Laser cooling of Barium atoms: a
numerical computation
J.E. van den Berg
Bachelor thesis May - August 2007
University of Groningen
3000
Number of atoms
in ground state
2500
2000
1500
1000
500
0
0
200
400
600
Final velocity [m/s]
800
Supervisor: dr. L. Willman, KVI
1
1000
Contents
1 Introduction
3
2 Description of the problem
2.1 Principles of laser cooling
2.2 Raman transitions . . . .
2.3 Theoretical model . . . .
2.3.1 The Hamiltonian .
2.3.2 Density matrix . .
2.4 Optical Bloch equations .
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3 Numerical simulations in Matlab
3.1 Solving the Optical Bloch equations . . . . . . . . .
3.2 The algorithm . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Outline . . . . . . . . . . . . . . . . . . . . .
3.2.2 End velocity after cooling for a specific initial
3.2.3 Building a table with final velocities . . . . .
3.2.4 Making the final velocities spectrum . . . . .
3.2.5 Summary of the computation procedure . . .
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velocity
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4 Results
4.1 Crossing the resonance . . . . . . . .
4.2 The amount of cooling power . . . .
4.3 The amount of repumping power . .
4.4 Detunings of the lasers . . . . . . . .
4.5 Comparison with experimental data
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5 Summary
22
5.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A Appendix - Matlab files
A-1 liouville4.m . . . .
A-2 coolatom.m . . . .
A-3 computevfinal.m .
A-4 velocityspectrum.m
A-5 mbdistr.m . . . . .
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24
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28
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35
2
1 Introduction
In the third year of their physics studies, students have to do a three month Bachelor
research project to complete their Bachelor programme. This report is the result of
the Bachelor project done at the Kernfysisch Versneller Instituut Groningen (KVI).
The topic is a numerical simulation of a laser cooling experiment for barium atoms.
This experiment is currently done at KVI and will be followed by the laser cooling of
radium to search for physics beyond the standard model.
At KVI the TRIµP (Trapped Radioactive Isotopes: µicro-laboratories for Fundamental Physics) group is doing very sensitive experiments that may lead to physics
beyond the standard model. One of the experiments will be looking for a possible
permanent electromagnetic dipole moment (EDM) of the radium atom. Such an EDM
can only be seen with very precise measurements. Therefore the radium atoms have
to be cooled and trapped before they can be studied. To learn how to do this, first
an experiment is performed in which barium atoms are cooled and trapped. A complete description of this experiment can be found in the thesis of U. Dammalapati [4].
Barium is very similar to radium, yet it is not radioactive which is very convenient.
The laser cooling technique is used to slow down and cool the barium atoms. Laser
cooling of barium is not trivial because a large leak rate into metastable D-states exists.
This means that not only a cooling laser is needed, but also several repump lasers to
bring the atoms back in the cooling cycle. It is useful to have a numerical simulation
of the experiment to be able to see what effects the different experimental parameters
have on the cooling of the atoms. In my bachelor project, I wrote the Matlab scripts
that do the simulation and I looked at the outcome with different input parameters.
The outline of this bachelor thesis is as follows: first the principles of laser cooling
and the theoretical framework are discussed in Chapter 2, then the algorithm and
Matlab scripts are treated in Chapter 3, in Chapter 4 some results are shown and in
Chapter 5 a summary of my bachelor project is given. The Matlab scripts are printed
in Appendix A.
3
2 Description of the problem
2.1 Principles of laser cooling
In laser cooling experiments the goal is to slow down atoms by shining laser light on
them. When the atoms slow down, they lose kinetic energy and hence the temperature
of the atoms, which is a measure of kinetic energy, drops and the atoms get cooled.
When the velocity of the atoms is low enough, they can be trapped in a Magneto
Optical Trap (MOT) where they can be studied in precision experiments as they have
almost zero velocity. A full treatment of laser cooling can be found in [2], but a short
introduction is given here.
The principle of laser cooling is based on momentum transfer from the laser beam
to the atoms. If the atoms travel in a beam with a certain direction, the laser beam
should be counter propagating. If the laserlight has the right frequency to be absorbed
by the atom, photons will not only transfer their energy E = h̄ω to the atom, but
also their momentum p~ = h̄~k, where ω is the photon angular frequency and k = ω/c
the wavevector pointing in the direction of the laser beam. This momentum is in the
opposite direction of the momentum of the atom, so the atom is kicked back a little bit.
When the photon was absorbed by the atom, the atom ended up in an excited state
above the ground state. This is not a stable equilibrium so for a two-level system the
atom will decay to the ground state again after a small time. In this decay, a photon is
emitted in a random direction. This gives the atom a momentum kick in the opposite
direction of the photon. When the atom is back in the ground state, another photon
from the laser beam can be absorbed and the cycle starts again. If this cycle is done
many times, then there is no preferred direction in which the atom emits the photons
in the decay. This means that the average momentum change due to the spontaneous
emission is equal to zero, but the absorbed photons all transfer their momentum in the
same direction opposite to the velocity of the atoms. The total effect on the atoms is
thus a decrease in momentum ∆~
p = nh̄~k with n the number of absorped photons and
~
the direction of k being opposite to the direction of the atoms. The change in velocity
is then
nh̄~k
= nvr
(2.1)
∆~v =
m
with m the mass of the atom. For one photon (n = 1) we call this the recoil velocity
vr and for Ba this is
vr = h̄k/m = 0.0052 m/s
(2.2)
If the atom is not a two-level system, but has other states to which it can decay
from the excited state, the atom can be lost from the cooling cycle. If the atom stays
in this other state, the cooling laser beam is not on resonance with the possible atomic
4
Figure 2.1: The level scheme of Ba. Only the relevant states for the laser cooling are
shown. The dashed lines represent the Raman transitions.
Transition
|2i − |1i
|2i − |3i
|2i − |4i
|2i − |5i
Wavelength [nm]
553.7
1500.4
1130.6
1107.8
A2k [106 s−1 ]
119
0.25
0.11
0.0031
Lifetime
8.4 ns
4 µs
9 µs
320 µs
Table 2.1: The transitions in barium and their decay strengths. Data taken from [3].
5
transitions and the atom cannot absorb the photons anymore. It will not cool any
more until it decays further to the ground state. In Barium and other earth-alkali
these other states are metastable which means that the atoms can stay there for even
a second or more. Compared to the lifetime of the excited state in Barium of 8 ns this
is very long.
The level scheme of Ba is shown in Fig. 2.1. Barium has a 1 S0 ground state and a 1 P1
excited state and the transition has a wavelength of 553 nm. This transition is a potential cooling transition. Excited barium atoms decay to the metastable states 1 D2 , 3 D2
and 3 D1 , with branching ratios given in Table 2.1. The leak rate for these metastable
state is 1:330(30), as can be calculated from the data in Table 2.1. This means that
on average after 330 cooling cycles the atom ends up in one of the metastable states.
