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9 changes: 2 additions & 7 deletions docs/src/resources/bibliography.bib
Original file line number Diff line number Diff line change
Expand Up @@ -11,7 +11,6 @@ @book{Gardiner-Zoller2004
@book{Nielsen-Chuang2011,
title = {Quantum Computation and Quantum Information: 10th Anniversary Edition},
ISBN = {9780511976667},
url = {http://dx.doi.org/10.1017/CBO9780511976667},
DOI = {10.1017/cbo9780511976667},
publisher = {Cambridge University Press},
author = {Nielsen, Michael A. and Chuang, Isaac L.},
Expand All @@ -28,8 +27,7 @@ @article{Jozsa1994
pages = {2315--2323},
year = {1994},
publisher = {Taylor \& Francis},
doi = {10.1080/09500349414552171},
URL = {https://doi.org/10.1080/09500349414552171},
doi = {10.1080/09500349414552171}
}

@article{gravina2024adaptive,
Expand All @@ -43,15 +41,13 @@ @article{gravina2024adaptive
year = {2024},
month = {Apr},
publisher = {American Physical Society},
doi = {10.1103/PhysRevResearch.6.023072},
url = {https://link.aps.org/doi/10.1103/PhysRevResearch.6.023072}
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},
Expand All @@ -64,7 +60,6 @@ @article{Tanimura1989

@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},
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2 changes: 2 additions & 0 deletions docs/src/resources/bibliography.md
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Expand Up @@ -2,5 +2,7 @@
CurrentModule = QuantumToolbox
```

This page is generated by [`DocumenterCitations.jl` with author-year style](https://juliadocs.org/DocumenterCitations.jl/stable/gallery/#author_year_style).

```@bibliography
```
4 changes: 2 additions & 2 deletions docs/src/users_guide/HEOM.md
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@@ -1,6 +1,6 @@
# [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:
The hierarchical equations of motion (HEOM) approach was originally developed by [Tanimura1989](@citet) 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),
Expand Down Expand Up @@ -28,6 +28,6 @@ J_{\textrm{U}}(\omega)=\frac{2 \Delta^2 W \omega}{(\omega^2 - \omega_0^2)^2 + \o

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).
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](@citet).

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.
2 changes: 1 addition & 1 deletion docs/src/users_guide/states_and_operators.md
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Expand Up @@ -172,7 +172,7 @@ z = thermal_dm(5, 0.125)

fidelity(x, y)
```
Note that the definition of [`fidelity`](@ref) here is from [Nielsen-Chuang2011](@cite). It is the square root of the fidelity defined in [Jozsa1994](@cite). We also know that for two pure states, the trace distance (``T``) and the fidelity (``F``) are related by ``T = \sqrt{1-F^2}``:
Note that the definition of [`fidelity`](@ref) here is from [Nielsen-Chuang2011](@citet). It is the square root of the fidelity defined in [Jozsa1994](@citet). We also know that for two pure states, the trace distance (``T``) and the fidelity (``F``) are related by ``T = \sqrt{1-F^2}``:

```@example states_and_operators
tracedist(x, y) ≈ sqrt(1 - (fidelity(x, y))^2)
Expand Down
2 changes: 1 addition & 1 deletion docs/src/users_guide/two_time_corr_func.md
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Expand Up @@ -9,7 +9,7 @@ With the `QuantumToolbox.jl` time-evolution function [`mesolve`](@ref), a state
```
where ``\mathcal{G}(t, t_0)\{\cdot\}`` is the propagator defined by the equation of motion. The resulting density matrix can then be used to evaluate the expectation values of arbitrary combinations of same-time operators.

To calculate two-time correlation functions on the form ``\left\langle \hat{A}(t+\tau) \hat{B}(t) \right\rangle``, we can use the quantum regression theorem [see, e.g., [Gardiner-Zoller2004](@cite)] to write
To calculate two-time correlation functions on the form ``\left\langle \hat{A}(t+\tau) \hat{B}(t) \right\rangle``, we can use the quantum regression theorem [see, e.g., [Gardiner-Zoller2004](@citet)] to write

```math
\left\langle \hat{A}(t+\tau) \hat{B}(t) \right\rangle = \textrm{Tr} \left[\hat{A} \mathcal{G}(t+\tau, t)\{\hat{B}\hat{\rho}(t)\} \right] = \textrm{Tr} \left[\hat{A} \mathcal{G}(t+\tau, t)\{\hat{B} \mathcal{G}(t, 0)\{\hat{\rho}(0)\}\} \right],
Expand Down
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