Aarhus University

Department of Physics and Astronomy

Master’s thesis

Building a high resolutionimaging system for detecting,addressing and manipulating

single, optically trapped atoms

Author:Nicolai A. Haugaard

20114020

Supervisor:Jacob F. Sherson

April 10, 2017

1 Abstract

1.1. English AbstractIn this master’s project a high resolution imaging system has been built and installedin the HiRes experiment at the Aarhus University. The imaging system providessingle site resolution of trapped atoms in optical lattices as well as a flexible methodfor addressing and manipulating single atoms and generate arbitrary conservativeoptical potential landscapes.Four of the so-called Digital Micromirror Devices (DMD) have been implementedin the imaging system for spatial light modulation. Two of them are placed in theFourier plane of the imaging system. A new, fast and robust way of interfacingwith the Fourier DMDs has been developed, which reduces the duration of theso-called phasemapping-procedure by a factor of 48.At the time of writing, the final alignment of the high resolution imaging systemwith respect to the HiRes experiment is being carried out.

1.2. ResuméI dette specialeprojekt beskrives konstruktionen og installationen af et billedsystemmed høj opløsningsevne i HiRes eksperimentet ved Aarhus Univeristet. Billedsys-temet muliggør både billedlig opløsning af enkelte atomer fanget i optiske gitre,adessering samt manipulation af enkelte atomer og generering af arbitrære konser-vative optiske potentiallandskaber.Fire såkaldte DMDer (Fra engelsk: Digital Micromirror Devices) er implementereti billedsystemet med henblik på rumlig modulation af lys. To af disse er placeret idet såkaldte Fourier-plan af billedsystemet. En ny, hurtig og robust metode til at

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interagere med Fourier-DMDerne er blevet udviklet, som reducerer varigheden afden såkaldte fasekort-procedure med en faktor 48.I skrivende stund udføres den sidste finjustering af højopløsnings-billedsystemetsposition relativt til HiRes experimentet.

1.3. AacknowledgementsFirst of all, I would like to thank my supervisor Jacob Sherson for letting me bedo this project in the HiRes group. A special thanks also go to my friend RomainMüller for acting as my day-to-day supervisor, for always being ready to answertricky questions and for always being up for an interesting discussions related tothis project or other relevant subjects. Finally, I would like to thank the rest ofthe "lab-wizards": Robert Heck, Ottó Elíasson, Jens Schultz and Aske Thorsen forsupport and for always being helpful throughout this project.

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Contents

1. Abstract 11.1. English Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Aacknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Introduction 6

3. The High Resolution Experiment 93.1. Overview of the HiRes Experiment . . . . . . . . . . . . . . . . . . 10

3.1.1. 2D MOT and 3D MOT . . . . . . . . . . . . . . . . . . . . . 103.1.2. Magnetic traps and microwave evaporation . . . . . . . . . . 133.1.3. Dipole traps and evaporation cooling . . . . . . . . . . . . . 133.1.4. Transport to the science chamber . . . . . . . . . . . . . . . 153.1.5. The science chamber . . . . . . . . . . . . . . . . . . . . . . 15

3.2. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3. Single spin addressing in the science chamber . . . . . . . . . . . . 18

4. The high resolution imaging system 194.1. Imaging theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1. Linear imaging systems . . . . . . . . . . . . . . . . . . . . . 194.1.2. Resolution of an imaging system . . . . . . . . . . . . . . . . 204.1.3. Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2. Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.1. The high resolution objective . . . . . . . . . . . . . . . . . 274.2.2. Viewport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.3. Coating of the high resolution viewport . . . . . . . . . . . . 28

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4.3. Preliminary test of the objective . . . . . . . . . . . . . . . . . . . . 30

5. High resolution board and spatial light modulation 385.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2. Spatial Light Modulators . . . . . . . . . . . . . . . . . . . . . . . . 385.3. Digital Micromirror Devices . . . . . . . . . . . . . . . . . . . . . . 395.4. DMDs in the imaging plane . . . . . . . . . . . . . . . . . . . . . . 425.5. DMDs in the Fourier plane . . . . . . . . . . . . . . . . . . . . . . . 43

5.5.1. Beamshaping with DMDs . . . . . . . . . . . . . . . . . . . 445.5.2. Phasemaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5.3. Generation of phasemaps . . . . . . . . . . . . . . . . . . . . 495.5.4. Unwrapping and interpolating the phasemap . . . . . . . . . 505.5.5. Corrected phasemaps . . . . . . . . . . . . . . . . . . . . . . 515.5.6. Fast phasemaps . . . . . . . . . . . . . . . . . . . . . . . . . 535.5.7. Raspberry Pi Hack . . . . . . . . . . . . . . . . . . . . . . . 555.5.8. C++ software for generating patches and extracting phases . 595.5.9. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.5.10. Amplitude map . . . . . . . . . . . . . . . . . . . . . . . . . 645.5.11. Holograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6. Alignment of the high resolution board 706.1. Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2. Cage systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.3. Alignment of the DMDs . . . . . . . . . . . . . . . . . . . . . . . . 756.4. Laser sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5. Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.6. Molasses arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.7. Pilot Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7. Initial tests and installation of the HiRes Board 857.1. Initial testing of the HiRes board . . . . . . . . . . . . . . . . . . . 857.2. Installation of the board . . . . . . . . . . . . . . . . . . . . . . . . 907.3. Pilot beam and removing of the HiRes board . . . . . . . . . . . . . 98

8. Action with atoms 998.1. Fluorescence signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2. Decay channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.3. Detecting fluorescence signals . . . . . . . . . . . . . . . . . . . . . 1038.4. Two-body light induced collisions . . . . . . . . . . . . . . . . . . . 104

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8.5. Status of the HiRes experiment . . . . . . . . . . . . . . . . . . . . 107

9. Conclusion and outlook 111

A. Raspberry Pi - Software hack 114

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2 Introduction

Nearly 100 years ago, Satyendra Nath Bose and Albert Einstein described theBose-Einstein distribution for identical bosons for the first time. In contrast to thebehaviour of fermions described by the Fermi-Dirac statistics, the Bose-Einsteindistribution predicts that an unrestricted amount of indistinguishable bosons couldpopulate the same state at low temperatures thus forming a so-called Bose-EinsteinCondensate (BEC).For many years, this remained as a theoretical postulate, since temperatureslow enough for crossing the phase transition to a BEC could not be achievedexperimentally.First in 1995 experimental realization of BECs was achieved [9] [18]. This openeda new door to the quantum world and since then tremendous progress has beenmade within the research field of ultra cold atoms. Not only have multiple researchgroups produced BECs, but they have also started to probe and manipulate theseultracold gases.Besides investigating the basic quantum nature of BECs, it has also been shownthat BECs can be loaded into optical lattices [25] with unitary filling[44]. Hereatoms can be probed while a phase transition into a so called Mott insulator takesplace[10] and be addressed individually [50].Ultra cold atoms loaded in optical lattices are still of great interest since they havebeen proposed and used as a platform for different applications and experimentsthat branch into different areas of physics. It has been suggested that it constitutesa promising candidate for a quantum simulator that for instance can be used forsimulating condensed-matter systems and electron fluids in magnetic fields [13].It has also been proposed that it constitutes a candidate for a scalable quantumcomputer [51]. Both a quantum simulator and a quantum computer are of greatinterest, because they are expected to be able to perform complex simulations

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and calculations, that otherwise would require a huge amount of computationalpower[38].Among the research groups exploring the exciting quantum world via Bose-Einsteincondensates in optical lattices is the high resolution group (HiRes) at AarhusUniversity. The HiRes experiment is a relatively new experiment that has beenable to produce BECs since 2013. This master’s project is an experimental projectthat has been carried out in the HiRes group. In this project, the high resolutionimaging system of the HiRes experiment has been built. The imaging systemwill provide single site resolution as well as a flexible method for addressing andmanipulation of single and multiple atoms trapped in optical lattices and generateconservative optical potential landscapes. Four Digital Micromirror Devices (DMD)have been implemented in the system, since they have proven to be a versatile toolfor spatial light modulation in atomic physics [46] [47] [52]. Two of them are placedin the Fourier plane. For using the Fourier DMDs, the relative phases between themicromirrors have to be measured. In this project a new and fast technique hasbeen developed for mapping these phases, which has proven to be 48 times fasterthat previously used methods.The content of this thesis is structured in the following way:

• Chapter 3: An introduction to the HiRes experiment is given, and the dif-ferent steps required for cooling atoms and crossing the BEC phase transitionare explained. The Bose-Hubbard model, which describes the dynamics ofcold atoms in optical lattices, and the scheme for addressing single atoms arealso presented.

• Chapter 4: This chapter contains relevant theory, design criteria of thehigh resolution system and preliminary tests of the so-called high resolutionviewport and high resolution objective.

• Chapter 5: This chapter explains the basics of the Digital MicromirrorDevices and how they are used in the high resolution system. The fasttechnique for mapping the phases is presented in great detail.

• Chapter 6: A detailed description is given of how the optics and DMDs asso-ciated with the high resolution system were aligned on the optical breadboardknown as the HiRes board.

• Chapter 7: Initial testes of the HiRes board are presented. A detaileddescription is given of how the HiRes board, and thus the high resolutionimaging system, was installed in the HiRes experiment.

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• Chapter 8: This chapter introduces physical concepts related to the furtheruse of the HiRes experiment. The concepts include fluorescence signals andtwo-body light induced collisions. Finally the current status of the experimentis described.

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3 The High Resolution Experiment

A gas of particles at room temperature can be thought of as hard spheres bouncingaround off each other. The temperature is related to the average kinetic energy ofthe atoms, and lowering the energy will reduce the temperature. The wave-particleduality, which was formulated roughly one hundred years ago by Louis de Broglie,states that matter has wave-like properties. The wavelength of these matter-waves,also known as the de Broglie-wavelength λDB, is given by the following equation:

λDB =√

2π~2

mkbT(3.1)

where m is the particle mass, kB is Boltzmanns constant and T is the tempera-ture. As the atoms are cooled, their wave-like nature becomes apparent due to anincreasing de Broglie-wavelength. First the atoms can be described as small indi-vidual wave packets, but as the wavelength becomes comparable to the interatomicdistance, these matter waves start to spatially overlap. If the particles in questionare bosons, they will eventually populate the ground state at a critical temperatureand below. This phenomenon can be described by a macroscopic wavefunction andis called a Bose-Einstein condensate (BEC). BECs are a prerequisite to investigatequantum phenomena with ultra cold atoms. The structure of this chapter is asfollows: Section 3.1 gives an overview of the experiment and the basic theoreticalconcepts in the cooling scheme. Future experiments will rely on optical latticeswhich are described in section 3.2. Finally, single spin addressing, which alsoconstitutes a key feature of future experiments, is introduced in section 3.3.

