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Merge pull request #28 from NCKU-QFort/dev/QO-support
Support for QuantumOptics.jl type operators
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,53 @@ | ||
| # [Drude-Lorentz Spectral Density](@id Boson-Drude-Lorentz) | ||
| ```math | ||
| J(\omega)=\frac{4\Delta W\omega}{\omega^2+W^2} | ||
| ``` | ||
| Here, ``\Delta`` represents the coupling strength between system and the bosonic environment with band-width ``W``. | ||
|
|
||
| ## Matsubara Expansion | ||
| With Matsubara Expansion, the correlation function can be analytically solved and expressed as follows: | ||
| ```math | ||
| C(t_1, t_2)=\sum_{l=1}^{\infty} \eta_l \exp(-\gamma_l (t_1-t_2)) | ||
| ``` | ||
| with | ||
| ```math | ||
| \begin{aligned} | ||
| \gamma_{1} &= W,\\ | ||
| \eta_{1} &= \Delta W\left[-i+\cot\left(\frac{W}{2 k_B T}\right)\right],\\ | ||
| \gamma_{l\neq 1} &= 2\pi l k_B T,\\ | ||
| \eta_{l\neq 1} &= -2 k_B T \cdot \frac{2\Delta W \cdot \gamma_l}{-\gamma_l^2 + W^2}. | ||
| \end{aligned} | ||
| ``` | ||
| This can be constructed by the built-in function [`Boson_DrudeLorentz_Matsubara`](@ref): | ||
| ```julia | ||
| Vs # coupling operator | ||
| Δ # coupling strength | ||
| W # band-width of the environment | ||
| kT # the product of the Boltzmann constant k and the absolute temperature T | ||
| N # Number of exponential terms | ||
| bath = Boson_DrudeLorentz_Matsubara(Vs, Δ, W, kT, N - 1) | ||
| ``` | ||
|
|
||
| ## Padé Expansion | ||
| With Padé Expansion, the correlation function can be analytically solved and expressed as the following exponential terms: | ||
| ```math | ||
| C(t_1, t_2)=\sum_{l=1}^{\infty} \eta_l \exp(-\gamma_l (t_1-t_2)) | ||
| ``` | ||
| with | ||
| ```math | ||
| \begin{aligned} | ||
| \gamma_{1} &= W,\\ | ||
| \eta_{1} &= \Delta W\left[-i+\cot\left(\frac{W}{2 k_B T}\right)\right],\\ | ||
| \gamma_{l\neq 1} &= \zeta_l k_B T,\\ | ||
| \eta_{l\neq 1} &= -2 \kappa_l k_B T \cdot \frac{2\Delta W \cdot \zeta_l k_B T}{-(\zeta_l k_B T)^2 + W^2}, | ||
| \end{aligned} | ||
| ``` | ||
| where the parameters ``\kappa_l`` and ``\zeta_l`` are described in [J. Chem. Phys. 134, 244106 (2011)](https://doi.org/10.1063/1.3602466). This can be constructed by the built-in function [`Boson_DrudeLorentz_Pade`](@ref): | ||
| ```julia | ||
| Vs # coupling operator | ||
| Δ # coupling strength | ||
| W # band-width of the environment | ||
| kT # the product of the Boltzmann constant k and the absolute temperature T | ||
| N # Number of exponential terms | ||
| bath = Boson_DrudeLorentz_Pade(Vs, Δ, W, kT, N - 1) | ||
| ``` |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,57 @@ | ||
| # [Lorentz Spectral Density](@id doc-Fermion-Lorentz) | ||
| ```math | ||
| J(\omega)=\frac{\Gamma W^2}{(\omega-\mu)^2+W^2} | ||
| ``` | ||
| Here, ``\Gamma`` represents the coupling strength between system and the fermionic environment with chemical potential ``\mu`` and band-width ``W``. | ||
|
|
||
| ## Matsubara Expansion | ||
| With Matsubara Expansion, the correlation function can be analytically solved and expressed as follows: | ||
| ```math | ||
| C^{\nu}(t_1, t_2)=\sum_{l=1}^{\infty} \eta_l^\nu \exp(-\gamma_l^\nu (t_1-t_2)) | ||
| ``` | ||
| with | ||
| ```math | ||
| \begin{aligned} | ||
| \gamma_{1}^{\nu} &= W-\nu i \mu,\\ | ||
| \eta_{1}^{\nu} &= \frac{\Gamma W}{2} f\left(\frac{iW}{k_B T}\right),\\ | ||
| \gamma_{l\neq 1}^{\nu} &= \zeta_l k_B T - \nu i \mu,\\ | ||
| \eta_{l\neq 1}^{\nu} &= -i k_B T \cdot \frac{\Gamma W^2}{-(\zeta_l k_B T)^2+W^2},\\ | ||
| f(x) &= \{\exp(x) + 1\}^{-1}, | ||
| \end{aligned} | ||
| ``` | ||
| where ``\zeta_l=(2 l - 1)\pi``. This can be constructed by the built-in function [`Fermion_Lorentz_Matsubara`](@ref): | ||
| ```julia | ||
| ds # coupling operator | ||
| Γ # coupling strength | ||
| μ # chemical potential of the environment | ||
| W # band-width of the environment | ||
| kT # the product of the Boltzmann constant k and the absolute temperature T | ||
| N # Number of exponential terms for each correlation functions (C^{+} and C^{-}) | ||
| bath = Fermion_Lorentz_Matsubara(ds, Γ, μ, W, kT, N - 1) | ||
| ``` | ||
|
|
||
| ## Padé Expansion | ||
| With Padé Expansion, the correlation function can be analytically solved and expressed as the following exponential terms: | ||
| ```math | ||
| C^\nu(t_1, t_2)=\sum_{l=1}^{\infty} \eta_l^\nu \exp(-\gamma_l^\nu (t_1-t_2)) | ||
| ``` | ||
| with | ||
| ```math | ||
| \begin{aligned} | ||
| \gamma_{1}^{\nu} &= W-\nu i \mu,\\ | ||
| \eta_{1}^{\nu} &= \frac{\Gamma W}{2} f\left(\frac{iW}{k_B T}\right),\\ | ||
| \gamma_{l\neq 1}^{\nu} &= \zeta_l k_B T - \nu i \mu,\\ | ||
| \eta_{l\neq 1}^{\nu} &= -i \kappa_l k_B T \cdot \frac{\Gamma W^2}{-(\zeta_l k_B T)^2+W^2},\\ | ||
| f(x) &= \frac{1}{2}-\sum_{l=2}^{N} \frac{2\kappa_l x}{x^2+\zeta_l^2}, | ||
| \end{aligned} | ||
| ``` | ||
| where the parameters ``\kappa_l`` and ``\zeta_l`` are described in [J. Chem. Phys. 134, 244106 (2011)](https://doi.org/10.1063/1.3602466) and ``N`` represents the number of exponential terms for ``C^{\nu=\pm}``. This can be constructed by the built-in function [`Fermion_Lorentz_Pade`](@ref): | ||
| ```julia | ||
| ds # coupling operator | ||
| Γ # coupling strength | ||
| μ # chemical potential of the environment | ||
| W # band-width of the environment | ||
| kT # the product of the Boltzmann constant k and the absolute temperature T | ||
| N # Number of exponential terms for each correlation functions (C^{+} and C^{-}) | ||
| bath = Fermion_Lorentz_Pade(ds, Γ, μ, W, kT, N - 1) | ||
| ``` |
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