Read through Section 2: Theory below and answer the following questions:

(a) If an object having a magnetic moment (but no angular momentum) is free to move, how will it move in the presence of the magnetic field?
(b) If the object is given angular momentum – parallel or anti-parallel to its magnetic moment – and is placed it in a magnetic field, how will it move?
(c) Assuming thermal equilibrium, which of the states illustrated in Fig. 4, has the most atoms?
(d) Use Eqs. (17) and (18) to calculate the ratio of the numbers of atoms in ${}^2S_{1/2}$ states given by $f = 2, m_f = +2$ and $f= 2, m_f = +1$. For your calculation assume a field of 1 Gauss, typical for this experiment.

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).

In this experiment you will study the phenomenon of optical pumping with a vapor of rubidium atoms. Specifically, your objectives for this experiment include the following:

• to observe optical pumping of a vapor illuminated by photons of appropriate energy and circular polarization;
• to use a “flipping” external magnetic field to indirectly observe the alignment of magnetic dipole moments with the field;
• to use changes in the optical pumping signal to determine the direction and magnitude of the Earth's magnetic field;
• to de-excite pumped atoms using radio-frequency magnetic fields;
• to measure the energy level splitting due to the Zeeman effect as a function of magnetic field;  
• to observe Larmor precession of the magnetic dipole moments and to measure the Larmor precession frequency as a function of field;
• to measure the gyromagnetic ratio of Rb-87; and
• to use and understand the magnetic fields produced by Helmholtz coils.

Consider a loop of positive current I whose path encloses area A, as in Fig. 1.

Figure 1: Magnetic dipole moment of a current loop.

The area enclosed by the loop may be considered a vector where the magnitude is the area and the direction is given by the right-hand rule. (With the fingers pointing in the direction of positive current flow, the thumb points in the direction of the resulting area vector, which is the same direction as the magnetic dipole moment vector.) The magnetic dipole moment is therefore given by 

If we place a magnetic dipole moment in an external magnetic field, the dipole will experience a torque given by 

where $\vec{B}$  is the magnetic field. This torque tends to twist the dipole to align parallel to the field; this is the lowest energy state. Therefore, it takes energy

to twist the dipole away from the orientation parallel to the field.

NOTEBOOK:

• If an object having a magnetic moment (but no angular momentum) is free to move, how will it move in the presence of the magnetic field?
• If the object is given angular momentum – parallel or anti-parallel to its magnetic moment – and is placed it in a magnetic field, how will it move?

To help us answer the second question above, carefully consider Fig. 2.

Figure 2 : Torque acting on an object with angular momentum and magnetic moment in a magnetic field (after Eisberg and Resnick, Quantum Physics).

The magnetic field acts on the magnetic dipole moment to produce a torque, given by Eq. (2). This torque also acts on the angular momentum (since the angular momentum and magnetic dipole moment are here assumed to remain co-linear). Thus, the torque on the angular momentum gives rise to a change in the angular momentum $\vec{dL}$  during the tim e$dt$  such that

The change$dL$  causes $\vec{L}$  to precess through an angle ωdtωdt $\omega dt$ , where $\omega$  is the precession angular velocity. Note from Fig. 2 that

such that

where

where $\gamma$  is called the gyromagnetic ratio.  

If available (ask an instructor or TA), take a few minutes to use the TeachSpin Magnetic Torque apparatus to explore these ideas. This apparatus consists of a cue ball with a small magnet embedded at its center. 

1. Turn on the air supply and the current to the Helmholtz coils. Without spinning the ball, observe how it is affected by the magnetic field. How is the ball’s magnetic dipole moment aligned relative to the ball’s black handle? - Turn down the magnetic field and twist the black handle to spin the ball. (I.e., give the ball some angular momentum.) What is the direction of the angular momentum relative to its magnetic dipole moment?
2. While the ball is spinning, turn on the magnetic field. How does the ball move now?
3. Reverse the direction of the ball’s angular momentum. How does the ball’s motion change in the $B$  field
4. Try to explain what you observe by referring to Fig. 2.

Rubidium atoms have co-linear (in this case, oppositely directed)  $\vec{\mu}$ and $\vec{L}$  related to each other by

where $\mu_B = e\hbar/2m_e \approx 5.7883 \times 10^{-9} ~\textrm{eV/G}$  is the Bohr magneton (the quantum of magnetic dipole moment) $\hbar$  is Planck’s constant divided by 2π, and $g$  is adimensionless g-factor. Thus, rubidium atoms are expected to behave just as our hypothetical classical object above when subjected to a magnetic field.  The exact form of a g-factor for a quantum mechanical system depends on what forms of angular momentum are relevant to the system, but they can be calculated analytically.

NOTE: While this model is useful, it has some limitations. The presence of a well-defined vector precessing around $\vec{B}$ implies knowledge of the x- and y- components of the vector, but quantum mechanics tells us that these components are really indeterminate.

The Larmor precession frequency is given – in analogy with the classical case – by Eq.(8), where the gyromagnetic ratio is the quantum-mechanical equivalent of ${\mu}/L$  (from Eq.(9)), namely

Note that the g factor is purely quantum mechanical with no classical analog.  The g factor relates the magnetic dipole moment to the relevant angular momentum quantum numbers, and is system dependent.  For instance, for an electron in an atom with no nuclear spin angular momentum, the Landé g-factor relates an electron's spin and angular momentum to its magnetic moment.  

