Download optical pumping of rubidium

OPTICAL PUMPING OF RUBIDIUM
1. INTRODUCTION
Optical pumping [123] is a process used in high frequency spectroscopy,
developed by A. Kastler who was awarded the Nobel Prize for this in 1966.
In this experiment, anomalous Zeeman transitions in the ground state of
87
85
+
-
Rb with σ - and σ -circular-polarized pumping light are observed
Rb and
and measured.
2. THEORY
a) Energy level scheme
Rubidium is an alkali metal in the first main group of the periodic table.
The low energy states of this element can be described very well using
Paschen notation, i.e.
2S+1
LJ.
The rubidium atom consists of a spherically symmetrical atomic residue with
orbital spin 0, and one optically-active electron with an orbital angular
momentum 0,1,2... and an electron spin of
½.
The fine structure of the
energy states is illustrated in Fig. 1. It is caused by the half-integral
electron spin, which leads to a multiplicity of 2, i.e. a doublet system.
The ground state is an S-state; the orbital angular momentum here is L = 0.
The spin-orbit coupling results in a total angular momentum quantum number
J =
½.
into a
Due to this coupling, the first excited level with L = 1 splits up
2
P 1/2 and a
2
P 3/2 state. Both states can be easily excited in a gas
2
discharge. During the transitions from the first excited states
2
P 3/2 into the ground state
all
alkali
atoms
is
P 1/2 and
2
S 1/2 , the doublet D 1 and D 2 characteristic of
emitted.
For
rubidium,
the
wavelengths
of
the
transitions are 794.8 nm (D 1 line) and 780 nm (D 2 line).
The hyperfine interaction, caused by the coupling of the orbital angular
momentum J with the nuclear spin I, leads to the splitting of the ground
state and the excited states. The additional coupling provides hyperfine
levels with a total angular momentum F = I ± J. For
spin I = 3/2, the ground state
87
Rb with a nuclear
2
S 1/2 and the first excited state
up into two hyperfine levels with the
quantum
numbers
F = 1
2
P 1/2 split
and
F = 2.
Page 1 of 24
Compared with the transition frequency of 4 x 10
14
Hz between
2
P 1/2 and
2
S 1/2 , the resulting splitting of the ground state and the excited states
is
much
6.8 x 10
smaller.
9
For
the
ground
state,
this
hyperfine
splitting
of
Hz is approx. 5 powers of 10 smaller than the fine structure
splitting.
Fine structure
splitting
Fig. 1
Hyperfine structure
splitting
Zeeman
splitting
Scheme of energy levels for
87
Rb
In the magnetic field, an additional Zeeman splitting (Fig. 1) into 2F + 1
sub-levels respectively is obtained. For magnetic fields of approx. 1 mT,
the transition frequency between neighbouring Zeeman levels of a hyperfine
state is 8 x 10
splitting.
The
6
Hz, i.e. another 3 powers of 10 smaller than the hyperfine
energy
respective
frequency
relationships
between
the
individual states are of particular significance in understanding optical
pumping.
Page 2 of 24
b) Optical pumping
The process can be explained in more detail using the energy level scheme
of
87
The
Rb (Fig. 1) as a reference:
transitions
from
2
S 1/2
2
to
P 1/2
and
2
P 3/2
are
electrical
dipole
transitions. They are only possible if the selection rules ∆m F = 0 or
∆m F = ±1 have been fulfilled.
The transitions between the Zeeman levels are detected using a method
discovered
by
A.
Kastler
in
1950
which
will
be
described
in
the
following [1]. The D 1 line emitted by the rubidium lamp displays such a
high
degree
of
Doppler
broadening
that
it
can
be
used
to
induce
permissible transitions between the various Zeeman levels of the
all
2
S 1/2 and
2
P 1/2 states. If an absorption cell located in a weak magnetic field and
filled with rubidium vapour is irradiated with the σ
+
circularly-polarized
component of the D 1 line, the absorption taking place in the cell excites
the
various
Zeeman
levels
which
are
higher
by
∆m F = +1.
However,
excited states decay spontaneously to the ground state and re-emit π, σ
the
+
or
-
light in all spatial directions, in accordance with the ∆m F = 0 or
∆m F = ±1 selection principle.
σ
The irradiating, circularly-polarized light effects a polarization of the
atomic vapour in the absorption cell. This can be interpreted as follows:
during the process of absorption, the polarized light transmits angular
momentum to the rubidium atoms. The rubidium vapour is polarized and thus
magnetized macroscopically.
Without optical irradiation, the difference between the population numbers
of the various Zeeman levels in the ground state is infinitesimally small,
due
to
its
low
irradiation with σ
energy
+
spacing
in
thermal
equilibrium.
However,
the
light results in a strong deviation from the thermal
equilibrium population. In other words, the population distribution changes
as a result of optical pumping. The F = 2, m F = +2 level has the largest
population probability — represented in Fig. 1 by bolder print — as the
angular momentum selection principle prevents excitation from this level to
higher ones. A Zeeman state with the m F quantum number 3 does not exist in
the first excited state. In other words, the irradiation of the absorption
+
cell with the D 1 σ
line generates a new population distribution in the
Page 3 of 24
equilibrium
between
optical
pumping
and
the
simultaneous
relaxation
processes. These are caused by depolarizing collisions of rubidium atoms
with the walls of the vessel, or by interatomic collisions. In the pumping
equilibrium,
the
population
probabilities
decrease
in
the
order
m F = 2, 1, 0, -1, -2 in the F = 2 state, and in the order m F = -1, 0, +1 in
the F = 1 state.
In the case of the magnetic fields used here, the position of the energy
states can be calculated with the Breit-Rabi formula [4,5]
For
F = I ±
Ε ( F, m F
)
1
2
4m F
∆Ε
∆Ε 
1 +
= −
+ µ N g I Βm F ±
ξ + ξ2
2( 2I + 1 )
2 
2I + 1
where ξ =
gJ µ B − gIµ N
∆Ε
1
2


