Table of Contents
Optical Pumping Winter 2024
Optical pumping is an experimental technique by which specially prepared photons are used to repeatedly excite an ensemble of Rb atoms in a way which drives the atoms into a specific atomic energy state from which the atom can no longer be excited by the photons. Once you learn how the technique works and how to apply it to Rb atoms in a vapor cell you will use it as a tool to make measurements of the energy of the Zeeman splitting as a function of magnetic field. You will be able to test the predictions of the quantum mechanical model for Zeeman splitting for a range of magnetic fields which covers both the small field approximation and larger fields at which larger order terms become important.
 |
| A vapor cell containing rubidium gas |
Teaching Points
Optical pumping is an extremely rich subject for an instructional lab. The physics governing how the technique works and its experimental uses are readily accessible for undergraduate physics majors. The apparatus we are using is both sophisticated and capable of making very precise measurements. In fact there is so much going on that it has the potential to become overwhelming in the context of a three day instructional lab experiment. For the purpose of this course we have defined the following teaching points as being the primary focus of the lab. Keeping them in mind while working in the lab will help you to get the most out of the experience.
- Learning what the optical pumping technique is and how it works. This will include.
- Understanding atomic energy levels and transitions.
- Understanding what the Zeeman effect is.
- Understanding how a beam of right circularly polarized light can be used to prepare an ensemble of Rb atoms in a known, non equilibrium distribution of energy states.
- Learning how to use the technique of optical pumping to:
- Make a precision measurement of magnetic fields.
- Explore the atomic energy level structure of an atom.
- Make measurements of the energy of Zeeman splitting.
A Note About Uncertainties
The apparatus and technique which you will be working with are capable of very precise measurements. While it may seem counter intuitive, this actually makes it more difficult to understand and account for your uncertainties. Most of your measurements are readings off of an oscilloscope, or DMM where the dominant uncertainty is the reading error of the instrument which is going to be very small. The main factors which contribute to limiting how well you know what you have measured are more subtle and complex. For example in this experiment we will assume that the Rb atoms we are studying are in an uniform magnetic field, but at some level there are gradients in the field strength through out the sample. Assessing the impact of this effect requires careful and detailed investigation, which can take longer than the time you have for the lab.
Since we want you to focus your time and efforts on the Teaching Points given above, we are going to limit the amount of uncertainty analysis for this lab. This is NOT to say that the uncertainties are unimportant. It is simply an acknowledgement of the fact that you have a finite amount of time to do the lab and we have to pick and choose which aspects of the experiment you should focus on. For this reason we do not ask that you do a full uncertainty analysis, meaning that we will not expect you to calculate and propagate uncertainties.
However you should always consider how well you know the value of anything which you measure and record in your notebook. Anytime you are making a measurement, ask yourself what is limiting how well I know this number. For example if you are reading a voltage off of a DMM, you cannot know that voltage better than the number of digits provided. Not only should you record an uncertainty for a measurement, you should include a note on what that uncertainty represents. We will expect to see this in your lab notebook.
Modern research
UChicago Physics Professor David DeMille uses the optical pumping technique as an integral part of his research. https://demillegroup.psd.uchicago.edu/
Since the zero-field crossing resonances where pumping occurs in Rb are very sensitive to local magnetic fields, such setups can be used to measure minute magnetic fields. Previously SQUID (Superconducting QUantum Interference Device) magnetometers were best for such measurements, but they tend to be bulkier or require extensive cooling. (See Moving magnetoencephalography towards real-world applications with a wearable system.)
Optically pumped Rubidium is also of interest in the field of quantum optics. The German group Laboratoire Kastler Brossel have published papers on storing and retrieving images in Rb85, with a long-term goal of creating a memory for a quantum communications network. (See Spatially addressable readout and erasure of an image in a gradient echo memory.)
More broadly, the dependency of the polarization of atomic emission spectra on local magnetic field is being used by astronomers to gather information about the conditions on the surface of stars. (See Discovery of Ground-state Absorption Line Polarization and Sub-Gauss Magnetic Field in the Post-AGB Binary System 89 Her.)
Pre-lab Preparation
Before your coming to lab on the first day of this experiment, you should do some background research. Much of ths information you need is in this wiki, but you may need to turn to outside resources as well. Wikipedia is your friend here. Ideally you should attempt to understand on a conceptual level the following:
- What is Optical Pumping and how does it work for Rubidium?