The average change of velocity is then from Eq. 2.2 only ∆v = 330vr = 1.7 m/s. For
an atomic beam coming from an oven at 800 K with an average speed of 220 m/s this
is not much cooling. To improve this number, the atoms need to be brought back into
the cooling cycle. This can be done with additional lasers tuned at the transitions between the metastable states and the 1 P1 state. These lasers pump the atoms from the
metastable states to the excited state where they have a bigger probability to decay
to the ground state than to decay in a metastable state again. In this way, the cooling
cycle can continue.
We want to investigate the effect of this repumping on laser cooling. The results are
directly useable in current and future experiments.
2.2 Raman transitions
The barium level scheme is in a so-called Λ-configuration, indicating the shape with
a ground state and metastable states that all share the same common excited 1 P1
upper state. In Λ-configuration the multiple lasers are not acting independently on
the atoms. The picture sketched above with the repump lasers not interacting with
the cooling laser is not complete. When two lasers that share the same upper state
have equal detunings, the transition can go directly between the two lower states. A
transition like this is called a Raman transition. They are indicated in Fig. 2.1.
In Barium, when the detuning of one of the repump lasers is equal to the detuning
of the green laser with respect to the upper state, atoms will be transported between
the ground state and the metastable state without reaching the upper state. This is a
completely coherent process, with no spontaneous emission. A photon at wavelength
λ1 is absorbed and a photon with wavelength λ2 is created by stimulated
emission
in the direction of the laser beam. The momentum transfer h̄ k~1 − k~2 6= 0, but
the momentum transfer is exactly opposite if the process goes backwards. That means
that no momentum transfer from the laser beam to the atoms occurs, hence no cooling
takes place. In spectroscopy where the 1 S0 -1 P1 transition is probed, no fluorescence is
seen when Raman transitions occur and therefore this is also called a dark resonance.
Raman transitions are a major difficulty in laser cooling of barium. Each atom sees
the lasers detuned differently due to the Doppler effect, since the beam of Ba atoms
6
from the oven has an initial Maxwell-Boltzmann distribution of speeds [2]
v2
v3
f (v) = 4 exp − 2
2v
2v
(2.3)
with v = vrms depending on temperature.
During cooling the velocities also change and the doppler shift is also present. It
is therefore not possible to just detune the lasers in a way that they don’t produce
Raman transitions. Computer simulations which take the Raman transitions as well
as the velocity change of the atoms into account are a useful tool to get insight in
these processes going on.
2.3 Theoretical model
We consider laser cooling of barium for an even isotope with nuclear spin I = 0. In this
case we can use a 4-level system for the calculations. The approach can be adapted to
the case I 6= 0 by including the hyperfine states.
The atom-laser interaction is calculated by using the Optical Bloch Equations. For
this part we followed the framework given in [6] which was set up for Ba ion spectroscopy. Instead of keeping the detunings of the lasers constant, we allow for a
change due to Doppler shift. The force on an atom is calculated from the spontaneous
scattering rate from the 1 P1 state. With this, we can calculate the change in velocity.
Since we start with a thermal velocity distribution of atoms from an effusive oven we
have to integrate over the full velocity range. On the next pages we describe these
steps in detail.
2.3.1 The Hamiltonian
The Hamiltonian of the atoms itself with eigenstates |1i, |2i, |3i and |4i indicating the
1
S0 , 1 P1 , 1 D2 and 3 D2 states respectively is defined as
Ĥatom |ai = h̄ωa with a = 1, 2, 3, 4
(2.4)
We only consider here the four most important states and neglect the 3 D1 state because
the probability for the atoms to decay to that state is two orders of magnitude lower
than for the other two metastable states (see Table 2.1). Of course this state may be
included in a straightforward way at costs of longer calculation times.
In the limit of large laser intensities, the lasers can be treated as classical electric
~ =E
~ b sin (ωb t) with b the wavelength used as an index to indicate the different
fields E
lasers. Indices 1-4 are used for internal atomic structure, and the wavelengths are used
for the laser fields. This does not implicate that the laser is perfectly on resonance, it
may be detuned as we see later on. Assuming only electric dipole interaction of the
7
atoms with the field, we can write the interaction Hamiltonian as:
~ 12 · E
~ 553 i |2ih1|e−iω553 t − |1ih2|eiω553 t +
Ĥint = − { D
~ 23 · E
~ 1500 i |2ih3|e−iω1500 t − |3ih2|eiω1500 t +
D
~ 24 · E
~ 1130 i |2ih4|e−iω1130 t − |4ih2|eiω1130 t }
D
(2.5)
~ 12 , D
~ 23 and D
~ 24 are the dipole matrix elements for the transitions between the
where D
upper state |2i and lower states |1i, |3i, |4i respectively. The non-resonant transitions
~ 12 · E
~ 1500 are neglected, as well as the oscillations much higher than the optical
like D
frequencies (rotating wave approximation).
Adding both Hamiltonians gives the total Hamiltonian in matrix form:


Ω12 +iω553 t
0
0
ω12
2 e
Ω23 −iω1500 t
Ω24 −iω1130 t 
 Ω12 e−iω553 t 0
2
2 e
2 e

(2.6)
Ĥ = h̄ 
Ω
23 +iω1500 t

 0
e
ω
0
23
2
Ω24 +iω1130 t
0
ω24
0
2 e
where the zero energy is set to |2i and ω12 = ω1 − ω2 , ω23 = ω3 − ω2 and ω24 = ω4 − ω2 .
The Rabi oscillation frequencies are defined as
(2.7)
h̄Ω23
~ 12 · E
~ 553
= D
~ 23 · E
~ 1500
= D
h̄Ω24
~ 24 · E
~ 1130
= D
(2.9)
h̄Ω12
(2.8)
2.3.2 Density matrix
In density matrix formalism the density matrix ρ̂ is defined as
X
ρ̂ =
ρab |aihb|
(2.10)
a,b=1,2,3,4
The trace Tr (ρ̂) = 1 which means that the probability of finding the atom in any of
the eigenstates is 1. The off-diagonal elements describe the coherences between two
states. The time evolution of the density matrix is expressed in the Liouville equation
i
i h
dρ̂
= − Ĥ, ρ̂ + L̂damp (ρ̂)
dt
h̄
(2.11)
where L̂damp is of the form
L̂damp (ρ̂) = −
i
1 Xh †
†
†
Ĉm Ĉm ρ̂ + ρ̂Ĉm
Ĉm − 2Ĉm ρ̂Ĉm
2 m
(2.12)
where the first two terms describe the decay from the excited states and the third term
describes the decay into the lower levels. The finite linewidth of the lasers can also be
8
included. The terms Ĉpq are:
Ĉ21
=
Ĉ23
=
Ĉ24
=
Ĉ553
=
Ĉ1500
=
Ĉ1130
=
p
A21 |1ih2|
p
A23 |3ih2|
p
A24 |4ih2|
p
2Γ553 |1ih1|
p
2Γ1500 |3ih3|
p
2Γ1130 |4ih4|
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
where A2k are the Einstein coefficients forP
the spontaneous decay. They are related to
the lifetime of the excited state as τ −1 = k A2k and can be found in Table 2.1. The
Γ’s are the linewidths of the lasers.
