improve math (\hat{} for operator)

cailixun · cailixun · commit 38326dafd8fa · 2025-06-17T23:20:36.000+08:00
diff --git a/docs/src/users_guide/time_evolution/brmesolve.md b/docs/src/users_guide/time_evolution/brmesolve.md
@@ -1,8 +1,6 @@
 # [Bloch-Redfield master equation](@id doc-TE:Bloch-Redfield-master-equation)
 
-## Introduction
-
-The Lindblad master equation introduced earlier is constructed so that it describes a physical evolution of the density matrix (i.e., trace and positivity preserving), but it does not provide a connection to any underlying microscopic physical model.
+The [Lindblad master equation](@ref doc-TE:Lindblad-Master-Equation-Solver) introduced earlier is constructed so that it describes a physical evolution of the density matrix (i.e., trace and positivity preserving), but it does not provide a connection to any underlying microscopic physical model.
 The Lindblad operators (collapse operators) describe phenomenological processes, such as for example dephasing and spin flips, and the rates of these processes are arbitrary parameters in the model.
 In many situations the collapse operators and their corresponding rates have clear physical interpretation, such as dephasing and relaxation rates, and in those cases the Lindblad master equation is usually the method of choice.
 
@@ -18,58 +16,58 @@ For a full derivation of the Bloch Redfield master equation, see e.g. [Cohen_Tan
 Here we present only a brief version of the derivation, with the intention of introducing the notation and how it relates to the implementation in `QuantumToolbox.jl`.
 
 
-## Brief Derivation and Definitions
+## [Brief Derivation and Definitions](@id doc-TE:Brief-Derivation-and-Definitions)
 
-The starting point of the Bloch-Redfield formalism is the total Hamiltonian for the system and the environment (bath): ``H = H_{\rm S} + H_{\rm B} + H_{\rm I}``, where ``H`` is the total system+bath Hamiltonian, ``H_{\rm S}`` and ``H_{\rm B}`` are the system and bath Hamiltonians, respectively, and ``H_{\rm I}`` is the interaction Hamiltonian.
+The starting point of the Bloch-Redfield formalism is the total Hamiltonian for the system and the environment (bath): ``\hat{H} = \hat{H}_{\rm S} + \hat{H}_{\rm B} + \hat{H}_{\rm I}``, where ``\hat{H}`` is the total system+bath Hamiltonian, ``\hat{H}_{\rm S}`` and ``\hat{H}_{\rm B}`` are the system and bath Hamiltonians, respectively, and ``\hat{H}_{\rm I}`` is the interaction Hamiltonian.
 
-The most general form of a master equation for the system dynamics is obtained by tracing out the bath from the von-Neumann equation of motion for the combined system (``\dot\rho = -i\hbar^{-1}[H, \rho]``). In the interaction picture the result is
+The most general form of a master equation for the system dynamics is obtained by tracing out the bath from the von-Neumann equation of motion for the combined system (``\frac{d}{dt}\hat{\rho} = -i\hbar^{-1}[\hat{H}, \hat{\rho}]``). In the interaction picture the result is
 
 ```math
-    \frac{d}{dt}\rho_S(t) = - \hbar^{-2}\int_0^t d\tau\;  {\rm Tr}_B [H_I(t), [H_I(\tau), \rho_S(\tau)\otimes\rho_B]],
+    \frac{d}{dt}\hat{\rho}_{\textrm{S}}(t) = - \hbar^{-2}\int_0^t d\tau\;  \textrm{Tr}_{\textrm{B}} [\hat{H}_{\textrm{I}}(t), [\hat{H}_{\textrm{I}}(\tau), \hat{\rho}_{\textrm{S}}(\tau)\otimes\hat{\rho}_{\textrm{B}}]],
 ```
 
-where the additional assumption that the total system-bath density matrix can be factorized as ``\rho(t) \approx \rho_S(t) \otimes \rho_B``.
+where the additional assumption that the total system-bath density matrix can be factorized as ``\hat{\rho}(t) \approx \hat{\rho}_{\textrm{S}}(t) \otimes \hat{\rho}_{\textrm{B}}``.
 This assumption is known as the Born approximation, and it implies that there never is any entanglement between the system and the bath, neither in the initial state nor at any time during the evolution.
 *It is justified for weak system-bath interaction.*
 
