You will perform this experiment much like a guided research project, with weekly meetings with your TA (and sometimes the lab staff) to evaluate your progress and make sure you are on the right track.  Because of the remote nature of the course you will not be able to come to the lab to work with the physical apparatus. As such, the goal of this lab is for you gain experience in all of the other aspects of experimental research, which include the following:

• Starting out with a literature search to learn about the physics of the phenomena being studied as well as appropriate experimental techniques that can be applied.
• Evaluating data (taken by the staff following the plan you develop) in order to assess your understanding of the experiment, and then modifying your plans to take additional data as needed.
• Drawing and reporting on conclusions from your data.

Every experiment is different.  When you did the gamma cross sections experiment the technique employed was very straightforward and easy to understand.  Source, absorber, detector, count pulses for different absorber thicknesses.  Much of the challenge in that experiment involved figuring out how to extract the needed information from the data, and in this process collecting multiple sets of data.  For this experiment you need a solid understanding of the physics of optical pumping, understanding of the technique being employed to make measurements, and familiarity with operating the apparatus. As such there is a lot more up front research which must be done, and almost certainly you are going to have to wrap your minds around some concepts that you have not yet encountered.  Although it may seem a bit overwhelming at first there is plenty of time to perform this research and assimilate the new concepts. 

Optical Pumping is a technique which uses light to manipulate the electronic spin state of atomic systems such as an ensemble of Rb atoms in a gas. Optical pumping can be used to study the behavior of atomic energy states, it is also used to create the inverted energy state populations needed for gas lasers.

In this lab you will learn how to create the conditions necessary to optically pump Rb atoms. By manipulating the local magnetic field in which the Rb atoms reside, you can then use an RF resonance technique to measure energy differences in the Zeeman splitting of the first excited state of Rb.

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.

Before coming to lab it will be very helpful for you to understand what we mean by optical pumping. Read over the Optical Pumping section on this wiki page. You should be able to answer the following questions before starting the experiment.

• For the D1 line of Rb which of the Zeeman states are the pumped states for the case when the pump photons are right circularly polarized? Why?
• If we changed the photon polarization from right circular to left circular would pumping still occur? If not why not? If yes, which Zeeman states would be the pumped states?
• If the Rb atoms are in an environment where there is no net external magnetic field will our optical pumping technique still work? Why or why not?

If you do not understand the answers to these questions before coming to lab you will use a lot of valuable time figuring out the concepts involved in order to have any idea what you are doing with the apparatus.

Doing this type of background research before starting work in the lab with apparatus is one of the experimental skills which we are teaching in this course. By the third quarter you will be expected to do your own prelab research into things like the subject of the experiment, apparatus and techniques with minimal guidance from us. For fall quarter we are giving you guidance on what to read up on before coming to lab. Ideally you should come to lab ready to start with the Apparatus section of the wiki page. If you are not ready to do so you are already behind.

This fall quarter one of the primary goals of the course is to help you develop some of the fundamental skills which are needed in order to do experimental physics. One of those skills is making sure that you understand what your apparatus and technique are doing in terms of the physics of what you are trying to measure or study. In this experiment you are given a source of light, a narrow bandpass filter, a linear polarizer and a quarter wave plate. You are told that using these components in a certain way will allow you to change the spin state of Rb atoms in a vapor cell. As an experimenter you need to test your apparatus to ensure that everything behaves as it should, otherwise how can you have confidence that your interpretation of the results of the experiment is correct. In this lab we will guide you through the process of using your knowledge of physics to go beyond relying on information provided in instruction manuals in order to study how your experiment works.

In this lab you will learn some common techniques for using electromagnetic radiation and precise control of magnetic fields to manipulate and probe atomic energy states.

You will gain experience conducting a resonance experiment.

This lab will test your understanding of E&M and 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.

In this section we present a qualitative description of the phenomena of Optical Pumping. If you are interested in a more complete description check out our Theory of Optical Pumping page. Simply stated, Optical Pumping is a process by which an atom is caused to repeatedly absorb and emit photons in such a way that the atom eventually ends up in an energy state which does not allow it to continue emitting and absorbing photons.

In this lab you will optically pump an ensemble of Rb atoms by placing them in a precisely controlled magnetic field and illuminating them with 794.8nm photons which have been right circularly polarized. Absorption of the right circularly polarized photons drives the Rb atoms into a pumped state which cannot then absorb a right circularly polarized photon. A photodetector is used to record the intensity of the light which passes through the Rb vapor. As more of the atoms are driven into the pumped state fewer photons are absorbed by the gas and thus more light reaches the photodetector. Changes in the opacity of the vapor are correlated with the degree of pumping which has occurred.