Equation 2.11 can be written as
dρ̂
= Lρ̂ (t)
dt
(2.19)
with
i
i h
Ĥ, ρ̂ + Ldamp (ρ̂)
(2.20)
h̄
The total system can be transformed into the rotating frame of the laser light to get


∆553
Ω12 /2 0
0
 Ω12 /2 0
Ω23 /2 Ω24 /2 

Ĥ ′ = h̄ 
(2.21)
 0

Ω23 /2 ∆1500 0
0
Ω24 /2 0
∆1130
Lρ̂ = −
with ∆553 = ω553 − ω12 , ∆1500 = ω1500 − ω23 and ∆1130 = ω1130 − ω24 specifying the
detunings of the lasers with respect to the atomic transitions.
2.4 Optical Bloch equations
The Liouville equation (2.19) can be seen as a system of linear transformations by
writing
ρ
~ = (ρ11 , ρ12 , . . . , ρ43 , ρ44 )
(2.22)
and
d~
ρ X
Mij ρ
~j
=
dt
j
(2.23)
where ρab = ha|ρ̂|bi. These equations are the Optical Bloch equations. The matrix M
is completely determined by L and it contains all the physical parameters needed to
describe the interaction of the atoms with the laserlight. Equation 2.23 can be written
in linear form as
X
dρrs
=
Mrs,kj ρkj
(2.24)
dt
kj
9
For an n-level system the matrix M is of size n × n and is called the Liouville matrix.
The matrix M is defined as:
X
∼†
i ∼
†
(2.25)
(Cm )rk Cm
Mrs,kj = −
H rk δjs − H js δrk +
js
h̄
∼
where the effective Hamiltonian H is introduced as
∼
i X †
′
Ĉ Ĉm
H = Ĥ −
2h̄ m m
(2.26)
This set of complex linear differential equations can be solved numerically on a computer.
10
3 Numerical simulations in Matlab
With the theoretical framework in place, the next step of implementing it in a computer
program is described in this chapter. For the simulation of laser cooling, the Optical
Bloch equations have to be solved in order to obtain information of the state of the
atoms during the time they spend in the laser beam. The rate of cooling can be easily
calculated from the rate of scattered photons by spontaneous emission from the 1 P1
state. This rate is:
X
ρ22
N=
= ρ22
A2k
(3.1)
τ
k
and with (2.1) we get for the amount of cooling per unit of time dt
X
dv = N vr dt = ρ22 vr dt
A2k
(3.2)
k
Since the scattering rate depends on laser detuning, where the most important one
is the Doppler shift, we have to choose for appropriately small time steps dt in a
numerical integration.
3.1 Solving the Optical Bloch equations
Two strategies were used to solve the Optical Bloch equations. The first approach was
also used by [6, 7] and consists of numerically integrating the differential equation (2.23)
by using the standard Matlab functions for solving ordinary differential equations
(ODEs). These standard ODE-solvers use smart time scaling for the problem so that
they include transient effects on small timescales. This allows one to see the oscillatory
transient effects on the populations. However, this method comes at great cost: it takes
very long to calculate.
We want to give an estimate of the time needed for the calculation. In the experiment
performed in the TRIµP group, the atoms coming from the oven travel 60 cm in the
laserbeam to reach the end of the apparatus, where measurements take place. For
a speed of order 100 m/s this means that they spend some 6 ms interacting with
the laser. The decay rates A2k on the other hand produce effects on the nanosecond
scale which will be computed by the ODE-solver. Taking into account that the whole
Maxwell-Boltzmann distribution of velocity must be calculated, for which the detuning
parameters are all different due to doppler shift, it turned out that it takes hours or
even days to compute a whole spectrum. This method was therefore abandoned after
some struggeling with it.
11
The other approached that was taken after the problems with numerical integration
is more suitable for the problem. The solution to the differential Liouville equation
(2.23)
d~
ρ X
Mij ρ
~j
=
dt
j
is
ρ
~ (t) = eMt ρ
~ (0)
(3.3)
where M is the Liouville matrix. This method involves taking the exponential of the
matrix M , for which Matlab has the function expm. The advantage of this method is
that values of ρ can be calculated for any time, and that therefore the effects on small
timescales need not be computed. This can be justified in this specific problem of
simulating laser cooling since the atomic system in laser cooling is always close to the
steady state solution and transient effects on the timescale of the Rabi frequencies or
the detunings have a small amplitude. On the other hand, the possibility to calculate
these small timescale effects still exists.
3.2 The algorithm
3.2.1 Outline
For all the initial velocities in the Maxwell-Boltzmann (MB) distribution up to 1000
m/s, the final velocities and the population of the ground state at the end of the
experimental apparatus at 60 cm are computed. This data is used as a lookup table.
Then a large number of atoms with random initial velocities are drawn from the
MB-distribution and their final velocity is looked up from the table. From this, a
histogram is made showing how many atoms are in a particular velocity class. Finally
this histogram is adjusted so that it only shows atoms that are in the ground state at
the end of the 60 cm. All the Matlab-scripts are printed in Appendix A.
3.2.2 End velocity after cooling for a specific initial velocity
To make a table with final velocities related to initial velocities the script coolatom.m
is used. As input this takes all the parameters that are in the Liouville matrix:
• Rabi frequencies Ω12 , Ω23 and Ω24 in units of 106 rad/s
• Laser detunings ∆553 , ∆1500 and ∆1130 given in MHz
• Laser linewidths Γ553 , Γ1500 and Γ1130 given in MHz
• The decay rate constants A21 , A23 and A24 given in 106 s−1
• The initial velocity of the atom v given in m/s.
The output consists of
12
• The final velocity v
• The final population in the ground state ρ11
• The distance the atoms travelled in the laser beam. This is smaller than 60 cm
only if the atoms reach zero velocity before they arrive at the end.
• The final population in the excited state ρ22
• The amount of cooling per timestep
Initially the atom is in the ground state as it comes from the oven. The effective
detuning for the lasers as the atoms see them is given by the formula for the Doppler
shift:
v
(3.4)
∆i,eff = 2π ∆i +
λi
where i is an index for the wavelengths 553, 1500 and 1130 nm and λ is given in units
of µm.
With these paramaters the script liouville4.m (modified from [6]) is called which
gives back the Liouville matrix as defined in (2.25). The exponential of this matrix
times a timestep ∆t is taken and this is then multiplied with the initial ρ
~ (0). From
the newly obtained density matrix and the original one the average population of the
excited state ρ22 is calculated. This gives then the recoil velocity according to (3.2).
The new velocity is then v ′ = v − ∆v. Then the cycle starts again: a Liouville matrix
with adjusted detunings for the new velocity v ′ is exponentiated and multiplied with
the vector ρ~ obtained from the previous cycle. This continues until the atom reaches
zero velocity or it reaches the end of the 60 cm apparatus. A timestep ∆t = 10 µs
is used. In this time the velocity change (see Fig. 4.1) and therefore the change of
Doppler shift is a small fraction of the natural linewidth of the transition. Thus the
changes in the parameters of the Liouville equation are small. In 10 µs, the maximum
recoil is ∆tvr /2τ = 3.1 m/s or equivalently a Doppler shift for the cooling laser of
δ = 3.1/0.553 = 6 MHz.
