Download Double resonance spectroscopy on the cesium atomic clock transition

Double resonance spectroscopy on the cesium atomic clock
transition
"
!
Thomas Olsen
Toke Lynæs Larsen
12th May 2004
Supervisors:
Carlos Leonardo Garrido Alzar
Eugene Polzik
QUANTOP Laboratory
Niels Bohr Insitute
University of Copenhagen
i
CONTENTS
Contents
1 Introduction
1.1 Why is this interesting? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Double resonance spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
2 Theory
2.1 Energy levels of cesium . . . . . . . . . . . . . .
2.1.1 Fine structure . . . . . . . . . . . . . . .
2.1.2 Hyperfine structure . . . . . . . . . . . .
2.1.3 Zeeman splitting . . . . . . . . . . . . .
2.2 Interaction with electromagnetic fields . . . . .
2.2.1 Electric dipole transitions . . . . . . . .
2.2.2 Magnetic dipole transitions . . . . . . .
2.2.3 Transition lines . . . . . . . . . . . . . .
2.3 Population rate equations for three level system
2.3.1 Short time behavior . . . . . . . . . . .
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2
2
3
3
4
6
7
7
8
9
10
3 The experiment
3.1 Setup . . . . . . . . . . . . .
3.1.1 Cells . . . . . . . . . .
3.1.2 Laser . . . . . . . . .
3.1.3 Microwaves . . . . . .
3.1.4 DC magnetic field . .
3.1.5 Imaging and detection
3.2 Measuring procedure . . . . .
3.3 Safety considerations . . . . .
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11
11
11
13
13
14
14
14
15
4 Results
4.1 Relaxation . . . . . . . . . . . . . . . . . .
4.1.1 Relaxation without laser . . . . . .
4.1.2 Relaxation with laser . . . . . . . .
4.1.3 The un-coated cell . . . . . . . . .
4.1.4 Concluding remarks on relaxation
4.2 The Zeeman effect . . . . . . . . . . . . .
4.2.1 Determining the magnetic field . .
4.2.2 Zeeman shift of the 0-0 transition .
4.3 Transition lines . . . . . . . . . . . . . . .
4.3.1 Accounting for reference spectrum
4.3.2 Transition rates . . . . . . . . . . .
4.4 Improvements . . . . . . . . . . . . . . . .
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16
16
17
19
21
21
22
23
25
26
27
27
29
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5 Conclusion
30
A Various physical constants
31
B Physical constants for cesium-137
32
C Microwaves
33
D Exact solution of two level problem
36
1 INTRODUCTION
1
1
Introduction
Direct measurements of frequencies in the microwave and radio regime is complicated by the fact
that all objects at room temperature emit black-body radiation in this region of the frequency
spectrum. This means that if one wants to examine a radio or microwave frequency related to some
physical event, it will be very difficult to separate it from the background radiation. There exists
different ways to bypass this problem and the one we shall consider, is called double resonance
spectroscopy. The basic idea behind this method is to couple a microwave or radio frequency to
an optical frequency which is easier to measure.
The ground state of cesium-133 is split into two hyperfine energy levels due to a non-zero
nuclear spin, and the frequency related to the transition between them is located in the microwave
region. We shall apply the method of double resonance spectroscopy to this transition.
Our objectives will be
• Sensitive measurements of magnetic fields using the Zeeman splitting of the transition lines.
• Determination of the Landé g-factor gJ by examining the Zeeman shift of the clock transition.
• Measuring transition rates between the hyperfine levels.
• Examining how spin collisions affect the spectra and calculate collisional rates.
1.1
Why is this interesting?
It is of course always interesting to derive atomic properties, but the examination of the hyperfine
transition in cesium is also important for other reasons. The current definition of the second is
based on this transition, as its frequency is defined as 9.192631770 GHz. When a magnetic field
is present the transition is split into several closely spaced transitions, but the frequency of one of
these will change very slowly with magnetic field and this can be used to measure the second.
The construction and working of cesium atomic clocks requires accurate knowledge of this
transition. We will not try to get an absolute measurement of the frequency, but rather examine
how the transition lines behave when a magnetic field is present. This is relevant, because an
atomic clock could take into account the non-zero magnetic field.
1.2
Double resonance spectroscopy
Application of the method was first reported in 1952 [1] after it had been proposed in 1949. The
principles behind double resonance spectroscopy applied to the ground state of cesium shall now
briefly be explained.
We have a small cell containing a dilute gas of cesium-133 at room temperature. To begin
with, the atoms are in thermal equilibrium and thus populate the two hyperfine energy levels
almost equally. A laser is used to excite the atoms of the upper ground state level. This is shown
schematically in Fig. 1, where also a numbering scheme for the levels is defined (the notation
will be explained in section 2.1.2). A small part of the excited atoms will decay by spontaneous
emission, emitting fluorescent light. The power of the fluorescence is proportional to the number
of F 0 = 5 atoms, and by measuring the amount of fluorescence, we can deduce the relative number
of atoms in the F 0 = 5 state.
After a short while, an equilibrium is reached, where only a small fraction of the total number
of atoms are in the F = 4 and F 0 = 5 levels, and thus only a low amount of fluorescence is
detected. If we now apply a microwave field resonant with the hyperfine transition, the atoms will
re-enter the F = 4 and thereby the F 0 = 5 state, and a larger amount of fluorescence is detected.
By scanning a range of microwave frequencies we can extract information about the resonances.
2
2 THEORY
F’=5
Fluorescence
Laser
F=4
Microwav e
F=3
Figure 1: Schematic representation of double resonance spectroscopy applied to the ground state
of cesium. The two levels of the ground state are the F = 3 and F = 4 levels.
2
Theory
We shall in this section present the theory needed to understand the experimental results. First we
consider an isolated cesium atom and describe its energy-levels. Then we see how these energylevels split up in a DC magnetic field and how a cesium atom responds when subjected to an
oscillating electromagnetic field. Finally we will make some model system describing the time
evolution of a collection of cesium atoms under the two radiation fields of double resonance spectroscopy.
2.1
Energy levels of cesium
Cesium is the 55th element and one of the alkali metals. Having a melting point of 28.44◦ C, it is
one of the lowest melting elemental metals. The only stable isotope is cesium-133, and we shall
solely be concerned with this.
In the ground state the electron-configuration of cesium is [Xe]6s, where [Xe] denotes the
ground state configuration of xenon 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 . Thus the atom consists of 54 electrons in closed shells (ideal-gas configuration) and one valence electron. As the
energy required to excite the inner (or core) electrons is much higher than that needed to excite
the valence electron, all low lying excited states will involve only changes in the configuration of
the single valence electron.
The total electron orbital angular momentum of the atom is represented by the operator L
and the total electron spin of the atom by S. As usual the different components of these operators
do not commute and thus cannot be determined simultaneously. A set of commuting operators
however, are their squares L2 , S 2 and their projections on an arbitrary quantization axis Lz , Sz
(the z-axis is chosen to coincide with this axis). These quantities are described by the quantum
numbers L, S, mL and mS respectively, defined so if |LSmL mS i denotes a simultaneous eigenstate
of these operators, the eigenvalue equations for L2 and Lz are
L2 |LSmL mS i = ~2 L(L + 1)|LSmL mS i,
Lz |LSmL mS i = ~mL |LSmL mS i
(1)
(2)
and similar for S 2 and Sz .
The quantity L (S) will in general be the sum of the orbital (spin) angular momentum for each
of the 55 electrons, but as the 54 core-electrons constitute closed shells, which have no orbital or
spin angular momentum, it reduces to that of the single valence electron. Thus S = 21 , because
the electron has spin s = 21 and L = 0, 1, 2, 3, . . . , n − 1, where n = 6, 7, 8, . . . is the principal
quantum number of the unpaired electron. n can only assume values starting from 6, because the
core electrons take up the first 5 shells.
The possible energies of the cesium-atom are determined as the eigenvalues of the Hamiltonian,
which we write as
Hatom = H0 + Hf s + Hhf s .
(3)
The three different terms will be described in the following.
3
2 THEORY
A simple approach is the central-field approximation, described by the Hamiltonian H0 , in
which the unpaired electron sees a potential, that is the Coulomb potential of the nucleus partly
shielded by the core electrons, and it is assumed, that this potential is spherical symmetrical.
Thus the Hamiltonian will commute with L and S, and the energy-eigenstates can be chosen to be
eigenstates of L2 , Lz , S 2 and Sz as well. The potential however is not a pure Coulomb potential
and the energy will depend on both n and L. For a fixed value of n and L there is a degeneracy
in all the possible values of mL and mS .
The ground state of cesium will have n = 6 and L = 0. The first excited state (which is the
only one we shall consider) has n = 6 and L = 1.
2.1.1
Fine structure
The fine structure, contained in the addition of Hf s , is
between the spin and orbital angular momentum of the
The effect of this is to shift the energy of all levels and
degeneracy.
The total electron angular momentum J of the atom
momentum
J = L + S.
reached by introducing an interaction
electrons as well as relativistic effects.
to split some, thus lifting some of the
is the sum of spin and orbital angular
(4)
The quantum numbers associated with this operator are J and mJ . The addition rules for angular
momentum dictates that J can assume the values
½ 1
L=0
2
J = L + S, . . . , |L − S| =
L ± 12 L 6= 0.
It is found, that the energy-levels are split according to their value of J. There is still a degeneracy
in the possible values of mJ .
The ground state has L = 0, so J = 21 and it will not be split. The first excited state with
L = 1 will be split into one state with J = 21 and one with J = 32 . In spectroscopic notation the
ground state is written 2 S1/2 , while the two fine structure levels J = 21 and J = 23 are written
2
P1/2 and 2 P3/2 respectively.1
2.1.2
Hyperfine structure
Many nuclei (including that of Cs-133) has a non-zero spin angular momentum I with associated
quantum numbers I and mI . The total angular momentum F of the atom is
F = I + J.
(5)
F = I + J, . . . , |I − J|.
(6)
The same addition rule applies as before
We can go one step further than the fine structure and introduce the hyperfine structure with the
extra term Hhf s . Now the interaction between the electron angular momentum and the nuclear
spin is taken into account. The fine structure levels split up further according to their value of F .
For each hyperfine level, there is still a degeneracy of 2F + 1 different values of mF .
Cesium-133 has a nuclear spin of I = 27 . The ground state is split in two hyperfine levels, one
with F = 27 − 21 = 3 and one with F = 72 + 12 = 4. The state 2 P1/2 is split in two levels with
F 0 = 3, 4, while the state 2 P3/2 is split in four levels with F 0 = 2, 3, 4, 5. We have here introduced
the notation F 0 to denote hyperfine levels of excited states.
We have not been concerned with the actual values of the energies in this discussion. One can
in general say, that the different terms in H are related to each other as H0 À Hf s À Hhf s . In
Fig. 2 the different fine and hyperfine levels of the ground and first excited states are shown along
with some of the energy differences.
