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Journal of the Optical Society of America B

Semiclassical calculation of the power saturation of the Kerr effect in Rb vapor. II: the effect of incomplete optical pumping

Zachary H. Levine

Published Apr 24, 2025• 1 citation
Publisher

Abstract

Rubidium vapor cells play a key role in contemporary quantum optics since they have strong nonlinear interactions with a low-power incident laser light. The simplest nonlinear interaction is the Kerr effect. Although most calculations concentrate on the unsaturated Kerr effect, the saturated Kerr effect has been measured in many materials including 85Rb and 87Rb vapor. Moreover, entanglement-generating vapor cells often operate in the saturated regime. Here, the calculation of the saturated Kerr effect for rubidium vapor at the D2 line uses a density matrix formulation that includes two hyperfine-split levels and all magnetic sublevels in each of the ground and excited states to account for incomplete optical pumping: the transit time of the atoms through the beam is too short to reach a final population distribution. The calculation shows a rapid qualitative change at low optical power, in semiquantitative agreement with experimental data.

DOI
10.1364/josab.553632

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Rubidium vapor cells play a key role in contemporary quantum optics since they have strong nonlinear interactions with a low-power incident laser light. The simplest nonlinear interaction is the Kerr effect. Although most calculations concentrate on the unsaturated Kerr effect, the saturated Kerr effect has been measured in many materials including 85Rb and 87Rb vapor. Moreover, entanglement-generating vapor cells often operate in the saturated regime. Here, the calculation of the saturated Kerr effect for rubidium vapor at the D2 line uses a density matrix formulation that includes two hyperfine-split levels and all magnetic sublevels in each of the ground and excited states to account for incomplete optical pumping: the transit time of the atoms through the beam is too short to reach a final population distribution. The calculation shows a rapid qualitative change at low optical power, in semiquantitative agreement with experimental data.
The optimal detuning of the pump light at the rubidium D2 line is calculated and experimentally validated. A three-level optically end-pumped model in a hot rubidium vapor cell is demonstrated, which takes into account the hyperfine structure of the ground state, various collisional processes, and the nonlinear absorption of the pump light along the cell length. By developing a constrained optimization model, we calculate the optimal detuning of the pump light that maximizes the average excited state atom concentration in the cell. Experimentally, a 1529 nm probe light at higher levels, which is attenuated to the weak-probe limit, is introduced to accurately characterize the excited state atom concentration. Coupling with the experimental results, the variation of the pump light optimal detuning with the temperature and the pump light intensity at the unsaturated pumping is analyzed. At the optimal detuning point, the theoretical decay function of the pump light percentage along the vapor cell length almost coincides regardless of the temperature and the pump light intensity changes. Moreover, the theoretical fitted pump light decay function at optimal detuning is predicted. The fitted pump light percentage exiting the vapor cell is 4.75%. A fast implementation method to achieve the maximal average excited state atom concentration in the experiment is demonstrated.
The potential to precisely control both the linear and nonlinear index of refraction through optical manipulation of the atomic states has recently pushed warm alkali vapors to the forefront of research in the field of quantum sensors, quantum memories, and quantum fluids of light. Rubidium (Rb) vapor in centimeter-scale glass cells or millimeter-scale MEMS cells has proven to be a very promising platform for these applications, yet only a handful of research works have been dedicated to the investigation of the (non)linear refractive index of Rb vapor. We present results of theoretical calculations of the (non)linear refractive index of warm Rb vapor, based on the optical Bloch equations for 6-level Rb atoms interacting with a probe laser. They are compared to the experimental results obtained using an interferometric technique, showing excellent quantitative agreement. A Kerr nonlinear refractive index n2n_2n2​ of up to 10−410^{-4}10−4 cm2^22/W is obtained. Python scripts for all theoretical calculations presented in this work are provided, including the refractive index calculation, that can readily be used in practical implementations for simulating the (non)linear refractive index of Rb vapor including the effects of Doppler broadening, transit time broadening, pressure broadening, saturation, optical pumping, and spin-exchange collisions.
The potential to precisely control both the linear and nonlinear index of refraction through optical manipulation of the atomic states has recently pushed warm alkali vapors to the forefront of research in the field of quantum sensors, quantum memories, and quantum fluids of light. Rubidium (Rb) vapor in centimeter-scale glass cells or millimeter-scale micro-electro-mechanical system (MEMS) cells has proven to be a very promising platform for these applications, yet only a handful of research works have been dedicated to the investigation of the (non)linear refractive index of Rb vapor. We present results of theoretical calculations of the (non)linear refractive index of warm Rb vapor, based on the optical Bloch equations for 6-level Rb atoms interacting with a probe laser. They are compared to the experimental results obtained using an interferometric technique, showing very good quantitative agreement. A Kerr nonlinear refractive index n2 of up to −1.2×10−4 cm2/W is obtained. Python scripts for all theoretical calculations presented in this work are provided, including the refractive index calculation, that can readily be used in practical implementations for simulating the (non)linear refractive index of Rb vapor including the effects of Doppler broadening, transit time broadening, pressure broadening, saturation, optical pumping, and spin-exchange collisions.
The Kerr effect in atomic vapor may be regarded as the power saturation of the susceptibility. Hence the saturable Kerr effect is intimately tied to the standard Kerr effect. Here, we calculate the saturable Kerr effect without free parameters using a two-level system and find good agreement with experimental results. Our approach permits a direct comparison of theory to experiment without an extrapolation to low power as required by previous approaches. An experimentally observed asymmetry between red and blue detuning led us to consider a model with one ground state and three excited states. Such an asymmetry cannot be described by a two-state model. The model predicts about 25% of the observed asymmetry.
Abstract We study non-linear absorption of intense monochromatic light through a dense natural rubidium (Rb) vapour. We measure transmission through a 10 cm long heated vapour cell for atom densities up to 3 × 1019 m−3 and saturation parameters up to 104, for linear and circular polarisation, close to resonance on the 87Rb D2 F = 1 → F′ = 0, 1, 2 transition. The strong absorption at low intensity is frustrated by an interplay of optical non-linearities (saturation and optical pumping) and non-linear effects due to the high atom density (collisional broadening and collisional depumping). To understand the results of the transmission measurements, we developed a model that incorporates these non-linear effects into the optical absorption. The model takes into account the absolute line strengths of all transitions from both hyperfine levels of the ground state of both isotopes of naturally abundant Rb. Doppler and collisional broadening are included in the Voigt profiles for the resonances. We show the effect of each of the non-linear processes on the calculation results of the model, and from comparison with experiment we conclude that all non-linear effects are necessary for a quantitative agreement.
The relative intensities of the Doppler-free signals of 85Rb and Cs D2 lines in saturation spectroscopy were calculated theoretically, taking into account the optical pumping process in a four-level system. The optical pumping effect in saturation spectroscopy is important when atoms with long-lived lower states are pumped by a weak beam. The theoretical results calculated from a simple analysis without any knowledge of the saturation parameters were found to be in good agreement with the experimental results on Rb and Cs D2 lines previously obtained using semiconductor laser.