Β
(I)
F
= Total angular momentum
I
= Nuclear spin
J
= Angular momentum of the electron shell
m F = Magnetic quantum number of the total angular momentum F
g I = g-factor of the nucleus
g J = g-factor of the electron shell
∆Ε = Hyperfine structure spacing
µ B = Bohr magneton
µ N = Nuclear magneton
B
= Magnetic flux density
When irradiated with σ
+
pumping light, the Zeeman levels within a hyperfine
state which have positive quantum numbers m F become enriched at the expense
of the levels with negative quantum numbers. The result is a population
which deviates from the thermal equilibrium population. In the ground state
87
Rb the level F = 2, m F = +F has the greatest population.
When irradiated with a linearly polarized alternating magnetic field of the
correct frequency, more transitions take place from a higher Zeeman level
m F to the next lowest level with m F - 1 than in the other direction. With
Page 4 of 24
-
σ
pumping light, the situation is just the opposite, as Zeeman levels with
negative quantum numbers predominate.
The frequency f of these transitions is always
f(m F
↔ mF
− 1) = ±
µ NgIΒ
h
1

4m F

2
∆Ε  
1 +
+
ξ + ξ 2 
2 h  
2I + 1



4( m F − 1 )

−  1 +
ξ + ξ2
2I + 1

1 
2 
 
 

