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Copy file name to clipboardExpand all lines: 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 author-year style](https://juliadocs.org/DocumenterCitations.jl/stable/gallery/#author_year_style).
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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:
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.
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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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```
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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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