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

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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

To cite this version:

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,

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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

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

Universit´e Nice Sophia Antipolis, CNRS,

Laboratoire de Physique de la Mati`ere Condens´ee UMR 7336, Parc Valrose, 06100 Nice, France

(Dated: November 16, 2017)

Abstract

We present results of a method for calculating optical pumping rates in87Rb, as described in [1]. We use a mixture of circularly polarised light (σ+ and σ−) on the (Fg = 1 ↔ Fe= 0) transition to populate an angular momentum state in the Fg = 1 hyperfine structure level of the ground state, with magnetic quantum number Mg= 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.

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I. INTRODUCTION

In a separate publication [1], we have described a simplified method for quantitative calcula-tions of the evolution of state populacalcula-tions 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 MathematicaTM) [2], and the obtained

results.

The specific case studied is:

• Rb87_{, with nuclear spin I = 3/2}

• Preparation of the state 5s2_S

1/2, Fg = 1, Mg = 0.

• Pumping with an adjustable mixture of σ+ _{and σ}− _{circularly polarized light on the}

transition Fg = 1 ↔ Fe = 0 on the D2-line (upper fine-structure state 5p 2P3/2).

• No π-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 Mg = +1, Mg = 0,

and Mg = −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:

d dtG(t) = 1 9      −2B 0 A B 0 A B 0 −2A      R G(t) . (1)

Here, A and B are the proportions of σ+ _{and σ}− _{polarizations respectively (A + B = 1).}

G(t) is a vector with the populations in the three states as its components, and R is the intensity dependent scattering rate:

R = Γ

2

(I/Isat)

1 + (I/Isat) + (2∆/Γ)2

. (2)

Γ is the natural linewidth of the excited states (for all states belonging to the term 5p 2_P 3/2,

(Γ = 2π × 6.0666 MHz [3]), I the intensity, Isat the saturation intensity (Isat = 1.669

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is the detuning (∆ = 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:

d dtG(t) = 1 18      −2 0 1 1 0 1 1 0 −2      R G(t) . (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 ≡√A2_{− AB + B}2_):

G+1(t) = nh B − B e2 C9 R t+ C 1 + e2 C9 R t i G+1(0) + A−1 + e2 C9 R t [G−1(0) + G+1(0)] o e− 1 C 9 R t 2 C G0(t) = h BC1 − e2 C9 R t G−1(0) + AC 1 − e2 C9 R t G+1(0) − C2_{(1 + e}2 C₉ R t_{) [G} −1(0) + G+1(0)] + 2C2e1 C9 R t[G₋₁(0) + G₊₁(0) + G₀(0)] i e− 1 C 9 R t 2 C2 G−1(t) = nh A − A e2 C9 R t+ C 1 + e2 C9 R t i G−1(0) + B−1 + e2 C9 R t [G−1(0) + G+1(0)] i e− 1 C 9 R t 2 C . (4)

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

MathematicaTM_.

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 Mg 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 Γ. This

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� ��

FIG. 1. Relative populations in the three sub-levels, as functions of time in microseconds, computed from Eq. 4. Dashed red line G1, full green line G0, full blue line G−1.

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

Mg = +1 and 0.25 in the two others. The total population is thus normalized to unity

(G−1 = G0 = 1/4 and G+1 = 1/2). Furthermore, we set the intensity to a fifth of the

saturation intensity (I = Isat/5 ⇒ R ≈ Γ/10), the detuning to zero, and the proportions of

the polarizations as A = B = 1/2. The population of the optically pumped level, G0, 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 (Fg = 1 ↔ Fe = 0)-transition, using a mixture of σ+

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and σ− 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_of_sigma_and_sigma-_light_for_the_D2_transition_in_ Rb-87_Fg_1-_Fg_0_-_Mathematica_code/3750549.

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