# Optical pumping in rubidium-87 - A mixture of σ + and σ − circularly polarized light on the (F g = 1 ↔ F e = 0) transition in the D2-line

Fred Atoneche, Anders Kastberg

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Fred Atoneche, Anders Kastberg. Optical pumping in rubidium-87 - A mixture of σ + and σ − circularly polarized light on the (F g = 1 ↔ F e = 0) transition in the D2-line. 2016, 10.6084/m9 figshare.3750702.v3. hal-01635973

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Submitted on 16 Nov 2017

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Optical pumping in rubidium-87 — A mixture of $\sigma^{+}$ and $\sigma^{-}$ circularly polarized light on the ($F_{g} = 1 \leftrightarrow F_{e} = 0$) 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 a mixture of circularly polarised light ($\sigma^{+}$ and $\sigma^{-}$) on the ($F_{g} = 1 \leftrightarrow F_{e} = 0$) transition to populate an angular momentum state in the $F_{g} = 1$ hyperfine structure level of the ground state, with magnetic quantum number $M_{g} = 0$ — that is a clock state. The method is shown to function well for the studied system. Explicit analytical equations for all level populations as functions of time, detuning, intensity, relative intensities of the polarization, 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$_{1/2}$, $F_g = 1$, $M_g = 0$.

* Pumping with an adjustable mixture of $\sigma^+$ and $\sigma^-$ circularly polarized light on the transition $F_g = 1 \leftrightarrow F_e = 0$ on the D2-line (upper fine-structure state 5p $^2$P$_{3/2}$).

* No $\pi$-component of the polarisation, 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 $M_g = +1$, $M_g = 0$, and $M_g = -1$. These three states will constitute our state space, and as derived in [*to be published*], the state population can be described by the matrix:

$$ \frac{d}{dt} \mathbf{G}(t) = \frac{1}{9} \begin{pmatrix} -2B & 0 & A \\ B & 0 & A \\ B & 0 & -2A \end{pmatrix} R \mathbf{G}(t). \quad (1) $$

Here, $A$ and $B$ are the proportions of $\sigma^+$ and $\sigma^-$ polarizations respectively ($A + B = 1$). $\mathbf{G}(t)$ is a vector with the populations in the three states as its components, and $R$ is the intensity dependent scattering rate:

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

$\Gamma$ 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). In the case of an equal mixture of the two polarizations, $A = B = 1/2$, and the matrix (1) reduces to:

$$ \frac{d}{dt} \mathbf{G}(t) = \frac{1}{18} \begin{pmatrix} -2 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & -2 \end{pmatrix} R \mathbf{G}(t). \qquad (3) $$

## A. Solution of the evolution matrix equations

The evolution equation (Eq. 1) is simple to solve analytically. The state populations in terms of the the parameters A and B are (with the definition $C \equiv \sqrt{A^2 - AB + B^2}$):

$$ \begin{align} G_{+1}(t) &= \left\{ \left[ B - B e^{\frac{2C}{9}Rt} + C \left( 1 + e^{\frac{2C}{9}Rt} \right) \right] G_{+1}(0) \right. \nonumber \\ & \qquad \left. + A \left( -1 + e^{\frac{2C}{9}Rt} \right) [G_{-1}(0) + G_{+1}(0)] \right\} \frac{e^{-\frac{1C}{9}Rt}}{2C} \nonumber \\ G_{0}(t) &= \left[ BC \left( 1 - e^{\frac{2C}{9}Rt} \right) G_{-1}(0) + AC \left( 1 - e^{\frac{2C}{9}Rt} \right) G_{+1}(0) \right. \nonumber \\ & \qquad \left. - C^2 \left( 1 + e^{\frac{2C}{9}Rt} \right) [G_{-1}(0) + G_{+1}(0)] \right. \nonumber \\ & \qquad \left. + 2C^2 e^{\frac{1C}{9}Rt} [G_{-1}(0) + G_{+1}(0) + G_{0}(0)] \right] \frac{e^{-\frac{1C}{9}Rt}}{2C^2} \nonumber \\ G_{-1}(t) &= \left\{ \left[ A - A e^{\frac{2C}{9}Rt} + C \left( 1 + e^{\frac{2C}{9}Rt} \right) \right] G_{-1}(0) \right. \nonumber \\ & \qquad \left. + B \left( -1 + e^{\frac{2C}{9}Rt} \right) [G_{-1}(0) + G_{+1}(0)] \right\} \frac{e^{-\frac{1C}{9}Rt}}{2C}. \tag{4} \end{align} $$

The mathematical expressions are lengthy, but they are explicit and exact, and are easily handled by desktop mathematical packages. In [2] we provide an annotated code for this in Mathematica™.

## III. RESULTS

### A. Explicit solution

Solving Eq. 1 gives the evolution of the three involved Zeeman states given by Eq. 4. In these equations, the subscripts refer to the value of the $M_g$ quantum number. 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
<figure><img src="image_2.png" /><figcaption>FIG. 1. Relative populations in the three sub-levels, as functions of time in microseconds, computed from Eq. 4. Dashed red line $G_1$, full green line $G_0$, full blue line $G_{-1}$.</figcaption></figure>

means that Eq. 4 will be equally valid for all alkali isotopes with the same nuclear spin, such as: $^7$Li, Na, $^{39}$K, and $^{41}$K.

## B. Numerical results

Figure 1 shows an example of evolutions of the populations of the involved levels, based on Eq. 4. In this example, we have assumed that the initial populations are 0.5 in the level $M_g = +1$ and 0.25 in the two others. The total population is thus normalized to unity ($G_{-1} = G_0 = 1/4$ and $G_{+1} = 1/2$). Furthermore, we set the intensity to a fifth of the saturation intensity ($I = I_{sat}/5 \Rightarrow R \approx \Gamma/10$), the detuning to zero, and the proportions of the polarizations as $A = B = 1/2$. The population of the optically pumped level, $G_0$, grows monotonically towards 1, as all other levels are depleted.

## IV. CONCLUSION

Using the method developed in [1], we have calculated the state population evolution for optical pumping to a clock state on the ($F_g = 1 \leftrightarrow F_e = 0$)-transition, using a mixture of $\sigma^+$
and $\sigma^{-}$ circularly polarized light. The method works well when applied to this system and yields exact and explicit expressions for the level populations. 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_a_mixture_ofclash_and_clash-light_for_the_D2_transition_in_Rb-87_Fg_1--_Fg_O_-_Mathematica_code/3750549.

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