Simulations of the Zeeman Slower for the Al Catraz Experiment

Simulations of the Zeeman slower for the Al41Catraz experiment

M.G. Bearda

KVI-Theory Group & Computer Science Intelligent Systems group

Under supervision of: L.W. Wansbeek, R.G.E Timmermans, M. Biehl

KVI University of Groningen November 29, 2010 Contents

1 Introduction 4 1.1 Objectives...... 4 1.2 Outline ...... 4

2 Experimental Setup 5 2.1 Oven...... 6 2.2 Compression ...... 6 2.3 Zeemanslower ...... 7 2.4 Deﬂection ...... 8 2.5 MagneticOpticalTrap(MOT) ...... 8

3 Theory 9 3.1 Dopplereffect...... 9 3.2 Emissionrate ...... 9 3.3 CollisionsofCalciumatoms ...... 10 3.3.1 Collision rate for all velocities and positions ...... 10 3.3.2 Collision probability given the begin velocity of an atom...... 10 3.4 Effectsofinputparametersonresults ...... 11 3.4.1 Magneticfield...... 12 3.4.2 Detuningofthelaser...... 12 3.4.3 Laserpower...... 12 3.4.4 Focusofthelaserbeam ...... 12

4 Simulation 13 4.1 Basicrandomnumbergenerating ...... 13 4.1.1 Uniformly distributed pseudo-random values ...... 13 4.1.2 Gaussian distributed pseudo-random values ...... 13 4.2 Distributions of random values using invertible functions...... 14 4.2.1 Randompointonasphere...... 14 4.2.2 Calculatingwaitingtime...... 16 4.3 Distributions of random values using the numerical inverse...... 16 4.4 Parallelization...... 17 4.5 Comparisonoldandnewapproach ...... 17 4.5.1 Keepingtherandomvalues ...... 17

5 Results 19 5.1 Comparing the eﬀect of detuning the laserlight ...... 20 5.2 Laserbeamfocus...... 21 5.3 Collisionprobability ...... 21

1 CONTENTS

6 Recommendations 24 6.1 Maincoil ...... 25 6.2 Peak...... 26 6.3 Dip...... 27 6.4 Laserpower...... 28 6.5 Detuning ...... 29 6.6 Focusinglaserbeam ...... 30 6.7 Bestsettingscombined...... 31

7 Dankwoord 33

A Design 34

B Pseudo code 35 B.1 Pseudocodefortheleader...... 36 B.2 Pseudocodefortheworker ...... 36

C Syntax of the conﬁguration ﬁle 38 C.1 Groups...... 41

2 CONTENTS

Abstract

For the Al 41Catraz experiment I simulated the Zeeman slower. With a new approach for the timestep and extensive search for the best parameters I came up with in improvement with a factor of approximately 5 × 102. The recommended settings are shown in Table 0.1.

Variable van Ditzhuijzen Mollema Bearda [van Ditzhuijzen, 2002] [Mollema, 2008] Main current 4.0 A 4.0 A 4.0 A Peakcurrent 6.0A 0.0A 0.0A Dip current 6.0 A 0.0 A 5.6 A Laserpower 30mW 40mW 30mW Laser Detuning −370 MHz −415 MHz −550 MHz Radius of laser beam at z = 0 m 4 mm 8 mm 1 mm Radius of laser beam at z =0.9 m 8 mm 8 mm 8 mm Acceptance fraction 19.9% <0.1% 35.8%

Table 0.1: Settings for the Zeeman slower

3 1 INTRODUCTION

1 Introduction

The Al 41Catraz experiment[van Ditzhuijzen, 2002, de Haan, 2006] is performed at the Kernfysisch Versneller Instituut (KVI), an experimental and theoretical physics institute of the University of Groningen. The Al 41Catraz experiment was started in 2001 by Hoekstra[Hoekstra, 2005] and was till 2008 under the care of Mollema[Mollema, 2004, Mollema, 2008] . The main goal of the experiment is to determine the concentration of the 41Ca isotope in a natural sample. In natural samples the concentration of 41Ca atoms is in the order of 10−14, so it is very important to reach a high level of eﬃciency. The trapping of the 41Ca isotope has two main application options. The 41Ca isotope has a lifetime of roughly 105 years. So it is a candidate to take over where carbon-dating looses its precision[Plastino et al., 2001]. Another application is due to the natural low concentration of the 41Ca isotopes. This suggests that it might be possible to use the 41Ca isotope as a tracer in biomedical research into bones. For more details of these applications see [Mollema, 2004]. Many goals for the experiment have been achieved, like trapping atoms of the 40,42,43,44,46,48Ca isotopes [Hoekstra et al., 2005]. Nevertheless, the 41Ca isotope has yet not been trapped. The motivation for this research, is to look how the set of parameters to improve the results in the experiment. These are found by doing Monte Carlo simulations of the Zeeman slower.

1.1 Objectives

The simulations study the parameters of the Zeeman slower, because it is thought that the most calcium atoms are lost in this phase of the trapping of the atom. For these simulations a simulator- platform is built that can be controlled with a conﬁguration ﬁle. No knowledge of the underlying programming language is needed to do simulations of the calcium atoms or any other particles that might be of future interest.

1.2 Outline

First, in Section 2 all five parts of slowing the atoms down are briefly described. In Section 3 the theory behind the slowing of atoms is explained. In this section the formulas that are used in the simulation are introduced. In Section 4 the computer model of the Zeeman slower is described. This section also contains a justification on the random number generator that is used, accompanied with an explanation why this random number generator is good enough to model the process of slowing the calcium atoms down. Simulation results acquired with the created program are given in Section 5, leading up to the recommendations formulated in Section 6. Finally the design of the simulation platform, the pseudo code of different parts of the platform and the syntax of the configuration file is described in the appendices.

4 2 EXPERIMENTAL SETUP

2 Experimental Setup

1 1 1 3 3 S0 P1 D2 S1 P0,1,2

4 671.8 1.2 10 7

3 272.2 2180 2.7 10 5 300 2 422.7 2.2 10 8 457.0 96 40 1 657.3 2 10 3 0

singlet triplet

Figure 2.1: Energy level diagram of the 40Ca-atom. In this experiment the transition of 422nm is used. Source: [Wansbeek, 2006]

In the Al41Catraz experiment precion measurements are done on calcium atoms, therefore the atoms have to be slowed down to velocities near 0 m/s. For slowing down the atoms we use that when an atoms absorbs light the momentum of the light is added to the momentum of the atom. So when light, of a frequency that an atom is willing to absorb, is shined at an atom in the opposite direction it travels, the atom is slowed down. To know what frequency of light calcium is willing to aborb we have to look at the electronic conﬁguration of calcium. Calcium has two valance electrons in the outer s-shell. Any good book about atomic physics (like [Foot, 2005]) learns us that atoms with two valance electrons in the 2 1 outer s-shell has an electron conﬁguration of [Ar]4s S0. All the relevant levels and states for 40Ca are shown in Figure 2.1. For slowing down atoms a stable transition between two states is 1 1 needed. The S0 → P1 at 422.79 nm seems to be a good choice because there is only a neglectable −5 1 small leak of the order 10 to the 3d D2 level.

Figure 2.2: Schematic overview of the Al41Catraz experiment. Source: [Wansbeek, 2006]

In Table 2.1 the relevant properties of the diﬀerent calcium isotopes are given. A schemetic overview of the experiment is shown in Figure 2.2. The ﬁve stages of the experimental setup are explained

5 2 EXPERIMENTAL SETUP

Isotope Abundance Nuclear Spin Frequency shifts [MHz] Half-life I F=I-1 F=I F=I+1 39 0.86sec 3/2 shiftsnotmeasured 40 96.94% 0 41 1.03 · 105 year 7/2 303 248 154 42 0.65% 0 391 43 0.14% 7/2 675 633 554 44 2.09% 0 771 45 163days 7/2 1054 998 930 46 0.004% 0 1159 47 4.54days 7/2 1423 1361 1292 48 0.1874% 0 1513

Table 2.1: Properties of the diﬀerent calcium isotopes. Source: [van Ditzhuijzen, 2002] in the following sections, for a more thorough description of the phases I recommend to read the thesis of Wansbeek [Wansbeek, 2006].

