A22. Optical Pumping
This is an experiment in atomic physics and magnetic resonance. Circularly
polarised light has the ability to polarise (or optically pump) atoms in a
vapour by selectively populating one of the Zeeman levels in a magnetic field.
This changes the amount of light absorbed by the vapour. You will observe
this effect with Rb atoms, and you will then observe changes in transmission
of the Rb vapour due to the redistribution of atoms among the ground state
Zeeman levels arising from resonant radiofrequency radiation incident on
the vapour. This will be used to measure various properties of the atoms and
their interaction with radiation.
1 Introduction and Background Theory
The rules governing transitions between atomic levels can be exploited to populate selected
levels preferentially. Induced transitions from such overpopulated levels can then be used
either as a diagnostic, as in the rubidium magnetometer, or to provide a source of coherent
electromagnetic emission, as in a laser.
As a simple model, consider the atomic level system shown in figure 1. This hypothetical atom
has a single electron outside closed shells. The ground state and first excited state are shown,
each with spin (S) equal to 1/2 and orbital angular momentum (L) equal to zero. In zero
external magnetic field there will therefore be (2S+1)=2 degenerate levels in each state, with MJ
= ±1/2. These levels are separated by an energy 2BB when a magnetic induction B is applied
to the system, B being the Bohr magneton.
Initially the levels in the ground state will be equally populated. If the system is irradiated with
circularly polarised photons along the direction of the applied magnetic field with each photon
having an energy corresponding to the energy difference between the two states, then this must
result in both an excitation of some of the atoms into the upper state and, because circularly
polarised photons carry angular momentum, a change in the angular momentum of the atom.
Hence, if circularly polarised light is used, the population in the ground state level with MJ =
1/2 can be lifted into the excited state level with MJ = +1/2. The population in the MJ = +1/2
ground state level cannot however be excited because angular momentum has to be conserved
and no suitable excited state is available. As the excited state +1/2 level can decay
spontaneously to the ground state +1/2 level as well as to the 1/2 level, the net effect is that the
+1/2 ground state level becomes more fully populated while the 1/2 ground state level
becomes nearly completely depopulated. Only some process which produces a transition
between the two closely spaced ground state levels can restore the equilibrium.
To see how this model applies to a real atomic system, consider the level structure of Rb-85
shown in figure 2. Rubidium is in the same group of the periodic table as sodium and potassium
and has the same atomic structure, one electron outside a closed shell with zero orbital angular
momentum (5s state). Rb-85 has nuclear spin (I) of 5/2, which couples to the electronic spin
A22 Optical Pumping
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(S) of 1/2 to give possible values of the total angular momentum quantum number, F, of 3 or 2
for the ground state. The energies of the states with these two values of F are slightly different
due to the fact that in one of them the nuclear magnetic moment is aligned parallel to the
magnetic field arising from the magnetic moment of the electron, while in the other it is
antiparallel. This 'hyperfine' splitting is small compared with typical atomic line separations
because the nuclear magneton is about 1800 times smaller than the Bohr magneton.
The first excited, L=l, p state of rubidium has a total electronic angular momentum, J, of 1/2 or
3/2, which adds to the nuclear spin to produce the effect shown in figure 2. In figure 2, the 5p
2
P1/2 F=3 state corresponds to the +1/2 excited state in figure 1. Hence this state can be
populated by irradiating the atoms with circularly polarised photons with a wavelength of 795
nm. Note that this wavelength is characteristic of the separation of the principal electronic
energy levels and therefore has the same value for both Rb-85 and Rb-87.
If the total angular momentum quantum number F is a good quantum number with which to
describe the system then the electronic and spin magnetic moments, rigidly coupled together,
can take up 2F+l possible orientations with respect to an external magnetic field. Hence there
are 2F+1 values of the magnetic quantum number MF associated with each F. Normally these
MF levels are degenerate, but they separate in an applied magnetic field as shown in figure 2.
The energy shift in each magnetic sublevel is linear with B until the applied field is high
enough for the nuclear and electronic spins to begin to decouple (c.f. break-down of LS
coupling).
According to the classical theory of the Zeeman effect, the electromagnetic radiation emitted by
an atom in a magnetic field is linearly polarised when viewed perpendicular to the field and
circularly polarised when viewed along the field. Hence if transitions between the states are to
be induced by the circularly polarised pumping light in this experiment, which is necessary
because the transitions have to involve a change in angular momentum, the magnetic field
applied to split the magnetic levels must be in the same direction as that in which the light is
propagating.
The transitions which take place when the circularly polarised light is applied are determined by
the selection rules for MF which state that MF = 1 This means that, for the D1 line at 795
