NOTE: We will rely heavily on spectroscopic notation in this experiment to identify different energy states. Recall that a state may be labeled as n2S+1LJ, where n is the principal quantum number,  { $S$  is the spin quantum number, [Math Processing Error]L { $L$  is the (letter corresponding to) the orbital quantum number, and J is the total angular momentum [Math Processing Error](J→=L→+S→) { $(\vec{J} = \vec{L} + \vec{S})$ . |

Ignoring hyperfine effects for the moment, the lowest energy level in which the single electron of rubidium outside the closed shells can reside is the  { $5^2\mathrm{S}_{1/2}$  state. In this experiment, we will concentrate only on transitions between this state and the next higher energy state, [Math Processing Error]52P3/2 { $5^2\mathrm{P}_{3/2}$ . Transitions between these states occur via the emission or absorption of a photon of wavelength 780.0 nm. Adding in hyperfine interactions, we find that the { $5^2\mathrm{S}_{1/2}$  and [Math Processing Error]52P3/2 { $5^2\mathrm{P}_{3/2}$  states split into two and four closely separated states (respectively) which are differentiated by quantum number [Math Processing Error]F { $F$ , where [Math Processing Error]F→=J→+I→ { $\vec{F} = \vec{J} + \vec{I}$  is the total atomic angular momentum. As Rb-87 and Rb-85 have different intrinsic nuclear spins ([Math Processing Error]I=3/2 { $I = 3/2$  and [Math Processing Error]I=5/2 { $I = 5/2$ , respectively), the allowed values of F are different for the two isotopes. The energy level diagram for both species of rubidium, including the hyperfine structure, is shown in Figure 1. { ${/download/attachments/232095803/fig_1.png?version=1&modificationDate=1576513968000&api=v2}$ Figure 1: Energy level diagrams for 87Rb and 85Rb. (Note that the vertical spacing of the energy levels is not drawn to scale.)

The quantum mechanical selection rules determining the allowed transitions are

{ $\Delta F = 0, \pm 1$  (but not [Math Processing Error]0→0 { $0 \rightarrow 0$ ), { $\Delta L = 0, \pm1$ , and

{ $\Delta S = 0$ . In this experiment you will measure the magnitude of the hyperfine splitting between the { $5^2\mathrm{P}_{3/2}$  substates. You will do so by measuring the difference in frequency of photons associated with transitions between the [Math Processing Error]52S1/2 { $5^2\mathrm{S}_{1/2}$  states and the hyperfine (F’) states. As an example, refer to Fig. 2 which illustrates the energy difference between the F’ = 2 and F’ = 3 hyperfine levels of the [Math Processing Error]52P3/2 { $5^2\mathrm{P}_{3/2}$  state. { ${/download/attachments/232095803/fig_2.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 2: Energy level diagram for 87Rb showing the frequencies of the hyperfine splittings. 

2.3 Doppler broadening

Atoms which are in motion relative to an observer will have Doppler shifted emission and absorption energies. For example, if an excited atom with energy { $E_1$  moving towards an observer with velocity [Math Processing Error]v≪c { $v \ll c$  makes a transition to a lower energy state [Math Processing Error]E2 { $E_2$  by emitting a photon, that photon will be observed to have a higher energy, [Math Processing Error]E′ { $E'$ , than the energy of the atomic transition [Math Processing Error](E1−E2) { $(E_1-E_2)$  as given by

{ $E' = (E_1 - E_2) \left(1 + \dfrac{v}{c}\right)$

(3)

Similarly if the same observer wishes to excite the atom by hitting it with a photon, the observer would have to use photons with energy slightly lower than the transition to be excited.

