An Optical Mask for Atomic Interferometry Experiments

An Optical Mask for Atomic

Interferometry Experiments

Simon Coop

a thesis submitted for the degree of

Master of Science

at the University of Otago, Dunedin,

New Zealand.

2013

Abstract

This thesis presents work performed to obtain an optical mask for conducting matter-wave interfer-ometry experiments with ultra-cold rubidium-85 atoms. The optical mask is essentially an absorptivediffraction grating made of light, and it can imprint a periodic density pattern on a cloud of atoms witha period of half the wavelength of the light used. The mask has an analogous effect to quickly passinga diffraction grating through the cloud, removing atoms located at the nodes of the grating (thoughthe optical mask depumps atoms to a different hyperfine ground state rather than actually removesthem). The mask should be useful in performing precision measurements of physical constants suchas the fine-structure constant α, and acceleration due to gravity g. The thesis briefly expounds majorhistorical developments in atom interferometry and laser cooling. Prerequisites for an optical maskinclude a functioning magneto-optical trap, and the ability to perform polarisation-gradient coolingon trapped atoms. The theory of these two techniques is reviewed and the principle of the the opticalmask is explained. The experimental apparatus that was constructed to realise the optical mask isdescribed, and technical developments made along the way are presented. Finally, aspects of the ex-periment relevant to the optical mask are characterised. The experiment can reliably produce samplesof cold atoms at temperatures of around 10 µK, and can use the cold atoms as an optical frequencyreference accurate to∼ 1 MHz. Clear evidence that the cloud of atoms is being density-modulated bythe optical mask is presented. Improvements required to make the experiment into a fully-functioninginterferometer are also discussed.

i

Acknowledgements

There’s a quote attributed to Isaac Newton “If I have seen further it is by standing on ye sholders ofGiants.” I won’t claim to have made the same impact as Newton, but if this thesis is any achievementat all, it would not have been possible without Mikkel. While I never actually stood on his shoulders,he is quite tall. Mikkel, your (as far as I can tell) endless patience and knowledge, make you aninspiration for me, both as a scientist and a person.

I’d probably still be trying to turn on the laser if it wasn’t for Peter, without the benefit of yourexperience and humour I’d never have finished this. I wouldn’t have made it much further if it weren’tfor Andrew and Tzahi. Protips from you guys made life a lot easier.

Though much less frequent than it should’ve been, coffee with Fung provided lively discussion oflife, the universe and everything. The excuse for some sunshine and non-HEPA-filtered air was nicetoo.

Alicia, it’s not normal to learn Spanish as well as science in a physics lab. From you I learnt largeamounts of both!

Peter and Richard, you made my regular trips to the mechanical workshop an absolute pleasure. Inyour professional, efficient approach to your work, your ingenuity, and your lively humour.

Sandy, your remarkable ability to navigate the labyrinthine university bureaucracy with ease, andyour bowl of chocolates both contributed significantly!

All my friends in Dunedin, in and outside of the physics department, you made my time at Otagopretty awesome. Thanks!

iii

Contents

1 About this Thesis 1

2 Introduction 32.1 Atom Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Laser-Cooled Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Background 73.1 One-Dimensional Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.2 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 σ+ − σ− Polarisation Gradient Cooling . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.2 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Atomic Interference using a Resonant Optical Standing Wave . . . . . . . . . . . . . 16

4 The MARIE Experimental Apparatus 214.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Rubidium Dispenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Lasers and Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.4.1 Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4.2 Tapered Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4.3 Experiment Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.5 Magnetic Field Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5.1 Quadrupole Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5.2 Quenching Field Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.5.3 Compensating Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 Making Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6.1 Photomultiplier Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6.2 Photodiode for Atom Measurement . . . . . . . . . . . . . . . . . . . . . . 32

4.6.3 PIXIS CCD Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

v

4.6.4 Video Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.7 LabVIEW Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.7.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.7.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.7.3 MATLAB Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.7.4 Controlling the Optical Mask . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Characterising MARIE 39

5.1 Counting Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Magneto-Optical Trap and Optical Molasses . . . . . . . . . . . . . . . . . . . . . . 41

5.2.1 Getting Cold Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2.2 Measuring the Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3 Measuring the Laser Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Optical Mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4.1 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4.2 Saturation with the Optical Mask . . . . . . . . . . . . . . . . . . . . . . . 49

5.4.3 Quenching Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.4.4 Interference in the Optical Mask . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Summary and Future Work 53

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A Using MARIE’s LabVIEW program 55

A.1 Description of MARIE’s LabVIEW program features . . . . . . . . . . . . . . . . . 55

A.2 List of Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B Derivation of Time-of-flight Equation 59

References 61

vi

Chapter 1

About this Thesis

This thesis presents work performed between March 2011 and June 2012 related to the constructionand characterisation of an ‘optical mask’ for atom interferometry experiments. The idea for the opticalmask is from [1]. It is hoped that this optical mask will provide New Zealand’s first absolute precisiongravimeter.

The basic principle of the optical mask is that it uses an optical standing wave to act as a diffractiongrating for a cloud of cold atoms. To make one we then need an apparatus that can trap and coolatoms, the optical mask, and a way of measuring the effect of the optical mask on the cold atoms.

Chapter 2 briefly discusses major historical developments in the fields of matter-wave interferom-etry and laser cooling.

Chapter 3 provides background information on laser cooling techniques used in this experiment,and then goes on to discuss the mechanism of the optical mask, and how it could be used to providea gravimeter.

Chapter 4 describes the experimental apparatus in detail. The equipment used and a computerprogram developed for the experiment are discussed.

Chapter 5 discusses some results in characterising the experiment. Including how the temperatureof the atoms was measured, and evidence the optical mask was working.

The appendices contain information that might prove useful to a future student working on theexperiment.

It should be noted that some of the apparatus discussed in Chapter 4 was not constructed by mealone, but the apparatus is presented in whole so that this thesis forms a complete description of theexperiment. When I started most of the apparatus used to make the magneto-optical trap was alreadyinstalled. Improvement of the system to obtain a functioning magneto-optical trap was performed in

2 About this Thesis

collaboration with another student. Beyond this, all of the content in this thesis is the result of myown work, except for the compensating coils in Section 4.5.3 which were designed and constructedby another student.

Chapter 2

Introduction

2.1 Atom Interferometry

Interferometry, the idea of traversing two or more waves along different paths in a device and thenrecombining them to observe their interference, has long been a tool for researchers to make precisionmeasurements of physical phenomena. A famous historical example is the Michelson-Morley exper-iment of 1887, an optical interferometer designed to measure the effect of the luminiferous ether onthe speed of light [2]. The unambiguous null result of this experiment played a major role in refutingthe theory of the ether and the development of the special theory of relativity. Another example isFroome’s 1958 measurement of the speed of light using a two-path microwave interferometer. At thetime it was the most accurate measurement of c ever made (his result was c = 299 792 500 ± 100

ms−1, compared with the modern definition of 299 792 458 ms−1) [3].

With the advent of quantum mechanics and the discovery of the wave nature of matter, it was re-alised particles could be used for interference measurements. As early as 1927, experiments wereconducted that showed electron diffraction [4]. Then in 1936 crude experiments showed neutrondiffraction from crystals. This effect was soon exploited to gain quantitative insights on crystal struc-tures that were unobtainable with conventional x-ray crystallography [5].

Interferometers require some kind of coherence between the waves. The interference signal arisesbecause of a phase difference between waves that have travelled alternate paths in the interferometer,so the waves must be initially coherent. With this in mind, any quantum mechanical degree of freedomcan be treated as a wave (e.g. position, momentum, spin, etc), so to interfere quantum particles theymust be first localised in some space to provide the initial coherence. For example, electrons areprepared in the same spin state in the Stern-Gerlach experiment, or atoms in an atomic beam (i.e.localised in momentum space) can be made to interfere by diffracting off a grating [4].

Neutral atoms are good candidates for matter-wave interference. They have large optical cross-sections compared to neutrons or electrons, sources are cheap to produce, and properties such asmass and magnetic moment can be selected over a large range. With the invention of laser cooling in

4 Introduction

the 70’s and 80’s, preparing samples of atoms with very narrow momentum distributions has becomecommonplace, making them ideal for interference experiments [4].

In a generic interferometer, some mechanism coherently splits the waves to send them down differ-ent paths. For example, in a light interferometer this can be done with a diffraction grating or a 50/50beamsplitter. As mentioned above, atoms can be diffracted off a grating. One very early example ofthis is the diffraction of helium atoms from a LiF crystal surface, first done by Estermann and Stern in1930. The periodicity of the crystal lattice lets it act as a phase grating for the atoms. Modern exper-iments can make use of nanofabricated structures to make transmission gratings, which can diffractatomic beams provided the transverse velocity of the beam is low enough. [4]

Another way of making diffraction gratings for atoms is to use optical standing waves, these haveseveral advantages over gratings made of matter: they can be switched on and off very fast, they havea tunable period, and the phase can be shifted easily. Far-off resonant light can be used to make phasegratings for atoms, and on-resonant light can make an absorption grating (such as the one describedin this thesis). [4]

Since the early 90’s, atom interferometry has been instrumental in making many precision measure-ments. In 1991, Kasevich and Chu made an interferometer that could measure gravity to a precisionof 3×10−8 g, [6] and in 1993 Chu made an interferometer that could measure ~/m for cesium atoms(proportional to the fine-structure constant α) to an accuracy of 10−7. [7]

More recent experiments include an accurate measurement of G, Newton’s gravitational constant,using a 500 kg mass moved between two atom interferometers [8], a measurement of the difference ingravitational acceleration between 85Rb and 87Rb atoms as a test of Einstein’s equivalence principle(they were unable to measure a difference) [9]. An experiment in Paris can continuously measuregravitational acceleration three times a second to an accuracy of 1 µGal (1 Gal = 1 cm/s2) [10].

2.2 Laser-Cooled Atoms

It has been thought for centuries that light exerts a force on matter, as long ago as 1619 Kepler sug-gested that the shape of comet tails was the result of solar radiation pressure. Radiation pressure wasfirst quantitatively described with the advent of Maxwell’s equations of electromagnetism in the 19th

century. The first experiments decisively proving that light exerts pressure on matter were performedindependently by Lebedev [11] and Nichols and Hull [12] in 1900 and 1901 respectively. Einstein’sdescription of radiation absorption and emission in 1917 explained a mechanism for this pressure[13]. The next big development was the suggestion (by Townes, [14]) and then implementation (byMaiman, [15]) of the laser in 1958 and 1960 respectively.

The intense, coherent light source that is the laser catalysed the field of atom manipulation. In themid-70’s several researchers suggested the idea of using lasers to reduce the random thermal velocities

Introduction 5

of a sample of atoms - what is now known as laser cooling [16] [17]. Although much earlier, in 1968,Letokhov suggested using the optical dipole force to trap atoms (see [18] for reference).

The first experiments on laser-cooled atoms were performed in 1978 first by Wineland, Drullinger,and Walls [19], and then Neuhauser et al [20]. In their experiments both groups laser cooled trappedions, rather than neutral atoms.

The first experiments that clearly showed laser-cooled neutral atoms were performed in 1981 atMoscow’s Institute for Spectroscopy by Andreev et al (see [18] for reference). Over the early-to-mid-80’s, Phillips and Metcalf perfected a device they called the Zeeman slower, which uses a spatially-varying magnetic field to keep a sample of atoms in resonance with a laser beam as it cools. Theysuccessfully produced the first sample of ‘stopped’ atoms with a temperature of less than 100 mKusing this device [18].

In 1985 Steven Chu and his group laser cooled a sample of sodium atoms to 240µK using whatis now known as Doppler cooling - a standard technique of laser cooling [21]. Shortly after thedemonstration of magnetic traps and optical dipole traps for neutral atoms in 1986 and 1987, Raab,Prentiss, Cable, Chu and Pritchard successfully used this radiation-pressure cooling technique toimplement a magneto-optical trap, and held atoms in the trap for around two minutes in 1987 [22][23]. A magneto-optical trap uses a combination of a magnetic field and radiation pressure to containatoms to a small region inside a vacuum chamber, a description is given later in this thesis.

While Chu was working on his magneto-optical trap, Phillips found his laser cooling system wasproducing much lower temperatures than what was theoretically predicted to be possible [18]. Thiswas the first instance of sub-Doppler cooling. It was later realised that all the Doppler cooling modelshad assumed simple two-level atoms, but real atomic structure is much more complex than that.Sub-Doppler cooling that exploited atoms with multi-level structure was theoretically explained byDalibard and Cohen-Tannoudji in 1989 [24]. This sub-Doppler cooling regime could produce atomicsamples with temperatures approaching the one photon recoil limit (∼0.4µK for 85Rb).