2.1 Oven

An oven where the solid calcium sample is heated to about 670◦ K. At this temperature the calcium evaporates and leaves the oven in all directions with velocities varying according to the Boltzmann velocity distribution [van Ditzhuijzen, 2002]

3 2 v −v f(v)= e 2µ2 (2.1) 2µ4

kB T with µ = where kB is Boltzmann’s constant, T is the temperature and M is the mass of q M the particle. The velocity distribution is plotted in Figure 2.3.

0.0016

0.0014

0.0012

0.001

0.0008 f(v)

0.0006

0.0004

0.0002

0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 v in m/s

◦ Figure 2.3: Velocity distribution for 40Ca with 670 K

2.2 Compression

In the compression phase two lasers perpendicular to the intended direction focus the beam of calcium atoms. This phase has not been installed in the experimental setup. Atoms that enter the

6 2 EXPERIMENTAL SETUP

Zeeman slower (next pahse) with large transverse velocities will not reach the end of the Zeeman slower. So the intention was that to decrease the transverse velocity of the atoms. The compression phase is designed such that the atoms leave this phase with a longitudinal velocity of 50-1600m/s and a transverse velocity varying from 0 till 5 m/s. This is just right velicity for traveling through the beam without hitting the inside of the tube[van Ditzhuijzen, 2002]. A schematic view of this process is showed in Figure 2.4.

Figure 2.4: Schematic overview oven and compression phase. Source: [van Ditzhuijzen, 2002]

2.3 Zeeman slower

A Zeeman slower is used to slow down the atoms from speeds up to 1000 m/s [Wansbeek, 2006] to 50 m/s. This is done by excitement of the atoms with photons. A laser can exert a force on an atom if the energy needed for an atomic transition is the same as 1 1 the energy of the photon. In our case we use the S0 → P1 transition. So when an atom, that is 1 1 in the S0-state, absorbs a photon with a frequency of 422.79 nm, it is excited to the P1-state. With the absorption of the photon the atom gets an impulse of the photon. Because the laser sends light in the opposite direction of the atoms the atoms slow down. From the excited state two things can happen; spontaneous and, due to the existence of the photon ﬁeld, also stimulated emission can occur. When the photon excites an atom spontaneously, the photon goes in a random direction1. Since lots of these random directions occur (about 3 · 105 as measured in the simulation) the net impulse due to absorption is close to 0. Combined with impulses that excited the atom the net force, due to excitement and spontaneous emission, slows

1As explained by Wansbeeks [Wansbeek, 2006] the actual distribution is not a random direction. But since the distribution is still symmetric in the x, y and z axis, a random direction is, for this calculation, equivalent

7 2 EXPERIMENTAL SETUP the atom down. When stimulated emission occurs the atom goes in the same direction as the photons in the photon field, so the atom gets an impulse in the opposite direction as when it was excited. Since both these impulses have the same magnitude but opposite directions, the net force due to excitement and stimulated emission is zero. Because of the Doppler effect the atoms see a different frequency depending on his velocity. Ab- sorbtion rate for an atom is strongly dependent on the difference between the required and the seen frequency. The laser has a low line width to focus the laser energy to the desired frequency. Since the Doppler shift is larger than the line width of the laser, compensation for this Doppler shift of the seen frequency by the atom is needed. This is done by using the Zeeman effect. This effect uses magnetic fields to change the energy levels and therefore the transition frequency. The magnetic field in the slower is designed to keep the energy levels in resonance with the Doppler effect. Equations for these effects are introduced in Chapter 3.

2.4 Deﬂection

To make sure that the photons for slowing down the atoms in the Zeeman slower stage do go trough the MOT, the beam of atoms is deﬂected in the deﬂection phase toward the MOT. This is done by using photons of the laser but in this stage two lasers make an angle of 60◦, with respect to the longitudinal direction, so that resulting angle for the atom is 30◦.

2.5 Magnetic Optical Trap (MOT)

The MOT is the ﬁnal phase where a combination of 6 lasers and a quadrupole magnetic ﬁeld slows the atom down to 0 m/s. The isotope of the trapped atoms is determined by looking up the frequency shift in Table 2.1.

8 3 THEORY

3 Theory

In this section the models and theory around the slowing down of the atoms is introduced. The resulting formulas are used in the simulations.

3.1 Doppler eﬀect

The atoms that come out of the oven have a velocity around 650 m/s (see Figure 2.3). A laser is directed at the atom beam. Since light, just like sound, behaves like a wave, the wavelength of the observed light by the atoms is Doppler shifted. Wansbeek [Wansbeek, 2006] showed that taking only the forward velocity (vz) into account gives a good approximation for the Doppler shift

δ = k0vz (3.1) where k0 is wave number of the light and vz the forward component of the velocity of the particle.

3.2 Emission rate

Initial State Final State Initial State Final State Excited state Excited state

Ground state Ground state (a) Absorption of a photon (b) Spontaneous emission Initial State Final State Excited state

Ground state (c) Stimulated emission

Figure 3.1: Atom-photon interactions

Imagine an atom that has two possible states; ground and excited, with ∆E the diﬀerence of en- ergies between the ground and excited state. The atom can go from ground state to excited state by absorbing an atom with energy equal ∆E. When the atom goes from excited state to ground state an photon is emitted with energy equal to ∆E. Emitting a photon can happen spontaneous but also by stimulation of another photon. All the cases are shown in Figure 3.1. The emission rate is used for calculating the population of the ground en excited state. In [Wansbeek, 2006] the transition rate

s0γ/2 B12W¯ (ω)= (3.2) 1+(2δ/γ)2 between both states is derived. Here W¯ (ω) is radiative energy, γ the line width of the transition and δ the shift of the laser light frequency. The on-resonance saturation parameter

P/IS s0 = (3.3) 2πr2

9 3 THEORY is a dimensionless quantity, where P is the power of the laser in mW and r the radius in cm of the 2 photon ﬁeld and IS is the saturation parameter. For this case IS is 59.9 mW/cm . This equation leads to the stimulated emission rate

s0γ/2 B12 = (3.4) ~ 2 1+ 2 δL+kv−µB |B|/h¯ h γ i here the detuning is the combination of the 3 parts discussed in the previous Sections: δL for µB |B~ | the detuning of the laser, kv is for the Doppler detuning and − h¯ is the detuning due to the magnetic ﬁeld.

3.3 Collisions of Calcium atoms

One important assumption made in all simulations and previous reports [van Ditzhuijzen, 2002, Hoekstra, 2005, Wansbeek, 2006]: the collisions between atoms in the tube can be neglected. In this section I derive a formula that gives the collision rate given the velocity and positions of atoms in the tube. This collision rate for all positions and velocities in the tube, can then be used to calculate the collision probability for an atom given the velocity at the begin of the tube.

3.3.1 Collision rate for all velocities and positions

For calculating the collision rate Γ I make the following assumptions:

• forces that are diﬀerent for the x and y direction (like gravity) can be neglected, therefore discs can be used, with radius r = x2 + y2 and center (0, 0,z), for which two points in the disc are indistinguishable, p

• the velocity of an atom is constant in a disc of volume Vdisc =2πrdrdz and

• the velocity in both vx and vy can be neglected since the velocity in the forward direction is for most atoms much higher.

Since I assume that the atom makes a straight path through the disc I can calculate the volume 2 that the atom traveled through by using the volume of a cylinder dz2πratom. I can also calculate the volume density ρ(z, r, v) = n(z, r, v) ∗ Vatom/Vdisc where n(z, r, v) is the number of atoms per 4 3 second that travel through the disc with velocity v±dv/2. Vatom is the volume of the atom 3 πratom and V is the volume of the disc dV =2πrdrdz. disc ′ ′ |vz−vz| ′ The number of atoms, with velocity v , seen over the distance of the disc is dz ρ(z, r, v ). This combines to the collision rate

Z,R,∞ |v − v′ | Γ(z, r, v) = n(z, r, v) n(z, r, v′) z z ρ(z, r, v′)dV dv (3.5) Zz=0,r=0,v′=0 dz with Z and R the length and the radius of the tube, respectively.