Atoms in a gas at a temperature { $T$  will have random velocities given by the Boltzmann distribution,

{ $P(v)dv = \left(\dfrac{M}{2\pi kT}\right)^{1/2}\exp{\left(-\dfrac{Mv^2}{2kT}\right)}dv$

(4)

where k is the Boltzman constant and M is the mass of the atom. This distribution of velocities leads to a corresponding distribution of Doppler-shifted photon energies for any given atomic transition. The resulting spread in photon energies, about the transition energy, is referred to as Doppler broadening. In terms of frequency, ν, the full-width at half-maximum of the Doppler-broadened line is given by

{ $\Delta v_{1/2} = 2\dfrac{v_0}{c}\left(\dfrac{2kT}{M} \ln 2 \right)^{1/2} = \delta \sqrt{\ln 2}$

(5)

where { $v_0$  is the un-shifted frequency of the transition and [Math Processing Error]δ { $\delta$  is called the linewidth parameter. Later we will identify Doppler-broadened spectral features in terms of the final energy level of the transition. For example, the notation 87Rb(_F _= 2) refers to the Doppler-broadened feature that contains all allowed transitions from any of the 52P3/2(F' = 0, 1, 2 or 3) states to the 52S1/2(_F _= 2) state. In all, there are four Doppler broadened features: 87Rb(_F _= 1), 87Rb(_F _= 2), 85Rb(_F _= 2), 85Rb(_F _= 3).

These questions should be answered in your lab notebook between day 1 and day 2 of your experiment. You can work on them in lab if you make enough progress on the experiment, but don't work on them at the expense of getting your optics aligned.

Question 1:  Use the quantum mechanical selection rules to determine the allowed hyperfine transitions for 87Rb(_F _= 2).

Question 2:  For Rb atoms at room temperature, T = 300 K, calculate the line width due to Doppler-broadening in units of frequency. 

Question 3:  Compare this line width to the frequency separations for appropriate the hyperfine transitions from Fig. 1. Do you expect to be able to resolve these hyperfine features based on the results of this comparison?

2.4 Doppler-free saturated absorption spectroscopy

A simple linear absorption spectroscopy experiment is shown at the top of Fig. 3a. The intensity of a monochromatic beam of laser light, passing through a vapor cell, is measured as a function of the laser frequency. When the frequency of the laser matches the frequency of an allowed atomic transition, the photons will excite the atoms in the vapor and the intensity of the transmitted beam will be diminished. A plot of beam absorption as a function of laser frequency will show Doppler broadened absorption features like those seen in the bottom of Fig. 3a.

{ ${/download/attachments/232095803/fig_3.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 3: Comparison of (a) linear saturated absorption spectroscopy to (b) Doppler-free saturated absorption spectroscopy.

For the case of rubidium atoms at room temperature, Doppler broadening of the spectral lines makes the hyperfine features un-resolvable using linear absorption spectroscopy. However, Doppler-free saturated absorption spectroscopy, as illustrated in Fig. 3b, allows one to resolve features normally obscured by the effect of thermal Doppler broadening.

Doppler free saturated absorption spectroscopy uses two overlapping, counter-propagating beams – from the same laser in our case – which we call the probe beam and the pump beam. The intensity of the probe beam is monitored as a function of laser frequency just as is done in linear absorption spectroscopy. In the absence of a pump beam, the probe beam would produce a Doppler broadened linear absorption spectrum. The effect of the pump beam depends on the velocity of the atom along the z-axis.

{ ${/download/attachments/232095803/fig_4.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 4: Doppler shifting of probe and pump beams.

First consider the case illustrated in Fig. 4a, where an atom has non-zero velocity relative to the laser, along the z-axis. Since the two beams are counter-propagating one beam will always appear, to the atom, to be red-shifted and the other blue-shifted. Thus, any given transition cannot simultaneously be in resonance with both the probe and pump beams.

Now consider the case in which the atom has zero velocity relative to the laser, along the z-axis as illustrated in Fig. 4b. Since there is no Doppler shift, the atom sees both probe and pump beam photons at the same frequency. Thus, it is possible for a given transition to be simultaneously in resonance with both beams. If the laser is tuned to the frequency of a transition and the pump beam is intense enough, it will keep most of the atoms in the vapor in the excited state. (I.e. the pump beam saturates the vapor.) This saturation reduces the number of atoms available to absorb photons from the probe beam and so there is less attenuation of the probe beam than there would be in the absence of a pump beam.