1 The spectroscopic notation for an atom described by quantum numbers L, S and J is 2S+1 X . 2S + 1 is the
J
spin multiplicity or the possible values of mS and X is a (capital) letter corresponding to the value of L chosen in
the usual scheme (0 = S, 1 = P , 2 = D, 3 = F ,. . . )
4
2 THEORY
F’=5
251 MHz
F’=4
2
201 MHz
P3/2
F’= 3
151 MHz
F’= 2
352 THz
F’=4
1180 MHz
2
P1/2
F’=3
335 THz
F=4
2
S1/2
9193 M Hz
F=3
H0
Hfs
Hhfs
Figure 2: The fine structure and hyperfine structure of the ground state and first excited state in
cesium with some of the relevant energy differences. The vertical scale is not correct.
2.1.3
Zeeman splitting
The different mF sublevels of a given hyperfine level correspond to different orientations of angular
momentum in space. With no external fields applied, there is no preferred direction, and for a
given F there will be a degeneracy in the 2F + 1 values of mF . If a DC (static) magnetic field is
applied, a direction is defined, and the degeneracy is lifted. This is the so called Zeeman effect.
The different atomic angular momentums (L, S and I) each posses a magnetic dipole moment
proportional to itself. Thus the total magnetic dipole moment of the atom can be written as
µ=
µB
(gL L + gS S + gI I)
~
(7)
Where µB is the Bohr-magneton and the different g’s are the Landé g-factors of each of the
angular momenta. The electronic g-factors are to a first approximation gL = 1 and gS = 2, while
the nuclear g-factor of cesium is a few orders of magnitude smaller gI ≈ −0.0004. The interaction
between the atom and the DC magnetic field B is described by the interaction Hamiltonian
HB = −µ · B = −
µB
(gL Lz + gS Sz + gI Iz )B
~
(8)
where we have further set the z-axis to be parallel to the magnetic field so B = Bẑ.
If the magnetic field is sufficiently small (. 1000 Gauss), HB will be small compared to Hhf s
and HB can be treated as a perturbation to this. The perturbed eigenstates will be the same as
the unperturbed hyperfine states, but will be shifted in energy according to their value of mF as
∆EJIF mF = µB gF mF B = KF mF B,
(9)
where gF is the Lande-factor of the total angular momentum given by
gF = gJ
F (F + 1) − I(I + 1) + J(J + 1)
F (F + 1) + I(I + 1) − J(J + 1)
+ gI
2F (F + 1)
2F (F + 1)
(10)
5
2 THEORY
and gJ (the g-factor of J) is
gJ = gL
J(J + 1) + S(S + 1) − L(L + 1)
J(J + 1) − S(S + 1) + L(L + 1)
+ gS
.
2J(J + 1)
2J(J + 1)
(11)
Experimentally determined values of the g factors gI , gS , gL (independent of state) and gJ (depends
on state) are given in appendix B along with other relevant physical data for cesium.
If HB is of the same magnitude as Hhf s , one can seldom obtain an algebraic solution. However,
for the two levels of the ground state of cesium an exact solution, the Breit-Rabi formula, can be
found. The stationary states are still the same, but their energy now shift with respect to the
unperturbed hyperfine levels according to [8]
r
´
4mF
hν0 ³
1+
∆EF mF = gI µB BmF + ²
x + x2 − 1 ,
(12)
2
2I + 1
where hν0 are the energy-difference between the two (unperturbed) hyperfine levels,
x=
and
²=
½
(gJ − gI )µB B
hν0
(13)
+1 F = I + 1/2 = 4
−1 F = I − 1/2 = 3.
An expansion of Eq. (12) to first order in B will, with the use of Eq. (10), return exactly Eq.
(9) as required. The splitting of the different sublevels of the cesium ground state for varying
magnetic fields are shown in Fig. 3.
m F=4
m F=3
m F=2
m F=1
m F=0
15
m F=-1
m F=-2
Energy / h [GHz]
10
m F=-3
F=4
5
m F=-4
0
m F=-3
F=3
m F=-2
m F=-1
m F=0
m F=1
m F=2
m F=3
-5
0
500
1000
1500
2000
2500
3000
Magnetic field [gauss]
3500
4000
4500
5000
Figure 3: The energy of the different mF sublevels of the ground state of cesium as a function of
magnetic field. The zero of the vertical scale is arbitrarily set to the F = 3 level at no magnetic
field. The energy of the mF = −4 sublevel has been evaluated by another formula [8].
6
2 THEORY
2.2
Interaction with electromagnetic fields
Up to this point we have only considered isolated atoms and atoms subjected to a stationary
external magnetic field. We shall now expand the description to interactions with general electromagnetic fields.
An electromagnetic field can be described in terms of the electrostatic potential V (r, t) and the
vector potential A(r, t).2 When no free charges are present, it is convenient to use the Coulomb
gauge where we set ∇ · A = 0 and V = 0 [2]. The field is then described by the vector potential
alone.
The simplest propagating electromagnetic field is a monochromatic plane wave, which can be
described by the vector potential
A(r, t) = A cos(k · r − ωt + δ)ê,
(14)
where ê is the polarization vector, k the wavevector, ω the angular frequency, δ the phase and A
the amplitude. This will give rise to the electric and magnetic fields
∂A
= −ωA sin(k · r − ωt + δ)ê,
∂t
B(r, t) = ∇ × A = A sin(k · r − ωt + δ)k × ê.
E(r, t) = −
(15)
(16)
We see, that the electric field vector will be parallel to ê, while the magnetic field vector will be
orthogonal to ê and the direction of propagation.
A more general field can be described as a superposition of plane waves with different frequencies
Z ∞
A(r, t) = ê
A0 (ω) cos(k · r − ωt + δ(ω))dω,
(17)
0
where A0 (ω) the amplitude of A in the frequency domain and δ(ω) the phase corresponding to
A0 (ω). The electric and magnetic fields can still be found as in equations Eq. (15) and Eq. (16).
An electron subjected to a vector potential A, will to first order in A have the Hamiltonian [5]
Hint = −i~
e
A · ∇.
m
(18)
In most cases Hint will be much smaller than Ha tom + HB and can be treated as a perturbation.
We consider the probability of making a transition between two eigenstates of Ha tom + HB
under the perturbation. Using first order time-dependent perturbation theory [5] it is found, that
the transition rate from an initial state |φi i to a final state |φf i is
¯
¯2
πe2 2
A0 (ωf i )¯hφf | exp(ik · r)ê · ∇|φi i¯
2
2m
¯
¯2
πe2
= 2 2 ρ(ωf i )¯hφf | exp(ik · r)ê · ∇|φi i¯ ,
m ωf i ²0
Rf i =
(19)
(20)
where ωf i is the frequency corresponding to the energy difference between |φi i and |φf i and ²0 the
permeability of vacuum. ρ(ω) is the spectral density at frequency ω. It has units of J m−3 Hz −1
and is equal to I(ω)/c where I(ω) is the intensity at frequency ω.
The operator exp(ik·r)ê·∇ appearing in the inner product of Eq. (19) can be Taylor-expanded
exp(ik · r)ê · ∇ = (1 + (ik · r) +
1
(ik · r)2 + . . .)ê · ∇.
2!
(21)
When the wavelength of the electromagnetic radiation is much larger than the characteristic
distance of the atom (a few Å), we have k · r ¿ 1 and it will be a good approximation to consider
only the first non-vanishing term of Eq. (21).3
2 In
this section r is the position and −i~∇ the momentum of the valence electron.
of the terms can vanish when the inner product of Eq. (19)
3 Some
7
2 THEORY
2.2.1
Electric dipole transitions
If we retain only the leading term of Eq. (21), the matrix element appearing in Eq. (19) can be
rewritten as
mωf i
hφf |ê · D|φi i,
(22)
hφf |ê · ∇|φi i =
~e
where D = −er is the electric dipole operator, so this approximation is known as the electric
dipole approximation. This matrix element can be simplified using the Wigner-Eckart theorem
[3], but we will not go into detail with this, since it would require us to introduce the concept of
irreducible tensor operators. We simply state that the matrix element will be proportional to a
Clebsch-Gordan coefficient. This coefficient will be zero unless the difference in orbital angular
momentum associated with the initial and final states fulfill4
∆L = ±1
(23)
∆mL = 0, ±1.
(24)
Transitions with ∆mL = 0 is induced when the polarization vector is parallel with the axis of
quantization and these are called π-transitions. Transitions with ∆mL = ±1 is induced when the
polarization vector is perpendicular to the quantization axis and these are called σ-transitions.
When considering the transition between the F = 4 level and the F 0 = 5 level, the electric
dipole approximation will be relevant. The laser we use to excite the atoms has a polarization
vector parallel to the quantization axis, and the magnetic quantum number mF will not change
in the transitions induced by the laser.
2.2.2
Magnetic dipole transitions
If the matrix element in Eq. (22) vanishes, we have to include the next term in Eq. (21). Using
only the second term of Eq. (21), the matrix element to be evaluated is
hφf |i(k · r)(ê · ∇)|φi i.
(25)
This can be split into two terms. One representing the so called magnetic dipole approximation
and one representing the electric quadrupole approximation. We will only be concerned with
magnetic dipole transitions. When this kind of transitions are considered, we will employ the
interaction Hamiltonian
0
= −µ · B,
(26)
Hint
where µ is the total atomic dipole moment of Eq. (7) and B the magnetic field of Eq. (16).
Again using the Wigner-Eckart theorem, one can obtain the selection rules for magnetic dipole
transitions
∆F = 0, ±1
∆mF = 0, ±1.
(27)
(28)
We now consider transitions between the two hyperfine levels of the ground state of cesium.
These states both have L = 0, so transition between them are not electric dipole allowed. The
magnetic dipole approximation will then be relevant. We consider a transition from |F = 3 mF i
to |F = 4 m̃F i with the transition rate
Rm̃F ,mF =
¯
¯2
πe2
ρ(ωf i )¯hF = 4 m̃F |(k · r)ê · ∇|F = 3 mF i¯ .
m2 ωf2 i ²0
(29)
By applying the Wigner-Eckart theorem and using the magnetic dipole approximation, the matrix
element can be shown to be proportional to the square of a Clebsch-Gordan coefficient, so
Rm̃F ,mF ∝ |h4 m̃F |3 1 mF qi|2 ,
(30)
4 ∆L = 0 cannot be excluded using the Wigner-Eckart theorem but the matrix element will vanish using the
parity selection rule.
8
2 THEORY
where q depends on the direction on of the magnetic field. If the magnetic field vector is parallel
to the axis of quantization, q = 0 and we get π transitions (∆mF = 0). If the magnetic field
is perpendicular to the quantization axis, q = ±1 and we have σ transitions (∆mF = ±1). In
general the magnetic field may have components in both directions, and thus excite both type
of transitions. However the two components need not have the same magnitude, and a factor
depending of this will enter Eq. (30).
For a given type of transition (π or σ) the Clebsch-Gordan coefficient contains all information
about the transition rate relative to some arbitrary fixed rate, since the other factors appearing
in equation Eq. (29) are common to all transitions of this type.5 For example
2.2.3
¯
¯
¯ h4 2|3 1 2 0i ¯2
RmF =2;mF =2
¯
¯
=¯
¯ .
¯
RmF =0;mF =0
h4 0|3 1 0 0i ¯
(31)
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Transition lines
When we do the actual measurement we will be interested in the different transitions between the
two hyperfine levels of the ground state. These are shown in Fig. 4.