The atomic beam is assumed to be unidirectional with no 3D-effects taken into
account. Furthermore it is assumed that all the atoms are in the laser beam without
being spread in transverse directions. This is a useful approximation because in the
actual experiment, the final velocity spectra are measured for atoms in the center of
the beam by means of a lens that has its focal point in the center of the beam. In the
experiment the laser beam is focussed to match the divergence of the beam. At the
oven it has a 1 mm radius and at the end a radius of 3 mm. This means a decrease in
laser intensity because the intensity is divided over a larger solid angle. Because the
~ 2 and the sustained area, and
intensity is related to the square of the electric field |E|
~
the Rabi frequency is proportional to E (2.7), the Rabi frequency decreases linearly
with distance. This feature is included in the simulation.
13
3.2.3 Building a table with final velocities
The script computevfinal.m stores for initial velocities ranging from 0 to 1000 m/s
the final velocity. For this, it calls coolatom.m for each of the initial velocities. All
the final velocities and final ground state populations are stored in a MAT-file with
filenames describing the set of parameters. This script is meant to be the starting
point for making a velocity spectrum. All the parameters can be set at the beginning
of the script and it can scan one parameter. For example a sequence of 10 different
detunings for the green laser can be specified, and for each of these detunings the final
velocities will be calculated. A plot showing the final velocity and amount of cooling
versus initial velocity is also saved.
3.2.4 Making the final velocities spectrum
Building a histogram with the numbers of atoms in the ground state versus final
velocity is done in velocityspectrum.m by taking random velocities from a MaxwellBoltzmann distribution and looking up the final velocity from the tables built as
described in the previous paragraph. The random drawing from the MB-distribution
is done using the rejection method [1] in mbdistr.m. The data from the table files is
interpolated with a spline function to increase the resolution from 1 m/s to 0.1 m/s.
In this way a histogram containing the number of atoms in all states per final velocity
bin is made. To leave only the atoms in the ground state that can be measured in the
experiment the histogram values must be multiplied by the fraction ρ11 . For this a
reverse lookup is done: for all the velocities in the histogram, the corresponding initial
velocities are found. With this initial velocity the final ρ11 is known from the table
and the histogram value can be multiplied with it. When this procedure gives multiple
initial velocities all ending up at the same final velocity, the mean of these velocities
is used.
The histogram is then plotted as the final velocity spectrum, displaying the number
of atoms in the ground state as a function of their final velocity. Also here multiple
spectra can be plotted whilst scanning one of the parameters.
3.2.5 Summary of the computation procedure
The procedure described above can be summarized as follows:
1. Run computevfinal.m to get final velocities for initial velocities 1 m/s to 1000
m/s. This script calls:
• coolatom.m to compute the final velocity for one initial velocity. This script
calls
– liouville4.m to obtain the Liouville matrix with each time a different
detuning
2. The final velocities are now stored in .mat files with filenames describing the
input parameters.
14
3. Run velocityspectrum.m to make the final velocity spectrum. The .mat files
are loaded and for random drawn initial velocities from the MB-distribution the
final velocity is looked up to make a histogram. This script calls
• mbdistr.m which gives random numbers drawn from the MB-distribution.
15
4 Results
A number of different spectra were made using sets of parameters that seemed reasonably achievable in the experiment. In general the calculated spectra showed what
was observed in the experiment. In this chapter a few plots are shown to indicate the
possibilities of the scripts. A general picture is shown on the front page where the
thick line shows the initial Maxwell-Boltzmann distribution and the thin line the final
distribution of cooled atoms.
4.1 Crossing the resonance
In Fig. 4.1 the amount of cooling per timestep is plotted as a function of time. At
t = 0 the atoms are at the oven and as time passes, they have travelled towards the
end of the apparatus. The atoms spend this time in the cooling and repumping laser
beams and are therefore slowed down. This cooling changes the doppler shift and this
means that the amount of cooling will vary along the path: when the lasers are on
resonance with the atoms the atoms are cooled more than off resonance. The figure
shows two peaks with a lorentzian profile. The one to the left is the resonance of the
1500 nm laser and the one to the right is for the 1130 nm repump laser which are both
on resonance at different speeds of the atoms. This plot also tells that most of the
cooling only takes place in a part of the apparatus and that it may be more efficient
to broaden the lasers than to add more length to the apparatus.
4.2 The amount of cooling power
In Fig. 4.2 the final velocity spectra are plotted for a set of different Rabi frequencies
of the cooling transition. The green laser is detuned for a velocity of 210 m/s and it
can be seen that atoms were cooled from that velocity class. As the cooling power
increases, the peak shifts to lower velocities until it reaches some saturation level for
which no additional cooling happens. In this configuration, Ω12 > 50 × 106 rad/s does
not cool more but it only removes more atoms from the ground state to one of the
excited states. This indicates that the repump intensities are not high enough or that
they are out of resonance.
4.3 The amount of repumping power
In Fig. 4.3 the final velocity spectrum for different 1500 nm repumping laser Rabi
frequencies Ω23 . The major feature in this plot is that for low repumping power,
16
Velocity loss in 10 microseconds
0.3
0.25
0.2
0.15
0.1
0.05
0
0
500
1000
1500
2000
2500
Time in microseconds
3000
3500
Figure 4.1: Amount of cooling per time step in terms of velocity loss. Parameters:
Ω12 = 50, Ω23,24 = 5 × 106 rad/s. ∆553 = −380, ∆1500 = −150, ∆1130 =
−160 MHz. Initial velocity is 210 m/s.
the atoms all end up in the metastable 1 D2 state and therefore they are invisible in
this plot. The plot also shows that the peak velocity where the most atoms are, is
shifted from approximately 100 m/s to 50 m/s by a change in Rabi frequency from 1
to 10 × 106 rad/s. In Fig. 4.4 again the total number of atoms in the ground state up
to a velocity of 100 m/s is plotted as a function of Ω23 . Clearly this shows saturation
starting from 10 × 106 rad/s. With the other parameters fixed, increasing the repump
laser intensity further more does not increase the amount of cooling. Calculations for
the other repump laser at 1130 nm show comparable results.
4.4 Detunings of the lasers
Changing the detunings of the lasers must happen in such a way that all the lasers
are still on resonance for the same velocity class. When for example the cooling laser
is detuned for a velocity of 100 m/s and the repump lasers for 500 m/s, there will be
no repumping for the atoms that were cooled and thus not much cooling will happen
at all. The lasers should be detuned in such a way that the cooling is most. The
repump lasers may be slightly red detuned with respect to the cooling laser because
they can start their work after some cooling has been done. Changing the detuning of
the cooling laser basically selects another velocity class that will be cooled. Selecting a
part of the Maxwell-Boltzmann distribution with many atoms also cools many atoms,
but maybe not to very low speeds. Selecting a part of the MB-distribution with low
initial velocities and fewer atoms cools less atoms, but yields lower final speeds.
