OPTICAL PUMPING OF RUBIDIUMphyweb.physics.nus.edu.sg/~L3000/Level3manuals... · electron spin,...

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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 87Rb and 85Rb with σ+- and σ--circular-polarized pumping light are observed

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+1LJ.

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. Due to this coupling, the first excited level with L = 1 splits up into a 2P1/2 and a

2P3/2 state. Both states can be easily excited in a gas

discharge. During the transitions from the first excited states 2P1/2 and 2P3/2 into the ground state

2S1/2, the doublet D1 and D2 characteristic of

all alkali atoms is emitted. For rubidium, the wavelengths of the

transitions are 794.8 nm (D1 line) and 780 nm (D2 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 87Rb with a nuclear

spin I = 3/2, the ground state 2S1/2 and the first excited state 2P1/2 split

up into two hyperfine levels with the quantum numbers F = 1 and F = 2.

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Compared with the transition frequency of 4 x 1014 Hz between 2P1/2 and 2S1/2, the resulting splitting of the ground state and the excited states

is much smaller. For the ground state, this hyperfine splitting of

6.8 x 109 Hz is approx. 5 powers of 10 smaller than the fine structure

splitting.

Fig. 1 Scheme of energy levels for 87Rb

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 106 Hz, i.e. another 3 powers of 10 smaller than the hyperfine

splitting. The energy respective frequency relationships between the

individual states are of particular significance in understanding optical

pumping.

Fine structure splitting

Hyperfine structure splitting

Zeeman splitting

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b) Optical pumping

The process can be explained in more detail using the energy level scheme

of 87Rb (Fig. 1) as a reference:

The transitions from 2S1/2 to 2P1/2 and 2P3/2 are electrical dipole

transitions. They are only possible if the selection rules ∆mF = 0 or

∆mF = ±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 D1 line emitted by the rubidium lamp displays such a

high degree of Doppler broadening that it can be used to induce all

permissible transitions between the various Zeeman levels of the 2S1/2 and 2P1/2 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 D1 line, the absorption taking place in the cell excites

the various Zeeman levels which are higher by ∆mF = +1. However, the

excited states decay spontaneously to the ground state and re-emit π, σ+ or σ- light in all spatial directions, in accordance with the ∆mF = 0 or

∆mF = ±1 selection principle.

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 energy spacing in thermal equilibrium. However, the

irradiation with σ+ 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, mF = +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 mF quantum number 3 does not exist in

the first excited state. In other words, the irradiation of the absorption

cell with the D1 σ+ line generates a new population distribution in the

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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

mF = 2, 1, 0, -1, -2 in the F = 2 state, and in the order mF = -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 21

IF ±=

( )( )

2

1

2FFINF 1I2

m41

2mg

1I22m,F

ξ+ξ

++

∆Ε±Βµ+

+∆Ε

−=Ε

where Β∆Ε

µ−µ=ξ NIBJ gg

(I)

F = Total angular momentum

I = Nuclear spin

J = Angular momentum of the electron shell

mF = Magnetic quantum number of the total angular momentum F

gI = g-factor of the nucleus

gJ = g-factor of the electron shell

∆Ε = Hyperfine structure spacing

µB = Bohr magneton

µN = Nuclear magneton

B = Magnetic flux density

When irradiated with σ+ pumping light, the Zeeman levels within a hyperfine

state which have positive quantum numbers mF 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 87Rb the level F = 2, mF = +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

mF to the next lowest level with mF - 1 than in the other direction. With

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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

( )h

g1mmf IN

FFΒµ

±=−↔

( )

ξ+ξ

+

−+−

ξ+ξ

++

∆Ε+

2

1

2F2

1

2F

1I2

1m41

1I2

m41

h2 (II)

The result is an average frequency spacing of approx. 8 MHz mT-1 between

neighbouring Zeeman levels in the ground state 2S1/2 for 87Rb. 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 ∆EF = hf between the

Zeeman levels in the ground state of 87Rb amounts to 3 x 10-8 eV. For the

induced emission, the transition of the atom between two Zeeman levels with

∆mF = 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 ∆Eopt = 1.5 eV. By observing this

absorption directly, one thus achieves an amplification of

1.5/(3x10-8) = 5 x 107, which is obtained without incurring.