- What does the atomic energy level structure look like for the D1 line of Rb, both with and without Zeeman Splitting.
- What is the Zeeman Effect and why is Zeeman splitting necessary to optically pump Rb?
Understanding these concepts will help you to better understand what you are doing in the lab.
This lab will help you explore your understanding of electricity and magnetism, as well as the basic quantum mechanics of single electron atomic transitions.
Atoms may occupy only discrete energy states, and if the atoms are in thermal equilibrium, the relative numbers of atoms occupying each state is given by the Boltzmann distribution. If we shine light of the appropriate energy and polarization on those atoms, some atoms will be excited to a higher energy state. If the atoms have no easy escape path from this excited state, then the population of atoms will no longer be Boltzmann-distributed. Such an inverted population is described as optically pumped.
In this experiment we will look at the behavior of rubidium-87 (Rb-87) atoms in a vapor cell, illuminated by light from a rubidium lamp. The rubidium atoms have an intrinsic net magnetic dipole moment and therefore will react to an applied magnetic field and exhibit energy level splitting due to the Zeeman effect. These atoms also posses angular momentum and can transition from energy level to energy level only through interactions with light that conserve the total angular momentum of the atom and absorbed/emitted photon system.
Theory of Optical Pumping
An overview of the basic principles of Optical Pumping can be found here, Theory of Optical Pumping.
Optical Pumping Apparatus
A description of the Optical Pumping apparatus can be found here, Optical Pumping Apparatus.
Exercises
Before you begin to collect the bulk of the data, you will complete a number of specific tasks, each of which is focused on a skill or technique which you need to understand in order to complete the experiment. Successfully completing these tasks, as determined by the instructors during the lab, will count for a total of 75% of the grade of this lab.
Completing these exercises will likely take most of the first one or two days of lab. Go slowly, and make sure you understand each step!
Initial Setup and Vertical Field Cancellation - 5 points
Start by turning on the control console; the power switch is located on the upper left (as viewed from the front of the apparatus) corner on the back of the unit. It will take 10 to 20 minutes for the vapor cell to warmup and come to equilibrium. During this warmup period the signal from the photodiode will be changing as the vapor pressure in the cell increases. This is a good time to make sure that all of the controls on the console are set to default values. Remember, there there likely have been other students working on this same apparatus and who knows what state they left it in.
Default Control Console Settings
Detector Amplifier Section
- Gain = 2
- Gain Multiplier = 1
- Time Constant = 10ms
- Meter Multiplier = 1
- Offset Fine = Set to its midpoint
- DC Offset = Anything is fine for now.
Horizontal Magnetic Field Section
- Horizontal Magnetic Field = 0
Vertical Magnetic Field Section
- Vertical Magnetic Field = 0
Horizontal Magnetic Field Sweep Section
- Time = 1s
- Start/Reset = Reset
- Single/Continuous = Continuous
- Range = 0 (Full CCW)
- Start Field = Max (Full CW)
- Recorder Offset = 0 (Full CW)
- Magnetic Field Modulation
- Amplitude = 0
- Start Field / MOD = Start Field
- Input = No Connection
Cell Heater / Controller
- Do not make any adjustments.
RF Amplifier
- Gain = 0
- Input RF Modulation = No Connection
- Output = No Connection
- Input = No Connection
Align the Detector
If you have put the control console into the default settings as described above, there should be no external magnetic fields being generated by the Helmholtz coils, and the Rb atoms are now sitting in the ambient magnetic field of the lab.
Optical pumping is very sensitive to the presence of external magnetic fields. You want to make sure that there are no magnets near the apparatus. You might be surprised by how many common items contain magnets. (For example, your cell phone likely has some strong magnets in it.) Don't forget to take your cell phone out of your pocket.
We can only control the magnetic field along two axes – vertical and the component of the horizontal plane which is parallel to the axis of the optical rail – using the Helmholtz coils. To completely zero the magnetic field in the vapor cell, you will need to rotate the whole apparatus so that the optical rail is aligned with the horizontal component of the ambient field in the room. Do a rough alignment using a compass; there is no need to be super-precise as we will fine tune this alignment in a moment.
Zero The Vertical Component Of $B_{room}$
Once you have roughly aligned the optical axis of the apparatus with the horizontal field in the room, use the Vertical Magnetic Field control on the console to cancel out the vertical component of the ambient field. There is a magnet mounted in a gimbal which you can use as a 3-D compass. Hold it near the vapor cell, inside the volume enclosed by the vertical coils. Now turn up the current to the vertical coils and watch how the magnet behaves. You can easily turn up the vertical field enough to reverse it.