In Fig. 4.5 the spectra are shown for different detunings of the cooling laser with
17
Number of atoms in ground state
1
5
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30
50
100
3500
3000
2500
2000
1500
1000
500
0
180
190
200
210
Final velocity [m/s]
220
230
Figure 4.2: Final velocity spectrum with different Rabi frequencies Ω12 (indicated
in units of 106 rad/s). Parameters: Ω23,24 = 5 × 106 rad/s. ∆553 =
−380, ∆1500 = −150, ∆1130 = −160 MHz
all other parameters constant. The peaks are shifted to lower velocities for smaller
detunings. The higher peaks are just below 100 m/s, that is where the repump lasers
are detuned for and the combined effect of cooling and repumping lasers is most
efficient.
Changing the detuning of the 1500 nm with the other lasers fixed shows a resonance
crossing. In Fig. 4.6 the number of atoms up to 100 m/s is shown as function of ∆1500 .
A peak is visible from which the optimal detuning can be read.
4.5 Comparison with experimental data
The plots looks very similar to what was measured in the experiment [5] as shown in
Fig. 4.7 although the experimental data are more like one smooth peak enveloping the
individual sharp peaks as shown for example in Fig. 4.3. This is an effect of the spread
in laser beam intensity: not all the atoms see the same intensity due to the geometry
of the beams and the apparatus. To get a simulation result that looks more like the
experiment, one should integrate over multiple intensities. However, the scope of this
thesis is more on looking at the influences of the physical parameters than on fitting
it to experimental data and therefore this integration was not performed.
18
Number of atoms in ground state
2500
1
2
3
4
5
10
15
20
25
2000
1500
1000
500
0
0
50
Final velocity [m/s]
100
150
Figure 4.3: Final velocity spectrum with different Rabi frequencies Ω23 , that are shown
in the legend in units of 106 rad/s. Parameters: Ω12 = 124, Ω24 = 20 × 106
rad/s. ∆553 = −190, ∆1500 = −80, ∆1130 = −80 MHz.
Number of atoms in
ground state up to 100 m/s
10000
5000
0
0
5
10
15
20
Green transition Rabi frequeny Ω23 [106 rad/s]
25
Figure 4.4: Total number of atoms in ground state up to 100 m/s scanning Rabi frequencies Ω23 . Parameters: Ω12 = 124, Ω24 = 20 × 106 rad/s. ∆553 =
−190, ∆1500 = −80, ∆1130 = −80 MHz
19
Number of atoms in ground state
−300
−260
−220
−200
−180
−160
−140
−120
−100
1200
1000
800
600
400
200
0
50
100
150
Final velocity [m/s]
200
250
Figure 4.5: Final velocity spectrum with different 553 nm detunings indicated in the
legend in MHz. Parameters: Ω12 = 50, Ω23,24 = 5 × 106 rad/s. ∆1500 =
−60, ∆1130 = −90 MHz.
Total number of atoms in
ground state up to 100 m/s
10000
8000
6000
−200
−150
−100
−50
1500 nm laser detuning [MHz]
0
Figure 4.6: Total number of atoms in ground state up to 100 m/s scanning 1500 nm
detunings ∆1500 . Parameters: Ω12 = 124, Ω23 = 18, Ω24 = 15 × 106 rad/s.
∆553 = −190, ∆1130 = −80 MHz.
20
Figure 4.7: Results from the experiment. Fluorescence is a measure of population of
the ground state. The points connected with a line are the initial MaxwellBoltzmann distribution. The other points are the final velocity spectrum
after cooling. [5]
21
5 Summary
This bachelor thesis reports what was done during a three month research project
in the TRIµP group. First, a general introduction on laser cooling was given. For
the specific cooling of barium, the Λ-configuration and dark Raman transitions were
explained. Because the Ba-atoms can leave the cooling cycle by means of decaying
into a metastable state, repump lasers are needed for the cooling process to continue.
This however makes Raman transitions possible in which the atoms oscillate between
the ground state and the metastable states without ever entering the cooling cycle.
Therefore the detunings of the repump lasers have great influence on the final amount
of cooling.
In Chapter 2 a theoretical description of the problem was given in terms of Optical Bloch Equations and the Liouville equation. With density matrix formalism a
straightforward way to calculate the time evolution of the velocity of atoms in the laser
beam was given in matrix form. In Chapter 3 this was extended with an algorithm to
use these theoretical equations to make a final velocity spectrum showing the number
of atoms per final velociy class. This chapter also describes how this was implemented
in a set of Matlab scripts that are printed in Appendix A.
In Chapter 4 some outcomes of the simulations are shown. The influence of cooling
and repumping power, as well as laser detunings on the final velocities of the atoms
are plotted.
The goal of the simulations was to get insight in the parameters that play a role in
laser cooling of barium. This goal was achieved: the relevant physics are included and
the results show where one should look for the optimal parameters in the experiment.
In addition to the specific barium case, the simulations may be extended to more level
systems or to other atoms or ions such as radium.
In addition to the above, my time in the TRIµP group was also a great opportunity to
see how a research group works in real life. I learned a lot about the other experiments
going on and I really liked my stay at KVI.
5.1 Acknowledgements
I would like to thank dr. Lorenz Willman for supervising my bachelor thesis and
for explaining me a lot about laser cooling, prof.dr. Klaus Jungmann for making me
enthusiastic about fundamental physics and showing me many other interesting things
happening at the KVI and Subhadeep De for his input from the experiment. The
TRIµP group as a whole I thank for listening to my talks in the meetings and the
feedback they gave. It was fun having Oscar and Klaas as office mates and I liked
22
the fruitful discussions I had with them and Wilbert (thanks for letting me use your
computer) and Lotje.