Laser cooling has come much further more recently, and has found a myriad of real-world applica-tions. In 1995 laser cooling was instrumental in creating the world’s first Bose-Einstein condensate.Laser cooling has also been useful in making atomic clocks orders of magnitude more accurate. [25].

6 Introduction

Chapter 3

Background

This chapter reviews the theory of some basic laser cooling techniques that were used in the exper-iment. The last section is a description of the optical mask, discussing how it works, and how it couldbe applied in a gravimetric measurement.

3.1 One-Dimensional Doppler Cooling

The first kind of laser cooling to be suggested - and conceptually the simplest - is Doppler cooling[16] [17]. Consider an atom whose internal structure can be approximated as consisting of a groundstate and a single excited state. This approximation is valid in the case of an atom in the path oflow-intensity laser light which is tuned close to the frequency difference between the two states. Insuch a case an atom is known as a two-level atom.

In the presence of resonant light a two-level atom in the ground state will absorb a photon. Thishas two effects: firstly the atom will jump up to the excited state, storing the energy of the photon;secondly the atom will absorb the momentum of the photon, giving the atom a ‘kick’ in the directionthe photon was travelling. After a short time the atom will spontaneously fall back down to the groundstate, emitting a photon in a random direction and again giving the atom a momentum kick, but ina direction opposite to the direction of the emitted photon. Over time, these momentum kicks fromphoton emission will cancel themselves out but the momentum kicks from absorption are always inthe same direction. This has the net effect of a force on the atom in the direction of the laser beam.This force is known as the radiation pressure force.

For Doppler cooling, the atom is placed in the path of two co-axial counter-propagating laser beamstuned slightly below atomic resonance (i.e. red detuned). Experimentally this can be achieved justby passing the two laser beams through a cloud of gaseous atoms in a vacuum chamber. If the atomhas a velocity towards one of the laser sources, the Doppler effect will cause the laser frequency toincrease in the atom’s rest frame and thus increase the atom’s absorption probability from that laserbeam. The atom will thus feel a net force in the direction opposite to its motion and will slow down.The force on the atom from either laser beam is given by Eq. 3.1 [25]

8 Background

F± = ±1

2~k

II0

Γ

1 + II0

+ 4(

∆∓|ωD|Γ

)2

(3.1)

where I is the laser intensity, I0 is the saturation intensity for the transition, Γ is the naturallinewidth of the transition ( 2π× 6.1 MHz for the 85Rb D2 transition), ∆ is the laser detuning fromresonance, k is the laser wavevector, and ωD is the Doppler shift of the laser in the atom’s rest frame(ωD = 2π × νatom/λ).

The rigorous derivation of Eq. 3.1 is quite long, so it will not be reproduced here. It can, however,be justified as being reasonable. For a stationary atom in an on-resonance light field (i.e. ∆ = 0 andνatom = 0), the force becomes F = 1

2~k[ II0 Γ/(1 + II0

)], which is simply the momentum per photon~k, times the scattering rate. The scattering rate can be interpreted as the rate at which photons areabsorbed and then reemitted by the atom. As the light intensity becomes very large (i.e. I →∞), thescattering rate approaches Γ/2. This is exactly the expected result: The excited state decays at rateΓ, and at high intensity the atom has a 50% probability of being in the excited state. Conversely, atlarge detunings the force goes to zero, which is also expected as the atom does not strongly absorblight that is far from resonance.

In general the total force on the atom is not simply the sum of the forces from each of the laserbeams. There can be sequential effects where one laser beam excites the atom and the other causesstimulated emission, leading to large velocity-independent changes in the atom’s speed. However inthe case where the light intensity is low enough such that stimulated emission is not important, thetotal force on the atom is simply the sum of the contributions from each laser beam, represented bythe solid line in Figure 3.1. [25].

In the absence of any other effects, the temperature of the atom would decrease to zero Kelvin.However, the effect of spontaneous emission must be included. Every time a photon is absorbed, it isemitted a random time later in a random direction. The average velocity imparted by these momentumkicks is zero, but the rms value is finite. The atom can be viewed as undergoing a random walk inmomentum space. The average velocity of the atom is still zero, but the rms velocity slowly grows,which is the same as heating the atom. Thus the final temperature is determined by an equilibriumbetween this heating mechanism and the cooling effect described above. This minimum achievabletemperature is called the Doppler cooling limit, and is roughly given by [18]:

TD ≈~Γ

2kB(3.2)

where kB is Boltzmann’s constant. For example, for rubidium this temperature is about 150µK.

This technique can easily be generalised to three dimensions by the use of three orthogonal pairsof counter-propagating laser beams (i.e. two counter-propagating beams along each of the x-, y-, and

Background 9

−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Atom Velocity (Γ/k)

Forc

e(h

kΓ)

Figure 3.1: This figure shows the force on an atom moving in one dimension due to F+ and F−(dotted lines), and Ftotal, which is the sum of the two contributions (solid line). These plots arecalculated with I

I0= 0.1 and ∆ = −Γ.

z-axes), instead of just two as described above. This provides a damping force to the atom’s motionin all directions. It has been shown that once caught in the intersecting laser beams, the atom hasa diffusive Brownian-like motion. This coupled with the fact Doppler cooling acts very much likeviscous friction has led to use of the term ‘optical molasses’ [18]. It should be noted that opticalmolasses does not actually trap neutral atoms, as there is no restoring force for atoms displaced fromthe centre of the beams. The atoms are slowed immensely but do eventually escape. A technique fortrapping atoms is discussed in the next section.

3.2 Magneto-Optical Trap

3.2.1 Description

A magneto-optical trap (MOT) uses Doppler cooling and a weak inhomogeneous magnetic fieldto trap neutral atoms into a small region. The principle is essentially Doppler cooling as describedabove, but the magnetic field Zeeman-splits the energy levels for an atom not at the centre of the trap.

10 Background

The Zeeman splitting causes a spatially-dependent difference in the absorption efficiency of eachlaser beam, and a restoring force is introduced. The basic structure of the trap is shown in Fig. 3.2:There are three orthogonal pairs of counter-propagating laser beams, which provide Doppler coolingin three dimensions; and two current loops which are arranged to produce a ‘spherical quadrupole’magnetic field [22]. The magnetic field is zero at the centre of the trap, and increases linearly in everydirection away from the centre (for small distances).

Figure 3.2: A three dimensional magneto-optical trap. The ‘spherical quadrupole’ magnetic field isgenerated by two coils with current flowing in opposite directions. Circularly polarised laser light isindicated by the red arrows. The origin is taken as the centre of the atom cloud.

The trap can be most easily understood in one dimension, with results applicable to the actualthree-dimensional trap. Consider the atom represented in Figure 3.3. It has an angular momentumJ = 0 ground state, and an angular momentum J = 1 (mJ = −1, 0,+1) excited state. In a weakmagnetic field described by B(y) = by the atom’s energy levels are Zeeman-split by an amount∆E = µBgmJB = µBgmJby, where b is the gradient of the magnetic field, µB is the Bohr mag-neton, and g is the appropriate g-factor for the atomic state [26]. Now introduce circularly-polarisedcounter-propagating laser beams as indicated in Fig. 3.2. The laser beams are tuned below the B = 0

resonance frequency, so the atom at y < 0 will be closer to resonance with the σ+ laser beam, andwill scatter more photons from this beam. The atom will thus feel a net force towards the origin.Similarly for an atom at y > 0. This also works in the other two dimensions, so an atom is alwaysbeing pushed towards the centre of the trap. If the experimental parameters are chosen appropriately,the motion of the atoms can also be damped by Doppler cooling as described above. The atoms arethus cooled and trapped.

Background 11

Figure 3.3: Energy levels of a two-level atom at y < 0 in Figure 3.2. The laser is red-detuned fromresonance with B = 0, Zeeman splitting causes the atom to be resonant with the laser polarisationthat pushes it back to the centre of the trap.

3.2.2 Practical Considerations

In a real-life experiment, there may be many atoms being cooled simultaneously, and there is theadded effect that a photon spontaneously emitted by one atom can be absorbed by another. Thisself-heating limits the density of trapped atoms in a MOT to ∼ 1011 cm−3 [25].

In my experiment, the transition used for cooling and trapping 85Rb was the D2 F = 3→ F ′ = 4

transition (see Figure 3.4). This is a closed transition, but there is still occasionally unavoidable off-resonant scattering to the F ′ = 3 or F ′ = 2 excited states, either of which can decay to the F = 2

ground state. This state is dark to the cooling laser and any atom in this state is no longer trapped.To thwart this effect a weak ‘repump’ laser tuned to the F = 2 → F ′ = 3 transition must be used.The direction is unimportant as the rate of excitation to the F = 3 excited state is small compared tothe rate of excitation to the F = 4 excited state, consequently the force from the repump laser is alsosmall.

3.3 σ+ − σ− Polarisation Gradient Cooling

This section describes σ+ − σ− and not lin⊥lin polarisation gradient cooling (PGC) as this wasthe technique used in the experiment. σ+ − σ− is practically very easy to implement as it uses thesame laser polarisations as a magneto-optical trap (described in section 3.2), therefore the same opticsand laser source can be used. This section only reviews why σ+ − σ− polarisation gradient coolingworks, and does provide a complete derivation. A detailed description can be found in [24].

12 Background

52S1/ 2

52P 3/ 2

780.241 368 271(27) nm384.230 406 373(14) THz

12 816.546 784 96(45) cm- 1

1.589 049 139(38) eV

1.264 888 516 3(25) GHz

1.770 843 922 8(35) GHz

3.035 732 439 0(60) GHz

F = 3

F = 2

100.205(44) MHz

20.435(51) MHz

83.835(34) MHz

113.208(84) MHz

120.640(68) MHz

63.401(61) MHz

29.372(90) MHz

F' = 4

F' = 3

F' = 2F' = 1

Figure 3.4: Hyperfine structure of the 85Rb D2 transition. The relative sizes of splittings are indicativeonly and should not be compared. Image and data from [26].

3.3.1 Description

σ+ − σ− polarisation gradient cooling uses two counter-propagating laser beams - as in Dopplercooling - to induce motion-dependent population of atomic ground states. Because of this motion-

Background 13

sensitive state population, each laser beam is absorbed with different efficiency giving rise to avelocity-dependent radiation pressure force which damps an atom’s motion. For 85Rb, the damp-ing force from PGC is much larger than from Doppler cooling, meaning much lower temperaturescan be acheived. However the capture velocity (the maximum initial velocity an atom can have if itis to be caught in the laser beams) is much smaller than in Doppler cooling.

The name ‘polarisation gradient cooling’ comes from the properties of the laser beams used. Bothlaser beams are circularly polarised, with one beam left-circularly polarised and the other right-circularly polarised. The beams add to give a light field with linear polarisation at every point, but theactual direction of polarisation changes as one moves along the propagation axis (see Fig. 3.5).

Figure 3.5: Polarisation gradient in σ+−σ− cooling. The two circularly polarised beams add at everypoint along the z-axis to give a linearly polarised light field that rotates in space. Image from [24].

σ+ − σ− cooling requires an atom with ground-state angular momentum Jg ≥ 1, so will beexplained using the simplest atom which can undergo this kind of cooling; an atom with Jg = 1 andexcited-state angular momentum Je = 2 (see Fig. 3.6). The description can easily be extended toatoms with larger ground-state angular momentum.

Say we have a stationary atom subject to a σ+ − σ− laser field where both of the constituent laserbeams have a wavevector of magnitude k, in a position where the polarisation is in the y direction.With the quantisation axis taken to be in the same direction as the local polarisation, it is shown in[24] that the relative steady state populations of the three ground states |g0〉y, |g−1〉y, and |g+1〉y- eigenstates of Jy - are 9/17, 4/17, and 4/17 respectively (g indicates an atomic ground state, andthe numerical subscript is the projection of angular momentum along the quantisation axis, the lettersubscript outside the ket indicates the direction of the quantisation axis using the coordinate systemin Figure 3.5).

Now consider an atom moving at velocity v through the laser field in the z−direction. As the atommoves, the local laser polarisation will change direction. It is convenient to introduce a rotating framethat moves with the atom such that the polarisation direction is constant. In this frame the atom’s spin

14 Background

Figure 3.6: Atomic level transition scheme and corresponding Clebsch-Gordan coefficients for aJg = 1↔ Je = 2 atom. Image from [24].

axis will precess about the laser propagation axis, which is the same effect as if the atom were subjectto a magnetic field in that direction (the z-axis in Figure 3.5). This adds a term

Vrot = kvJz (3.3)

to the Hamiltonian describing atomic evolution in the moving rotating frame.

By making the assumption that the detuning of the laser from resonance is much larger than thenatural linewidth of the transition, and then supposing that the atom is moving slowly through thelight field, then Eq. 3.3 can be treated as a perturbation to the atom’s Hamiltonian [24].