3.3.2 Collision probability given the begin velocity of an atom

Most of the collisions that occur are not head on but are a grazing. Therefore, a large amount of the z momentum is transferred to momentum in the x and y direction. Thus, both atoms will hit the inside of the tube. So, I assume that all collisions lead to two atoms that do not reach the end of the tube. I want to know the fraction of the atoms that do not reach the end of the tube due to collisions.

10 3 THEORY

Consequently, I have to combine the probability of collisions in each disc with the probability of ﬁnding a atom in that disc given the begin velocity

Z,R,∞ n(z,r,v,v0) ′ Γ(v0) = Γ(z, r, v) dzdrdv , (3.6) Zz=0,r=0,v=0 n(z, r, v) with n(z,r,v,v0) the number of atoms per second that go through disc with velocity v and v0 at the beginning of the tube. Dividing the collision rate by the number of atoms per second that leave the oven with a given begin velocity n(v0) gives the collision probability for that given begin velocity

Γ(v0) p(v0) = . (3.7) n(v0)

Since I don not have mathematical formulas for n(z, r, v) and n(z,r,v,v0) simulations are needed to calculate the collision probability.

3.4 Eﬀects of input parameters on results

For slowing down the atoms to 50 m/s the right amount of impulses by photons have to be given. The time for a particle in ground state to interact with a photon is given by Equation 3.4. The goal it to change the parameters such that the value of B12 ( gets the best value for slowing the most atoms to ±50 m/s at the end of the tube. Increasing B12 leads to a higher excitement rate but also a higher stimulated emission rate. This gives two boundaries for the value of B12; too low and the atoms have traveled too far between excitements and therefore goes out of resonance. Too high and all atoms stay in resonance but the probability of stimulated emission increases in favor of spontaneous emission. This lowers the number of atoms that are slowed down because stimulated emission gives an impulse in the opposite direction of the impulse due to the absorption of the photon. This cancels the slowing down eﬀect. The value of B12 depends on diﬀerent variables so I write B12(P, r(z),δL, ~v, B~ (z)) of which

• P,δL are constants of the laser, • r(z) is the radius of the laser beam which, by focusing the beam, can depend on z,

• B~ (z) is the magnetic ﬁeld which also can depend on z and • v the velocity of the atom.

A theoretical approach for the best values to slow down the atoms is done by van Ditzhuijzen [van Ditzhuijzen, 2002]. Here recommendations are shown in Table 3.1. In this table the three currents (main, peak and dip) determine the magnetic ﬁeld.

Variable Recommended by van Ditzhuijzen [van Ditzhuijzen, 2002] Main current 4.0 A Peak current 6.0 A Dip current 6.0 A Laserpower 30mW Laser Detuning −370 MHz Radius of laser beam at z = 0 m 4 mm Radius of laser beam at z =0.9 m 8 mm

Table 3.1: Settings for the Zeeman slower

11 3 THEORY

3.4.1 Magnetic ﬁeld

One way to change the value of B12 is to change the magnetic field. The magnitude of the magnetic field depends on the forward position(z) in the tube with a slope that must be shallow enough to keep the atoms in resonance. The value of magnetic field at the beginning of the tube determines up to what velocity the atoms are slowed down. At the end of the tube the particles should not be in resonance any more; when they stay in resonance they will be slowed down to 0 m/s and will turn around. Therefore steep changes of magnetic field must be added near where the atoms reach the desired velocity. This gives a very narrow area of where the atoms are in resonance. Most atoms will therefore go out of resonance and are thus not slowed down anymore. The position of the peaks determine the end velocity of the atoms.

3.4.2 Detuning of the laser

The detuning of the laser is a constant of the laser and can not be changed certain positions in the tube. Therefore this value has to be chosen such that as many as possible atoms are slowed down to the desired velocity.

3.4.3 Laser power

The laser power is a constant as well. Therefore the same reasoning as for the detuning of the laser holds: choose the value such that as many as possible atoms are slowed down to the desired velocity.

3.4.4 Focus of the laser beam

Focusing the laser beam has three eﬀects:

• Atoms that have a too high velocity and are perpendicular to the forward direction, will not reach the end of the tube because they will hit the wall of tube before reaching the end. Therefore it is useless to put eﬀort in slowing down these particles. • More eﬀort is put in slowing down atoms that will reach the end of the tube because the radius of the photon beam at the beginning of the tube is lowered, therefore s0 will be higher at the beginning of the tube. • Atoms that absorb a photon at a certain distance from the center of the laser beam get an impulse toward the center of the beam since the photon is directed at the center of the beam (assumed that the focal point of the laser is behind the oven). This will result in a higher distribution of particles near the center of the laser beam and thus less atoms that hit the wall of the tube before reaching the end.

12 4 SIMULATION

4 Simulation

For the simulations a platform is constructed. This platform reads a configuration file and executes the commands given in this file. The platform does a simulation for each particle separately. This gives a good representation since nearly no atom interacts with other atoms (see Section 3.3 and Section 5.6).

4.1 Basic random number generating

On the basis of a simulation are random numbers. A computer cannot generate true random numbers; there is always a period when the same sequence of ’random’ numbers rises again. To make sure that the random numbers ﬁts the distribution requested a histogram of the random values is made. For the simulations random values are used taken from diﬀerent distributions. For each of these types a validation is done.

4.1.1 Uniformly distributed pseudo-random values

The most basic random method used is the uniformly distributed. In the c++ library cstdlib a function (rand()) is available that gives a new pseudo random value each time the function is called. This random value generator uses the formula: Xn+1 = (aXn + b) mod m. When the result of the algorithm is devided by the maximum value (RAND MAX) this function gives random values on the range [0..1]. One problem of this algorithm is that the random values reappear after the mth value is requested. Therefore the values from this algorithm are only considered random as long as the number of random values used is signiﬁcantly less than the period m. Since this value of m is near 4 × 109 and, looking at the simulations, the number of random values used is around 8 × 1011 the standard random value generator cannot be used. There- fore another random value generator is chosen: the Mersenne Twister random number generator [Matsumoto and Nishimura, 1998]. This random number generator has a period of 219937 − 1 random numbers, so we conclude that we can assume that the values from that random number generator are random.

4.1.2 Gaussian distributed pseudo-random values

The second group of random values used are Gaussian (also called normal) distributed values, for example for the begin velocity distribution. For the generation of these numbers the following algorithm [Box and Muller, 1958] is used:

if (phase == 0) do V1 = random(-1..1); V2 = random(-1..1); S = V12 + V22; while(S ≥ 1 OR abs(S) < 0); X=V1* sqrt(-2 * log(S) / S); else X=V2* sqrt(-2 * log(S) / S); end if phase = 1 - phase return X

The resulting random value distribution of the algorithm is sown in Figure 4.1. This distribution corresponds with a gaussian value distribution.

13 4 SIMULATION

Gaussian distribution with µ=0 σ=1 N=1.000.000

80000 µ=-0.000850405 σ=1.00011 70000 60000 50000 40000 30000 Occurrences 20000 10000 0

-5 -4 -3 -2 -1 0 1 2 3 4 5 Random value

Figure 4.1: Distribution of 1.000.000 values

4.2 Distributions of random values using invertible functions

For our simulations also other random distributions are used which are not uniformly distributed on the [0..1] range. We can use the uniform distribution if there is a probability distribution function(pdf) for this non-uniform distribution. A requirement for this pdf is that its cdf is invertible. The following algorithm is used to get random numbers of this distribution.