 The master equation above is non-Markovian, i.e., the change in the density matrix at a time ``t`` depends on states at all times ``\tau < t``, making it intractable to solve both theoretically and numerically.
-To make progress towards a manageable master equation, we now introduce the Markovian approximation, in which ``\rho_S(\tau)`` is replaced by ``\rho_S(t)``.
+To make progress towards a manageable master equation, we now introduce the Markovian approximation, in which ``\hat{\rho}_{\textrm{S}}(\tau)`` is replaced by ``\hat{\rho}_{\textrm{S}}(t)``.
 The result is the Redfield equation
 
 ```math
-    \frac{d}{dt}\rho_S(t) = - \hbar^{-2}\int_0^t d\tau\; {\rm Tr}_B [H_I(t), [H_I(\tau), \rho_S(t)\otimes\rho_B]],
+    \frac{d}{dt}\hat{\rho}_{\textrm{S}}(t) = - \hbar^{-2}\int_0^t d\tau\; \textrm{Tr}_{\textrm{B}} [\hat{H}_{\textrm{I}}(t), [\hat{H}_{\textrm{I}}(\tau), \hat{\rho}_{\textrm{S}}(t)\otimes\hat{\rho}_{\textrm{B}}]],
 ```
 
 which is local in time with respect the density matrix, but still not Markovian since it contains an implicit dependence on the initial state. By extending the integration to infinity and substituting ``\tau \rightarrow t-\tau``, a fully Markovian master equation is obtained:
 
 ```math
-    \frac{d}{dt}\rho_S(t) = - \hbar^{-2}\int_0^\infty d\tau\; {\rm Tr}_B [H_I(t), [H_I(t-\tau), \rho_S(t)\otimes\rho_B]].
+    \frac{d}{dt}\hat{\rho}_{\textrm{S}}(t) = - \hbar^{-2}\int_0^\infty d\tau\; \textrm{Tr}_{\textrm{B}} [\hat{H}_{\textrm{I}}(t), [\hat{H}_{\textrm{I}}(t-\tau), \hat{\rho}_{\textrm{S}}(t)\otimes\hat{\rho}_{\textrm{B}}]].
 ```
 
 The two Markovian approximations introduced above are valid if the time-scale with which the system dynamics changes is large compared to the time-scale with which correlations in the bath decays (corresponding to a "short-memory" bath, which results in Markovian system dynamics).
 
-The Markovian master equation above is still on a too general form to be suitable for numerical implementation. We therefore assume that the system-bath interaction takes the form ``H_I = \sum_\alpha A_\alpha \otimes B_\alpha`` and where ``A_\alpha`` are system operators and ``B_\alpha`` are bath operators.
+The Markovian master equation above is still on a too general form to be suitable for numerical implementation. We therefore assume that the system-bath interaction takes the form ``\hat{H}_{\textrm{I}} = \sum_\alpha \hat{A}_\alpha \otimes \hat{B}_\alpha`` and where ``\hat{A}_\alpha`` are system operators and ``\hat{B}_\alpha`` are bath operators.
 This allows us to write master equation in terms of system operators and bath correlation functions:
 
 ```math
-\begin{split}\frac{d}{dt}\rho_S(t) =
+\begin{split}\frac{d}{dt}\hat{\rho}_{\textrm{S}}(t) =
 -\hbar^{-2}
 \sum_{\alpha\beta}
 \int_0^\infty d\tau\;
 \left\{
-g_{\alpha\beta}(\tau) \left[A_\alpha(t)A_\beta(t-\tau)\rho_S(t) - A_\beta(t-\tau)\rho_S(t)A_\alpha(t)\right]
+g_{\alpha\beta}(\tau) \left[\hat{A}_\alpha(t)\hat{A}_\beta(t-\tau)\hat{\rho}_{\textrm{S}}(t) - \hat{A}_\beta(t-\tau)\hat{\rho}_{\textrm{S}}(t)\hat{A}_\alpha(t)\right]
 \right. \nonumber\\
 \left.
-+ g_{\alpha\beta}(-\tau) \left[\rho_S(t)A_\alpha(t-\tau)A_\beta(t) - A_\beta(t)\rho_S(t)A_\beta(t-\alpha)\right]
++ g_{\alpha\beta}(-\tau) \left[\hat{\rho}_{\textrm{S}}(t)\hat{A}_\alpha(t-\tau)\hat{A}_\beta(t) - \hat{A}_\beta(t)\hat{\rho}_{\textrm{S}}(t)\hat{A}_\beta(t-\alpha)\right]
 \right\},\end{split}
 ```
 
-where ``g_{\alpha\beta}(\tau) = {\rm Tr}_B\left[B_\alpha(t)B_\beta(t-\tau)\rho_B\right] = \left<B_\alpha(\tau)B_\beta(0)\right>``,
-since the bath state ``\rho_B`` is a steady state.
+where ``g_{\alpha\beta}(\tau) = \textrm{Tr}_{\textrm{B}}\left[\hat{B}_\alpha(t)\hat{B}_\beta(t-\tau)\hat{\rho}_{\textrm{B}}\right] = \langle\hat{B}_\alpha(\tau)\hat{B}_\beta(0)\rangle``,
+since the bath state ``\hat{\rho}_{\textrm{B}}`` is a steady state.
 
