat{\Theta}\hat{\rho}_{N-1}</math>.

The statistical nature of the HEOM approach allows information about the bath noise and system response to be encoded into the equationequations of motion. doctoringIt cures the infinite energy problem of Kubo's SLEstochastic Liouville equation by introducing the relaxation operator ensuringthat ensures a return to equilibrium.

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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|pmid=25877565|bibcode=2015JChPh.142n4110T|s2cid=24328605|authorlink=Yoshitaka Tanimura}}</ref><ref>{{Cite journal|last1=Tanimura|first1=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|pmid=9905511|bibcode=1991PhRvA..43.4131T|issn=1050-2947}}</ref>

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===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 GPUs <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 by David Wilkins and Nike Dattani.<ref>{{cite journal |last1=Wilkins |first1=David |author2= Dattani, Nike|title=Why quantum coherence is not important in the Fenna-Matthews-Olsen Complex |journal=Journal of Chemical Theory and Computation |year=2015 |volume=11 |issue=7 |pages=3411–9 |doi=10.1021/ct501066k |pmid=26575775 |arxiv=1411.3654 |s2cid=15519516 |url=https://doi.org/10.1021/ct501066k}}</ref> The [[nanoHUB]] version provides a very flexible implementation.<ref>{{Cite journal|url=https://nanohub.org/resources/16106/relax|doi = 10.4231/D3RF5KH7G|year = 2017|last1 = Kreisbeck|first1 = Christoph|last2 = Kramer|first2 = Tobias|title = Exciton Dynamics Lab for Light-Harvesting Complexes (GPU-HEOM)}}</ref> An open source parallel CPU implementation is available from the [[Klaus Schulten|Schulten]] group.<ref>url=https://www.ks.uiuc.edu/Research/phi/</ref> The HierarchicalEOM.jl<ref>{{Cite journal |url=https://www.nature.com/articles/s42005-023-01427-2 | last = Huang | first = Yi-Te |author2= Kuo, Po-Chen |author3= Lambert, Neill |author4= Cirio, Mauro |author5= Cross, Simon |author6=Yang, Shen-Liang |author7= Nori, Franco |author8= Chen, Yueh-Nan | year = 2023 |title=An efficient Julia framework for hierarchical equations of motion in open quantum systems | journal = Communications Physics | volume = 6 | issue = 1 | pages = 313 | doi = 10.1038/s42005-023-01427-2}}</ref> is an open source software package written in [[Julia (programming language)]] and provides a user-friendly implementation on both CPUs and GPUs.

==See also==