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Figure 3.1.: Shows the HiRes setup. Figure borrowed from [37].

3.1. Overview of the HiRes ExperimentA brief overview of how the HiRes experiment crosses the BEC phase transition for87Rb atoms will be given in the following sections. A standard technique for coolinga gas below the so-called Doppler temperature is the magneto-optical trap (MOT).The atoms are first trapped and cooled in two and then three dimensions in the2D- and 3D-MOT chambers. By moving the anti-Helmholtz coil pairs used in the3D-MOT, the atoms are transferred to the cube chamber where forced microwaveevaporative cooling is performed to increase the phasespace density. The atomsare finally loaded into an optical dipole trap which is transported into the finalchamber (also denoted as the science chamber), where a crossed dipole trap is made.Here the phase transition to a BEC is crossed by lowering the power of the trapand thus invoking an additional phase of evaporative cooling. A schematic drawingof the HiRes setup is presented in figure 3.1.The working principle of the MOT is explained in section 3.1.1. In section 3.1.2 themagnetic trap and microwave evaporation is explained. The principle of the dipoletrap is explained in 3.1.3 and finally the science chamber is explained in 3.1.5

3.1.1. 2D MOT and 3D MOTMagneto-optical trapping is a widely used technique for precooling atoms. In thefollowing the MOT will be explained in line with [21].Since a photon carries a momentum along the direction of propagation given byp = ~k, where k is the wave-vector of the photon, it is obvious that the velocity ofan atom will change during an absorption event: An atom that absorbs a photon,

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will get a momentum transfer in the propagation direction of the absorbed photon.In the case where a photon is always absorbed from the same direction, a netmomentum will be transferred to the atom as a photon is isotropically re-emitted.Setups for optical cooling consists of non-interacting pairs of counter propagatinglaser beams. One pair is required for each dimension in which cooling is needed.A schematic drawing can be seen in figure 3.2a for cooling in 3D. Each pair hasthe same frequency which is red-detuned with respect to the atomic transition.For simplicity’s sake, the cooling scheme will be explained for only one dimension

(a) (b) (c)

Figure 3.2.: (a) Shows the configuration of the three pairs of counter propagatingbeams, that are used for molasses cooling. (b) Shows an atom movewith velocity v in the +z-direction. The gray arrow indicates the energyof the non-Doppler shifted beams, while the red arrows indicate theenergy of the Doppler shifted beams as seen by the atom. The atomsees the beam propagating in the ±z-direction as having ω∓. (c) Showsthe working principle in a MOT. The figure originates from [21].

which is taken to be the z-axis. A two-level atom is assumed. A stationary atomwill experience an equal radiation force from each beam and thus no net force willbe exerted on the atom. Due to the Doppler effect, a moving atom will, see onebeam as being closer to resonance. This will increase the absorption rate and hencethe radiation force. This concept is shown in figure 3.2b, where an atom is movingin the positive z-direction. Seen from the atom’s frame of reference, the beams,originally with frequency ω, are shifted by the Doppler shift, kv, to ω+ = ω + kv

and ω− = ω− kv for the beam propagating in the negative and positive z-directionrespectively. Mathematically put, the net force, which will be denoted Fmolasses, isgiven by

Fmolasses = F+(ω − ω0 − kv)− F−(ω − ω0 + kv)− = −αv (3.2)

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where F+ and F− are the forces exerted by the beam propagating in the positiveand negative z-direction respectively. Their magnitide is given by the productbetween the photon momentum and the scattering rate. v is the velocity, and α isgiven by [21]:

α = 4~k2 I

Isat

−2∆/Γ[1 + (2∆/Γ)2]2

(3.3)

with I being the intensity, Isat being the saturation intensity, ∆ = ω−ω0 being thelaser detuning and Γ being the natural linewidth. In the denominator a factor ofI/Isat has been omitted since I has to be much smaller that Isat for this descriptionto hold[21]. It is seen that in order for Fmolasses to damp the velocity, the detuninghas to be negative i.e. red-detuned. The technique is called optical molasses becausethe atoms experience a frictional force just as particles moving in a viscous fluid.If the hyper fine structure of the atom is not spectroscopically resolvable by thelaser, the lowest reachable temperature with this cooling technique is the Dopplertemperature, where an equilibrium between the molasses cooling and recoil hasbeen reached:

TD = ~Γ2kB

. (3.4)

For 87Rb this amounts to 146 µK[45]. When the hyperfine levels are resolvable,temperatures below the Doppler temperature can be achieved due to polarizationgradient cooling [17]. This is also known as sub-Doppler cooling.The magnetic field used in the MOT is generated by two coils in an anti-Helmholtzconfiguration. Figure 3.2c shows a schematic version of the MOT. The correspondingquadrupole magnetic field manipulates the scattering force of the lasers. Assumingthat the atomic transition is a J = 0 to J = 1 transition, the Zeeman effect causesthe J = 1 state to split into the three sublevels denoted MJ = 0,± 1. Due to theconfiguration of the coils, the generated magnetic field is 0 in the middle of thetrap and varies linearly in the spatial vicinity. The two counter propagating laserbeams have σ− and σ+ polarization as seen by the atom.The MOT works in the following way: When an atom moves to the region withz > 0, the ∆MJ = −1 becomes closer to resonating with the laser beams. Dueto selection rules, a photon from the σ− beam will be absorbed. The resultingscattering force pushes the atoms back towards the center. This is similar for theatoms in the z < 0 region, but now photons from the σ+ beam is absorbed. Theenergy of the M = −1 state is higher than the energy of the M = 1 state in thez < 0 region because the magnetic field points in the opposite direction comparedto the +z-direction, which is taken as the quantization axis.

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The theoretical cooling limit is the recoil temperature, which is the heating due toa single emission event[45]:

TR = ~2k2

kbm(3.5)

which is 362 nK for 87Rb. Typically ∼ 160 µK is reached with the MOT setup[37].After the MOT-phase, the magnetic field is turned off and pure molasses coolingis applied. Sub-Doppler temperature is the reached via polarization-gradient cooling.

3.1.2. Magnetic traps and microwave evaporationAfter having cooled the atoms in the 3D MOT and by molasses cooling the atomsare trapped magnetically. The high current coils around the cube chamber (as seenin figure 3.1) generate the magnetic field for trapping. The shift in energy of anMF level due to the Zeeman effect is given by

∆E = gFµBmF |B| (3.6)

where gF is the Landé factor, which depends on the internal fine structure,µB is the Bohr magneton. In 87Rb the |F = 1,mF = −1〉, |F = 2,mF = 1〉 and|F = 2,mF = 2〉 of the ground state are the low field seeking states because theirenergy is proportional to the magnetic field, which makes them trappable. Theatoms are optically pumped into the |F = 2,mF = 2〉 state and the magnetic fieldgradient is ramped up to the order of 150Gcm−1. Typically 109 atoms are trappedin the magnetic trap[37]. The atoms are then moved in the magnetic trap to thecube chamber.In the cube chamber a microwave evaporation sequence is used to cool the atomseven further.The working principle is to lower the potential by a radio frequencysignal (RF), the RF signal addresses the transition between the low seeking state|F = 2,mF = 2〉 to the a high seeking state |F = 1,mF = 1〉. By detuning theRF signal thr proper way, only the hottest atoms leave the trap. The remainingatoms re-thermalize and the mean temperature, which is given by the Boltzmann-distribution, is lowered. Figure 3.3 shows the principle in the microwave evaporation.The detuning of the RF signal is gradually lowered. In the end, roughly 5× 108

atoms are trapped with a temperature in the order of 30 µK.

3.1.3. Dipole traps and evaporation coolingFrom the magnetic trap, the atoms are loaded into an optical trap. The physicsbehind the optical trapping of neutral atoms will be explained in the following, in

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Figure 3.3.: Shows the principle in microwave evaporation. The red arrow representsthe RF signal that couples the low field seeking state with the highfield seeking state. Courtesy to [37].

line with [26]. The electric field, E, from a laser with frequency ω will induce anoscillating atomic dipole moment given by

p(r) = α(ω)E(r) (3.7)

where α(ω) is the frequency dependent complex polarizability. The correspondinginteraction potential of the dipole, Udip, and the scattering rate, Γsc, are given by

Udip = − 12ε0c

Re(α)I(r) (3.8)

andΓsc(r) = 1

~ε0cIm(α)I(r) (3.9)

where I(r) is the intensity of the light field. The imaginary part of the polarizabilityenters the expression for the scattering rate since it describes absorption, and ascattering event can be thought of as an absorption event followed by a reemissionevent. An expression for α can be derived via a semiclassical model; and, usingthis expression in the equations above, the following equations are obtained

Udip(r) = −3πc2

2ω0

(Γ

ω0 − ω+ Γω0 + ω

)I(r) (3.10)

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and

Γsc(r) = 3πc2

2~ω30

(ω

ω0

)3(

Γω0 − ω

+ Γω0 + ω

)2

I(r) (3.11)

where Γ is the natural linewidth ω0 is the frequency of the atomic transition. Thedetuning of the light is defined as ∆ = ω − ω0. When trapping atoms, a deeppotential as well as a low scattering rate is desired since the latter corresponds tothe heating of the atoms. From equation 3.10 and 3.11 it is seen that Udip ∝ I(r)

∆and Γdip ∝ I(r)

∆2 . Long lifetimes are therefore achieved by using high power lasersand large detunings.The temperature of the atoms trapped in the optical dipole trap is reduced byevaporative cooling. The concept is similar to the microwave evaporation explainedearlier. Through a lowering of the potential the atoms with energy larger thanthe trap depth leave the trap. This corresponds to cutting away the high energytail of the Boltzmann distribution. The remaining atoms rethermalize via elasticcollisions, which results in a lower average temperature.

3.1.4. Transport to the science chamber

If experiments are to be carried out in the science chamber, the atoms are transferredto the science chamber before they are cooled evaporatively. The transfer is carriedout by shifting the focus of the longitudinal dipole beam from the cube chamberto the final chamber. Here the atoms are loaded into a crossed dipole trap, andcooled. In the end, a BEC consisting of in a few 105 atoms is created [28].

3.1.5. The science chamber

The experiments are carried out in the science chamber of the HiRes. My work hascontributed significantly to building the tools needed for performing experiments inthis chamber. In the following the design will be explanined based on the explanationgiven in [37]. The chamber is shown in figure 3.4 and has four viewports in thehorizontal plane, a bottom viewport and a top viewport, which is not shown.Two of the horizontal viewports are used for medium resolution imaging and willnot be discussed further here. The last two viewports are for the optical lattice.The optical lattice ports have a coating on the vacuum side that is transmittivefor 780 nm light but reflective for light in the range from 790 nm to 1064 nm. Thismakes it possible to create retroreflective lattices of various wavelengths.High resolution imaging and projection of arbitrary potentials onto the atomshappens via the bottom viewport and is explained further in section 4.2.2 and 4.2.3.