Figure 4: The different possible transitions between the two hyperfine levels of the ground state.
The red lines are π-transitions with ∆mF = 0 and the blue lines are σ-transitions with ∆mF = ±1.
The splitting between sublevels is greatly exaggerated compared to the distance between the two
hyperfine levels.
To first order in B the energy difference between neighboring mF sublevels is by Eq. (9)
proportional to B with a constant of proportionality KF . The values of K3 and K4 we shall use
are [10]
(32)
K3 = 350.98 kHz Gauss−1
−1
K4 = 349.86 kHz Gauss
.
(33)
From these numbers and Fig. 4 it is seen, that 12 of the 14 σ-transitions will two by two have
almost equal frequencies. When we observe the σ-transitions in the experiment, the frequency
difference between such a pair is less than our frequency resolution, and we effectively only observe
14 − 12/2 = 8 σ-transitions. The K’s can be treated as just having one value, the mean of the two
K = 350.42 kHz Gauss−1 .
(34)
5 A (ω ) is assumed to have the same value for all transitions. Strictly this will not be true when a DC magnetic
0
fi
field is applied, since there will be a zeeman splitting of energy levels and ωf i will take slightly different values for
different transitions. However A0 (ω) will vary slowly in the frequency region of interest, and we will consider it as
constant.
9
2 THEORY
With 7 π-transitions and 8 σ-transitions, we thus expect to observe a total of 15 transition lines.
A indexing scheme for these has been introduced in Table 1 where we have also stated the square
of the Clebsch-Gordan coefficients of Eq. (30) associated with each transition. As is evident from
Fig. 4, the frequency difference between neighboring lines will be ∆ν = KB.
Index
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Transition
L−4,−3
L−3,−3
L−3,−2 , L−2,−3
L−2,−2
L−2,−1 , L−1,−2
L−1,−1
L0,−1 , L−1,0
L0,0
L1,0 , L0,1
L1,1
L2,1 , L1,2
L2,2
L3,2 , L2,3
L3,3
L4,3
Type
σ
π
σ
π
σ
π
σ
π
σ
π
σ
π
σ
π
σ
2
CG
1
1
4
3
1
4 , 28
3
7
3
15
28 , 28
15
28
3
5
14 , 14
4
7
3
5
14 , 14
15
28
15
3
28 , 28
3
7
1
3
4 , 28
1
4
1
Table 1: Transition lines. First column defines the indices of the transition lines, second column
describes which mf sublevels are involved in the transition, third column is the type of transition
(π or σ) and fourth column states the square of the Clebsch-Gordan coefficient associated with
the transition. We denote the transition (F = 3, mF ) → (F = 4, m0F ) by Lm0F ,mF .
The clock transition is the mF = 0 ↔ mF = 0 transition and its frequency with no magnetic
field is defined as
ν0 = 9.192631770 GHz.
(35)
We shall also refer to it as the 0-0 transition.
The mF = 0 sublevels will not be shifted to first order in B, which is also seen from Fig. 3.
From the Breit-Rabi formula Eq. (12) with mF = 0 we get the energy shift of the clock transition
to second order in B
(gJ − gI )2 µ2B B 2
= K0 B 2 ,
(36)
∆E (2) =
2hν0
with [10]
K0 = 0.42745 kHz Gauss−2 .
2.3
(37)
Population rate equations for three level system
We want to make a model describing the behavior of our system subjected to the microwave and
the laser fields. A full quantum mechanical treatment is reserved for masochists, and we shall not
be concerned with this.6 We shall treat the different sublevels of a given F level as one, and only
be concerned with the population of this level. We thus have three different levels F = 3, F = 4
and F 0 = 5 and the fraction of atom in each of these states is denoted f3 , f4 and f5 respectively.
We assume that atoms can change state in 4 different ways: Absorbtion, stimulated emission,
spontaneous emission and by collision. The rate at which atoms escape a given state, is proportional to the population of that state. The time evolution of the population of the three different
6 The method that could be used is stated in Appendix D, where the two level problem is solved exactly. This
is not the whole story though, as coherences will play a major role when considering populations of atoms.
10
2 THEORY
levels are assumed described by the equations
¡
¢
f˙3 (t) = R34 f4 (t) − f3 (t) + Γ34 f4 (t) − Γ43 f3 (t) + C35 f5 (t),
¡
¢
¡
¢
f˙4 (t) = R45 f5 (t) − f4 (t) + R34 f3 (t) − f4 (t) + A45 f5 (t) − Γ34 f4 (t) + Γ43 f3 (t),
¡
¢
f˙5 (t) = R45 f4 (t) − f5 (t) − A45 f5 (t) − C35 f5 (t).
(38)
(39)
(40)
These are the population rate equations. The R’s denote the rate of absorbtion and stimulated
emission, A (the Einstein A coefficient) is the rate of spontaneous emission and the Γ’s are the
rate atoms change state by collisions. The C is the rate at which atoms are transferred from the
5 state to the 3 state and we shall treat is a a phenomenological constant and not be concerned
with its cause. The transition is electrical dipole forbidden, and the energy involved is too large
to involve a direct collisional decay.
We have not included spontaneous emission of the 4 and 5 levels to the 3 level, since the rates
are very small compared to the other terms appearing in the equations. Furthermore, collisions
that bring atoms to or from the 5 state can be neglected, because atoms at room temperature
does not carry enough kinetic energy to bring about the energy change.
The R’s are given theoretically by Eq. (19), but we shall not use this formula to calculate
them. Sometimes the Einstein B-coefficient is used instead of R. It is defined as
Bf i =
Rf i
ρ(ωf i )
(41)
and is a property of the energy levels alone. It can be shown [4] to relate to A as follows
Aif =
~ωf3 i
Bf i
π 2 c3
(42)
where ωf i is again the angular frequency associated with the transition.
The total decay rate of the 5 state is given by [8]. We assume it to be equal to A45
A45 = 32.815 · 106 s−1 .
(43)
If no laser field is present the characteristic lifetime of the 5 state will be
T5 = A−1
45 = 30 ns.
(44)
From Eq. (42) we can estimate the Einstein B coefficient to
B45 = 8 · 1021 m3 J −1 s−2
(45)
In the experiment, we use a laser with a (center) intensity of I ≈ 130 mW cm−2 and a bandwidth
of 5 M Hz corresponding to an averaged spectral density of
ρ45 ≈
2.3.1
130 mW cm−2
≈ 8.7 · 10−13 J m−3 Hz −1
c 5 M Hz
(46)
Short time behavior
Although the equations are not to hard to solve, the exact solution will not be particularly informative. Instead we shall make an approximation.
The terms containing R45 and A45 in the population rate equations describe the coupling
between levels 4 and 5 and are typically much larger than the rest of the terms. So at very short
timescales we effectively have the two level system
¡
¢
(47)
f˙4 (t) = R45 f5 (t) − f4 (t) + A45 f5 (t),
¡
¢
f˙5 (t) = R45 f4 (t) − f5 (t) − A45 f5 (t),
(48)
11
3 THE EXPERIMENT
to which the solution is
³
f5 (t) = f5 (0) −
´
R45
R45
e−(A45 +2R45 )t +
.
A45 + 2R45
A45 + 2R45
(49)
We see that it approaches equilibrium with a characteristic time
Teq,45 =
1
1
=
≈ 0.2 ns.
A45 + 2R45
A45 + 2B45 ρ45
(50)
A transition is called saturated if the two involved states are equally populated when the
system reaches equilibrium. The fraction in the lower state is always a bit greater than in the
upper state, since spontaneous emission removes atoms from the upper level and absorbtion and
stimulated emission work at the same rate. For practical purposes though we can say the states
F = 4 and F 0 = 5 are equally populated if the spectral density of the laser exceeds the value [4]
ρ45,sat =
A45
≈ 2 · 10−15 J m−3 Hz −1 ,
2B45
(51)
which is well below the spectral density we use. In short terms this means, that when equilibrium
is reached between the states 4 and 5 after a period of 0.2 ns, the two levels will be equally
populated. Thus when we are applying the laser, the two upper states can effectively be regarded
as one state on the timescale of the experiment (typically ms). Furthermore, when we switch off
the laser, the 5 population will decay with a characteristic lifetime of 30 ns. This is also very
short on our timescale, so at any relevant times, the 5 state will be empty, and we have another
two level system.
3
The experiment
This section concerns the actual experiment. First we describe the setup and its different parts
and then we describe how the measuring was conducted. At the end is also a short comment about
the possible security risks involved.
3.1
Setup
A sketch of the experimental setup is shown in Fig. 5 and in the following we shall briefly explain
the different components.
3.1.1
Cells
We used two cells (A) during the experiment, a coated cell and an un-coated cell. Both cells
contained a low pressure cesium gas. Small amounts of solid cesium were deposited in both cells,
so the gas could be assumed saturated. From [8] we have, that the vapor pressure of cesium at
25◦ C is 1.3 · 10−6 torr.
The walls were in both cases quartz, but the coated cell had a thin layer of paraffin on the
inside. The un-coated cell was formed as a cylinder with a length of 3.5 cm and a diameter of
2.1 cm. When we used this cell, the laser beam was directed along its axis of symmetry. The
coated cell was cubic with a side length of 3.6 cm, rounded corners and circular faces with diameter
2.2 cm. When we used this cell, the beam was sent through the centers of two opposing faces.
The un-coated cell was only used for comparison, and all double resonance spectroscopy measurements were conducted using the coated cell. A picture of the two cells is shown in Fig. 6.
12
3 THE EXPERIMENT
?
6= 5
>
0
4
>
3
1
1
1
<
<
27
89:
;
8
Figure 5: A sketch of the experimental setup. The components are: A: Cell, B: Laser generation
device, C: λ/2-plate, D: Frequency generator, E: Antenna, F: Helmholtz coils, G: DC power supply,
H: Lens, I: Photodetector, J: Oscilloscope.
Figure 6: The two cells used in the experiment. To the left the un-coated cell and to the right the
coated cell. On the walls of the un-coated cell the deposited cesium can be seen as impurities and
in the coated cell, the solid cesium is seen in the "arm" to the bottom right. The scales of the two
images are not the same.
3 THE EXPERIMENT
3.1.2
13
Laser
To produce the laser beam we used a tunable diode laser, which had the property that the frequency
could be controlled. Saturation absorbtion spectroscopy was used to lock the frequency at the
transition F = 4 ↔ F 0 = 5. The tuning was achieved at another setup in the laboratory (B)
and we used a beam-splitter and a couple of mirrors to guide the laser beam to our experiment.
The beam we received was already plane polarized in some arbitrary direction and by sending
it through a properly adjusted λ/2-plate (C), we made the polarization horizontal. Thus the
polarization vector was parallel to the DC magnetic field and only (or mainly) π-transitions would
be induced.
The power of the laser was measured to 1.8 mW and its diameter to 3 mm. The intensity can
then be estimated to 130 mW cm−2 . The bandwidth, was 5 M Hz, and the spectral density can
be estimated to 8.7 · 10−13 J m−3 Hz −1 . 7 .