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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: Overview diagram of the entire experimental setup; position specifications are measured from the left edge of the optical riders

1 Rubidium high frequency lamp 2 Lens 3 Interference filter 4 Linear polarization filter 5 λ/4 plate 6 Lens 7 Helmholtz coil 8 Absorption chamber 9 Absorption cell 10 HF coils 11 Lens 12 Silicon photodetector

Fig. 3: Schematic representation of the experiment set-up

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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 D1 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 filter (4) and a

λ/4 plate (5) are used to separate the σ+ circularly-polarized component of

the D1 line from the emission spectrum of the rubidium lamp. The D1 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. 70 °C 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-

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polarized field; the component with the correct direction of rotation

(corresponding to ∆mF = ±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 2S1/2, F = 2, the Zeeman

level with mF = +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 ∆mF = -1 are possible, and a new

equilibrium settles in, where now the state 2S1/2, F = 2, mF = -2 is the

most populated. The circular polarization of the pumping light radiation

(σ+ or σ-) relative to the magnetic field orientation is decisive in

determining whether ∆mF = +1 or ∆mF = -1 transitions occur. Thus when

right-handed circularly-polarized light is used for irradiation parallel to

the magnetic field, it produces ∆mF = +1 transitions, but for irradiation

antiparallel to the field it causes ∆mF = -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

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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,mF) - E(F,mF-1) = hf0 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 mF = +2 to +1, +1 to 0, 0 to –1,

-1 to –2 and in the F = 1 state from mF = -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 population

difference between the Zeeman levels involved each time. When σ+ light is

used for pumping, the transition F = 2, mF = +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- light is

radiated into the absorption cell instead of σ+ light, or the magnetic

field direction of the Helmholtz coils is reversed while the polarization

is retained, the transition F = 2, mF = -2 to –1 appears with the greatest

intensity.

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Fig. 5 Absorption spectrum 87Rb with σ+ light

f) Zeeman splitting in 85Rb

A further special feature of the anomalous Zeeman effect is displayed in

the multiplicity of the Zeeman energy levels in isotopes with different

nuclear spins. In natural rubidium the isotopes 87Rb and 85Rb occur in the

ratio 72:28. While 87Rb has nuclear spin 3/2, 85Rb 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 87Rb and F = 3 and F = 2

for 85Rb. Due to the different nuclear g factors the hyperfine splitting is

6.8 x 109 Hz for 87Rb and 3.1 x 109 Hz for 85Rb.

This, however, means that for the same magnetic field the Zeeman splitting

is of different magnitude for the two isotopes. While the splitting for 87Rb in the ground state has an average value of 8 MHz for a magnetic field

of 1 mT, for 85Rb 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 87Rb split into 2F + 1 = 5 or

2F + 1 = 3 Zeeman levels respectively, while the splitting for 85Rb has 7

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or 5 levels respectively. Correspondingly 4 + 2 lines occur in the 87Rb

absorption spectrum (Fig. 5) and 6 + 4 lines in the 85Rb spectrum (Fig. 6).

Fig. 6 Absorption spectrum 85Rb 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

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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 Icoil ≈ 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.

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b) Observation and measurement of Zeeman splitting in 87Rb

i) Finding the absorption signal

▬ For 87Rb 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)

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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

(fA = 9.0 MHz, fE = 9.2 MHz)

▬ Switch the function generator to period 100 ms (fast sweep).


▬ Set the Helmholtz coil current I ≈ 0.791 A and vary it until an

absorption signal can be seen on the oscilloscope screen.

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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 xA = 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 horizontal deflection of the oscilloscope with the

channel I VOLTS/DIV. knob to xE = 9.0 scale divisions.

Fine adjustment:


▬ 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 fA and stop frequency fE 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.


▬ Wait until the absorption spectrum has been completely recorded.

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▬ 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 fA and the stop frequency fE.

▬ Check the Helmholtz coil current I.

Second run:

▬ Set the quarter wavelength panel to σ--polarization.


▬ 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 fA and the stop frequency fE.

▬ Check the Helmholtz coil current I.

▬ Tabulate x and U values in Table 1.

Setting:

Oscilloscope:


Storage mode

Trigger: Auto

Channel I: >500 mV/DIV. (DC)


Recording range: 1.0 division – 9.0 divisions

Timebase: 200 S/s

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Function generator:

Stop: 9.2 MHz

Start: 9.05 MHz

Period: 10 s (slow sweep)

Amplitude: 12½ o’clock

Attenuation: 20 dB

Measuring example:

Fig. 8 Absorption spectrum 87Rb with σ+ light (top) and σ- light (bottom).

IPhoto

0.2 µA 9.10 9.15

9.10 9.15 f [MHz]

f [MHz]

1 2

3

4

5

6

2

3 4 5 6

1

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No. x [Div] U [mV] No. x [Div] U [mV]

1 1

2 2

3 3

4 4

5 5

6 6

Table 1: Position x, amplitude U of the 87Rb absorption lines.

left side: Measurement with σ+ light, I = __________ A

right side: Measurement with σ- light, I = __________ A

iii) Evaluation

Determining the nuclear spin I of 87Rb

The irradiated alternating magnetic field makes possible transitions

with the selection rules ∆F = 0, ±1 and ∆mF = ±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 87Rb thus has the value I=3/2.

Determining the transition frequencies.

