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‎QuantumToolbox.jl/time_evolution/rabi.qmd‎

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---
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title: "Vacuum Rabi oscillation"
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author: Li-Xun Cai
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date: 2025-01-14 # last update (keep this comment as a reminder)
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engine: julia
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---
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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
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In this notebook, the usage of `QuantumToolbox.sesolve()` and `QuantumToolbox.mesolve()` will be demonstrated with Jaynes-Cummings model to observe Rabi Oscillation in the isolated case and the dissipative case. In dissipative case, a bosonic environment would interact with JC model's cavity and two-level atom.
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## `using` Packages
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```{julia}
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using QuantumToolbox
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using CairoMakie # For high-quality plotting
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```
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## Introduction to Jaynes-Cumming model
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Jaynes-Cummings (JC) model, simplest quantum mechanical model for light-matter interaction, describes an atom interacting with electromagnetic field. To simplify the interaction at the time of the model proposal, JC model considered a two-level atom interacting with a single bosonic mode (or you can consider it a single-mode cavity), i.e., a two-dimensional Hilbert space interacting with a Fock space. <!-- (with finite truncation $N$) -->
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The Hamiltonian of JC model is given by
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$$
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\hat{H}_\text{tot} = \hat{H}_{\text{a}} + \hat{H}_{\text{c}} + \hat{H}_{\text{int}}
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$$
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where
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- $\hat{H}_{\text{a}} = \frac{\omega_\text{a}}{2} \hat{\sigma}_z$: Hamiltonian for atom alone
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- $\hat{H}_{\text{c}} = \omega_\text{c} \hat{a}^\dagger \hat{a}$: Hamiltonian for cavity alone
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- $\hat{H}_{\text{int}} = \Omega \left( \hat{\sigma}^\dagger + \hat{\sigma} \right) \cdot \left( \hat{a}^\dagger + \hat{a} \right)$: Interaction Hamiltonian for coherent interaction
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with
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- $\omega_\text{a}$: Frequency gap of the two-level atom
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- $\omega_\text{c}$: Frequency of the cavity's EM mode (This is not specified whether to be in resonance with atom or not.)
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- $\Omega$ : Coupling strength between the atom and the cavity
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- $\hat{\sigma}_z$ : Pauli's $z$ matrix. Equivalent to $|e\rangle\langle e| - |g\rangle\langle g|$
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- $\hat{a}$ : Annihilation operator in Fock space <!-- $N$-truncated -->
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- $\hat{\sigma}$ : Lowering operator of atom. Equivalent to $|g\rangle\langle e|$
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By applying rotating wave approximation (RWA), the counter rotating terms ($\hat{\sigma} \cdot \hat{a}$ and its hermition conjugate) which rotate considerably faster than the others in the interaction picture is ignored, yeilding
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$$
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\hat{H}_\text{tot} \approx \hat{H}_{\text{a}} + \hat{H}_{\text{c}} + \Omega \left( \hat{\sigma} \cdot \hat{a}^\dagger + \hat{\sigma}^\dagger \cdot \hat{a} \right)
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$$
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### Usage of `QuantumToolbox` for JC model in general
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Since the packages are already imported with `using` at the beginning, we hereby start by defining some parameters and system Hamiltonian.
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```{julia}
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N = 15 # Fock space truncated dimension
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ωa = 1
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ωc = 1 * ωa # this frequency considers cavity and atom are IN resonance
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ωc_ = 0.8 * ωa # this frequency considers cavity and atom are OFF resonance
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σz = sigmaz() ⊗ qeye(N) # order of tensor product should be consistent throughout
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a = qeye(2) ⊗ destroy(N)
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Ω = 0.05
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σ = sigmam() ⊗ qeye(N)
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Ha = ωa / 2 * σz
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Hc = ωc * a' * a # the symbol `'` after a `QuantumObject` act as †
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Hc_ = ωc_ * a' * a
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Hint = Ω * (σ * a' + σ' * a)
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Htot = Ha + Hc + Hint
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Htot_ = Ha + Hc_ + Hint
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print(Htot)
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println("\n####################\n")
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print(Htot_)
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```
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## Isolated case
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For the case of JC system being isolated, i.e., with no interaction with the surrounding environment, the system's time-evolution is governed solely by Schrödinger equation $\hat{H}|\psi\rangle = \partial_t|\psi(t)\rangle$. Using `QuantumToolbox.sesolve()` is ideal for pure state evolution.