1. get the probability distribution function p(x) 2. calculate the cumulative distribution function P (x)= p(x)dx R 3. Normalize this function to the range [0..1] ; N(x)= P (x)/P (xmax)= y 4. Invert the resulting function N −1(y)= x 5. Get a random number r from a function that generates random numbers uniformly on the range [0..1] 6. Use this number to get the random number of the desired range. N −1(r)

4.2.1 Random point on a sphere

Since the spontaneous emission of a photon occurs in a random direction a random point on a sphere is needed. A naive approach would use two angles θ ∈ [0, π) and φ ∈ [0, 2π) and set the random point to (cos θ sin φ, cos θ cos φ, sin θ)

This is distribution is wrong, because the radius of a disc of the sphere is set by cos θ. This radius only depends on θ. Since θ is uniformly distributed the same probability for θ = 0 (top disc) π holds for θ = 2 (center disc). For these two values the radius of the discs are respectively 0 and

14 4 SIMULATION

6.28 12.57

5.50 11.00

4.71 9.42

3.93 7.85 θ ) θ 3.14 ) d 6.28 θ O( 2.36 O( 4.71

1.57 3.14

0.79 1.57

0.00 0.00 0.00 0.79 1.57 2.36 3.14 0.00 0.79 1.57 2.36 3.14 θ θ (a) pdf for angle θ (b) cdf for angle θ

1.00 3.14

0.88

0.75 2.36

0.62

0.50 θ 1.57

Normalized 0.38

0.25 0.79

0.12

0.00 0.00 0.00 0.79 1.57 2.36 3.14 0.00 0.12 0.25 0.38 0.50 0.62 0.75 0.88 1.00 θ Normalized (c) normalized cdf for angle θ (d) inverted normalized cdf for angle θ

Figure 4.2: Random point on a sphere

1. Therefore the circumference of the top disc is 0 and the circumference of the center disc is 2π. π So the chance for picking θ = 2 should be much larger, otherwise there is a too large number of ’ random’ points near the top and bottom. To prevent an unfair distribution we split the circle in discs with a circumference 2πr = 2π sin θ (Figure 4.2a) and use the proposed algorithm to get a random distribution for points on the circle. We do this by integrating over θ (Figure 4.2b) and normalizing this function (Figure 4.2c). Now we have a function that maps [0, π) to [0, 1] if we invert this function to map [0, 1] to [0, π)(Figure 4.2d) we can draw a random number, put it in this function and get a θ that ﬁts the distribution we want. Now the random point on the sphere is calculated with the new distribution method for θ and the same formula is used: (cos θ sin φ, cos θ cos φ, sin θ). Figure 4.3 shows this is indeed a correct distribution.

Correct distribution Naive distribution

Figure 4.3: Top and side view of a correct and naive distribution of random points on a sphere

15 4 SIMULATION

4.2.2 Calculating waiting time

Since both formulas for spontaneous and stimulated emission are exponential functions for the probability for going to the other state, we transform these formulas into occupation formulas. − t For exponential decay functions, which can be written as P (t)= γe γ , the occupation formula is − t D(t)= e γ . This occupation formula has values varying from [0..1] where 1 is fully occupied and 0 is empty. This range is the same as for random numbers. So to get the time from the formula we solve the formula for t(D) with D is a random number. The result is shown in Figure 4.4

1 0.9 0.8 0.7 0.6 0.5 0.4 Random Value 0.3 0.2 0.1 0 0 5e-09 1e-08 1.5e-08 2e-08 Time

Figure 4.4: Randomized time picking using a exponential occupation function

t(D)= − ln(γD) (4.1)

In the case of spontaneous emission the value of γ is equal to A as deﬁned in Section 3.2. And for the case of stimulated emission the value of γ is equal to B21.

4.3 Distributions of random values using the numerical inverse

For the simulations also distributions of random values are used for which the cumulative distribution function is not invertible. For example the Boltzmann velocity distribution (Equation 2.1). For these functions an algorithm for a numerical inverse is implemented. This algorithm uses binary search to get an approximation of the value (x) for which the function (f) gives the desired result (targetValue). Required is that the function has a range from 0 to a certain maximum (f : [0 : max] → [0 : 1]) and is increasing. These two conditions hold for a normalized cdf.

x = maxValue/2 step = maxValue/4 do functionValue = f(x) if (targetValue ≥ functionValue) x += step else x -= step endif step = step/2 while (abs(targetValue-functionValue) ≥ ǫ ) return x

Where ǫ is a parameter that determines the allowed error in the function value. Usually this value is chosen to be 10−6.

16 4 SIMULATION

4.4 Parallelization

Variable Number Length of the tube 10 Main current values × 20 Atoms × 4,000 Total 800,000

Table 4.1: Number of simulations done for main current

In order to obtain good statistics from the simulations, I use a large numbers of measurements to reduce the error (in each simulation setup). For example, the number of simulations needed, looking into the main coil current for the magnetic ﬁeld is shown in Table 4.1. About 5,000 simulations can be done in one minute. Hench approximately 160 minutes are needed for simulating the main coil current. Since the latest computers have multiple (2 or 4) processing units it is worthwhile to look into parallelisation of the code. Parallelisation can easily be done because we assumed that all atoms can be simulated indepen- dently. Each core gets its own part of atoms to simulate and reports the results back to the main process that makes the plots. The algorithm used to parallelize the simulations is explained in Appendix B.

4.5 Comparison old and new approach

Because the stimulated emission time depends on the magnetic ﬁeld value which changes signiﬁ- cantly in certain regions, van Ditzhuijzen[van Ditzhuijzen, 2002] and Wansbeek [Wansbeek, 2006] both introduced a time step dt in their simulations. Although it is a valid idea to introduce this dt they claimed that the time waiting for one of the actions was not correct. If the emission time was larger than dt, they let the particle travel for dt time and draw a new random number to calculate a new waiting time. This leads to discarding random numbers that gives large time steps and requesting new random numbers until a random number is given for which a time step less than dt. The result was that dt became a control parameter in stead of a precision parameter. In the search of the best value (too low would lead to stronger random number picking, too high would not give enough precision) they came up with dt = 10−6 s.

4.5.1 Keeping the random values

The new approach focuses on keeping the random value the generator has given, not looking at how bad the value may be. At each point the new time for stimulated emission is calculated using the same random value. Since the population formula for the particle works for a full occupation of a state, the time in the current state has to be remembered. As long as both the stimulated an spontaneous times are not in the current reach a new step of dt is done. In pseudo code this gives:

get random values rndSpon and rndStim timeInState = 0; while (not ﬁnished) if (excited state & in photon ﬁeld) calculate dtSpon using rndSpon calculate dtStim using rndStim and current position if (dtSpon < timeInState + dt & dtSpon < dtStim) do spontaneous emission timeInState = 0 get new random values rndSpon and rndStim else if (dtStim < timeInState + dt & dtStim < dtSpon)

17 4 SIMULATION

do stimulated emission timeInState = 0 get new random values rndStim and rndSpon else particle travels for dt and does no transition timeInState = timeInState + dt end if else ..... other for options: inside or outside photonField & excited or groundState end if end while

To look why dt = 10−6 s became a control parameter rather than en precision parameter, I looked at the random values that were ’picked’ by the algorithm: when a spontaneous emission occured, the random value that was used to calculate te time step form spontaneous emission was logged. And the same for stimulated emission. The results for the old and new approach are shown in Figure 4.5. It is clear that for the old algorithm at values lower than dt = 10−6 s, more random values close to 0 are used than close to 1. In the new algorithm this eﬀect does not occur.

0.16 1 0.028 0.9 0.14 0.026 0.024 0.8 0.12 0.8 0.022 0.1 0.02 0.7 0.08 0.018 0.6 0.06 0.6 0.016 0.014 0.5 0.04 0.012 0.02 0.4 0.01 0.4 0 0.008 0.3 0.2 0.2 0.1 0 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-8 10-7 10-6 10-5 10-4 10-3 10-2 timestep [s] timestep [s]

(a) Old algorithm (y axis: [fraction]) (b) new algorithm (y axis: [fraction])

Figure 4.5: Random numbers used for calculating the winning time step dtStim or dtSpon

18 5 RESULTS

5 Results

All simulations in this section are done, unless stated otherwise, with Mollema’s default settings; Imain =4A,Ipeak =0A,Idip =0 A, P =40 mW, δL = -415 MHz. I want to investigate what settings will give the best performance. Before doing this, the simulation has to be validated. In this section I attempt to reproduce the experimental results with the simulation. The experimental results are measured and described in the thesis of Mollema [Mollema, 2008].