Understanding the details of how Optical Pumping works in this case is analogous to the quantum description of the Hydrogen atom and requires a conceptual understanding of:

• Quantum numbers associated with atomic energy states such as n, l, s and $m_{s}$.
• Electromagnetic transitions between atomic energy states and their associated selection rules.
• Behavior of magnetic dipoles in an external magnetic field.
• Hyperfine and Zeeman effects.

In the next section we provide a qualitative description of the physical processes involved in optical pumping.

Rubidium is a Hydrogen like atom, in the sense that its ground state it has just a single electron in its outermost shell. Thus the structure of the lower energy states is very similar to that of Hydrogen which also has only one electron in its outermost shell. Figure X below shows the energy level structure for the ground and first two excited states of Rb.

This diagram uses spectroscopic notation to denote the values of the quantum numbers of the various energy states. As a refresher the relevant quantum numbers are:

• N - The principle quantum number denoting which orbital the electron is in.
• L - The orbital angular momentum of the electron.
• S - The spin angular momentum of the electron.
• J - The sum of the orbital and spin angular momenta of the electron J = L + S.
• F - The hyperfine quantum number which is the sum of the electron momentum J and the nuclear angular momentum I. F = J + I.
• $m_{F}$ - The magnetic quantum number which denotes the orientation of F with respect to the quantization axis defined by an external magnetic field B.

As indicated on this diagram transitions between the first excited state $^{2}P_{1/2}$ and the ground state $^{2}S_{1/2}$ absorb (in the case of excitation) or emit (in the case of de-excitation) a photon with a wavelength ($\lambda}) of 794.8nm. Likewise transitions between the ground and second excited states involve photon wavelengths of 780.0nm.

For an ensemble of Rb atoms at 50ºc the number of atoms in the ground and first excited states at any given moment is determined by the Boltzman distribution, however there will be more atoms in the ground state than are in the first excited state. This is the equilibrium state of the ensemble. This equilibrium state is the result of a balance between physical processes which drive atoms from the ground state into an excited state, such as the thermal motion of the atoms in the gas, and processes which cause excited atoms to de-excite, such as the natural lifetime of the excited state and thermal interactions.

We can produce a non-equilibrium distribution of energy states by shining 780.0nm light into the vapor. Some of the 780.0nm photons will be absorbed by Rb atoms in the ground state, thus putting them into the first excited state. These same 780nm photons will also stimulate atoms in the first excited state to de-excite back to the ground state, and thus simply shining 780nm light on the ensemble of atoms alone will not significantly alter the state of the ensemble.

There are however additional contributions to the various energy states of Rb. Referring back to fig X we see that both the ground and first excited states split into additional energy sub-states. Rb nuclei have an intrinsic spin and corresponding magnetic moment which interacts with the electron's magnetic moment resulting in two hyperfine states. If the atom is in an external magnetic field, such as the Earths magnetic field for example, there will be additional splitting of the hyperfine states due to the interaction of the electron magnetic moment with the external field which is known as the Zeeman effect. Thus in the presence of an external magnetic field, electronic transitions between the ground and first excited states are in fact transitions from the different Zeeman levels of the excited and ground states. Although there are a lot of possible combinations of transitions between ground and first excited state zeeman levels, many of these combinations are “forbidden”. In this context saying that a particular transition is “forbidden” simply means that the probability of that transition occurring is 0 or very near 0. You have probably encountered what are called quantum selection rules for electro magnetic transitions which we summarize below.

As an example lets walk through one possible sequence of transitions for the case where the light being absorbed is right circularly polarized.

In fig X above we show a set of 4 possible electronic transitions (a), (b), (c), and (d).

( a ) : For purposes of this example we pick an arbitrary starting point where the atom is in the $^{2}S_{1/2}(f = 1, m_{f}=0)$ ground state. This atom absorbs a right circularly polarized photon. Right circular polarized is another way of saying the photon carries +1 unit of angular momentum, so upon absorption the $m_{f}$ quantum number must increase by +1. For the sake of this example lets assume the f quantum number increases by 1, since $m_{f}$ must increase by 1 this puts the atom in the $^{2}P_{1/2}(f = 2, m_{f}=0)$ state.

( b ) : When the atom de-excites it is not required to lose +1 unit of angular momentum, that requirement only exists for absorption because of the circular polarization of the incident light. So if we take the case where the f quantum number does not change the atom could decay to any one of three possible states $^{2}S_{1/2}(f = 2, m_{f}=0, 1, or 2)$. All three of these states are possible, but lets look at what happens if the atom decays to the $m_{f} = 1$ state.

( c ) : If the atom now absorbs another right circularly polarized photon it would only be able to be excited to the $^{2}P_{1/2}(f = 2, m_{f}=2)$ state because of the requirement that $\Delta m_{f} = +1$.

( d ) : At this point the atom can de-excite to more than one possible ground state, one possibility is that it ends up in the $^{2}S_{1/2}(f = 2, m_{f} = 2)$ state.