The precession cone angle (the angle between $\vec{\mu}$  and $\vec{B}$  ) of a classical magnetic moment can have any value. However, in the quantum mechanical case of rubidium atoms, the precession cone angle (in this simplified model) can take on only certain values. These discrete states are derived from proper vector addition of the angular momentum contributions of the electron spin angular momentum $\vec{S}$  , the electron orbital angular momentum $\vec{L}$ , and the nuclear spin angular momentum $\vec{I}$ . Associated with these vectors are quantum numbers ${s}$ , $\ell$ , and $i$ . Be careful! “Spin angular momentum” or “spin magnetic moment” are often replaced by the shorter “spin” which is ambiguous.

The ground state of Rb-87 in particular is denoted by quantum numbers in the form 2S1/2. To extract the relevant quantum numbers from this spectroscopic notation, we use the following guide:

• The superscript “2” means that 2_s _+ 1 = 2; therefore, _s _= 1/2. This is true for rubidium, since there is one un-paired electron in the outer orbital.
• The letter “S” means that $\ell$  = 0. * The subscript “1/2” denotes the value of j = $\ell$  + s.  Since $\ell$  = 0 in the ground state, j = s = 1/2.

In addition to these electronic contributions, the nucleus of Rb-87 has nuclear spin angular momentum i = 3/2.

For the general case of an atom, the total angular momentum is given by the sum of the electron orbital-, the electron spin-, and the nuclear spin angular momentum vectors, denoted by capital letters,

all in units of $\hbar$ . For the special case of rubidium in the ground state where _L _= 0,

In the absence of an externally applied magnetic field, the angular momentum of an isolated electron has no preferred direction. The same may be said of an isolated rubidium nucleus. However, when the electron and nucleus are bound together in an atom, each magnetic dipole moment “sees” the magnetic field of the other and the two precess around their vector sum as in Fig. 3.

Figure 3: The $\vec{I}$ and $\vec{S}$ vectors precess about $\vec{F}$.

Using the rules for adding the quantum numbers for such a combination, we get the following two possibilities for the quantum number f:

The f = 1 state corresponds to $\vec{I}$  and $\vec{S}$  anti-parallel; this is the lower energy state for magnetic dipoles. The _f _= 2 state corresponds to $\vec{I}$  and $\vec{S}$  parallel; this is the higher energy state for magnetic dipoles. Note that the _f _= 1 and _f _= 2 states have different energies due to relative orientation of $\vec{I}$  and $\vec{S}$ , independent of any external magnetic field. This energy difference is called hyperfine splitting.

Subjected to an external magnetic field – which is weak compared to the internal fields produced by the nuclear and electron spin magnetic moments – the two possible vectors _F _= 1 and _F _= 2 will precess around the applied B field with specific quantized orientations relative to the B field (the quantization axis). These precession cones are shown in Fig. 4 below.

Figure 4: Shown here are cones of precessing $\vec{\mu}$  and $\vec{L}$  for spin 3/2 (nucleus) and spin 1/2 (electron) giving $m_f$  = -2, -1, 0, 1, and 2 for f = 2, and $m_f$  = -1, 0, +1 for f = 1.

Fig. 4 shows that for f= 2 there are 5 distinct projections of $\vec{\mu}$  onto $\vec{B}$ . From Eq.(3), it follows that there are 5 distinct energies associated with these quantized states. For the _F _= 1 state, there are three distinct energies.  The quantum number $m_f$  corresponds to the possible projections of a total angular momentum F onto the applied B field, and may take integer values between -f and +f. A similar analysis may be done for the excited Rb-87 state. This addition is more complex, since for this case the electron orbital angular momentum is $\ell$ = 1 and three vectors must be properly added in all possible orientations.

The appearance of these distinct energy states in the presence of a magnetic field is the Zeeman effect. The splitting of the hyperfine energy levels into finer Zeeman levels is illustrated in Fig. 5.

Figure 5: Energy level diagram for Rb-87, showing hyperfine splitting and Zeeman splitting.

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

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

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

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

Though these many energy states are all possible, they are not equally likely to be populated. The atoms will naturally want to tend toward the lower energy states, but each atom possess some thermal energy which leads to excited states.

Let us consider an ensemble of a large number of Rb atoms. In thermal equilibrium, the ratio of numbers $N_1/N_2$ of atoms in any two energy states is given by the Boltzmann distribution

where $E_1$ and $E_2$ are the energies of the states containing $N_1$ and $N_2$ atoms, respectively,  $T$ is the absolute temperature, and $k_B$ is the Boltzmann constant.

NOTEBOOK:

* Assuming thermal equilibrium, which of the states illustrated in Fig. 4, has the most atoms?
* Use Eqs. (17) and (18) to calculate the ratio of the numbers of atoms in ${}^2S_{1/2}$ states given by $f= 2, m_f = +2$ and $f = 2, m_f = +1$. For your calculation assume a field of 1 Gauss, typical for this experiment.

For large numbers of atoms, we may consider the cones of precessing magnetic moments to be nearly uniformly filled with the magnetic dipole moment vectors. The vector sum of the atomic magnetic moments is well represented by a single vector $\vec{M}$ , the net magnetic dipole moment, lying along the direction of the $B$  field as in Fig. 6 below.

Figure 6: The cone of precessing magnetic moments is nearly uniform and the vector sum is the net magnetic dipole moment  $\vec{M}$ .

If the net magnetic dipole moment is non-zero, the ensemble is said to be polarized.

The optical portion of the apparatus is shown in Fig. 7 below.

Figure 7: Optical pumping apparatus

A radio frequency oscillator excites a small bulb containing rubidium vapor. The unit is enclosed in a box that maintains an elevated temperature.