The result is an average frequency spacing of approx. 8 MHz mT
neighbouring Zeeman levels in the ground state
2
S 1/2 for
-1
(II)
between
87
Rb. This means
that in a magnetic field of 1 mT, irradiation with a high frequency of
8 MHz can induce transitions between ground state levels. In the case of
the magnetic fields used here, the energy spacing ∆E F = hf between the
Zeeman levels in the ground state of
87
Rb amounts to 3 x 10
-8
eV. For the
induced emission, the transition of the atom between two Zeeman levels with
∆m F = 1 is accompanied by the emission of the inducing HF quantum as well
as one having the same energy. In view of the very small number of atoms
available at a vapour pressure of 10
-5
mbar, the total energy released in
this process is so low that a direct detection is not possible; a higher
vapour pressure leads to a broadening of the lines due to collisions. If
the HF field were observed directly, the signals would be masked entirely
by noise. The process developed by A. Kastler requires that, for every act
of induced emission, the population equilibrium between optical pumping and
relaxation
be
changed
by
1
atom.
This
allows
the
absorption
of
an
additional pumping photon with an energy ∆E opt = 1.5 eV. By observing this
absorption
-8
directly,
one
thus
achieves
an
amplification
of
7
1.5/(3x10 ) = 5 x 10 , which is obtained without incurring.
Page 5 of 24
c) Experiment arrangement
Fig.
2
is
an
overview
diagram
of
the
experimental
set-up.
Fig.
3
schematically illustrates the individual optical and magnetic components
and the optical path.
Fig. 2:
1
2
3
4
5
6
7
8
9
10
11
12
Overview diagram of the entire experimental setup; position
specifications are measured from the left edge of the optical riders
Rubidium high frequency lamp
Lens
Interference filter
Linear polarization filter
λ/4 plate
Lens
Helmholtz coil
Absorption chamber
Absorption cell
HF coils
Lens
Silicon photodetector
Fig. 3:
Schematic representation of the
experiment set-up
Page 6 of 24
A high-frequency rubidium lamp (1) serves as the light source. A quartz
cell filled with 1 to 2 mbars of Argon and containing a small amount of
rubidium is located in the electromagnetic field of a resonant circuit coil
of an HF transmitter (60 MHz, 5W) which is integrated in the lamp housing.
In order to prevent an excessive rate of rise of the Rb vapour pressure
during HF excitation and thus the spontaneous inversion of the D 1 line, the
lamp
temperature
is
maintained
at
approx.
130
°C
by
means
of
an
electronically controlled thermostat. For this, the lamp is in thermal
contact with the electrically heated base plate. A temperature sensor and a
two-point controller in the supply device control the temperature of this
plate and thereby the flow of heat between the lamp and the lamp housing.
The lamp is located at the focus of the lens (2).
An
interference
filter
(3),
a
linear-polarization
λ/4 plate (5) are used to separate the σ
+
filter
(4)
and
a
circularly-polarized component of
+
the D 1 line from the emission spectrum of the rubidium lamp. The D 1 σ
light thus generated is focused on to the absorption cell (9) with a lens
(6) having a short focal length. The absorption cell (9) also contains
rubidium. It is located in a water bath, the temperature of which is
maintained
at
approx.
°C
70
with
a
circulation
thermostat.
At
this
temperature, the rubidium vapour pressure in the absorption cell settles at
approx. 10
-5
mbar. The absorption chamber (8), together with the absorption
cell (9) is located in a pair of Helmholtz coils (7), so that the magnetic
field necessary for the Zeeman separation can be generated. The axis of the
magnetic field coincides with the optical axis of the experiment set-up.
The transmitted light is focused onto a silicon photodetector (12) by means
of another lens (11) with a short focal length. The detector signal is
proportional
to
the
intensity
of
the
radiation
passing
through
the
absorption cell and is fed to the Y input of an oscillograph, an XY
recorder or an interface, via a downstream current-to-voltage converter. HF
coils (10) are positioned on both sides of the absorption cell. A function
generator serves as the HF source. Its frequency is swept around by a
gradually rising, saw-tooth voltage. This voltage is simultaneously used
for
the
X-deflection
of
the
registering
system.
As
the
HF
coils
are
oriented perpendicularly with respect to the Helmholtz coil field, the
linearly
polarized
HF
coil
field
generates
the
circularly-polarized
HF