Using first-order perturbation theory, the term in Eq. 3.3 has no first-order effect on the energies ofthe different eigenstates of the atom where the quantisation axis is taken in the y−direction. However,the wavefunction is changed to first-order. The wavefunction of the perturbed |g0〉y state is

|g0〉y = |g0〉y +kv√

2(∆′0 −∆′1)(|g−1〉y + |g+1〉y) (3.4)

where ∆′m is the light shift of the mth magnetic sublevel. The light shift is negative for red-detunedlight, and is approximately proportional to the Rabi frequency squared (so ∆′1 = ∆′−1). By comparingat the relative strengths of the π transitions from |g0〉 and |g±1〉 in Figure 3.6, we can say

∆′0 =4

3∆′1 (3.5)

Similarly to Eq. 3.4:

|g+1〉y = |g+1〉y −kv√

2(∆′1 −∆′0)|g0〉y (3.6a)

|g−1〉y = |g−1〉y −kv√

2(∆′1 −∆′0)|g0〉y (3.6b)

These equations show that a moving atom initially in the state |g0〉y is ‘contaminated’ by the othertwo ground states |g+1〉y and |g−1〉y (and vice versa).

Background 15

The next step is finding the relative steady-state populations of the eigenstates |g0〉z , |g+1〉z and|g−1〉z . This can be accomplished if we exploit the fact that the expectation value of Jz is proportionalto the relative occupancy of its eigenstates. To this end, we first find the expectation value of Jz withthe quantisation axis in the y−direction (see Appendix A of [24]):

y〈g0|Jz|g0〉y =2~kv

∆′0 −∆′1(3.7a)

y〈g+1|Jz|g+1〉y = y〈g−1|Jz|g−1〉y =~kv

∆′1 −∆′0(3.7b)

These new |gm〉y states have the same relative populations as the |gm〉y states. Weighting theexpectation values of Jz (Eq. 3.7) by the relative populations of the |gm〉y states, one can calculatethe steady-state expectation value of Jz as

〈Jz〉st =2~kv

∆′0 −∆′1

(9

17− 2

17− 2

17

)=

40

17

~kv∆′0

(3.8)

where Eq. 3.5 has been used. Thus an atom moving in a σ+ − σ− laser field has an average Jzproportional to the velocity of the atom, and hence the two eigenstates |g±1〉z of Jz have differentsteady-state populations. Writing Π+1 and Π−1 to represent these populations, from Eq. 3.8 we get〈Jz〉st = ~(Π+1 −Π−1). So:

Π+1 −Π−1 =40

17

kv

∆′0. (3.9)

∆′0 is negative for red-detuned light, so it follows that if the laser beams are red-detuned from atomicresonance, and the atom is moving in the positive z direction, the |J−1〉z state has a larger populationthan the |J+1〉z state.

Absorption of a σ+ photon increases the eigenvalue of Jz by 1, and absorption of a σ− photondecreases it by 1. Say our laser beams are configured to match those shown in Figure 3.5. Figure3.6 shows that an atom in the |g−1〉z state is six times more likely to absorb a σ− photon than a σ+

photon, while the opposite is true for an atom in the |g+1〉z state. Eq. 3.9 shows that if an atom ismoving in the positive z direction with red-detuned light, then the |g−1〉z state is more populated thanthe |g+1〉z state. This atom thus scatters more photons travelling in the negative z direction, and thereverse is true for an atom travelling in the opposite direction. Hence whichever way the atom moves,its motion is opposed and it slows down.

3.3.2 Practical Considerations

Due to the small capture velocity of PGC, for effective cooling one must start with pre-cooledatoms. This can be accomplished with a MOT, which Doppler cools the atoms enough for the PGCto be effective.

16 Background

PGC is assumed to take place in zero external magnetic field. In the presence of an externalmagnetic field the atoms will still be cooled, but to some non-zero mean velocity. In other wordsthe velocity distribution of an ensemble of atoms will narrow, but the mean velocity will not be zero[27]. An external magnetic field will add to the ficticious magnetic field generated by atomic motionthrough the light field: the atom will be cooled to a point where these two magnetic fields add to zero,which is some non-zero velocity.

Because of the detrimental effect of external magnetic fields, the MOT quadrupole magnetic fieldmust therefore be turned off for the duration of PGC, and any stray magnetic fields must be eliminatedalso (such as Earth’s magnetic field, fields from nearby equipment, etc). The method used in thisexperiment for compensating for stray magnetic fields is described in Section 4.5.3.

The optimum parameters for PGC are a compromise. The derivation of PGC assumes the atom isat steady-state, i.e. the interaction time is long compared to the optical pumping time [25]. Howeverif there is no MOT quadrupole magnetic field, the atoms are no longer trapped and will eventuallydiffuse away. The PGC step of cooling must be long enough such that the atoms have reached the low-est possible temperature, but must be short enough such that the atoms have not moved a significantdistance. Parameters and results from this experiment are described in section 5.2.

3.4 Atomic Interference using a Resonant Optical Standing Wave

The long term goal for this experiment is to accurately measure interference between atomic mo-mentum eigenstates, for the purpose of precision measurements of quantities such as the fine-structureconstant α, and acceleration due to gravity g. This section explains the principle of an atomic inter-ferometer, and how it can be practically realised.

As mentioned in the introduction, atoms can be made to interfere by passing them through a diffrac-tion grating. The kind of diffraction grating used for the experiment described in this thesis is called anoptical mask. The optical mask implemented for this thesis is comprised of two counter-propagatinglaser beams resonant with an open atomic transition. The two laser beams interfere such that the lightintensity along the propagation axis is periodic, with the period being half the wavelength of the lightused. Atoms in a particular electronic ground state interact with the mask and are pumped into adifferent ground state unless they are close to the nodes of the standing wave where the light intensityis very low. The atoms that were not at the nodes of the mask are pumped into a far off-resonantinternal state. Once in this state, the atoms are decoupled from the light field and no longer interactwith the laser beam, they have effectively been ‘absorbed’ by the optical mask. The density of atomsremaining in the original state is now spatially modulated, and looks very similar to how it would ifthe atoms had passed through a absorption grating. This is shown in Figure 3.7.

An alternative to the absorptive optical mask considered here is to use a standing wave of far off-resonant light. This creates a phase grating for the atoms which can also diffract atoms into different

Background 17

Rb atoms in theF=3 ground state

85

Laser Laser

Figure 3.7: Operation of the optical mask. The two lasers form a periodic intensity pattern across thecloud of atoms. Atoms not at the nodes of the standing wave are depumped to the F = 2 groundstate, atoms at the nodes stay in the F = 3 ground state as they see very little light.

momentum states. In the limit of short pulses, this is known as as Kapitza-Dirac scattering. For moreinformation see page 1059 of the review [4].

Our experiment uses 85Rb, which has an energy level structure convenient for interferometry ex-periments. Figure 3.4 shows the hyperfine structure of the D2 transition utilised in this experiment.The F = 3 → F ′ = 4 transition is closed: an atom excited to the F ′ = 4 state can only decay backdown to the F = 3 ground state. This means this transition can be used for trapping, cooling anddetecting the atoms. The F = 3 → F ′ = 3 and F = 3 → F ′ = 2 transitions are open and can beused in the optical mask. The meaning of this is explained in the next paragraph.

Figure 3.8 shows what happens to an atom not at a node of the optical mask. Say the atoms start inthe F = 3 ground state, and the optical mask is resonant with the F = 3 → F ′ = 2 transition. Anatom in the presence of this light will move up to the F ′ = 2 excited state. Provided the light intensityis low enough such that stimulated emission is not significant, it will fall into the F = 2 ground statewith a probability of 0.79, and into the F = 3 ground state with probability 0.21. Once in the F = 2

ground state it will stay there as the optical mask is about 3 GHz off-resonance from any transitionstarting in the F = 2 ground state, and there is no light except that from the optical mask present atthis stage. Now say we have a cloud of atoms subject to an optical mask pulse. If the mask is turnedon for a time equal to several pumping cycles, most of the atoms not at the nodes of the optical maskwill end up in the F = 2 ground state. Atoms at the nodes do not see any light and stay in the F = 3

18 Background

ground state. The atom cloud now has a periodic density pattern of atoms in different states, with theperiod being half the wavelength of the light used for the optical mask. We use a laser beam resonantwith the F = 3 → F ′ = 4 transition to detect the atoms, so atoms in the F = 2 ground state can beconsidered gone from the system, i.e. absorbed by the optical mask.

F = 2

F = 3

F' = 3

F' = 2

0.79 0.440.21

0.56

Figure 3.8: Relative branching ratios from different 85Rb hyperfine excited states. Branching to theF = 2 ground state is much stronger from the F ′ = 2 excited state than from the F ′ = 3 excited state.Values were calculated from data in [25] by assuming equal population in all magnetic sublevels, andaveraging over all 3 possible light polarisations.

Besides modulating the density pattern, the optical mask has another effect on the atoms knownas the “optical Stern-Gerlach effect” in which the standing wave transfers momentum to the atoms.For sufficiently short pulses, the process is coherent [4]. The amount of momentum transferred isproportional to the gradient of the electric field (i.e. the dipole force) [28]. This coherent transfer ofmomentum to the atoms is effectively a beamsplitter, diffracting each atom into different momentumstates which evolve coherently, i.e. with a well-defined phase relationship.

After a single optical mask pulse, the density-modulated atoms do not quite constitute an interfer-ometer. While they are very cold, the atoms still have a Maxwell-Boltzmann velocity distribution andcan be thought of as an incoherent mixture of plane waves. While each atom is coherent with itself,there is no definite phase relationship between different atoms.

To obtain a fully-functioning interferometer, a second optical mask pulse is used. Applied at a timeT after the first pulse, it quenches all atoms except those with an integer multiple of a momentum thatlets them pass through both optical mask pulses without being depumped into the dark state. Thisis shown in Figure 3.9. Atoms that survive the first pulse disperse due to their initial momentumdistribution, atoms that survive both pulses must have a momentum such that they moved an integernumber of wavelengths along the optical mask during the time between the pulses. After this secondpulse the remaining atoms are coherent, and form an interference pattern that can be measured (notethat the uncertainty principle does not significantly contribute to broadening of fringes in the interfer-

Background 19

ence pattern [1]). While Figure 3.9 shows a classical picture of the possible interference patterns, onecan measure interference patterns that are classically not possible, as in [29].

An appropriate value for T , the time between optical mask pulses, has been experimentally shownto be of the order 50 µs [29]. If the pulses are too close together then there will be less selection oftransverse momentum states, if they are too far apart then less atoms will survive overall as they arebeing depumped due to stray light fields and they will also eventually move out of the optical maskbeams.

Figure 3.9: Classical trajectories for atoms after the optical mask. The horizontal axis is time, andthe vertical axis is distance along the optical mask. The black lines represent classical trajectoriesof atoms. After the first mask atoms are localised in position space but not momentum space. Theydiffuse out for time T , then a second mask ‘absorbs’ all atoms except those with an integer multipleof a particular momentum. Image from [29].

The optical mask could be applied to gravimetry: If the optical mask propagation axis is vertical,the atomic interference pattern will fall between pulses. A third optical mask can be applied at time2T , by measuring the number of atoms that survive this third mask as a function of the phase ofthe mask, the interference pattern can be ‘mapped’. By mapping the interference pattern around 2T ,the gravitational acceleration of the atoms can be determined very accurately. This technique can bemade even more accurate by measuring around times that are harmonics of the interference pattern,i.e. at times t = T (N + 1)/N . The higher-frequency interference pattern means smaller movementcan be resolved. [9]

20 Background

Chapter 4

The MARIE Experimental Apparatus

Figure 4.1: Photo of MARIE’s vacuum chamber and axes definition used in this thesis.

This chapter describes the instruments and equipment used to make the experiment, so that the laterchapter on characterising the experiment can be understood.

22 The MARIE Experimental Apparatus

4.1 Overview

The experiment was named Measuring Atomic Resonances in an Interferometric Experiment,which coincidentally forms the acronym MARIE. MARIE is a typical cold-atoms experiment: Avacuum chamber with windows to allow laser light to enter, an internal source of alkali metal atoms,surrounding optics, cameras/observation instruments, and coils of wire for generating desired mag-netic fields.

The entire experiment is controlled at the top level by a LabVIEW program which is basicallya highly configurable arbitrary function generator. Data is collected on an oscilloscope, saved in abinary file on a computer and then analysed with MATLAB.