Several optical elements are mounted on separate stands and inserted between the light source and cell.

• A lens is used to gather the light and direct it toward the vapor cell and detector.
• An interference filter is used to allow only the desired D1 line of the Rb ($\lambda$  = 794.8 nm) to pass. * A linear polarizer and a quarter-wave retardation plate are attached and oriented so as to form a “circular polarizer”.

Of the two vapor cells available, we will use only the one containing Rb-87. The cell is a transparent glass vessel that contains some Rb-87 metal and neon, a buffer gas. A heating element and fan, located below the table, raises the temperature of the vapor cell to the level necessary (40-50°C) to maintain the desired vapor pressure of Rb in the cell. Check that the vapor cell is in its clear plastic box centered in the Helmholtz coils, so that the light enters and leaves the flat, circular windows.

The Vertical (or “V”) coils produce field in the up-down direction and are principally used to cancel the vertical component of the Earth's magnetic field.

The Horizontal (or “H”) coils (the inner layers on the aluminum forms) produce a field in the north-south direction, the same direction as the light beam (defined to be the z-axis). These coils are used to apply varying horizontal fields during the B-flip experiments. They are also used to cancel the Earth's horizontal field while observing Larmor precession.

The “Y” coils are smaller than either the V or H coils, and generate a field in the east-west direction, perpendicular to the light path. They are used to produce the field around which Larmor precession takes place.

The “Z” coils are wrapped in a single layer on the outer surface of the H coils, and are used in switching the field direction during the Larmor precession experiments.

The radio frequency (RF) coils are wrapped directly on the vapor cell support, and carry the radio-frequency current that produces the oscillating magnetic field causing stimulated transitions between Zeeman energy levels. The RF coils are not in the Helmholtz configuration.

The RF oscillator provides the variable frequency signal which, when passed through the small coils immediately next to the rubidium cell, produces an oscillating B field to excite transitions between the Zeeman levels.

A photo transistor is used as a light detector. Its current generates a potential difference across a resistance that must be small to preserve high frequency response. Hence we use an operational amplifier as a current follower. (See Fig. 8.) This device presents zero impedance to the source current. The potential variations follow those of the current with a time lag of much less than a millisecond. Room light is suppressed by the small acceptance angle at the detector’s input.

Optical pumping changes the transparency of the vapor cell only slightly. Thus, periodic pumping induces only a small AC signal superimposed on a large DC value. To observe the action of pumping and de-pumping, it is necessary to remove the DC and amplify the AC component. These functions are accomplished by the DC offset and amplifier.

A digital storage oscilloscope is provided for measurements of pumping signals. The scope’s images may be sent to the computer using the OpenChoice Desktop program. Once the desired trace appears on the scope, press the run/stop switch and launch the Data Transfer software. The data points are saved as tab-delimited text files.

A panel is provided which controls the currents supplied to the coil array. Its schematic diagram is given in Fig. 9. This box controls several voltage supplies and generators.

Figure 9: Field switching panel

This generator is used for the following two purposes:

•  For B-Flip, it provides the horizontal coils with a square-wave current to change the magnetic field in the north-south direction.
•  For Larmor precession it is used to operate a relay that switches the current between the Z coils and the Y coils.

Located in the left hand portion of the blue power supply, this generator is used for RF de-pumping and provides a DC voltage plus an optional additional AC voltage swept symmetrically around the DC voltage. The amplitude and frequency of the sweep are adjustable. While taking RF data, the DC field required for resonance is adjusted to be close to the desired value and then the field is swept through resonance periodically.

The right hand portion of the blue power supply provides DC for the vertical coils, enabling cancellation of the Earth's vertical field.

We wish to study the interaction of light with Rb-87 atoms in a magnetic field. Specifically, we will use right or left circularly polarized light of just the energy necessary to induce the D1 transition in Rb-87 shown in Fig. 4. D1 transitions are accompanied by emission or absorption of a photon with wavelength, $\lambda$  = 794.8 nm in the near infrared.

It should be noted that right circularly polarized photons carry one quantum of angular momentum, $\hbar$ , pointed in the direction of travel of the photons. Let us track the states of a RB-87 atom which starts in the 2S1/2, f = 2 state (see Fig. 5), absorbing a circularly polarized, $\lambda$  = 794.8 nm photon described above, moving to an excited state, re-radiating a photon and finally returning to some f = 2 state, as follows.

As a starting point let us arbitrarily choose an atom in the 2S1/2, f = 2 state, and having  $m_f$  = 0.

NOTEBOOK:  As a result of absorbing one of the prepared photons, to which state will the atom go?

The atom in the excited state is unstable and decays to a 2S1/2, f = 2 state through electric dipole radiation, for which the selection rules are  $\Delta m_f$  = -1, 0, or +1.

NOTEBOOK:

* To which states may this atom decay? Find this state both in the angular momentum vector model (Fig. 3) and in the energy level diagram (Fig. 4).
* Following the same atom through successive excitations and decays, to which state will the atom tend? Find this state in both the vector model and energy level diagram.
* For each excitation, how is the angular momentum of the atom affected by the angular momentum of the absorbed photon?
* How does the orientation of the atom change by these absorptions and re-emissions?

As the ensemble of atoms absorbs and re-emits many of these specially prepared photons, the population is driven to a higher average energy, which is a non-Boltzmann distribution. This situation is analogous to pumping water out of the ground to a higher potential energy, and thus is referred to as optical pumping.