field necessary for the Zeeman transitions. The linearly polarized HF field
can
be
divided
into
one
right-handed
and
one
left-handed
circularly-
Page 7 of 24
polarized
field;
the
component
with
the
correct
direction
of
rotation
(corresponding to ∆m F = ±1) induces transitions in the rubidium vapour of
the absorption cell.
The process makes it possible to observe the absorption in terms of its
dependence on the HF irradiation.
d) Observation of the pump signal
Without HF irradiation, an equilibrium is reached in the absorption cell
while pumping with σ
+
light; in the ground state
2
S 1/2 , F = 2, the Zeeman
level with m F = +2 is the most populated. The absorption of the cell is
constant and thus so is the light intensity observed in transmission. If
the circular polarization of the pumping light is now quickly changed from
+
-
σ
to σ , light is absorbed more strongly. With the irradiation from σ
light,
absorption
transitions
with
∆m F
=
equilibrium settles in, where now the state
-1
are
possible,
and
a
-
new
2
S 1/2 , F = 2, m F = -2 is the
most populated. The circular polarization of the pumping light radiation
(σ
+
-
σ )
or
relative
to
the
magnetic
field
orientation
is
decisive
in
determining whether ∆m F = +1 or ∆m F = -1 transitions occur. Thus when
right-handed circularly-polarized light is used for irradiation parallel to
the magnetic field, it produces ∆m F = +1 transitions, but for irradiation
antiparallel to the field it causes ∆m F = -1 transitions. Instead of the
quick
switchover
from
σ
+
to
-
σ
pumping
light,
which
is
experimentally
difficult to realize, a direction changeover of the magnetic field can be
used. This change in the magnetic field direction can be easily carried out
by changing the polarity of the Helmholtz coils with a switch. The signals
observed
when
this
is
done
(Fig.
4)
are
a
clear
indication
that
the
absorption cell is being pumped optically.
Fig.4
Above: voltage across the Helmholtz coil pair with respect to time
Below: Pumping light signal with respect to time
Page 8 of 24
e) HF transitions between Zeeman levels in the ground state
Two processes can be used to observe the transitions between Zeeman levels
in the ground state. In one case, the magnetic field can be slowly changed
at constant frequency. With increasing magnetic field strength the energy
spacing between neighbouring Zeeman levels increases. If this spacing is
exactly
E(F,m F )
-
E(F,m F-1 )
=
hf 0
and
if
f0
is
the
frequency
of
the
irradiating high frequency, induced transitions between the corresponding
Zeeman levels take place and the absorption for the pumping light changes.
If the HF irradiation is strong enough, population equilibrium of the
corresponding Zeeman levels is forced and, as a result, a deviation from
pumping equilibrium.
In the other case, the frequency of the irradiating HF field can be slowly
varied at constant magnetic field strength. Here, the absorption changes
each time ∆E = hf reaches the spacing between neighbouring Zeeman levels.
We will make use of the latter method here.
The spacing between neighbouring Zeeman levels in the hyperfine state F = 2
increases monotonically in the order m F = +2 to +1, +1 to 0, 0 to –1,
-1 to –2 and in the F = 1 state from m F = -1 to 0, 0 to +1. As a result of
this, the individual absorption signals appear in the corresponding order
with increasing frequency of the HF irradiation. The relative position
between the absorption transitions of the F = 2 state and those of the
F = 1 state is dependent on the strength of the applied magnetic field. The
intensity
of
the
absorption
lines
is
determined
by
the
difference between the Zeeman levels involved each time. When σ
population
+
light is
used for pumping, the transition F = 2, m F = +2 to +1 appears with the
greatest intensity (Fig. 5). Here we are dealing with a special case of the
anomalous Zeeman effect. If the polarization is changed, i.e. σ
radiated into the absorption cell instead of σ
+
-
light is
light, or the magnetic
field direction of the Helmholtz coils is reversed while the polarization
is retained, the transition F = 2, m F = -2 to –1 appears with the greatest
intensity.
Page 9 of 24
Fig. 5
f) Zeeman splitting in
Absorption spectrum
87
Rb with σ
+
light
85
Rb
A further special feature of the anomalous Zeeman effect is displayed in
the multiplicity of the Zeeman energy levels in isotopes with different
87
Rb and
nuclear spins. In natural rubidium the isotopes
87
ratio 72:28. While
85
Rb occur in the
85
Rb has nuclear spin 3/2,
Rb has a nuclear spin 5/2.