Play with the current to the vertical coils while observing the effect on the 3-D magnet and convince yourself that you understand what is happening. Then, set the current to create roughly zero field along the vertical axis. You will fine tune this cancellation in a later step.
Observe And Optimize The Zero Field Depumping Signal - 5 points
At this point you should have roughly aligned the optical axis of the apparatus with the horizontal component of the ambient magnetic field in the room, and used the vertical Helmholtz coil to cancel out the vertical component of this field. The final step in the setup is to use the horizontal Helmholtz coil to generate a time varying magnetic field along the horizontal axis which sweeps back and forth across the point where it precisely cancels out the horizontal component of the ambient magnetic field present in the lab. We will refer to this time varying magnetic field as the Horizontal Sweep Field $B_{H}$. Since the vertical coil has been used to cancel $B_{AV}$, each time $B_{H}$ passes through the point where it cancels out $B_{AH}$ the total net field $B_{Net}$ seen by the Rb atoms in the vaporcell will be zero and as a result the magnitude of the Zeeman Splitting will also be zero. In the absence of Zeeman Splitting there can be no pumping effect so the ensemble of atoms will “depump” and the transparency of the vaporcell will decrease as more atoms are available to absorb the light passing through it. This results in a reduction of the amount of light reaching the photo-detector whose signal you are monitoring on the scope.
Note that the concept of Zero Field Depumping is not as difficult to understand as it first appears. It is just one of those things which combines several concepts in a way which is difficult to explain. Do not be surprised if you struggle at first with understanding and being able to visualize what is happening. Keep working on it and you will find that once you get it, it will seem pretty straightforward.
- In the Horizontal Magnetic Field Sweep section of the control console flip the Start/Reset switch to the Start position. Make sure that the Range is turned all the way up (Full CW) and the Start Field is turned all the way down (Full CCW).
- Connect the Recorder Output to one channel of the scope and the Detector Output to the other channel. Setup the scope to trigger on the Recorder Output signal. Keep in mind that the period of the sweep is about 1 second, so make sure the time base of the scope is set accordingly.
At this point you should see a sawtooth waveform on the scope from the Recorder Output. This sawtooth represents the current which is being sent to the Horizontal Sweep Coils. As the current to these coils increases so does the magnetic field $B_{sweep}$ it produces. If you have properly aligned the detector with $B_{AH}$ and canceled $B_{AV}$, $B_{sweep}$ should at some point in its sweep cancel out $B_{AH}$. When this happens the Rb gas will depump and you will observe a decrease in the signal from the photo-detector as the transparency of the vaporcell suddenly increases. Once you observe this effect you can proceed to optimize the depumping effect through an iterative process as follows.
- While watching the photo-detector signal on the scope, make small adjustments to the Vertical Field control to fine tune the cancellation of $B_{AV}$. If your adjustment causes the drop in the signal from the photo-detector to get bigger you have improved the cancellation of $B_{AV}$. If your adjustment makes the drop in the signal smaller you have made the cancellation of $B_{AV}$ worse. Continue to adjust the Vertical Field control until you have made the drop in the signal from the photo-diode as large as you can. Note how the drop in the signal from the photo-diode also gets narrower as you improve the cancellation of $B_{AV}$.
- Now make small adjustments to the orientation of the apparatus on the horizontal axis by rotating it a tiny amount. Just as with the vertical field, the drop in the signal from the photo-diode will get bigger and narrower as you improve the cancellation of $B_{AH}$. Continue making small adjustments to the orientation of the apparatus until you have made the detector signal as large and narrow as possible.
- Go back to step 1 and repeat the process for the vertical field control. Continue going back and forth between steps 1 and 2 until the drop in the signal from the detector is as large and narrow as possible.
Save a screenshot of the optimized depumping signal, and be prepared to talk about how sensitive the signal was to field changes.
View the zero field crossing on the scope
This link takes you to a video which gives an overview on how to use the digital scope to see the de-pumping signal.
Left vs Right vs 50/50 Mixture of Circular Polarizations - 5 points
Once you have a nice clear zero field depumping signal we can consider the following question.
In the description for how optical pumping works we illustrate how a beam of right circularly polarized light will drive the ensemble of Rb atoms into a state where they can no longer absorb any more right circularly polarized light.
