- Notation following [1, 2].
$$\begin{gathered} \mathscr{H}= \sum_{q = -1}^1 \sum_{i = g, e} \mu_B g_i \mathbf{F}^i \cdot \mathbf{B} \ \mathscr{L}[\rho]=\sum_{q = -1}^1 \frac{\Gamma_{q}}{2}\left(2 \hat{\Sigma}{ q^{\prime}} \rho \hat{\Sigma}{q}^{\dagger}-\hat{\Sigma}{q}^{\dagger} \hat{\Sigma}{q} \rho -\rho \hat{\Sigma}{q}^{\dagger} \hat{\Sigma}{q} \right) \ J_{q}=0 \ \Gamma_{q} = 1 \ \hat{\Sigma}{q}^{\dagger}=\sum{m_g=-F_g}^{F_g} C_{m_g, q}^{F_e, F_g} \hat{\sigma}{F_e m_g+q, F_g m_g} \ \hat{\sigma}{F_e m_g+q, F_g m_g}=\left|F_e m_g+q \right> \left< F_g m_g\right| \ C_{m_g, q}^{F_e, F_g} = \left< F_g, m_g;1, q|F_e, m_g+q\right> \end{gathered}$$
To calculate the radiation power at
, where the electric field operator is defined as
$$\begin{gathered} \hat{\mathbf{E}}^{+}(\mathbf{r})= \mu_{0} \omega_{0}^2 \sum_{q = -1}^1 \mathbf{G} \left(\mathbf{r}, 0, \omega_{0} \right) \cdot \hat{\mathbf{e}}{q}^{*} \wp \hat{\Sigma}{j q} \ \mathbf{G}\left(\mathbf{r}, \omega_0\right)=\frac{e^{\mathrm{i} k_0 r}}{4 \pi k_0^2 r^3}\left[\left(k_0^2 r^2+\mathrm{i} k_0 r-1\right) \mathbb{1} + \left(-k_0^2 r^2-3 \mathrm{i} k_0 r+3\right) \frac{\mathbf{r} \otimes \mathbf{r}}{r^2}\right] \end{gathered}$$
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)
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
Here, (un)primed numbers are for the (excited)ground states. For $^{3} {D}{1} \rightarrow ^{3} P{ J^{zz} }$, where
$$ \left| \left< {}^{3}{D}{1} || \mathbf{d} || {}^3{P}{J^{\prime}} \right> \right|^2 \propto (2J^{\prime}+1)(2\cdot 2+1)\left| \left\lbrace\begin{array}{ccc} 2 & 1 & 1 \ J^{\prime} & 1 & 1 \ \end{array}\right\rbrace \right|^2 $$
This gives the ratio
The decay rate will be proportional to the cube of the frequency or the inverse cube of the wavelength. The transition wavelengths of
, which results in decay rates very close to [2].
We now consider the hyperfine structure. The matrix element has the same form as before.
$$ \begin{aligned} \left| \left< {}^{3}{D}{1}, F||\mathbf{d}|| {}^3{P}{J^{\prime}}, F^{\prime} \right> \right|^2 =& (2J^{\prime} + 1)(2 \cdot 2 + 1)(2F^{\prime} + 1)(2 \cdot 1 + 1) \ & \times \left| \left\lbrace\begin{array}{ccc} 2 & 1 & 1 \ J^{\prime} & 1 & 1 \ \end{array}\right\rbrace \left\lbrace\begin{array}{ccc} 1 & J^{\prime} & 1 \ F^{\prime} & F & 9/2 \end{array}\right\rbrace\right|^2 \left| \left< L=2||\mathbf{d}|| L^{\prime}=1\right> \right|^2 \end{aligned} $$
-
D. A. Steck, Quantum and Atom Optics (2022).
-
A. Asenjo-Garcia, H. J. Kimble, D. E. Chang, Proc. Natl. Acad. Sci. U.S.A. 116, 25503–25511 (2019).
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B. Zhu, J. Cooper, J. Ye, A. M. Rey, Phys. Rev. A. 94, 023612 (2016).
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CollectiveSpins.jl
theoretical description page. -->
To (locally) reproduce this project, do the following:
- Download this code base. Notice that raw data are typically not included in the git-history and may need to be downloaded independently.
- 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.