In the pumped state the atoms are unable to absorb subsequent photons and the vapor cell becomes more transparent. Thus, an increase of the transmitted light intensity is an indicator of optical pumping.

Before we make measurements with the apparatus, let us begin by running through the different settings to understand how the Helmholtz coils produce a field and how the photodetector measures the vapor cell transparency.

In order to observe a steady signal, the rubidium lamp must warm up and stabilize before taking data. The bulb’s color is determined by its temperature, which is in turn controlled by the voltage from a power supply. Pale blue-white means the bulb is too cool and will produce a small signal. Deep red means it is too hot and will produce a noisy signal. The correct color is purple (a mixture of red and blue).

The lamp power supply is connected to a voltage regulator which prevents voltage (and therefore temperature and intensity) drift. Similarly, the rubidium atoms in the vapor cell should be warmed above room temperature to form a sufficient vapor.

A delicate, small gimbaled magnet (MagnaProbe) is provided to check the magnetic field directions. Use the MagnaProbe to observe the direction of the Earth's field in the room. As you bring the probe near to the center of the Helmholtz coils, you should see no change in the field direction if no current is running through the coils.

NOTEBOOK: What is the direction of the Earth's field in the Optical Pumping room? Does it point purely north? Does it have any east-west component or angular inclination?

Turn on the control panel and set the two switches to “B-flip”. Increase the current to the vertical coils by rotating the knob on the lower right of the panel. You should see the current meter reading increase.

You can obtain a more accurate reading of the current by using the built-in voltmeter and selecting “V” on the input switch. A 1 Ω resistor (with a 1% tolerance) has been placed in series with each set of coils and therefore the voltage across these resistors gives a measure of the current through each coil. (Recall from Ohm's law that 1 A flowing through a 1 Ω resistor produces a voltage drop of 1 V: V = IR. Therefore, for example, a reading of 0.50 V on the meter corresponds to a current of 0.50 A through the coil.)

As you increase the current through the vertical coils, use the MagnaProbe to observe the change in the net field near the center of the coils.

NOTEBOOK:

How does the net magnetic field direction change as you increase the current to the V coils? What is the current range that you can supply?

You should observe that the net field gradually becomes more and more vertical (pointing up) as the current increases. (If the net field becomes more downward pointing, you will need to reverse the electrical connector (banana plug) to reverse the direction of the current.)

Return the current in the V coil to zero.

Turn on the external function generator (located above the blue panel) and make sure that the square wave option is selected. Set the frequency to about 0.1 Hz and rotate the amplitude knob clockwise to the maximum value.

In “B-flip” mode, the current flowing through the horizontal coils is not controlled by the internal power supply, but instead is provided by this external function generator as a square wave, symmetric about zero volts. For this reason, the horizontal current dial (located in the lower left of the panel, next to the vertical current dial) will not deflect. (This dial will be used later in the experiment.) To read the current to the coils, one can again use the built-in voltmeter, this time selecting H on the input switch. Note, however, that when the generator is flipping its field quickly, the voltmeter cannot respond and so values can be read only when the frequency of flipping is very low (e.g. less than one Hz.)

Again, place the MagnaProbe near the center of the coils and observe the effect on the field direction.

NOTEBOOK:

Describe the behavior of the net field direction when the amplitude is maximum. Is it constant in time?
Using the built-in voltmeter, what is the maximum current supplied to the horizontal coil? Is it indeed symmetric about zero? Decrease the current amplitude and comment on how the net field direction changes.
Return the amplitude to maximum and increase the generator frequency. Does the field orientation change as expected?

Set the function generator to zero amplitude and the frequency to about 5 Hz.

Use a meter stick to check that the heights of all optical components (filter, circular polarizer, detector) are equal to the fixed height of the Rb vapor cell.

Remove the interference filter and the detector from the optical rail. With your eye at the position of the detector, check for bright and uniform illumination of the vapor cell. Replace the detector and filter.

At the moment (with no applied magnetic field) there is little optical pumping occurring. Therefore, the detector is sensing a nearly constant intensity through the vapor. To find and optimize the detector signal, complete the following steps:

• Connect the output of the detector to the input of the current follower, and connect the output of the current follower to channel 1 of the scope.
• Connect the cable from the back of the function generator (the “SQ” output) to channel 2 of the scope. (This output is a square wave of constant amplitude which is in sync with the output voltage sent to the horizontal coils.)
• In the trigger menu, select the following:
  • Type: Edge
  • Source: CH2
  • Slope: Falling
  • Mode: Normal
  • Coupling: HF Reject
• Adjust the trigger level until you find a stable trace of the square wave on channel 2. Adjust the time axis so that one or two full periods are visible.
• Under the channel 1 menu, set the coupling to “ground” and move the trace so that the resulting horizontal line is centered on the screen.
• Return the coupling to “DC” and now adjust the “offset” knob on the current follower until the line is again centered on the screen. If you do not immediately see the channel 1 signal, you may need to increase the voltage scale on channel 1.

NOTEBOOK:

Which way does the trace move when your hand blocks the light? On the scope, which direction indicates greater transmitted light intensity?
Adjust the detector height and angle to maximize the light level. You may now need to decrease the voltage scale on channel 1 (i.e. zoom in) to see the small effects that each adjustment makes. Continue to keep the signal centered by adjusting the DC offset knob.

Once the signal is optimized, you may still notice a small amount of drift up or down in the baseline intensity. If the signal begins to drift off screen, use only the DC offset knob on the current follower to return the signal back to the center. Make small changes in the DC offset in order not to lose the trace.

