# Optical pumping in rubidium-87 - Impure polarization (a mixture of σ +, σ − and π light) on the (F g = 2 ↔ F e = 3) transition in the D2 line

Anders Kastberg, Fred Atoneche

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Anders Kastberg, Fred Atoneche. Optical pumping in rubidium-87 - Impure polarization (a mixture of σ +, σ − and π light) on the (F g = 2 ↔ F e = 3) transition in the D2 line. 2016, 10.6084/m9 figshare.3859260.v3. hal-01635977

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Submitted on 16 Nov 2017

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Optical pumping in rubidium-87 — Impure polarization (a
mixture of σ+, σ− and π light) on the (Fg = 2 ↔ Fc = 3) transition
in the D2 line

Fred Atoneche

*Laboratory of Research on Advanced Materials and Non-linear Sciences,*

*Department of Physics, Faculty of Science,*

*University of Buea, P.O. Box 63, Buea, Cameroon*

Anders Kastberg

Université Nice Sophia Antipolis, CNRS,

Laboratoire de Physique de la Matière Condensée

UMR 7336, Parc Valrose, 06100 Nice, France

(Dated: November 16, 2017)

Abstract

We present results of a method for calculating optical pumping rates in $^{87}$Rb, as described in
[1]. We use light with an impure polarization (i.e. a mixture of $\sigma^{+}$, $\sigma^{-}$ and $\pi$ light) to populate
a stretched angular momentum state in the $F_{g} = 2$ hyperfine structure level of the ground state.
The method is shown to function well for the studied system. Absolute scale level populations as
functions of time, detuning, intensity, relative intensity of polarizations, and starting parameters
are derived.
# I. INTRODUCTION

In a separate publication [1], we have described a simplified method for quantitative calculations of the evolution of state populations in an atomic system undergoing optical pumping. In this short communication, we provide details for a specific case, including the evolution matrix, a link to the used programming code (using Mathematica™) [2], and the obtained results.

The specific case studied is:

* Rb$^{87}$, with nuclear spin $I = 3/2$

* Preparation of the state 5s $^2$S$_{1/2}$, $F_g = 2$, $M_g = +2$.

* Pumping with a mixture of $\sigma^+$, $\sigma^-$ and $\pi$ polarizations of light on the transition $F_g = 2 \leftrightarrow F_e = 3$ on the D2-line (upper fine-structure state 5p $^2$P$_{3/2}$).

* No external magnetic field, and low saturation.

For the underlying theory, and also for some nomenclature, we refer to [*to be published*].

# II. EVOLUTION MATRICES

We consider atoms populating a statistical mixture of the Zeeman states $M_g = +2$, $M_g = +1$, $M_g = 0$, $M_g = -1$, $M_g = -2$ of the $F_g = 2$ level. These five states will constitute our state space, and as derived in [*to be published*], the state population can be described by a matrix obtained from the addition of three matrices corresponding to $\sigma^+$, $\sigma^-$ and $\pi$ polarizations. The three matrices for $\sigma^+$, $\sigma^-$ and $\pi$ are weighted by the relative light intensities $c_+$, $c_-$, and $c_\pi$ respectively ($c_+ + c_- + c_0 = 1$).

The three matrices are:

$$ M_+ = \frac{1}{225} \begin{pmatrix} 0 & 50 & 6 & 0 & 0 \\ 0 & -50 & 48 & 9 & 0 \\ 0 & 0 & -54 & 27 & 6 \\ 0 & 0 & 0 & -36 & 8 \\ 0 & 0 & 0 & 0 & -14 \end{pmatrix} \quad (1) $$
$$ M_{-} = \frac{1}{225} \begin{pmatrix} -14 & 0 & 0 & 0 & 0 \\ 8 & -36 & 0 & 0 & 0 \\ 6 & 27 & -54 & 0 & 0 \\ 0 & 9 & 48 & -50 & 0 \\ 0 & 0 & 6 & 50 & 0 \end{pmatrix} \qquad (2) $$

$$ M_{\pi} = \frac{1}{225} \begin{pmatrix} -50 & 8 & 0 & 0 & 0 \\ 50 & -56 & 27 & 0 & 0 \\ 0 & 48 & -54 & 48 & 0 \\ 0 & 0 & 27 & -56 & 50 \\ 0 & 0 & 0 & 8 & -50 \end{pmatrix}. \qquad (3) $$