23
A Appendix - Matlab files
A-1 liouville4.m
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function L= l i o u v i l l e 4 ( O12 , O23 , O24 , D553 , D1500 , D1130 , g553 , g1500 , g1130 , A21 , A23 ,
A24 )
% LIOUVILLE4 calculates the 4- level Liouville matrix
%
This script makes the Liouville matrix for a four level system in
%
lambda - configuration. The level scheme is shown below:
%
%
__________ |2> 1P1
%
|
\ \
%
|
\ \ 1500
%
|553
\ \ O23 ,A23 ,D1500 ,l1500
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|O12 ,A21 \ \ ________|3> 1D2
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|D553 , l553\
%
|
1130 \ ________|4> 3D2
%
|
O23 ,A24 ,D1130 , l1130
%
|
%
| __________ |1> 1S0
%
%
Required input parameters:
%
- Rabi frequencies O12 , O23 , O24 in units of 10^6 rad/s
%
- Laser detunings including doppler shift D553 , D1500 , D1130 in MHz
%
- Laser linewidths g553 , g1500 , g1130 in MHz
%
- Einstein A coefficients/ decay constants A12 , A23 , A24 in 10^6/s
%
%
Output consists of the 16 x16 Liouville matrix
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25 z =4;
26 E=eye ( z ) ; % eye(z) is identity matrix of dimension z
27 i i = (1 : z ) ’ ∗ o nes ( 1 , z ) ; %[ 1 1 1 1; 2 2 2 2; 3 3 3 3; 4 4 4 4]
28
29 i 1=reshape ( i i ’ , 1 , z ˆ 2 ) ;
% first index [1 1 1 1 2 2 2 3 3 3]
30 i 2=reshape ( i i , 1 , z ˆ 2 ) ;
% second index [1 2 3 4 1 2 3 1 2 3]
31 % Coherent hamiltonian
32 H=[D553 O12/2 0 0 ; O12/2 0 O23/2 O24/2 ; 0 O23/2 D1500 0 ; 0 O24/2 0 D1130 ] ;
33 % Relaxation
34 C1=sqrt ( A21 ) ∗E ( : , 1 ) ∗E ( 2 , : ) ; %A21|1 > <2|
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C2=sqrt ( A23 ) ∗E ( : , 3 ) ∗E ( 2 , : ) ; %A23 |3 > <2|
C3=sqrt ( A24 ) ∗E ( : , 4 ) ∗E ( 2 , : ) ; %A24 |4 > <2|
C4=sqrt ( 2 ∗ g553 ) ∗E ( : , 1 ) ∗E ( 1 , : ) ; % g553 |1 > <1|
C5=sqrt ( 2 ∗ g1500 ) ∗E ( : , 3 ) ∗E ( 3 , : ) ; % g1500 |3 > <3|
C6=sqrt ( 2 ∗ g1130 ) ∗E ( : , 4 ) ∗E ( 4 , : ) ; % g1130 |4 > <4|
CC=C1 ’ ∗ C1+C2 ’ ∗ C2+C3 ’ ∗ C3+C4 ’ ∗ C4+C5 ’ ∗ C5+C6 ’ ∗ C6 ;
% Effective Hamiltonian
H=H−i /2∗CC;
Hc = conj (H) ; %Hc is the complex conjugate of the Hamiltonian
L=−i ∗ (H( i 1 , i 1 ) . ∗ E( i 2 , i 2 )−Hc ( i 2 , i 2 ) . ∗ E( i 1 , i 1 ) ) ;
% " feeding " terms
f e e d i n g = C1 ( i 1 , i 1 ) . ∗ C1 ( i 2 , i 2 )+C2 ( i 1 , i 1 ) . ∗ C2 ( i 2 , i 2 )+C3 ( i 1 , i 1 ) . ∗ C3 ( i 2 , i 2 )+C4 ( i 1
, i 1 ) . ∗ C4 ( i 2 , i 2 )+C5 ( i 1 , i 1 ) . ∗ C5 ( i 2 , i 2 )+C6 ( i 1 , i 1 ) . ∗ C6 ( i 2 , i 2 ) ;
49 L = L + f e e d i n g ;
25
A-2 coolatom.m
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function [ v , rho11 , d i s t a n c e , rho22 , r e c o i l t i m e ] = co o la to m ( O12 , O23 , O24 , D553 , D1500
, D1130 , g553 , g1500 , g1130 , A21 , A23 , A24 , v , maxdist , t )
% COOLATOM calculates the cooling of one atom with initial velocity v
%
This function calculates the final velocity of an atom at the end of
%
the experimental apparatus. This is done by making a Liouville matrix
%
with the desired parameters and then calculating the density matrix by
%
exponentiating: rho(t) = expm(L*t)*rho (0)
%
%
Input parameters that are required:
%
- Rabi frequencies O12 , O23 , O24 in units of 10^6 rad/s
%
- Laser detunings D553 , D1500 , D1130 in MHz
%
- Laser linewidths g553 , g1500 , g1130 in MHz
%
- Einstein A coefficients/ decay constants A12 , A23 , A24 in 10^6/s
%
- Initial velocity of the atom v in m/s
%
- The length of the apparatus maxdist in m
%
- The timestep t used in the equation above
%
%
Output consists of:
%
- Final velocity v in m/s
%
- Final population of the ground state rho11
%
- The distance travelled in m
%
- Final population of the excited state rho22
%
- The amount of recoil per timestep: a vector recoiltime in m/s
%
%
See also LIOUVILLE4
lambda553 = . 5 5 3 7 ;
lambda1500 = 1 . 5 0 0 4 ;
lambda1130 = 1 . 3 0 0 6 ;
recoiltime = [ ] ;
vr = 0 . 0 0 5 2 ;
% Start in groundstate
r ho 1 1 = 1 ;
r ho 2 2 = 0 ;
rho = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ’ ;
distance = 0;
co mpleter un = 0 ;
%The initial Rabi frequencies at the oven
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O12i = O12 ;
O23i = O23 ;
O24i = O24 ;
while ˜ co mpleter un
r 0 = rho ;
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rho = expm( l i o u v i l l e 4 ( O12 , O23 , O24 , 2 ∗ pi ∗ ( D553+v/ lambda553 ) , 2 ∗ pi ∗ ( D1500+v/
lambda1500 ) , 2 ∗ pi ∗ ( D1130+v/ lambda1130 ) , g553 , g1500 , g1130 , A21 , A23 , A24 ) ∗ t )
∗ r0 ;
r ho 1 1 = rho ( 1 ) ;
r ho 2 2 = r e a l ( rho ( 6 ) ) ;
r e c o i l = ( A21+A23+A24 ) ∗ ( r e a l ( r 0 ( 6 ) )+r ho 2 2 ) /2∗ t ∗ vr ;
recoiltime = [ recoiltime ; recoil ] ;
v = v − recoil ;
d i s t a n c e = d i s t a n c e + ( v +0.5∗ r e c o i l ) ∗ t ∗1 e −6;
O12 = O12i ∗ 0 . 6 / ( 3 ∗ d i s t a n c e ) ;
O23 = O23i ∗ 0 . 6 / ( 3 ∗ d i s t a n c e ) ;
O24 = O24i ∗ 0 . 6 / ( 3 ∗ d i s t a n c e ) ;
i f d i s t a n c e > maxdist | v <= 0
i f distance > 0.6
v = v + ( d i s t a n c e −maxdist ) / ( v +0.5∗ r e c o i l ) ∗ r e c o i l / t ;
end
co mpleter un = 1 ;
end
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A-3 computevfinal.m
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% COMPUTEVFINAL saves final velocities for a set of initial velocities
%
For a set of input parameters , this program calculates the final
%
velocities of atoms with initial velocities from 1 p to vmax m/s by
%
calling COOLATOM.M for each initial velocity. In this way a table
%
relating initial velocity to final velocity is created and saved.
%
%
All variables can be set below and are described as:
%
- Rabi frequencies O12 , O23 , O24 in units of 10^6 rad/s
%
- Laser detunings D553 , D1500 , D1130 in MHz
%
- Laser linewidths g553 , g1500 , g1130 in MHz
%
- Einstein A coefficients/ decay constants A12 , A23 , A24 in 10^6/s
%
- The length maxdist of the apparatus in m
%
- The timestep tstep in microseconds
%
- The maximum velocity vmax in m/s
%
The parameter list can be directly copied & pasted to VELOCITYSPECTRUM.M
%
%
One of the parameters can be scanned by assigning a vector to it. Then
%
line 39 in the script must be changed accordingly.
