# Optical pumping in rubidium-87 - Circular polarization on the (F g = 2 ↔ F e = 3) transition in the D2-line, in the presence of a magnetic field

Fred Atoneche, Anders Kastberg

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Fred Atoneche, Anders Kastberg. Optical pumping in rubidium-87 - Circular polarization on the (F g = 2 ↔ F e = 3) transition in the D2-line, in the presence of a magnetic field. 2016, 10.6084/m9 figshare.3979227.v2. hal-01635979

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Submitted on 16 Nov 2017

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Optical pumping in rubidium-87 — Circular polarization on the
($F_g = 2 \leftrightarrow F_e = 3$) transition in the D2-line, in the presence of a
magnetic field

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 circularly polarized light to populate a stretched angular momentum state in the
$F_g = 2$ hyperfine structure level of the ground state. We include in the analysis an applied weak
magnetic field. The method is shown to function well for the the studied system. Absolute scale
level populations as functions of time, detuning, intensity, magnetic field 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 $5s \ ^2\text{S}_{1/2}$, $F_g = 2$, $M_g = +2$.

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

* Pure polarization, and low saturation.

* Applied external magnetic field.

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 $M_g = +2$, $M_g = +1$, $M_g = 0$, $M_g = -1$, $M_g = -2$. These five states will constitute our state space, and as derived in [*to be published*], the state population can be described by the matrix:

$$ \frac{d}{dt} \mathbf{G}(t) = \frac{1}{225} \begin{pmatrix} 0 & 50 R_{+1} & 6 R_0 & 0 & 0 \\ 0 & -50 R_{+1} & 48 R_0 & 9 R_{-1} & 0 \\ 0 & 0 & -54 R_0 & 27 R_{-1} & 6 R_{-2} \\ 0 & 0 & 0 & -36 R_{-1} & 8 R_{-2} \\ 0 & 0 & 0 & 0 & -14 R_{-2} \end{pmatrix} \mathbf{G}(t). \quad (1) $$

$\mathbf{G}(t)$ is a vector with the populations in the five states as its components. $R_{+1}$, $R_0$, $R_{-1}$, and $R_{-2}$ are the intensity dependent scattering rates:

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

The scattering rate $R_{+2}$ does not enter into the optical pumping problem, since the transition from $M_g = +2$ is a cycling one. $\Gamma$ is the natural linewidth of the excited states (for all
states belonging to the term 5p $^2$P$_{3/2}$, ($\Gamma$ = 2$\pi$ × 6.0666 MHz [3]), I the intensity, $I_{sat}$ the saturation intensity ($I_{sat}$ = 1.669 mW/cm$^2$ for the case with a cycling transition and a coupling coefficient of one).

In eq. 2, $\Delta_i$ is the state dependent detunings. A consequence of the external field is that the five sub-levels will no longer be degenerate. The state dependent Zeeman shifts will in turn make the detunings different for all transitions.

We limit this calculation to a case with relatively weak magnetic fields, where Paschen-Back effects do not have to be taken into account (for $^{87}$Rb, this means that the field cannot be much stronger than 0.005 T.). In this case, the Zeeman shifts are:

$$ 
\begin{aligned}
E_Z &= g_F M_F \mu_B B, \\
g_F &= g_J \frac{F(F+1) - I(I+1) + J(J+1)}{2F(F+1)}.
\end{aligned} 
\quad (3) $$

Here, B is the amplitude of the magnetic flux density, and $g_J$ and $g_F$ are the gyromagnetic ratios for the fine structure and hyperfine structure, respectively.

In the case of $^{87}$Rb, we have $g_J$ = 2.00233113 for 5s $^2$S$_{1/2}$, and $g_J$ = 1.3362 for 5p $^2$P$_{3/2}$ [3]. For the hfs-levels involved, this in turn means that $g_F$ = 0.500583 ≈ 1/2 for $F_g$ = 2 in the ground state, and $g_F$ = 0.6681 ≈ 2/3 for $F_e$ = 3 in the excited state. The Zeeman shifted detunings then become:

$$ \Delta_i = \omega - \left( \omega_0 + \frac{E_{Zi_{lower}} - E_{Zi_{upper}}}{\hbar} \right) \quad (4) $$

In this equation, $E_{Zi_{lower}}$ and $E_{Zi_{upper}}$ are the energy Zeeman shifts in the lower and upper states pertaining from the lower state $M_i$, $\omega_0$ is the resonance angular frequency for B = 0, and $\omega$ is the angular frequency of the monochromatic light field.

