Add first version of rabi.qmd

QuantumToolbox.jl/time_evolution/rabi.qmd

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+---
+title: "Vacuum Rabi oscillation"
+author: Li-Xun Cai
+date: 2025-01-14 # last update (keep this comment as a reminder)
+
+engine: julia
+---
+
+Inspirations taken from this [QuTiP tutorial](https://nbviewer.org/urls/qutip.org/qutip-tutorials/tutorials-v5/time-evolution/004_rabi-oscillations.ipynb) by J.R. Johansson, P.D. Nation, and C. Staufenbiel
+
+In this notebook, the usage of `QuantumToolbox.sesolve` and `QuantumToolbox.mesolve` will be demonstrated with Jaynes-Cummings model to observe Rabi oscillations in the isolated case and the dissipative case. In dissipative case, a bosonic environment would interact with the cavity and two-level atom in JC model.
+
+## Introduction to Jaynes-Cumming model
+Jaynes-Cummings (JC) model, a simplest quantum mechanical model for light-matter interaction, describes an atom interacting with an external electromagnetic field. To simplify the interaction, JC model considered a two-level atom interacting with a single bosonic mode (or you can consider it a single-mode cavity).
+
+The Hamiltonian of JC model is given by
+
+$$
+\hat{H}_\text{tot} = \hat{H}_{\text{a}} + \hat{H}_{\text{c}} + \hat{H}_{\text{int}}
+$$
+
+where 
+
+- $\hat{H}_{\text{a}} = \frac{\omega_\text{a}}{2} \hat{\sigma}_z$: Hamiltonian for atom alone
+- $\hat{H}_{\text{c}} = \omega_\text{c} \hat{a}^\dagger \hat{a}$: Hamiltonian for cavity alone
+- $\hat{H}_{\text{int}} = \Omega \left( \hat{\sigma}^\dagger + \hat{\sigma} \right) \cdot \left( \hat{a}^\dagger + \hat{a} \right)$: Interaction Hamiltonian for coherent interaction
+
+with
+
+- $\omega_\text{a}$: Frequency of the two-level atom
+- $\omega_\text{c}$: Frequency of the cavity's EM mode (This is not specified whether to be in resonance with the atom or not.)
+- $\Omega$         : Coupling strength between the atom and the cavity
+- $\hat{\sigma}_z$ : Pauli-$Z$ matrix. Equivalent to $|e\rangle\langle e| - |g\rangle\langle g|$
+- $\hat{a}$        : Annihilation operator of single-mode cavity <!-- $N$-truncated -->
+- $\hat{\sigma}$   : Lowering operator of atom. Equivalent to $|g\rangle\langle e|$
+
+By applying [rotating wave approximation (RWA)](https://en.wikipedia.org/wiki/Rotating-wave_approximation), the counter rotating terms ($\hat{\sigma} \cdot \hat{a}$ and its Hermitian conjugate), which rotate considerably faster than the others in the interaction picture, are ignored, yielding
+
+$$
+\hat{H}_\text{tot} \approx \hat{H}_{\text{a}} + \hat{H}_{\text{c}} + \Omega \left( \hat{\sigma} \cdot \hat{a}^\dagger + \hat{\sigma}^\dagger \cdot \hat{a} \right)
+$$
+
+### Usage of `QuantumToolbox` for JC model in general
+
+##### import:
+```{julia}
+using QuantumToolbox
+import QuantumToolbox: ⊗, sigmaz, sigmam, destroy, qeye, basis, fock, n_thermal, mesolve, sesolve, versioninfo
+using CairoMakie
+import CairoMakie: Figure, Axis, lines!, axislegend, xlims!, display
+```
+
+```{julia}
+N = 2 # Fock space truncated dimension
+
+ωa = 1
+ωc = 1 * ωa    # considering cavity and atom are in resonance
+σz = sigmaz() ⊗ qeye(N) # order of tensor product should be consistent throughout
+a  = qeye(2)  ⊗ destroy(N)  
+Ω  = 0.05
+σ  = sigmam() ⊗ qeye(N)
+
+Ha = ωa / 2 * σz
+Hc = ωc * a' * a # the symbol `'` after a `QuantumObject` act as adjoint
+Hint = Ω * (σ * a' + σ' * a)
+
+Htot  = Ha + Hc + Hint
+
+print(Htot)
+```
+
+## Isolated case
+For the case of JC model being isolated, i.e., no interaction with the surrounding environment, the time-evolution is governed solely by Schrödinger equation $\hat{H}|\psi(t)\rangle = \partial_t|\psi(t)\rangle$. Using `QuantumToolbox.sesolve` is ideal for pure state evolution.