%
%
See also COOLATOM
cl ear a l l ;
O12 = [ 0 : 1 : 4 ] ;
O24 = 5 ;
O23 = 5 ;
D553 = −380;
D1130 = −160;
D1500 = −150;
g553 = 1 ;
g1500 = 0 . 0 0 5 ;
g1130 = 0 . 0 0 5 ;
A21 = 1 . 1 9 e2 ;
A23 = 0 . 2 5 ;
A24 = 0 . 1 1 ;
maxdist = 0 . 6 ;
tstep = 10;
vmax = 1 0 0 0 ;
for O12 = [ 0 : 1 : 4 ]
v f i n a l = zeros ( 1 , vmax ) ;
r 1 1 f i n a l = zeros ( 1 , vmax ) ;
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r 2 2 f i n a l = zeros ( 1 , vmax ) ;
vspectrum = zeros ( 1 , vmax ) ;
%
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for v i n = 1 : 1 : vmax−1
[ vend , r11end , d i s t a n c e , r22end , r e c o i l t i m e ] = co o la to m ( O12 , O23 , O24 , D553 ,
D1500 , D1130 , g553 , g1500 , g1130 , A21 , A23 , A24 , vin , maxdist , t s t e p ) ;
% Uncomment to see the time development of the cooling .
if vin == 210
figure (8)
plot( tstep *[1:1: length( recoiltime)], recoiltime)
title(’recoil versus time ’);
xlabel(’ time in microseconds ’);
ylabel(’ Velocity loss in 10 microseconds ’);
end
v f i n a l ( v i n ) = vend ;
r 2 2 f i n a l ( v i n ) = r 2 2 end ;
r 1 1 f i n a l ( v i n ) = r 1 1 end ;
end
f i l e n a m e = [ ’ spectrum /O12− ’ i n t 2 s t r ( O12 ) ’ O23− ’ i n t 2 s t r ( O23 ) ’ O24− ’
i n t 2 s t r ( O24 ) ’ D553− ’ i n t 2 s t r ( D553 ) ’ D1500− ’ i n t 2 s t r ( D1500 ) ’ D1130− ’
i n t 2 s t r ( D1130 ) ’ g 5 5 3− ’ i n t 2 s t r ( g553 ) ’ g 1 5 0 0− ’ i n t 2 s t r ( g1500 ∗ 1 0 0 0 ) ’
g 1 1 3 0− ’ i n t 2 s t r ( g1130 ∗ 1 0 0 0 ) ’ d i s t − ’ i n t 2 s t r (1 0 0 ∗ maxdist ) ’ t − ’
int2str ( tstep ) ] ;
f i l e n a m e 1 = [ f i l e n a m e ’ . mat ’ ] ;
save ( f i l e n a m e 1 , ’ v f i n a l ’ , ’ r 1 1 f i n a l ’ , ’ r 2 2 f i n a l ’ ) ;
v s c a t t e r f i g = fi gure ;
plot ( v f i n a l , ’ . ’ ) ;
t i t l e ( ’ Where do t h e i n i t i a l v e l o c i t i e s go t o ? ’ ) ;
xlabel ( ’ I n i t i a l v e l o c i t y [m/ s ] ’ )
ylabel ( ’ F i n a l v e l o c i t y /amount o f c o o l i n g [m/ s ] ’ ) ;
hold on
plot ( ( 1 : vmax )−v f i n a l , ’ . r ’ ) ;
legend ( ’ F i n a l v e l o c i t y ’ , ’ Amount o f c o o l i n g ’ ) ;
hold o f f
filename2 = [ filename ’ . f i g ’ ] ;
saveas ( v s c a t t e r f i g , filename2 ) ;
cl os e a l l
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A-4 velocityspectrum.m
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% VELOCITYSPECTRUM computes the final velocity specrum after cooling
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This program uses the data calculated by COMPUTEVFINAL to build the
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spectrum showing the number of atoms in the ground state as function of
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final velocity.
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All variables can be set below and are described as:
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- Rabi frequencies O12 , O23 , O24 in units of 10^6 rad/s
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- Laser detunings D553 , D1500 , D1130 in MHz
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- Laser linewidths g553 , g1500 , g1130 in MHz
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- Einstein A coefficients/ decay constants A12 , A23 , A24 in 10^6/s
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- The length maxdist of the apparatus in m
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The parameter list can be directly copied & pasted from COMPUTEVFINAL.M
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One of the parameters can be scanned by assigning a vector to it. Then
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other lines in the script must be changed accordingly ( lines 57 and
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from 118 onwards ). Plots are made and shown with legends that must be
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verified.
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To calculate the spectrum , random initial velocities are drawn from
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See also MBDISTR , COMPUTEVFINAL
O12 = [ 1 : 4 5 : 5 : 1 0 0 ] ;
O24 = 5 ;
O23 = 5 ;
D553 = −380;
D1130 = −160;
D1500 = −150;
g553 = 1 ;
g1500 = 0 . 0 0 5 ;
g1130 = 0 . 0 0 5 ;
A21 = 1 . 1 9 e2 ;
A23 = 0 . 2 5 ;
A24 = 0 . 1 1 ;
maxdist = 0 . 6 ;
tstep = 10;
vmax = 1 0 0 0 ;
natoms = 1 0 0 0 0 0 ;
integral = [ ] ;
peakenh = [ ] ;
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44 m = 1 3 8 ∗ 1 . 6 6 e −27;
45 k = 1 . 3 8 e −23;
46 Temp = 8 0 0 ;
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48 vmean = sqrt ( k∗Temp/m) ;
49 v = 1 : 0 . 1 : 1 0 0 0 ;
50 mb2 = v . ˆ 3 . / ( 2 ∗ vmean ˆ 4 ) . ∗ exp(−v . ˆ 2 . / ( 2 ∗ vmean ˆ 2 ) ) ;
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52 fi gure ( 1 )
53 hold on
54 fi gure ( 7 )
55 hold on
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57 for O12 = [ 1 : 5 1 0 : 1 0 : 1 0 0 ]
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f i l e n a m e = [ ’ spectrum /O12− ’ i n t 2 s t r ( O12 ) ’ O23− ’ i n t 2 s t r ( O23 ) ’ O24− ’
i n t 2 s t r ( O24 ) ’ D553− ’ i n t 2 s t r ( D553 ) ’ D1500− ’ i n t 2 s t r ( D1500 ) ’ D1130− ’
i n t 2 s t r ( D1130 ) ’ g 5 5 3− ’ i n t 2 s t r ( g553 ) ’ g 1 5 0 0− ’ i n t 2 s t r ( g1500 ∗ 1 0 0 0 ) ’
g 1 1 3 0− ’ i n t 2 s t r (1 0 0 0 ∗ g1130 ) ’ d i s t − ’ i n t 2 s t r (1 0 0 ∗ maxdist ) ’ t − ’
i n t 2 s t r ( t s t e p ) ’ . mat ’ ] ;
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da ta = load ( f i l e n a m e ) ;
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mbdist = mb2 ;
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v f = da ta . v f i n a l ;
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r 1 1 = da ta . r 1 1 f i n a l ;
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v f s p l i n e = s pl i ne ( 1 : 1 0 0 0 , vf , 1 : 0 . 1 : 1 0 0 0 ) ;
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r 1 1 s p l i n e = s pl i ne ( 1 : 1 0 0 0 , r11 , 1 : 0 . 1 : 1 0 0 0 ) ;
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i n i t i a l v s = mbdistr ( natoms ) ; %get natoms random velocities from MB
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f i n a l v s = zeros ( 1 , length ( i n i t i a l v s ) ) ;
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for i = 1 : length ( i n i t i a l v s )
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f i n a l v s ( i ) = v f s p l i n e ( round ( i n i t i a l v s ( i ) ∗ 1 0 ) +1) ;
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end
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[ mbrnd , x ] = h i s t ( i n i t i a l v s , 2 0 0 ) ;
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[ n , x ] = hist ( f i n a l v s , 2 0 0 ) ;
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% Now find for each final velocity in the histogram the corresponding
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% initial velocity. Then multiply the number of atoms at that final
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% velocity , with the fraction rho11 to keep only the ground state atoms.