The corresponding angular momentum quantum numbers of the hyperfine levels
are F = 3/2 + 1/2 = 2 and F = 3/2 – 1/2 = 1 for
for
87
Rb and F = 3 and F = 2
85
Rb. Due to the different nuclear g factors the hyperfine splitting is
6.8 x 10
9
Hz for
87
Rb and 3.1 x 10
9
Hz for
85
Rb.
This, however, means that for the same magnetic field the Zeeman splitting
is of different magnitude for the two isotopes. While the splitting for
87
Rb in the ground state has an average value of 8 MHz for a magnetic field
of 1 mT, for
85
Rb and the same field the corresponding value is 5 MHz.
Both pumping light source and absorption cell contain the natural isotope
mixture. By setting the appropriate frequency on the function generator the
Zeeman structures of the two isotopes can be observed separately. The
hyperfine states with F = 2 and F = 1 for
87
Rb split into 2F + 1 = 5 or
2F + 1 = 3 Zeeman levels respectively, while the splitting for
85
Rb has 7
Page 10 of 24
or 5 levels respectively. Correspondingly 4 + 2 lines occur in the
absorption spectrum (Fig. 5) and 6 + 4 lines in the
Fig. 6
Absorption spectrum
85
87
Rb
85
Rb spectrum (Fig. 6).
Rb with σ
+
light
3. PROCEDURE
a) Observation of the pumping signal
i) Optical adjustment of the experiment setup and warming up the system
take
several
hours.
Ask
the
laboratory
officer
in
charge
of
the
experiment to prepare the apparatus in the morning so that you can
perform the experiment in the afternoon the same day.
Settings:
Operation device for optical pumping:
Supply voltage:
U = 19.0 V
Temperature setting:
θ ≈ 118 °C
Circulation thermostat:
Temperature setting:
65 °C
Page 11 of 24
ii) Measuring
▬
Switch on the oscilloscope’s storage mode by pressing and holding
the oscilloscope STOR. ON/HOLD pushbutton for about 1 second.
▬
Set the Helmholtz coil current I coil ≈ 0.8 A.
▬
Press and hold the SINGLE pushbutton for about 1 second to switch to
SINGLE mode.
▬
Start oscilloscope storage mode by briefly pressing the oscilloscope
button RES. (RESET)
▬
Toggle the two-way switch back and forth.
Settings:
Oscilloscope:
Operating mode:
Storage mode
Trigger:
Auto
Y-deflecting, Channel II:
200 mV/DIV. (DC)
Timebase:
1 s/DIV
Polarization Filter
Angle:
0°
Quarter wavelength panel:
Angle:
+45° or -45°
I/U Converter:
Toggle switch:
DC
Measuring example:
Fig. 7
Transmitted intensity as a function of time for
multiple rapid reversals of the Zeeman magnetic field.
Page 12 of 24
b) Observation and measurement of Zeeman splitting in
87
Rb
i) Finding the absorption signal
▬
For
87
Rb the frequencies of the Zeeman transitions in the ground
state in a Helmholtz field of 1.2 mT (coil current 740 mA) are
around 8 MHz. When the setup is optimally adjusted, the absorption
signal reaches an amplitude of at least 40 mV:
▬
Set the polarization filter to 0° and the quarter wavelength panel
to +45° or -45°.
▬
Set the desired operating mode and frequency range on the function
generator.
▬
Start the function generator by pressing the button labelled MANUAL.
▬
Vary
the
Helmholtz
coil
current
until
the
maximum
(negative)
absorption signal appears on the oscilloscope.
▬
If necessary, set the toggle switch of the I/U converter to AC or
use the offset potentiometer of the I/U converter to bring the
signal back to the middle of the oscilloscope screen.
▬
Maximize the absorption signal by changing the operating parameters
of the rubidium high-frequency lamp.
Settings:
Polarization filter:
Angle
0°
Quarter wavelength panel:
Angle:
+45° or -45°
I/U converter:
Toggle switch:
DC
Oscilloscope:
Operating mode:
X-Y mode
Channel I:
500 mV/DIV. (DC)
Channel II:
20 mV/DIV. (DC)
Page 13 of 24
Function generator:
Function:
: (sine)
Amplitude:
Middle position
Attenuation:
20 dB
DC-offset:
0 V (DC button pressed)
Sweep button:
Pressed
Mode*:
‘Cu
Stop*:
8.5 MHz
Start*:
7.5 MHz
Period*:
ca. 100ms (fast sweep)
* Press button and set desired value with knob
ii) Measuring
Preparation
Finding the signal:
▬
Operate the oscilloscope in the X-Y mode by pressing button DUAL for
about 1s until XY appears on the screen.
▬
Switch off the oscilloscope storage mode.
▬
If necessary, switch off the sensitivity of oscilloscope channel I.
▬
Set the toggle switch of the I/U converter to DC.
▬
Set
the
start
and
stop
frequencies
on
the
function
generator
(f A = 9.0 MHz, f E = 9.2 MHz)
▬
Switch the function generator to period 100 ms (fast sweep).
▬
Start the function generator by pressing the button labelled MANUAL.
▬
Set the Helmholtz coil current I ≈ 0.791 A and vary it until an
absorption signal can be seen on the oscilloscope screen.
