Hierarchical equations of motion: Difference between revisions

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The '''Hierarchicalhierarchical equations of motion''' (HEOM) technique derived by [[Yoshitaka Tanimura]] and [[Ryogo Kubo]] in 1989,<ref name=TanimuraKubo>{{Citation | last = Tanimura | first = Yoshitaka |author2= Kubo, Ryogo | year = 1989 |

authorlink = Yoshitaka Tanimura | author2link = Ryogo Kubo |title=Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath | journal = J. Phys. Soc. Jpn. | volume = 58| issue = 1 |pages= 101–114 | doi = 10.1143/JPSJ.58.101 | bibcode = 1989JPSJ...58..101T | doi-access = free }}</ref> is a non-perturbative approach developed to study the evolution of a density matrix <math> \rho(t)</math> of quantum dissipative systems. The method can treat system-bath interaction non-perturbatively as well as non-Markovian noise correlation times without the hindrance of the typical assumptions that conventional Redfield (master) equations suffer from such as the Born, Markovian and rotating-wave approximations. HEOM is applicable even at low temperatures where quantum effects are not negligible.

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 = -i \left(\frac{i}{\hbar}\hat{H}^{\times}_A + n\gamma\right) \hat{\rho}_n - {1i\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 }}</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 |arxiv=1407.1811 }}</ref>

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-VernonFeynman–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 = -i (\hat{H}_A + n\gamma) \hat{\rho}_n - {1\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 = -i \left(\frac{i}{\hbar}\hat{H}^{\times}_A + n\gamma \right) \hat{\rho}_n - {1i\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{n\gamma}{\beta} \big( \hat{V}^{\times} - i \frac{\beta\hbar\gamma}{2} \hat{V}^{\circ }\big) </math>

:<math> \hat{\Theta} = -\frac{n\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 Stochasticstochastic Liouville Equationequation 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 hierachy by the final term:

<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>.

:<math>

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.

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

</math>

The closing line of the hierarchy is thus:

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

<!--

===Arbitrary spectral density and low temperature correction===

It was pointed out by Dattani ''et al.'' in 2012 that the HEOM method can be employed as long as the bath correlation function is written as a sum of exponentials,<ref name=Dattani>{{Citation | last = Dattani | first = Nike | year = 2012 |title=Analytic Influence Functionals for Most Open Quantum Systems | journal = Quantum Physics Letters | volume = 1|pages= 35–45 | url=http://www.naturalspublishing.com/files/published/464k51t1luip94.pdf }}</ref> and therefore an arbitrarily complicated spectral distribution function <math>J(\omega)</math> can be fitted to any of the forms listed in Table 1 of<ref name=Dattani>{{Citation | last = Dattani | first = Nike | year = 2012 | |title=Analytic Influence Functionals for Most Open Quantum Systems | journal = Quantum Physics Letters | volume = 1|pages= 35–45 | url=http://www.naturalspublishing.com/files/published/464k51t1luip94.pdf }}</ref> whose bath response function can analytically be written as a sum of exponentials, and then the HEOM can be applied for that spectral density at arbitrary temperature. In a subsequent paper,<ref name=Dattani2>{{cite arxiv | last = Dattani | first = Nike | year = 2012 |title=Optimal representation of the bath response function & fast calculation of influence functional coefficients in open quantum systems with BATHFIT 1 | arxiv = 1205.4651 |mode=cs2 }}</ref> it was suggested that the bath response function be fitted directly to a sum of exponentials rather than fitting the spectral density to one of the forms in Table 1 of<ref name=Dattani>{{Citation | last = Dattani | first = Nike | year = 2012 |title=Analytic Influence Functionals for Most Open Quantum Systems | journal = Quantum Physics Letters | volume = 1|pages= 35–45 | url=http://www.naturalspublishing.com/files/published/464k51t1luip94.pdf }}</ref> and then calculating the bath response function as a sum of exponentials analytically.

HEOM can be derived for a variety of spectral distributions i.e. Drude,<ref name=IshizakiTanimura>{{Citation | last = Ishizaki | first = Akihito |author2= Tanimura, Yoshitaka |

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"/>. The equations of motion in this case can be written in the form

<math>

Line 72 ⟶ 88:

</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|pmid=25877565|bibcode=2015JChPh.142n4110T|s2cid=24328605|authorlink=Yoshitaka Tanimura}}</ref><ref>{{Cite journal|lastlast1=Tanimura|firstfirst1=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><br />

-->

===Computational cost===

When the [[open quantum system]] is represented by <math>M</math> levels and <math>M</math> baths with each bath response function represented by <math>K</math> exponentials, a hierarchy with <math>\mathcal{N}</math> layers will contain:

:<math>

\frac{\left(MK + \mathcal{N}\right)!}{\left(MK\right)!\mathcal{N}!}

</math>

matrices, each with <math>M^2</math> complex-valued (containing both real and imaginary parts) elements. Therefore, the limiting factor in HEOM calculations is the amount of [[RAM]] required, since if one copy of each matrix is stored, the total RAM required would be:

:<math>

16M^2\frac{\left(MK + \mathcal{N}\right)!}{\left(MK\right)!\mathcal{N}!}

</math>

[[bytes]] (assuming double-precision).

===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>

==See also==

* [[Quantum Mastermaster Equationequation]]

* [[Open quantum system]]

* [[Fokker-PlanckFokker–Planck equation]]

* [[Quantum dynamical semigroup]]

* [[Quantum dissipation]]