Parameter Figure Figure Figure Figure 5.4(a) 5.4(b) 5.5(a) 5.5(b) Main Current [A] 4 4 4 0,2,4,6 Peak Current [A] 6,0,0 0 0 0 Dip Current [A] 6,6,0 0 0 0 Beam power [mW] 40 15,25,30,35 40 40 Laser Detuning [MHz] −415 −415 −435, −415, −395 −415

Table 5.1: Settings for the measurements done by Mollema

Power: 0.5mW Radius: 0.7mm Detuning: δ Probe Photons

Zeeman slower 60 degrees

Ca atoms Photons

Power: 40.0mW Radius: 8.0mm Detuning: δ = −415MHz S 95 cm

Figure 5.1: Setup for measuring the velocity distribution of the atoms.

Figure 5.2 shows the velocity distribution of the atoms at the end of the slower given the input parameters shown in Table 5.1. These velocity distributions are determined by shining with another laser (probe laser) on the atomic beam at the end of the slower. Figure 5.1 shows the setup of the measurement. The extra intensity, relative to the intensity of the light emitted with only using the slower laser, is measured as a function of the detuning of the probe laser. The probe laser makes an angle of θ = 60◦ with the beam. Therefore Mollema states that

δprobe = kvL cos θ (5.1) where δprobe is the detuning of the probe laser, k is the wave number of the laser light and vL is the longitudinal velocity component of the atoms in the beam. Therefore he concludes that

1 MHz detuning of the probe laser ∼ 1.18vL. (5.2)

In this way, Mollema makes a one-to-one identification between detuning of the laser and the longitudinal velocity, and thus the velocity distribution at the end of the slower. I do not think that this one-to-one identification can be made. The probability for an atom to interact with a photon with a certain wavelength is not a delta function but is a Lorentzian distribution, as shown in Equation 3.4. This Lorentzian distribution has a peak at the same position as Equation 5.1, but also has a line width. Therefore atoms with a small velocity difference to the resonance velocity are also slowed down.

19 5 RESULTS

54a 54b 100 100 400 15 406 25 80 466 80 30 35 60 60

40 40

Intensity [arb. un.] 20 Intensity [arb. un.] 20

0 0 0 -25 -50 -75 -100 -125 -150 0 -25 -50 -75 -100 -125 -150 Detuning [MHz] Detuning [MHz]

0 20 40 60 80 100 120 0 20 40 60 80 100 120 Velocity [m/s] Velocity [m/s] (a) Peak and dip current (b) Laser power 55a 55b 100 100 395 000 415 200 80 435 80 400 600 60 60

40 40

Intensity [arb. un.] 20 Intensity [arb. un.] 20

0 0 0 -25 -50 -75 -100 -125 -150 0 -25 -50 -75 -100 -125 -150 Detuning [MHz] Detuning [MHz]

0 20 40 60 80 100 120 0 20 40 60 80 100 120 Velocity [m/s] Velocity [m/s] (c) Detuning (d) Mail current

Figure 5.2: Measurements done by Mollema [Mollema, 2008]. Correct axis are included (see text)

5.1 Comparing the eﬀect of detuning the laserlight

55a 1.2 395-sim 415-sim 435-sim 1 395-exp 415-exp 435-exp 0.8

0.6

Intensity [arb. un.] 0.4

0.2

0 0 -20 -40 -60 -80 -100 -120 -140 Detuning [MHz]

Figure 5.3: Comparing simulation and measurement [Mollema, 2008]

20 5 RESULTS

First the peak positions are checked so I compared (Figure 5.3) the detuning of the laser (Figure 5.5(a) in the thesis of Mollema[Mollema, 2008]) with the simulation to validate the simulator. Since both y-axis represent additional intensity with an arbitrary scale, I can rescale both images such that the highest peak of the measurement and the simulation align. The other peaks are scaled accordingly to their class (measurement or simulation). Compared with the simulation the peaks of the measurement are at the same position and the value falls in the error margin of the simulation. Although the peaks are close enough to accept the simulation, the width of the peaks do not correspond to the measured values. This means that the velocity distribution of the slowed down atoms is more spread in the measurement than in the simulation. I do know the explanation for this diﬀerence, but I can think of two possible reasons;

• a eﬀect that broadens the velocity distribution is not simulated, • Mollema only mentions stabilizing the laser used to slow down the atoms. He left open whether he also did this for the probe laser. When the laser is not stabilized the frequency has a larger line width. This can be compared by adding a gaussian to the laser frequency; although setting the detuning to -60 HMz also a part of the photons have a detuning of -80 MHz. This line width has not been simulated but I expect when adding the line width would broaden each peak.

If only the second reason is true the simulations are valid because the peak positions of the simulation correspond to the peak positions of the measurement and the width of the peaks can be ignored because this is an effect of the measurement and not a property of the velocity distribution at the end of the tube. Also the first possibility is plausible but this is not a reason to reject the simulations. Still the peak positions very well correspond to the measurement. The width of the peaks is approximately 35 HMz. Using the statement of Mollema 5.2 as a thumb rule this corresponds with a velocity of 30 m/s. This broadening of the velocity distribution is likely (and thus assumed) to be equal for all end velocities. So we can still compare peak heights of two simulations. Therefore we conclude that the simulations is close to the measurement when using the values the simulation gives, the values might be lower then the measurement results. But a comparison of two peak heights of the simulation can be used because the effect that should broaden the peaks is equally absent in both simulation results.

5.2 Laser beam focus

As stated in Section 3.4.4 atoms that reach the end of the tube are strongly related with the begin velocity in the direction perpendicular to the forward direction. Therefore it is useful to focus the laser beam. Simulations are done to check what focus of the beam can be used. For each point (z, r) I looked into the probability of reaching the end of the tube, where z is the position forward and r is the distance to the center of the tube. This is done by tracking the path of each particle and registering for each point it traveled through whether it has reached the end. Results are shown in Figure 5.4. The results show that it is best to focus the beam such that the radius in the begin of the tube is 1.0 mm and at the end of the tube 8.0 mm.

5.3 Collision probability

In Section 3.3.1 is explained that I cannot calculate the collision probability of atoms because mathematical formulas for n(z, r, v) and n(z,r,v,v′) are not available. But I can use simulations to give an approximation for these functions. Since I cannot simulate all inﬁnitesimal parts of dz, dr and dv I use step sizes and assume that function is smooth in each cell. Also I cannot calculate the exact fraction of atoms that go through a cell, so I simulate 1000 atoms and look for each begin velocity and calculate for each cell the number of atoms that go through the cell. See Figure 5.5 (a) for the position distribution for atoms with begin velocity 400 m/s . Finally the velocity at a forward position in the tube is simulated

21 5 RESULTS

Failed fraction by traveled through point Succeeded fraction by traveled through point 0.009 1 0.009 1 0.008 0.8 0.008 0.8 0.007 0.6 0.007 0.6 0.006 0.006 0.4 0.4 0.005 0.005 0.004 0.2 0.004 0.2 0.003 0 0.003 0 0.002 0.002 0.001 0.001 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Position from start [m] Position from start [m]

(a) Fail probability (y axis: [m] distrance to the cen- (b) Succeeded probability (y axis: [m] distrance to ter of the tube) the center of the tube) Accepted fraction by traveled through point 0.009 1 0.008 0.8 0.007 0.6 0.006 0.4 0.005 0.004 0.2 0.003 0 0.002 0.001 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Position from start [m]

Figure 5.4: Probabilities for positions in the tube where an atom has traveled through

Constant Value atoms 5 × 2.514 s−1 [Hoekstra, 2005] zmin 0.0 m ∆z 0.01 m zmax 1.0 m rmin 0.0 m ∆r 0.00025 m rmax 0.008 m vmin 0.0 m/s ∆v 10 m/s vmax 3000.0 m/s

Table 5.2: Constants used for simulations to approximate n(z, r, v) in the same way as the position, see Figure 5.5 (b) for the velocity distribution. The results of both simulations are combined in n(z,r,v,v0) assuming that the velocity does not depend on the oﬀ axis position r in the tube. In Table 5.2 the values for the constants are shown. Combining this fraction with the number of atoms per second that leave the oven n(v0) I get n(z,r,v,v0). Now summing over all begin velocities for each cell gives the rate of atoms

vmax n(z, r, v)= n(z,r,v,v0) v0X=vmin in each cell with a certain velocity. Looking at Figure 5.6 I can see that the probability of colliding with other particles is the most for particles leaving the oven at 100 m/s with a probability of 1 × 10−9. Since this is a very low probability I can conclude that it is valid to assume that no collisions occur in the tube.