Page 14 of 24
Oscilloscope storage mode:
▬
Switch on the storage mode of the oscilloscope.
▬
Press the Start button of the function generator.
▬
Adjust the horizontal deflection of the oscilloscope with the knob
Y-Pos I to x A = 1.0 scale division.
▬
Press the Stop button of the function generator.
▬
Press and hold CH I – VAR pushbutton until the VAR-LED located above
the VOLTS/DIV. knob is lit.
▬
Adjust
the
channel I
horizontal
deflection
of
the
oscilloscope
with
the
VOLTS/DIV. knob to x E = 9.0 scale divisions.
Fine adjustment:
▬
Start the function generator by pressing the button labelled MANUAL.
▬
Switch the function generator to period 10 s (slow sweep).
▬
Switch the vertical deflection of the oscilloscope to sensitive.
▬
Turn the quarter wavelength panel back and forth
between
+45° and
-45° and check whether all lines of the absorption spectrum appear
on the oscilloscope screen.
▬
If necessary, readjust the Helmholtz coil current I or the start
frequency f A and stop frequency f E accordingly.
Procedure
+
Note: at σ -polarization the absorption line with the lowest frequency
is the most intense.
First run:
▬
Set the quarter wavelength panel to σ -polarization.
▬
Start the function generator by pressing the button labelled MANUAL.
▬
Wait until the absorption spectrum has been completely recorded.
+
Page 15 of 24
▬
Stop
recording
of
the
absorption
spectrum
by
pressing
the
oscilloscope button HOLD.
▬
Determine the position x of the absorption lines on the oscilloscope
screen.
▬
Determine the amplitude U of the absorption lines.
▬
Check the start frequency f A and the stop frequency f E .
▬
Check the Helmholtz coil current I.
Second run:
▬
Set the quarter wavelength panel to σ -polarization.
▬
Start the function generator by pressing the button labelled MANUAL.
▬
Wait until the absorption spectrum has been completely recorded.
▬
Stop
-
recording
of
the
absorption
spectrum
by
pressing
the
oscilloscope button HOLD.
▬
Determine the position x of the absorption lines on the oscilloscope
screen.
▬
Determine the amplitude U of the absorption lines.
▬
Check the start frequency f A and the stop frequency f E .
▬
Check the Helmholtz coil current I.
▬
Tabulate x and U values in Table 1.
Setting:
Oscilloscope:
Operating mode:
X-Y mode
Storage mode
Trigger:
Auto
Channel I:
>500 mV/DIV. (DC)
Channel II:
10 mV/DIV. (DC)
Recording range:
1.0 division – 9.0 divisions
Timebase:
200 S/s
Page 16 of 24
Function generator:
Stop:
9.2 MHz
Start:
9.05 MHz
Period:
10 s (slow sweep)
Amplitude:
12½ o’clock
Attenuation:
20 dB
Measuring example:
IPhoto
9.10
f [MHz]
9.15
5
3
6
4
2
1
9.10
0.2 µA
f [MHz]
9.15
4
1
6
2
3
5
Fig. 8
Absorption spectrum
Rb with σ
and σ light (bottom).
87
+
light (top)
Page 17 of 24
No.
x [Div]
U [mV]
No.
1
1
2
2
3
3
4
4
5
5
6
6
Table 1:
x [Div]
87
Position x, amplitude U of the
U [mV]
Rb absorption lines.
σ
light, I = __________ A
right side: Measurement with σ
light, I = __________ A
left side: Measurement with
+
-
iii) Evaluation
Determining the nuclear spin I of
87
Rb
The irradiated alternating magnetic field makes possible transitions
with the selection rules ∆F = 0, ±1 and ∆m F
= ±1. The available
frequency range only permits ∆F = 0. Thus, between the nuclear spin I
and the number of transitions n there exists the relationship
n = 2F + + 2F - = 4I
(III)
We can observe six lines. According to equation (III), the nuclear
spin of
87
Rb thus has the value I=3/2.
Determining the transition frequencies.
The position x on the oscilloscope screen allows us to determine the
frequencies f M of the absorption lines using the formula
fM = fA + (fE − fA
)
x − xA
(IV)
xE − xA
x A = Start of scale (set with Y-POS. I)
x E = End of scale (set with channel I VOLTS/DIV. knob)
f A = Start frequency
f E = Stop frequency
+
-
Summarize the results in Table 2 as the quantities f M (σ ) and f M (σ ).
The standard deviation for these quantities is σ = 0.002 MHz.
Page 18 of 24
No.
+
fM(σ ) [MHz]
-
fM(σ ) [MHz]
F
mF(1)↔mF(2)
1
2
2 ↔ 1
2
2
1 ↔ 0
3
2
0 ↔-1
4
1
1 ↔ 0
5
2
-1↔-2
6
1
0 ↔-1
87
Table 2:
f [MHz]
Rb transition frequencies at B = __________ mT
Assignment of the transitions
The following numerical values can be found in the literature:
µ N = 5.05083(4) x 10
-27
JT
-1
[9]
µ B = 9.27410(7) x 10
-24
JT
-1
[9]
g J = 2.002332(2)
h
[6]
= 6.6260(5) x 10
-34
Js
[9]
∆E/h = 6834.682614(1) MHz
[8]
g I = -1.827231(2)
[7]
Using (II), we obtain within the hyperfine state F = 2 four transitions
which have frequencies
fF = 2 = −0.01393 MHz ×
B
mT
(
+ 3417.34 MHz  1 + m F ξ +

