deletions | additions       diff --git a/section_Introduction_It_is_common__.tex b/section_Introduction_It_is_common__.tex  index 4263382..1573b26 100644  --- a/section_Introduction_It_is_common__.tex  +++ b/section_Introduction_It_is_common__.tex 
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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 \cite{Zapka_1983}. 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 SRS 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}.)  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 \textbf{section} . \textbf{It is a good idea to insert an explanation of Doppler spectroscopy (as you did above), since your readers are unlikely to already know what it is. But in addition, you also need to include some references to relevant papers ---- such as ones you yourself read and consulted in preparation for this experiment? --- where your reader can learn more, or where they can find where the technique was first used. I have randomly chosen a paper to cite by using Authorea's search CrossRef function (searching on the term ``Doppler Spectroscopy of Rb''), but you should replace it with one you actually used. }  To understand Doppler broadening, consider a vapor cell at room temperature. Since the atoms have some temperature, they are moving at some velocity whose distribution (ignoring the natural Lorentzian profile) can be exactly described by the Maxwell Boltzmann distribution. When an atom is moving towards or away from the laser beam, it sees photons of a higher or lower frequency as being the correct frequency to excite the electrons—so, different frequencies of light interact with different velocity classes of atoms. In the atom frame of reference, only photons with the correct frequency are absorbed, but in the lab frame of reference, the Doppler shift added to the frequency of the laser is the frequency of light that the atoms absorb. We measure in the lab frame, so we see a broadened curve.   An atom with zero velocity, which is in what is referred to as the zero velocity class, is the one which will absorb light at the exact frequency of the transition in the lab frame. This atom is either not moving, or is moving perpendicular to the laser beam. To be exact, the atom's velocity in the direction of the laser beam has to be within the Lorenztian profile of the transition.An example of a fit to a negative Gaussian is shown in Fig.~\ref{fig:NegativeGaussian}.  The effects of Doppler broadening can be removed when using a technique known as saturated absorption spectroscopy. In saturated absorption spectroscopy, two counter-propagating beams interact with the rubidium cell. While not on resonance, both beams interact with separate velocity classes. On resonance, both lasers interact with the zero velocity class. However, one beam (known as the pump beam) is much stronger than the other (known as the probe beam). Since the pump beam is so much stronger, it interacts with the atoms much more strongly than the probe beam, and so the atoms in the zero velocity class are excited by the pump beam but not the probe beam. If we monitor the transmission of the probe beam, the result is an increase in the transmission for the zero velocity class. To summarize, off resonance the transmission of the probe beam is exactly like Doppler spectroscopy since the pump and probe beam interacts with different velocity classes. As the two laser beams pass through resonance, they interact with the same atoms causing a decrease in absorption for the probe beam. Since this interference only happens for the zero velocity class atoms, the shape of the increase in transmission is the Lorenztian profile.