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Figure 3.4.: Shows a top view and a side view of the science chamber. a) indicatesmedium resolution ports and b) are the retroreflective viewports usedfor the optical lattice. On the side view, the high resolution viewportis seen. Figure taken from [37].

The window of the top viewport is placed at an angle of 15° to avoid internalreflections that might degrade the trapping potential of the optical lattice.

3.2. Optical latticesIn the science chamber, the atoms are loaded into optical lattices. Lattices aregenerated by standing waves formed by a pair of counter-propagating beams withwavelength λ = 1064 nm. Due to interference between the beams, the lattice periodis λ/2 = 532 nm. One, two or three orthogonal beam pairs are need to generate a1-, 2- or 3-dimensional lattice. The lattice potential, which is formed by each pair,is on the form [12]:

Ulat(x) = U0 sin2(kx) (3.12)

where k = 2π/λ and U0 is the lattice depth.The so-called Bose-Hubbard model describes the dynamics of interacting bosons inoptical lattices and the corresponding Bose-Hubbard Hamiltonian has the form [49]

H = −J∑〈i,j〉

a†i aj +∑i

(εi − µ)ni +∑i

U

2 ni(ni − 1) (3.13)

where a†i and aj are the creation and annihilation operator, ni = a†i ai is the numberoperator, 〈i,j〉 is the sum over the nearest neighbouring lattice sites, J is the tunnelcoupling between adjacent sites, U is the interaction energy, εi is the externalpotential energy at site i and µ is the chemical potential.

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Figure 3.5.: Shows the limit of a shallow lattice (black) to the left where the matterwave is delocalized over the entire lattice. To the right, a deep lattice isshown. The gray area corresponds the the probability density. Figureborrowed from [16].

It is seen that the the Hamiltonian consists of three terms. The first term describesthe kinetic energy which arises from atoms hopping from site i to the neighboursite j with the hopping rate J/~. The second term describes the potential energydue to an external field. The last term describes the on-site interaction for eachpair of atoms at site i.In the limit of a shallow lattice, which corresponds to J U , the tunneling isdominating and the atoms are delocalized over the the lattice. The ground state ofthe many-body system, |Ψ〉, can be described by one matter wave which can beapproximated by a product of Bloch waves in the lowest state [16]:

|Ψ〉 ∝NL∑i=1

a†i

N |0〉 (3.14)

where N is the number of atoms and NL is the number of lattice sites. The atomsare not correlated but in the same state. The number fluctuation on each latticesite follows a Poissonian distribution.In the limit of a deep lattice, where U J , the atoms are localized at the latticesite and since the tunneling is suppressed number of particles per site is fixed. Thisstate is known as the Mott-insulator. For reasonable filling of the lattice, the fillingcan be described by ν = N/NL and the ground state is given by a product of Fockstates[16]:

|Ψ〉 ∝NL∏i=1

(a†i)ν|0〉 (3.15)

This state is highly correlated and the on-site filling is constant. In the case ofν = 1, each site has a unitary filling. The two limits are schematically depicted infigure 3.5.In the case of ν > 1, the atoms will undergo two body collisions in the presenceof light, leaving the trap vacant or with unitary filling. Two-body light inducedcollisions will be explained in greater detail in section 8.4.

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3.3. Single spin addressing in the science chamberIn the science chamber, single 87Rb atoms trapped in optical lattices will beaddressed. In the HiRes experiment, an off-resonant laser of circular polarizationwith a wavelength of 787.5 nm is used. The addressing scheme is based on thescheme already implemented in Munich [49] and will be explained in the following:The laser beam is focused onto a single atom and causes a differential light shiftbetween the ground state, |0〉 = |F = 1,mF = 1〉, and the excited state, |1〉 =|F = 2,mF = 2〉. The shift of each state is given by [26]:

Udip(r) = πc2Γ2ω3

0

(2 + PgFmF

∆2,F+ 1− PgFmF

∆1,F

)I(r) (3.16)

where I(r) is the intensity, gF is the Landé factor, P describes the polarization ofthe laser (P = 0,± 1 for linear and circular, σ±, polarization of the light) and thedetunings ∆1,F and ∆1,F are the energy splittings between the ground state andthe center of the hyperfine split in the two excited levels.For the choice of wavelength in the HiRes experiment, a pure σ+ polarization of thelight will only shift the energy level of the excited state[49] and hence the potentialdepends on the mF -states. The energy shift brings the transition into resonancewith a global oscillating microwave field at 6.8GHz.Since the global field oscillates, the probability of transferring the atom from |0〉 to|1〉 also oscillates. Assuming that the atom is a two- level system, the probabilityof finding the atom in the excited state at a given time, t, is given by [35]:

P|1〉(t) = 12

(χ

Ω

)2[1− cos(Ωt)] (3.17)

where χ is the Rabi frequency and Ω =√χ2 + ∆2 is the generalized Rabi frequency

that depends on the detuning of the laser. The probability of finding the atom inthe ground state is trivially given as P|0〉(t) = 1−P|1〉(t). It is noted, that the largerthe detning, the smaller the probability of finding the atom in the excited state.The optimal duration of the microwave pulse can be found by solving equation 3.17for the time that maximizes P|1〉. It has been proven that a 95% spin flip fidelitycan be achieved [49].A laser resonant with |1〉 can then be used to heat the spin-flipped atoms out ofthe trap.

18

4 The high resolution imaging system

In this chapter, the high resolution imaging system will be introduced. The firstpart of the chapter is dedicated to introduce relevant concepts from imaging theory.The second part of the chapter will be explaining the design criteria associated withthe high resolution system. Finally some preliminary tests of the ‘high resolution’objective will be presented.

4.1. Imaging theory

The foundation of imaging theory was laid in the later part of the the 19th centuryby Lord Rayleigh through his work on lenses and resolution power and by ErnstAbbe through his work of developing microscopes. Even though imaging theoryhas not changed much since then, it is still relevant. Imaging systems based onthe principles developed by Rayleigh and Abbe have become widespread since thecamera became a common property a few decades ago.Creating arbitrary potentials for trapping and manipulating atoms and achivingsingle site resolution are some of the cornerstones in this project. Here imagingtheory is also extremely relevant, since a high quality imaging systems is a require-ment in order to realize this. Unless otherwise stated, section 4.1.1-4.1.3 are basedon [19].

4.1.1. Linear imaging systems

Linear systems are often used to model the imaging process. In a linear systeman input signal, u(x,y), maps uniquely into an output signal, g(x,y), via a linear

19

operator, O[]. This can be put mathematically as

g(x,y) = O [u(x,y)] , (4.1)

where x and y are continuous spatial coordinates. The response of an imagingsystem to a point source is called the point-spread function (PSF). This correspondsto the case where the input signal is a delta function at (x1,y1). The output isdefined as

h(x,y;x1,y1) = O [δ(x− x1,y − y1)] (4.2)

where h(x,y;1 ,y1) is the PSF.Taking advantage of the shifting property of the delta function, it is possible toexpress an arbitrary input signal as

u(x,y) =∫ ∞−∞

∫u(x1,y1)δ(x− x1,y − y1)dx1dy1 (4.3)

which enables the following rewriting of equation 4.1:

g(x,y) = O [u(x,y)]

=∫ ∞−∞

∫u(x1,y1)O [δ(x− x1,y − y1)] dx1dy1

=∫ ∞−∞

∫u(x1,y1)h(x,y;x1,y1)dx1dy1

(4.4)

Equation 4.4 shows that if the response of the imaging system to a point source isknown, we are able to characterize the system. Equation 4.4 is also recognized as aconvolution of the input signal with the PSF, which can be written as

g(x,y) = h(x,y) ∗ u(x,y). (4.5)

This result reduces to a simple product in Fourier space:

G(fx,fy) = H(fx,fy)U(fx,fy) (4.6)

where G, H and U are the Fourier transforms of the output signal, PSF and inputsignal, respectively. H is also known as the transfer function.

4.1.2. Resolution of an imaging systemImagine an imaging system consisting of a thin, perfect lens placed at z = 0 in thepresence of monochromatic light. An incident light wave field is denoted U(x,y,0).It can be shown that the wave field just after the lens can be expressed as

U ′(x,y,0) = U(x,y,0)P (x,y)e−ik2f (x2+y2) (4.7)

20

Figure 4.1.: Shows a lens configuration used for image formation. An object isplaced at z = −d1.

where f is the focal length of the lens and P (x,y) is the so called pupil functionthat only includes light within the lens aperture:

P (x,y) =1, if (x,y) is inside the aperture

0, otherwise(4.8)

If an object is placed at z = −d1, the corresponding wave field is denoted U(x1,y1,−d1), see figure 4.1.The goal is to determine the wave field at U(x0,y0,d0). These two wave fields canbe related via equation 4.4:

U(x0,y0,d0) =∫ ∞−∞

∫h(x0,y0,x1,y1)U(x1,y1,− d1)dx1dy1 (4.9)

In order to find the PSF, assume that the object is a point source. In that caseU(x1,y1, − d1) is a delta function. This corresponds to having a spherical waveoriginating in the point (x1,y1, − d1). From equation 4.7 we know how the lenschanges the wave field. The field at the focal point can be calculated via the Fresnelapproximation. In this particular case, where the object is described by a deltafunction, the output wave field corresponds to the point-spread function, which isgiven by the following equation since the propagation is limited by the pupil:

h(x0,y0) =∫ ∞−∞

∫P (λd0fx,λd0fy)e−i2π(x0fx+y0fy)dfxdfy (4.10)

where P (λd0fx,λd0fy) is the scaled pupil function. Equation 4.10 is recognized asthe 2D Fourier transformation of the scaled pupil function. The spatial frequenciesin each direction are denoted fx and fy. From equation 4.9 and 4.10 it is evidentthat the final image is just the perfect image convoluted with the PSF. In general,

21

Figure 4.2.: Shows the numerical aperture (NA) of a lens.

this is a smoothing of the perfect image that attenuates or blurs details in theoutput image.The intensity of the image of a point source is given by

I(x0,y0) =∣∣∣∣h(x0

M,y0

M

)∣∣∣∣2 (4.11)

where M = d0/d1 is the magnification of the system. Carrying out the calculationgives the following intensity distribution for the PSF[37]:

IPSF (r,z = 0) =J1

(2πrNA

λ

)πrNA

λ

2

(4.12)

where J1 is the first Bessel function, r =√x2 + y2 and NA is the numerical aperture.