and
within
the
hyperfine
)
1 2
ξ2
(
− 1 + ( m F − 1 )ξ +
state
F
=
1
two
)
1 2
ξ2
(V)


transitions
with
the
frequencies
fF = 1 = 0.01393 MHz ×
(
B
mT
+ 3417.34 MHz  1 + m F ξ +

where
)
1 2
ξ2
ξ = 4.10243(4) × 10 − 3 ×
(
− 1 + ( m F − 1 )ξ +
)
1 2
ξ2
(VI)


B
mT
Page 19 of 24
B ≈ 1.64 x I(A) mT
Start with the above approximate B value. Vary B in equations (V) and
(VI)
until
the
mathematically.
measured
Record
the
transition
value
+
of
frequencies
B
and
are
compare
reproduced
the
measured
-
transition frequencies f M (σ ) and f M (σ ) with the calculated transition
frequencies f in Table 2.
+
-
also
change.
When changing from σ - to σ -pumping light, the amplitude U of the
absorption
lines
Compare
the
intensities
of
the
transitions belonging to hyperfine state F = 2 and F = 1 in Table 3.
No.
+
-
U(σ )
U(σ )
[mv]
[mV]
m F (1)↔m F (2)
No.
+
-
U(σ )
U(σ )
[mV]
[mV]
m F (1)↔m F (2)
1
2 ↔ 1
4
1 ↔ 0
2
1 ↔ 0
6
0 ↔-1
3
0 ↔-1
5
-1↔-2
Table 3:
87
Rb transition intensities at B = __________ mT.
left side: F = 2, right side: F = 1
Page 20 of 24
c) Observation and measurement of Zeeman splitting in
85
Rb
Repeat all procedures in (b) with the following settings:
Helmholtz coil current I = 0.838 A
Oscilloscope:
Operating mode:
X-Y mode
Storage mode
Trigger:
Auto
Channel I:
>500 mV/DIV. (DC)
Channel II:
10 mV/DIV. (DC)
Recording range:
1.0 division — 9.0 divisions
Timebase:
200 S/s
Function generator:
Stop:
6.55 MHz
Start:
6.35 MHz
Period:
10 s (slow sweep)
Amplitude:
12½ o’clock
Attenuation:
20 dB
Measuring example:
IPhoto
6.40
6.45
6.50
7
5
3
8
9
f [MHz]
10
6
4
2
1
6.40
0.2 µA
1
2
6.45
3
6.50
f [MHz]
5
4
7
6
9
8
10
Fig. 9
Absorption spectrum
Rb with σ
and σ light (bottom)
85
+
light (top)
Page 21 of 24
No.
x [Div]
U [mV]
No.
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
Table 4:
x [Div]
85
Position x, amplitude U of the
U [mV]
Rb absorption lines.
σ
light, I = __________ A
right side: Measurement with σ
light, I = __________ A
left side: Measurement with
+
-
Assignment of the transitions
The following numerical values can be found in the literature:
µ N = 5.05083(4) x 10
-27
µ B = 9.27410(7) x 10
-24
JT
JT
-1
[9]
-1
[9]
g J = 2.002332(2)
h
= 6.6260(5) x 10
[6]
-34
Js
[9]
∆E/h = 3085.732439(5) MHz
[8]
g I = -5.39155(2)
[7]
Using (II), we obtain within the hyperfine state F = 3 six transitions
which have frequencies
fF = 3 = −0.004199 MHz ×
B
mT