The position x on the oscilloscope screen allows us to determine the

frequencies fM of the absorption lines using the formula

( )AE

AAEAM xx

xxffff

−

−−+= (IV)

xA = Start of scale (set with Y-POS. I)

xE = End of scale (set with channel I VOLTS/DIV. knob)

fA = Start frequency

fE = Stop frequency

Summarize the results in Table 2 as the quantities fM(σ+) and fM(σ

-).

The standard deviation for these quantities is σ = 0.002 MHz.

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No. fM(σ+) [MHz] fM(σ-) [MHz] f [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

Table 2: 87Rb transition frequencies at B = __________ mT

Assignment of the transitions

The following numerical values can be found in the literature:

µN = 5.05083(4) x 10-27 JT-1 [9]

µB = 9.27410(7) x 10-24 JT-1 [9]

gJ = 2.002332(2) [6]

h = 6.6260(5) x 10-34 Js [9]

∆E/h = 6834.682614(1) MHz [8]

gI = -1.827231(2) [7]

Using (II), we obtain within the hyperfine state F = 2 four transitions

which have frequencies

( ) ( )( )

ξ+ξ−+−ξ+ξ++

×−==

212F

212F

2F

1m1m1MHz34.3417

mTB

MHz01393.0f

(V)

and within the hyperfine state F = 1 two transitions with the

frequencies

( ) ( )( )

ξ+ξ−+−ξ+ξ++

×==

212F

212F

1F

1m1m1MHz34.3417

mTB

MHz01393.0f

(VI)

where mTB

10)4(10243.4 3 ××=ξ −

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B ≈ 1.64 x I(A) mT

Start with the above approximate B value. Vary B in equations (V) and

(VI) until the measured transition frequencies are reproduced

mathematically. Record the value of B and compare the measured

transition frequencies fM(σ+) and fM(σ

-) with the calculated transition

frequencies f in Table 2.

When changing from σ+- to σ--pumping light, the amplitude U of the

absorption lines also change. Compare the intensities of the

transitions belonging to hyperfine state F = 2 and F = 1 in Table 3.

No. U(σ+)

[mv]

U(σ-)

[mV]

mF(1)↔mF(2) No. U(σ+)

[mV]

U(σ-)

[mV]

mF(1)↔mF(2)

1 2 ↔ 1 4 1 ↔ 0

2 1 ↔ 0 6 0 ↔-1

3 0 ↔-1

5 -1↔-2

Table 3: 87Rb transition intensities at B = __________ mT. left side: F = 2, right side: F = 1

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c) Observation and measurement of Zeeman splitting in 85Rb

Repeat all procedures in (b) with the following settings:

Helmholtz coil current I = 0.838 A

Oscilloscope:


Storage mode

Trigger: Auto

Channel I: >500 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:

Fig. 9 Absorption spectrum 85Rb with σ+ light (top)

and σ- light (bottom)

1

2

3 4

5 6 7 8

9 10

1 2 3

4 5

6

7

8

9

10

0.2 µA

6.40 6.45 6.50 f [MHz]

6.40 6.45 6.50 f [MHz]

IPhoto

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No. x [Div] U [mV] No. x [Div] U [mV]

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 10

Table 4: Position x, amplitude U of the 85Rb absorption lines.

left side: Measurement with σ+ light, I = __________ A

right side: Measurement with σ- light, I = __________ A

Assignment of the transitions

The following numerical values can be found in the literature:

µN = 5.05083(4) x 10-27

JT-1 [9]

µB = 9.27410(7) x 10-24

JT-1 [9]

gJ = 2.002332(2) [6]

h = 6.6260(5) x 10-34

Js [9]

∆E/h = 3085.732439(5) MHz [8]

gI = -5.39155(2) [7]

Using (II), we obtain within the hyperfine state F = 3 six transitions

which have frequencies

( )

ξ+ξ−+−

ξ+ξ++

×−==

212

F

212

F

3F

1m32

1m32

1MHz866.1542

mTB

MHz004199.0f

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and within the hyperfine state F = 2 four transitions with the

frequencies

( )

ξ+ξ−+−

ξ+ξ++

×==

212

F

212

F

2F

1m32

1m32

1MHz866.1542

mTB

MHz004199.0f

where mTB

10)9(08341.9 3 ××=ξ −

No. fM(σ+) [MHz] fM(σ-) [MHz] f [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

Table 5: 85Rb transition frequencies at B = __________ mT

No. U(σ+)

[mV]

U(σ-)

[mV]

mF(1)↔mF(2) No. U(σ+)

[mV]

U(σ-)

[mV]

mF(1)↔mF(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: 85Rb transition intensities at B = __________ mT.

left side: F = 3, right side: F = 2

of 24

Literature

[1] A. Kastler: Journal de Physique, 11 (1950) 255

[2] H.Kopfermann: Über optisches Pumpen an Gasen, Sitzungsberlchte der

Heidelberger Akademie der Wissensschaften, Jahrgang 1960, 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

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