What would happen if instead of right circularly polarized light we used left circularly polarized light? Would the optical pumping effect still take place?
What would happen if we prepared the light in a 50/50 mixture of right and left polarized states? Would optical pumping still take place?
Now, figure out how to use the apparatus to test your predictions.
Keep notes on how you changed the chirality of the incoming light and the effect on the rubidium.
Calibrating the Horizontal Field Sweep - 5 points
At this point you have the apparatus setup and optimized to sweep back and forth through the Zero Field Depumping Condition. For the experiments which you will perform using this apparatus you will need to know the precise values of the magnetic fields which you are generating in the different Helmholz coils surrounding the vaporcell. The current to the vertical coil is easy to measure directly by inserting an ammeter into the signal path. The horizontal sweep coil current however is constantly changing in time as part of the sweep. The signal which you view on the scope is a voltage which is proportional to the amount of current going through that coil, but you need to determine the constant of proportionality so that you can convert from voltage readings on the scope to coil current values. A description of the process to perform this calibration is given in the section Calculating The Coil Currents on the Optical Pumping Apparatus wiki page.
Be prepared to discuss how you found your conversion factors, and to estimate their reasonability.
Measure Magnetic Field In The Lab - 5 points
Once you have fine tuned the vertical and horizontal fields to make the zero crossing signal on the scope as narrow as possible you have all the information you need to calculate the ambient horizontal and vertical magnetic fields in the lab at the point where the Rb vapor cell sits. The main ambient field in the lab is the Earth's magnetic field. There are likely to be other sources of magnetic fields in the lab (such as the magnetic blackboard and AC electrical lines in the room with the apparatus). You should, however, be able to confirm that the dominant field is the Earth's (to within about a factor of 50%). Doing this calculation at this point will help to ensure that you fully understand what all of the currents, coils and fields are doing.
You should:
Make the calculations to find the components of the magnetic field within the apparatus.
Obtain an estimate of the Earth's expected magnetic field in the vicinity
- There are nice tools for doing this on the NOAA website.
Zeeman Splitting
You now know enough to move on and begin a detailed study of the Zeeman splitting of atomic Rubidium (Rb). This experiment breaks down into two parts:
Zeeman splitting in the weak field approximation - In the presence of a weak magnetic field the energy difference between adjacent Zeeman states is linearly proportional to the net magnetic field ($B_{net}$). You can use optical pumping and a technique called RF depumping to directly measure the energy of the Zeeman splitting as a function of $B_{net}$. The weak field approximation refers to the fact that when you use quantum mechanics to calculate the energy of the Zeeman effect you end up using a Taylor series expansion and in the case of small $B_{net}$ you drop all but the first term which is linear. This solution works well for “small” magnetic fields.
Zeeman splitting in the strong field - As $B_{net}$ increases the small field approximation eventually breaks down, and you need to include the higher order terms from the Taylor series expansion. When you do this the solution becomes non-linear and the different Zeeman states become individually resolvable. The number of different zeeman energies you observe are related to the magnitude of the nuclear spin of the atom. So by increasing $B_{net}$ until you can resolve the individual Zeeman states you can make a direct measurement of the nuclear spin.
RF depumping and measurement of Zeeman splitting vs $\mathbf{B_{net}}$
Small Field Approximation
At this point, you should be familiar with the concept of optical pumping and comfortable setting up and using the apparatus to observe the depumping signal while sweeping $B_{net}$ through the zero field condition.
Now we want you to extend your knowledge of the apparatus and the technique to include RF depumping which can be used to directly measure the energy of the Zeeman splitting for a given $B_{net}$. Recall that there is one more set of coils on the apparatus which we have not yet used. These are the RF coils which are connected to the output of a function generator. By sending an oscillating current to these coils, we produce an oscillating magnetic field which is orthogonal to the sweep field generated in the horizontal coils. When the energy of the RF photons, $E=\hbar \nu$, matches the energy difference between Zeeman energy levels, they will drive electronic transitions between the Zeeman states. If the gas is in the pumped state when we turn on this RF field, electrons which are in the pumped state will be driven into a different state and gas will be depumped.
To begin, set up the apparatus so that it is sweeping through the zero field condition and so that the zero field depumping signal occurs at approximately the mid-point of the horizontal field sweep. Set the Horizontal Sweep Range to its maximum value so that the Rb atoms are seeing the largest range of $B_{net}$ possible.
