[Docs] Update documentation for time evolution (qutip#241)

* update documentation for time evolution

* separate `sesolve` and `mesolve` in docs

Author: ytdHuang

docs/make.jl

@@ -6,6 +6,8 @@ using Documenter
 
 DocMeta.setdocmeta!(QuantumToolbox, :DocTestSetup, :(using QuantumToolbox); recursive = true)
 
+const DRAFT = false # set `true` to disable cell evaluation
+
 const MathEngine = MathJax3(
     Dict(
         :loader => Dict("load" => ["[tex]/physics"]),
@@ -33,6 +35,12 @@ const PAGES = [
         "Tensor Products and Partial Traces" => "users_guide/tensor.md",
         "Time Evolution and Dynamics" => [
             "Introduction" => "users_guide/time_evolution/intro.md",
+            "Time Evolution Solutions" => "users_guide/time_evolution/solution.md",
+            "Schrödinger Equation Solver" => "users_guide/time_evolution/sesolve.md",
+            "Lindblad Master Equation Solver" => "users_guide/time_evolution/mesolve.md",
+            "Monte-Carlo Solver" => "users_guide/time_evolution/mcsolve.md",
+            "Stochastic Solver" => "users_guide/time_evolution/stochastic.md",
+            "Solving Problems with Time-dependent Hamiltonians" => "users_guide/time_evolution/time_dependent.md",
         ],
         "Solving for Steady-State Solutions" => "users_guide/steadystate.md",
         "Symmetries" => [],
@@ -66,7 +74,8 @@ makedocs(;
         assets = ["assets/favicon.ico"],
         mathengine = MathEngine,
         size_threshold_ignore = ["api.md"],
-    )
+    ),
+    draft = DRAFT,
 )
 
 deploydocs(; repo = "github.com/qutip/QuantumToolbox.jl", devbranch = "main")

docs/src/users_guide/time_evolution/intro.md

@@ -1,3 +1,31 @@
 # [Time Evolution and Quantum System Dynamics](@id doc:Time-Evolution-and-Quantum-System-Dynamics)
 
-This page is still under construction, please visit [API](@ref doc-API) first.
+**Table of contents**
+
+```@contents
+Pages = [
+    "intro.md",
+    "solution.md",
+    "sesolve.md",
+    "mesolve.md",
+    "mcsolve.md",
+    "stochastic.md",
+    "time_dependent.md",
+]
+Depth = 2:3
+```
+
+## [Introduction](@id doc-TE:Introduction)
+
+Although in some cases, we want to find the stationary states of a quantum system, often we are interested in the dynamics: how the state of a system or an ensemble of systems evolves with time. `QuantumToolbox` provides many ways to model dynamics.
+
+There are two kinds of quantum systems: open systems that interact with a larger environment and closed systems that do not. In a closed system, the state can be described by a state vector. When we are modeling an open system, or an ensemble of systems, the use of the density matrix is mandatory.
+
+The following table lists the solvers provided by `QuantumToolbox` for dynamic quantum systems and the corresponding type of solution returned by the solver:
+
+| **Equation** | **Function Call** | **Returned Solution** |
+|:-------------|:------------------|:----------------------|
+| Unitary evolution, Schrödinger equation | [`sesolve`](@ref) | [`TimeEvolutionSol`](@ref) |
+| Lindblad master eqn. or Von Neuman eqn. | [`mesolve`](@ref) | [`TimeEvolutionSol`](@ref) |
+| Monte Carlo evolution | [`mcsolve`](@ref) | [`TimeEvolutionMCSol`](@ref) |
+| Stochastic Schrödinger equation | [`ssesolve`](@ref) | [`TimeEvolutionSSESol`](@ref) |

docs/src/users_guide/time_evolution/mcsolve.md

@@ -0,0 +1,5 @@
+# Time Evolution and Quantum System Dynamics
+
+## [Monte-Carlo Solver](@id doc-TE:Monte-Carlo-Solver)
+
+This page is still under construction, please visit [API](@ref doc-API) first.

docs/src/users_guide/time_evolution/mesolve.md

@@ -0,0 +1,9 @@
+# Time Evolution and Quantum System Dynamics
+
+## [Lindblad Master Equation Solver](@id doc-TE:Lindblad-Master-Equation-Solver)
+
+This page is still under construction, please visit [API](@ref doc-API) first.
+
+### Von Neumann equation
+
+### The Lindblad master equation

docs/src/users_guide/time_evolution/sesolve.md

@@ -0,0 +1,5 @@
+# Time Evolution and Quantum System Dynamics
+
+## [Schrödinger Equation Solver](@id doc-TE:Schrödinger-Equation-Solver)
+
+This page is still under construction, please visit [API](@ref doc-API) first.

docs/src/users_guide/time_evolution/solution.md

@@ -0,0 +1,9 @@
+# Time Evolution and Quantum System Dynamics
+
+## [Time Evolution Solutions](@id doc-TE:Time-Evolution-Solutions)
+
+This page is still under construction, please visit [API](@ref doc-API) first.
+
+### Solution
+
+### Multiple trajectories solution

docs/src/users_guide/time_evolution/stochastic.md

@@ -0,0 +1,7 @@
+# Time Evolution and Quantum System Dynamics
+
+## [Stochastic Solver](@id doc-TE:Stochastic-Solver)
+
+This page is still under construction, please visit [API](@ref doc-API) first.
+
+### Stochastic Schrodinger equation

docs/src/users_guide/time_evolution/time_dependent.md

@@ -0,0 +1,5 @@
+# Time Evolution and Quantum System Dynamics
+
+## [Solving Problems with Time-dependent Hamiltonians](@id doc-TE:Solving-Problems-with-Time-dependent-Hamiltonians)
+
+This page is still under construction, please visit [API](@ref doc-API) first.