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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

Zachary H. Levine, Ziran Du

Published Oct 30, 2023• 6 citations
Publisher

Abstract

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.

DOI
10.1364/josab.503903

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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.
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.
Atomic vapors make flexible and useful nonlinear optical media when light is detuned far from resonances. However, few direct measurements of their nonlinear coefficients have been performed to date. We present a measurement of the Kerr coefficient for various detunings to both the red and blue of the D2D2D2 line in hot atomic rubidium vapor. Saturation effects complicate the interpretation of the standard z-scan method, but a simple saturation model agrees well with our data. We find values of ${n}_{2}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}{10}^{\ensuremath{-}7}{\mathrm{cm}}^{2}/\mathrm{W}$ for 1 GHz detunings from the D2D2D2 line.
We have studied two-photon resonant three-photon ionization of atomic sodium vapor. A dye laser tuned to either the 4d or 5s two-photon allowed transition was used to excite the atom and a second more intense laser (λ = 7000 Å) was used for ionization. The ion yield was measured as a function of resonant laser intensity and detuning, the nonresonant laser intensity, and the temporal overlap of the exciting and ionizing laser pulses. We have found1 that at high laser intensities the photoionization yield decreases as the nonresonant laser intensity is increased. When tuned to the 5s level a value of ~−0.6 was obtained for the exponential index even with the resonant laser tuned to maximize the ion yield. A theoretical model including the effects of Stark shift of the ground state, power broadening of the intermediate level due to saturation of the two-photon transition and by rapid transitions to the ionization continuum was used to predict the experimental results. Good agreement between theory and experiment required the use of a photoionization cross section for the 5s level which was ~20 times greater than the accepted value. Results for photoionization from the 4d level are also presented.
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.
We demonstrate a scheme to measure the saturable nonlinearity of atomic vapor by mapping its nonlinear response function onto a light beam profile. Our analysis shows that a part of a nonlinear optical solution solved in a model governing the nonlinear beam dynamics in atomic vapor can be used to perform this measurement, even in the presence of large absorption. A desired beam profile is achieved by an evolution of a well-known structured beam, namely the Airy beam. Using this simple yet effective method, we retrieve the saturable nonlinear response function of rubidium (Rb) atomic vapor in experiment, and employ it in light propagation simulation that reproduces well observed nonlinear dynamics, which nevertheless cannot be fitted in a strong nonlinear regime with an ideal Kerr approximation. Our method is applicable to a broad spectrum of materials featured with saturable nonlinearities.