## A. Solution of the evolution matrix equations

The evolution equations (Eq. 1) can be solved analytically. Explicit expressions, as functions of all involved parameters (e.g. magnetic field) 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. If a numerical method is instead used for the solutions, the convergence will be very fast. The problem is easily handled by desktop mathematical packages. In [2] we provide an annotated code for this in Mathematica™.
<figure><img src="image_2.png" /><figcaption>FIG. 1. Relative populations in the five sub-levels, as functions of time in microseconds, for the laser frequency $\omega = \omega_0$. This has been computed by a solution of Eq. 1. 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

#### A. Explicit solution

Solving eq. 1 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 scattering rate $R$.

Figure 1 shows an example of an evolution of the populations of the involved levels, based on eq. 1, for the specific laser frequency $\omega = \omega_0$ (corresponding to zero detuning for all transitions in the limit of zero field). In this example, we have assumed that the initial populations are the same in all five levels, and that the total population is normalized to unity ($G_{-2} = G_{-1} = G_0 = G_{+1} = G_{+2} = 0.2$). Furthermore, we set the intensity to a tenth of the saturation intensity ($I = I_{sat}/10$), and the magnetic field intensity to 0.002 T. The population of the optically pumped level, $G_{+2}$, grows monotonically towards 1, as all other levels are depleted. Note that the growth of the population of level $G_{+2}$ towards one is slower than that in case 1, where there was no external magnetic field.
<figure><img src="image_3.png" /><figcaption>FIG. 2. Relative populations in the five sub-levels, as functions of detuning in units of the natural linewidth, Γ, for the optical pumping time $t = 100$ µs. This has been computed by a solution of Eq. 1. 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>

Figure 2 shows the population as function of detuning (laser frequency) for the specific time $t = 100$ µs after the onset of optical pumping. All other parameters are the same as in fig. 1. The population of the optically pumped level, $G_{+2}$, is closed to unity for an optimal detuning. The other four levels have a varying degree of depletion as functions of detuning. Figures 3, 4, 5, 6, and 7, are surface plots of the level populations as functions of both time and detuning. Figure 3 is for $M_g = -2$, fig. 4 for $M_g = -1$, fig. 5 for $M_g = 0$, fig. 6 for $M_g = +1$, and fig. 7 for $M_g = +2$. All other parameters are the same as in figs. 1 and 2.

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 circularly polarized light and in the presence of a magnetic field. 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_circularly_polarized Light_for_the_D2_transition_in_Rb-87_Fg_2--_Fg_3_with_an_additional_magnetic_field_-_Mathematica_code/3978684.

[3] D. A. Steck, *Rubidium 87 D Line Data* (2001), URL http://steck.us/alkalidata/rubidium87numbers.pdf.
<figure><img src="image_4.png" /><figcaption>FIG. 3. Relative populations in sub-levels $M_g = -2$, as function of detuning in units of the natural linewidth, $\Gamma$, and time in units of microseconds.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 4. Relative populations in sub-levels $M_g = -1$, as function of detuning in units of the natural linewidth, $\Gamma$, and time in units of microseconds.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>FIG. 5. Relative populations in sub-levels $M_g = 0$, as function of detuning in units of the natural linewidth, $\Gamma$, and time in units of microseconds.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 6. Relative populations in sub-levels $M_g = +1$, as function of detuning in units of the natural linewidth, $\Gamma$, and time in units of microseconds.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>FIG. 7. Relative populations in sub-levels $M_g = +2$, as function of detuning in units of the natural linewidth, $\Gamma$, and time in units of microseconds.</figcaption></figure>