Finally let us examine how both of these cases affect the absorption of the probe beam as the laser frequency is scanned across a Doppler-broadened transition as shown in Fig. 3b. As the laser frequency approaches the atomic transition frequency, the probe beam will be absorbed only by that fraction of the atoms in the vapor whose velocity along the z-axis Doppler shifts the probe beam onto resonance. These atoms cannot simultaneously be in resonance with both beams, and so the probe beam will measure the Doppler broadened absorption profile until the laser matches the frequency of an atomic transition. At that point, { $f' = f''$  and both beams can excite the same population of atoms having zero velocity parallel to the beams. The more intense pump beam will saturate the vapor and the absorption of the probe beam will decrease. As the laser frequency continues past the transition frequency the probe beam will again return to sampling the Doppler broadened spectrum. The result is a dip in the Doppler-broadened spectrum at the Doppler free frequency of the transition. The width of the Doppler free line is just the natural line width of the transition. Recall that the natural line width is related to the lifetime of the excited state by the uncertainty principle as

{ $\Gamma = \hbar / \tau$

(6)

where { $\tau$  is the lifetime of the excited state and [Math Processing Error]Γ { $\Gamma$  is the natural line width (in units of energy).

2.5 Crossover frequencies

Consider the case when the frequency of the laser { $f_L$  is tuned to the midpoint between two transitions with frequencies a and b so that [Math Processing Error]fL=12(fa+fb) { $f_L = \frac{1}{2} (f_a + f_b)$ . There will be a subset of atoms whose velocity Doppler shifts the photons in one beam (pump or probe) into resonance with transition a while shifting the photons in the other beam (probe or pump) into resonance with transition b. At these frequencies referred to as crossover frequencies, a dip in the Doppler broadened absorption spectra will appear. Thus, an atom with two transitions will exhibit three Doppler-free features, two transitions and one crossover. An atom with 3 transitions will have 3 crossover frequencies as illustrated in Fig. 5. { ${/download/attachments/232095803/fig_5.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 5: Energy level diagram for 85Rb showing allowed transitions to the F = 3 ground state (solid lines) along with the associated crossover frequencies (dashed lines).

3 Experimental procedure

3.1 Apparatus

A single laser is used to produce both the pump and probe beam and is controlled by a small electronics rack. A photodiode is used to monitor the intensity of the probe beam after its passage through the vapor cell and this output is fed to anto an oscilloscope.

A block diagram of the electronics is given in Fig. 6 and a more detailed description is given below.

{ ${/download/attachments/232095803/fig_6.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 6: Electronics block diagram.

[A] Grating feedback diode laser

The laser used is a grating feedback diode laser which provides a very narrow bandwidth output and which is continuously tunable over the spectral range of the 52P3/2 « 52S1/2 transition of Rb. The central element in the design is a diode laser with a peak power output of 120 mW at 780 nm. The diode is housed in a temperature controlled aluminum head with built-in collimating optics.

CAUTION:  This laser is powerful enough to cause serious eye injury.  Use the special IR absorbing goggles provided.

The bandwidth of this diode laser is about 20 MHz which is too broad to resolve the spectral features of interest. A diffraction grating is used both to narrow the bandwidth of the diode laser and to provide the tuning mechanism.

The output of the diode laser is directed onto the diffraction grating mounted in front of the collimating lens at the Littrow angle. The Littrow angle is the angle at which the first order diffraction from the grating will be fed back into the laser cavity. In this configuration the lasing cavity consists of the rear facet of the diode laser and the front surface of the grating, typically a distance of a few centimeters. (Laser bandwidth is inversely proportional to the length of the lasing cavity. For more details read chapter 4 in the second edition of “Experiments in Modern Physics” by Melissinos.) This technique reduces the bandwidth of the laser to less than 1 MHz.

By changing the distance between the diode and the grating, the laser frequency can be varied. This distance is controlled by means of a piezo-electric crystal mounted behind the diffraction grating. The thickness of a piezo-electric crystal varies in proportion to the voltage applied to it. By using a function generator to apply a triangle wave to the piezo, we change its thickness linearly and periodically in time. This change in thickness corresponds to a variation of laser frequency over a range of up to ~20GHz.