4.2 Vacuum Chamber

Figure 4.1 shows the heart of MARIE: a cylindrical vacuum chamber (14 cm radius, 17 cm depth),with 10 cm radius windows on both of the flat sides, and eight smaller flat faces arranged radiallyin an octagonal pattern around the cylinder. Seven of these eight sides have windows in them. Fourof these windows are 2 cm in radius, and the other three are 3.2 cm in radius. The windows arenot anti-reflection coated, and light transmission at normal incidence was measured to be 93% perwindow. The entire chamber is oriented such that the cylinder axis is parallel to the ground (seeFigure 4.2 for a schematic). Pressure is kept below the measurement capability of an MKS I-Magcold-cathode vacuum gauge (less than 10−11 Torr) with a combination of a Varian StarCell ion pumprunning continuously, and a titanium sublimation pump (TSP) which we ran whenever pressure in thechamber was measurable. The ion pump was installed as far as possible from the main chamber as itemits quite a strong magnetic field, which can interfere with experiments. Experiments take place inthe centre of the cylindrical region of the system, which is connected to the two pumps by a series ofpipes coming out the bottom of the cylinder.

4.3 Rubidium Dispenser

Rubidium for experiments is sourced from a SAES alkali metal dispenser, commercially producedfor coating applications. It consists of a small amount of rubidium chromate with a reducing agent,such that when the compound is heated the metal is reduced and pure rubidium is released from thedispenser as a vapour. The source is mounted on a wire, so that it can be heated by passing a currentthrough the wire. The dispenser sits behind a 5 cm-long narrow copper tube pointing directly at thecentre of the experiment trapping region, collimating the vapour to increase the trapping efficiencyand decrease the trap load time. Current to heat the dispenser was controlled by MARIE’s LabVIEWprogram.

The dispenser is powered by an Agilent 6553A power supply running in voltage-programmedcurrent mode. The output current is proportional to an input voltage from the computer. Typically

The MARIE Experimental Apparatus 23

Figure 4.2: Schematic of the laser beam and instrument configuration at the vacuum chamber, fromtwo different perspectives. Axes shown are the same as in Figure 4.1. All of the features are explainedin the text. Each feature is shown in the orientation that is most convenient. The letters label the entryof different laser beams: O labels the optical mask beams, C the cooling beams, D the detectionbeam, and R the repump laser. (a) A side-on view of the vacuum chamber. Two of the cooling beamsand the detection beam are perpendicular to the page. (b) A top-down view.

when ‘on’ a current of 5.3 A was used to heat the dispenser.

To minimise the dead-time waiting for the trap to load each time an experiment was run, the trapwas turned on as quickly as possible after each experiment. This meant most of the atoms fromthe previous experiment were re-trapped, minimising the amount of time the dispenser needed to beturned on.

For a further description and characterisation of the rubidium dispenser see [30].

4.4 Lasers and Optics

Light sources are two frequency-locked and temperature stabilised laser diode systems. One diodeprovides repump light, and the other is amplified and split into three beams to provide light for trap-ping and cooling the atoms, detecting the atoms, and the standing wave required for the optical mask.


4.4.1 Diodes

MARIE uses two Sharp Microelectronics GH0781JA2C temperature-stabilised laser diodes. Onediode is locked above the 85Rb D2 F = 3 → F ′ = 4 transistion. It provides a seed to a taperedamplifier, the output of which is passed through three acousto-optic modulators (AOMs). This isshown schematically in Figure 4.3. The locking system used for the two diodes is described elsewhere[31]. It can provide light with a centre frequency stable to 1 MHz, with a full-width half-maximumlinewidth (FWHM) of 2-4 MHz.

The other diode is locked above the F = 2 → F ′ = 3 transition and is used for the repumplaser. The output of the diode is passed through an Isomet AOM which is used to shift it to down toresonance, and is also used to switch off the light when it is not needed. The output of the AOM isdelivered directly to the experiment.

4.4.2 Tapered Amplifier

The tapered amplifier (TA) used in the experiment is an Eagleyard Photonics EYP-TPA-0780-01000-3006-CMT03-0000. This TA was assembled as part of another student’s Honours project, see[32] for more information on construction and characterisation. The TA is prone to multi-mode whennot aligned properly ∗.

The TA is seeded with about 40 mW of light from the laser diode. It uses about 2 A of current toamplify the seed beam up to about 800 mW.

4.4.3 Experiment Laser Beams

The 800 mW output beam of the TA is passed through three AOMs to shift the light to providethree different laser beams at different frequencies for different purposes (Figure 4.3). After eachAOM the light is coupled into a single-mode polarisation-maintaining optical fibre for delivery to theexperiment. See Figure 4.2 for a diagram of the configuration of the beams at the vacuum chamber.Due to the poor-quality ouput mode of the TA, coupling efficiency into the fibres is quite low, about20-30%. Using the fibres is necessary to have Gaussian-shaped beams to use in the experiment. Thelaser is above the F = 3→ F ′ = 4 resonance, so AOMs are used to lower the frequency of the laserto appropriate values.

The AOMs serve a second purpose of providing a mechanism for switching the laser beams on andoff when required. Light deflection could be cut to below detectable levels in less than 200 ns.

Cooling BeamLight for the MOT and PGC is from the same fibre. The light frequency needs to be about 8-15

∗A good guide for aligning it can be found at http://www.eagleyard.com/fileadmin/downloads/app notes/AppNote TPA 2-0.pdf


Figure 4.3: Schematic of one of the laser sources for the experiment. Optics such as mirrors, wave-plates, optical isolators, etc are omitted for clarity. The names and purposes of the different beamsare explained in the text.

MHz below the F = 3 → F ′ = 4 resonance for the MOT stage of the experiment, and about 18-25MHz below resonance for the PGC stage. This means the laser frequency needs to be changed duringan experiment. This is accomplished using the locking system (see [31] for an explanation of howthis works). At the end of the MOT stage, the locking system changes the laser frequency by about-10 MHz within 50 µs. None of the other beams derived from the same diode are being used duringPGC, so this frequency shift does not affect any other part of the experiment.

After the fibre the light is split into six beams, aligned to make the three orthogonal pairs of beamsrequired for magneto-optical trapping. All are circularly polarised by passing them through polarisingbeam splitters followed by quarter-wave plates before entering the vacuum chamber. Two of the MOTbeams enter the vacuum chamber horizontally through the large windows, and the other four beamsenter at 45 angles relative to the ground through four of the smaller windows (see Figure 4.2). Thecombination of all the beams provides a damping force for the atoms in three dimensions.

The cooling beam fibre outputs light with a 1/e2 diameter of 7.5 mm. Just before entering thevacuum chamber the beams are expanded by a factor of four, so the 1/e2 beam diameter is 3 cm.Four of the six cooling beams enter through the smaller windows on the vacuum chamber, which are4 cm in diameter. Inevitably there is light scattered by clipping the edges of these windows. Thisscattered light results in quite a large background on the photomultiplier tube (PMT, described inSection 4.6.1). With 75 mW coming directly out of the fibre, peak light intensity at the experimentchamber is about 3 mW/cm2 for each of the six beams, so the peak total light intensity in the trappingregion is about 18 mW/cm2.

Detection BeamThe purpose of the detection beam is to measure fluorescence from atoms in the F = 3 ground

state. It is on-resonance with the cycling F = 3 → F ′ = 4 transition. We need a separate detectionbeam for two reasons: One being that the cooling beams scatter lots of light onto the PMT as men-tioned above. The detection beam therefore only passes through a large vertical window in the sideof the vacuum chamber, scattering very little light onto the PMT. The other reason is that the cooling


beams are tuned 15 or 25 MHz below resonance, and limitations in the locking system mean it cannoteasily be set to also lock such that the cooling beams are 0 MHz detuned. For the brightest signalfrom the atoms we want to detect them with on-resonant light.

The detection beam is coupled in to the same path as the MOT x-beams through a polarising-beamsplitter cube. At the cube its 1/e2 diameter is 2.1 mm. It is polarised such that it exits throughonly one side of the cube, and it is then passed through the same optics as the MOT beams to expandit by a factor of four and circularly polarise it. The result is that the detection beam forms a circularly-polarised travelling wave. With 30 mW coming out of the fibre, the peak intensity in the vacuumchamber is 1.5 mW/cm2.

Figure 4.4: Schematic of the optics used to generate the optical mask.

Optical Mask BeamThe final beam used in the experiment is used to provide the standing wave to induce atomic

interference (described in Section 3.4). This beam is resonant with the F = 3 → F ′ = 2 transition,and it is switched in the same way as the detection beam.

The transverse mode of this beam was required to be high-quality to ensure optimal interferencebetween the laser beams. See Figure 4.4 for a diagram of the optics used to generate the optical mask.At the output of the fibre, there was a Glan-Thompson polariser to provide a very clean and constantpolarisation to a Wollaston prism, which then splits the beam to provide the two counterpropagatingbeams for interference. After the Wollaston prism only metal mirrors are used so polarisation ismaintained. Polarising beamsplitters are placed at the windows of the vacuum chamber to correctfor any slight change in polarisation that might have occurred as a result of reflecting off the mirrors,and in the case of one of the beams, passing through a half-wave plate as the Wollaston prism outputsbeams with orthogonal polarisations while we want both beams of the standing wave to have the samepolarisation. The relative power of the two beams was controlled by changing the angle of the Glan-Thompson polariser before the Wollaston prism. The standing wave in this experiment is linearlyhorizontally polarised.


With the experiment in this state, the phase of the optical mask cannot be varied. However this canbe made possible by putting an electro-optic modulator in the path of one of the optical mask beams.This would be required for gravitometric measurements to be carried out, as described in Section 3.4.

4.5 Magnetic Field Coils

There were three sources of magnetic fields required for the experiment. Quadrupole coils for theMOT, compensating coils to cancel Earth’s and other stray magnetic fields, and a coil to quenchparticular atomic states that are unintentionally dark to the standing wave laser beam.

Figure 4.5: Schematic of the circuits used to switch the (a) quadrupole and (b) quenching magneticfield coils.

4.5.1 Quadrupole Coils

The coils used to generate the quadrupole magnetic field for the MOT are mounted either side of thevacuum chamber in a quasi-anti-Helmholtz configuration (the distance between the coils is slightlymore than the radius of each coil). The axes of the coils are parallel with the ground (see Figure4.2). They have an inner radius of 10.5 cm and an outer radius of 14 cm. Each coil contains 168turns of wire. They are powered by an Agilent 6543A power supply which provides 5.8 A current.They generate an estimated magnetic field gradient of 7.3 G/cm in the y−direction, and 3.2 G/cmin the x− and z−directions near the centre of the trap. PGC requires zero magnetic field, so thequadrupole coils need to be turned off sufficiently quickly so the atoms do not move significantly


from the centre of the beams. Cutting off the power supply suddenly will cause a large voltagespike across the coils due to their self-inductance. To deal with this, the circuit in Figure 4.5 (a) wasconstructed. The coil current is switched off using a IXYS IXFN100N50P N-channel MOSFET, andan ST Microelectronics transient voltage suppression diode (TVS) absorbs the energy stored in thecoils. The DG642 chip switches the MOSFET gate from the 15V supply (13.6V after the resistor,12V is needed for saturation) to ground within 500 ns, draining the intrinsic capacitor as quickly aspossible.

Assuming that the moment the MOSFET is switched off, 5.8 A of current is going through theTVS, then the peak power dissipation is 5.8 A × 300 V = 1740 W. The TVS datasheet says thatfor a current dissipation time of 150 µs, the maximum peak power dissipation is about 2500 W. Wetherefore operate the TVS within its safe operating range.

Figure 4.6: Quenching coil switching on and off, and quadrupole coils switching off. The switch-offtime of the quenching coil is about 50 µs, and 150 µs for the quadrupole coils. Measured with a LEMHEME PR200 current probe. The quadrupole coil data is very noisy during the coil switch-off, ifaveraged the decay was approximately linear.

4.5.2 Quenching Field Coil

There is an important phenomenon that can adversely affect the performance of the optical mask:the internal state of the atoms themselves. For the best signal we want all the atoms in the cloudto react in the same way to the optical mask, however for the case of the D2 F = 3 → F ′ = 3

and F = 3 → F ′ = 2 transistions there are states which cannot be excited despite the presence ofa linearly-polarised resonant laser beam (‘dark states’, see Figure 4.7). For the F = 3 → F ′ = 3

transistion this is themF = 0 sublevel (wheremF is the projection of the total angular momentum onthe quantisation axis) and for the F = 3→ F ′ = 2 transition it is the mF = ±3 magnetic sublevels.


This means many atoms in the anti-nodes of the optical mask will not be excited and decay to theF = 2 ground state, giving an artificially high number of atoms that survive the optical mask. Tomitigate this effect, a magnetic field is applied in a direction perpendicular to the light polarisation.

