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docs/make.jl

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@@ -16,7 +16,7 @@ const DOCTEST = true # set `false` to skip doc tests
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# generate bibliography
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bib = CitationBibliography(
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joinpath(@__DIR__, "src", "resources", "bibliography.bib"),
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style=:authoryear,
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style=:numeric,
2020
)
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# generate changelog

docs/src/resources/bibliography.md

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CurrentModule = QuantumToolbox
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```
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This page is generated by [`DocumenterCitations.jl` with numeric style](https://juliadocs.org/DocumenterCitations.jl/stable/gallery/#numeric_style).
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```@bibliography
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```

docs/src/tutorials/lowrank.md

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# [Low rank master equation](@id doc-tutor:Low-rank-master-equation)
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In this tutorial, we will show how to solve the master equation using the low-rank method. For a detailed explanation of the method, we recommend to read the article [gravina2024adaptive](@cite).
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In this tutorial, we will show how to solve the master equation using the low-rank method. For a detailed explanation of the method, we recommend to read the article [gravina2024adaptive](@citet).
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As a test, we will consider the dissipative Ising model with a transverse field. The Hamiltonian is given by
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docs/src/users_guide/HEOM.md

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# [Hierarchical Equations of Motion](@id doc:Hierarchical-Equations-of-Motion)
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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:
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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:
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```math
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\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),
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Here, ``\Delta`` represents the coupling strength between the system and the bosonic bath with band-width ``W`` and resonance frequency ``\omega_0``.
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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).
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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).
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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.

docs/src/users_guide/states_and_operators.md

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fidelity(x, y)
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```
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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}``:
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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}``:
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```@example states_and_operators
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tracedist(x, y) ≈ sqrt(1 - (fidelity(x, y))^2)

docs/src/users_guide/two_time_corr_func.md

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```
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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.
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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
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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
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```math
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\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],

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