The frequency of the diode laser is also linearly related to its forward bias current. The laser diode current supply can be used to adjust the laser's output frequency.

In addition to the grating feedback angle and forward bias current, other parameters affect the output wavelength of the laser. The lasing action requires having an integer number of half-wavelengths in the cavity. Since the cavity length depends on temperature the output wavelength is also sensitive to temperature. A thermoelectric cooler built into the laser housing will keep the laser at a stable temperature.

[B] Laser Controller

The electronics modules controlling the operation of the laser are housed together in a rack which supplies power to the modules and allows them to communicate with one another via a backplane.

The Laser Controller module, the right most module in the rack, is configured to display information from the other modules. In particular,

• Tact1 - The actual temperature of the laser head;
• Tset1 - The setpoint temperature for the laser head;
• Imax - The upper current limit for the diode laser; and
• Iact - The actual current of the diode laser.     

There is a keyed Mains on/off switch on the lower right corner of the Laser Controller which controls the power to the modules. Located just above the Mains on/off switch are the Module on and off buttons.

[C] DC110 Current Control

The Current Control provides the forward bias current to the laser diode. To prevent damage to the diode, an upper limit to the current (Imax) has been set. The on/off switch controls power to the module. Both Imax and Iact are displayed on the Laser Controller. Other setting on the module should be as follows.

[D] DTC110 Temperature Control

The Temperature Control drives the thermoelectric cooler built into the laser housing. The correct temperature set point for your diode is already programmed into the controller (Tset1) and should not be changed. Once the temperature controller is turned on the laser temperature will reach equilibrium in about 5 minutes. Both the set point temperature (Tset1) and the actual temperature (Tact1) are displayed on the Laser Controller. Leave the temperature controller on until you are finished for the day.

[E] SC110 Scan Control

When turned on, the Scan Control will send a sawtooth voltage to the piezo which modulates the laser cavity length. The Output should be connected to the Piezo input on the back of the laser head ([A] in Fig. 6). The Trigger output should be connected to channel 2 of the scope. The On/Off switch will power the module on and off. 

[F] Photodiode

The intensity of the pump beam is monitored with a ThorLabs photodiode whose current output is converted to a voltage and displayed on the oscilloscope. The photodiode has on/off and gain controls on the side of the housing. The photodiode should be turned off at the end of the day.

[G] Digital oscilloscope

The signal from the photodiode and the output of the function generator should be displayed on channels 1 and 2 of the scope. Trigger the scope on the output of the function generator. The scope digitizes the signal from the photodiode allowing the absorption spectra to be captured and sent to the computer. Your instructor will show how to do the transfer.

Rubidium vapor cell

The rubidium is contained in a glass cell. Note that rubidium is a highly reactive element and should not be exposed to air. Be careful not to break the vapor cell.

3.2 Procedure

3.2.1 Laser safety

Note that the beam from the laser will cause permanent damage if it strikes your eye. If the laser is on and you are in the room you must wear the protective goggles provided. Additionally, anyone entering the room while the laser is on must be wearing goggles as well.

CAUTION: Always wear the protective laser goggles provided when the laser is on.

The laser beam is not harmful to the skin, so it is safe to operate with your hands in the beam. However, you should remove any reflective objects – such as watches and rings – from your hands and wrists before working on the optics table to prevent accidental deflection of the beam out of the plane of the table. 

3.2.2  Overview of the optical setup

{ ${/download/attachments/232095803/fig_7.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 7: The optical components and beam path used for saturated absorption spectroscopy.