2

+ 1542.866 MHz   1 +
mF ξ + ξ2 
3

 
1 2
2

( m F − 1 ) ξ + ξ 2 
− 1 +
3


1 2



Page 22 of 24
and
within
the
hyperfine
state
F
=
2
four
transitions
with
the
frequencies
fF = 2 = 0.004199 MHz ×
B
mT
1 2
1 2

2
2


( m F − 1 ) ξ + ξ 2  
+ 1542.866 MHz   1 +
− 1 +
mF ξ + ξ2 
3
3



 

where ξ = 9.08341(9) × 10 − 3 ×
+
fM(σ ) [MHz]
No.
-
fM(σ ) [MHz]
F
mF(1)↔mF(2)
1
3
3 ↔ 2
2
3
2 ↔ 1
3
2
2 ↔ 1
4
3
1 ↔ 0
5
2
1 ↔ 0
6
3
0 ↔-1
7
2
0 ↔-1
8
3
-1↔-2
9
2
-1↔-2
10
3
-2↔-3
85
Table 5:
No.
B
mT
+
-
U(σ )
U(σ )
[mV]
[mV]
f [MHz]
Rb transition frequencies at B = __________ mT
m F (1)↔m F (2)
No.
+
-
U(σ )
U(σ )
[mV]
[mV]
m F (1)↔m F (2)
1
3 ↔ 2
3
2 ↔ 1
2
2 ↔ 1
5
1 ↔ 0
4
1 ↔ 0
7
0 ↔-1
6
0 ↔-1
9
-1↔-2
8
-1↔-2
10
-2↔-3
Table 6:
85
Rb transition intensities at B = __________ mT.
left side: F = 3, right side: F = 2
Page 23 of 24
Literature
[1]
A. Kastler: Journal de Physique, 11 (1950) 255
[2]
H.Kopfermann:
Über
Heidelberger
Akademie
optisches
der
Pumpen
an
Gasen,
Wissensschaften,
Sitzungsberlchte
Jahrgang
1960,
der
3.
Abhandlung
[3]
J. Recht, W. Klein: LH-Contact 1 (1991) pp. 8-11
[4]
G. Breit, I. Rabi: Phys. Rev. 38 (1931) 2002
[5]
Kopfermann, H.: Kernmomente, 2nd ed., pp. 28ff
[6]
S. Penselin: Z. Physik 200 (1967) 467
[7]
C.W. White et al.: Phys. Rev. 174 (1968) 23
[8]
G. H. Fuller et al. : Nuclear Data Tables A5 (1969) 523
[9]
B. N. Taylor et al.: Rev. Mod. Phys. 41 (1969) 375
Page 24 of 24