nm, upward transitions can take place from the ground state from all levels except F=3, MF
=+3. Hence this is the level that becomes overpopulated.
In the experiment all transitions to the 5p 2P3/2 excited state are suppressed, in order to exclude
unwanted transitions, by inserting a D2 line blocking filter at 780 nm into the light path. This is
in fact accomplished by using a narrow band pass filter at 795 nm.
The arrangement of the apparatus is shown in figure 3. The main elements are a rubidium
discharge lamp producing strong emission at 795 nm, a quarter wave plate and polariser
producing left-handed circularly polarised light, an absorption cell containing rubidium vapour,
and finally a diode light detector and amplifier. A long solenoid, whose axis is directed along
the optical axis in the direction of propagation of the light, can be used to apply a static
longitudinal uniform field, B0, to the absorption cell. The whole system is surrounded by a set
of cylindrical mumetal shields. These not only cut out all external magnetic fields but also have
the effect of making the internal solenoid producing B0 appear infinitely long.
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When the absorption cell is put into the light beam the signal falls as the resonance radiation is
absorbed. However, as atoms in the absorption cell are transferred into the F=3, MF=+3 level
in the ground state by the action of the polarised light then the signal from the diode will begin
to rise again because there are fewer atoms in absorbing ground state levels. In effect the
pumping process 'polarises' the atoms in the absorption cell. However, if the atoms are now
transferred from the overpopulated level ground state back to other levels by some mechanism
then the light will once again be absorbed and the signal will fall. Hence an increase in the
absorption of the light by the cell is a measure of the relaxation processes taking place within it.
In practice there will always be some relaxation processes counteracting the pumping
procedure. Whether or not these have a significant effect on the signal depends on the time
constants associated with them. A rapid redistribution between adjacent levels can be achieved
by applying a radiofrequency signal at the resonance frequency for transitions of the type 5s
2
S1/2 F=3 (MF = +3  MF = +2), which immediately allows pumping to resume. Such a
transition takes place at a frequency of E/h, where h is Planck's constant and E is the energy
level separation due to the applied magnetic field. The value of E can be derived from IJ
coupling theory, according to which the magnetic moment associated with a nuclear spin
angular momentum I coupled to an electron with a magnetic moment of one Bohr magneton B
and spin 1/2, the combined system being in state F,MF, is
(MF)=  MF B [F(F + 1) + (1/2)(1/2 + 1) - I(I + 1)] / [(1/2)2F(F+ 1)] 
MF I [F(F + 1) + I(I + 1) - (1/2)(1/2 + 1)] / 2IF(F + 1)
(1)
For Rb-85, which has a nuclear moment of 1.3482±0.0005 nuclear magnetons, E=[(MF)
(MF1)]B0, and the corresponding transition frequency is 4.6619 MHz/mT.
It has already been stated that the applied static magnetic field must be perpendicular to the
plane of polarisation of the circularly polarised light. The radio frequency (RF) field used to
induce transitions between the magnetic sublevels must also be circularly polarised if it is to
induce transitions which involve a change in angular momentum. This circular polarisation can
be achieved by simply applying a sinusoidally varying RF magnetic field perpendicular to the
beam direction using a pair of small Helmholtz coils. This produces a linearly polarised
oscillating field which is equivalent to the sum of two circularly polarised fields rotating in
opposite senses.
The action of the RF field can also be understood from a classical point of view. The angle
which the total magnetic moment (F) and individual angular momentum vectors make with the
static applied field is determined by the magnetic quantum number MF which describes the
state. The interaction between the tilted magnetic moment of the atom and the applied field
produces a torque on the atomic angular momentum which results in a gyroscopic precession
around the magnetic field direction. It is easy to show that the angular frequency of this
precession, the Larmor frequency, (torque)/(angular momentum), divided by 2, is equal to the
E/h calculated above. At resonance therefore the vector representing the RF field is rotating at
the same frequency as that at which the angular momentum vector is processing. In the frame
in which these two vectors are stationary the magnetic moment experiences a torque exerted by
the RF field which induces precession around the RF field direction, causing the direction of
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the magnetic moment with respect to the applied static field to change cyclically. In other
words, the time averaged populations of the MF = +3 and MF = +2 states are equalised but the
ensemble of atoms all precess in phase.
In practice the fact that other depolarisation effects are present means that the angular
momentum vectors of different atoms which were previously aligned become 'dephased' with
respect to each other on a timescale comparable with the relaxation time constants. The width
of the resonance should be inversely related to the lifetime defined by these time constants,
which can be measured by applying the RF field in bursts. At the onset of the burst a coherent
precession of the magnetic moments of many atoms will be set up around the direction of the
