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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
To cite this version:
Anders Kastberg, Fred Atoneche. Optical pumping in rubidium-87 - Impure polarization (a mix-ture 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�
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
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 in 87Rb, as described in [1]. We use light with an impure polarization (i.e. a mixture of σ+, σ− and π light) to populate a stretched angular momentum state in the Fg = 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 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 5s2S
1/2, Fg = 2, Mg = +2.
• Pumping with a mixture of σ+, σ− and π polarizations of light on the transition Fg = 2 ↔ Fe = 3 on the D2-line (upper fine-structure state 5p2P3/2).
• 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 Mg = +2, Mg = +1, Mg = 0, Mg = −1, Mg = −2 of the Fg = 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 σ+, σ− and π polarizations. The three matrices for σ+, σ− and π are weighted by the relative light intensities c+, c−, and cπ respectively (c++ c−+ c0 = 1).
The three matrices are:
M+= 1 225 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 (1)
M− = 1 225 −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 (2) Mπ = 1 225 −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 . (3)
The total evolution is given by: d
dtG(t) = M R G(t) = (c+Mσ+ + cπMπ+ c−Mσ−) R G(t) . (4)
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 = Γ 2
(I/Isat)
1 + (I/Isat) + (2∆/Γ)2
. (5)
Γ is the natural linewidth of the excited states (for all states belonging to the term 5p 2P 3/2, (Γ = 2π × 6.0666 MHz [3]), I the intensity, Isat the saturation intensity (Isat = 1.669 mW/cm2 for the case with a cycling transition and a coupling coefficient of one), and ∆ is the detuning (∆ = 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 func-tions of all involved parameters (e.g. relative intensity distribufunc-tions for the different polar-izations) 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 MathematicaTM.
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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 G0, dashed red line G−1, full red line G−2.
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 Γ.
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 cπ = 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 = G0 = G+1 = G+2 = 0.2). Furthermore, we set the intensity to a fifth of the saturation intensity (I = Isat/5 ⇒ R ≈ Γ/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: G−2(t → ∞) = 0.000083 G−1(t → ∞) = 0.00042 G0(t → ∞) = 0.0040 G+1(t → ∞) = 0.032 G+2(t → ∞) = 0.96 . (6) IV. CONCLUSION
Using the method developed in [1], we have calculated the state population evolution for optical pumping to a stretched state on the Fg = 2 ↔ Fe = 3 -transition, using a mixture of σ+ and σ− and π 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.