Figure 4.7: Relative transition strengths between different magnetic sublevels in the 85Rb D2 transi-tion for linearly-polarised light. Dark states are the mF = 0 state for the F = 3→ F ′ = 3 transition,and themF = ±3 states for the F = 3→ F ′ = 2 transition. The F = 3→ F ′ = 4 and F = 1→ F ′

transitions are not shown. Data from [25].


The effect can be explained as follows: An atom within the F = 3 ground state manifold |ψ〉 canbe described as being in a superposition of eigenstates of Fz:

|ψ〉 =∑

mF

cmF |mF 〉ze−iEmFt/~ (4.1)

Where EmF is the energy of the |mF 〉z eigenstate. In the case of no magnetic field, the |mF 〉z statesare degenerate and the exponential term can be factored out. In this case the exponential term onlycontributes an overall phase which is physically insignificant. This means if we prepare an atom ina particular |mF 〉y state (i.e. an eigenstate of Fy), there is no time dependence and it will stay in itsinitial state. However, in the presence of a magnetic field in the z−direction, the |mF 〉z states areZeeman-split, the energies are no longer degenerate and an atom initially in the |mF 〉y state becomesa time-dependent superposition of the |mF 〉z states. This mixing means an atom that initially startsin a particular magnetic sublevel will change state with time, and will eventually be in a state coupledto the standing wave laser beam.

The quenching coil in this experiment is a single 7-cm-radius coil mounted above the vacuumchamber to provide a magnetic field in the z−direction (the standing wave is linearly polarised in they−direction, see Figure 4.2 for the position of the quenching coil). The coil is powered by an Agilent6553A power supply. This power supply is rated to supply a maximum of 15 A, but was measured toovershoot and supply up to 18 A for a short time when the circuit switch is turned on. The distancefrom the centre of the coil to the centre of the MOT is 15 cm. It is switched using the circuit inFigure 4.5 (b), which functions in a similar way to the quadrupole coil switch. It is dangerous to haveunnecessarily high voltages, and the current was found to decay sufficiently quickly with a 10 V TVS(timing of the experiment is described in Section 5.4.1). This meant an IRF 3202 MOSFET couldbe used rather than the large and expensive IXYS MOSFET used for the quadrupole coils. The IRF3202 has a lower saturation voltage so the gate can be driven directly from the optocoupler. The mostimportant characteristic of the quenching coil is that it turns on as quickly as possible after PGC, sothat the experiment can be started while the atoms are still in the centre of the trap. The current wasmeasured to reach 100% of maximum value 2 ms after being turned on (see Figure 4.6). Atoms at10µK will move about 500nm in this time due to expansion, and 10µm due to gravity, so they arestill very much in the centre of the experiment region. The coil was estimated to dissipate 110 W ofpower when turned on, so the coil will burn if left on indefinitely. For this reason the coil is fusedsuch that if it is left on, the fuse will blow, keeping the coil and experiment safe (5 A household fusewire was found to work best: it was found to not blow after 50 ms but would blow after 1 s, with thepower supply set to 15 A current). When running, the experiment repetition rate was about once everythree seconds. The quenching coil is on for about 3 ms each cycle, so average power consumptionis about 110 mW. The on-axis magnetic field produced 15 cm away from the coil with 3.2 A currentwas measured to be 1.0 Gauss, so 18 A is estimated to produce a field of 5.6 G.

The effect of the quenching coil on atoms in the optical mask is characterised in Section 5.4.3.


4.5.3 Compensating Coils

As discussed in Section 3.3, in the presence of a magnetic field atoms are cooled about some non-zero velocity i.e. their velocity distribution narrows about some non-zero value. For optimum coolingwe must therefore compensate for any stray magnetic fields - such as Earth’s, or fields from nearbyinstruments - in the centre of the vacuum chamber. To do this three orthogonal pairs of large coilswere installed around the experiment. Each pair was placed such that the vacuum chamber centre ishalfway between them, and a perpendicular line drawn from the centre of the coil passes through thecentre of the vacuum chamber. The three pairs of coils produce fields in three orthogonal directions,such that the magnetic field at the centre of the vacuum chamber can be controlled at will.

Each coil pair is powered by a single Agilent 3615A power supply running in constant currentmode. The x− and z−coil pairs are estimated to produce a field of 1.7 Gauss/A at the trap centre,and the y−direction coils 1.3 Gauss/A. The optimal currents were found to be Ix = 0.19 A, Iy =

0.67 A, Iz = −0.23 A. This gives the compensating magnetic field as Bc = 0.32x + 0.87y −0.39z G. The y−axis points 17 clockwise from true north. Rotating our axes such that y → y′ nowpoints north: Bc

′ = 0.56x′ + 0.74y′ − 0.39z′ G. From [33], the magnetic field in Dunedin due tothe Earth is BE

′ = 0.08x′ + 0.18y′ + 0.59z′ G. Clearly, in the x′− and y′−directions at least, thereis a major contributor to the local magnetic field besides Earth. This is thought to be the nearby ionpump, or perhaps current-carrying wires in nearby instruments.

4.6 Making Measurements

4.6.1 Photomultiplier Tube

Quantitative measurements of atomic fluorescence were made using a Hamamatsu H9858-20 pho-tomultiplier tube (PMT). An imaging system above the vacuum chamber centre images an estimated1.0% of the solid angle of a point source at the chamber centre onto the PMT. The light is passedthrough a 780 nm bandpass filter to remove background such as room light. The gain of the PMTis controlled by a voltage on one of the pins. This gain control voltage should be above 250 mVand below 900 mV. Note that this is not the actual voltage applied to the PMT, which has an internalamplifier to generate the high voltages required for charge multiplication. The PMT is affected bymagnetic fields, so it is wrapped in µ-metal as much as possible, and the quadrupole and quenchingmagnetic field coils are turned off when taking important measurements.

To calibrate the PMT, a narrow laser beam with known power was shone on to the sensor. Thelaser entered the vacuum chamber from the bottom, exited through the top, passed through the lensand bandpass filter to fall onto the PMT. The laser power was measured before entering the chamber,and then imperfect transmission through the entry window was corrected for. This determined thecurrent produced by a known light power in the vacuum chamber, without needing to measure theabsorption of the lens and filter. Data is shown in Table 4.1. Note that there is a systematic errorin this measurement as it assumes transmission through the system is independent of the angle of


incidence; the calibration laser had a diameter of about 2 mm and so was transmitted only throughthe centre of optics in the system and is incident perpendicular to the surfaces. Light from the MOTis incident across the entire imaging lens. The lens is 70 mm in diameter and is 152 mm from thecentre of the MOT, so light at the edge of the lens is incident 13 from normal. Surfaces reflect moreat higher angles of incidence so this means that the number of atoms is underestimated.

Gain (mV) A/W

250 0.818300 2.83400 63.2500 314600 1200700 3290

Table 4.1: PMT output current to calibration laser power ratio as a function of gain voltage (laserpower in the vacuum chamber, before the chamber window, lens, and filter). Note the PMT collects1% of the solid angle of the MOT light, so to determine the output light power of atoms in the MOT,measure PMT current to determine light power before the filter and lens, and then multiply by 100.

The PMT is powered by an Agilent E3615A power supply with 3 V. Noise properties could notbe distinguished whether the PMT was powered by batteries or the power supply, so there was nodetectable mains noise coming through the power supply. The output of the PMT is connected to anSRS SR570 low-noise current preamplifier. The output of the preamplifier is observed on a TektronixTDS 3054B oscilloscope. Data is transferred to the computer the LAN.

4.6.2 Photodiode for Atom Measurement

While the MOT is building we wish to monitor the number of atoms trapped, so the experimentcan be started once a threshold has been reached. The oscilloscope can be used to monitor the PMToutput and trigger the computer to start the experiment once the trap contains the desired number ofatoms. However, data transfer from the oscilloscope takes around 200 ms, so there can be a largevariation in the initial number of atoms due to the slow sampling rate. The solution to this was tomake a dedicated photodiode and amplifier to monitor the trap size and then trigger the computerthrough a digital input at exactly the right moment.

A ThorLabs PD100 photodiode was used. It was mounted near the top of one of the large windowson the vacuum chamber, behind a telescope which collects about 0.8% of the solid angle of the lightemitted by atoms in the MOT and images it onto the diode. The circuit used as the amplifier andtrigger is shown in Figure 4.8. The photodiode does not need to be reverse-biased as high speedresponse is not needed in this situation. Biasing also increases noise from the diode †; given the veryhigh gain of the amplifier circuit it is critical to eliminate as much signal noise as possible.

†See, e.g. http://sales.hamamatsu.com/assets/applications/SSD/photodiode technical information.pdf


Figure 4.8: Circuit used to amplify photodiode signal and trigger computer. All op-amps are poweredwith ±15 V.

Referring to Figure 4.8: the transimpedence amplifier converts the photodiode current to a voltageand amplifies it, the signal is then amplified again by an inverting amplifier, a 50 Hz notch filteris needed to remove mains noise from the signal, an RC filter is used to smooth out the signal,which is then compared to a reference voltage with an analogue comparator. When the amplifiervoltage is greater than the reference voltage, it pulls a digital input on the computer low, triggeringthe experiment. The atom number can thus be controlled by the reference voltage. The referencevoltage is proportional to the number of atoms as there is very little background from the lasers (atypical Vref used was 1-3 V, while the background from the lasers is less than 50 mV).

Rather than calculate the amount of light incident on the photodiode and the gain of the amplifiercircuit, it was easier to calibrate the photodiode by comparing it to the signal from the already-calibrated PMT. See Section 5.1.

4.6.3 PIXIS CCD Camera

We used a Princeton Instruments PIXIS 1024 CCD camera to perform time-of-flight (TOF) mea-surements of the atom cloud. The basic idea is that the atom cloud is left to expand for a brief time,and then the postion distribution is measured by the camera. By doing this repeatedly for differentexpansion times, the velocity distribution of the atoms can be calculated. The cloud temperature canbe calculated from the velocity distribution, as explained in Appendix B.

The camera looked through a mirror up at the MOT from underneath the vacuum chamber. A two-lens imaging systems was placed in front of the camera to image the cloud on the CCD detector. Af = +500 mm lens and a f = +150 mm lens were used, arranged to make the magnification of thesystem equal to 0.3.

To calibrate and make sure the camera would focus on the MOT atom cloud, the distance fromthe telescope to the vacuum chamber centre was carefully measured. A mirror was placed in front ofthe telescope, and a ruler with half-millimetre marks was placed after the mirror. Care was taken toensure the ruler was exactly the same distance from the telescope as the MOT centre. The ruler was


illuminated with 780 nm light, and the telescope lenses were moved until the ruler was in sharp focus.By measuring the distance in pixels between marks on the ruler, it was found the conversion factor ofpixels to millimetres is 23.38 pixels/mm. Moving the ruler by ±1 cm changed the conversion factorby less than 0.5% (1 cm is thought to be larger than possible error in the distance measurement).

Pixel size on the detector is 13 µm × 13 µm. The measured magnification is therefore 23380 px/m× 13×10−6 m/px = 0.3039. This is very close to the calculated magnification of the imaging system.

The camera can be triggered with a digital output on the computer. Once triggered, the shutterbegins opening and will remain open for a pre-programmed exposure time. The time from the triggerpulse to the shutter being completely open is between 3 and 4 ms. For this reason the camera istriggered 5 ms before it is needed to photograph something.

The atoms fluoresce in the presence of resonant light, and the camera view is dark otherwise. Ashort laser pulse is therefore used as a ‘flash’ so that the atoms are only imaged at exactly the desiredtime, and the slow opening/closing of the camera shutter is not a problem.

4.6.4 Video Camera

For qualitative observations of trapped atoms, a CCD video camera was installed in the experimentto enable real-time viewing of the MOT on a TV. This is useful in optimising various parameters, asit was found a symmetrical cloud usually corresponded to a symmetrical velocity distribution. Thusto roughly optimise laser beam powers and compensating magnetic fields, these parameters can beadjusted until the MOT cloud looks symmetrical. Fine-tuning is done by directly measuring the effectof changing a parameter on the cloud shape and temperature with the PIXIS camera (temperaturemeasurements are described in section 5.2).

4.7 LabVIEW Control System

Automatic data collection is a necessary part of any modern atomic physics experiment. Precisetiming requirements and large parameter spaces mean manual control is impossible. For this reasona LabVIEW program was developed to control MARIE. This program can prepare a sample of coldatoms and take measurements repetitively without any user input. It is highly customisable, and canoutput arbitrary waveforms on 8 digital and 8 analogue channels simultaneously.