We are now ready to begin investigating optical pumping. Set up the magnetic field as follows:

• Return the function generator to maximum amplitude. Set the frequency to a few hertz (say, 2-3 Hz).
• Turn the current in the vertical coil down to zero. (Remember, however, that there is still a small component downward due to the Earth’s vertical field.

Now, the total field is swept from north, through vertical to south, never going through zero, as shown in Fig.10.

You should now observe the pumping signal on channel 1. You should see a series of exponentials whose beginnings coincide with the start of each half-cycle of the square wave.

NOTEBOOK and ANALYSIS:

Which part of the signal corresponds to increased transmission, and which corresponds with decreased transmission?
Which part of the signal corresponds to pumping and why? What is happening on the other part of the signal? (Is this _de-pumping _(i.e. removing electrons from the pumped state) or something else?)
Use the scope to measure the characteristic (i.e. 1/e) times for both. (A rough estimation, 1/e ~ 1/3, is good enough here.)
Adjust the square wave frequency. Does this affect the characteristic times? What noticeable effect(s) does this have?

Remove the polarizer/quarter wave plate combination.

NOTEBOOK and ANALYSIS:

What happens to the pumping signal? Why? (Hint: How is the angular momentum of the photons affected by the polarizer/quarter wave plate combination?)

Replace the polarizer/quarter wave plate combination.

Verify that the oscilloscope is attached to the computer via a serial cable and open the “Scope Capture” software on the desktop. The program should blink and initialize.

On the scope, find the best settings for displaying both all the features on channels 1 and 2 and select “Run/Stop” to freeze the image. Through the software, capture a scope trace of both channel 1 and channel 2. Both channels should appear on screen. 

NOTEBOOK and ANALYSIS:

Save the scope trace as a tab-delimited file and note the filename in your notebook. You will want to reproduce this image in your report.

With the current in the vertical coils still set to zero, slowly reduce the current in the horizontal coils, while observing the effect on the heights of the pumping signal for both halves of the cycle.

NOTEBOOK and ANALYSIS:

How do the amplitudes of each half-cycle change as you reduce the applied horizontal B field?

It is interesting to try to explain your observations in light of quantum mechanics. Figure 11 shows the fields for both half-cycles.

Figure 11: (left) The total B field as the applied field is swept from north to south with the precession cones of the magnetic dipole moments superimposed. (right) The Earth’s field separately.

As we have noted earlier, the atomic magnetic dipole moments (and therefore, angular momenta) may be viewed as precessing about the total magnetic field. As the field sweeps from north to south, the precessing atoms follow. At the northern and southern extremes, $\vec{M}$  doesn't point purely along the horizontal direction, but makes an angle with the light path. Recall that the prepared photons carry one unit of angular momentum, directed along the light path. If the total magnetic field and the direction of the light path are not parallel, then the polarization of the photons – as measured in the frame of reference of the magnetic moments – is not considered to be a pure state, but instead is a superposition of both left and right circular polarizations. The probability that a photon appears with a given polarization is related to how aligned (or not) the magnetic field and light path are.

Suppose, for example, that the light is prepared so that photons are right circularly-polarized. Consider the following three scenarios:

• If the total magnetic field is parallel to the light path, the atoms will view the incident photons as being right circularly-polarized with 100% probability, and left circularly-polarized with 0% probability.
• If the total magnetic field is anti-parallel to the light path, the atoms will view the incident photons as being right circularly-polarized with 0% probability, and left circularly-polarized with 100% probability.
• If the total magnetic field is perpendicular to the light path, the atoms will view the incident photons as being right circularly-polarized with 50% probability, and left circularly-polarized with 50% probability.

NOTEBOOK and ANALYSIS:

Considering the above discussion about photon polarization, which part of the signal corresponds to the

Continue varying the horizontal field until you find evidence for no pumping during one of the half-cycles.

NOTEBOOK and ANALYSIS:

What is your evidence that no pumping is taking place during that half-cycle?

Is light absorption taking place during this same half-cycle? Explain your answer.Record the current in the horizontal coils at this special setting. The uncertainty in this field measurement may be estimated by varying the current over the range where there is no notable signal change.

What is the relationship between the field produced by the horizontal coils at this setting and the Earth’s horizontal field?

Again, find the best scope settings for displaying the features on channels 1 and 2 and select “Run/Stop” to freeze the image. Again, capture a scope trace and save.

NOTEBOOK and ANALYSIS:

Save the scope trace as a tab-delimited file and note the filename in your notebook. You will want to reproduce this image in your report.

Return the square wave to its maximum value. Observe the pumping signal while slowly increasing the applied vertical field through its entire range. Note that at some point the Earth’s vertical field will be canceled by the opposing, applied vertical field. In this case the field’s reversal from north to south would be as shown in Fig. 12 below.

Figure 12: Flipping the horizontal B field in the absence of a vertical field. The total field passes through B = 0.

NOTEBOOK and ANALYSIS:

Do you see any evidence for a special value of applied vertical field? Explain what you observe. (Hint: In Case III the total B field passes through zero as it switches from north to south. How would the atomic magnetic moments react to the changing B field in Case III?)

Use these observations to measure the current which creates a field which cancels the Earth’s vertical B field.

In the above B-flip cases, you determined the currents to the Helmholtz coils which create fields equal in magnitude, but oppositely directed, to the Earth's magnetic field. We now wish to convert these current values to true magnetic field values.

