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Introduction
Saturated absorption spectroscopy is a common technique used to measure the rest-frame (Doppler-free) electronic energy states of complex atoms at room temperature. This avoids the need for costly cooling techniques to reduce the thermal motions of the gas sample to measure hyperfine splittings. Here, we demonstrate a setup that utilizes a single non-commercial external cavity diode laser to measure hyperfine splittings of the 5P_3/2 state of two isotopes of Rubidium. We also measure the splittings of the ground state of Rb, and include a demonstration of the Doppler-broadening of the spectral lines that prevent simple absorption spectra to observe hyperfine structure.
Theory of Saturated Absorption
The purpose of this experiment is to measure the hyperfine splittings of Rubidium. The hyperfine splittings are present due to coupling between the total electron angular momentum and the nuclear angular momentum. However, these splittings are incredibly small, and are obscured by the Doppler broadening present in normal absorption spectroscopy, which for Rubidium is on the order of 500MHz. Saturated Absorption spectroscopy is one such method.
The logic behind saturated absorption spectroscopy can be demonstrated by examining the figure below.
Two counter-propagating beams ("Pump" and "Probe") from the same laser are introduced to a gas cell (we then let the beam path define the z axis). Since the individual atoms are moving with a nonrelativistic Gaussian distribution of velocity v, they will be subject to Doppler-shifted photons from either beam. Either beam will be partially absorbed by exciting those atoms inside the sample where the frequency of the laser f_0 and the frequency of a transition f_t obey the relation
with c as the speed of light.
Next, examine only one of the beams. By measuring the fraction of its light absorbed by the cell as a function of frequency (proportional to the number of atoms excited by the beam), we would expect to see a Gaussian profile with frequency width
where k_b is Boltzmann's constant,T is the temperature of the gas, m is the mass of an atom, and v_t is the rest frame frequency of the transition.
Since the two beams are counterpropagating, they will interact with classes of atoms that are moving with opposing velocities. That means that one beam does not effect the other's absorption by the sample. There are two exceptions to this case. The first occurs when the laser is tune directly to the rest-frame resonance of a transition. In this case, both beams are now interacting with the same class of atoms - those with zero z-velocity. Then, if the probe beam is much weaker than the pump beam, the pump beam will saturate these atoms, allowing the probe beam to pass through the sample without interacting. In this case, the absorption of the probe beam drops, and resonances within the Doppler-broadened profile can be seen.
The second case occurs when the laser is tuned to a frequency v_L halfway between two resonances v_1 and v_2. Here, both beams can interact with the same two group of atoms, one redshifted and one blueshifted, as shown below.
When this occurs, the same dip in the absorption of the probe beam seen at a resonance is also observed. This means that "false" dips will be seen positioned halfway between each hyperfine resonance.
We will also measure the effect of temperature on Doppler broadening, which is due to the atoms traveling with velocities from the Maxwell-Boltzmann distribution, and the ground state hyperfine splitting using normal absorption spectroscopy. In this case, the splitting is larger than the broadening observed at room temperature.
Experimental Setup
The setup for normal absorption spectroscopy is shown below. The laser passes through the cell once, exciting the atoms at the correct energies while being absorbed. It is read by a diode, which is then measured by an oscilloscope. Calibration is performed by a Fabry-Perot Interferometer, which sees part of the beam deflected from the main beam path by a microscope slide. A Fiber Spectrometer is also used to determine the approximate laser wavelength. The laser is tuned by a piezo pushing on the diffraction grating, which is driven by a triangle wave provided by a power supply. This setup was used for temperature based Doppler broadening measurements and ground state splitting measurements. To change the temperature in the cell, the cell is placed inside an oven heated by heater wire.
The saturated absorption spectroscopy setup is seen in the following two images. The laser (housed in the cardboard box to attempt to limit disturbances caused by air currents) passes through the cell twice, first as the pump and then as the probe. A third beam is deflected off and passes through the cell once, used for the background subtraction. The previous setup is expanded. Wave plates are used to ensure the polarization of the beam is optimal for absorption, and so that the probe beam can be deflected by the polarizing beam splitter to be measured. A neutral density filter is used to attune the strength of the returning beam. All outputs are read by an oscilloscope.
Caution is used in adjusting the beam height so that the absorption beam does not cross the two saturated absorption beams.
Results
The temperature dependent Doppler broadening was plotted. The slope was determined to be 2.96+/-0.04 * 10^7 Hz/sqrt(K), within one sigma of the expected value, 2.987 *10 ^ 7 Hz/sqrt(K). This produced a relatively good chi-squared value of 0.858, and the expected slope lies within the error bars on all points. The inhomogeneous broadening was found to be about 1MHz. Errors were found from the calibration and peak fitting method.
The ground state splitting was found by splicing together two data sets, since the mode hope free tuning range was smaller than the 10 GHz required to obtain the entire spectrum. The third peak was included in both spectra as the reference point. Since the current supplied to the laser had to be adjusted in order to shift the frequency enough to obtain the fourth peak, the voltage measured in the second spectrum (of the third and fourth peaks) was very slightly higher than the first. The height of the third peak in the first spectrum was used to normalize the second spectrum. The x-values were unchanged in splicing the two peaks.
The measured splittings are seen below. All values are found to be within one sigma of the expected value. Errors were from calibration and peak fitting.
The measured excited state hyperfine splittings are shown in the following plots and tables.The background subtraction was imperfect, so a further subtraction in software was performed, leading to peaks that were not resolved very well. However, Lorentzian functions were fitted to all six peaks, and averaging of five spectra were used to determine the spacings. The crossover peaks were taken as independent measurements, since they are known to be exactly between the two real peaks.
85 Rubidium Saturated Absorption Spectrum
85 Rubidium Hyperfine Splittings
Both the 87Rb and 85Rb produced results within error of the expected values. However, due to instabilities in the laser, an extra systematic error was introduced. Small jumps of approximately 10-20MHz were observed. Since data was taken on an oscilloscope, measurements known to contain such a jump were not used in our analysis, but smaller fluctuations were unable to be detected before analysis. However, since these fluctuations appeared to be random, the averaging of five spectra for each isotope produced results close to what was expected.
87 Rubidium Saturated Absorption Spectrum
87 Rubidium Hyperfine Splittings
Conclusion
The goal of this experiment was to determine whether or not a homemade laser could be used to determine the hyperfine splittings. Due to instabilities in the laser, the precision to which the splittings could be determined was not as high as we initially expected. However, all of the errors from the fitting process were within the experimental error due to instability, indicating that this is a viable method, and with improvements to the laser, the correct splittings could be obtained. Ultimately, we were able to determine the correct splittings to within our experimental error, but with lower precision than we had expected.
References
1. Zhou, Leo. Characterizing Hyperfine Structures in 87Rb and 85Rb with Doppler-Free Spectroscopy, MIT Department of Physics. Preprint Copy (2013)
2. Jacques et. al., Non-linear Spectroscopy of Rubidium: An Undergraduate Experiment. European Journal of Physics, 30 (5), 921-934 (2009)
3. Steck, Daniel Adam. “Rubidium 85 D Line Data”. Oregon Center for Optics and Department of Physics, University of Oregon. 2 May 2008.
4. Steck, Daniel Adam. “Rubidium 87 D Line Data”. Oregon Center for Optics and Department of Physics, University of Oregon. 2 May 2008.
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