4.7.1 Hardware

Data is output using a National Instruments PCI-6733 High-Speed Analogue Output data acqui-sition card. It has 8 digital I/O ports and 8 analogue output ports with 16-bit resolution. It claimedto have a maximum output sample rate of 1 MHz, but it was found attempting to run much above200 kHz would result in a program crash (a possible cause was the computer not being able to loadsamples onto the card fast enough, but this was never proven correct). However, this did not prove to


be a problem as the fastest output rate needed was 100 kHz. As we will see later, temporal resolutiondown to 1 ns was required to control the optical mask, but this was accomplished using an externalarbitrary function generator triggered by the LabVIEW program.

Figure 4.9: Screenshot of MARIE’s LabVIEW interface.

4.7.2 Operation

The LabVIEW program controls the dispenser, magnetic field switches, and AOMs, and triggersthe camera, function generator, and oscilloscope. Optionally, the program can execute MATLABscripts before and after running the experiment.

The main feature of the program is the lines of outputs which can be seen in the lower half of Figure4.9. For each line, the user specifies the output value for each of the 16 output channels and the lengthof time the outputs will have that value. The time of any output and the value of an analogue outputcan be made to change each run if the user wants to scan over a range of parameters.

A block diagram of operation of the program is shown in Figure 4.10. Once a user clicks ‘Go!’,the first thing that happens is the program executes a MATLAB script, which is usually used to set


instrument parameters for that run. See the next subsection for details. Next, LabVIEW reserves thehardware for MARIE’s program to stop another program using it, and outputs the values on the firstline. The length of time these values are output for depends on mode selection by the user: In ‘TimedMOT’ mode the program will simply output the first line for the specified number of seconds, thencontinue with the experiment. In ‘Measured MOT’ mode it will output the values on the first lineuntil it receives a trigger from the photodiode circuit seen in Section 4.6.2. Once the time is up, or theMOT has reached the desired size, the program outputs the rest of the values at the times specified.After the outputs are finished, the program can execute another MATLAB script. Once it has finishedthe number of runs it was programmed to do, it will go back to an idle state. If told to run repetitively,it can be made to change the value of an analogue output each run, or change the length of an outputeach run.

All the measurements described in this thesis were made possible using this LabVIEW program.A full description of each feature in Figure 4.9 and a full list of outputs is given in Appendix A.

4.7.3 MATLAB Scripts

One very useful feature of the LabVIEW program is the ability to execute MATLAB scripts. Theprogram can execute an arbitrary MATLAB script before and after a running an experiment sequence.Recent versions of MATLAB have support for the VISA communication standard, so it can be used toprogram and download data from instruments that support this standard. VISA was used to downloadmeasurements from the oscilloscope over a TCP/IP connection, and as explained in the next section,it was also used to program the behaviour of the optical mask over a GPIB connection.

4.7.4 Controlling the Optical Mask

Controlling the optical mask required generating pulses much shorter than the sample period ofthe LabVIEW program (the authors in [29] use 800 ns pulses). For characterisation it was useful togenerate pulses from 200 ns to 100 µs. An Agilent 33250A Arbitrary Function Generator (AFG) wasused to make short pulses, it can generate pulses with picosecond resolution. It could be programmedbefore each experiment run using a MATLAB script and then triggered with an external pulse fromthe computer.

The arrangement of the computer, optical mask AOM, and AFG is shown in Figure 4.11. Whilea completely arbitrary waveform could be generated in MATLAB and then downloaded to the AFGover the GPIB connection, this was quite slow (several seconds). For some purposes, as explainedlater, it was useful to generate a short optical mask pulse and then a short time later a much longerpulse. Because of the long download time it was faster to use the AFG’s built-in settings for the firstpulse, and then use the computer to generate the longer pulse.

By measuring the optical mask light intensity with a fast photodiode, it was empirically found thatthe optical mask pulses were symmetrical only for particular pulse lengths. For example, a 540 ns


Figure 4.10: Block diagram showing conceptual operation of MARIE’s LabVIEW program.

pulse from the AFG produced a 540 ns light pulse followed by a ∼ 1 µs-long tail at about half theamplitude. A 536 ns pulse produced no such tail. It was never confirmed, but this effect is thought tobe due to an impedence mismatch between the OR gate (shown in Figure 4.11) and the optical maskAOM driver. ‘Good’ pulse lengths, i.e. pulse lengths that produce symmetrical light pulses are givenin Table 4.2.


Figure 4.11: Instruments used to control the optical mask AOM.

Pulse Length (ns)

120145265400536670809950

Table 4.2: Optical mask pulse lengths that do not produce a tail. This table is mainly included forreference for a future student. For longer pulses, the tail is short compared to the pulse length so isnot seen as significant. Due to the finite rise and fall time of the AOM, pulses 120 ns and 145 ns longdo not reach maximum amplitude.

Chapter 5

Characterising MARIE

5.1 Counting Atoms

The number of atoms in the MOT can be counted by measuring their fluorescence and calculatingthe scattering rate. The scattering rate can be interpreted as the number of photons an atom scattersfrom a laser beam every second, and for a two-level atom is given by [34]:

R =1

2

IIsat

Γ

1 + IIsat

+ 4(

∆Γ

)2 (5.1)

where I is total laser intensity, Isat is the saturation intensity for the transition, Γ is the naturallinewidth of the transition (for the D2 transition in 85Rb this is 2π× 6.1 MHz), and ∆ is the detuningof the laser from resonance.

The number of atoms is then:

N =4π

Ω

C

GER(5.2)

where Ω is the solid angle of the MOT light collected by the PMT (for this experiment, Ω/4π =

0.010), C is the PMT current, G is the responsivity of the PMT (see Table 4.1), and E is one photonenergy (E = hc/λ). So the number of atoms is the power incident on the PMT (corrected forabsorption in lens and filter), divided by the fraction of emitted light collected by the PMT, dividedby the power emitted by a single atom.

Figure 5.1 shows the PMT current signal from a cloud of cooled atoms for different initial Vref ’s.To make this figure the photodiode measured MOT light scattered from the atoms that was 8 MHzred-detuned, then the atoms were cooled (procedure described in Section 5.2) and all the lasers wereturned off. The detection laser was then turned on for 50 µs to provide the signal to the PMT. In otherwords, Figure 5.1 shows the number of atoms trapped for different Vref ’s, with the cooling lasers 8MHz detuned. The right-hand vertical axis was calculated using Eq. 5.2 and Table 4.1.

40 Characterising MARIE

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5PMT

Current(×10−8A)

Vref (V)

0

76

153

229

306

382

458

535

611

688

Number

ofAtoms(×103)

Figure 5.1: Average PMT current over 50 µs detection pulse as a function of photodiode referencevoltage Vref . Error bars show absolute variation in 5 measurements. Right-hand vertical axis showsnumber of atoms calculated corresponding to the fluorescence signal. For these data, the detectionlaser was on-resonance with the F = 3→ F = 4 transition and was a circularly polarised travellingwave. The saturation intensity for this situation is 1.67 mW/cm2 [26]. The detection laser intensitywas about 1.5 mW/cm2.

In order to measure the light intensity inside the vacuum chamber of the detection beam, the fluo-rescence of a cloud of cold atoms from a 50 µs detection pulse was measured as a function of lightpower at the output of the detection beam fibre. This is shown in Figure 5.2. To work out the lightintensity, Eq. 5.1 was fit to the data with ∆ = 0 (The detuning of the detection beam was measuredto be zero using the technique discussed in Section 5.3), and letting I = αP , where I is the lightintensity, P is the light power coming out of the fibre, and α is the only fitting parameter. α acts asa scale factor that lets us convert from something we can easily measure (the light power at the fibreoutput) to a parameter that we need to know for data analysis (the light intensity experienced by theatoms). The solid line in Figure 5.2 has α = 0.0472 cm−2.

Characterising MARIE 41

0 0.5 1 1.5 2 2.50

5

10

15

20

25Fluorescence

(a.u)

Light Intensity (mW/cm2)

0 10.596 21.192 31.788 42.384Fibre output power (mW)

Figure 5.2: Fluorescence of a cloud of cold atoms subject to a 50 µs detection beam pulse, as afunction of detection beam power/intensity. The top x−axis was measured, and the bottom x−axiswas calculated using the method discussed in the text. Error bars show peak-to-peak variation overfour measurements. The solid line is Eq. 5.1 fitted to the mean of the data with Isat = 1.67 mW/cm2.

5.2 Magneto-Optical Trap and Optical Molasses

5.2.1 Getting Cold Atoms

Obtaining cold atoms is a two-step process: first a magneto-optical trap (MOT) to collect the atomsin the centre of the trap, and then polarisation gradient cooling (PGC) to cool the atoms. The optimumparameters for PGC can be experiment-dependent, as they depend on many variables. A literaturesurvey found the optimum length of PGC was somewhere between 5 and 10 ms, and optimum laserdetuning during PGC was somewhere greater than 25 MHz below resonance [35] [36]. We found ourPGC worked well even with the cooling lasers only 18 MHz detuned from resonance, but worked bestwith about 25 MHz detuning. The procedure we found to be optimum for cooling atoms is shown inFigure 5.3.

This experiment is able to routinely produce samples of atoms at temperatures of 8-12 µK.


Figure 5.3: Optimum parameters for cooling atoms. Frequencies refer to red-detuning of coolinglaser from resonance. Intensities refer to total intensity of cooling laser inside the vacuum chamberThe time taken for the number of atoms in the MOT to reach the desired number depends on howmany atoms are re-trapped from a previous experiment, and how hot the dispenser wire is.

5.2.2 Measuring the Temperature

In this experiment, the temperature of the atom cloud was measured using the time-of-flight tech-nique. Once cooled, the atoms are left to expand freely for a small amount of time and then the cloudis imaged by flashing the MOT laser beam on for a short time, and the atom cloud can be seen usingthe calibrated PIXIS camera. The density of atoms in the cloud is proportional to fluorescence, thecamera can measure fluorescence as a function of position and hence can measure the cloud densityas a function of position. By doing this repeatedly for different times, the speed of expansion can bemeasured. Classically, the temperature of a gas can be related to the velocity distribution of its con-stituent particles. Therefore a measurement of the rate of expansion of the cloud can be interpreted asa measurement of its temperature.

Assuming a Maxwell-Boltzmann velocity distribution, the density of the cloud along in the x − yplane at time t should be proportional to (derivation in Appendix B):

N(x, y) =Av3

0

√π

r20 + t2v2

0

exp(− x2 + y2

r20 + t2v2

0

)(5.3)

where v0 =√

2kT/m, T is the temperature of the atoms, m is the mass of an atom, k is Boltzmann’sconstant, r0 is the 1/e width of the cloud at t = 0, and A =

(m

2πkT

)3/2. The origin is taken as thecentre of the cloud. Note that this equation describes a Gaussian with a well-defined width.

To obtain a measurement of N , the atoms were prepared as described in Figure 5.3. t is taken aszero at the moment the laser intensity reaches 0 at the end of PGC. Once the laser was off the camerashutter was opened, then the atom cloud was left to expand for a short time. At the desired time, thecooling lasers were briefly (1 ms) turned on to make the atoms fluoresce and provide a ‘flash’ for the


camera∗. Doing this for a range of different expansion times provided a measurement proportional toN(x, y) as a function of t (the quantum efficiency of the camera is unknown). The cloud is imagedfrom directly below, so the temperature in the z−direction cannot be measured using this technique.Example camera data is shown in Figure 5.4 (a).

Integrating N(x, y) in one direction to get a one-dimensional signal in the other provides the bestsignal-to-noise ratio. For example, in the x−direction:

N(x) =

∫ ∞

−∞N(x, y)dy

=Av3

0π√r2

0 + t2v20

exp(− x2

r20 + t2v2

0

)(5.4)

So as not to drown the signal with noise, in practice N(x, y) was only integrated over the narrowslice where the signal is non-zero, as indicated in Figure 5.4 (a). This obtained data that looked likeFigures 5.4 (b) and 5.4 (c). Gaussian functions of the form

y = c1exp(−(x− c2)2

c3

)+ c4 (5.5)

were fit to the data. Comparing Eq. 5.5 to Eq. 5.3, we can see that c1 = αAv30

√π/(r2

0 + t2v20) (where

α is an unknown constant relating the density of the atoms to the magnitude of the camera signal)and c3 = r2

0 + t2v20 . Measuring c1 and c3 for a range of different times gives two ways of measuring

the temperature of the atoms: seeing how the peak height (c1) changes with time, and seeing howthe width (c3) changes with time. Plotting c1 and c3 against time, and choosing r0 and T to best fitthe data provides a measurement of r0 and T . In practice ten photos of the atom cloud were takenfor expansion times from 5 ms to 23 ms. Each photo is of a different atom cloud. The cooling beam‘camera flash’ will affect the temperature of the cloud, so each measurement has to be of a differentexperiment realisation.