3.1.3
Microwaves
The frequency generator, a HP 8341B (D), could provide an AC output at very high frequencies
ranging from 10 M Hz to 20 GHz with a bandwidth less than one Hz. It was connected to the
microwave antenna (E) which created electromagnetic waves at the frequency of incoming current.
The HP 8341B could be set to scan a range of frequencies in a specified time interval, the scan
was to a high degree of precision linear in time. The power output was adjusted between −2 dBm
and 15 dBm. 8
The antenna is made of cobber and consists of a box cavity that is open at one end, a horn
mounted at the open end of the cavity and an exciting probe that is connected to the frequency
generator. The box cavity had the three side lengths, a = 22.86 mm, b = 10.16 mm and c =
34.76 mm and electromagnetic waves propagated from the open end (side lengths a and b). Two
pictures of the antenna are shown in Fig. 7.
Figure 7: The microwave antenna. Seen from the front (left) and from the side (right).
We mounted the antenna slightly above the setup in such a way, that it was pointed at the
cell. Furthermore we made sure, that the component of the magnetic field orthogonal to the the
direction of propagation, was parallel to the DC magnetic field. The magnetic field also had a
component along the direction of propagation, so both π and σ-transitions was induced. The form
of the propagating waves is derived in Appendix C.
7 The beam intensity was gaussian distributed with respect to both space and frequency. The intensity is
estimated by calculating the amplitude of a two-dimensional Gauss-function when the width and integrated value
(power) is known. The average spectral density is estimated by dividing by the bandwidth and velocity of light.
8 The unit dBm is defined so, that if x is the power in dBm and P the power in ordinary units (W for instance),
then
³ P ´
x
.
(52)
= 10log10
dBm
1 mW
14
3 THE EXPERIMENT
3.1.4
DC magnetic field
We used two identical circular coils (F) placed with their common axis of symmetry passing
through the cell and the detection equipment. The number of windings in each coil was 150
and the radius of the coils was 7.2 cm. The coils were placed in the Helmholtz configuration, in
which the separation of their centers is equal to their radius. By running a DC current supplied
by a DC-generator (G) through the two coils in the same direction, a stationary magnetic field
proportional to the current was generated. We used currents between 0 A and 1 A and a magnetic
field between 0 Gauss and 22 Gauss resulted. At the position of the cell the magnetic field was
very homogenous and directed along the axis of symmetry.
3.1.5
Imaging and detection
We detected fluorescence from the cell perpendicular to the laser beam. We used a lens (H) to
create an image of the cell (A) on the sensor of the photodetector (I). We could have tried to focus
the light at a point on the sensor, but as this would not provide sufficient sensitivity, it was more
reliable to spread the fluorescence on a larger part of the sensor.
The photodetector generated a voltage that was proportional to the power of light incident on
its sensor. We connected the photodetector to an oscilloscope (J), which recorded and displayed
the generated voltage versus time.
The linear dimensions of both cells were lo ≈ 2 cm, and we wanted to image that on the
quadratic sensor with side length li ≈ 0.5 cm. The lens we used had a focal length of f = 6 cm
and a diameter of 7 cm. We needed to calculate the distance from the cell to the lens so , and the
distance from the lens to the sensor si , that would give us the right magnification. The distances
are shown in Fig. 8. From [6] we took the following expression for the magnification
M=
si − f
f
li
=
=
,
lo
f
so − f
(53)
from which the distances could be found. We got si ≈ 7 cm and so ≈ 32 cm. When we set up
the photodetector, we used these distances to get an idea where the lens and the sensor should be
located. The best place for detection was then found by moving the photodetector around a bit
until we found the position where the strongest signal was received.
Figure 8: The distances involved in the imaging of the cell on the sensor. The cell is located to the
left, has a linear dimension of lo and a distance to the lens of s0 . The sensor of the photodetector
is located to the right, has linear dimension li and a distance to the lens of si . The lens has focal
lenght f .
3.2
Measuring procedure
The laser was controlled by a LabVIEW program. By acousto-optical modulation the laser beam
could be divided into pulses. From the computer we controlled four parameters, which are defined
in Fig. 9.
In all measurements we had, that the number of pulses was 10, the total width 510 ms and
the repeat period 50, 000.00 µs. The only parameter we actually varied was the pulse width. In
F
B
L
A
@
L
AN
@
B
L
A
@@
@
AB@
AC@
AD
AE
ABF
ACF
AD
G
HI
KL
MFF
J
3 THE EXPERIMENT
15
Figure 9: The four adjustable parameters controlling the laser pulse. The pulse width (0.2 s on
the figure) is the duration of each individual laser pulse and the repeat period (0.3 s on the figure)
is the period between these pulses. A number of pulses (3 on the figure) are collected in a group,
which has some duration (1.2 s on the figure). The vertical axis is the laser intensity and the lower
level corresponds to the laser being off.
some cases we wanted a continuous laser pulse for 500 ms and this was achieved by setting the
pulse width to 49, 999.90 µs.
The oscilloscope, frequency generator and laser were triggered by the same program in such
a way that the frequency scan started simultaneously with the laser. In all experiments where
microwaves was involved, the scanning time of the frequency generator was 400 ms.
When the oscilloscope received the trigger signal, it started recording the signal from the
photodetector. A single recording by the oscilloscope will be denoted a sweep, and such a sweep
could be obtained every 510 ms. As the data is quite noisy, we averaged a number of sweeps to
get a smoother set of data. The oscilloscope was set to automatically average a given number of
sweeps. All data presented below are averages of 20 sweeps unless otherwise stated and from now
on we will simply denote an average of 20 sweeps, a "measurement". A typical measurement is
shown in Fig. 10 along with a single sweep.
The measurements had to be done in darkness since the photodetector was very sensible to
background light. We were not able to achieve complete darkness and we got a small background
signal, but this could easily be subtracted when we processed the data.
We used a computer to collect the measurements from the oscilloscope along with the latest
sweep.
3.3
Safety considerations
Laser The laser we used was of such an intensity, that it might damage the human eye upon
exposure. All free propagating lasers was kept at about 10 cm above table level (well below
eye-level), so you would have to be rather unlucky to damage an eye.
Microwave At most, the microwave was run with a power of 30 mW , which in no way posed
any health threat, especially since the output was diverging.
Cesium As cesium is both less volatile and less poisonous than mercury and only present in very
small amounts, the largest danger connected with the breaking of one of the cells would have
been to collect the splinters.
Thus we concluded, that the experiment was relative harmless.
16
4 RESULTS
0.1
Single sweep
Average of 20 sweeps
0.08
Voltage [mV]
0.06
0.04
0.02
0
-0.02
-0.05
0
0.05
0.1
0.15
0.2
0.25
Time [s]
0.3
0.35
0.4
0.45
0.5
Figure 10: A typical sweep and an average of 20 sweeps denoted "a measurement". The average
has been shifted up by an amount of 0.03 mV for clarity. This spectrum was recorded with a DC
current of 40 mA, a microwave power of 15 dBm and a pulse width 49, 999.90 µs, corresponding
to continuous laser radiation for 500 ms. The average is seen to reduce the noise significantly.
4
Results
In this section we shall state and describe the results of the experiment along with some of the
considerations we made concerning the derivations. First we examine how the atoms evolve in
time when microwaves are not present. This will give us some basic knowledge of the behavior
of the atoms and we will be able to determine collisional rates. We then examine the Zeeman
effect by measuring how the atomic resonances respond when we change the DC current and thus
the DC magnetic field. This will provide us with a very precise method of determining magnetic
fields, and from this the Landé factor gJ can be derived. Finally the relative transition rates for
the hyperfine transition are estimated by measuring how the height of the resonance lines vary
with the microwave power.
All data are processed and presented with the program gnuplot.
4.1
Relaxation
We examine how the atoms of the coated cell responds when we apply only the laser and how they
behave when no external radiation at all is present. This will provide us with some information
of the effect of the collisional processes. A single measurement on the un-coated cell is done for
comparison.
With the term "relaxation" we will understand events at which atoms change state by processes
not related to radiation. This is mostly due to atoms colliding with the walls of the cell. Atoms
colliding with each other will not contribute much, since we are working with a very dilute cesium
gas. The coating of the coated cell is there to prevent atoms making a spin-flip (change of state)
when they collide with the cell walls.
Whether or not the laser is on, an equilibrium will be reached between the three levels. The
equilibrium composition will depend on the coating of the cell and on whether the laser is on or
17
4 RESULTS
off, but not on the power of the laser, as long as it is above saturation level.
As argued in section 2.3.1, saturation of the laser transition will happen much faster (less than
ns) than all other processes. Thus when we apply the laser, half of the atoms that was previously
in the F = 4 state, will instantaneously populate F 0 = 5. When the laser is turned off again, the
atoms in the state F 0 = 5 will decay to F = 4 in a matter of 30 ns, which is still much faster than
the other processes. Thus we can treat the system by a two-level formalism at all relevant times,
since F = 4 and F 0 = 5 are equally populated when the laser is on and F 0 = 5 is empty when the
laser is off.
The voltage recorded by the oscilloscope is proportional to the power received by the photodetector. The power will be composed of the fluorescence emitted by the atoms and some constant
background radiation due to other light sources in the room. The background is easily subtracted
as it can be found as the voltage, when the laser is turned off. The remaining signal is due solely to
the fluorescence and will be proportional to the fluorescence power. The constant of proportionality is the product of the detector efficiency and a geometric factor that depends on the setup. As
the fluorescence only spans a narrow range of frequencies, the energy associated to each photon
is nearly constant, and the fluorescence power is proportional to the number of atoms making a
transition from the F 0 = 5 level, which in turn is proportional to the fraction of atoms in this
level. We can thus write the voltage U at time t as
U (t) = Ef5 (t) + U0
(54)
where E is a collection of constants, f5 (t) is the fraction of the total number of atoms that are in
the F 0 = 5 state and U0 is a constant term describing the background radiation. U0 might vary
from measurement to measurement, depending on whether or not a lamp in the laboratory was
on or off, but it is assumed constant during one measurement.
When no external field is present, atoms will tend to statistical (thermal) equilibrium, where
the fraction of atoms in a given state with energy E is proportional to the Boltzmann factor
exp(−E/kb T )[12]. The ratio of atoms in two different states will be
N1
= exp(−hν/kb T ),
N2
(55)
where ν is the frequency corresponding to the energy difference. The population ratio between
a sublevel of F = 3 and a sublevel of F = 4, will have a value of 1.0015 at room temperature
(300 K), when we use ν = 9.2 Ghz. For our purpose the two sublevels are practical equally
populated. In contrast, optical excited levels are nearly empty, since Eq. (55) will give a value
of 10−25 if we take ν to be in the infrared (850 nm). At statistical equilibrium all atoms thus
populate the ground state. Due to the different degeneracies of the two hyperfine levels (9 for
F = 4 and 7 for F = 3), they are not equally populated, but contain the following fractions of the
total number of atoms
7
7
=
,
7+9
16
9
9
f4 =
=
.
