ROUGH DRAFT authorea.com/96205

# Defining energy levels in Rubidium 87 and 85 using Doppler and Doppler-Free Spectroscopy: Emily and Liam 2nd draft

Abstract

We performed both Doppler and subDoppler spectroscopy on the $$5^{2}S_{1/2}$$$$5^{2}P_{3/2}$$ transition of rubidium 85 ($$^{85}$$Rb) and rubidium 87 ($$^{87}$$Rb). We fit the Doppler spectroscopy curves to Maxwell Boltzmann velocity distributions to determine the temperature of the cell. We also fit the subDoppler spectroscopy curves to extract the transition energies and hyperfine structure of the $$5^{2}S_{1/2}$$ F=2 ground state to the $$5^{2}P_{3/2}$$ F=1, F=2, and F=3 excited states of $$^{85}$$Rb, the $$5^{2}S_{1/2}$$ F=3 ground state to the $$5^{2}P_{3/2}$$ F=2, F=3, and F=4 excited states of $$^{85}$$Rb, the $$5^{2}S_{1/2}$$ F=1 ground state to the $$5^{2}P_{3/2}$$ F=0, F=1, and F=2 excited states of $$^{87}$$Rb, and the $$5^{2}S_{1/2}$$ F=2 ground state to the $$5^{2}P_{3/2}$$ F=1, F=2, and F=3 excited states of $$^{87}$$Rb. The calculated difference between the transition from the F=2 excited state of the $$5^{2}S_{1/2}$$ state of rubidium 87 to the $$5^{2}P_{3/2}$$ state and the transition from the F=3 excited state of the $$5^{2}S_{1/2}$$ state of rubidium 85 to the $$5^{2}P_{3/2}$$ state, was found to be $$1.062 x 10^{9}$$ Hz $$\pm$$ $$1.6 x 10^{7}$$ Hz, compared to the value from (Daniel Adam Steck 2001) of $$1.22039 x 10^{9}$$ Hz $$\pm$$ $$2 x 10^{4}$$ Hz. Our results for Doppler Spectroscopy were not quite consistent with theory within uncertainty, but there are many possible reasons for this slight discrepancy. The uncertainty in the Doppler profile fitting is most likely due to an incomplete model of the transmission curve. While we modeled the as a Maxwell Boltzmann Gaussian distribution, the actual curve is a convolution between the six Lorentzian line profiles and the Gaussian distribution. The uncertainty in the hyperfine fitting is likely mainly due to the fact that we modeled our frequency scan as linear but it is not actually linear.

# Introduction

It is common knowledge that atoms are made up of protons, neutrons, and electrons. Electrons within the atom exist in different excited states, or energy levels, which differ with the number of protons and neutrons within in atom. The Coulomb potential is the reason for the confinement of electrons, and we must consider the various interactions taking place between the electrons and protons in the atom to think about the Coulomb potential. Hydrogen is simple—it only has one interaction, the interaction between its one electron and one proton. Helium has two electrons and two protons, so it has six interactions. Lithium, with three protons and three electrons, has fifteen interactions, and so on and so forth.

Apart from the Coulomb interaction, there are two other main factors can effect the energy levels of electrons within an atom: internal and external fields and spin-spin interaction between all of the fermions in the atom. Both internal and external fields must be taken into account as these fields perturb the energy levels. Electrons, protons, and neutrons all possess intrinsic spin while electrons also have orbital angular momentum. All of these interactions also perturb the energy levels. The spin-spin interaction between the outermost electron and the nucleus results in hyperfine structure. The experimental result is a splitting of the energy levels that can usually only be observed at cold temperatures. Temperature and laser power can cause broadening of the absorption curve, but does not shift the levels ((Melissinos 2003) pp 236-237).

We can measure the energy levels of the atom by exciting the electrons with a laser at the appropriate wavelength for each energy level, which is known as spectroscopy . A laser is sent through a cell of the element we are trying to do spectroscopy on (in this case rubidium) and the transmission of the laser is measured with a photodiode. The photodiode has a response due to incident light, known as responsivity. The photodiode produces a current given a certain amount of incident light. That current then gets converted to a voltage. For the Doppler spectroscopy we used an amplifier to convert the current to a voltage. While not ideal, this was the equipment we had at the time. For the subDoppler spectroscopy, which uses far less power in the transmission beam, the oscilloscope, which has an input impedance of $$1 M\Omega$$, was suitable to convert the current to voltage. This input impedance was too large for the Doppler spectroscopy experiment.

If we had a cell at zero temperature, and the rubidium was still a gas, as the frequency of the laser is scanned, the transmission through the vapor cell would produce Lorentzian dip. Off resonance the transmission should be $$100\%$$. As the laser frequency passes through the resonance, the laser transmission drops. The resulting Lorentzian shaped dip is a direct result of the Heisenberg uncertainty principle. The inverse of the full width half maximum of the Lorentzian profile (in angular frequency space) is the average lifetime of the excited state. As the temperature increases from zero kelvin, the peak will broaden about this central frequency. Doppler broadening, described below, is actually a convolution between the zero temperature Lorentzian profile and the Maxwell Boltzmann Gaussian shaped velocity profile. In the Doppler spectroscopy experiment, we fit to a negative Gaussian profile because there are six different Lorentzian shaped profiles that are simultaneously Doppler broadened. (An example of a fit to a negative Gaussian is shown in Fig. \ref{fig:NegativeGaussian}. This data will be thoroughly discussed in the Results section.) The resultant complicated Voight profile is extremely difficult to fit. We therefore expect the fitted temperature to be larger than room temperature since we ignore this convolution. This is what we find, see Discussion.