See Figure 5.5 (a) for measurements of c3, and Figure 5.5 (b) for measurements of c1.

Some comment should be made on the two different ways of measuring the temperature. Measuringc3 is more accurate - it is less susceptible to shot-to-shot cooling beam amplitude noise, and atomnumber fluctuations. When plotted against t2 as in Figure 5.5 (a), it provides easy interpretationof the initial size of the cloud (zero crossing of the line), and the temperature (proportional to thegradient). It does however rely on accurate calibration of the pixel-to-length conversion explained inSection 4.6.3†. c1 however does not depend at all on the calibration of the camera, only on how thepeak brightness changes with time. The very good agreement between these two methods, as shown

∗The cooling beams were used for this purpose, rather than the detection beams, simply because the detection beamwas not installed when the temperature measurement technique was developed. It was found to work quite well so therewas no need to change it.

†Although this was checked in two different ways, which were found to agree, as discussed in Section 4.6.3


Figure 5.4: a) Example raw data from the PIXIS camera showing a fluorescence measurement of acloud of atoms 5 ms after the trap has been turned off. Dashed white lines show ‘slices’ integratedto get projections along b) x−axis and c) y−axis. Red lines are Gaussian fits to the data. Axes showorientation of photo relative to experiment axes.

in Figure 5.5, is a strong indication the temperature is being measured correctly.

5.3 Measuring the Laser Frequency

One thing cold atoms are very useful for is as an absolute frequency reference. There is verylittle Doppler broadening of the transition spectrum, so the frequency can be measured to an accuracylimited by power broadening, the natural linewidth of the transition, and the linewidth of the light usedfor the measurement. For an atom at 10 µK, the Doppler broadening should be about 200 kHz, whichis much less than the natural linewidth of the transition (Γ = 6.1 MHz for the 85Rb D2 transition).

The cooled atoms can be used to measure the frequency of the detection laser. This was practicallyaccomplished by subjecting a cloud of cold atoms to a 50 µs pulse from the detection laser andmeasuring the fluorescence. This was done for a range of detection laser AOM frequencies with thelaser locked, so fluorescence as a function of AOM frequency could be measured. Data is shown inFigure 5.6. A clear peak can be seen at 140 MHz. The detection beam AOM deflects light into the-1st order, so Figure 5.6 tells us our laser is locked 140 MHz above the F = 3 → F ′ = 4 transition.The centre frequency of the AOM used for the measurements was 150 MHz, and light power wasrecalibrated after each adjustment of the AOM frequency, so that the same light power was used forall measurements.


0 1 2 3 4 5 6

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2x 10

−6

t2(s2)

r2 0+t2v2 0(m

2)

Tx = 7.6µK

Ty = 7.7µK

r0x = 0.25mm

r0y = 0.25mm

(a)

0 0.005 0.01 0.015 0.02 0.0252

4

6

8

10

12

14

16

Tx = 7.7µK

Ty = 8.1µK

r0x = 0.23mm

r0y = 0.23mm

(b)

Time (s)

αAv3 0

√π/√r2 0

+t2v2 0

Figure 5.5: Five measurements of (a) the width of the atom cloud and (b) the ‘peak height’ as givenin Eq. 5.3, as a function of the free-expansion time, with measurements from five separate days.Errorbars show peak-to-peak variation in measurements. Solid lines are best fits to the mean of allthe measurements. Blue data are measurements in the x−direction and red data are measurements inthe y−direction. Temperatures and radii given correspond to the solid lines. The size of the variationin different days’ measurements corresponds to a temperature variation of about ±1µK.


110 120 130 140 150 160 1700

5

10

15

20

25

AOM Frequency (MHz)

Fluorescence

Signal(a.u.)

Figure 5.6: Fluorescence signal vs. detection laser AOM frequency. The solid blue line is Eq. 5.1fitted to the mean of the data with I = 1.5 mWcm−2, it has a FWHM of 8.3 MHz. The peak isat 140.0 MHz. The red dashed line is a Lorentzian fit to the mean of the data where the width waschosen by a least-squares algorithm, and has a FWHM of 11.6 MHz. Error bars show peak-to-peakvariation in 10 measurements.

The blue solid line in Figure 5.6 is Eq. 5.1 with I = 1.5 mW/cm2, and has a FWHM of 8.3 MHz,the only fitting performed was the amplitude and position of the peak. The red line is a Lorentzianwhere the width was chosen by fitting to the data, and has a FWHM of 11.6 MHz. The broader widthof the measured data can be explained by the finite laser linewidth: this measurement was effectivelya convolution of the laser and atomic spectral widths. The laser has a linewidth measured to be 3MHz, so the red line has a width almost exactly what one would expect. The width of the peakis also consistent with the atoms being the temperature measured using the time-of-flight techniquedescribed earlier.

Once the laser frequency is known, the cooling laser and optical mask laser frequencies can bechosen appropriately: The cooling laser should be about 15 MHz red-detuned from the F = 3 →F ′ = 4 transition, and the optical mask should be resonant with the F = 3 → F ′ = 2 or F ′ = 3

transition.


5.4 Optical Mask

As discussed in Section 3.4, the optical mask should act as a comb of transmission slits. Atoms notat the nodes need to be pumped quickly into the F = 2 ground state, and atoms near the nodes shouldstay in the F = 3 ground state. This section describes how the optical mask was characterised andconfirmed to be functioning correctly.

To characterise the optical mask, a mask was applied to a cloud of cold atoms in the F = 3 groundstate. For all cases the optical mask light was resonant with the F = 3 → F ′ = 2 transition. Thefluorescence was then measured after the mask with a pulse from the detection beam. This effectivelyperforms a measurement of how many atoms were lost to the F = 2 ground state as a result of themask (i.e. measures the ‘absorption’ of the mask). This absorption was measured for a range ofdifferent mask parameters to ensure it was working as expected.

5.4.1 Noise

There are two large sources of noise in characterising the optical mask:1) The initial number of atoms can fluctuate. While the photodiode trigger was made so that the

initial number of atoms was always the same, it was found there are still fluctuations of about ±10%.The effect of these fluctuations could be minimised by measuring the number of atoms during eachrun, and comparing the measured signal to the initial number of atoms.

2) There is background noise on the PMT from scattered detection beam light and fluorescencefrom stray atoms (i.e. atoms that fluoresce due to the detection beam but were not in the opticalmask). This background could be minimised by depumping all the atoms with a long optical maskpulse, and then measuring fluorescence. This measured background can then be subtracted from othermeasurements.

To minimise noise, a particular detection pulse sequence was developed. Figure 5.7 shows raw datafrom a typical run of the experiment. A typical run of the experiment went as follows (letters indicatecorresponding events in the figure):

• Trap and cool the atoms as in Section 5.2.

• Apply an optical mask pulse (100 ns – 100 µs) (b).

• Measure fluorescence from a detection pulse (50 µs) (d).

• Repump all the atoms to the F = 3 ground state with a 50 µs pulse from the repump laser (50µs) (e).

• Measure fluorescence from a detection pulse. This is effectively determining the total numberof atoms present (50 µs) (f).

• Subject the atoms to a long (150 µs) optical mask pulse. This depumps them all to the F = 2

ground state so they don’t fluoresce due to the detection laser. (g)


• Measure light from a detection pulse. Since all the atoms in the region of the optical mask havebeen depumped, this measures background due to stray laser light and fluorescence from strayatoms (50 µs) (h).

• The cooling laser and quadrupole magnetic field is turned on again to re-trap atoms (i).

0 0.2 0.4 0.6 0.8 1

x 10−3

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Time (s)

OscilloscopeVoltage(V

)

(a)

(b)(c)

(d)

(e)

(f)

(g)(h)

(i)

Figure 5.7: Typical PMT signal from oscilloscope. The letters indicate different events: (a) Repumplaser turned off. (b) Short optical mask pulse. (c) Quenching coil turned off. (d) Detection pulse.(e) Repump pulse. The large signal is due to repump light scattering onto the PMT, not from atomicfluorescence. (f) Detection pulse. (g) Long optical mask pulse. (h) Detection pulse. (i) Cooling laserand quadrupole magnetic field turned on.

Notice that there are 50 µs gaps between pulses in Figure 5.7. The PMT amplifier has a low-passfiltering effect. The cut-off frequency of the low-pass filter can be increased, but at the expense ofnoise. Fluorescence was measured by finding the area of each peak, and then dividing by the lengthof the laser pulse used to generate that peak. This finds the average oscilloscope voltage over thelength of the pulse, which can then be converted to a light power. The 50 µs gaps enable the signal togo back to zero between laser pulses, so the area of one pulse is not contaminated by another.

The parameter of interest is how many atoms are transmitted through the optical mask. Havingmeasured the fluorescence signal at (d) (call it s), the total fluorescence from the atoms at (f) (call it


n), and the background at (h) (call it b), the proportion of atoms p transmitted through the mask isthen given by:

p =s− bn− b (5.6)

As confirmation of this technique, p was measured for different optical mask lengths. It was foundto average to ∼ 0.99 for no optical mask, and ∼ 0.01 for long optical mask pulses. See Section 5.4.4for quantitative data. Ideally, it would average to exactly 1 for no optical mask pulse, and exactly0 for long pulses. This non-ideality did not affect conclusions made in this thesis, but could haveimplications for use of the optical mask in an interferometer. Further improvement is needed with thedetection scheme.

5.4.2 Saturation with the Optical Mask

To minimise the effect of variations in optical mask light intensity, it was kept well above saturation.Figure 5.8 shows the proportion of atoms remaining in the F = 3 ground state after an optical maskpulse, as a function of light power coming out of the optical mask fibre. This is proportional to theoptical mask light intensity. There is very little decrease in transmission above around 20 mW. Forall optical mask measurements light power was kept around 30 mW.

5.4.3 Quenching Magnetic Field

As mentioned in Section 4.5.2, there are atomic states dark to the linearly-polarised optical masklaser. However these states can be quenched using a magnetic field. In order to ensure the quenchingfield is working, a cloud of cold atoms was subjected to a optical mask pulse for different currentsthrough the quenching magnetic field coil. The quenching coil was turned on just after the polarisationgradient cooling step of the experiment, and was quickly turned off after the optical mask but beforethe detection step.

The data in Figure 5.9 shows the proportion of atoms remaining in the F = 3 ground state after a536 ns optical mask pulse, for different quenching coil currents. In this case one of the beams of theoptical mask were blocked, so that it formed a travelling wave instead of a standing wave. This wasto eliminate any artifacts that might have been due to interference between the laser beams. A clearincrease in depumping efficiency can be seen as the current increases.

5.4.4 Interference in the Optical Mask

The ultimate goal of this project was to make a cloud of cold atoms with a periodic density pattern.This section presents strong evidence that this goal was successfully achieved, and thus the experimentis very close to forming a functioning interferometer.

Figure 5.10 shows the proportion of atoms remaining in the F = 3 ground state after an opticalmask pulse, for different pulse lengths. The chance of an atom surviving the optical mask clearly


0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Proportionofatomsremaining

Output power of optical mask fibre (mW)

Figure 5.8: Proportion of atoms left in F = 3 ground state after a 536 ns optical mask pulse, fordifferent optical mask light powers (power is proportional to light intensity). Error bars show peak-to-peak variation over four measurements. Crosses show the mean of the data.

decreases as the duration of the pulse increases. To check for interference between the optical maskbeams this measurement was performed for two different cases: the red data in Figure 5.10 survivalfor when all of the light power was directed along just one of the optical mask beams, and the bluedata shows survival for when there was equal power in both optical mask beams. Total light powerwas the same in both cases.

With all the power in one beam, the light forms a travelling wave with constant intensity. All atomsin the cloud will experience the same light field. However, with equal light power in both beams thelight will interfere, causing a spatially-varying light intensity along the beams. An atom’s chance ofbeing depumped to the F = 2 ground state will therefore depend on its position in the beam. Atomsin the nodes should have a very small chance of being depumped, so survival of atoms should increasewith equal light power in both optical mask beams. This is exactly what is shown in Figure 5.10; forthe same length pulse, more atoms survive the standing wave than the travelling wave, so this must bedue to atoms in the nodes of the standing wave not being depumped, and forming a periodic densitypattern of atoms in the F = 3 ground state. This effect is clearly visible above the noise in themeasurements (the errorbars show absolute variation in 10 measurements). Survival of atoms in the


0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Current (A)

Proportionofatomsafter

536nsTW

Figure 5.9: Proportion of atoms left in F = 3 ground state after a 536 ns travelling-wave opticalmask pulse, for different quenching coil currents. Errorbars show peak-to-peak variation over 10measurements. Crosses show the mean of the data.

standing wave is up to three times the survival of atoms in the travelling wave.