Helmholtz coils, by definition, have their spacing equal to their radius. A consequence of their geometry is that the magnetic field near the center of the coil assembly is very uniform and is calculable from their geometry and current. For Helmholtz coils, (in which a = 2_b_ and c « a), the on-axis field can be calculated, to good precision, from the formula

where  $\mu_0$  is the permeability of free space, $N$ is the number of turns in each coil,  $I$ is the coil current, and  $a$  is the coil radius as shown in Fig. 13.

In order to convert from current to field, we must measure collect some information about the Helmholtz coils.

NOTEBOOK:

Measure the dimensions a, b and c of the H coils and estimate uncertainties. Count the number of turns of wire N.

Measure the dimensions_a_, b and c of the V coils (which may be different from the H coil) and estimate uncertainties. Count the number of turns of wire N.

Also, while we're at it, let's also measure the dimensions of the Z coils (wrapped on the outside of the H coils) and the Y coils (the smaller coils perpendicular to both H and V) since we'll need those later.

NOTEBOOK

Measure the dimensions a, b and c of the Z coils and estimate uncertainties. Count the number of turns of wire N.
Measure the dimensions a, b and c of the Y coils and estimate uncertainties. Count the number of turns of wire N.

We now wish to compare our measured values for the Earth's field against the measurements of others. Note, however, that the measurement of our local field is actually a superposition of the Earth's field and any stray fields due to electronics (like computers, cell phones, lights and even the wiring in the wall), metallic objects (like pipes, chairs or metal doors) or other magnets (like the MagnaProbe). While some precautions have been taken (Did you notice that you have only wooden chairs and that the light switch was moved outside the room?) some sources remain. In addition, the value of the Earth's field varies depending on one's location and can even change over time.

NOTEBOOK and ANALYSIS

Use Eq.(19) to compute the magnitudes of the horizontal and vertical components of the Earth's magnetic field. From the horizontal and vertical components, determine the total magnitude of the total Earth's magnetic field.

Look up the values of the horizontal (N/S) and vertical (down/up) components of the Earth's magnetic field and compare to the values you calculate. (The National Oceanic and Atmospheric Administration (NOAA) has a fairly easy to navigate site where this information is available.)

In the B-flip experiment, we saw that when the rubidium atoms of the vapor cell were subject to a strong external magnetic field, the vapor developed a net magnetization as the individual magnetic dipole moments aligned with (and precessed about) this field. We also inferred that these individual magnetic moments could exist in different orientations (more or less aligned with the field) that were related to different energy and angular momentum states. When the vapor was subject to an incident beam of circularly polarized photons (with uniform energy and angular momentum), we saw that this vapor could be “optically pumped”, that is to say, that we could invert the population of occupied energy states from the natural one given by the Boltzmann distribution to one where most atoms were in an excited state (and a state where there were no allowed transitions for the incident photons to excite).

In this part of the experiment, we will place our rubidium vapor into a pumped state as before, but we will now look for special circumstances of de-pumping.

De-pumping may be achieved by any mechanism that removes atoms from their highest angular momentum state $m_f$ = +2, to which the light has pumped them. Recall that the angular momentum states are separated in energy by the presence of the applied magnetic field. This Zeeman splitting is quantified in Eq. (17). If we supply oscillating electromagnetic radiation of the correct frequency and polarization, we can cause stimulated emission from the $m_f$  =+2 state down to the $m_f$  = +1 Zeeman energy level. These de-pumped atoms are then again available for pumping.

The population of electrons in each possible energy states is given by the Boltzmann distribution for a given temperature. If there is a population inversion (that is, if more electrons populate the higher, less energetically-favored states than predicted by Boltzmann), some electrons are expected to transition down to an unoccupied lower energy state, emitting a photon in the process.

Normally, this timing of this transition is random and the photon will emerge with random direction and phase; this is spontaneous emission.

However, if the atoms in the excited states are subject to incident photons with energy and angular momentum equal to the difference between the states, the incident photon will induce the transition and the photon which is emitted will have the same energy, angular momentum, phase and direction as the incident photon; this is stimulated emission. (Stimulated emission is key, for example, in a laser. A system is placed into a population inversion and photons are bounced back and fourth inside a lasing cavity causing stimulated emission and producing an amplified, monochromatic and coherent beam.)

Stimulated emission can also be achieved without direct photons. If pumped atoms are placed inside an electromagnetic field which is oscillating at a frequency corresponding to the energy difference between states,

and in a direction perpendicular to the magnetic field, then the electromagnetic field itself (which carries energy and angular momentum) can stimulate the transition.

NOTEBOOK:

For magnetic fields of about 0.5 Gauss, use Eqs. (17) and (20) to predict the radio frequency required to induce transitions from the _m_f = +2 to the _m_f = +1 state.

We shall apply a radio-frequency sine wave to the small coils surrounding the Rb vapor cell. Connect the output of the RF generator to the small coils nearest the vapor cell. Be sure that the 100 Ω resistor is mounted in series with the RF coils to limit the current drawn from the RF generator.

In practice it is difficult to match the radio frequency energy precisely to the energy of the Zeeman splitting. We shall therefore use the technique of sweeping (modulating) the magnetic field in order to vary the Zeeman splitting through a range of values. The blue power supply gives a DC current with an optional sweep (a range of currents around the DC value). If we apply the DC plus the sweep current to the H coils, the total field which the atoms see (and therefore the total magnitude of Zeeman splitting) will modulate a small amount around the DC value. If the radio frequency value corresponds to an energy within this range of Zeeman splitting values, stimulated emission will occur as shown in Fig. 14.