RF field vector. This will result in periodic reversal of the angular momentum vectors with
respect to the direction of the static B field, that is to say, periodic occupation of all the F=3
levels, which results in periodic availability of atoms for optical pumping and therefore a
periodic variation in the signal from the detector. The period of this variation will be related to
the strength of the RF field in the same way that the RF frequency is related to the static B field.
Thus, if the RF field amplitude is 2 T, the period should be given by 1/T = 4.6619 kHz. The
factor 2 difference arises because only half the oscillating field amplitude contributes to each
sense of circular polarisation.
The 'periodic availability for pumping' mentioned in the previous paragraph means that the
pumping light is periodically absorbed, leading to an oscillating signal. It also means that some
rof the processing atoms are removed. Not all of these return in phase to the F = 3, MF = +3
level, so the pumping process contributes to the decay of the oscillations.
Other relaxation processes that lead to a decay of the oscillating signal arise through a loss of
coherence between the phase of the precession of the magnetic moments. Relaxation processes
which contribute to this loss of coherence include collisions between Rb atoms, collisions
between Rb atoms and gas molecules in the cell, collisions with the wall of the cell, loss of
atoms into the stem of the cell, and the effect of inhomogeneities in the magnetic field across
the cell.
2 Apparatus
A schematic of the apparatus is shown in Fig. 1.
Fig.1. Schematic of the apparatus for studying Optical Pumping. You should check this
carefully yourself.
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a)
Rubidium Lamp. This is in the box covered in black cloth on the wooden optical table.
Inside is a bulb filled with rubidium vapour, a heater to heat the stem of the bulb, silicone oil,
two high power heaters to heat the oil and a tuned circuit to excite the rubidium.
The bulb needs to be kept under oil to maintain a constant temperature, it is very unstable when
the temperature varies. The oil is heated to around 115 C by the two large heaters and the
smaller stem heater. This means the outside of the box gets to around 100 C, so be careful.
The temperature of the outside of the box is monitored using a thermocouple.
b)
RF Power Supply. This is a big metal box with a fan on the top. It supplies around
15W of RF at about 67MHz to excite the rubidium vapour.
c)
Photodiode Detector. This is at the back of the optical table (inside the mu-metal
shields) and has a switch on top for changing between AC and DC coupling.
d)
RF Helmholtz Coils. A pair of Helmholtz coils in the middle of the optical table
connected in series. Diameter 10cm; Separation 5cm; 9 turns each; Combined resistance 47.2
ohms.
e)
Absorption Cell. This is a glass bulb in a protective black foam case with rubidium
vapour inside. It sits in the centre of the optical table inside the RF Helmholtz coils.
f)
Circular Polariser. This comes immediately after the lens in front of the bulb and
consists of a quarter wave plate and a linear polariser. Do not adjust it as it takes quite a long
time to align the two plates.
g)
Filter. This is an interference filter which transmits light at just the right wavelength
(795 nm) to excite the correct rubidium transition in the absorption cell.
h)
Long Solenoid. This is a solenoid inside a mu-metal magnetic shield. The mu-metal
makes the solenoid appear to be infinitely long. Length of coil 1.07m; Radius 0.19m; 830
turns; Resistance 15.3 ohms.
i)
Signal Generator. For supplying the RF coils. It is capable of generating pulses of RF
and modulating the frequency automatically. Make sure you learn how to operate this early on
as it is a complicated piece of equipment which is capable of giving different types of signal.
j)
Dual Power Supply. Capable of two independent outputs. One used as a constant
current source for supplying the solenoid. The other can be used as a constant voltage source for
backing off the DC output firom the photodiode (so that small changes on a large constant
background signal can be seen). Again, make sure that you learn early on how to operate this
supply properly.
k)
Oscilloscope. Digital storage scope with cursor control.
l)
Power Supplies. Photodiode power supply with ±12V outputs. Low power lamp stem
heater supply, capable of delivering a maximum of around 10W. Higher power supply capable
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of delivering about 40W for powering high power heaters (called 50W heaters because this is
the maximum they can be run at).
The wooden optical bench can be slid in and out of the magnetic shield for easy access to the
absorption cell and photodiode switch. Be careful when removing though as the oil in the bulb
box can be spilled out of a vent hole in the top, i.e. keep it horizontal and move it carefully.
Always use the setup with the bulb box nearest the open end of the solenoid as excess fields
from the power supply cables can affect the solenoid field. There are sockets mounted on the
front of the optical table for easy connection to the equipment mounted on it. The socket for
the RF coils is underneath the table in the middle.
The set up is microphonic, i.e. it is very sensitive to vibrations, so when experiments are being
carried out try not to knock it about. The oil in the bulb moves about and takes a little time to
settle down. Also if the optical table is pulled out when the bulb is on, the oil needs time to
settle down again.
3 Experiment