The numerical aperture is defined as the sine of the angle between the optical axisand the most extreme ray, which is sketched on figure 4.2:

NA = sin (θ) = sin(

arctan(D

2f

))≈ D

2f (4.13)

where D and f are the diameter of the lens and focal length, respectively.The profile of equation 4.12, and thus of a point source, is the so called Airy disk,which is illustrated in figure 4.3, where the intensity has been scaled in order tomake the outer rings more visible.In broad terms, the resolution of an imaging system describes to what extent thesystem can discriminate between small details in the object to be imaged. Theresolution depends on the NA of the lens and therefore also the Airy disk. Afrequently used measure for resolution is the so called Rayleigh criterion that statesthat the minimum resolvable distance between two point sources corresponds towhen the center of the Airy function of the first point source coincides with thefirst minimum of the Airy function generated by the second point source[23]. This

22

Figure 4.3.: Shows an Airy disk. The intensity has been modified in order to makethe outer rings more visible.

is shown in figure 4.4, where two Airy functions are seen. The dashed line showsthat the center of the Airy function to the right coincides with the first minimumof the Airy function to the left. In a real image, the red curve, which is the sumof the two Airy functions, would be the obtained intensity profile. This limitingminimum distance between the two intensity profiles can mathematically be put as

rmin = 1.22 λ

2NA (4.14)

Depth of field, ∆z, is a concept that describes how far an object can be placedaway from the focal point of a lens. It depends on the NA and the wavelength inthe following way:

∆z = λ

NA2 (4.15)

4.1.3. AberrationsAccording to the Rayleigh criteria, the performance of an imaging system is almostindistinguishable from a perfect system if the optical path difference (OPD) orphase difference between the central beam and any other beam within the apertureis smaller than or equal to λ/4. If an ideal wave front propagates through anon-perfect imaging system, different parts of the beam will pick up a differencein phase due to OPDs i.e. the wave front becomes distorted. This in calledaberration. Aberrations could arise from many different things, for instance defectsand imperfections in lenses and mirrors used in a setup, a defocus, a tilt, etc. Ifone wants to include aberrations in the mathematical description of the imagingsystem, the pupil function has to be modified in the following way[19]:

PA(x,y) = P (x,y)eikφA(x,y) (4.16)

where P (x,y) is the pupil function and φA(x,y) is the phase error introduced byaberrations. Zernike polynomials in polar coordinates can be used to describe the

23

-1.5 -1 -0.5 0 0.5 1 1.5 2

Distance [ µm]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Inte

nsity [arb

. units]

The Rayleigh criterion

Figure 4.4.: Shows the Rayleigh criterion. The blue curves represent the two pointsources. The black dashed line marks the first minimum of the pointsource to the left and the center of the point source to the right. Thered curve shows the sum of the two signals.

24

Table 4.1.: Zernike Polynomials, inspired by [19] and [39].k n m Polynomial Physical meaning1 0 0 1 Piston2 1 -1 2ρ sin(θ) y-tilt3 1 2ρ cos(θ) x-tilt4 2 0

√3(2ρ2 − 1) defocus

5 -2√

6ρ2 sin(2θ) astigmatism, 06 2

√6ρ2 cos(2θ) astigmatism, 45

7 3 -1√

8(3ρ2 − 2)ρ sin(θ) vertical coma8 1

√8(3ρ2 − 2)ρ cos(θ) horizontal coma

9 -3√

8ρ3 sin(3θ) vertical trefoil10 3

√8ρ3 cos(3θ) oblique trefoil

... Higher order terms

different types of aberrations in the case where the pupil is circular. This is doneby expressing φa(x,y) in terms of the Zernike polynomials, zk(ρ,θ),

φA(ρ,θ) =K∑k=1

wkzk(ρ,θ), (4.17)

where ρ is the radial coordinate within the unit circle, θ is the polar angle andthe polynomials are orthogonal and normalized within the circle of unit radius, wkis a fitting coefficient. The summation runs over the different polynomials. Eachpolynomial is of the form:

zk(ρ,θ) = Rmn (ρ) cos(mθ) (4.18)

with n and m being non-negative integers and n describing the degree of thepolynomial Rm

n (ρ) that contains no power of ρ less than m. The k in equation 4.17is the so-called Noll index and it maps the n and m indices[39]. A list over the firstZernike Polynomials and their physical meaning is given in table 4.1. The samepolynomials have been plotted in figure 4.5, where they are labeled by the Nollindex and each row has the same n. It should be noted that the first three termsdo not affect the imaging quality, since a piston gives a global phase shift and a tiltshifts the position in the focal plane.

25

Figure 4.5.: Shows the first 10 Zernike polynomials. They are listed after the Nollindex. Each row has the same n. See table 4.1 for more information.

26

4.2. Design CriteriaThe overall goal of the high resolution system is to be able to detect single atomstrapped in optical lattices as well as address these single atoms and generatearbitrary trapping potentials. Light associated with these operations will enter thefinal chamber via the high resolution viewport introduced in section 3.1.5. Thebeams used to create the optical lattice have a wavelength of 1064 nm, which resultsin a lattice spacing of 532 nm. The following sections will focus on the differentdesign criteria.

4.2.1. The high resolution objectiveThe high resolution objective is custom made and has a field of view of 100 µmand a focal length of 5mm. For light with wavelengths in the range 780 nm to790 nm, the objective is designed to perform diffraction limited. The objective hasa numerical aperture of NA = 0.7. The minimum resolvable distance between twopoints to be imaged is given by equation 4.14 and amounts to rmin ≈ 680 nm. Thisdistance is immediately larger than the spacing between two neighboring atomstrapped in the lattice. This could potentially constitute a problem. However, inthe HiRes experiment atoms in highly regular lattices are to be imaged, whereeach lattice site is either occupied by a single atom or vacant. The principle isshown in figure 4.6: To the left, three neighboring lattice sites are occupied. Theobserved signal is close to constant over these three sites. To the right, the site inthe middle is vacant. A clear detectable dip in the signal is seen. This difference inobserved signal can be used to reconstruct the atom distribution in the lattice eventhough the distance between the atoms are lower than rmin; hence the Rayleighresolution is not the limiting factor in the setup. A test of the performance of thehigh resolution objective is documented in the end of this chapter.

4.2.2. ViewportConstraints are put on the flatness of the viewport, since deformations can lead toaberrations and limit the performance of the optical system. Therefore great effortwas put in to characterize the flatness of the viewport. The viewport was mountedin a test flange, as seen in figure 4.7. The screws were enumerated and tightenedto 13Nm. The flatness was measured with commercial interferometer (µPhase1000, Trioptics). Within a defined region of interest (ROI) with a diameter ofroughly 20mm, the peak-valley value (PV) was measured to 0.4λ after subtractinga polynomial of second order that describes the tilt and defocus. To a first

27

-1.5 -1 -0.5 0 0.5 1 1.5

Distance [ µm]

0

0.2

0.4

0.6

0.8

1

Inte

nsity [

arb

. u

nits]

(a)

-1.5 -1 -0.5 0 0.5 1 1.5

Distance [ µm]

0

0.2

0.4

0.6

0.8

1

Inte

nsity [arb

. units]

(b)

Figure 4.6.: Shows the observed signal from three occupied lattice sites (a) and theobserved signal when the site in the middle is vacant (b).

order approximation, the power term (quadratic term) can be compensated byrepositioning the objective. The flatness profile is shown in figure 4.8a. Bysystematically tightening 6 of the screws in steps of 1Nm to 18Nm the flatnesswas increased to a PV of 0.2λ which is below the criteria for a diffraction limitedsystem. The profile is shown in 4.8b.When the viewport was mounted in the final chamber, the screws were tightenedin the same way, which should reproduce the same flatness.

4.2.3. Coating of the high resolution viewport

As described in [37], the coating has a transmission window for light with awavelength in the range 780 nm to 790 nm and for 940 nm while it reflects light at912 nm and 1064 nm. This is needed in order to generate optical lattices at differentwavelengths, the so-called superlattices.The coating can induce an angle dependent phase shift between s- and p-polarizedlight. The shift can mainly be described by a quadratic term which represents adefocus. This is compensated by changing the thickness of the glass substrate whichgives a residual phase difference that amounts to 0.3λ for s- and p-polarized lightfor all angles. The compensation thickness was chosen to be optimal for 780 nmsince it will be possible to correct the 787 nm and 940 nm wavefronts, which willbe explained in detail later.

28

Figure 4.7.: Shows the viewport mounted in a test-flange. The flange has beenmounted on the end of a vacuum tube.

(a) (b)

Figure 4.8.: Shows the flatness profile of the ROI of the viewport before tighteningthe screws (a) and after tightening the screws (b). Fringe patterns arevisible due to mechanical vibrations in the laboratory.

29

Figure 4.9.: Shows the main part of the test setup for the preliminary test of thehigh resolution objective.

4.3. Preliminary test of the objectiveTo be confident that the high resolution objective performs well enough, a pre-liminary test was carried out before building it into the HiRes experiment. Thesetup is seen on figure 4.9: A 780 nm beam comes out of the outcoupler and passesthrough a half-wave plate, which allows us to rotate the polarization. It thenpasses through a glass plate which is placed on a mount (Gimbal Mount, Ø50.8mm,Thorlabs) that can be tilted around the x- and y-axis. The glass plate is coatedwith the same coating as the viewport. After the glass plate, the high resolutionobjective is placed. The objective is mounted on a piezo stepper (PiFoc 625, PI)that will translate the objective along the z-axis in steps of 100 nm1. The beamthen propagates through an imaging system that focusses the beam onto a CMOScamera (uEye, IDS). On the glass substrate a target plate was placed which hasholes with diameters in the range 60 nm-100 nm.First the magnification of the system was calculated. This was done by taking animage of three holes in the target plate with the test setup. The image is seenin figure 4.10. The pixel size of the uEye is Spix = 5.3 µm. Knowing this andmeasuring the distance in pixels between the center of the three holes gives the

1The precision of the PiFoc goes down to 10 nm

30

Figure 4.10.: Shows an image of a small part of the target plate

physical distance between the holes in the plane of the CCD. The manufacturerof the plate (ETH Zürich) provided a high resolution image of the plate with aphysical scale, which is shown in figure 4.11, so the actual distance between theholes can be calculated as well. The fraction between the distances in the imagingplane and the corresponding actual distances yield the magnification, which wasfound to be M = 204.9Since the holes in the target plate are smaller than the wavelength, they effectivelyconstitute point sources in our setup. One small hole in the target plate was chosenas a point source. The outcome of this test will thus reveal with which qualityatoms can be imaged in the final chamber. The test is subdivided into three parts.The first part tests the position of the objective and the tilt with respect to the glassplate. The next part explores the effect of the coating for different polarizationsand the final part investigates how an actual fluorescence signal is imaged.