Figure 14: The resonance condition with swept B field.

Set the two toggle switches on the control panel to the “RF” position and turn off the function generator. In this mode, the internal power supply of the control panel provides a “DC + sweep” to the H coils and a constant current to the V coils.

1. The current level to the V coils is again controlled by the vertical current knob in the lower right corner of the panel. - The DC current level to the H coils is controlled by the horizontal current knob in the lower left corner of the panel. - A symmetric sawtooth (“sweep”) component is added when the “SWEEP ON” toggle switch is in the up position. - The amplitude of the sweep is controlled by the knob to the right of the DC current knob. - The period of the sweep is controlled by the knob to the left of the vertical ammeter knob.

Set the DC level in the horizontal coils to about half-maximum. Turn on the sweep and increase its amplitude to its maximum (knob turned all the way clockwise). Turn the vertical coil current to zero.

Leave the detector plugged into channel 1, but now use the selectable input cable on the front of the control panel in channel 2. Select “H” on the dial and trigger on this signal. (Note… the sweep amplitude is small and your trigger level must be set within that sweep range. Use the known DC level as a guide for where to move your trigger point and slowly move it up and down in that region until you find a steady trace on the scope.)

Turn on the RF generator and select a frequency in the range you calculated earlier. You may read the frequency directly from the device, as it has a built-in display.

Slowly vary the frequency on the generator until a pumping signal appears.

NOTEBOOK:

Sketch the “H” coil voltage and detector signal to scale. Where along the “DC + sweep” horizontal trace on the scope does the pumping signal occur? What does this tell you about the total magnetic field at that point of the sweep?

NOTE: Check that the pumping signal is lost when the RF supply is turned off. This step is important, since it is possible that the sweep could cause the B field to switch from north to south if the horizontal coils’ DC current is small enough. In this case pumping would occur due to B-flip, not due to RF stimulated emission. 

Adjust the frequency up or down slightly (without losing the pumping signal).

NOTEBOOK:

Does the pumping signal occur sooner or later within the cycle as you increase the frequency? As you decrease the frequency? Over what range of frequencies is a pumping signal visible?

Recenter the pumping signal and slowly increase the current to the vertical field. 

NOTEBOOK:

With the vertical coils set to oppose the Earth's magnetic field, does increasing the current increase or decrease the magnitude of the net magnetic field? How does the timing position of the pumping signal within the sweep cycle change and is this consistent with your expectations?

Now that we've established that our pumping signal is real and behaves as expected, we want to measure pairs of points (net magnetic field and resonance frequency) at which de-pumping occurs. To do so, keep the DC value of our horizontal field constant and vary only the vertical field and frequency.

Beginning with the resonance signal found above, do the following:

• Modifying only the vertical field current or frequency, center your pumping signal.
• Without varying the DC component of the horizontal coil current, turn down the amplitude of the sweep. (You may need to adjust your oscilloscope trigger level.)
• As you decrease the amplitude, you may find that the pumping signal is no longer centered. Stop and re-center (changing only V current or frequency).
• Continue to decrease the amplitude, re-centering as necessary, until the amplitude is as small as you can manage while keeping the pumping signal visible.

At this point, you can now make a stable measurement of the current in the H and V coils and the frequency of the oscillating fields. 

NOTEBOOK:

Record your (now fixed) DC H coil current. Construct a data table for V current and frequency and enter your first data point.

You may now vary the V field by a small amount. As the resonance condition shifts, “chase” the pumping signal by varying the frequency to re-center. Repeat this process for the full range of V currents available, measuring the current in the V coil and radio frequency value for each resonance found. In addition, you should collect a few resonance points for negative values of the V current. To do so, reverse the direction of the current through the V coil by reversing the polarity of the double banana plug where it attaches to the coils.

NOTEBOOK:

What shape should a plot of RF resonant frequency vs. V field have? (Hint: Draw a vector diagram showing all the B fields seen by the Rb atoms as you vary the V field. Recall that the magnitude of Zeeman energy splitting is given by the magnitude of the _total _B field.)

From Eqs. (17) and Eq.(20), we find that

or, equivalently,

where $B_H$ ,$B_V$ , and$B_{EW}$  are the magnitudes of the total magnetic field in the horizontal, vertical, and east-west directions (respectively), and where the subscripts “E” and “app” refer to the contributions from the Earth's and the applied (Helmholtz coil) fields (respectively). (NOTE: we expect the east-west component to be effectively canceled out by placing the table at a small angle, but include it here for completeness.)

NOTEBOOK:

Plot your data (radio frequency versus V current) as you go. Choose points intelligently, making sure that you cover a wide range on both sides of the minimum.

ANALYSIS:

Plot radio frequency versus applied vertical field and fit to the form  $f = A\sqrt{(B_{\textrm{V,E}}-B_{\textrm{V,app}})^2+(B_{\textrm{H,E}}-B_{\textrm{H,app}})^2}$ , where $B_{H,app}$  is fixed, and where $A$ , $B_{V,E}$ ,and  $B_{H,E}$ are allowed to float. (This assumes $B_{EW}$  = 0.)

How does the prefactor $A$  compare to the expected value from Eq.(22)? Does your value of Rubidium's g-factor $g_f$  match the expected value of $g_f$  depicted in Fig. 5?

Compare your extracted values for the Earth’s magnetic field components with those you measured in the B-flip experiments and with literature values.