Familiarise yourself with the apparatus. If necessary, consult a demonstrator or technician.

The temperature and colour of the discharge lamp are important. Learn how to control
these.
3.1 Starting The Bulb
See appendix for extra instructions on this (especially on avoiding instabilities).
Turn on the stem heater to 20V (constant current) and the 50W heater to 2.5A (constant
voltage). Heat up the bulb for about 20 minutes or until the box reaches about 55C (The value
will depend on the ambient temperature). Switch on the RF. If the bulb does not start, switch
off and try again when the box gets hotter. The bulb is on when there is a purple glow from the
back of the box. When the RF lamp power is switched on the power increases slowly (over a
few seconds) so the bulb will not strike immediately, even if the box is hot enough. The power
can be seen increasing on the power meter.
DO NOT LEAVE THE RF SUPPLY ON WHEN THE BULB IS NOT LIT.
When the bulb has lit, turn on the photodiode power supply and the digital volt meter (DVM).
The DVM shows the DC light level from the bulb. Use the smallest absorption cell to begin
with. When the light level reaches about 4V (this may take quite a while), turn the 50W heater
power supply down to exactly 1A and the stem heater supply down to 13V. The light level will
fall initially, then rise again and settle down to a value around 4V. There are certain
temperatures at which the bulb will be unstable. If the temperature has settled down to one of
these, adjust the stem heater voltage (always down rather than up) until the light becomes
stable. The light is unstable when the trace from the photodiode on the oscilloscope oscillates
A22 Optical Pumping
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with quite a high frequency (a few kHz). The bulb does not like changing temperatures, that is
why the output is sometimes unstable as it heats up.
Even though the bulb likes steady temperatures, it rarely stays at one. The light level will tend
to increase, so, reduce the stem heater a little. If the level starts to drop too much, increase the
stem heater. The light level will only ever change very slowly so only adjust the heater by
small amounts (i.e. 0.5 to 1.0 V). With a thermometer placed just under the foam insulation of
the bulb box, the ideal working temperature is just below 100 C . Do not adjust the 50W
heaters.
NEVER LET THE BULB GET > approx. 120°C – IT IS NOT GOOD FOR IT AND IT
BEHAVES QUITE UNPREDICTABLY AFTERWARDS!

