# Optical pumping in rubidium-87 - Linear polarization on the (F g = 2 ↔ F e = 2) transition in the D2 line

Fred Atoneche, Anders Kastberg

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Fred Atoneche, Anders Kastberg. Optical pumping in rubidium-87 - Linear polarization on the (F g = 2 ↔ F e = 2) transition in the D2 line. 2016, 10.6084/m9 figshare.3756801.v2. hal-01635975

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Submitted on 16 Nov 2017

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Optical pumping in rubidium-87 — Linear polarization on the
($F_g = 2 \leftrightarrow F_e = 2$) 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 linearly polarized light on the ($F_g = 2 \leftrightarrow F_e = 2$) transition to populate an angular
momentum state in the $F_g = 2$ hyperfine structure level of the ground state, with magnetic quantum
number $M_g = 0$ — that is a clock state. The open decay path to $F_g = 1$ necessitates repumping
light, and this is accordingly included in the calculation. The method is shown to function well
for the studied system. Absolute scale level populations as functions of time, detunings, intensities
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$_{1/2}$, $F_g = 2$, $M_g = 0$.

* Pumping with linearly polarized light on the transition $F_g = 2 \leftrightarrow F_e = 2$ on the D2-line (upper fine-structure state 5p $^2$P$_{3/2}$).

* Repumping with linearly polarized light on the transition $F_g = 1 \leftrightarrow F_e = 2$.

* Pure polarizations, no external magnetic field, and low saturation.

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

# II. EVOLUTION MATRIX

We consider atoms populating a statistical mixture of the Zeeman states $M_g = +2$, $M_g = +1$, $M_g = 0$, $M_g = -1$, and $M_g = -2$ of the $F_g = 2$ ground state hyperfine level, and also the states $M_g = +1$, $M_g = 0$, and $M_g = -1$ of the $F_g = 1$ level. These eight states will constitute our state space, and as derived in [*to be published*], the state population can be described by the matrix in Eq. 2. In this equation, $G(t)$ is a vector with the populations in the eight states as its components. $R_2$ is the intensity dependent scattering rate for the pumping transition (from $F_g = 2$), and $R_1$ that for the repumping transition (from $F_g = 1$):

$$R_{1,2} = \frac{\Gamma}{2} \frac{(I_{1,2}/I_{\text{sat}})}{1 + (I_{1,2}/I_{\text{sat}}) + (2\Delta_{1,2}/\Gamma)^2} \quad (1)$$

$\Gamma$ 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]). $I_2$ and $I_1$ are the intensities for the pumping and repumping transitions respectively, and $I_{\text{sat}}$ is the saturation intensity ($I_{\text{sat}} = 1.669$ mW/cm$^2$ for the case with a cycling transition and a coupling coefficient of one). $\Delta_2$ and $\Delta_1$ are the detunings for the two transitions ($\Delta_2 = \Delta_2 = 0$ for all calculations in the present article).
$$ \frac{d}{dt} \mathbf{G}(t) = \frac{1}{144} \begin{pmatrix} -32 R_2 & 2 R_2 & 0 & 0 & 0 & 6 R_1 & 0 & 0 \\ 8 R_2 & -11 R_2 & 0 & 0 & 0 & 3 R_1 & 12 R_1 & 0 \\ 0 & 3 R_2 & 0 & 3 R_2 & 0 & 9 R_1 & 0 & 9 R_1 \\ 0 & 0 & 0 & -11 R_2 & 8 R_2 & 0 & 12 R_1 & 3 R_1 \\ 0 & 0 & 0 & 2 R_2 & -32 R_2 & 0 & 0 & 6 R_1 \\ 24 R_2 & 3 R_2 & 0 & 0 & 0 & -27 R_1 & 4 R_1 & 0 \\ 0 & 3 R_2 & 0 & 3 R_2 & 0 & 9 R_1 & -32 R_1 & 9 R_1 \\ 0 & 0 & 0 & 3 R_2 & 24 R_2 & 0 & 4 R_1 & -27 R_1 \end{pmatrix} \mathbf{G}(t). \quad (2) $$

## A. Solution of the evolution matrix equations

The evolution equation (Eq. 2) is simple to solve analytically. Explicit expressions, as functions of all involved parameters 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™.

## III. RESULTS

Solving eq. 2 gives the evolution of the eight 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 $\Gamma$. This means that eq. 2 will work equally well for all alkali isotopes with the same nuclear spin, such as: $^7$Li, Na, $^{39}$K, and $^{41}$K, if the value of $\Gamma$ is changed.
<figure><img src="image_2.png" /><figcaption>FIG. 1. Relative populations in all eight sub-levels, as functions of time in microseconds. This has been computed by a solution of Eq. 2. Full blue line $F_g = 2$, $M_g = +2$; dashed blue line $F_g = 2$, $M_g = +1$; full green line $F_g = 2$, $M_g = 0$; dashed red line $F_g = 2$, $M_g = -1$; full red line $F_g = 2$, $M_g = -12$; full magenta line $F_g = 1$, $M_g = +1$; full cyan line $F_g = 1$, $M_g = 0$; dashed magenta line $F_g = 1$, $M_g = -1$.</figcaption></figure>

Figure 1 shows an example of evolutions of the populations of all eight involved levels, based on solutions of eq. 2. In figures 2 and 3, we show the same data, but with the evolutions for $F_g = 2$ and $F_g = 1$ separate, for improved clarity. Note that the scale of the y-axis is different in fig. 3.

In this particular example, we have taken the initial populations in the five levels of the $F_g = 2$ hyperfine structure level of the ground state as $G_{2,+2} = G_{2,+1} = 0.2$, $G_{2,0} = 0$, and $G_{2,-1} = G_{2,-2} = 0.1$; and those in $F_g = 1$ as $G_{1,+1} = 0.2$, and $G_{1,0} = G_{1,-1} = 0.1$. Thus, the total population is normalized to unity. Furthermore, we have set the intensity of linearly polarized light for the $F_g = 2 \leftrightarrow F_e = 2$ to a fifth of the saturation intensity ($I = I_{sat}/5 \Rightarrow R \approx \Gamma/10$) while the light intensity for the repumping transition is set to one tenth of the saturation intensity. The detuning for both transitions is set to zero. As shown in figures 1, 2 and 3, the population of the optically pumped level, $G_{2,0}$, grows monotonically towards 1, as all other levels are depleted.
<figure><img src="image_3.png" /><figcaption>FIG. 2. Relative populations in the five sub-levels of $F_g = 2$, as functions of time in microseconds. This has been computed by a solution of Eq. 2. Full blue line $M_g = +2$, dashed blue line $M_g = +1$, full green line $M_g = 0$, dashed red line $M_g = -1$, full red line $M_g = 12$.</figcaption></figure>

IV. CONCLUSION

Using the method developed in [1], we have calculated the state population evolution for optical pumping a state on the $F_g = 2 \leftrightarrow F_e = 2$-transition, using linearly polarized light, and with added repumping 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.

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

<footnote>[2] A. Kastberg and F. Atoneche (2016), URL https://figshare.com/articles/Optical_pumping_in_rubidium-87_with_linear_polarization_for_the_the_D2-transition_in_Rb87_Fg_2-_Fe_2_and_a_repumper_-_Mathematica_code/3756612.</footnote>
<figure><img src="image_4.png" /><figcaption>FIG. 3. Relative populations in the five sub-levels of $F_g = 1$, as functions of time in microseconds. This has been computed by a solution of Eq. 2. Full magenta line $M_g = +1$, full cyan line $M_g = 0$, dashed magenta line $M_g = -1$.</figcaption></figure>

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