Here we wish to perform the following process:

1. Cancel all external magnetic fields so that the magnetic dipole moments have no preferred orientation.  - Apply a magnetic field purely in the z-direction (along the light path). - Switch off the z-field and turn on a new magnetic field perpendicular to the light path (in what we will call the y-direction).

NOTEBOOK and ANALYSIS:

What will happen to the atoms while the Z field is on?

How will the net magnetic dipole moment move while the Y field is on (and Z field is off)?

The precession of the light cone about the y-axis varies the degree of alignment of the ensemble along the light path. Therefore, the transparency of vapor cell will change as the incident light sees a more or less pumped gas of atoms. We can observe this oscillation in the transparency on the scope and measure the frequency of the Larmor precession.

In Larmor mode, a power supply in the control panel sends current alternately to the Z and Y coils. This switching is controlled by by a relay driven by the function generator’s square wave and the amplitudes of the currents are controlled by the Y and Z knobs on the right of the middle panel. LEDs next to the knob indicate which coil is currently energized and an audible click can be heard as relay switches between the two fields.

To set up the apparatus in Larmor mode, do the following:

• Turn off the RF generator (if still on), and switch the “SWEEP ON” toggle to the down position.
• Set the two toggle control switches to “LARMOR”.
• Turn the function generator back on. Adjust the amplitude to maximum, then turn the knob back down about one turn. (The exact magnitude of the function generator is not critical, but we want neither full amplitude, nor low amplitude.)
• Adjust the frequency of the generator to about 0.5 Hz. You should hear an audible click about once per second as the coils switch on and off.

Let us now try to find the predicted Larmor signal. We expect to see the transparency oscillation begin when the Y coil is turned on (since this is when the ensemble of atoms established by the Z coil will begin to precess about the y-axis. However, due to grounding issues in the control panel, we cannot trigger our scope directly on the Y coil current. Instead, we will again use the “SQ” output pulse from the rear of the function generator as a time-synced alternate signal. Connect the detector to Channel 1 and the square wave output from the rear of the function generator to Channel 2. Trigger on the falling edge of Channel 2.

Note that when the scope triggers (once every few seconds), you should hear a click and observe that the Y LED is lit. Thus, we see that the “low” cycle of the square wave corresponds to the Y coil on. (And the “high” cycle therefore corresponds to the Z coil on.)

However, at the moment you should find that the detector signal is flat and featureless because the atoms are seeing a net field made up of the Earth's field and the applied Z or Y fields; it is never fully aligned along the z-axis, nor does it ever precess purely about the y-axis. To cancel the Earth's field, let us use the H and V coils to provide counter-directed fields.

To help you find this signal, here is a general procedure:

• Adjust the H and V currents to match the values obtained above (in either B-flip or RF depumping) which correspond to the Earth's field values. (If you switched the polarity of the V coil during RF, remember to switch it back.)
• Slowly tweak the H and V currents until you see a strong, periodic de-pumping signal every half-cycle
• Zoom in on your signal to observe the high frequency oscillations

No Larmor signal, H and V currents too far from canceling Earth's field.

Weak signal, keep adjusting fields in small increments.

Strong Larmor signal, ready to start collecting data.

Center the scope’s trigger point on the screen and expand the time base so that at least 10 ms is visible after the trigger. When the Earth's field is canceled and the Y and Z coils are set to full amplitude, you should see a feature on the detector trace where the signal jumps to a higher voltage (lower transparency). Center this feature on screen and expand the time base of the scope to zoom in. You should begin to see high frequency oscillation that damps out after a few hundred microseconds. To maximize this signal, make small adjustments to the V and H coils until the amplitude of these oscillations is as large as possible.

NOTEBOOK and ANALYSIS:

Even when optimized, the Larmor oscillation signal will eventually decay away. What might cause this decay?

Closer view of strong Larmor signal.  Note the difference in the time scale.

We now wish to measure the frequency of this oscillation as a function of applied Y field. We expect these variables to be related by Eq. (8). Note that you will have to reduce the switching frequency to almost zero to measure the DC current in the Y coils with the DC meter.

NOTEBOOK and ANALYSIS:

For as many values of the Y current as possible, save stable scope traces by transferring to the computer. In each case, measure the frequency. (This can be done, for example, by measuring the time it takes to complete a certain number of periods.)
For the report, plot the pairs of points ( $B_Y$  and E $f$ ) and fit to an appropriate function to find $\gamma$ , the gyromagnetic ratio of Rb-87. (Note that theory predicts a linear relationship given by Eq. (8). You may wish to include an additional constant offset. What would cause this additional term?)

So far in this experiment, we have seen frequency appear in two (seemingly) different ways: the (Larmor) frequency of precession and the resonant radio frequency of stimulated emission.

ANALYSIS:

Using the value of $\gamma$  from the Larmor fit above, compute the rubidium atom's g-factor $g_f$  from Eq.(10). How does this value compare to the value you computed above in RF-depumping (Sec. 5) and to the value predicted by Fig. 5?

From the data collected in Sec. 5 above, choose one pair of values: total magnetic field (including the Earth's contribution) and radio frequency at resonance. Using this value of magnetic field and your value of $\gamma$  from the fit just above, calculate the expected Larmor precession frequency at that field. Compare your RF value to the Larmor frequency and comment on your findings.

As the TA looks over your notebook, they will be looking for general good practices and for specific things. Be prepared to point to (and explain) the appropriate places in your lab notebook where the following information is contained.

All rubric items carry the same weight. The final grade for the analysis (75% of the experiment grade) will be assigned based on the average (on a 4.0 scale) over all rubric items. Missing elements will receive a score of 0.