( f ) : The atom is now pumped because it cannot absorb another right circularly polarized photon due to the face that it would have to go to a $m_{f} = 3$ state of which there are none. So unless some other process, such as thermal agitation or stimulated emission by an RF frequency photon, drives the atom into some other ground state, the atom is stuck in the pumped state.

The sequence of excitations and de-excitations described above is just one of many possible permutations of allowed transitions. Not all atoms in the ensemble will end up in the pumped state, but some will end up there. The rate at which atoms are driven into the pump state is in part determined by the intensity of the light source. At the same time energy exchanges due to random thermal collisions between atoms will tend to depump atoms out of the pumped state. Equilibrium is reached when the rate of pumping equals the rate of depumping.

For Rb-87, the relation between the applied magnetic field and energy splitting is

$E = -\vec{\mu}\cdot\vec{B} = g_f \left(\frac{e}{2m_e}\right)\vec{F}\cdot\vec{B} = g_f \left(\frac{e\hbar}{2m_e}\right)Bm_F$,

where $\mu_B = e\hbar/2m_e \approx 5.7883 \times 10^{-9}$ eV/G is the Bohr magneton. $g_f$ relates the atom's magnetic dipole moment to its quantum numbers, and is given by

$g_f = g_j\frac{f(f+1) + j(j+1) - i(i+1)}{2f(f+1)}$.

Here $g_j$ (the Landé g-factor) relates the electron's contribution to the total magnetic moment, and is given by

$g_j = 1+\frac{j(j+1) + s(s+1) - \ell(\ell+1)}{2j(j+1)}$.

The energy level shifts are therefore given by

$\Delta E = \mu_Bg_fm_fB \approx (5.7883 \times 10^{-9} eV/G)g_f m_f B$.

Note that this energy difference represents the shift up or down from the un-split (hyperfine) level characterized by quantum number $g_f$.

In this section we describe the apparatus and help you obtain an initial signal. In the image above, from left to right you see the main apparatus, the control console and digital scope, a function generator, DC power supply, a couple DMMs some compasses and a computer.

The main apparatus consists of a Rb vapor cell in a temperature controlled oven situated in the center of a pair of Helmholz coils all sitting atop an optical rail. Mounted to the optical rail from left to right we have:

• An amplified photodiode detector.
• Focusing lens.
• The vapor cell - Helmholz coil assembly.
• Quarter waveplate.
• Linear polarizer.
• Narrow bandpass interference filter.
• Collimating lens.
• Light source.

The light source uses a radio frequency (rf) oscillator to excite Rb atoms thereby producing light at wavelengths needed to pump the Rb atoms in our vapor cell. Light from the source is collimated before passing through the narrow bandpass filter which only passes light from the D1 transition.

The filtered photons then pass through a linear polarizer and a quarter waveplate whose optical axes are oriented such that light which emerges is right circularly polarized. A quarter waveplate will convert linearly polarized light into circular polarization. If you send unpolarized light through a quarter waveplate you will get a mixture of left and right circular polarized light out. By placing a linear polarizer in front of the quarter waveplate, and setting the correct angle between the optical axes of the two, you can produce an output beam ranging from pure right circular polarization, to pure left circular polarization to anything inbetween. To produce a beam of right circularly polarized photons the linear polarizer should be set to an angle of 45° and the quarter waveplate should be set to 0° in their holders. If you do not remember how polarizers and quarter waveplates work, google them to refresh your memory.

The TeachSpin Optical Pumping control console is pictured above. Controls on the console are grouped based on function and from left to right consist of:

• RF Amplifier - Not used and therefor not hilighted in the photo.
• Cell Heater/Controller - Controls and indicates the temperature of the vapor cell. It automatically starts when the power switch is turned on and requires no user input. The temperature display will stabilize at 50°C in about 10 minutes.
• Magnetic Field Modulation - Not used and therefor not hilighted in the photo.
• Horizontal Magnetic Field Sweep - These controls are used to setup a time varying current to the horizontal Helmholz coils. Note that the BNC output labeled Recorder Output on the lower panel is connected to channel 1 of the scope and is used to monitor the current going to the coils.
• Vertical Magnetic Field - This 10-turn potentiometer is used to set the current going to the vertical Helmholz coils.
• Horizontal Magnetic Field - This 10-turn potentiometer sets a constant current to the horizontal Helmholz coil. This current is in addition to any current being provided by the Horizontal Magnetic Field Sweep controls.
• Detector Amplifier - The output from the photodetector on the optical rail comes here. The user can amplify and add an offset voltage to the detector signal, and filter out high frequency noise. The final signal is then sent to channel 2 on the scope.