This is the main result of my project, and I want to emphasise that it shows the optical mask isworking correctly. The only difference in the experiments used to obtain the two sets of data inFigure 5.10 is the composition of the optical mask: the red line shows the survival of atoms after atravelling wave, and the blue line show atoms after a standing wave. The increased survival shownby the blue line must be due to atoms at the nodes of the standing wave not being depumped to theF = 2 ground state, and so the density of atoms in the F = 3 ground state must have the diffractiongrating-like pattern explained in Section 3.4.

Note: As vindication of this conclusion, since I finished working on the experiment another studenthas used it to successfully measure atomic interference. The experiment consisted of two standingwave pulses, followed by a third pulse scanned in time after the second pulse. The survival of atomsafter this third pulse shows clear oscillatory behaviour as a function of the delay between the secondand third standing wave pulses. This shows that the optical mask is definitely working as expected.


0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Length of Pulse (µs)

Proportionofatomsremainingafter

pulse

(a)

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Length of Pulse (µs)

Proportionofatomsremainingafter

pulse

(b)

Figure 5.10: Proportion of atoms remaining in the F = 3 ground state after an optical mask pulse.Red data shows survival for a one-beam optical mask (a travelling wave), blue data a two-beamoptical mask (a standing wave). Error bars show peak-to-peak variation in ten measurements. (b)simply shows a magnified region of (a).

Chapter 6

Summary and Future Work

6.1 Conclusion

This thesis presented a successful attempt to construct and characterise an optical mask for atomicinterferometry experiments. The work presented here should form a foundation for many interest-ing interferometry experiments, hopefully culminating in a precision measurement of local g, theacceleration due to gravity.

The introduction gave a brief history of major developments in the related fields of atom interferom-etry and laser cooling. This provided a gentle transition into more technical aspects of the backgroundphysics needed in this project: basic theory of laser cooling and trapping. Specifically, I described thefundamentals of Doppler cooling, magneto-optical traps, and polarisation gradient cooling. Alongthe way practical aspects of implementing these techniques were discussed. The last section of thebackground chapter explained how to make an optical mask, and most importantly, provided somemotivation for doing so.

Chapter 4 described the anatomy of the experiment in detail. As more detail can already be foundelsewhere, I only briefly described the vacuum chamber and pumps used, how the rubidium dispenserdelivers atoms to the experiment, and the laser sources/tapered amplifier. In more detail I describedthe magnetic field coils, the quadrupole coils which were necessary for the magneto-optical trap,and the quenching coil which was critical to making the optical mask work properly. Both of thesecoils have high-performing electronic circuits to switch them on and off quickly and reliably. Thecompensating coils were successfully exploited to obtain samples of very cold atoms. Chapter 4 alsodescribed the four instruments for observing trapped atoms: a photomultiplier tube for quantifyingthe number of atoms in the experiment, a PIXIS CCD camera that was successfully used to accuratetemperature measurements, a highly linear photodiode trigger to ensure the initial atom number ineach experiment run was repeatable, and a video camera for qualitative observations of the atoms.The former three are used for quantitative measurements and their characterisation was described.

54 Summary and Future Work

The LabVIEW program developed for this experiment enabled a range of different measurements tobe performed automatically. It can trap and cool atoms and then run a measurement, and it can do thisrepetitively for a range of different parameters with no user input. The program was instrumental inmeasuring the temperature of cooled atoms, characterising the photodiode trigger module, accuratelydetermining the laser frequency, and characterising the optical mask. The versatility of this programshould ensure it remains useful well into the future.

The experiment can reliably produce samples of cold atoms with temperatures from 8-12 µK. Thistemperature is consistent with literature values, and was measured using two independent parameters,suggesting that the temperature is indeed being measured correctly.

The main tangible result of this project is the optical mask. There is strong evidence presented herethat the optical mask is writing density patterns on clouds of cooled atoms, and so it should be usefulfor interference experiments.

6.2 Future Work

As mentioned previously, the experiment as it was presented here does not constitute a fully-functioning interferometer as in [29]. To be able to reproduce their results the experiment needs away to control the phase of the optical mask. This could be done by installing an electro-optic mod-ulator in one of the arms of the optical mask. For gravitational measurements the optical mask willneed to propagate vertically - not horizontally as it does currently.

There is a small non-ideality in the detection scheme that might need correcting before interferom-etry experiments can be performed, as discussed in Section 5.4.1.

Appendix A

Using MARIE’s LabVIEW program

This section is intended as reference for a future student using MARIE’s LabVIEW program.

A.1 Description of MARIE’s LabVIEW program features

Every label appearing in the graphical user interface is listed below with an explanation of itspurpose and how to use it. Text in bold is the label/name of a feature in the main program window.

Counter ParametersCounter(s) - This tells LabVIEW what clock source to use for timing, you probably don’t need tochange this.Rate - This is the sampling rate of the digital and analogue outputs. The data sheet of the card saysit should be able to go up to 1 MHz, but running above 200 kHz seems to result in an error. I alwaysused it at 100 kHz.Analogue Channel Parameters Features in this box tell LabVIEW the analogue outputs to use andthe maximum and minimum voltages that are allowed to be outputted. ±10 V is the physical limitfor this card. You shouldn’t need to ever change the physical channels option (this card only has 8analogue outputs).Digital Channel Parameters Same as the analogue channel parameters except I used the last digitalchannel as an input. This card only has 8 digital channels so it shouldn’t be necessary to change theseoptions either. MOT Selecta tells the program whether to wait on the first line of outputs (the MOTstage)for the trigger from the photodiode (Measured MOT), or just wait for an alotted amount oftime (Timed MOT). The time to wait is the MOT time (s).Execute Post-Function? tells LabVIEW whether or not to run a MATLAB function file as soon asthe current experiment run has finished. This is normally the data collection step, i.e. LabVIEW usesa MATLAB function to transfer data from the oscilloscope. The name and location of the functionis given in the Post-Function and Folder boxes respectively. The text in Datafile name is passed asan argument to the function, and gives the name of the output data file. Execute Pre-Function? isbasically the same thing, except it runs immediately before the experiment. I normally used this tochange settings on the arbitrary function generator.

56 Using MARIE’s LabVIEW program

Abort? stops program execution. One of the bugs of this program is that it does not do so imme-diately. As far as I can tell it should stop straight away, but it can take several seconds before theprogram will stop running. Check the outputs are what you think they are once it does stop.Set Output to Zero When Done? Once execution is completed, the program will normally outputwhatever the last values were. Toggle this switch to instead only output zeros after the last step.Run Experiment starts program execution.# Runs tells the program how many times to run. Loop number tells you what runs the experimentis up to (note counting starts from zero).Experiment Running! tells you whether the program is currently running or not, it lights up when itis running.The Time column controls how long each row is output for (except the first row, which is controlledby the MOT Selecta as previously discussed. The value in the boxes is number of samples that rowis ouput for, so the duration of the output is the number of samples divided by the sampling rate.dt tells the program how much to change the corresponding row in the Time column each run. Forexample, if you put ‘10’ in a box in dt, and set the program to run 5 times, it will add 10 to the valuein the Time column each run. This is so you can scan across a range of parameters without needingto change the time manually.Ramp to this value? If two adjacent analogue output rows are different values, normally the programwill just jump the output to the new value at the appropriate time. However sometimes it is useful tolinearly ramp to the new value, which is the purpose of this feature. Select a button for the programto ramp from the previous value to the value in the row corresponding to the button pressed. This willtake place in as many samples as indicated in the same row in the Time column.Analogue Outputs sets the voltage of each of the eight outputs at the corresponding Time in thesame row. Digital Outputs is the same thing, except ‘on’ sets the output to 5 V and ‘off’ sets it to 0V.

Some miscellaneous information that might be useful:

• Make sure there are no other LabVIEW programs running simultaneously, if they use the samehardware there’ll be resource conflicts and neither program will work correctly.

• If the program crashes during an experiment run, the output card will just continue to output themost recent value, this can cause problems like having the dispenser turned on continuously, bewary of this!

• A small bug in the program is that all the output vectors (e.g. the time vector, analogue outputsvectors, etc) need to be the same length.

A.2 List of Outputs

This section lists which instruments were connected to which outputs of the computer, at the timeI finished working on the experiment.

Using MARIE’s LabVIEW program 57

Analogue Outputs

• AO 0 Controls current in the rubidium dispenser

• AO 1 Sets the frequency of the laser through the locking system (changes the voltage on theFM input of the locking AOM).

• AO 2 Sets the amplitude of the detection laser through the AM input of the AOM, not normallyused.

• AO 3 Controls how big the MOT gets before the program is triggered (Vref in Figure 4.8)

• AO 4 Amplitude of the MOT beams, used in PGC.

• AO 5 Frequency of the MOT beams AOM. This AOM was found to ‘leak’ when switched off,so the light is deliberately misaligned from the fibre input by changing the AOM frequencywhen it is not needed.

• AO 6 Turns the quenching coil on and off, used essentially as a digital switch.

• AO 7 Turns the standing wave on and off, also used as a digital switch.

Digital Outputs

• DO 0 Detection AOM switch

• DO 1 Quadrupole coils switch

• DO 2 PIXIS camera trigger

• DO 3 Repump AOM switch (note this is inverted relative to the rest of the AOM switches: 5 Vis off, 0 V is on)

• DO 4 Arbitrary function generator trigger

• DO 5 MOT AOM switch

• DO 6 Oscilloscope trigger

• DO 7 Is actually a digital input. Program trigger from photodiode circuit.

58 Using MARIE’s LabVIEW program

Appendix B

Derivation of Time-of-flight Equation

Assuming an isotropic Gaussian atom cloud with a Maxwell-Boltzmann velocity distribution, wecan write the initial cloud distribution as the product of its velocity distribution and initial spatialdistribution:

N(r,v)d3νd3r = A1

r30π

3/2exp

(−ν2x + ν2

y + ν2z

ν20

)d3ν

×exp

(−r2x + r2

y + r2z

r20

)d3r (B.1)

where m is atomic mass, νx, νy, and νz are speeds, and rx, ry, and rz are distances along the x, y,

and z directions respectively. A =(

m2πkT

)3/2, ν0 =√

2kTm is the most probable velocity, and r0 is

the 1/e width of the cloud at t = 0. T is temperature.

We want to find what the cloud looks like at time t. Our camera is below the cloud, looking upalong the z-axis. So we can image the cloud projected on to the x−y plane. First we need to transformfrom velocity to spatial xyz-coordinates. Using Newtonian mechanics to relate the two coordinatesystems:

x = rx + νxt (B.2)

y = ry + νyt (B.3)

z = rz + νzt−1

2gt2 (B.4)

i.e. an atom’s position at time t is given by its initial position plus its velocity times t. Rearranging:

νx = (x− rx)/t (B.5)

νy = (y − ry)/t (B.6)

νz = (z − rz +1

2gt2)/t (B.7)

60 Derivation of Time-of-flight Equation

Therefore the differential d3ν can be transformed to dxdydz as

d3ν = dνxdνydνz (B.8)

=dxdydz

t3(B.9)

We can now rewrite B.1 as:

N(r, x, y, z; t)d3rdxdydz =1

t3A

1

r30π

3/2

×exp(−(x− rx)2 + (y − ry)2 + (z − rz)2

t2ν20

)dxdydz

×exp

(−r2x + r2

y + r2z

r20

)d3r (B.10)

Integrating B.10 over r to get an expression for the density of the cloud as a function of (x, y, z; t)

(working left as exercise for reader. Completing the square is useful here, trust me):

N(x, y, z; t) =

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞N(r, x, y, z; t)d3r

=Av3

0

(r20 + t2v2

0)3/2exp

(−x

2 + y2 +(z + 1

2gt2)2

r20 + t2v2

0

)(B.11)

Projecting onto the x− y plane by integrating over z:

N(x, y; t) =

∫ ∞

−∞N(x, y, z; t)dz

=Av3

0

√π

r20 + t2v2

0

exp(− x2 + y2

r20 + t2v2

0

)(B.12)

It is easiest in MATLAB to fit to one-dimensional data, so the cloud is projected onto either thex− or y−axis then a Gaussian is fitted to the measured data to get the temperature in the respectivedirection. The equation to fit the data to is:

N(x; t) =

∫ ∞

−∞N(x, y; t)dy

=Av3

0π√r2

0 + t2v20

exp(− x2

r20 + t2v2

0

)(B.13)

One can simply swap y for x to change direction. A series of photos taken of the atom cloud atdifferent times can be used to find r0 and T .

The above derivation is adapted from a similar calculation in [37]. In this paper the authors calcu-late the expected fluorescence from a cloud of cold atoms falling through a light sheet.

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An Optical Mask for Atomic Interferometry Experiments

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