+
+For the context of [Rabi problem](https://en.wikipedia.org/wiki/Rabi_problem), we set the initial state $\psi_0 = |e\rangle \otimes |0\rangle$ where $|e\rangle$ is the excited state of atom and $|0\rangle$ is the vacuum state of cavity.
+
+```{julia}
+e_ket = basis(2,0) 
+ψ0 = e_ket ⊗ fock(N, 0)
+
+tlist = 0:2.5:1000 # a list of time points of interest
+
+# define a list of operators whose expectation value dynamics exhibit Rabi oscillation
+eop_ls = [
+    a' * a,                      # number operator of cavity
+    (e_ket * e_ket') ⊗ qeye(N), # excited state population in atom
+]
+
+sol = sesolve(Htot , ψ0, tlist; e_ops = eop_ls)
+print(sol)
+```
+
+Compare the dynamics of $| e \rangle\langle e|$ alongside $a^\dagger a$
+```{julia}
+n = real.(sol.expect[1, :])
+e = real.(sol.expect[2, :])
+fig_se = Figure(size = (600, 350))
+ax_se = Axis(
+    fig_se[1, 1],
+    xlabel = L"time $[1/\omega_a]$", 
+    ylabel = "expectation value", 
+    xlabelsize = 15, 
+    ylabelsize = 15,
+    width = 400,
+    height = 220
+)
+xlims!(ax_se, 0, 400)
+lines!(ax_se, tlist, n, label = L"$\langle a^\dagger a \rangle$")
+lines!(ax_se, tlist, e, label = L"$P_e$")
+axislegend(ax_se; position = :rt, labelsize = 15)
+display(fig_se);
+```
+
+In the above plot, the behaviour of the energy exchange between the atom and the cavity is clearly visible, addressing the Rabi problem.
+
+## Dissipative case
+
+In contrast to isolated evolution, a factual system interacts with its surrounding environments, resulting in energy/particle exchange. We are currently interested in observing JC model's Rabi oscillation with the addition of interaction with the external EM field. 
+
+We start by reviewing the interaction Hamiltonians between the EM environment and atom/cavity
+
+- Atom: $$
+        \hat{H}_{\text{a}}^\text{int} = \sum_l \alpha_l \left( \hat{b}_l + \hat{b}_l^\dagger \right) \left( \hat{\sigma} + \hat{\sigma}^\dagger \right)
+        $$
+- Cavity: $$
+        \hat{H}_{\text{c}}^\text{int} = \sum_l \beta_l \left( \hat{b}_l + \hat{b}_l^\dagger \right) \left( \hat{a} + \hat{a}^\dagger \right)
+        $$
+
+where for the $l$-th mode
+
+- $\alpha_l$ is the coupling strength with the atom
+- $\beta_l$ is the coupling strength with the cavity
+- $\hat{b}_l$ is the annihilation operator
+
+Following the RWA approach previously mentioned and the standard procedure of [Born-Markovian approximation](https://en.wikiversity.org/wiki/Open_Quantum_Systems/The_Quantum_Optical_Master_Equation), we obtain $\kappa$, the cavity dissipation rate, and $\gamma$, the atom dissipation rate. Therefore, the time evolution of the dissipative JC model can be described by the [Lindblad master equation](https://en.wikipedia.org/wiki/Lindbladian)
+
+$$
+\dot{\hat{\rho}} = -\frac{i}{\hbar} [\hat{H}, \hat{\rho}] + \sum_{i = 1}^4 \mathcal{D}[\sqrt{\Gamma_i} \hat{S}_i]\left(\hat{\rho}\right)
+$$
+
+where $\sqrt{\Gamma_i} \hat{S}_i$ are the collapse operators, given by 
+
+|$i$| $\Gamma_i$ | $\hat{S}_i$ |
+|---| :--------: | :---------: |
+|1|$\kappa \cdot n(\omega_c, T)$|$\hat{a}^\dagger$|
+|2|$\kappa \cdot [1 + n(\omega_c, T)]$|$\hat{a}$|
+|3|$\gamma \cdot n(\omega_a, T)$|$\hat{\sigma}^\dagger$|
+|4|$\gamma \cdot [1 + n(\omega_a, T)]$|$\hat{\sigma}$|
+
+with $n(\omega, T)$ being the Bose-Einstein distribution for the EM environment and 
+$$
+\mathcal{D}[\hat{\mathcal{O}}]\left(\cdot\right) = \hat{\mathcal{O}} \left(\cdot\right) \hat{\mathcal{O}}^\dagger - \frac{1}{2} \{ \hat{\mathcal{O}}^\dagger \hat{\mathcal{O}}, \cdot \}
+$$ 
+being the Lindblad dissipator.
