README_old.md

DecayDynamics

Single particle, multi level

• Notation following [1, 2].

$$\begin{gathered} \mathscr{H}= \sum_{q = -1}^1 \sum_{i = g, e} \mu_B g_i \mathbf{F}^i \cdot \mathbf{B} \ \mathscr{L}[\rho]=\sum_{q = -1}^1 \frac{\Gamma_{q}}{2}\left(2 \hat{\Sigma}{ q^{\prime}} \rho \hat{\Sigma}{q}^{\dagger}-\hat{\Sigma}{q}^{\dagger} \hat{\Sigma}{q} \rho -\rho \hat{\Sigma}{q}^{\dagger} \hat{\Sigma}{q} \right) \ J_{q}=0 \ \Gamma_{q} = 1 \ \hat{\Sigma}{q}^{\dagger}=\sum{m_g=-F_g}^{F_g} C_{m_g, q}^{F_e, F_g} \hat{\sigma}{F_e m_g+q, F_g m_g} \ \hat{\sigma}{F_e m_g+q, F_g m_g}=\left|F_e m_g+q \right> \left< F_g m_g\right| \ C_{m_g, q}^{F_e, F_g} = \left< F_g, m_g;1, q|F_e, m_g+q\right> \end{gathered}$$

$C_{m_g, q}$ is the Clebsch–Gordan coefficient, $\hat{\mathbf{e}}_q$ is the polarization vector in Cartesian coordinate. $\Gamma_q$ does not depend on a specific choise of the hyperfine state but depends on the fine structure state.

To calculate the radiation power at $\mathbf{r}$, evaluate

$$\begin{gathered} I(\mathbf{r}) = \left< \psi \right| \hat{\mathbf{E}}^{-} (\mathbf{r}) \cdot \hat{\mathbf{E}}^{+} (\mathbf{r}) \left| \psi \right> \end{gathered}$$

, where the electric field operator is defined as

$$\begin{gathered} \hat{\mathbf{E}}^{+}(\mathbf{r})= \mu_{0} \omega_{0}^2 \sum_{q = -1}^1 \mathbf{G} \left(\mathbf{r}, 0, \omega_{0} \right) \cdot \hat{\mathbf{e}}{q}^{*} \wp \hat{\Sigma}{j q} \ \mathbf{G}\left(\mathbf{r}, \omega_0\right)=\frac{e^{\mathrm{i} k_0 r}}{4 \pi k_0^2 r^3}\left[\left(k_0^2 r^2+\mathrm{i} k_0 r-1\right) \mathbb{1} + \left(-k_0^2 r^2-3 \mathrm{i} k_0 r+3\right) \frac{\mathbf{r} \otimes \mathbf{r}}{r^2}\right] \end{gathered}$$

Cascaded multilevel

Schematics

flowchart TD
    A(3D1, F = F_1, level-1) -->|2 μs| B(3P1, F = F_2, level-2) --> |20 μs| C(1S0, F = F_3, level-3)

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Branching ratio

Calculations can be found in notebooks/branching_ratio_calculation.jl.

From the reference [1], equation (7.283), the reduced dipole moment of the transition between two different fine structure states are given by

$$\begin{aligned} \left< J||\mathbf{d}|| J^{\prime}\right> & \equiv \left< L S J||\mathbf{d}|| L^{\prime} S J^{\prime} \right> \\ & = \left< L||\mathbf{d}|| L^{\prime} \right> (-1)^{J^{\prime}+L+1+S} \sqrt{\left( 2 J^{\prime}+1 \right) (2 L+1)} \left\lbrace \begin{array}{ccc} L & L^{\prime} & 1 \\ J^{\prime} & J & S \\ \end{array} \right\rbrace \end{aligned}$$

Here, (un)primed numbers are for the (excited)ground states. For $^{3} {D}{1} \rightarrow ^{3} P{ J^{zz} }$, where $J^{zz} \in (0, 1, 2)$ and $J = 1$, $L=2$, $L'=1$, $S=1$ are fixed. The ratio between squared dipole matrix elements of different ${}^3{P}_{J^{zz}}$ states will be

$$ \left| \left< {}^{3}{D}{1} || \mathbf{d} || {}^3{P}{J^{\prime}} \right> \right|^2 \propto (2J^{\prime}+1)(2\cdot 2+1)\left| \left\lbrace\begin{array}{ccc} 2 & 1 & 1 \ J^{\prime} & 1 & 1 \ \end{array}\right\rbrace \right|^2 $$