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```{julia}
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# define initial state ψ0
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ψ0 = basis(2,0) ⊗ fock(N, 1)
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ψ0 = ψ0 / norm(ψ0)
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tlist = 0:2:200 # a list of time points of interest
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# define a list of operators whose expectation value dynamics exhibit Rabi oscillation
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eop_ls = [
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a' * a, # number operator of cavity Fock space
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σ' * σ # excited state population in atom
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]
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sol = sesolve(Htot , ψ0, tlist; e_ops = eop_ls)
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sol_ = sesolve(Htot_, ψ0, tlist; e_ops = eop_ls)
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print(sol)
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println("\n####################\n")
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print(sol_)
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```
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```{julia}
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# Create the first figure for expectation value of photon number
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n = real.(sol.expect[1, :])
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n_ = real.(sol_.expect[1, :])
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fign = Figure(size = (600, 350))
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axn = Axis(
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fign[1, 1],
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title = "⟨n⟩",
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xlabel = "time [1/ωa]",
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ylabel = "expectation value",
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xlabelsize = 15,
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ylabelsize = 15,
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width = 400,
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height = 220
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)
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lines!(axn, tlist, n, label = "resonant")
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lines!(axn, tlist, n_, label = "off resonant")
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axislegend(axn; position = :rt, labelsize = 12)
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display(fign);
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```
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```{julia}
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# Create the second figure for excited state population
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e = real.(sol.expect[2, :])
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e_ = real.(sol_.expect[2, :])
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fige = Figure(size = (600, 350))
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axe = Axis(
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fige[1, 1],
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title = "⟨e⟩",
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xlabel = "time [1/ωa]",
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ylabel = "probability",
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xlabelsize = 15,
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ylabelsize = 15,
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width = 400,
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height = 220
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)
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ylims!(axe, 0, 1)
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lines!(axe, tlist, e, label = "resonant")
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lines!(axe, tlist, e_, label = "off resonant")
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axislegend(axe; position = :rb, labelsize = 12)
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display(fige);
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```
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## Dissipative case
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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.
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We start by reviewing the interaction Hamiltonians between the EM environment and atom/cavity
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- Atom: $$
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\hat{H}_{\text{a}}^\text{int} = \sum_l \alpha_l \left( \hat{b}_l + \hat{b}_l^\dagger \right) \left( \hat{\sigma} + \hat{\sigma}^\dagger \right)
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$$
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- Cavity: $$
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\hat{H}_{\text{c}}^\text{int} = \sum_l \beta_l \left( \hat{b}_l + \hat{b}_l^\dagger \right) \left( \hat{a} + \hat{a}^\dagger \right)
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$$
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where for the $l$-th mode
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- $\alpha_l$ is the coupling strength with the atom
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- $\beta_l$ is the coupling strength with the cavity
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- $\hat{b}_l$ is the annihilation operator
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Following the RWA approach previously mentioned and the standard procedure of Born-Markovian master equation, we obtain $\kappa$, the cavity dissipation rate, and $\gamma$, the atom dissipation rate.
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Therefore, the time evolution of the dissipative JC model can be described by the master equation
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$$
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\dot{\hat{\rho}} = -\frac{i}{\hbar} [\hat{H}, \hat{\rho}] + \sum_{i = 1}^4 \mathcal{D}[\sqrt{\Gamma_i} \hat{S}_i]\left(\hat{\rho}\right)
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$$
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where the $(\Gamma_i, \hat{S}_i)$ pairs are
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1. $\kappa \cdot n(\omega_c, T), \hat{a}^\dagger$
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2. $\kappa \cdot (1 + n(\omega_c, T)), \hat{a}$
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3. $\gamma \cdot n(\omega_a, T), \hat{\sigma}^\dagger$
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4. $\gamma \cdot (1 + n(\omega_a, T)), \hat{\sigma}$
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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.