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% rho11 is only indexed with the initial velocity , hence the conversion
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% from final to initial velocity is needed .
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for k = 3 : length ( n )
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done = 0 ;
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% final velocties in the histogram are integers , so when doing
%find () it may not find the corresponding initial velocity because
%that is a floating point. Then we should widen the search .
%So if vfinal == n gives nothing , try n-1 < vfinal < n+1
%The factor 5 comes in because the histogram contains 200 bins for
%1000 velocities.
wider = 0 ;
while ˜ done
v i n i t i a l i n d e x = round (mean( find ( k∗5−wider < v f s p l i n e ( 2 : end−10) &
v f s p l i n e ( 2 : end−10) < k∗5 +wider ) ) ) ;
i f ˜ isnan ( v i n i t i a l i n d e x )
done = 1 ;
else
wider = wider +1;
end
end
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r11final = r11spline ( vinitialindex ) ;
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n( k ) = n( k ) ∗ real ( r 1 1 f i n a l ) ;
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end
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fi gure ( 1 )
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plot ( x , n , ’+−−r ’ ) ;
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fi gure ( 7 )
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plot ( x , n . / mbrnd , ’+−−k ’ ) ;
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sum( n )
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[ peak , i n d i c e s ] = max( n ( 2 : 4 4 ) ) ;
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i f i n d i c e s == 1
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peak = 0 ;
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end
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r11spline ( indices )
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peakenh = [ peakenh peak / ( mbdist ( ( i n d i c e s ) ∗50+1)∗ length ( i n i t i a l v s ) ) ] ;
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i n t e g r a l = [ i n t e g r a l sum( n ( 2 : 4 4 ) ) ] ;
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117 end
118 hold o f f
119 fi gure ( 1 )
120 legend ( i n t 2 s t r ( [ 1 : 5 1 0 : 1 0 : 1 0 0 ] ’ ) )
121 fi gure ( 6 )
122 plot ( [ 1 : 5 1 0 : 1 0 : 1 0 0 ] , i n t e g r a l ) ;
123 t i t l e 1 = s p r i n t f ( ’ Enhancement o f c o o l i n g . \ n 553nm l i n e w i d t h %d MHz. 1500 nm
d e t u n i n g %d MHz. 1130 nm d e t u n i n g %d MHz. \ n ’ , g553 , D1500 , D1130 ) ;
124 t i t l e 2 = s p r i n t f ( ’ Rabi f r e q u e n c i e s \\Omega {12} = %d , \\Omega { 2 3 , 2 4 } = %d \\
t i m e s 10ˆ6 rad / s . ’ , O12 , O23 ) ;
125 t i t l e 3 = [ t i t l e 1 t i t l e 2 ] ;
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xlabel ( ’ Rabi f r e q u e n c y \\Omega {12} [ 1 0 ˆ 6 rad / s ] ’ ) ;
% title( title3 )
legend ( ’ I n t e g r a l under peak up t o 220 m/ s ’ , ’ Enhancement a t peak v a l u e ’ )
fi gure ( 1 )
t i t l e 1 = s p r i n t f ( ’ F i n a l v e l o c i t y spectrum . 553 nm d e t u n i n g −300:20: −100 MHz.
1500 nm d e t u n i n g %d MHz. 1130 nm d e t u n i n g %d MHz. \ n ’ , D1500 , D1130 ) ;
t i t l e 2 = s p r i n t f ( ’ Rabi f r e q u e n c i e s \\Omega {12} = %d , \\Omega { 2 3 , 2 4 } = %d \\
t i m e s 10ˆ6 rad / s . 553 nm l i n e w i d t h %d MHz. ’ , O12 , O23 , g553 ) ;
title3 = [ title1 title2 ];
title ( title3 )
xlabel ( ’ F i n a l v e l o c i t y [m/ s ] ’ ) ;
legend ( i n t 2 s t r ( l 5 5 3 ’ ) ) ;
ylabel ( ’ Number o f atoms i n ground s t a t e ’ ) ;
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A-5 mbdistr.m
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function numbers = mbdistr ( natoms )
% MBDISTR draws random numbers from Maxwell - Boltzmann distribution
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The random numbers are generated via the rejection method algorithm.
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For the velocity distribution the MB - distribution from an oven is used
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with the atoms travelling in one direction.
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Input parameter required:
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- natoms : how many numbers are drawn from the distribution
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Output consists of the vector numbers containing natom random
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velocities.
m = 1 3 8 ∗ 1 . 6 6 e −27;
k = 1 . 3 8 e −23;
Temp = 8 0 0 ;
vmean = sqrt ( k∗Temp/m) ;
mb2 = @( v ) v . ˆ 3 . / ( 2 ∗ vmean ˆ 4 ) . ∗ exp(−v . ˆ 2 . / ( 2 ∗ vmean ˆ 2 ) ) ;
f = @( v ) 0 . 0 0 7 ∗exp(−v / 5 0 0 ) ; % function always bigger than mb2
g = @( v ) 3.5∗(1 − exp(−v / 5 0 0 ) ) ; % integral of f from 0 to v
numbers = zeros ( 1 , natoms ) ;
for i = 1 : length ( numbers )
rejected = 1;
while r e j e c t e d
u = 3∗rand ;
v = −500∗log(1−u / 3 . 5 ) ;
w = f ( v ) ∗rand ;
i f w <= mb2( v )
rejected = 0;
end
end
numbers ( i ) = v ;
end
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Bibliography
[1] Numerical recipes, chapter 7.3. Cambridge University Press, 1992.
[2] Laser Cooling and Trapping. Springer, 1999.
[3] A. Bizzarri and M. C. E. Huber. Transition probabilities from the 6s6p 1 P1 resonance level of neutral
barium. Phys. Rev. A, 42(9):5422–5424, Nov 1990.
[4] Umakanth Dammalapati. Metastable D-State Spectroscopy and Laser Cooling of Barium. PhD thesis,
University of Groningen, 2006.
[5] Subhadeep De. Doing a PhD on the cooling experiment, KVI Private communication.
[6] Hilmar Oberst. Resonance fluorescence of single barium ions. Master’s thesis, Universität Innsbruck,
1999.
[7] Christoph Raab. Interference experiments with the fluorescence light of Ba+ ions. PhD thesis, Universität
Innsbruck, 2001.
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