22 5 RESULTS

Distance from center at position in the tube, with v0 = 0400 m/s Foreward velocity distribution at position in the tube, with v0 = 0400 m/s 0.008 2000 3000 2000 0.007 1500 2500 1500 0.006 1000 1000 0.005 2000 0.004 500 1500 500 0.003 0 1000 0 0.002 0.001 500 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Position from start [m] Position from start [m]

(a) Oﬀ axis position distribution (y axis: [m] dis- (b) Velocity distribution (y axis: [m/s]) trance to the center of the tube)

Figure 5.5: Distributions of atoms at forward position z given begin velocity v0= 400m/s

Collisions rate for begin velocity Collisionrate for begin velocity 10000 1e-008

100 1e-009 1e-010 1 1e-011 0.01 1e-012 0.0001 1e-013

1e-006 1e-014 Collision rate 1e-015

Collisions per second 1e-008 1e-016 1e-010 1e-017 1e-012 1e-018 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 Velocity at the beginning of the tube [m/s] Velocity at the beginning of the tube [m/s] (a) Collision rate (b) Collision probability

Figure 5.6: Collision rate and probabilty given begin velocity

23 6 RECOMMENDATIONS

6 Recommendations

The goal of the Zeeman slower is to get as much as possible atoms at z =0.95 m from the oven at v = 50 m/s. Possible parameters to tune and their eﬀect on the velocity at 0.95 m are described in Section 3.4. In this section I looked at results of simulations where the parameters are tuned to give the best results keeping in mind that the simulation does not fully match the measurements. First measurements are done with the setup Mollema[Mollema, 2008] used. For all settings two plots are made:

• Velocity distribution of the atoms that are slowed down at 0.95 m • The fraction of atoms that reached the end of the tube and have a velocity lower than 200 m/s at 0.95 m

The first plot is made to see what the effect of the parameter is on the velocity of the slowed down atoms at 0.95 m. The second plot is made to see what fraction of atoms is slowed down by the Zeeman slower. For the parameters that influence the magnetic field a third plot is made of the magnetic field value at different positions in the tube. In the concluding Section 6.7, a combination of parameters is simulated. For this simulation only atoms that have the desired velocity[Wansbeek, 2006] of 40 m/s to 60 m/s are accepted. The combination of parameters, with the best settings found in the first sections, leads to a recommendation for the settings of the Zeemanslower in Table 6.1.

24 6 RECOMMENDATIONS

6.1 Main coil

Velocity distribution at 0.95m

1000 450 400 350 800 300 250 200 600 150 100 50 400 0

200

0 1 2 3 4 5 6

Main current [A]

(a) Velocity distribution (y axis: [m/s]) Fraction of atoms accepted at 0.95m 0.5 0.45 6 0.12 0.4 0.1 5 0.08 0.35 0.06 0.3 4 0.04 0.25 3 0.02 0 0.2

-0.02 Accept fraction 2 0.15 1 0.1 0 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 1 2 3 4 5 6 Position from start [m] Main current [A] (b) Magnetic ﬁeld in the tube due for main current (c) Fraction accepted atoms (y axis: [A])

Figure 6.1: Eﬀect of main current with laser power = 40 mW, Ipeak =0A,Idip = 0 A, δ = −415 MHz and no focus

Figure 6.1 shows that increasing the main current increases the range of atoms that are slowed down. Figure 6.1 also shows that there is a maximum number of atoms that can be slowed down given the other parameters. This is clearly shown in the velocity distribution. For main current higher than 5.0 A part of the atoms are slowed down into two groups, one near 100 m/s and one group near 600 m/s. The second group of atoms are slowed down but loose resonance at velocity of around 600 m/s, this is, as will be shown later, because of the power of the laser which can not slow all atoms down.

25 6 RECOMMENDATIONS

6.2 Peak

Velocity distribution at 0.95m

1000 350

300

800 250 200

150 600 100

50 400 0

200

0 1 2 3 4 5 6

Peak current [A]

(a) Velocity distribution (y axis: [m/s]) Fraction of atoms accepted at 0.95m 0.4

6 0.08 0.38 0.07 5 0.06 0.36 0.05 0.04 4 0.03 0.34 0.02 3 0.01 0 0.32

2 -0.01 Accept fraction 0.3 1 0.28 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.26 0 1 2 3 4 5 6 Position from start [m] Peak current [A] (b) Magnetic ﬁeld in the tube due for peak current (c) Fraction accepted atoms (y axis: [A])

Figure 6.2: Eﬀect of peak current with laser power = 40 mW, Imain =4A,Idip = 0 A, δ = −415 MHz and no focus

Figure 6.2 shows that the peak current has no (signiﬁcant) eﬀect on both the velocity at the and of the tube nor the acceptance rate therefore I conclude that it is useless to set this parameter with these standard settings.

26 6 RECOMMENDATIONS

6.3 Dip

Velocity distribution at 0.95m

1000 350

300

800 250 200

150 600 100

50 400 0

200

0 1 2 3 4 5 6

Dip current [A]

(a) Velocity distribution (y axis: [m/s]) Fraction of atoms accepted at 0.95m 0.35

6 0.08 0.3 0.07 0.06 5 0.05 0.25 0.04 0.03 4 0.02 0.2 0.01 0 3 -0.01 0.15 -0.02

2 -0.03 Accept fraction 0.1 1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 1 2 3 4 5 6 Position from start [m] Dip current [A] (b) Magnetic ﬁeld in the tube due for dip current (y (c) Fraction accepted atoms axis: [A])

Figure 6.3: Eﬀect of dip current with laser power = 40 mW, Imain =4A,Ipeak = 0 A, δ = −415 MHz and no focus

Figure 6.3 shows that the dip current has a strong eﬀect on the velocity at the and of the tube and therefore also on the acceptance rate. Increasing the dip current lowers the velocity of the slowed down atoms but also lowers the number of atoms that keep in resonance. This is clearly shown at 3.0 A, here the velocity is less (more near the desired velocity of 50 m/s) than at 2.0 A but the number of atoms that reach that velocity is also lower. Therefore another parameter has te be found that keeps the atoms in resonance.

27 6 RECOMMENDATIONS

6.4 Laser power

Velocity distribution at 0.95m

1000 350

300

800 250 200

150 600 100

50 400 0

200

0 5 10 15 20 25 30 35 40

Laser power [mW]

(a) Velocity distribution (y axis: [m/s]) Fraction of atoms accepted at 0.95m 0.4

0.35

0.3

0.25

0.2

0.15 Accept fraction 0.1

0.05

0 0 5 10 15 20 25 30 35 40 Laser power [mW] (b) Fraction accepted atoms

Figure 6.4: Eﬀect of power of the laser with Imain =4A,Ipeak =0A,Idip = 0 A, δ = −415 MHz and no focus

The laser power can divide the atoms in several groups; the ones that are not slowed down, the ones that start to slow down but lose resonance, and the ones that are slowed down and keep in resonance. Looking at Figure 6.4 we can see the three groups; up to 8 mW no atoms are slowed down, from 11 mW to 17 mW the atoms are divided in two groups and from 20 mW all atoms are slowed down. So we conclude that when, by tuning the other parameters, more atoms are slowed down, new measurement with the laser power have to be done to look for what value all atoms keep in resonance.