Becoming comfortable with sitting down to a new experiment with new apparatus and understanding how things work and making sense of your signals is one of the main goals of this course. As a third year physics major you know enough physics to start making sense of what is going on with your experimental apparatus.

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.

Conceptually you are using the controls on the console to do two things.

1. Create magnetic fields along the horizontal axis of the optical rail and along the vertical axis by sending current to the Helmholz coils.
2. Amplify and filter the signal from the photodetector so that you can view it conveniently on the scope.

That's pretty much it. Nothing complicated, you just need to familiarize yourself with the controls. So lets walk you through the process of figuring out how to use the controls.

What we want to do with the apparatus is equally simple in concept. We want to use magnetic fields generated by the the helmholz coils to cancel out the ambient magnetic field present in the lab where the Rb vapor cell is located. Then we want to use those same coils to create a very specific known magnetic field and watch how the intensity of the light passing through the vapor cell changes as we change that know field.

There are four 10-turn potentiometers which control the currents to the horizontal and vertical coils. Turn all of them fully CCW to zero. In the Horizontal Magnetic Field Sweep control section there are two toggle switches, flip both down so that they are in the Reset and Continuous settings.

There should now be no external magnetic fields being generated by the coils and the Rb atoms are now sitting in the ambient magnetic field of the basement 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. There should be a compass on the table, take it and move it around the apparatus, look for any unexpected deflections of the needle which would indicate the presence of something magnetic. Don't forget to take your cell phone out of your pocket.

We can only control the field along two axes, vertical and horizontal along the axis of the optical rail, using the Helmholz coils. To completely zero the magnetic field in the vapor cell you want 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 the compass, no need to be super precise as we will fine tune this alignment in a moment.

Now that 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. Holding 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 vine tune this cancellation in a later step.

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 (RF = Radio Frequency) 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 RF oscillating current to these coils we produce an RF oscillating magnetic field which is orthogonal to the sweep field generated in the Horizontal coils. When the energy of the RF photons $E=h f$, where $h$ is planck's constant and $f$ is frequency, 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 setup the apparatus so that it is sweeping through the zero field condition, and 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 40kHz sine wave with an amplitude of ~0.5Vpp. 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. Now 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 85Rb and 87Rb, 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 $B_{net}$'s.

Your analysis will be evaluated based on the following rubric. Each item below is graded on a 0-4 point scale:

• 4 – Good (A): completes all listed tasks and provides appropriate context; thinks carefully about data and analysis; addresses all concerns raised by the results (where appropriate).
• 3 – Adequate (B): misses one or more minor element or lacks appropriate context; leaves a problem or ambiguity unaddressed; does not present analysis clearly enough.
• 2 – Needs improvement (C): omits or mishandles one or more item which renders the analysis fundamentally incorrect or incomplete; presents results in an incorrect or unclear way.
• 1 – Inadequate (D): omits or mishandles multiple items or treats them at an insufficient level; presentation is overall muddled or inaccurate; flaws in logic or process.
• 0 – Missing (F): omits all elements or makes no meaningful attempt.

All rubric items carry the same weight. The final grade for the analysis will be assigned based on the average (on a 4.0 scale) over all rubric items.

W. B. Hawkins, "Orientation and Alignment of Sodium by Means of Polarized Resonance Radiation," Phys. Rev. 98, 478 (1955).

W. Franzen, "Spin Relaxation of Optically Aligned Rubidium Vapor," Phys. Rev. 115, 850 (1959).

A. L. Bloom, "Optical Pumping," Sci. Am. 203, 72 (1960).

R. L. de Zafra, "Optical Pumping," Am. J. Phys. 28, 646 (1960).

R. Benumof, "Optical Pumping Theory and Experiments," Am. J. Phys. 33, 151 (1965).

M. Nagel and F. E. Haworth, "Advanced Laboratory Experiments on Optical Pumping of Rubidium Atoms--Part I: Magnetic Resonance," Am. J. Phys. 34, 553 (1966).

S. G. Kukolich, "Time Dependence of Quantum State Amplitudes: Demonstrated by Free Precession of Spins," Am. J. Phys. 36, 420 (1968).

All three sets of coils are wound with AWG #20 (nominal diameter 0.032“) copper wire. Each pair of coils is connected in series.

These coils contain N = 20 turns on each of two coils. The wire has been wound 5 across and 4 layers deep. According to the manufacturer the coil radius is about 4.61” ± 0.01“. The spacing of the coils should match their radius.

These coils contain N = 154 turns on each of two coils. The wire has been wound 11 across and 14 layers deep. According to the manufacturer the coil radius is about 6.22” ± 0.01“. The spacing of the coils should match their radius.

These coils contain N = 11 turns on each of two coils. The wire has been wound 11 across in a single layer. According to the manufacturer the coil radius is about 6.46” ± 0.01“. The spacing of the coils is the same as for the Horizontal Field Coils.