7+9
16
f3 =
4.1.1
(56)
(57)
Relaxation without laser
We shall examine the approach to statistical equilibrium of the atoms of the coated cell in the
absence of the laser. We would like to have the atoms in their equilibrium distribution with the
laser on, then turn the laser off and measure how the populations of atoms evolve in time. This
is complicated by the fact, that without the laser on we cannot "see" anything. So instead we
shall turn off the laser for just a short period, the dark period τ , and then read off the voltage
immediately after it is turned on again.
By varying the laser pulse width we effectively get to vary the dark period, and the desired
time evolution can be collected from several measurements. A typical spectrum achieved in this
18
4 RESULTS
P
Y
QP
X
QP
W
Q
P
V
Q
d[
P
U
c
Q
b][
à
^_
P
T
\
QP
S
QP
U
P
R
P
g
j
g
j
R
QP
OP
QOR
P
P
P
P
P
P
U
P
P
P
P
P
P
QR
QR
QS
QT
Qe
QW
QX
QY
QZ
QR
fg
ij
kQV
h
way is shown in Fig. 11. The first peak (as is seen from the figure) will usually be higher than
the following, because the dark period there has been 10 ms longer, as there are 10 ms between
each group of 10 pulses.
Figure 11: A typical spectrum to measure the approach to statistical equilibrium. Here the dark
period is τ = 10 ms (marked on the graph). The sum of the pulse width (40 ms) and the
dark period (10 ms) is the repeat period (50 ms). The DC current during the measurement was
500 mA.
With a DC current of 500 mA we made several of such spectra for values of the dark period
ranging from 1 ms to 40 ms. The voltages right after the dark periods was then read off the
graphs. In Fig. 12 these voltages are plotted versus τ .
When the laser is turned off all atoms in F 0 = 5 will immediately decay to F = 4 and we have
an effective two level system with levels F = 3 and F = 4. The population rate equations Eq.
(38)-(40) reduce to
f˙4 (t) = Γ43 f3 (t) − Γ34 f4 (t) = Γ43 (1 − f4 (t)) − Γ34 f4 (t).
(58)
This equation has the solution
f4 (t) =
³
´
Γ43
Γ43
+ f4 (0) −
e−(Γ34 +Γ43 )t ,
Γ43 + Γ34
Γ43 + Γ34
(59)
where f4 (0) is the fraction of atoms in F = 4 at time t = 0. This function will approach the value
Γ43
Γ43 +Γ34 for t → ∞. Since this corresponds to statistical equilibrium, where the two states are
9
43
populated according to Eq. (56) and Eq. (57), we must have that Γ43Γ+Γ
and
= 16
34
f4 (t) =
³
9
9 ´ − 16 at
+ f4 (0) −
e 9 .
16
16
(60)
Inspired by this we fit a general exponential of the form
U (t) = Ae−Bt + C
(61)
19
4 RESULTS
0.1
Measurements
Fit
0.09
0.08
Voltage [mV]
0.07
0.06
0.05
0.04
0.03
0.02
0
5
10
15
20
25
Time [ms]
30
35
40
45
Figure 12: Approach to statistical equilibrium τ . The measured voltage at the end of a dark
period as a function of the duration τ of the dark period.
to the "dark" data displayed in Fig. 12. The fit is also displayed in the figure, and the correspondence between the fit and the experimental data, are seen to be excellent. The values of the
parameters A, B and C are
A = (−0.0834 ± 0.0018) mV,
B = (0.0437 ± 0.0017) ms−1 ,
C = (0.1049 ± 0.0019) mV. (62)
The voltage we detect right after probing the system with the laser corresponds to half of the
atoms previously in the F = 4 being in the F 0 = 5. By inserting one half of f4 (t) given by Eq.
(60) as f5 (t) in the expression Eq. (54) for the voltage and comparing with Eq. (61), we find that
16
(C − U0 ) = (0.369 ± 0.007) mV
9
9
Γ43 =
B = (24.6 ± 0.9) Hz
16
7
Γ34 =
B = (19.1 ± 0.7) Hz
16
´
9³ A
+ 1 = 0.111 ± 0.002
f4 (0) =
16 C − U0
E=2
(63)
(64)
(65)
(66)
where the estimated value U0 = (0.0010 ± 0.0001)mV has been used for the background radiation.
The characteristic time of the decay is then
Tno laser,coated =
4.1.2
1
= (22.9 ± 0.9)ms.
B
(67)
Relaxation with laser
We shall now investigate the approach to equilibrium with the laser on. This is much more
straightforward than the examination in the previous section, since we can follow the time evolution
of the F 0 = 5 population directly.
20
4 RESULTS
0.055
Measurement
Fit
0.05
Voltage [mV]
0.045
0.04
0.035
0.03
0.025
0.02
0
2
4
6
8
10
Time [ms]
Figure 13: Decay to equilibrium with laser. A general exponential, which has been fitted to the
data, is also displayed.
Running a DC current of 500 mA and with a continuous laser pulse we recorded a spectrum
of the first 10ms after the laser was applied. This is shown in Fig. 13.
Without the microwave field the time evolution of f3 reduces from that stated in Eq. (38) to
f˙3 (t) = Γ34 f4 (t) + C35 f5 (t) − Γ43 f3 (t).
(68)
By using the saturation assumption f4 = f5 and the fact that the sum of all three fractions is one,
we arrive at the following solution for f5 (t)
³
f5 (t) = f5 (0) −
´ 1
Γ43
Γ43
e− 2 (Γ34 +C35 +2Γ43 )t +
.
Γ34 + C35 + 2Γ43
Γ34 + C35 + 2Γ43
(69)
We fit a function of the form
(70)
U (t) = Ae−Bt + C
to the measurement and get the parameters
A = (0.02869±0.00002) mV,
B = (0.3100±0.0006) ms−1 ,
C = (0.02030±0.00002) mV. (71)
The fit is also displayed in Fig. 13 and it closely resembles the measurements. Comparing B with
the time constant of Eq. (69) and using previously determined values of Γ34 and Γ43 we find
C35 = (552 ± 1) Hz,
(72)
and the characteristic time of this decay is
Tlaser,coated =
1
= (3.23 ± 0.01)ms.
B
(73)
21
4 RESULTS
4.1.3
The un-coated cell
The previous measurements were all done on the coated cell. For reference we performed the same
measurements on the un-coated cell. We observed, that thermal equilibrium was already reached
with a dark period of 1 ms. As this was the lowest dark period we used, we are not able to
determine the collisional rate constants associated with the approach to statistical equilibrium.
We recorded a spectrum of the approach to equilibrium when the laser was on. As the voltage
change very little compared to the noise we needed to average 100 sweeps to get a smooth curve.
The result is plotted in Fig. 14 and the fit of a general exponential to the data is also included.
This figure can be compared to Fig. 13. From the fit, we find that the characteristic time of the
0.057
Measurement
Fit
0.056
0.055
Voltage [mV]
0.054
0.053
0.052
0.051
0.05
0.049
0
0.01
0.02
0.03
0.04
Time [ms]
0.05
0.06
0.07
Figure 14: Decay to equilibrium with laser for the atoms of the un-coated cell. Also shown is the
general exponential fitted to the measurement.
decay is
Tlaser,uncoated ≈ 20µs.
(74)
This is about 100 times faster than the corresponding time for the atoms of the coated cell.
4.1.4
Concluding remarks on relaxation
When we scan the system for resonant frequencies with microwaves, we want to have a greater
fraction of atoms in F = 3 than F = 4. If these are equally populated, resonant microwaves will not
change the populations and we will not see a signal. With the coated cell we obtain a population
9
difference, since when we apply the laser an equilibrium is reached where f3 = 1 − f4 (0) ≈ 10
by Eq. (66). The un-coated cell keeps the populations almost equal. So, if we were to use an
un-coated cell, we would have to apply optical pumping to deplete the population in the state
F = 4. A laser tuned to the transition F = 4, F 0 = 4 could have done the job, but it was not
necessary when we were using the coated cell.
We have found that the characteristic time of decay is 3.23 ms when the laser is on. When we
scan the resonances, we want each resonance to be hit under the same circumstances, so we can
4 RESULTS
22
compare them. This is assured if we adjust the frequency scan such that neighboring resonances
are separated by a time somewhat larger than the characteristic decay time.
It is worth noting that our detection laser actually plays an important role in the pumping
of the populations. It opens some decay channel from F 0 = 5 to F = 3 which results in a large
population difference at equilibrium.
4.2
The Zeeman effect
We are now ready to apply the microwaves to scan for resonances. We shall use the position
of the transition lines to deduce the magnitude of the magnetic field and then examine how the
frequency of the 0-0 transition changes with the magnetic field.
The obtained spectra are recorded as a function of time but the timescale can easily be translated into a frequency scale, since we know the rate at which frequencies are scanned and the start
frequency. It turns out, that we cannot exactly associate a given time with a frequency, since the
triggering of the oscilloscope and frequency generator is not completely synchronized. However
we still trust, that the frequency scan is accurately linear in time, so the rate at which frequencies
are scanned is very precise. We do not know, if the triggering error is constant from measurement
to measurement, but in a given measurement we can associate a time difference with a frequency
difference, from the corresponding scan rate.
As mentioned in section 2.2.3 we expect to observe 15 peaks and this is exactly what we observe,
when the frequency scan ranges over a wide enough frequency range. By varying the DC current
we observed, that the position of the peaks varied, just as we expect from the Zeeman effect. When
no DC current was applied however, we did not only see one transition line, as would be expected,
since at zero magnetic field, all the transitions should have the same associated frequencies. This is
due to the presence of background magnetic fields in the laboratory, primarily the earth magnetic
field. An example of this shift of position is shown in Fig. 15, where two spectra recorded at
different values of the DC current are shown.
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Figure 15: Example of the Zeeman effect. To the left a spectrum recorded with a DC current of
60 mA and to the right a spectrum recorded with 120 mA. Both spectra was obtained with a
microwave power of 15 dBm. The time scale has been translated to a frequency scale and only
the part containing transition lines is shown, but else the spectra have not been modified. The
splitting of the levels to the right is seen to be roughly twice that to the left. The 0-0 transition
has the largest line height and is situated just below 9193 M Hz.
With a fixed output power of 15 dBm and a scanning rate of 20 M Hz s−1 starting at
9188 M Hz, we recorded spectra with DC currents between 0 mA and 1 A. For low DC currents all 15 lines are present in the spectra and are thus easily identified. For higher DC currents
only some of the lines were present, and they could not simply be identified by counting. We
23
4 RESULTS
know however from section 2.2.3, that the 0-0 transition (index 8) changes its frequency very little
with the magnetic field and this transition could be used to identify the other transitions. From
each of the recorded spectra we read off the frequency of the transitions appearing. We have not
subtracted the background spectrum, since here we will not be interested in the actual height of
the lines.
4.2.1
Determining the magnetic field
To first order in the total magnetic field, the frequency difference between neighboring lines is
the same for all pairs of lines. The difference is from section 2.2.3 given by ∆ν = KB where
K = 350.42 kHz Gauss−1 .
For each value of the DC current we plotted the frequency of appearance of lines versus the line
index. An example of this is shown in Fig. 16. This is of course only interesting when more than
one line is present. It is confirmed, that there is indeed a constant frequency difference between
neighboring lines at a given magnetic field. This is evident from the figure, where a straight
line has been fitted to the measurements. The slope of this line will be our best estimate of the
frequency difference.