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### Solve for evolutions in dissipative case
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We can now define variables in `julia` and solve the evolution of dissipative JC model
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```{julia}
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# Bose-Einstein distribution
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function nb(_ω, _KT)
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if _KT < 0
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println("Temperatur should not go below absolute zero, returned 0")
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_n = 0.0
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else
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_n = (_KT == 0) ? 0 : (1 / (exp(_ω / _KT) - 1))
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end
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return _n
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end
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# Collapse operators for interaction with the environment with variable
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# dissipation rates, cavity frequency, and thermal energy of the environment
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cop_ls(_γ, _κ, _ωc, _KT) = (
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√(_κ * nb(_ωc, _KT)) * a',
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√(_κ * (1 + nb(_ωc, _KT))) * a,
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√(_γ * nb(ωa, _KT)) * σ',
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√(_γ * (1 + nb(ωa, _KT))) * σ,
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)
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```
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```{julia}
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# use the same ψ0, tlist, and eop_ls from isolated case
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γ = 4e-3
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κ = 0.005
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KT = 0 # for theoretical vacuum EM field
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# `mesolve()` only has one additional keyword argument `c_ops` from `sesolve()`
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sol_me = mesolve(Htot, ψ0, tlist, cop_ls(γ, κ, ωc, KT), e_ops = eop_ls)
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sol_me_ = mesolve(Htot_, ψ0, tlist, cop_ls(γ, κ, ωc_, KT), e_ops = eop_ls)
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print(sol_me)
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println("\n####################\n")
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print(sol_me_)
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```
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```{julia}
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# Create the first figure for expectation value of photon number
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n_me = real.(sol_me.expect[1, :])
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n_me_ = real.(sol_me_.expect[1, :])
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fign_me = Figure(size = (600, 350))
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axn_me = Axis(
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fign_me[1, 1],
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title = "⟨n⟩",
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xlabel = "time [1/ωa]",
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ylabel = "expectation value",
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xlabelsize = 15,
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ylabelsize = 15,
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width = 400,
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height = 220
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)
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lines!(axn_me, tlist, n_me, label = "resonant")
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lines!(axn_me, tlist, n_me_, label = "off resonant")
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axislegend(axn_me; position = :rt, labelsize = 12)
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display(fign_me);
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```
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```{julia}
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# Create the second figure for atom excited state population
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e_me = real.(sol_me.expect[2, :])
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e_me_ = real.(sol_me_.expect[2, :])
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fige_me = Figure(size = (600, 350))
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axe_me = Axis(
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fige_me[1, 1],
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title = "⟨e⟩",
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xlabel = "time [1/ωa]",
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ylabel = "probability",
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xlabelsize = 15,
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ylabelsize = 15,
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width = 400,
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height = 220
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)
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ylims!(axe_me, 0, 1)
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lines!(axe_me, tlist, e_me, label = "resonant")
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lines!(axe_me, tlist, e_me_, label = "off resonant")
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axislegend(axe_me; position = :rb, labelsize = 12)
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display(fige_me);
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```
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### Note for RWA
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Without applying RWA, things are more interesting in the \emph{Ultrastrong Coupling} (USC) regime, where $\Omega \sim \omega_c$. In that regime, the \emph{localized} master equation (the one we used for this tutorial) fails. As that phenomenon is worth itself an independent tutorial, we will pass that to [this tutorial for `liouvillian_generalized()`]().
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## Version Information
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```{julia}
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QuantumToolbox.versioninfo()
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```

‎QuantumToolbox.jl/toc.md‎

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date: "Last Update"
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contents:
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- "time_evolution/*.qmd"
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- "rabi/*.qmd"
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---
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The following tutorials demonstrate and introduce specific functionality of `QuantumToolbox.jl`.

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