+
+### Solve for evolutions in dissipative case
+
+We can now define variables in `julia` and solve the evolution of dissipative JC model 
+```{julia}
+# Collapse operators for interaction with the environment with variable dissipation rates 
+# and thermal energy of the environment. `n_thermal()` gives Bose-Einstein distribution
+cop_ls(_γ, _κ, _KT) = (
+    √(_κ * n_thermal(ωc, _KT)) * a', 
+    √(_κ * (1 + n_thermal(ωc, _KT))) * a, 
+    √(_γ * n_thermal(ωa, _KT)) * σ', 
+    √(_γ * (1 + n_thermal(ωa, _KT))) * σ, 
+)
+```
+
+```{julia}
+# use the same ψ0, tlist, and eop_ls from isolated case
+γ = 4e-3
+κ = 7e-3
+KT = 0 # for theoretical vacuum EM field
+
+# `mesolve()` only has one additional keyword argument `c_ops` from `sesolve()`
+sol_me  = mesolve(Htot,  ψ0, tlist, cop_ls(γ, κ, KT), e_ops = eop_ls)
+
+print(sol_me)
+```
+
+```{julia}
+n_me = real.(sol_me.expect[1, :])
+e_me = real.(sol_me.expect[2, :])
+
+fig_me = Figure(size = (600, 350))
+ax_me = Axis(
+    fig_me[1, 1],
+    xlabel = L"time $[1/\omega_a]$", 
+    ylabel = "expectation value", 
+    xlabelsize = 15, 
+    ylabelsize = 15,
+    width = 400,
+    height = 220
+)
+lines!(ax_me, tlist, n_me, label = L"\langle a^\dagger a \rangle")
+lines!(ax_me, tlist, e_me, label = L"$P_e$")
+axislegend(ax_me; position = :rt, labelsize = 15)
+display(fig_me);
+```
+
+From the above example, one can see that the dissipative system is losing energy over time and asymptoting to zero. We can further consider the near-vacuum environment with finite temperature.
+
+```{julia}
+sol_me_  = mesolve(Htot,  ψ0, tlist, cop_ls(γ, κ, 0.3 * ωa), e_ops = eop_ls) # replace KT with finite temperature
+
+n_me_ = real.(sol_me_.expect[1, :])
+e_me_ = real.(sol_me_.expect[2, :])
+fig_me_ = Figure(size = (600, 350))
+ax_me_ = Axis(
+    fig_me_[1, 1],
+    xlabel = L"time $[1/\omega_a]$", 
+    ylabel = "expectation value", 
+    xlabelsize = 15, 
+    ylabelsize = 15,
+    width = 400,
+    height = 220
+)
+lines!(ax_me_, tlist, n_me_, label = L"\langle a^\dagger a \rangle")
+lines!(ax_me_, tlist, e_me_, label = L"$P_e$")
+axislegend(ax_me_; position = :rt, labelsize = 15)
+display(fig_me_);
+```
+Despite the asymptotic behaviour persisting, one can see that they no longer approach zero and instead find a steady condition above zero. That is, the system eventually becomes thermalized by the environment.
+
+## Version Information
+```{julia}
+QuantumToolbox.versioninfo()
+```
+