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DieHenkels (talk | contribs) Language polished and simplified, definitions in proper order. There is still a problem with the frequency for the bath Hamiltonian (last term in H), see discussion. |
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The '''
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 = -
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).
== 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}\
By writing the density matrix in path integral notation and making use of
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 = -
where <math> \Theta </math> destroys system excitation and hence can be 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{
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.▼
▲
<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}
\\
\end{align}
</math>
The counter <math> n </math> provides for <math>n = 0</math> the system density matrix.
As with
:<math>
\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 = -
▲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 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>.
The
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===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.
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
<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|pmid=25877565|bibcode=2015JChPh.142n4110T|s2cid=24328605|authorlink=Yoshitaka Tanimura}}</ref><ref>{{Cite journal|
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===Computational
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>
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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>
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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]]
==See also==
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