 In the eigenbasis of the system Hamiltonian, where ``A_{mn}(t) = A_{mn} e^{i\omega_{mn}t}``,
 ``\omega_{mn} = \omega_m - \omega_n`` and ``\omega_m`` are the eigenfrequencies
-corresponding the eigenstate ``\left|m\right>``, we obtain in matrix form in the Schrödinger picture
+corresponding the eigenstate ``\left|m\rangle``, we obtain in matrix form in the Schrödinger picture
 
 ```math
 \begin{aligned}
@@ -94,7 +92,7 @@ corresponding the eigenstate ``\left|m\right>``, we obtain in matrix form in the
 ```
 
 where the "sec" above the summation symbol indicate summation of the secular terms which satisfy ``|\omega_{ab}-\omega_{cd}| \ll \tau_ {\rm decay}``.
-This is an almost-useful form of the master equation. The final step before arriving at the form of the Bloch-Redfield master equation that is implemented in QuTiP, involves rewriting the bath correlation function ``g(\tau)`` in terms of the noise-power spectrum of the environment ``S(\omega) = \int_{-\infty}^\infty d\tau e^{i\omega\tau} g(\tau)``:
+This is an almost-useful form of the master equation. The final step before arriving at the form of the Bloch-Redfield master equation that is implemented in `QuantumToolbox.jl`, involves rewriting the bath correlation function ``g(\tau)`` in terms of the noise-power spectrum of the environment ``S(\omega) = \int_{-\infty}^\infty d\tau e^{i\omega\tau} g(\tau)``:
 
 ```math
     \int_0^\infty d\tau\; g_{\alpha\beta}(\tau) e^{i\omega\tau} = \frac{1}{2}S_{\alpha\beta}(\omega) + i\lambda_{\alpha\beta}(\omega),
@@ -114,6 +112,7 @@ where ``\lambda_{ab}(\omega)`` is an energy shift that is neglected here. The fi
 where
 
 ```math
+\begin{aligned}
     R_{abcd} =  -\frac{\hbar^{-2}}{2} \sum_{\alpha,\beta}
     \left\{
     \delta_{bd}\sum_nA^\alpha_{an}A^\beta_{nc}S_{\alpha\beta}(\omega_{cn})
@@ -126,15 +125,17 @@ where
     -
     A^\beta_{ac}A^\alpha_{db} S_{\alpha\beta}(\omega_{db})
     \right\},
+\end{aligned}
 ```
 
 is the Bloch-Redfield tensor.
 
-The Bloch-Redfield master equation in this form is suitable for numerical implementation. The input parameters are the system Hamiltonian ``H``, the system operators through which the environment couples to the system ``A_\alpha``, and the noise-power spectrum ``S_{\alpha\beta}(\omega)`` associated with each system-environment interaction term.
+The Bloch-Redfield master equation in this form is suitable for numerical implementation. The input parameters are the system Hamiltonian ``\hat{H}_{\textrm{S}}``, the system operators through which the environment couples to the system ``\hat{A}_\alpha``, and the noise-power spectrum ``S_{\alpha\beta}(\omega)`` associated with each system-environment interaction term.
 
-To simplify the numerical implementation we often assume that ``A_\alpha`` are Hermitian and that cross-correlations between different environment operators vanish, resulting in
+To simplify the numerical implementation we often assume that ``\hat{A}_\alpha`` are Hermitian and that cross-correlations between different environment operators vanish, resulting in
 
 ```math
+\begin{aligned}
     R_{abcd} =  -\frac{\hbar^{-2}}{2} \sum_{\alpha}
     \left\{
     \delta_{bd}\sum_nA^\alpha_{an}A^\alpha_{nc}S_{\alpha}(\omega_{cn})
@@ -147,22 +148,23 @@ To simplify the numerical implementation we often assume that ``A_\alpha`` are H
     -
     A^\alpha_{ac}A^\alpha_{db} S_{\alpha}(\omega_{db})
     \right\}.
+\end{aligned}
 ```
 
-## Bloch-Redfield master equation in `QuantumToolbox.jl`
+## [Bloch-Redfield master equation in `QuantumToolbox.jl`](@id Bloch-Redfield-master-equation-in-QuantumToolbox-jl)
 
 ### Preparing the Bloch-Redfield tensor
 
 In `QuantumToolbox.jl`, the Bloch-redfield master equation can be calculated using the function [`bloch_redfield_tensor`](@ref). 
-It takes two mandatory arguments: The system Hamiltonian ``H`` and a nested list `a_ops` consist of tuples as `(A, spec)` 
-with `A::QuantumObject{Operator}` being the operator ``A_\alpha`` and `spec::Function` being the spectral density functions ``S_\alpha(\omega)``.
+It takes two mandatory arguments: The system Hamiltonian ``\hat{H}`` and a nested list `a_ops` consist of tuples as `(A, spec)` 
+with `A::QuantumObject` being the [`Operator`](@ref) ``\hat{A}_\alpha`` and `spec::Function` being the spectral density function ``S_\alpha(\omega)``.
 
-It is possible to also get the term for a bath `\alpha` directly using [`brterm`](@ref). This function takes only one hermitian coupling operator ``A_\alpha`` and spectral response function ``S_\alpha(\omega)``.
+It is possible to also get the ``\alpha``-th term for the bath directly using [`brterm`](@ref). This function takes only one Hermitian coupling operator ``\hat{A}_\alpha`` and spectral response function ``S_\alpha(\omega)``.
 
 To illustrate how to calculate the Bloch-Redfield tensor, let's consider a two-level atom
 