Now turn on the function generator and set it to produce a 40 kHz sine wave with an amplitude of ~0.5 V peak-to-peak. Turn on the output to channel 1 which should already be connected to the RF coils on the apparatus. The signal from the detector should now show additional depumping dips on both sides of the zero field point.
Spend a bit of time making sense of what you are seeing on the scope. Keeping in mind that the vapor cell we are using contains natural Rb which is a mixture of ${}^{85}$Rb and ${}^{87}$Rb, see if you can understand the following:
- Can you explain why you see the number of RF depumping dips which appear when the RF coils are energized?
- What is the significance of where along the horizontal sweep these RF depumping signals appear?
- Increase and decrease the RF frequency and see if you can explain how the RF depumping signals change in response.
- Increase and decrease the amplitude of the RF. Do you understand how the signals respond?
Once you understand how the system responds to the RF depumping, it is time to measure the magnitude of the energy splitting of the Zeeman levels as a function of $B_{net}$ over the widest possible range of $B_{net}$ values. The goal is to test the low energy approximation for Zeeman splitting. This approximation predicts a linear relationship between the energy of the Zeeman splitting and the net magnetic field which is related to the Lande g factor.
Strong Field Zeeman Splitting
To explore the Zeeman splitting at high magnetic fields, you will need to use the external power supply to power a set of concentric Helmholtz coils within the apparatus. The process goes roughly as follows:
- Set up your system such that you can observe the RF depumping dips
- Turn on the external power supply, but keep the voltage near zero.
- Increase the power from the external supply, which will shift the depumping dips on the scope
- Choose one of the two dips to follow, and increase the RF depumping frequency to better center the signal on the screen.
- Repeat steps 3 & 4 to track the behavior to stronger fields.
We expect that, at some point, we will observe the depumping dip start to broaden and split when in a sufficiently strong B field.
When you leave the lab
Since there will be other groups working on the apparatus, it is your responsibility to ensure that everything in the lab is in order when the next group arrives. The room should be tidy, and everything should be either put away or reset to it's default. If the lab room was in disarray when you arrived, you are still responsible for leaving it in the appropriate state for the next group.
Here are some general tips on things to check before you leave.
- Dial all vertical and horizontal field coil currents to zero.
- Set the sweep amplitude to 100%.
- Set the sweep offset to zero.
- Disconnect any function generators, multimeters, or external power supplies which you may have been using.
- Disconnect the scope from the apparatus and turn it off.
- Close any applications on the computer and don't forget to log out of any accounts which you may have opened.
- The computer may be left on; it will go into sleep mode.
- Reorient the apparatus to be square to the room.
- Place the compasses on the table next to the apparatus.
Analysis And Conclusions
Zeeman Splitting In The Small Field Approximation
Data Analysis and Calibration - 25 points
Present your full analysis of the magnitude of the energy splitting of both isotopes of Rb as a function of $B_{net}$ for small fields. This includes calibrating the measurements from the scope to get $B_{net}$ for each RF depumping resonance, plotting $\delta E$ vs. $B_{net}$ and fitting to a line to extract the value of the Lande-g factor for each isotope.
Do the following:
- Present your data for calibrating the horizontal sweep field. Include the measured values in tabular form, with proper units and uncertainties.
- Show your calculations for determining the calibration factor for the horizontal sweep field. Provide just enough description of how you collected the calibration data so that the reader can understand how the data were collected.
- Present your RF depumping data, including units and uncertainties on measured quantities. Include an example of a typical scope trace with relevant features in the spectrum identified. Make clear how you determined the values of Bnet relative to the zero field depumping signal.
- Show how you applied the horizontal sweep calibration.
- Present a plot of the calibrated data for $E$ vs $B_{net}$ for both isotopes. Include any fits you performed on the data.
- Show how you determined your experimental values for the Lande-g factors for both isotopes.
Conclusions - 25 points
Compare your measured values for the Lande-g factors for both isotopes of Rb to the values predicted by the theoretical model.
Show your calculation of the theoretical values of the Lande-g factors.
Comment on the degree of agreement between your experimentally determined values and the theoretical predictions.
Large Field Zeeman Splitting - 25 points
- Present your data from the scope traces in a publication quality form for each isotope.
- Include the experimental parameters such as RF settings, applied $\mathbf{B}$-Fields, etc. for each isotope.
- Identify the features in your scope data.
- Provide your measured values for the nuclear spins $(\mathbf{I})$ for both isotopes.
- Describe how you deduced the nuclear spins from your data.