28 6 RECOMMENDATIONS

6.5 Detuning

Velocity distribution at 0.95m

1000 400 350 300 800 250 200 600 150 100 50 400 0

200

100 0 -100 -200 -300 -400 -500

Detuning of the laser [MHz]

(a) Velocity distribution (y axis: [m/s]) Fraction of atoms accepted at 0.95m 0.35

0.3

0.25

0.2

0.15 Accept fraction 0.1

0.05

0 100 0 -100 -200 -300 -400 -500 Detuning of the laser [MHz] (b) Fraction accepted atoms

Figure 6.5: Eﬀect of detuning with laser power = 40 mW, Imain =4 A,Ipeak =0A,Idip = 0 A and no focus

Figure 6.5 shows that detuning of the laser has a strong eﬀect on the velocity of the atoms at the end of the tube and the number of atoms slowed down. Looking at the velocity distribution, detuning is best set around −270 MHz since then the average velocity at 0.95 m is closest to the desired velocity of 50 m/s. The acceptance rate increases strongly from −200 MHz to −400 MHz. For −450 MHz, the velocity at the and of the tube becomes higher than the acceptance velocity of 150 m/s and therefore no more atoms are accepted.

29 6 RECOMMENDATIONS

6.6 Focusing laser beam

Velocity distribution at 0.95m

1000 350

300

800 250 200

150 600 100

50 400 0

200

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

laser focus, radius at z=0 [m]

(a) Velocity distribution (y axis: [m/s]) Fraction of atoms accepted at 0.95m 0.5

0.45

0.4

0.35 Accept fraction

0.3

0.25 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 laser focus, radius at z=0 [m] (b) Fraction accepted atoms

Figure 6.6: Eﬀect of focussing the laser beam with laser power = 40 mW, Imain =4A,Ipeak =0A,Idip = 0 A and δ = −415 MHz

Figure 6.6 shows that focussing the laser beam does not eﬀect the velocity at the and of the tube, but strongly eﬀects the number of atoms that are slowed down. Therefore we conclude that no matter to what settings the other parameters are set it is best to focus the laser beam.

30 6 RECOMMENDATIONS

6.7 Best settings combined

Both the dip current (Section 6.3) and laser detuning (Section 6.5) have a strong eﬀect on the velocity of the slowed down atoms and the acceptance rate. Decreasing the detuning increases the number of atoms slowed down but also increases the velocity. We can compensate for the increase of velocity by increasing the dip current. Focussing the laser beam (Section 6.6) increases the number of atoms slowed down and does not aﬀect the velocity therefore I recommend to focus the beam to a radius of 1 mm at the beginning of the tube. The peak current does not contribute to the acceptance rate so we keep that to 0 A. The main current is at maximum value with 4.0A so won’t increase the main current above 4.0A. In Figure 6.8 the combination of detuning and dip current with these new settings is shown. About 35% of the atoms are slowed down to the range 40-60 m/s when setting dip current to 5.6A and setting the detuning to −550 MHz. Finally we looked again into to laser power since with these settings more atoms are slowed down. Simulations (Figure 6.7) show that the laser power is best set to at least 30 mW. All the settings I advise are shown in Table 6.1.

Fraction of atoms accepted at 0.95m

0.4

0.35

0.3

0.25

0.2 Accept fraction

0.15

0.1

0.05

0 5 10 15 20 25 30 35 40

Laser power [mW]

Figure 6.7: Eﬀect of power of the laser with Imain =4A,Ipeak =0A,Idip = 5.6 A, δ = −550 MHz and no focus

Table 6.1: Settings for the Zeeman slower

31 6 RECOMMENDATIONS

Looking at the region Mollema did his experiments (shown with arrows in Figure 6.8) it is clear why he did not ﬁnd many atoms. Changing the settings of the Zeeman slower to the advised values would increase the number of atoms with a factor of approximately 5 × 102. So given that with the current setup approximately 1 atom is seen every week, with the new settings approximately 3 atoms are seen every hour. Which makes determining the concentration of 41Ca atoms in a natural sample doable in a more acceptable timeframe.

Figure 6.8: Combination of laser detuning and dip current with laser power 40 mW, Imain =4A,Ipeak = 0 A. The arrows show the settings where Mollema[Mollema, 2008] did his experiments.

32 7 DANKWOORD

7 Dankwoord

Het heeft een behoorlijke tijd geduurd, maar hier ligt dan mijn Bachelor thesis voor informatica en natuurkunde. Daarmee wordt ook het einde van mijn Natuurkunde opleiding bereikt en ga ik verder met de Master Computational Science af te ronden. Allereerst wil ik Lotje Wansbeek bedanken voor haar geduld, advies en mij de kans te geven het onderzoek af te ronden na een lange afwezigheid. Daarnaast Michael Biehl omdat hij mijn begeleider vanuit Informatica wilde zijn. Verder gaat natuurlijk mijn dank uit naar de personen die er voor gezorgd hebben dat ik terug ben gekomen, Sjoerd Bearda, Joke Bearda, Luc Vlaming, Luc van Pevenage en Britta van der Pal, zonder jullie had dit verslag er nog niet gelegen. Daarnaast wil ik ook nog Wim Ottjes bedanken voor het mij attent maken op dit leuke onderzoek waarmee ik zowel de natuurkunde en informatica kant van mijn opleiding heb kunnen belichten. Tenslotte gaat mijn dank ook nog uit naar Pim Lubberdink en Wouter Lueks voor het oplossen van een aantal latex worstelingen.

33 A DESIGN

A Design

Figure A.1: Class design of ZeemanSlower

34 B PSEUDO CODE

B Pseudo code

The simulation is done according to the settings submitted in the configuration file. In this configuration file begin parameters are given. These variables are:

• the begin position of the particle • the begin velocity of the particle • the spin(S) of the particle • the mass of the particle • the nuclear spin of the particle (I) • the gravitational force • the wavelength of the laser • the detuning of the laser • the spontaneous emission rate of an excited atom • the length of the tube • the radius of the tube • the radius of the laser ﬁeld at the entry point of the tube for the particles • the radius of the laser ﬁeld at the end of the tube

All these parameters can be set to:

• a constant value • a (group of) uniform distributed random value(s) in a speciﬁed range • a (group of) Gaussian value(s) with a certain µ and σ

max−min • a group of N values in a range (min,max) with step size N . • a value from a group of values • a group of N logarithmic values in the range (min, max).

Of all these groups of values a complete super set of begin parameters is generated. This set is put in a queue. For each simulation the first parameter set is popped of the queue and is simulated. When the particle fails or succeeds and the next parameter set is simulated. The simulation is done by using the mentioned time determination. For each evaluation moment 4 things can happen according to the current state and position: Inside a photon field Outside a photon field Ground state Absorption Nothing Excited state Stimulated or Spontaneous Spontaneous Emission Emission

At the beginning and end of the simulation the state is sent to all output classes. All these output classes use the data to generate a histogram, a 2D-plot or a 3D-plot.

35 B PSEUDO CODE

B.1 Pseudo code for the leader

For the parallelization of the program the leader-worker-model is chosen. This is because only the leader has access to the file system so only the leader can read the configuration file and save the plots. This also gives the advantage that there is one process that coordinates what part of work each worker is simulating, so no double work is done. Each worker gets the first 500 begin parameters from the remaining work queue. When a worker is done with the 500 tasks the worker requests new work. The leader gives then the next 500 begin conditions until the queue is empty. Otherwise the leader sends a empty work queue message and the worker terminates.

