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American Journal of Physics

Interferometric measurement of the resonant absorption and refractive index in rubidium gas

Kenneth G. Libbrecht, Maxwell W. Libbrecht

Published Nov 22, 2006• 24 citations
Publisher

Abstract

We present a laboratory demonstration of the Kramers-Kronig relation between the resonant absorption and refractive index in rubidium gas. Our experiment uses a rubidium vapor cell in one arm of a simple Mach-Zehnder interferometer. As the laser frequency is scanned over an atomic resonance, the interferometer output is affected by variations of both the absorption and refractive index of the gas with frequency, all of which can be calculated in a straightforward manner. Changing the vapor density and interferometer phase produces a family of different output signals. The experiment was performed using a commercially available tunable diode laser system that was designed specifically for the undergraduate physics laboratory. As a teaching tool this experiment is reliable, fun, and instructive, while it also introduces the student to some sophisticated and fundamental physical concepts.

DOI
10.1119/1.2335476

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We present a laboratory demonstration of the Kramers-Kronig relation between the resonant absorption and refractive index in rubidium gas. Our experiment uses a rubidium vapor cell in one arm of a simple Mach-Zehnder interferometer. As the laser frequency is scanned over an atomic resonance, the interferometer output is affected by variations of both the absorption and refractive index of the gas with frequency, all of which can be calculated in a straightforward manner. Changing the vapor density and interferometer phase produces a family of different output signals. The experiment was performed using a commercially available tunable diode laser system that was designed specifically for the undergraduate physics laboratory. As a teaching tool this experiment is reliable, fun, and instructive, while it also introduces the student to some sophisticated and fundamental physical concepts.
We describe an experiment to be carried out by advanced students as part of their laboratory practice. Its aims are (i) to measure simultaneously the frequency dependence of both the refractive index n and the extinction coefficient of an atomic gas interacting with a resonant laser beam and (ii) to examine that n and satisfy the Kramers-Kronig relations. The medium to be probed consists of a dilute 85 Rb vapour exposed to collisions with a buffer gas. The D 2 resonance line is thereby broadened to such an extent that its hyperfine components merge into single Lorentzian dispersion and absorption profiles, respectively. The main features of the apparatus are a tunable laser diode, a Mach-Zehnder interferometer at nearly zero path difference, an array of photodiodes to measure the interference pattern as a function of laser frequency and Rb density in one of the interferometer arms, an absorption monitor, computer-controlled electronics to read out and analyse the data, and a video camera for visual control and adjustment of the interference pattern.
The goal of this experiment was to measure the hyperfine energy splittings of the ground to first excited state transitions in rubidium using saturated absorption spectroscopy. Using this technique, we measured these transition energy spectra by taking the difference of two photodiode outputs due to multiple beams of a single laser scanned over a range of frequencies and shone through a cell of Rb vapor. When the laser frequency was resonant with an atomic transition, photons of those frequencies were absorbed, leaving a dip in intensity of the beam measured at the photodiode. One of the two laser beams had its excitations saturated by a more intense, counter-propagating pump beam from the same laser. The pump beam saturated the absorption of the velocity equal zero atoms that the counter-propagating probe beam could have also interacted with due to no Doppler shift. At these frequencies the probe beam experienced less absorption that was measured by a photodiode. The photodiode output was calibrated to the change in frequency of the scanned laser using a Fabry-Pérot interferometer. The output of this interferometer served as a reference for how the laser frequency changed. Final results for the 87Rb F=1 spectrum accounted for the largest error with a maximum of 12.2% deviation from accepted energy spacing values due in part to systematic error. Other spectra measurements were in better agreement, possessing error ranging from 2.1% to 10.1% and some agreeing with accepted values to within our uncertainty.
We have experimentally demonstrated a general method to perform high-precision refractive index spectrum measurement using phase-shifting interferometry in a phase-tunable Mach–Zehnder interferometer. We observed coherent amplification on the refractive spectrum, across the Lamb dips of ground hyperfine states in a Doppler-broadened Rb87 vapor. Most importantly, the spectrum still has a large signal-to-noise ratio when the probe power is in the dark light level under which the method using a single interferogram fails. This experimental scheme is efficient for refractive spectrum measurements and could be extended for 2D phase imaging. It provides many potential applications for optical phase measurements in linear, nonlinear, and quantum optics.
We have developed a Mach-Zehnder atom interferometer which uses elastic diffraction of lithium atom on laser standing waves for the coherent manipulation of the atom wave. The interference signal is of sufficient quality to achieve a phase sensitivity near 10 milliradians/. With this apparatus, we have recently measured the index of refraction of several gases for lithium waves. The interference signals give access to the real and imaginary parts of () and we have tested the linear dependence of these two quantities with gas density. The measured values of () are in good agreement with calculations and they exhibit glory oscillations, as predicted by theory.
The dispersive properties of the atomic transition in the rubidium D2{D}_{2}D2​ line ($5{S}_{\frac{1}{2}}\ensuremath{-}5{P}_{\frac{3}{2}}$) at 780.0 nm are measured with a Mach-Zehnder interferometer when an additional coupling field at 775.8 nm is applied to an upper transition ($5{P}_{\frac{3}{2}}\ensuremath{-}5{D}_{\frac{5}{2}}$). This ladder-type system is observed to exhibit electromagnetically induced transparency together with a rapidly varying refractive index. A reduction in group velocity for the probe beam (vg=c13.2{v}_{g}=\frac{c}{13.2}vg​=13.2c​) is inferred from the measured dispersion curve with 52.5% suppressed absorption on resonance.
In this paper, we present a tutorial set-up based on an optical three-grating Mach–Zehnder interferometer. As this apparatus is very similar in its principle to the Mach–Zehnder interferometers used with matter waves (neutrons, atoms and molecules), it can be used to familiarize students with particle optics, and in our explanations, we use the complementary points of view of wave optics and particle optics. Finally, we have used this interferometer to measure the index of refraction of BK 7 glass for red light at 633 nm, with a technique equivalent to the one used to measure the index of refraction of solid matter for thermal neutrons. The dimensions of this interferometer and its cost make it very interesting for laboratory courses and the experiment described here can be reproduced by students.