Position and tilt

For a given tilt of the glassplate, the objective was translated 10 times along thez-axis in steps of 100 nm. For each step, an image was taken of the intensity profile.This is shown in figure 4.12. It is seen that the position of the objective is crucialfor the imaging quality and the precision with which it should be placed is on thesub micrometer scale.For each image, the two radii, r1 and r2, were defined from the center of the profile,where r1 corresponds to the lattice spacing, alat = 532 nm, in camera pixels:

r1 = alatM

Spix≈ 21Pixels (4.19)

and was chosen as r2 = 8√

2r1 ≈ 237Pixels. The ratio between the total intensities

31

Figure 4.11.: Shows the image of the test target provided by the manufacturer. Therectangle corresponds to 4.10

Figure 4.12.: Shows how the imaging quality of a point source changes as the focusof the objective is swept through. The sweep starts in the upper leftcorner and ends in the the lower right corner. The translation stepsize here is 100 nm.

32

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Distance [ µm]

0

50

100

150

200

Inte

nsity [

Arb

. va

lue

s]

Cutout through the center in the x-dim.

xLine

xFit

PSF

1 2 3 4 5

Distance [ µm]

0

50

100

150

200

Inte

nsity [A

rb. valu

es]

Cutout through the center in the y-dim.

yLine

yFit

PSF

Figure 4.13.: Shows the linecut in each dimension through the center of the 7th spotin figure 4.12. A Gaussian profile has been fitted to the linecut andfor comparison, a simulated point spread function has been plotted.

within these two radii was calculated. The optimal position for the objective wastaken as the position with the highest intensity ratio, Imax. A linecut through thecenter of the 7th spot (counted from the top left in figure 4.12) in each dimension isshown in figure 4.13. For comparison, a simulated point spread function has beenplotted as well. It it seen that the PSF and the linecut coincide, which indicatesthat the imaging system performs diffraction limited. A Gaussian profile has beenfitted to the linecuts and from the fit, the 1/e2 waist is extracted. For the 7th spotwaist amounts to wx = 0.451(4) µm and wx = 0.496(4) µm where the numbers inthe parenthesis are the standard deviation of the fit.Systematically, the tilt of the glass plate was then changed horizontally and vertically.For each tilt, the objective was swept through the focus and a new Imax wascalculated. Figure 4.14 shows the Imax as a function of the horizontal er verticaltilt denoted by H and V , respectively. Thorlabs has subsequently informed thatthe unit of H and V is 0.0062°. In the following, the values of H and V has to bemultiplied by this factor to be in physical units.First, the horizontal tilt was changed systematically, while the vertical tilt was fixedat V = 29. Then the vertical tilt was changed systematically for a fixed horizontaltilt of H = 19.It is seen that for H = 19 and V = 29, close to 100% of the light comes from aregion of a radius equal to that of a lattice site. This indicates a very focussed spot,when taken into account that the size of a lattice site is below Rayleigh resolution.The test gives an insight of the tolerance of the position and tilt of the objective.

33

H

5 10 15 20 25 30 35 40 45

I max

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Imax

as a function of horisontal tilt

(a)V

10 15 20 25 30 35 40 45

I max

0.75

0.8

0.85

0.9

0.95

Imax

as a function of vertical tilt

(b)

Figure 4.14.: Shows Imax as a function of horisontal tilt (a) and vertical tilt (b). Hand V are in units of 0.0062°

The effect of the coating

For each combination of H and V presented above, the objective was moved to theoptimal position. The polarization of the light was then rotated by rotating thehalf-wave plate. It was rotated 90° in steps of 10°.The goal was to investigate the influence of the coating (on the viewport and glassplate) has on the imaging quality for different polarizations of the light.For each rotation of 9°, the two radii, r1 and r2, were defined in the same way asbefore, and the ratio between the total intensities within these radii was calculated.After rotating the half-waveplate 90°, the minimum of the intensity ratios, Imin,was plotted as a function of the tilt. This is seen in figure 4.15.The minimum intensity is chosen as the important parameter here, because it setsa lower limit for the performance of the system. It is clearly seen that the coatinghas a negative effect of the performance of the imaging system, and that it dependson the polarization of the light.

Imaging of a fluorescence signal

As a final preliminary test, the half-wave plate was rotated 2× 360° in steps of 5°.This was done only for the optimal tilt (H = 19 and V = 29). For each step animage was taken of the profile. After completing the rotations, all the images weresuperimposed. Physically this imitates the signal expected to be detected from asingle atom trapped in the optical lattice. When the atoms emit a fluorescence

34

H

5 10 15 20 25 30 35 40 45

I min

0.55

0.6

0.65

0.7

0.75

0.8

Imin

as a function of horisontal tilt

(a)V

10 15 20 25 30 35 40 45

I min

0.6

0.65

0.7

0.75

0.8

Imin

as a function of vertical tilt

(b)

Figure 4.15.: Shows Imin as a function of horisontal tilt (a) and vertical tilt (b). Hand V are in units of 0.0062°

signal, the polarization is expected to be random, hence the rotation of the half-waveplate during the experiment. The resulting profile is seen in figure 4.16, where theintensity scale has been modified in order to make the lobes more visible.A cut through the center of the profile along each dimension is shown in figure 4.17,where they are compared to the expected point spread function. The fact that theintensity profile is as narrow as the PSF shows that the imaging system performsdiffraction limited.To get an idea of how a signal from an optical lattice with unitary filling wouldturn out, the added profile in figure 4.16 has been plotted multiple times with aseparation of alat. This is seen in figure 4.18a, where a ‘diagonal’ of lattice siteshas been kept vacant. This is to show that even though alat is smaller than thermin predicted by the Rayleigh criteria, it is still possible to distinguish vacantand occupied lattice sites, as mentioned previously. For comparison, a line cutthrough the bottom row of the lattice is shown in figure 4.18b. Here, also, the dipin intensity is clearly seen.

35

Added Profiles

-1.5 -1 -0.5 0 0.5 1 1.5

Width [ µm]

-1.5

-1

-0.5

0

0.5

1

1.5

Heig

th [

µm

]

0

20

40

60

80

100

120

Inte

nsity [

Arb

. U

nits]

Figure 4.16.: Shows the added intensity profiles. The square root of the intensityhas been taken to make the outer parts more visible.

-1.5 -1 -0.5 0 0.5 1 1.5

Width [ µm]

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Inte

nsity [arb

. units]

Profile in the x-direction

Profile

PSF

-1.5 -1 -0.5 0 0.5 1 1.5

Heigth [ µm]

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Inte

nsity [arb

. units]

Profile in the y-direction

Profile

PSF

Figure 4.17.: Shows the line cut through the center of the added profile in eachdimension. The profiles have been compared to the expected pointspread function.

36

Array of Profiles with vacant sites

5 10 15 20 25 30 35 40 45

Width [ µm]

5

10

15

20

25

He

igth

[µ

m]

(a)

200 400 600 800 1000 1200 1400 1600 1800

Width [ µm]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2In

ten

sity [

arb

. u

nits]

×10 4 Line cut through the array

(b)

Figure 4.18.: (a) Shows the expected signal from an optical lattice with unitaryfilling. A ‘diagonal’ of lattice sites has been kept deliberately vacant.(b) Shows a line cut through the bottom row of the lattice.

37

5High resolution board and spatial lightmodulation

5.1. Overview

In the HiRes experiment, neutral 87Rb atoms will be trapped in an optical latticewith a spacing of alat = 532 nm and since the goal is to have single site resolution,to perform single atom addressing and generate arbitrary conservative opticalpotentials, the high resolution imaging system has to be able to generate arbitrarywavefronts and control their position in the plane of the atoms.Light with a wavelength of 787.5 nm will be used for addressing single atoms and,for creating arbitrary potentials, light with a wavelength of 940 nm will be used.One of the key features of the HiRes board is thus a flexible way to shape 787.5 nmand 940 nm light beams. For archiving this feature, a type of spatial light modulatorscalled digital micromirror devices (DMD) are implemented on the HiRes board.In this chapter, a brief introduction to spatial light modulators is given in section 5.2followed by an introduction to DMDs in section 5.3. The DMDs can be used eitherin a so called direct configuration or a Fourier configuration, which is explained insection 5.4 and 5.5, respectively.A detailed explanation of the optics of the high resolution imaging system placedon an optical bread board (HiRes board) is given in chapter 6.

5.2. Spatial Light ModulatorsAs the name suggests, a spatial light modulator (SLM) is a type of device thatcan be used to spatially modulate light. Two simple examples of SLMs are thetransparencies used in old overhead projectors and the slides used in old slideprojectors. In each case light from a light source is modulated in order to produce

38

arbitrary images. For changing the projected image, these two methods are ex-tremely slow and have been outdated for many years. These days, technologies likeLiquid Crystal Displays (LCDs) and Digital Micromirror Devices (DMDs) are usedto dynamically modulate light. LCDs are for instance known from televisions andDMDs is a key technology in regular projectors, which are be found in e.g. offices,auditoriums, class rooms etc.In the HiRes experiment only DMDs will be used to generate arbitrary potentialswhich are projected on to atoms.Due to intrinsic properties of the liquid crystals in the LCDs, the crystals have to berefreshed to prevent charges from building up and to prevent leakage currents. Therefresh rate causes a flickering in the signal, which in turn can lead to heating ofthe trapped atoms depending on the trap frequency. This makes LCDs unsuitablefor projecting optical potentials onto trapped atoms[52].