Read configuration file Distribute global values from configuration file to the workers Initialize ParameterQueue from settings configuration file Send 500 begin parameters from the ParameterQueue to each worker while expect results process data from worker if worker requests new data then if ParameterQueue is not empty then Send beginConditions from the ParameterQueue to the worker else Send no work left endif endif end while Make plots

B.2 Pseudo code for the worker

get conﬁguration parameters from leader get own part of the ParameterQueue from leader while state = pop parameters from the ParameterQueue Vector focalPoint = new Vector(0,0,−rb/slope) state.failed = state.done = false update plots with current state while !state.failed and !state.done if groudState if inside photonField check for absorption else move particle with a timeStep end if else (excited state) if inside photonField check for spontaneous or stimulated emission else check for spontaneous emission end if end if if (outside tube or particle goes backwards) state.failed = true if (pos.z ≥ length) state.done = true end while update plots with current state Send results of plots to leader if ParameterQueue is empty then

36 B PSEUDO CODE

Request new beginConditions from the worker if received no work left then ﬁnalize else ﬁll ParameterQueue with the new received work endif endif end while

37 C SYNTAX OF THE CONFIGURATION FILE

C Syntax of the conﬁguration ﬁle

The conﬁguration ﬁle that commands the simulation platform what to do has three root commands:

• Distribution • Correlation • ZeemanSlower

These simulations can be given a list of plots. These are: a Histogram, a Plot and a 3d Plot. The set of parameters for each command and plot are shown in the following tables.

Table C.1: Parameters for Distribution simulation

Syntax Description Default value name Name of the simulation must be set value Random value distribution Random(0.0, 1.0, 1000)

Table C.2: Parameters for Correlation simulation

Syntax Description Default value name Name of the simulation must be set value Random value-distribution Random(0.0, 1.0, 1000) N Check for correlation with N previous values 2

Table C.3: Parameters for Zeeman slower simulation

Syntax Description Default value name Name of the simulation must be set dt Time step 1e−6 s Approach Whether to use Gjalt’s approach of Lotje’s Gjalt MagneticFieldFile File where the magnetic field values are grouped with z-position versus the magnetic field strength in Tesla CoilMain Current through the main coil for the magnetic 4.0 A used if Magnetic- field FieldFile is not set CoilPeak Current through the peak coil for the magnetic 0.0 A used if Magnetic- field FieldFile is not set CoilDip Current through the dip coil for the magnetic field 0.0 A used if Magnetic- FieldFile is not set Length Length of the tube 0.90 m TubeRadius Radius of the tube 0.0085 m RadiusAtBegin Radius of the laser field at the point where the 0.004 m atoms enter the Zeeman slower Continued on next page

38 C SYNTAX OF THE CONFIGURATION FILE

Table C.3 −− continued from previous page Syntax Description Default value RadiusAtEnd Radius of the laser field at the point where the 0.008 m atoms leave the Zeeman slower Wavelength Wavelength of the photons 422.79e−9 m LaserPower Power of the laser 40 mW Gravity The gravitational force [0 −9.81 0] m/s2 Gamma linewidth of the atomic transition (in units of 2 217.59 MHz π) Delta Detuning of the laser frequency (in units of 2 π) −415 MHz Mass The mass of the particle 6.68e−26 kg Temp Temperature of the Calcium that generates the 673.0 ◦K velocity distribution I The Nuclear spin of the particle 0.0 BeginPosition Begin position of the particle [0.0, 0.0, 0.0] m BeginPosition Begin velocity of the particle [0.0, 0.0, 650.0] m/s useBeginV Use customized velocity distribution or Boltz- TRUE mann velocity distribution measurePower Power of the measure laser 0.5 mW measureDetuning Detuning of the measure laser −100 MHz measureAngle Angle of measure laser w.r.t. the calcium beam 60◦ measureRadius Radius of the laser field of the measure laser 0.0007 m measureHitZ Position where the laser field hits the calcium 0.95 m beam

Table C.4: Parameters for Histogram output

Syntax Description Default value data What value to put in the histogram for each sim- must be set ulated begin state and the number of bins xType XaxiscanbeeitherLOGorLINEAR LINEAR condition Value is only put in the histogram if the condition none is met dataﬁle File to write the results(+“.csv”), simulation must be set (+“.sim”) and plots (+“.eps” for B/W and +“.ps” for color) title Title of the histogram “” gnuplotcmd Additional commands to give to gnuplot (plot “” tool that makes the plots) xlabel Labelforthexaxis “”

Table C.5: Parameters for plot output

Syntax Description Default value X Where on the x axis must the value put in the plot must be set for each simulated begin state and the number points Continued on next page

39 C SYNTAX OF THE CONFIGURATION FILE

Table C.5 −− continued from previous page Syntax Description Default value Y Where on the y axis must the value put in the must be set plot for each simulated begin state xType XaxiscanbeeitherLOGorLINEAR LINEAR yType YaxiscanbeeitherLOGorLINEAR LINEAR condition Value is only put in the plot if the condition is none met datafile File to write the results(+“.csv”), simulation must be set (+“.sim”) and plots (+“.eps” for B/W and +“.ps” for color) title Title of the plot “” gnuplotcmd Additional commands to give to gnuplot (plot “” tool that makes the plots) xlabel Labelforthexaxis “” ylabel Labelfortheyaxis “” mode For each point compute the average and no stan- avg2 dard deviation (avg0), standard deviation(avg1), different standard deviation for below and above average(avg2) or median and the first and third quartile(median) of the y values minx minimal value for x in the plot not set maxx maximal value for x in the plot not set

Table C.6: Parameters for 3d plot output

Syntax Description Default value X Where on the x axis must the value put in the plot must be set for each simulated begin state and the number of rows Y Where on the y axis must the value put in the plot must be set for each simulated begin state and the number of columns Z Where on the z axis must the value put in the must be set plot for each simulated begin state xType XaxiscanbeeitherLOGorLINEAR LINEAR yType YaxiscanbeeitherLOGorLINEAR LINEAR zType YaxiscanbeeitherLOGorLINEAR LINEAR condition Value is only put in the 3d plot if the condition none is met dataﬁle File to write the results(+“.csv”), simulation must be set (+“.sim”) and plots (+“.eps” for B/W, +“.ps” for color and +“ top.ps” for top view) title Title of the 3d plot “” gnuplotcmd Additional commands to give to gnuplot (plot “” tool that makes the plots) xlabel Labelforthexaxis “” ylabel Labelfortheyaxis “” zlabel Labelforthezaxis “” mode For each cell compute the average (avg), sum avg (sum) or median (median) of the y values minx minimal value for x in the plot not set Continued on next page

40 C SYNTAX OF THE CONFIGURATION FILE

Table C.6 −− continued from previous page Syntax Description Default value maxx maximal value for x in the plot not set miny minimal value for y in the plot not set maxy maximal value for y in the plot not set

C.1 Groups

Next to the root commands an extra root command is available: group. This command groups some parameters and can be included in the root commands and plots with the “Include” command. Before including also set of parameters with values can be set. When a group is included the parameters are replaced by the parameter value. For example:

Group “Plot1” { Histogram { DataFile “output/$(FILEPREFIX) Zeeman vx begin” Title “Begin velocity distribution” Xlabel “Velocity(m/s) in x-direction at t=0” Data v.z t=0.0 40 } }

ZeemanSlower { Name “Zeemantest1“ Param FILEPREFIX“test1” BeginVelocity [0.0,0.0,Gauss(650.0,200.0,200)] Include “Plot” } becomes due to the set Parameter:

ZeemanSlower { Name “Zeemantest1“ BeginVelocity [0.0,0.0,Gauss(650.0,200.0,200)] Histogram { DataFile “output/test1 Zeeman vx begin” Title “Begin velocity distribution” Xlabel “Velocity(m/s) in x-direction at t=0” Data v.z t=0.0 40 } }

41 REFERENCES

References

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[van Ditzhuijzen, 2002] van Ditzhuijzen, C. (2002). Monte Carlo simulations and atomic beam analysis for the Al41Catraz experiment: ultra-sensitive trace analysis of calcium. Diploma Thesis, University of Groningen. [Wansbeek, 2006] Wansbeek, L. (2006). Escape from Al41Catraz, Monte-Carlo simulations for the Al41Catraz experiment. Diploma Thesis, University of Groningen.