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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 }}</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 (\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>
== Hierarchical
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}\big[ {\ \hat{p}^2\over{2}} + \frac{1}{2}\hat{x}_{j}^{2} \big] </math>
Characterising the bath phonons by the spectral density <math> J(\omega) = \sum\nolimits_j c_j^{2}\delta(\omega - \omega_j)</math>
By writing the density matrix in path integral notation and making use of
:<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>
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>
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> \hat{A}^{\circ} \hat{\rho} = \hat{A}\hat{\rho} + \hat{\rho} \hat{A}</math>
As with the Kubo's
:<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 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.
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▲===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>
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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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