5.3. Digital Micromirror DevicesDigital Micromirror Devices are chips that consist of an array of aluminum mi-cromirrors. The size of the single mirror and the size of the mirror array variesfrom DMD-model to DMD-model[5]. Each mirror is a pixel and can individually betilted to pointing into one of two directions: One direction represents the "on"-stateof the pixel while the other direction represents the "off"-state. This means thatthe DMDs are basically binary devices, but despite DMDs being binary, ways toproduce a grayscale exist, which will be explained in detail later. A close up of aDMD is shown in figure 5.1a where it clearly can be seen that the mirrors can becontrolled individually. The schematic design of a single DMD mirror is shown infigure 5.1b. Each micromirror is attached at the center to a torsion hinge placedright under the mirror along the diagonal. Four spring tips make contact with theunderside of the mirror as well. Two electrodes are used for holding the mirror inone of the two states. A memory cell is placed underneath each mirror. The stateof each memory cell determines the position of the mirror but not in a direct way;the state of the memory cell can be set without affecting the mechanical state of thecorresponding mirror. The mirror state is set to the memory state when the mirrorreceives a so-called mirror clock pulse. This enables a group of mirror positions tobe preloaded to the memory blocks and then being set simultaneously by a singlemirror clock pulse. During the mirror clock pulse, it is not possible to write to thememory block[33]. The rotation of a mirror is created by manipulating electrostaticpotentials between the mirror electrodes (see figure 5.1b) and the corresponding

39

(a) (b)

Figure 5.1.: (a) Shows a close up of an array of micromirrors. Figure from [5]. (b)Shows the schematic design of a single DMD pixel. The lightblue squarerepresents the mirror, the torsion hinge is the lightgreen diagonal underthe mirror, the yellow squares are the spring tips and the ‘massive’ redblock is one of the two electrodes that control the state of the mirror.The figure is taken from [33].

memory block. The mirror rotation is stopped by the spring tips.The DMDs used in the HiRes experiment are the DLP Lightcrafter 6500 EvaluationModule (DLP6500) from Texas Instruments which is depicted in figure 5.2. Themodule consists of two main parts: A DLPC900 chip, which controls the DMD andenables pattern rates up to 9523Hz for binary patterns, and a DLP6500FYE chipwhich is the Digital Micromirror Device. The DLP6500FYE chip has a 1920× 1080square mirror array. An example of a small square array is given in figure 5.3awhere each square represents a mirror. Including the distance to the neighbouringmirror, each mirror has the dimension of 7.56 µm × 7.56 µm and can be tilted ±12°around the diagonal axis of the mirror, indicated by the blue dashed line in thefigure. Tilted one way for realizing the "on"-state and they other way for realizingthe "off"-state. An alternative to the square array is the diamond array, as shown infigure 5.3b. A DMD with diamond alignment has been tested in the HIRES-group[27], but is not used in the final setup.The DLP6500 can be operated in four different modes[4]. The two relevant modesfor this project are

• Pattern On-The-Fly Mode: The user can upload a sequence of patterns(BMP-images) to the internal memory of the DLP6500. The exposure time ofeach pattern can be set individually. The sequence of images can be loopedthrough a single time or repeatedly. The advancement from one pattern tothe subsequent pattern in the sequence can be invoked by an external trigger.

40

Figure 5.2.: The DLP Lightcrafter 6500 Evaluation Module. The DLP6500FYEchip (DMD) can be seen close to the center of the red print board. TheDLPC900 is on the backside of the green print board and is thus notvisible.

(a)(b)

Figure 5.3.: Shows two different alignments of the micromirrors in a DMD. Eachsquare represents a mirror while the blue dashed lines represent theaxis around which the mirror can tilt. (a) Is inspired by [2] and showsthe square alignment with P= 7.56 µm for the DLP6500 and (b) is thediamond alignment.

41

The DLP6500 can also produce a trigger signal when the pattern is changed.If the sequence consists of patterns with a bit depth of 1; 400 images canbe uploaded to the internal memory and the sequence can be played with apattern rate of 9.5 kHz.

• Video Mode: The DMD acts as a monitor. The video feed to the DMD canhappen via a HDMI-connection. For 24 bit depth video stream, the fame rateof the DMD is limited to 60Hz.

A USB connection is used to interface with the DLP6500 in the HiRes experiment.The only USB-protocol supported by the DLP6500 is the so-called USB 1.1 Humaninterface device-protocol which is designed for interacting with keyboards, controllersetc. A maximum of 64 bytes can be sent per data transfer through the HIDprotocol[3]. When transferring patterns to the DMD, e.g. in the Pattern On-The-Fly mode, the total transfer time depends on the number of patterns to transferand the compressibility of the patterns; the more complex a pattern is, the longertime it takes to transfer. The time it takes to transfer 400 patterns ranges from 20seconds to a minute.Deformation of the mirrors can occur due to applied stress, if a mirror remainsfor too long in the same state. To prevent this, the DLP6500 is designed with a"switching cycle" that switches the mirror to the opposite state and back again at10 kHz. This gives rise to a flickering signal from the DMD and hence the signal isnot truly static. It has been proven, that the switching cycle can be supressed by ahardware extension to the DLP6500 [30].On the HiRes board, four DMDs will be implemented. Two DMDs will be operatedat 787 nm and the other two at 940 nm. For each wavelength, one DMD will beimplemented in the imaging plane and one DMD will be implemented in the Fourierplane. In the following, the two ways of implementing DMDs will be explained.

5.4. DMDs in the imaging planeSetups where DMDs are placed in the imaging plane of an imaging system is knownas the direct configuration, and it is a more straightforward way to use a DMDcompared to placing it in the Fourier plane, which will become clear in the nextsections. Each mirror of the DMD will be directly imaged in the final image, onlylimited by the point spread function of the system.Since each mirror can be controlled individually, optical potentials of arbitrarygeometry can be generated by turning on the mirrors in a desired configuration.

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DMDs are binary devices and as a consequence, the intensity scale of the opticalpotential is also binary.The binarity can be circumvented by binning the mirrors in small subarrays calledsuperpixels. A greyscale can rise either from the use of a magnification system thatmaps a superpixel onto one pixel in the final image or by the use of a spatial filterplaced in the Fourier plane of the imaging system in such a way that it removesthe spatial frequencies corresponding to the details smaller than the size of thesuperpixel [27].If the superpixel is a square array of N mirrors, the resolution of the DMD willdecrease by a factor

√N in each dimension. The intensity scale rises from having a

variable number of mirrors within the superpixel in the "on"-state.The reflected power scales directly with the number of mirrors in the "on"-state;thus reducing the size of an arbitrary potential landscape will reduce the outputpower. It will later be showed, that the depth of potential is independent of theoutput power.In general, the direct DMD configuration is best suited for big structures withsharp edges and many features.

5.5. DMDs in the Fourier planeSetups where the DMDs are placed in the Fourier plane of an imaging system areslightly more abstract than the direct configurations. Here each mirror on theDMD corresponds to certain spatial frequency component in the final image, hencefor generating a final image, one has to calculate the Fourier transform of thisimage and model it on the DMD. This technique is very well suited for generatingsmall structures[52], since small features in real space are big in Fourier space. Thiscan intuitively be understood by considering that small features in real space aredescribed by high spatial frequencies which, in turn, will result in a wide Fouriertransform. A broad frequency spectrum makes it possible to use more of the DMDchip to generate one small feature. This technique will, however, be limited if thefeature becomes too detailed; for example, it would require very high frequenciesto describe a sharp edge. If the frequencies are too high to fit on the DMD chip,the feature in the final image will not be fully recreated.The fact that each mirror has a spatial extent, limits the precision with which eachspatial frequency can be defined. The need for precise definition of the frequenciesincreases with an increasing number of features in the final image, since each DMDpixel contributes to the intensity in each pixel in the final image.As explained in section 4.1, aberrations are basically changes in the phase picked

43

up by different parts of a beam as it propagates through an imaging system. WithFourier DMDs it is possible to control both the phase and the amplitude in thefinal image, hence it is possible to correct for aberrations and thus improving theimaging quality.First step towards being able to phase- and amplitude-correct wavefronts is to mapthe relative phase difference and relative intensity for each individual mirror. Inthe following, the principle of phase- and amplitude modulation with DMDs willbe presented as well as methods for mapping the relative phase and amplitude.

5.5.1. Beamshaping with DMDsIn the following, the way DMDs can be used to shape beams will be explainedin line with [52]: The working principle is based on a general diffraction grating.Assuming that a grating is illuminated by an incoming plane wave, diffractionorders will form in the far field. The directions at which orders will be formedfulfill the criteria that the OPD (optical path difference) from the different slitscorresponds to a whole number of wavelengths which is equal to demanding adifference in phase of a multiple of 2π. This phase difference can be related backto the slits, where there has to be a phase difference of n2π between neighbouringslits for the n’th diffraction order. Shifting the position of the slits will thus shiftthe phase of the outgoing wave front. The intensity of outgoing wave is relatedto the width of the slits in the grating. Arbitrary outgoing wavefronts can thusbe generated, if one has control over the position and the width of slits. This issketched in figure 5.4. In addition to changing the phase and amplitude of theoutgoing beam, the k-vector and hence the direction of propagation can also becontrolled. The relation between the incoming and outgoing beam is given by

θn = arcsin(λπ

d− sin(θi)

)(5.1)

where θn is the angle to the n’th order, θi is the angle of incidence, λ is thewavelength of light and d is the grating spacing. It is seen that changing the spacingwill change the direction of propagation.In the experimental setup, Fourier DMDs will be used as a reflective diffractiongrating. The ability to dynamically change the state of the mirrors enables thedynamic change of the width, position and spacing of the slits.

5.5.2. PhasemapsA phasemap is basically a map of the phase delays (also denoted as ‘relative phase’),∆φ, over the DMD pixels with respect to a reference pixel. The relative phase

44

Figure 5.4.: Shows the principle in beam shaping by modifying the width andposition of the slits in a grating. Figure borrowed from [52].

45

(a) (b)

Figure 5.5.: (a) Shows the chosen ROI on the DMD filled with a plane slit pattern.(b) Shows two patches. The patch to the left is the sampling path andthe path to the right (centered with respect to the ROI) is the fixedreference patch.

of a pixel can be viewed as an imaging systems response to the spatial frequencyrepresented by that specific pixel. Mapping these phases makes it possible to takethem into account when an arbitrary wavefront has to be generated.In the HiRes experiment, only a part of the DMD will be used, hence a ROI isdefined on the DMD. A plane grating is ascribed to this region, as shown in figure5.5a, where white represent pixels in the "on"-state.The relative phases are retrieved by subdividing the DMD into small patches. Atthe same time, two patches show the regular grating, the rest of the DMD pixelsare in the "off"-state. This is shown in figure 5.5b. The patch to the left is thesampling patch and the patch to the right, which corresponds to the center of theROI, is the reference patch. The reference patch is fixed while the sampling patchis shifted within the ROI. When the DMD is illuminated by a laser source, a beamwill be reflected from each of the two patches. This is seen in figure 5.6, wherethe incoming beam is modulated into a reference beam, BR and a sampling beam,BS. These are separated by a distance, d. A lens focuses the beams onto a CCD.After the lens, the beams have a relative angle of γ. On the CCD, these beams willoverlap and generate an interference pattern of fringes.At each position of the sample patch, the phase of the grating in the sample patch,φG, is shifted five times by an number of 2π/5 with respect to the phase of thegrating in the reference patch, which is taken to be zero, since only the relativephase is needed. This is seen in figure 5.7. As explained previously this gives riseto a phase shift of the outgoing wave front which in the end will cause the fringesin the interference pattern to shift.In a small region around the focus, the two beams can be described as plane waves,

46

γ

CCDBR

BS

z

x

d

f f

DMD

Figure 5.6.: Shows how two beams are generated by the two patches on the DMD.Br and BS are the beam from the reference- and sampling beam,respectively. The angle between the focused beams is denoted γ and dis the distance between the two patches. The figure is adapted from[52].