Make sure you understand how the circularly polarised light is produced.

You may want to run the system without the absorption cell in place to ensure that the
detector is working and not saturating and that the output from the lamp is stable. It might
help to plot the output as a function of time on a chart recorder so you have a record of the
stability.

Note the reduction in output when the absorption cell No.l is inserted. For the time being
ignore the other cells.

Calculate the field at the position of the absorption cell as a function of current in the
solenoid, given the characteristics of the solenoid. Measure the field with a Hall probe to
confirm the accuracy of your calculation (this might be difficult). The effect of the mumetal shield will be to make the solenoid look more like an infinitely long solenoid with
the same number of turns per unit length.
3.2 Observation of Resonance
Set the solenoid current at 44mA. With this field there should be a resonance at a frequency of
about 200kHz. Set the equipment up as figure 5.
Set the signal generator to a continuous output of 200kHz with amplitude 100mV. Set the
voltage source to the negative of the DVM reading. This will back off the DC offset from the
photodiode. Look at the result on the oscilloscope (DC coupled). Adjust the frequency of the
signal generator manually around 200kHz and watch the line on the scope dip. With a light
level of about 4V the dip is 30mV. This dip is due to the light being absorbed by the Rb-85
isotope. Change the frequency of the signal generator to 300kHz and scan manually either side.
The dip here is due to light being absorbed by the Rb-87 isotope.
The box with “add” on it in figure 5 is actually a little contraption with three BNC sockets on it.
This just adds two voltages together in series.
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3.3 Measurement Of Resonances
Use the setup in the previous section. Take a note of the solenoid current. Adjust the RF
frequency until the bottom of the absorption is reached. Plot a graph of resonant frequency
against DC solenoid field for a range of currents. Repeat for the other isotope.
The equation for finding the frequency for a given field is theoretically
f[Hz] = (B/h) g B[T] ,
with B = 9.274  10-24 J/T
h = 6.626  10-34 Js
where g = 1/3 for Rb-85 and g = 1/2 for Rb-87.
The equation to calculate the field inside the solenoid is
B [T] = 0 m I [A]
where m, the number of turns per unit length = 830 / 1.07m = 775.71/m.

Put a small resistor in series with the supply to the RF coils. Make sure you can find the
current IRF through the coils by measuring the voltage across the resistor.

Estimate the inductance of the RF coils from the data supplied and confirm your estimate
by making a measurement by some suitable method. At what frequency does the measured
inductance begin to deviate from the predicted value, and why does this happen (see
Appendix)? Express the RF magnetic field, BRF , in terms of IRF. For what value of IRF
will BRF be comparable with B0/1000?
Same thing as in 3.2
3.4 Measurement of Line Width
Use the setup above. Set the solenoid current to say, 20 mA and the RF signal amplitude to say,
5V. Find the centre of the absorption by adjusting the RF frequency and measure its depth.
Move the frequency down until the signal is half the depth of the centre of resonance, and take a
note of the frequency. Do the same moving the frequency higher than the resonant frequency.
The difference in the frequencies is the full width half maximum (FWHM) width in Hz.
Reduce the RF signal amplitude by small amounts calculating the width for each value. Plot a
graph of FWHM width against RF signal amplitude. Repeat for different values of Solenoid
current (e.g. 20, 50 and 100 mA).
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An alternative method of measuring the line width is to plot out the line. Set up the signal
generator so that it sweeps through the resonant frequency using the frequency modulation
mode. Feed the auxiliary output of the signal generator into the X-input of an X-Y plotter and
channel 2 of the scope. Connect the photodiode output with the DC offset backed off into the V-input of the X-Y plotter. Adjust the scan time of the signal generator to about 30 seconds
and plot a picture.
Plot signal amplitude and peak width (FWHM) as a function of IRF. Give a qualitative
explanation for your observations. Extrapolate the FWHM against current graph to zero current
and find the intercept, f0, on the FWHM axis. What can be deduced from this intercept? Note
that it is possible that the lifetime of a state, even if it is determined by collisions, may be
inversely dependent on the energy difference between it and the ground state.
3.5 Relaxation Times
Set the signal generator to give a burst of RF using the trigger mode. The generator asks for the
number of cycles in the burst: set this to give a burst of about 20ms. Set the photodiode
coupling to AC, this gives a higher AC gain and so makes the signal easier to observe. There is
still a DC offset from the photodiode so this can either be backed off using the voltage source
or removed by AC coupling the scope. Set the solenoid current and signal generator frequency
to give a known resonance and set the RF signal amplitude to about 1V. The setup is given in
figure 6.
Look at the shape of the graph for different RF signal amplitudes and different resonant
frequencies. As the resonant frequency and/or the RF signal amplitude are increased the shape
of the envelope of the signal changes.
How does what you see depend on the pulse duration, T ?
You should see a decaying oscillation on the output (if the RF frequency is slightly different
from the resonance frequency this may have 'beats' superimposed on it). Record an example of
this waveform, including on the same graph a plot of BRF against time (i.e. the shape, duration
and on/off time of the RF pulse). Plot the frequency  of the oscillation, its initial amplitude
and its decay time . against the amplitude BRF. Extrapolate this last graph to zero BRF to obtain
the intercept 0. How does 0 compare with 1/f0 in 3.3 above? Due to the depolarising effect
of the pumping light (see Introduction), 0 may be slightly shorter than expected. Field
inhomogeneities, B0, make little contribution to f0 but can make a significant contribution to
0. As B0 is proportional to B0 it is possible that a dependence of 0 on static field strength
will be found. For fixed BRF, plot  against B0.
To measure the frequency of the signal, hold the trace on the screen of the scope with about 10
cycles showing. Using the cursors, measure the length of the 10 cycles and hence find the
frequency.
To measure the decay time, use the same setup. Set the photodiode and the scope to AC
coupling rather than backing off the DC offset. Obtain the deepest absorption by fine tuning the
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frequency. Measure the depth of the first oscillation by holding the trace and using the cursors.
Multiply this depth by l/e and measure how long it takes the envelope to decay to this value.
This value is the decay constant assuming the envelope decays exponentially. Plot a graph of
decay constant against RF signal amplitude. Repeat for different RF signal amplitudes,
different solenoid currents and different absorption cells.
Remember that for high solenoid fields and large RF signal amplitudes, the envelope shape
distorts and makes it very hard to take any reasonable results. Another way of measuring the
decay time is to hold the trace of the ringing oscillations and plot a graph of the amplitudes of
each cycle with time. Then fit a curve to the data for all values of RF signal amplitude and
work out the decay constants from it. This method is very time consuming.