Fig. 7 shows the optical setup we will use to create the overlapping pump and probe beams in the rubidium vapor cell. The main components consist of the following:

• _Toptica laser: _This laser produces light at a wavelength of 780 nm with a power of ~120 mW. The laser can be set to scan back and forth across a narrow range of wavelengths.
• _Half waveplate: _The half waveplate is used to adjust the plane of polarization of the light from the laser. This waveplate is a birefringent crystal which has been cut and mounted so that light of a specified wavelength passing though the crystal will have its plane of polarization rotated. The amount of rotation of the polarization depends on the relative angle between the polarization plane of the incident light, and the optical axis of the waveplate.
• Polarizing beam splitter: The polarizing beam splitter is setup so that horizontally polarized light will pass straight through while vertically polarized light will be reflected.
• _Mirrors: _Mirrors are mounted on kinematic mounts allow for precise control of the beam path.
• 50/50 beam splitter: This beamsplitter passes 50% of light straight through while reflecting the other 50%.
• Rb vapor cell: This sealed glass cell contains a small quantity of rubidium with a natural mix of isotopes 85 and 87.
• Linear polarizer: The polarizer is used to control the intensity of the retro-reflected probe beam.
• Photo-detector: The photodiode produces a current proportional to the intensity of the light striking it.

{ ${/download/attachments/232095803/fig_8.png?version=1&modificationDate=1576513967000&api=v2}$Figure 8: The laser feeds two separate experiments; we use only the right half of the table for this experiment. As shown in Fig. 8, the Toptica laser is used for two different experiments. The combination of the half waveplate (a) and polarizing beamsplitter (b) are used to determine how much of the laser beam goes to each experiment. Rotating the half waveplate will vary the intensity of the beam going to the rubidium hyperfine experiment.

For the rest of this manual we will only be concerned with the beam going to the rubidium hyperfine experiment and will ignore the path leading to the optical trapping experiment.

3.2.3 Aligning the beams

Since the laser frequency is in the near infrared and you will be wearing goggles which block light at the frequency of the laser, the beam must be observed indirectly. The TAs and lab staff will instruct you in how to safely observe and align the laser beam using an infrared viewing card and a CCD video camera. 

Getting the probe and pump beams to overlap in the vapor cell is fairly straight forward with the optical geometery we are using. Here are some tips to keep you on the right track.

• Keep the beam at a constant height above the optical table. The beam coming out of the laser is likely not parallel to the surface of the table. However, you can control the height of the beam after it reflects off of the first mirror. Use a ruler to measure the height of the beam above the table at the first mirror. Then, use the ruler to make sure that the beam is always at that same height when it strikes all subsequent optical components.
• Direct the beam along the path indicated on the optical table as closely as possible.
• Use the half waveplate to vary the intensity of the beam going to the hyperfine experiment.
  • At full strength, the reflections of the beam may saturate the CCD camera and viewing card. In order to determine the location of the beam precisely it may be necessary to dim the beam.
  • On the other hand, observing fluorescence in the vapor cell is easier with a more intense beam.

3.2.4 Tuning the laser to resonance

Once you have the probe beam aligned with the pump beam and striking the photo-detector, you should tune the laser onto resonance with the rubidium hyperfine transitions.

The laser is designed to scan back and forth over a narrow range of frequencies. The frequency range over which the laser scans can be shifted up and down frequency by changing the current to the diode laser. The TAs and lab staff will instruct you on how to adjust the diode current.

To tune the laser onto resonance:

• Make sure that the output of the photo-detector is connected to channel 1 of the scope and the trigger output from the scan controller (see Fig. 6) is connected to channel 2. Trigger the scope on channel 2. The scan controller generates a positive square pulse on the trigger output at the beginning of each frequency sweep of the laser. Triggering the scope on this pulse allows us to stay synchronized with the frequency sweep of the laser.
• Dim the laser beam by adjusting the half waveplate and then adjust the gain on the photo-detector so that it is not saturated. You can tell the detector is saturated when its output is a fixed amplitude sine wave. It is useful to have channel 1 of the scope set to “AC coupling” for this experiment.
• Once you have a signal from the photo-detector which is not saturated, adjust the current to the laser diode until you observe the beam fluorescing the vapor in the cell. At this point you know that you are scanning over at least some of the allowed hyperfine transitions. Now, play with the laser diode current while watching the detector output on the scope. You want to obtain a signal which looks like Fig. 9. The challenge is finding the laser diode current sweet spot setting which allows the frequency to sweep continuously across the entire range. Due to various factors which compete with one another to determine exactly what frequency the laser runs at, discontinuous jumps in frequency can occur which appear on the scope as jagged glitches in the signal. These discontinuities are called “mode hops”. Although the laser diode has more than one current setting at which it will fluoresce the vapor, not all such settings will allow the laser to sweep over the full frequency range.
• Attempt to find a spectrum showing all four peaks (as shown in Fig. 9), but you must, at minimum, be able to identify the first two peaks – 85Rb(F = 3) and 87Rb(F = 2) – which will be used for further analysis.