φG = 0 φG = 152π φG = 2

52π φG = 352π φG = 4

52π

Figure 5.7.: Shows the sampling patch for 5 different phases. The grating is clearlyshifted from patch to patch.

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Figure 5.8.: Shows the five interference patterns as the phase of the grating in thesampling patch is changed. The shift of the fringes due to the phaseshift of the grating is seen. The virtual pinhole is marked by a redsquare in the center of the interference pattern. The square has beenenlarged to be more visible.

and the intensity of the interference pattern can be described as[52]:

I(x,z) = |BR(x,z) +BS(x,z)|2 = a2+b2+2ab cos(2πλ

((1− cos γ) z + sin γx) + ∆φ)

(5.2)where λ is the wavelength of the illumination profile and ∆φ is the relative phasebetween the two beams. On the CCD camera, a virtual pinhole is defined.A test setup corresponding to figure 5.6 was built and the five sample patcheswere in turn uploaded (together with a fixed reference patch) to the DMD. Thecorresponding five interference images are shown in figure 5.8. It is seen how thefringes are shifted as the phase of the grating is shifted. The small red square inthe center of each image is the virtual pinhole. In this figure, it has been enlargedto be more visible. The size of the pinhole is chosen in such a way that it is smallerthan the scale at which the intensity changes. The mean intensities within thepinholes are calculated, normalized and shifted to lie within the range of I ∈ [−1; 1]in arbitrary units.The relative phase, ∆φ, is extracted by fitting an equation on the form

f(x) = A cos (x−∆φ) (5.3)

to the calculated intensities, where the amplitude, A, and relative phase are fittingparameters. The amplitude should be redundant in the fit since intensities havebeen normalized, but it turned out that the fits were improved by including it as afit parameter. The intensities and a fit are seen in figure 5.9, where the calculatedintensities have been normalized and plotted as red dots as a function of the relativephase. The blue line is the fit. The phase could be extracted directly by a reading ofthe position of the peak, but in our case, it is extracted from the fit. The extractedvalue of the relative phase is attributed only to the center pixel of the samplingpatch.

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Phase [rad]0 1 2 3 4 5 6

Inte

nsity

[arb

. uni

ts]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Intensity as the slit pattern is changed

Figure 5.9.: Shows a fit to the five normalized and rescaled intensity values extractedfrom the five interference patterns shown in figure 5.8.

By shifting the position of the sample patch around the ROI defined on the DMDand repeating the above mentioned phase shifting sequence, it is possible to ascriberelative phases to the mirrors within the ROI and thus generating a phasemap.

5.5.3. Generation of phasemaps

In this section typical parameters used for generating phasemaps will be described.The round ROI defined on the DMD is kept as large as possible without exceedingthe dimensions of the DMD. The radius of the ROI is set to 500 pixels and thecenter coincides with center of the DMD. The size of the sampling and referencepatch is related to the intensity of the interference signal. The bigger the patches,the more intensity can be recorded. A size of 45× 45 pixels was found to give aclear signal. The size was chosen as an odd number on purpose in order to havea clearly defined center pixel to which the relative phase could be ascribed. Anoversampling of 15 pixels has been introduced to avoid mapping the relative phasesof all the ∼ 785× 103 mirrors within the ROI. The oversampling is the distancebetween two neighbouring sample patches. For this value of oversampling, thephase of 3503 mirrors will be mapped which requires 17520 patterns (similar tofigure 5.5b) to be uploaded to the DMD. The internal memory on the DLP6500

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can only contain 400 patterns, which means that 44 packets of 400 patterns haveto be uploaded and displayed in turn.The slits are generated by binary algorithm seen in algorithm 1, where the loopdefined in line 1 runs over all the pixels in the patch. Depending on the coordinatesof the pixel, (x,y), the pixel value is either white (1) or black (0) corresponding toa DMD pixel in the "on"- and "off"-state, respectively. In line 2, kx and ky are thek-vectors in the x- and y-direction, which determine the angle of the slits and aredefined as

ki = 2πdi, i = x,y (5.4)

where d is the grating spacing of the slits which is taken to be 20 pixels.

Algorithm 1 Plane slit generation1: for pixel(x,y) in patch do2: if |mod(kx · x+ ky · y,2π)-π| ≤ π/2 then3: pattern(x,y) = 14: else5: pattern(x,y) = 0

Extracting the phases, as explained in section 5.5.2, gives the phasemap presentedin figure 5.10. Trivially, the measured values are relative phases modulo 2π.Interpolation reveals the relative phase of the rest of mirrors in the ROI. However,interpolating between values of 0 rad → 0 rad or 0 rad → 2π rad yield differentvalues even though the points are identical. To circumvent this, the phasemapis unwrapped first. The unwrapping method will be explained in the followingsection.

5.5.4. Unwrapping and interpolating the phasemapUnwrapping the phase is essentially adding or subtracting multiples of 2π. Asimple approach has been taken in this project: First a phase value of the centerpixel is found by calculating the mean of the nearest neighbouring pixels. Thephasemap’s center row is then unwrapped along the x-axis (width) from the centerand outwards. For each pair of neighbouring sampling points the slope is calculated.Is it greater than a threshold, a multiple of 2π is added. If it is smaller than thethreshold with opposite sign, a multiple of 2π is subtracted. Finally, the sameprocedure is carried out for all columns along the y-axis (height), starting from thecenter. The method is shown in figure 5.11, where the red stars indicate measured

50

Width [pixels]

40 50 60 70 80 90

He

igh

t [p

ixe

ls]

10

20

30

40

50

60

Relative phase over the DMD

∆ φ

[R

ad

]

1

2

3

4

5

6

Figure 5.10.: Shows the extracted phases arranged in a phasemap. The dimensionsof the phasemap corresponds to 1/15 of the dimensions of the DMD.

phases, which are clearly modulo 2π. The blue circles indicate the unwrappedphases, the green dashed line indicates the center pixel and the black dashed linesindicate a multiple of 2π. To retrieve ∆φ for the rest of the mirrors in the ROI, thephasemap is linearly interpolated. The interpolation is chosen in such a way thatit compensates for the oversampling. The 3 rows in the middle of the phasemapsare first interpolated from the center and outwards. After that, the columns areinterpolated from the center row and outwards. A unwrapped and interpolatedversion of the phasemap presented in figure 5.12. Due to the interpolation, there isa 1 : 1 correspondence between the dimensions of the phasemap and the dimensionsof the DMD.Aberrations, which were introduced in section 4.1.3 as phase shifts due to imper-fections in an imaging system, are contained in the phasemap. By fitting Zernikepolynomials to the phasemap, one can thus characterize the imperfections in theimaging system. It is seen that the overall peak-valley (PV) value of the phasemapis ∼ 30 rad which roughly corresponds to 5λ. This a factor 20 higher than the λ/4that is required for a diffraction limited system. In the next section it is explainedhow the overall flatness of the phasemap can be increased.

5.5.5. Corrected phasemapsBy adding the relative phase from a phasemap, ∆φ(x,y), to the pixel at (x,y), onecompensates for aberrations and the overall flatness of the corresponding phasemapis increased. Algorithm 2 shows how phases are added.Adding the phase to a new set of patterns with corrected slits and obtaining the

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Width [pixels]

30 40 50 60 70 80 90

∆φ

[ra

d]

0

5

10

15

20

25

30

Unwrapping of the phase ∆φ

Measured phase

Unwrapped phase

Center position

Height [pixels]

10 20 30 40 50 60

∆φ

[ra

d]

0

5

10

15

20

25

30

Unwrapping of the phase ∆φ

Measured phase

Unwrapped phase

Center position

Figure 5.11.: Shows the principle in the unwrapping procedure. To the left; alongthe x-axis through the center. To the right; along the y-axis throughthe center. In each subfigure the green dashed line indicates the centerposition of the phasemap. The black horizontal dashed lines indicatea multiple of 2π. The red points are the measured phases, ∆φ, whichare clearly modulo 2π. The blue circles are the same phases afterunwrapping.

Figure 5.12.: Shows the unwrapped and interpolated phasemap. Each pixel in thephasemap represents the relative phase, ∆φ, of the correspondingDMD-mirror.

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Algorithm 2 Corrected slit generation1: for pixel(x,y) in patch do2: if |mod(kx · x+ ky · y + ∆φ(x,y),2π)-π| ≤ π/2 then3: pattern(x,y) = 14: else5: pattern(x,y) = 0

Figure 5.13.: Shows three different phasemaps. From left to right there is anuncorrected phasemap, a phasemap corrected once and a phasemapcorrected twice. It is clearly seen that slopes and tendencies areremoved by correcting the phasemaps so that the overall flatness isincreased.

corresponding phasemap can be carried out iteratively to increase the flatness of thephasemap. This is seen in figure 5.13, where three phasemaps are presented. Thefirst phasemap is uncorrected while the two other phasemaps have been correctedonce and twice, respectively. It is noted that the flatness is increased roughly bya factor of 20. This is a clear sign that the wave front has been corrected by theDMD. It is noted that the flatness of the phasemap corrected twice is approximately0.6 rad which amounts to 0.095λ.

5.5.6. Fast phasemapsFrom the previous sections it is evident that multiple phasemaps have to be takenin order to increase the flatness or if one wants to use the phasemap as a tool forcharacterizing the imaging system. The DLP6500 is controlled via a graphical userinterface (GUI) developed by Texas Instruments on a computer running Windows.Using the device in Pattern On-The-Fly mode in our case requires 44 packets ofpatterns to be uploaded manually and displayed on the DMD.This motivates for developing a fast and automatic method for obtaining phasemaps.Since no application programming interface (API) was provided to control theDLP6500, the first attempt to automate the phasemapping process was to develop a

53

bot that, by controlling the mouse and keyboard of the computer, deterministicallycould carry out the sequence. No speed improvement would be gained by a bot,since it would basically mimic a person operating the computer, but the intentionwas to let it run over night. The bot was written in Matlab where features tosimulate key strokes and mouse interaction were imported from the Java RobotClass. Even though the bot never ended up being completely robust, it was able totake a phasemap in roughly 7 to 8 hours. The unpredictable nature of the Windowsoperative system was the reason for this lack of robustness: For the bot to work, itwas extremely important that the right program was in focus at the right timesduring the sequence. The focus could be ‘stolen’ by pop-ups telling the user toinstall Windows updates, checking that the version of Windows used was licensedand so on. During the phasemapping sequence, it was discovered, that the uEyecamera, which was triggered by the DMD, was dropping frames, i.e. even though400 images were displayed on the DMD, less than 400 images were captured. Thenumber of dropped frames seemed to be correlated with the pattern rate of theDMD, which was then decreased. The bot was programmed to check if all 400images had been captured. If not it would replay the sequence instead of uploading400 new images. The duration of 7 to 8 hours is roughly on the same time scale asmentioned in [31].