 ```math
-    H = -\frac{1}{2}\Delta\sigma_x - \frac{1}{2}\varepsilon_0\sigma_z
+    \hat{H}_{\textrm{S}} = -\frac{1}{2}\Delta\hat{\sigma}_x - \frac{1}{2}\varepsilon_0\hat{\sigma}_z
 ```
 
 ```@setup brmesolve
@@ -179,15 +181,7 @@ CairoMakie.enable_only_mime!(MIME"image/svg+xml"())
 
 H = -Δ/2.0 * sigmax() - ε0/2 * sigmaz()
 
-function ohmic_spectrum(ω)
-    if ω == 0.   # dephasing noise
-        return γ1
-    elseif ω > 0 # relaxing noise  
-        return γ1 / 2 * ω / (2π)
-    else
-        return 0
-    end
-end
+ohmic_spectrum(ω) = (ω == 0.0) ? γ1 : γ1 / 2 * (ω / (2 * π)) * (ω > 0.0)
 
 R, U = bloch_redfield_tensor(H, [(sigmax(), ohmic_spectrum)])
 
@@ -199,18 +193,18 @@ Bloch-Refield tensor by passing the operators via the third argument `c_ops`
 like you would in the [`mesolve`](@ref) or [`mcsolve`](@ref) functions.
 For convenience, when the keyword argument `fock_basis = false`, the function [`bloch_redfield_tensor`](@ref) also returns the basis
 transformation operator `U`, the eigen vector matrix, since they are calculated in the
-process of calculating the Bloch-Redfield tensor `R`, and the `U` are usually
+process of generating the Bloch-Redfield tensor `R`, and the `U` are usually
 needed again later when transforming operators between the laboratory basis and the eigen basis.
 The tensor can be obtained in the laboratory basis by setting `fock_basis = true`,
-in that case, the transformation operator is not returned.
+in that case, the transformation operator `U` is not returned.
 
 ### Time evolution
 
 The evolution of a wave function or density matrix, according to the Bloch-Redfield
 master equation, can be calculated using the function [`mesolve`](@ref)
-using Bloch-Refield tensor in the laboratory basis instead of a liouvillian.
-For example, to evaluate the expectation values of the ``\sigma_x``,
-``\sigma_y``, and ``\sigma_z`` operators for the example above, we can use the following code:
+with Bloch-Refield tensor `R` in the laboratory basis instead of a [`liouvillian`](@ref).
+For example, to evaluate the expectation values of the ``\hat{\sigma}_x``,
+``\hat{\sigma}_y``, and ``\hat{\sigma}_z`` operators for the example above, we can use the following code:
 
 ```@example brmesolve
 Δ = 0.2 * 2 * π
@@ -245,14 +239,14 @@ fig, _ = render(sphere)
 fig
 ```
 
-The two steps of calculating the Bloch-Redfield tensor and evolving according to
+The two steps of calculating the Bloch-Redfield tensor `R` and evolving according to
 the corresponding master equation can be combined into one by using the function
 [`brmesolve`](@ref), in addition to the same arguments as [`mesolve`](@ref) and
 [`mcsolve`](@ref), the nested list of operator-spectrum
 tuple should be given under `a_ops`.
 
 ```@example brmesolve
-    output = brmesolve(H, ψ0, tlist, ((sigmax(),ohmic_spectrum),); e_ops=e_ops)
+sol = brmesolve(H, ψ0, tlist, ((sigmax(),ohmic_spectrum),); e_ops=e_ops)
 ```
 
-The resulting `output` is of the `struct` [`TimeEvolutionSol`](@ref) as [`mesolve`](@ref).
+The resulting `sol` is of the `struct` [`TimeEvolutionSol`](@ref) as [`mesolve`](@ref).