Is  a linear function of BRF? The slope should be the same as that obtained for frequency
against field in the previous section. Is it? Explain why this should be so.

Note also the return to the DC baseline level at the end of the RF burst. Estimate the time
constant associated with this and explain what is happening.

Repeat the above experiments for four widely spaced values of the static field B0 and try to
explain what you observe.
 Try different absorption cells and select the one with the longest decay time (relaxation
time).

A long relaxation time should correspond to a narrow line width. Now you have a cell with
a long relaxation time scan the line again which you scanned in 3.4. Is it any narrower?
What factors other than relaxation time would you expect might affect the line width?
4 Extended experiments
4.1 Identification of Transitions
When the solenoid current is pushed up so that the field exceeds 1 mT and the resonant
frequency is of the order of MHz, the line begins to split as the different transitions are no
longer quite at the same frequency. The 'weak field' approximation is now beginning to break
down. I and J can no longer be considered to be rigidly coupled to give F. This means that the
transitions between the different MF levels for the same F no longer all occur at exactly the
same frequency. For such high fields the solenoid current is of the order of Amps. The
solenoid is rated at about 2A but cannot be left on for long as it starts to heat up. When
bursting the frequency, the RF signal amplitude has to be quite high for any effect to be seen.
Instead the setup shown in figure 7 needs to be used. Up to a certain point the smaller the RF
signal amplitude, the narrower the line so the more the high-field splitting effects can be seen.
But, the larger the RF signal amplitude, the more the RF couples into either the absorption cell
forcing it to oscillate at high frequencies, or into the photodiode electronics causing high
frequency noise on the signal. In this case it is almost impossible to see the signal on the scope.
The X-Y plotter though has high frequency rejection so the signal can still be seen.
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Calculate where the resonant frequency should be for a solenoid current of 1A. Scan through
this frequency using the frequency modulation mode of the signal generator. Set the RF signal
size to 2V.
How do you explain the differing signal strengths? Look at figure 4 and assign quantum
numbers to the transitions you observe.
Ramsey [1] gives the magnetic energy associated with an energy level (F, MF) in an
intermediate magnetic field for the special case of J=1/2 as
W =  MF B0I /I ± (W/2)(l + 4 MFx/(2I+1) + x2)1/2
(2)
where
x = (I /I - B /J) B0/W
and W = ha(I+1/2), ha being the energy separation of the hyperfine 5s 2S1/2 F=2 and F=3
levels. According to Ramsey (eqn III 80),
ha =  [I BJ]/[I(J(J+1))1/2]
(3)
where, for Rb-85, I =1.3482 B me/MP (me and MP are the electron and proton mass
respectively), and BJ is the field at the nucleus due to the circulating electrons. According to
Wolf [2], in the ground state of rubidium, BJ  130T.

Confirm that equation (2) is equivalent to equation (1) when B0 is sufficiently small, if
J=1/2.