If we had a reliable independent measurement of the magnetic field, we could use the measured
frequency splitting to arrive at K. As K is a collection of atomic constants, we would be able to
deduce some atomic properties. However this is not the approach we shall employ, as we do not
have reliable, independent measurements of the magnetic field. Instead we shall take the value of
K for granted and use the frequency splitting to determine the magnitude of the magnetic field
at the different DC currents.
9195
Measurements
Linear fit
9194.5
9194
Frequency [MHz]
9193.5
9193
9192.5
9192
9191.5
9191
9190.5
9190
0
2
4
6
8
Line index
10
12
14
16
Figure 16: The frequency of the 15 lines at a DC current of 20 mA. As is evident, the frequency
difference between neighboring lines is a constant, an can be taken to be the slope of the straight
line fitted to the data. Uncertainties have not been included, since they practically zero on this
scale.
In Table 2 we have stated the magnetic field calculated at different DC currents, along with
the indices of lines appearing in the measurements.
24
4 RESULTS
Current [mA]
0
20
40
60
80
100
120
140
160
180
200
240
280
320
360
400
500
Lines appearing
1-15
1-15
1-15
1-13
2-12
3-11
4-11
5-10
5-10
5-10
6-10
6-9
6-9
7-9
7-9
7-9
7-8
Magnetic field [Gauss]
0.6938± 0.0008
0.9074± 0.0009
1.241 ± 0.002
1.659 ± 0.002
2.053 ± 0.002
2.493 ± 0.003
2.916 ± 0.002
3.309 ± 0.004
3.737 ± 0.005
4.176 ± 0.005
4.553 ± 0.008
5.409 ± 0.006
6.28 ± 0.01
7.11 ± 0.03
8.05 ± 0.03
8.86 ± 0.03
10.94 ± 0.03
Table 2: The magnetic field at different DC currents calculated from the Zeeman splitting. The
lines appearing has also been stated.
The magnetic field has been plotted as a function of current in Fig. 17. We see the effects of
magnetic fields other than that of the coils at low currents. If only the coils contributed we would
get a straight line through 0, but the earth and electrical components in the room also give rise
to magnetic fields, resulting in a total magnetic field of the form
(75)
B = BH + BE ,
where BH is the field of the Helmholtz coils, and BE is the background field. If we assume that
the direction and magnitude of BE is constant in time, the magnitude of the total field for a given
DC current I is
q
q
(76)
B = BH 2 + BE 2 + 2 cos(θ)BH BE = a2 I 2 + BE 2 + 2 cos(θ)aIBE ,
where a is the proportionality parameter between the current and BH and θ is the angle between
BH and BE . This function is fitted to data and plotted together with the calculated field at
different currents. The fit is good and the parameters determined are
(77)
BE = (0.690 ± 0.005) Gauss
−1
a = (0.0218 ± 0.0001)Gauss mA
◦
θ = (74.12 ± 0.02)
(78)
(79)
From [9] we have found, that the Earth magnetic field at the site of the experiment has a magnitude
of 0.50 Gauss and is pointing downwards to the north with an angle of 20.5◦ with vertical. We
estimate our DC magnetic field to be horizontal and pointing northwest, which will result in an
angle between our DC magnetic field and the Earth magnetic field of θ = 75.7◦ . The measured
magnitude of the background magnetic field is somewhat higher than the magnitude of the Earth
magnetic field, but there is a very good correspondence between the angles.9
9 Perhaps this could be explained by the Earth magnetic field magnetizing some ferromagnetic materials in the
laboratory, thus amplifying itself.
25
4 RESULTS
14
Measurements
Fit
12
Magnetic field [Gauss]
10
8
6
4
2
0
0
100
200
300
DC current [mA]
400
500
Figure 17: The calculated magnetic field as a function of the DC current. Uncertainties have not
been included, since these are practically zero on this scale. Also shown is a fit of Eq. (76).
4.2.2
Zeeman shift of the 0-0 transition
If we want to examine the Zeeman effect to second order in B, we can look at the peak representing
the 0-0 transition. This is not shifted in the first order theory. The shift to second order in B is
given by Eq. (36)
(gJ − gI )2 µ2B B 2
= K0 B 2 .
(80)
∆E (2) =
2hν0
We have measured the frequency for the 0-0 transition for currents up to 1000 mA, but we only
calculated the magnetic field at currents up to 500 mA, since at larger currents only the 0-0
transition line was present and no value of the line splitting could be obtained. Since the shift
we want to examine is quadratic in the magnetic field, it will be more prominent at high fields
and we want to estimate the field at currents higher than 500 mA. By the use of Eq. 76 and
the determined parameters, we calculate the magnetic fields and these are tabulated in Table 3.
A plot of the frequency of the 0-0 transition versus the magnitude of the total magnetic field is
Current [mA]
600
700
800
900
1000
Magnetic field [Gauss]
13.2 ± 0.1
15.5 ± 0.1
17.6 ± 0.1
19.8 ± 0.1
22.0 ± 0.1
Table 3: Estimated magnetic field for DC currents above 500 mA. The uncertainties are estimated
from the uncertainties on the parameters in Eq. (77)-(79).
shown in Fig. 18. A function of the form
ν(B) = a + bB 2
(81)
26
4 RESULTS
9192.85
Measurements
Expected
Fit
Frequency of 0-0 transition [Mhz]
9192.8
9192.75
9192.7
9192.65
9192.6
0
5
10
15
Magnetic field [Gauss]
20
25
Figure 18: Frequency of the 0-0 transition as a function of the magnetic field. We have not
included uncertainties in B.
has been fitted to the data. The fit is in good agreement with our experimental data, as is
also evident from the figure. The parameters are determined to (9192.6092 ± 0.0004)M Hz and
(b = 0.410 ± 0.003)kHz Gauss−2 .
The parameter a is the frequency of the 0-0 transition in the presence of no magnetic field. The
determined value of (9192.6092±0.0004)M Hz is near the expected value of 9192.631770 M Hz, but
there is a significant discrepancy. As we wrote previously we cannot make any absolute frequency
determinations so this is not disturbing.
From the calculated value of b we find by Eq. (80) the difference between the Landé g-factors
gJ and gI
h p
2ν0 b = 1.964 ± 0.007.
(82)
gJ − gI =
µB
The established value of gJ − gI is 2.003 and although our value is close to this, it is not consistent
with it. The expected value of b is stated in section 2.2.3 as 0.42745 kHz Gauss−2 . We plot the
expected behavior of the 0-0 transition frequency shifted downwards by an amount determined by
ν0 − a along with the measurements in Fig. 18. This line is actually seen to be more or less inside
the uncertainties of all the data points.
Since gI is much smaller than gJ , we could as well say, that we have determined the value of
gJ to gJ = 1.964 ± 0.007.
4.3
Transition lines
We will now vary the power of the frequency generator and see how the appearance of the spectra
will change. This will provide us with means to examine the relative transition rates of the 15
transition lines.
We recorded several spectra, where we varied the microwave power in integer steps between
15 dBm and 0 dBm. The frequency scan was conducted in the range 9.188 GHz to 9.196 GHz
and the DC current was 42 mA, both chosen in such a way, that all the 15 transition lines were
4 RESULTS
27
present. The resultant spectra all looked rather like the one in Fig. 10, except that the line height
depended strongly on the power. We also turned off the microwave completely and recorded a
reference spectrum.
4.3.1
Accounting for reference spectrum
The reference spectrum describes the equilibrium distribution of the atoms, when the laser is
on. When we apply the microwave field, we will therefore see a response superposed on this
equilibrium. The actual response of the atoms to the applied microwave field can then be found
by subtracting the reference spectrum from the spectra.
We would expect the reference spectrum to reach a constant level in a matter of ms, following
our discussion in section 4.1.2. However for these measurements, the voltage of the reference
spectrum was oscillating about a stationary value, with a constant frequency. This is also the case
of the actual spectra, see Fig. 10. The frequency of these oscillations were an estimated 24 Hz and
were probably caused by the way the oscilloscope collected data. The phase of these oscillations
were changing from measurement to measurement, so we could probably have gotten rid of them
by averaging a larger number of sweeps. The changing phase however complicated matters, as we
could not simply subtract the reference spectrum from the actual spectra and get a flat floor.
The way we bypass the problem is by fitting a general sinusoidal function to the part of the
reference spectrum, where the voltage should be constant (after 400 ms where the frequency scan
was finished). This function is then by a change of phase, mean value and amplitude (but not
frequency) fitted to the last 80 ms of the actual spectra, where these should resemble the reference
spectrum. The reason we change the mean value is, that the background level might be different
in different measurements, and the reason we change the amplitude is, that the phase may change
a varying amount per time unit, and thus not allowing the oscillations in different spectra to have
averaged to zero the same amount. Finally the fitted function is subtracted from the spectrum to
give a double resonance spectrum with a baseline at zero.
We have also subtracted the initial decay in the spectra displayed in Fig. 19 and on the front
page. This is not crucial since the interesting part of the spectra is the part where the microwaves
hit the resonances. We subtracted it for aesthetical reasons only.
An example of this fitting procedure is shown in Fig. 19, where both the original recorded
spectrum, the fitted function, and the resultant peaks on zero background is shown.
4.3.2
Transition rates
Each of the recorded spectra was corrected in this manner and the height of each line was read
off whenever possible. For some of the lowest powers, the lines were almost indistinguishable from
the noise and so any height measurement was almost impossible. The determined values of the
height of each of the π and σ lines are plotted in Fig. 21 and 20 respectively as a function of power
(in mW ).
The height of the peak (above the equilibrium position) is a measure of how many atoms have
been transferred to the F = 4 by the microwave at resonance. The transition rate from the mF
sublevel of F = 3 to the m̃F sublevel of F = 4 is given by Eq. (19).
The π and σ transitions are not caused by the same component of the microwave magnetic
field. As we do not know the exact relationship between the two components (other than that
they are of the same order of magnitude, see appendix C), we cannot compare the two types
of transitions. However for a given type of transition, we can compare the different transition
lines and their heights will be proportional to the intensity and the square of a Clebsch-Gordan
coefficient.
We thus expect the height of a given peak to vary linearly with the microwave power, an
assumption, which is not confirmed by the experiment. This is due to beginning saturation of the
system.
In table 4 we have tabulated theoretical and measured values of the transition probability for
the π transitions relative to the 0-0 transition. The theoretical values are obtained by taking the
28
4 RESULTS
Measurement
Fit
Corrected spectrum
0.05
0.04
Voltage [mV]
0.03
0.02
0.01
0
-0.01
0.05
0.1
0.15
0.2
0.25
Time [s]
0.3
0.35
0.4
0.45
0.5
Figure 19: Example of a spectrum. This was recorded at 14 dBm. We show the actual measurement, a fit to the reference spectrum and the measurement with subtracted reference spectrum.
0.022
2
4
6
8
10
12
14
0.02
0.018
0.016
Voltage [mV]
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
5
10
15
20
25
30
35
Power [mW]
Figure 20: The height of each of the π transitions as a function of microwave power. The numbers in the key refer to the transition index. Uncertainties have not been included and points
corresponding to the same transition have been connected by straight lines for clarity.