{ ${/download/attachments/232095803/fig_9.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 9: The Doppler-broadened absorption spectrum of natural rubidium.

3.2.5 Optimizing the absorption spectrum

Once the laser has been tuned so that it is able to sweep over the full hyperfine spectrum, zoom in on the 87Rb(F = 2) peak. Assuming that you have overlap of the pump and probe beams, there should be some Doppler-free dips in the spectrum. In order to see all six dips you will need to do the following:

• Maximize the overlap of the probe and pump beams. This can be done by adjusting the position of the probe beam while observing the signal on the scope. Adjust the probe beam to maximize the amplitude of the Doppler-free dips in the spectrum.
• Adjust the ratio of the intensities of the pump and probe beams. Ideally we want the pump beam to have just enough intensity to keep all of the atoms along the beam path in the excited state (i.e. saturated).
• Too much intensity in the pump beam results in power broadening of the spectral features which increases their line width. If you put too much power into the vapor cell, the stronger Doppler-free lines will become wide enough to overlap with some of their neighbors which can obscure the weaker lines. By adjusting the half waveplate you can observe how the lines broaden as the intensity is increased.
• For the probe beam we want it to be weak enough, relative to the pump beam, so that it does not contribute to either exciting the atoms or power broadening. You can adjust the intensity of the probe beam by rotating the linear polarizer.  

Adjust the 1/2 waveplate so that the optical power meter provided reads 2mW. To do this place the meters detector head in the beam path in front of the beam splitter. Note that the optical power meter provided is calibrated for a wavelength of 670nm. At our wavelength of 780 nm the meter is accurate enough for the purpose of obtaining approximately correct power.

Once you have a spectrum which shows all six Doppler free features, stop the scope and transfer the digitized waveform to the computer using the application Scope Transfer which is located on the computer desktop.

3.2.6 Using an interferometer to calibrate the frequency of the laser

Your absorption spectra will consist of photodiode voltage as a function of time as recorded by the digital scope. The vertical axis measures the intensity of the light passing through the vapor cell. As the piezo voltage varies linearly with time, and as the laser frequency varies with piezo voltage, we may convert the sweep times to changes in the frequency of the laser output. Ultimately what we want to measure are the energy differences between hyperfine levels, so all we need are frequency differences between features in the spectra. We can accomplish this by using a Michelson interferometer to measure the change in frequency of the laser as a function of time.

{ ${/download/attachments/232095803/fig_10.png?version=1&modificationDate=1576513967000&api=v2}$ Figure 10: Layout of the Michelson interferometer.

The geometry of the Michelson interferometer is shown in Fig. 10. The beams from the two arms of the interferometer will combine at the photodetector with varying degrees of constructive interference depending on their phase difference Δ_φ_. It can be shown that the phase difference depends on the difference in lengths of the two arms of the interferometer and the frequency of the light as

{ $\phi_1 - \phi_2 = \Delta \phi = \dfrac{4\pi f}{c}(L_1 - L_2)$

(7)

where { $f$  is the frequency of the light, [Math Processing Error]L1 { $L_1$  and [Math Processing Error]L2 { $L_2$  are the path lengths of the two arms of the interferometer, and [Math Processing Error]c { $c$  is the speed of light. From this relation, it can be shown that the frequency spacing of the interference maxima at the output of the interferometer is

{ $\Delta f = \dfrac{c}{2(L_1 - L_2)}$