This gives the ratio $\frac{5}{9}:\frac{5}{12}:\frac{1}{36}$. The actual decay rate will depends on the frequency of the transition. For the decay rate $\Gamma$ from $e$ to $g$,

$$ \Gamma = \frac{\omega_0^3}{3\pi \epsilon_0 \hbar c^3}| \left< g|\mathbf{d}|e \right>|^2. $$

The decay rate will be proportional to the cube of the frequency or the inverse cube of the wavelength. The transition wavelengths of ${}^3{P}_{J^{\prime}}$ are (2.60315, 2.7362, 3.06701) μm. The actual decay ratio will be

$$ \frac{5}{9}\frac{1}{2.6^3}:\frac{5}{12}\frac{1}{2.74^3}:\frac{1}{36}\frac{1}{3.07^3} = 0.597:0.385:0.018 $$

, which results in decay rates very close to [2].

We now consider the hyperfine structure. The matrix element has the same form as before.

$$ \begin{aligned} \left< F||\mathbf{d}|| F^{\prime}\right> & \equiv\left< J I F||\mathbf{d}|| J^{\prime} I F^{\prime}\right> \\ & = \left< J||\mathbf{d}|| J^{\prime}\right> (-1)^{F^{\prime}+J+1+I} \sqrt{\left(2 F^{\prime}+1\right)(2 J+1)}\left\lbrace\begin{array}{ccc} J & J^{\prime} & 1 \\ F^{\prime} & F & I \end{array}\right\rbrace \\ & = \left< L||\mathbf{d}|| L^{\prime}\right>(-1)^{J^{\prime}+L+1+S} \sqrt{\left(2 J^{\prime}+1\right)(2 L+1)}\left\lbrace\begin{array}{ccc} L & L^{\prime} & 1 \\ J^{\prime} & J & S \\ \end{array}\right\rbrace \\ & \times (-1)^{F^{\prime}+J+1+I} \sqrt{\left(2 F^{\prime}+1\right)(2 J+1)}\left\lbrace\begin{array}{ccc} J & J^{\prime} & 1 \\ F^{\prime} & F & I \end{array}\right\rbrace . \end{aligned} $$

$$ \begin{aligned} \left| \left< {}^{3}{D}{1}, F||\mathbf{d}|| {}^3{P}{J^{\prime}}, F^{\prime} \right> \right|^2 =& (2J^{\prime} + 1)(2 \cdot 2 + 1)(2F^{\prime} + 1)(2 \cdot 1 + 1) \ & \times \left| \left\lbrace\begin{array}{ccc} 2 & 1 & 1 \ J^{\prime} & 1 & 1 \ \end{array}\right\rbrace \left\lbrace\begin{array}{ccc} 1 & J^{\prime} & 1 \ F^{\prime} & F & 9/2 \end{array}\right\rbrace\right|^2 \left| \left< L=2||\mathbf{d}|| L^{\prime}=1\right> \right|^2 \end{aligned} $$

Summary plot:

Many particle, multilevel

Two particles with hyperfine states

$$\begin{gathered} \rho = \rho_1 \otimes \rho_2 \\ \end{gathered}$$

Questions

Reference

1. D. A. Steck, Quantum and Atom Optics (2022).

2. A. Asenjo-Garcia, H. J. Kimble, D. E. Chang, Proc. Natl. Acad. Sci. U.S.A. 116, 25503–25511 (2019).

3. B. Zhu, J. Cooper, J. Ye, A. M. Rey, Phys. Rev. A. 94, 023612 (2016).

4. CollectiveSpins.jl theoretical description page. -->

Activating this project

To (locally) reproduce this project, do the following:

0. Download this code base. Notice that raw data are typically not included in the git-history and may need to be downloaded independently.
1. Open a Julia console and do:

julia> using Pkg
julia> Pkg.add("DrWatson") # install globally, for using `quickactivate`
julia> Pkg.activate("path/to/this/project")
julia> Pkg.instantiate()

This will install all necessary packages for you to be able to run the scripts and everything should work out of the box, including correctly finding local paths.

You may notice that most scripts start with the commands:

using DrWatson
@quickactivate "DecayDynamics"

which auto-activate the project and enable local path handling from DrWatson.