29
4 RESULTS
0.016
1
3
5
7
9
11
13
15
0.014
0.012
Voltage [mV]
0.01
0.008
0.006
0.004
0.002
0
0
5
10
15
20
Power [mW]
25
30
35
Figure 21: The height of each of the σ transitions as a function of microwave power. The numbers in the key refer to the transition index. Uncertainties have not been included and points
corresponding to the same transition have been connected by straight lines for clarity.
ratio of the coefficients stated in table 1. The second column of experimental values have been
found by taking the ratio between the height of the transition line in concern and the height of
the 0-0 transition in the spectrum recorded at 15 dBm. The first column of experimental data are
obtained by making a weighted average of such ratios over all powers.10
We would expect symmetry about the 0-0 transition, but this is not what we observe. The low
energy transitions (2-6) are in very good agreement with the expected theoretical values, whereas
the high energy transitions (10-14) are somewhat lower than expected. We are not capable of
explaining this.
We could use the same procedure on the σ transitions, but this turns out to give results,
which do not at all correspond to the theoretical predictions. This is probably do to the fact
that σ-transitions can pump atoms between different sublevels. The microwave scan will thus
completely disturb the populations of atoms in the different sublevels and the transitions will not
be comparable.
4.4
Improvements
The noise present in all measurements could probably be reduced by averaging a larger number
of sweeps, but this would of course increase the time needed to do the experiment.
When we measured the magnetic field, the precision was limited by the broadening of the
lines. This width is primarily due to the Doppler shift experienced by moving atoms. The line
broadening due to the Doppler effect at 300K can be calculated [4]
r
2ν0 2RT
δνD =
log(2) ≈ 10 kHz,
(83)
c
M
10 A weighted average in this connection means, that the ratios has been weighted with their respectively uncertainties.
30
5 CONCLUSION
Line index
Theoretical
2
4
6
8
10
12
14
0.44
0.75
0.94
1
0.94
0.75
0.44
Measured
Average 15 dBm
0.46
0.52
0.72
0.75
0.90
0.91
1
1
0.84
0.86
0.62
0.69
0.34
0.43
Table 4: Transition rates for π transitions relative to the 0-0 transition. First column is line
index. Second column is the theoretical calculated values. Third column is the values obtained
by averaging over all powers, whereas the fourth column is the values obtained from the spectrum
recorded at 15 dBm alone.
which is about the line width we observe (15kHz). The broadening could be reduced by cooling
the atoms, for example by means of optical techniques, where temperatures below mK can be
reached.
The precision in determining the transition probabilities is limited by reading off the line
heights in the measurements. The signal to noise ratio could possibly be increased if a greater
population difference between the F = 3 and F = 4 levels is obtained. This could be achieved
either by means of optical pumping, where atoms are prepared in a specific state, or by using a cell
with a more efficient coating. Also the validity of the calculations depends on the system being
prepared in the same state before each line is scanned. When we scan all 15 lines in one sweep we
might disturb the system so the different transitions are not directly comparable.
Finally, instead of scanning frequencies one could make measurements at one frequency at a
time, thereby ensuring that the system could be prepared in the same state before each measurement. This would also mean that absolute values of the frequencies could be obtained.
5
Conclusion
The objectives of the experiment stated in the introduction has been fulfilled.
We found that the coating of our cell was efficient enough to keep a significant population
difference between the F = 3 and F = 4 levels. When the laser was turned on, equilibrium was
reached after 3 ms, and the F = 3 population at this point was f3 = 0.9. This was clearly enough
to obtain a signal when we applied resonant microwaves.
We calculated magnetic fields from the zeeman splitting with a relative precision of less than
3/1000. Not the best precision ever seen, but certainly better than the Hall probe used for
reference.
The Landé g-factor gJ = 2.003, is not within the uncertainty of our measured value of gJ =
1.964 ± 0.007. This is probably due to the fact that this measurement is based on the assumption
that the position of a frequency on the time scale of the oscilloscope did not change from one
measurement to another.
The relative transition rates of the π-transitions have been calculated and they are in good
agreement with the theory. The high energy transition rates are a bit lower than expected, which
perhaps is due to the way we scan the transitions.
From the beginning, a main objective has been to use a simple setup and still be able to extract
a lot of physics. Here we consider ourselves most successful as it has been applied to such different
subjects, such as measuring magnetic fields, determination of atomic properties and calculating
collision rates
A
A
VARIOUS PHYSICAL CONSTANTS
Various physical constants
The speed of light in vacuum
c = 299792458 m s−1
Boltzmann constant
kB = 1.38062 · 10−23 J K −1
Planck’s constant
h = 6.62620 · 10−34 J s
~ = h/2π = 1.05459 · 10−34 J s
The Bohr magneton
µB = 9.27410 · 10−24 J T −1
Permittivity of vacuum
²0 = 8.85 · 10−12 C 2 N −1 m−2
The charge of the electron
e = −1.60219 · 10−19 C
The mass of the electron
m = 9.10956 · 10−31 kg
31
B
B
PHYSICAL CONSTANTS FOR CESIUM-137
Physical constants for cesium-137
Atomic mass of cesium-133 [8]
m = 132.905451931 u
Frequency of the clock transition [8]
ν0 = 9.192631770 GHz
Landé g factors
gS = 2.002319
gL = 0.999996gI = −0.00039885395
Landé g factor for the ground state of cesium [8]
gJ = 2.00254032
First order Zeeman splitting between neighboring mF sublevels of the ground state [10]
K3 = 350.98 kHz Gauss−1
K4 = 349.86 kHz Gauss−1
Second order Zeeman shift of the 0 − 0 transition [10]
K0 = 0.42745 kHz Gauss−2
32
33
C MICROWAVES
C
Microwaves
We will describe how a microwave field can be generated using an antenna and a frequency
generator. The antenna is made of cobber and consists of a box cavity that is open at one
end, a horn mounted at the open end of the cavity and an exciting probe that is connected to the
frequency generator.
Different modes of electromagnetic radiation can exist in a cavity with conducting walls, depending on its dimensions. A long rectangular waveguide is considered and we look for a monochromatic field propagating in the longitudinal direction (z), represented by the generic form:
E(x, y, z, t) = E0 (x, y)ei(kz−ωt)
(84)
i(kz−ωt)
(85)
B(x, y, z, t) = B0 (x, y)e
where the x axes is taken to be parallel to the side of length a and the y axes is parallel to the side
of length b. It is understood that the real parts of Eq. (84) and Eq. (85) are the actual physical
fields
We will determine the amplitudes E0 and B0 which are written as:
E0 (x, y) = Ex (x, y)x̂ + Ey (x, y)ŷ + Ez (x, y)ẑ
B0 (x, y) = Bx (x, y)x̂ + By (x, y)ŷ + Bz (x, y)ẑ
(86)
(87)
The Maxwell Equations in the interior of a waveguide are (no current or charge):
∇·E=0
(88)
∇·B=0
(89)
∂B
∂t
∂E
∇ × B = µ0 ²0
∂t
∇×E=−
(90)
(91)
Inserting into Eq. (84) and Eq. (85) into Eq. (90) and Eq. (91) yields:
∂Ez
− ikEy
∂y
∂Ez
ikEx −
∂x
∂Ey
∂Ex
−
∂x
∂y
∂Bz
− ikBy
∂y
∂Bz
ikBx −
∂x
∂Bx
∂By
−
∂x
∂y
= iωBx
(92)
= iωBy
(93)
= iωBz
(94)
iω
Ex
c2
iω
= − 2 Ey
c
iω
= − 2 Ez
c
=−
Combining Eq. (92), Eq. (93), Eq. (95) and Eq. (96) gives:
³ ∂B
i
ω ∂Ez ´
z
Bx =
k
−
(ω/c)2 − k 2
∂x
c2 ∂y
³
i
∂Bz
ω ∂Ez ´
By =
k
+ 2
2
2
(ω/c) − k
∂y
c ∂x
³ ∂E
∂Bz ´
i
z
+
ω
Ex =
k
(ω/c)2 − k 2
∂x
∂y
³ ∂E
∂Bz ´
i
z
k
−ω
Ey =
(ω/c)2 − k 2
∂y
∂x
(95)
(96)
(97)
(98)
(99)
(100)
(101)
34
C MICROWAVES
so all the components can be determined if the z-components are known. Differentiating Eq. (98)
and Eq. (99) with respect to x and y and adding the two gives
³ ∂2
∂By
ik
∂2 ´
∂Bx
+
=
+
Bz
∂x
∂y
(ω/c)2 − k 2 ∂x2
∂y 2
(102)
using Eq. (89) and noting that differentiating Bz with respect to z just gives a factor of ik we get:
´
³ ∂2
∂2
2
2
+
+
(ω/c)
−
k
Bz = 0.
∂x2
∂y 2
(103)
The exciting probe is mounted in the direction of ŷ and this is the direction of the current density
that will initiate the radiation. Since the electric field is proportional to the current density, this
will also be the direction of the electric field and we have that Ex = Ez = 0.
Eq. (103) can be solved using separation of variables. Assuming a solution of the form:
Bz (x, y) = F (x)G(y)
(104)
and inserting in Eq. (103) we get
´
d2 F
d2 G ³
2
2
+
F
+
(ω/c)
−
k
FG = 0
dx2
dy 2
(105)
´
1 d2 G ³
1 d2 F
+
+ (ω/c)2 − k 2 = 0
2
2
F dx
G dy
(106)
G
Dividing by F G gives
The first and second terms depends only on x and y respectively and they must both be constant.
The problem is reduced to solving the two ordinary differential equations:
1 d2 F
= −kx2
F dx2
1 d2 G
= −ky2
G dy 2
(107)
(108)
with
−kx2 − ky2 + (ω/c)2 = k 2
(109)
The constants are chosen to be negative, because positive values would give rise to exponentials
that don’t fit the boundary conditions.
The solutions are:
F (x) = A cos(kx x) + B sin(kx x)
(110)
G(y) = C cos(ky y) + D sin(ky y)
(111)
The component of the magnetic field that are orthogonal to an interface must be continuous across
that interfaces and for a good conductor the electric field will be zero inside. Equation Eq. (90)
then says that the time-derivative of the magnetic field must be zero inside, so if it is zero to start
with, it will remain so. From this and the fact that Ez is zero, we can read of Eq. (98) and Eq.
(99) that ∂Bz /∂x and ∂Bz /∂y must be zero at the cavity walls.