The total evolution is given by:

$$ \frac{d}{dt} \mathbf{G}(t) = M R \mathbf{G}(t) = (c_{+} M_{\sigma^{+}} + c_{-} M_{\sigma^{-}}) R \mathbf{G}(t). \qquad (4) $$

$\mathbf{G}(t)$ is a vector with the populations in the five states as its components and $R$ is the intensity dependent total scattering rate:

$$ R = \frac{\Gamma}{2} \frac{(I/I_{\text{sat}})}{1 + (I/I_{\text{sat}}) + (2\Delta/\Gamma)^2}. \qquad (5) $$

$\Gamma$ is the natural linewidth of the excited states (for all states belonging to the term 5p $^2$P$_{3/2}$, ($\Gamma = 2\pi \times 6.0666$ MHz [3]), $I$ the intensity, $I_{\text{sat}}$ the saturation intensity ($I_{\text{sat}} = 1.669$ mW/cm$^2$ for the case with a cycling transition and a coupling coefficient of one), and $\Delta$ is the detuning ($\Delta = 0$ for all calculations in the present article).

## A. Solution of the evolution matrix equations

The evolution equation (eq. 4) is simple to solve analytically. Explicit expressions, as functions of all involved parameters (e.g. relative intensity distributions for the different polarizations) become very lengthy. With a modest mathematical program, and limitations in computer memory, it is computationally more economical to first set the parameters for the case in hand, and to then solve for just the numerical matrix. In that case the problem is easily handled by desktop mathematical packages. In [2] we provide an annotated code for this in Mathematica$^{TM}$.
<figure><img src="image_2.png" /><figcaption>FIG. 1. Relative populations in the five sub-levels, as functions of time in microseconds. This has been computed by a solution of Eq. 4. Full blue line $G_{+2}$, dashed blue line $G_{+1}$, full green line $G_0$, dashed red line $G_{-1}$, full red line $G_{-2}$.</figcaption></figure>

### III. RESULTS

Solving eq. 4 gives the evolution of the five involved Zeeman states. Note that the only specific atomic characteristic that is included in the analysis, which is different from another alkali atom with the same nuclear spin ($I = 3/2$), is the natural linewidth $\Gamma$.

Figure 1 shows an example of an evolution of the populations of the five involved levels, based on the solution of the evolution equation (Eq. 4), with the relative intensities set to $c_π = 0.025$, $c_+ = 0.95$ and $c_- = 0.025$. In this example, we have assumed that the initial populations are the same in all five levels, and that the total population normalized to unity ($G_{-2} = G_{-1} = G_0 = G_{+1} = G_{+2} = 0.2$). Furthermore, we set the intensity to a fifth of the saturation intensity ($I = I_{sat}/5 \Rightarrow R \approx \Gamma/10$), and the detuning to zero. The population of the optically pumped level, $G_{+2}$, grows monotonically towards 1, as all other levels are gradually depleted. For these particular parameters, the asymptotic values of the
populations are:

$$
\begin{align*}
G_{-2}(t \to \infty) &= 0.000083 \\
G_{-1}(t \to \infty) &= 0.00042 \\
G_0(t \to \infty) &= 0.0040 \\
G_{+1}(t \to \infty) &= 0.032 \\
G_{+2}(t \to \infty) &= 0.96. \tag{6}
\end{align*}
$$

IV. CONCLUSION

Using the method developed in [1], we have calculated the state population evolution for
optical pumping to a stretched state on the $F_g = 2 \leftrightarrow F_e = 3$ -transition, using a mixture
of $\sigma^+$ and $\sigma^-$ and $\pi$ light. The method works well when applied to this system and yields
level populations on an absolute scale. It is suitable and convenient for quickly estimating
population dynamics in an optical pumping experiment.

[1] F. Atoneche and A. Kastberg, Eur. J. Phys. **38**, 045703 (2017), URL http://stacks.iop.org/0143-0807/38/i=4/a=045703.

[2] A. Kastberg and F. Atoneche (2016), URL https://figshare.com/articles/Optical_pumping_with_arbitrarily_polarized Light_for_the_D2_transition_in_Rb-87_Fg_2_-_Fg_3_-_Mathematica_code/3858918.

[3] D. A. Steck, *Rubidium 87 D Line Data* (2001), URL http://steck.us/alkalidata/rubidium87numbers.pdf.