From 2eb730d926ed0af1771567a05b1951e3492d5cc6 Mon Sep 17 00:00:00 2001
From: Yi-Te Huang <y.t.d.huang@phys.ncku.edu.tw>
Date: Thu, 5 Dec 2024 14:04:49 +0900
Subject: [PATCH] add HEOM documentation page

---
 docs/make.jl                        |  1 +
 docs/src/resources/bibliography.bib | 29 +++++++++++++++++++++++++
 docs/src/users_guide/HEOM.md        | 33 +++++++++++++++++++++++++++++
 3 files changed, 63 insertions(+)
 create mode 100644 docs/src/users_guide/HEOM.md

diff --git a/docs/make.jl b/docs/make.jl
index 5512e8621..25d910cc1 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -49,6 +49,7 @@ const PAGES = [
             "Stochastic Solver" => "users_guide/time_evolution/stochastic.md",
             "Solving Problems with Time-dependent Hamiltonians" => "users_guide/time_evolution/time_dependent.md",
         ],
+        "Hierarchical Equations of Motion" => "users_guide/HEOM.md",
         "Solving for Steady-State Solutions" => "users_guide/steadystate.md",
         "Two-time correlation functions" => "users_guide/two_time_corr_func.md",
         "Extensions" => [
diff --git a/docs/src/resources/bibliography.bib b/docs/src/resources/bibliography.bib
index 3cddf83b5..3327c024b 100644
--- a/docs/src/resources/bibliography.bib
+++ b/docs/src/resources/bibliography.bib
@@ -46,3 +46,32 @@ @article{gravina2024adaptive
   doi = {10.1103/PhysRevResearch.6.023072},
   url = {https://link.aps.org/doi/10.1103/PhysRevResearch.6.023072}
 }
+
+@article{Tanimura1989,
+  title = {Time Evolution of a Quantum System in Contact with a Nearly Gaussian-Markoffian Noise Bath},
+  volume = {58},
+  ISSN = {1347-4073},
+  url = {http://dx.doi.org/10.1143/JPSJ.58.101},
+  DOI = {10.1143/jpsj.58.101},
+  number = {1},
+  journal = {Journal of the Physical Society of Japan},
+  publisher = {Physical Society of Japan},
+  author = {Tanimura,  Yoshitaka and Kubo,  Ryogo},
+  year = {1989},
+  month = jan,
+  pages = {101–114}
+}
+
+@article{Huang2023,
+  doi = {10.1038/s42005-023-01427-2},
+  url = {https://doi.org/10.1038/s42005-023-01427-2},
+  year = {2023},
+  month = {Oct},
+  publisher = {Nature Portfolio},
+  volume = {6},
+  number = {1},
+  pages = {313},
+  author = {Huang, Yi-Te and Kuo, Po-Chen and Lambert, Neill and Cirio, Mauro and Cross, Simon and Yang, Shen-Liang and Nori, Franco and Chen, Yueh-Nan},
+  title = {An efficient {J}ulia framework for hierarchical equations of motion in open quantum systems},
+  journal = {Communications Physics}
+}
diff --git a/docs/src/users_guide/HEOM.md b/docs/src/users_guide/HEOM.md
new file mode 100644
index 000000000..7844af929
--- /dev/null
+++ b/docs/src/users_guide/HEOM.md
@@ -0,0 +1,33 @@
+# [Hierarchical Equations of Motion](@id doc:Hierarchical-Equations-of-Motion)
+
+The hierarchical equations of motion (HEOM) approach was originally developed by Tanimura and Kubo [Tanimura1989](@cite) in the context of physical chemistry to "exactly" solve a quantum system (labeled as ``\textrm{s}``) in contact with a bosonic environment, encapsulated in the following total Hamiltonian:
+
+```math
+\hat{H}_{\textrm{total}} = \hat{H}_{\textrm{s}} + \sum_k \omega_k \hat{b}^\dagger_k \hat{b}_k + \hat{V}_{\textrm{s}} \sum_k g_k \left(\hat{b}_k + \hat{b}^\dagger_k\right),
+```
+
+where ``\hat{b}_k`` (``\hat{b}^\dagger_k``) is the bosonic annihilation (creation) operator associated to the ``k``th mode (with frequency ``\omega_k``), ``\hat{V}_{\textrm{s}}`` refer to the coupling operator acting on the system's degree of freedom, and ``g_k`` are the coupling strengths.
+
+As in other solutions to this problem, the properties of the bath are encapsulated by its temperature and its spectral density,
+
+```math
+J(\omega) = 2 \pi \sum_k g^2_k \delta(\omega - \omega_k).
+```
+
+In the HEOM approach, for bosonic baths, one typically chooses a Drude-Lorentz spectral density:
+
+```math
+J_{\textrm{DL}}(\omega) = \frac{4 \Delta W \omega}{\omega^2 + W^2},
+```
+
+or an under-damped Brownian motion spectral density,
+
+```math
+J_{\textrm{U}}(\omega)=\frac{2 \Delta^2 W \omega}{(\omega^2 - \omega_0^2)^2 + \omega^2 W^2}.
+```
+
+Here, ``\Delta`` represents the coupling strength between the system and the bosonic bath with band-width ``W`` and resonance frequency ``\omega_0``.
+
+We introduce an efficient `Julia` framework for HEOM approach called [`HierarchicalEOM.jl`](https://github.com/qutip/HierarchicalEOM.jl). This package is built upon `QuantumToolbox.jl` and provides a user-friendly and efficient tool to simulate complex open quantum systems based on HEOM approach. For a detailed explanation of this package, we recommend to read its [documentation](https://qutip.org/HierarchicalEOM.jl/) and also the article [Huang2023](@cite).
+
+Given the spectral density, the HEOM approach requires a decomposition of the bath correlation functions in terms of exponentials. In the [documentation of `HierarchicalEOM.jl`](https://qutip.org/HierarchicalEOM.jl/), we not only describe how this is done for both bosonic and fermionic environments with code examples, but also describe how to solve the time evolution (dynamics), steady-states, and spectra based on HEOM approach.