∂Bz
(x = 0, y) = 0
∂x
∂Bz
(x, y = 0) = 0
∂y
(112)
(113)
Since Bz is equal to F (x)G(y), we have B = D = 0. Setting AC = Bz,0 the result is:
Bz = Bz,0 cos(kx x) cos(ky y)
(114)
35
C MICROWAVES
kx and ky can only take discrete values since
∂Bz
(x = a, y) = −kx Bz,0 sin(kx a) cos(ky y) = 0
∂x
∂Bz
(x, y = b) = −ky Bz,0 cos(kx x) sin(ky b) = 0
∂y
(115)
(116)
implies that
nπ
mπ
,
ky =
,
n, m ∈ N0
(117)
a
b
The different propagation modes are described by the two integers m and n. When we have
a transverse electric field, one specific mode is denoted TEmn Equation Eq. (109) shows that
some values of kx and ky gives rise to complex values of k, which lead to dissipation. We want to
construct the cavity in such a way that only one mode can propagate freely. If k must be real we
have
³
´
(ω/c)2 > kx2 + ky2 = π 2 (m/a)2 + (n/b)2
(118)
kx =
The dimensions of the cavity we are using is, a = 22.86mm and b = 10.16mm. With a frequency
of 9.2Ghz we have (ω/c)2 = 3.7 · 104 m−2 and only the mode TE10 have real k
³
´
³
´
1.23 · 105 m−2 = π 2 (1/a)2 + (1/b)2 > (ω/c)2 > π 2 (1/a)2 + (0/b)2 = 2.47 · 104 m−2 . (119)
The mode TE00 should be investigated before the steps leading to Eq. (98)-Eq. (101) since these
are not defined for (ω/c)2 − k 2 = 0, but Eq. (92)-Eq. (97) will give Bz = 0 in this mode, and it
can be shown in general that transverse electromagnetic waves (TEMmn ) cannot exist in a hollow
waveguide. Therefore TE10 is called the principal mode.
For the principal mode kx is 0 so
Bz = Bz,0 cos(kx x)
(120)
From Eq. (98)-Eq. (101) and the fact that Ez is 0, we can calculate the other field components
(121)
By = 0
¡ πx ¢
−ik(π/a)
Bz,0 sin
Bx =
2
2
(ω/c) − k
a
Ex = 0
¡ πx ¢
iω(π/a)
Bz,0 sin
.
Ey =
2
2
(ω/c) − k
a
(122)
(123)
(124)
Inserting in Eq. (84) and Eq. (85) and taking the real part we get
¡ πx ¢
sin(kz − ωt)ŷ
a
¡ πx ¢
¡ πx ¢
k(π/a)
Bz,0 sin
B = Bz,0 cos
cos(kz − ωt)ẑ +
sin(kz − ωt)x̂
2
2
a
(ω/c) − k
a
E = E0 sin
with
E0 = −Bz,0
ω(π/a)
.
(ω/c)2 − k 2
(125)
(126)
(127)
The free parameter Bz,0 will be determined by the output power of the frequency generator.
It is worth noticing that the magnetic field will have a component in the direction of propagation. The amplitude of the two components are about the same
k(π/a)
Bx,0
=
≈ 0.98
Bz,0
(ω/c)2 − k 2
(128)
36
D EXACT SOLUTION OF TWO LEVEL PROBLEM
D
Exact solution of two level problem
We wish to describe the time evolution of a two level atom subjected to a time varying magnetic
field. Without the time-dependent magnetic field the stationary states of the system is denoted
|1i and |2i. They are simultaneously eigenstates of J2 , I2 , F2 and mF , that is the square of the
total electron angular momentum, the square of the nuclear spin, the square of the total angular
momentum and its projection on the z-axis. The Hamiltonian for the system without applied
time-dependent magnetic field is written as
H0 = E1 |1ih1| + E2 |2ih2|
(129)
in which E1 (E2 ) is the energy of the state |1i (|2i). The energy difference between the two
states is E2 − E1 ≡ ~ω21 . We shall choose ω21 positive, so that |1i is the lowest lying state. The
interaction Hamiltonian for the interaction of the atom with the applied magnetic field is to a first
approximation
Hint (t) = −µ · B(t)
(130)
where µ is the magnetic dipole moment operator and B(t) is the time-dependent magnetic field.
The total Hamiltonian is the sum of H0 and Hint .
As we consider a two state system, an arbitrary state |ψ, ti can of course be written as a
superposition of |1i and |2i with time-dependent coefficients. As is often done in problems involving
time-dependent Hamiltonians we write11
´
³
´
³
¯ ®
¯ψ, t = c1 (t) exp − iE1 t |1i + c2 (t) exp − iE2 t |2i
(131)
~
~
Inserting this expression for |ψ, ti in the time-dependent Schrödinger equation yields
∂
|ψ, ti
∂t
³ iE t ´
³ iE t ´
1
2
i~ċ1 exp −
|1i + i~ċ2 exp −
|2i
~
~
i~
=
(H0 + Hint )|ψ, ti ⇒
³ iE t ´
1
= c1 exp −
Hint |1i
~
³ iE t ´
2
+c2 exp −
Hint |2i
~
(132)
(133)
(134)
Where we have used the explicit form Eq. (129) of H0 . Multiplying Eq. (132) from the left
with by turn h1| and h2| and employing that these states are orthonormal we get the following
equations, that govern the time-development of the coefficients
i~ċ1
i~ċ2
= c1 h1|Hint |1i + c2 e−iω21 t h1|Hint |2i
iω21 t
= c1 e
h2|Hint |1i + c2 h2|Hint |2i
(135)
(136)
To calculate these matrix-elements, let us consider a oscillating magnetic field along the x-axis
B(t) = B cos(ωt)x̂
(137)
This magnetic field could be produced by an AC-current in a pair of coils or it could be part of some
electromagnetic radiation. The magnetic dipole moment operator will be a sum of contributions
from the nucleus and from the electrons. However, the magnetic dipole moment of the nucleus
is several orders of magnitude smaller than the magnetic dipole moment of the electrons of the
atom, and we can therefore safely neglect the nuclear contribution to the magnetic dipole moment
operator. Thus we can write
gµB
µ=−
J
(138)
~
e~
Where µB is the Bohr magneton 2mc
(m is the mass of the electron) and g is the Landé factor
g =1+
11 See
Sakurai [3] page 318-319
J(J + 1) + S(S + 1) − L(L + 1)
2J(J + 1)
(139)
D EXACT SOLUTION OF TWO LEVEL PROBLEM
37
Here J, S and L are as usual the quantum numbers associated with the corresponding operators
J2 , S2 and L2 . The interaction Hamiltonian reduces to
Hint (t) =
gµB B
gµB B
cos(ωt)Jx =
cos(ωt)(J+ + J− )
~
2~
(140)
Where we have written the x-component of J as half the sum of the ladder operators J+ and
J− . Let us furthermore restrict our considerations to a system with L = 0 and J = 21 (and thus
J = S). The two states are |1i = |mJ = + 21 i and |2i = |mJ = − 21 i. The matrix-elements of Hint
can now easily be calculated12
h1|Hint |1i
h1|Hint |2i∗
= h2|Hint |2i = 0
= h2|Hint |1i
gµB B
1
1
=
cos(ωt)hmJ = − |(J+ + J− )|mJ = + i
2~
2
2
gµB B iωt
gµB B
cos(ωt) =
(e + e−iωt )
=
2
4
(141)
(142)
(143)
(144)
Substituting these into Eq. (135) and Eq. (136) we arrive at
i~ċ1
=
i~ċ2
=
gµB B −i(ω21 −ω)t
(e
+ e−i(ω21 +ω)t )c2
4
gµB B i(ω21 +ω)t
(e
+ ei(ω21 −ω)t )c1
4
(145)
(146)
We will only consider the case of near-resonance, that is we assume ω21 ≈ ω.13 Then there are two
different time scales involved in these equations. The exponentials involving (ω21 + ω) oscillate
very rapidly compared to those involving (ω21 − ω) and we will assume these average to zero in the
time scale we are interested in. This is called the Rotating Wave Approximation since it amounts
to decomposing a linearly polarized field into two circularly polarized components. One with a
rotation frequency of ω21 and one with −ω21 . At resonance the component with −ω21 can be
safely neglected, so the atoms will simply experience a rotating field with an amplitude of B/2.
The equations are thus reduced to
i~ċ1
=
i~ċ2
=
gµB B −i(ω21 −ω)t
e
c2
4
gµB B i(ω21 −ω)t
e
c1
4
(147)
(148)
These coupled differential equations are easily solved (with the use of the substitution c̃1 = c1 ,
c̃2 = c2 e−i(ω21 −ω)t ). With the initial conditions c1 (0) = 1 and c2 (0) = 0, that is all the atoms
start in the lower level |1i, the solutions are
c1 (t)
c2 (t)
³ Ω ´´ ∆
³Ω ´
∆
t + i sin
t e−i 2 t
2
Ω
2
³Ω ´ ∆
ωr
= −i sin
t ei 2 t
Ω
2
=
³
cos
(149)
(150)
Where we have defined the Rabi-frequency ωr , a de-tuning factor ∆ and a frequency Ω as
gµB B
2~
∆ = ω21 − ω
p
Ω =
∆2 + ωr2
ωr
12 We
=
(151)
(152)
(153)
use the fact that hj 0 , m0 |J± |j, mi = (j ∓ m)(j ± m + 1)~δj 0 j δm0 ,m±1
coupled equations Eq. (145) and Eq. (146) can actually be solved exactly, but the solution is not very nice
13 The
p
38
D EXACT SOLUTION OF TWO LEVEL PROBLEM
The probability at time t to find the atom in either one of the states, are the norm squares of the
coefficients appearing in Eq. (131)
P1 (t) = |c1 (t)|2 = cos2
³Ω ´
³Ω ´
³ ω ´2
³ Ω ´ ³ ∆ ´2
r
t +
t =1−
t
sin2
sin2
2
Ω
2
Ω
2
³ ω ´2
³Ω ´
r
P2 (t) = |c2 (t)|2 =
t
sin2
Ω
2
We have oscillations between the two different states with a characteristic frequency
oscillations are called Rabi-oscillations.
(154)
(155)
Ω
2.
These
39
REFERENCES
References
[1] Jean Brossel and Francis Bitter: A New "Double Resonance" method for Investigating Atomic
Energy Levels. Application to Hg 3 P1∗ , Physical Review
[2] David J. Griffiths: Introduction to Electrodynamics, Third Edition, Prentice Hall 1999
[3] J. J. Sakurai: Modern Quantum Mechanics, Revised Edition, Addison Wesley Longman 1994
[4] Peter W. Milonni and Joseph H. Eberly: Lasers, John Wiley & Sons 1988
[5] B. H. Bransden and C. J. Joachain: Physics of Atoms and Molekyles, Second Edition, Prentice
Hall 2003
[6] Eugene Hecht: Optics, Fourth Edition, Addison Wesley 2002
[7] Richard L. Liboff: Introductory Quantum Mechanics, Fourth Edition, Addison Wesley 2003
[8] Daniel A. Steck: Cesium D Line Data, version 1.6 2003, http://steck.us/alkalidata
[9] DGRF/IGRF Geomagnetic Field Model 1945 - 2005
http://nssdc.gsfc.nasa.gov/space/model/models/igrf.html.
and
Related
Parameters,
[10] J. Vanier and C. Audoin: The Quantum Physics of Atomic Frequency Standards, IOP Publishing Ltd 1989
[11] S. Svanberg: Atomic and Molecular Spectroscopy, Second edition, Springer-Verlag 1992
[12] Charles Kittel and Herbert Kroemer: Thermal Physics, Second edition, W. H. Freeman and
Company 2002