Using a small amount of RF power, obtain an output similar to that shown in figure 4.

Do equations (2) and (3) correctly predict the separation of the peaks? Use equation (3) to
derive the elective theoretical value of ha for Rb-85 and compare this with the value you
derive from your observations.

Using the observed peak separation for Rb-87, find ha for Rb-87. Hence, by comparison
with the value of ha for Rb-85, find the value of I for Rb-87 from I for Rb-85.
4.2 Multiple quantum transitions
By applying a strong enough RF field it should be possible to excite transitions in which more
than one RF photon is involved. These transitions have to conserve both energy and angular
momentum and there are two general classes. Firstly, those in which MF =  n where n is the
number of quanta involved and can be any positive integer up to 2F. These should appear at
about the same frequencies as those which have already been used but should be narrower i.e.
you might see a narrow resonance siting on top of a broad resonance. Secondly, transitions
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between adjacent sub-levels, MF =  1, can involve multiple quanta. For a strictly
perpendicular crcularly polarised RF field only n-odd transitions are allowed (explain this). In
practice misalignments between fields relax this constraint.

Increase the RF power. This may result in the appearance of some double quantum
transitions between the peaks shown in figure 4. The amplitude of these will increase as
more RF power is applied. For example, two identical quanta absorbed in succession can
result in the transition F=3,MF=3  F=3, MF =l. The width 2 of double quantum peaks
tends to be less than that of single quantum peaks, 1 because, to first order, h(1+1) =
2h(2+2).

Look for multiple quantum transitions and report your findings.
separating Rb-85 and Rb-87 transitions.
Be careful about
4 References
1.
N.F.Ramsey, 'Molecular Beams' International Series of Monographs on Physics OUP Eqn
III 93.
2. Wolf, Physics of Atoms and Quanta
5 Appendix
A. Determination of RF magnetic field
It is quite important to know the value of BRF. Computing this from the current depends on
knowing the geometry of the coils. One way of checking that this geometry has been fed
correctly into the calculation is to compute the inductance of the coils and then measure it.
One way of carrying out this measurement is as follows.
R is stated in this manual
Next, put an additional known resistance r in series with the coil. Apply an AC voltage V0 of
variable frequency f across L+R and r. Measure V0 and Vr (the voltage across r), and the phase
angle  between them, as a function of f up to 1 MHz. The gradient of a graph of tan against
f should be a straight line passing through the origin with slope L/(R+r) times 2π. There may
be deviations from this straight line at high frequencies. What causes these?
The interpretation of the results from the optical pumping experiment depends on the
assumption that B0 and BRF are constant over the volume of the cell within which the
transitions are taking place. How big is this volume? What is the cross section of the light
beam doing the pumping? How big are the variations in BRF along the length of the light
beam within the Helmholtz coils, and what are the consequences of these variations, if any?
What is the field produced midway between a pair of Helmholtz coils as a function of distance
off-axis?
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B. How to switch on and stabilise Rb discharge lamp
This must be one of the most unstable and temperamental bits of kit in operation but this is
roughly how we got it started. The bulb box is better with its black insulation removed.




Switch on Stem heater to ~22V (in the red)
Switch on 15W heater to maximum
Wait until temperature reaches ~ 45-55oC and switch on RF supply.
Bulb should fire up within 10 secs (if not turn off RF and wait a few more minutes
before re-trying)
Do not worry about bulb instability during the warm up process – it normally recovers.



At about 110 C start to back off the stem heater slowly. Monitor the output on the
scope. You are aiming to keep the trace level. If it drops increase the stem by a tiny
amount and visa versa.
It should be possible by about 120 C to have a stable trace and NO STEM HEATER.
You are aiming for about 3 – 3.5 V output.
The bulb will be at its most stable with no stem heater and high 15W heater.
Recovery from instability can be tricky. If the trace is lower than 3 V try turning up the stem.
If you are above, try turning down the stem. Generally turning the stem off makes the trace
stable but fall dramatically. If you then slowly turn up the stem you should recover a stable
trace.
DO NOT EXCEED 123 C – THIS IS THE POINT OF NO RETURN – THE BULB WILL
BECOME DRAMATICALLY UNSTABLE AND YOU WILL NEED TO SWITCH OFF
EVERYTHING FOR ABOUT 30 MINUTES !
R C Thompson
September 1999
Corrections December 2005.
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