At this point, it was not clear if the frames were dropped due to a faulty camera,due to a faulty trigger signal from the DLP6500 or due to an unknown factor. Thecamera was replaced, but frames were still dropped, which could indicate a faultytrigger signal.

A workaround was thought of: Instead of using the DLP6500 in the Pattern On-The-Fly mode, it would be used in the video mode connected to an external videosource that could also provide a trigger signal for the camera. The good thingabout using the video mode is that the patterns no longer have to be uploaded 400at the time but instead have to be streamed as a video. This greatly reduces theneed for human interference during the phasemapping sequence. A disadvantage ofthis operational mode is a low frame rate which is limited to maximum 60Hz.

As a video source a Raspberry Pi 3B (RPI) was chosen. A Raspberry Pi is a cheapmicro processor and the chosen model supports 1080p video streaming througha HDMI-port. In addition to that, it has a 40-pin general purpose input/output(GPIO) header and can be controlled by a Linux based operating system (OS). Inour case a Linux distribution named Raspbian (Jessie) is used.

In the following section, it will be explained how the RPI was hacked and used inthe setup.

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5.5.7. Raspberry Pi HackThe overall goal for the RPI was to use it as a video source that can stream a videodirectly onto the DMD. For each frame uploaded to the DMD, a trigger signalshould be produced by the GPIO for triggering the camera. The GPIOs are able tooutput 3.3V.First, the RPI was connected to the network via an ethernet cable to a TP-Linkrouter. The computer used to operate the RPI was connected to the same router.For controlling the RPI from the computer, a Secure Shell (SSH) remote sessionwas initialized from the computer via the client program PuTTY. To limit thedata transfer over the SSH connection, the network drive of the HiRes group wasmounted on the RPI.No software solution for video streaming while producing trigger signals via theGPIOs exists, so the first approach was to write a program with those features inC++. For the video processing part, features from the Open Source Computer Visionlibrary (OpenCV) were imported, while the GPIO-control features were importedfrom a library called WiringPi (WPI). A quick test showed that a central processingunit-optimized (CPU-optimized) C++-program was only able to stream a video witha frame rate of roughly 0.5Hz. This would make the duration of the phasemappingsequence even longer than that of the bot.The approach was then changed to modify an already existing video player namedOMXplayer (OMXP) to include the needed features. The OMXP was chosenbecause it is an open source video player that has been hardware optimized tothe graphical processing unit (GPU) of the RPI[8]. Since the GPU of the RPIdoes not have its own RAM, a part of the system memory had to be allocated forthe GPU. The maximum amount, which is 512MB, was allocated to achieve thehighest possible frame rate.The source code of the OMXP had to be modified in order to produce the triggersignal (still generated by features in the WPI). The entire OMXP consists of morethan 17 k lines of C++ code divided into multiple files. No documentation forthe source code exists and only a few comments are left in the code, hence themodification process was somewhat influenced by trial an error.After consulting an online Raspberry Pi Forum, it was pointed out that it probablywould be difficult if not impossible to hack the OMXP in this particular way,since the OMXP does not have control over the frame rendering [6, 7]. Exampleswere also given on which functions to modify, if the hack was possible. Afterfailed attempts, a function was discovered that presumably deletes frames from theOMXP’s buffer. The function is declared asOMX_ERRORTYPE COMXCoreComponent::DecoderEmptyBufferDone and is located

55

in file OMXCore.cpp around line 1625.By connecting the RPI via HDMI to a regular monitor and playing videos atdifferent frame rates while having modified the found function in such a way thateach function call prints a message to the a Linux terminal, it was deduced thateach function call temporally coincides with a frame being rendered. The C++ codefor the trigger pulse was then added to the end of the function. The code snippetof the trigger pulse is given below:

1 // I n i t i a l i z i n g the GPIO−c on t r o l2 // Remember to d e f i n e bu f f e rCa l l I ndex in the top o f the f i l e3

4 i f ( bufferCallIndex>buffSize )5 6 // Generating the t r i g g e r pu l s e7 pinMode (8 , OUTPUT ) ;8 digitalWrite (8 , LOW ) ; // On9 delay ( timeDelay ) ;

10 digitalWrite (8 , HIGH ) ; // Off11 printf ( " Buf f IndexAfter5 = %d \n" , bufferCallIndex−5) ;12 13 bufferCallIndex++;

The variable buffCallIndex counts the number of function calls. Since testsshowed that the function was called a few times in the beginning of each videosequence without any frames being rendered on the DMD, the trigger pulse isfirst generated after the function has been called a number of times defined bybuffSize. For our setup buffSize = 4 was used. Line 7 activates pin 8 on theRPI for outputs, which was used for the trigger signal. Lines 8 and 10 generatethe trigger pulse of a duration of timeDelay which is given in milliseconds. Thenumber of function calls is then printed to the terminal and the buffCallIndex isadvanced by one. In appendix A, a more detailed step-by-step guide to hackingthe RPI is presented.To investigate the produced trigger signal, the test setup in figure 5.14 was built:The hacked RPI was connected via HDMI to a DLP6500 which was operated in theVideo mode and locked to the RPI video source. A laser source was continuouslyilluminating the DMD and the reflected beam was picked up by a photo diode(PDA36A-EC, Thorlabs) connected to an oscilloscope (DSO-1062D, Voltcraft).The trigger signal from the RPI was also connected to the oscilloscope. A videoconsisting of alternating white and black frames with a frame rate of 5Hz (equalto an exposure time of 200ms) was played on the RPI (and showed on the DMD)

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Photo diode

DMD

Figure 5.14.: Shows the test setup for investigating the trigger signal. The DMD iscontinuously illuminated. A Raspberry Pi (not shown) is connectedto the DMD via a HDMI cable and used to show a video consistingof alternately white and black frames while producing a trigger signal.The reflected beams by the white frames are picked up by a photodiode. The trigger signal and the signal from the photo diode areviewed on an oscilloscope (not shown).

while the duration of the trigger signal was set to 100ms. Only the white framesreflect light to the photo diode. The resulting signals from the DMD and thetrigger are seen on the screen shot of the oscilloscope presented in figure 5.15. TheDMD signal is blue and the trigger signal is yellow. It is clearly seen that theduration of the DMD signal is twice that of the trigger signal and that the twosignals are locked to each other, in this case in such a way that a falling edgeof the trigger signal lies in middle of the exposure time of a frame on the DMD.This was expected from the choice of frame rate and the duration of the triggerpulse. It is also noted that the DMD signal is noisy. The noise was not investigatedfurther in this setup, but a possible explanation could be an undersampling of the10 kHz switching signal explained previously. Further tests of the RPI hack showthat the one-to-one correspondence between deleting frames from the buffer andrendering frames on the DMD is not valid for videos with high bit rates. The bitrate increases with increasing complexity of the individual frames. This puts arestriction on the maximum size of the patches as well as the frame rate of thevideo. If the patch size is increased, the complexity of each frame increases.Taking actual phasemaps with the RPI hack requires that the 17520 patterns aremerged into a single video. Merging them into a video with a frame rate of 30Hzgives a bit rate of 8.67Mb/s for for both uncorrected and corrected slits, whichwas found to be low enough for the one-to-one correspondence between deletingframes from the buffer and rendering frames on the DMD to hold. The duration ofthese ‘patch videos’ are 10 minutes.The RPI hack was tested for phasemapping in the same setup used for testing the

57

Figure 5.15.: Shows the trigger signal from the Raspberry Pi (yellow) and the noisysignal from the DMD (blue).

Matlab bot. The uEye camera was now controlled by the uEye Cockpit whichis a part of the IDS Software suite. The Cockpit can change the settings of theuEye as well as capturing the data from the camera. The settings needed for thecamera depends on the specifications of the computer used. The overall goal forthe configurations of the settings is to increase the frame rate of the camera sothat it matches that of the patch video. This can be done in two ways: First thepixel clock of the camera, which basically determines the read out speed of thesensor cells in the camera[1], can be increased. Setting the pixel clock too high willhowever result in dropped frames since the data transmission from the camera willexceed the bandwidth of the data connection to the computer. The second way isto decrease the size of the camera chip that is read out.A combination of the two ways was chosen and the trigger of the camera was set toactivate on falling edges. After playing a patch video, it was discovered that thetrigger signal was too weak to activate the camera. The solution was to amplifythe trigger signal to 6.5V by an electronic circuit based on an ADG451B integratedcircuit, as seen in figure 5.16. In the figure, RPI represents the trigger signal fromthe Raspberry Pi, S represents the source voltage which also corresponds to thevoltage of the output trigger, denoted T.The camera was then triggered during a phasemapping sequence, but frames werestill dropped. Different settings of the uEye were tried with no change in theoutcome. Finally, the uEye-driver was updated on the computer, which solved theproblem of frames being dropped.

58

RPI

S

T

Figure 5.16.: Shows the amplifying circuit. RPI represents the trigger signal fromthe Raspberry Pi, S represents the voltage source, T represents theamplified trigger signal. The heart of the circuit is an ADG451Bintegrated circuit. A 301 W resistor and a 10 nF capacitor are alsoused.

The duration of a phasemapping sequence corresponds trivially to the duration ofthe patch video, which in our case was 10 minutes. This is a roughly a factor of 48faster than taking a phasemap manually or by using the Matlab bot. Its hard tocompare directly, since the bot was only used with a very low frame frame rate dueto the dropping of frames, but assuming optimistically that each of the 44 packetsof 400 patterns is uploaded to the DLP6500 in 20 seconds gives a total uploadingtime of approximately 15 minutes, which is a factor 3/2 slower than RPI hack.

5.5.8. C++ software for generating patches and extractingphases

Originally, Matlab scripts were used to generate the patch video and extractingthe relative phases. The scripts were not optimized and it took in the order of 1.5hours to produce a patch video and 30 minutes to extract the phases. The time itwould take to generate three corrected phasemaps (figure 5.13) would then be inthe order of 6.5 hours. To reduce the computational time and speed up the process,the scripts were rewritten in C++. In the following the two C++-programs will beexplained. Imaging processing constitutes a substantial part of both programs. Forthis routines from the OpenCV were imported.

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Patches

This program outputs a video where each frame is a patch (figure 5.5b). As input,it takes a text file containing the input parameters listed in the following form andorder:

• numOfPatches=17520• patchSize=45• slitSize=10• dimX=1920• dimY=1080• epsilon=0.000001• fps=30;• buffSize=10;• LUT=/path/LUTname.txt• thePhasemap=path/phasemapName.txt• outputVideoName=/path/videoName.avi

The five first points specify the total number of patches, the size of these patches,the slit size (where the grating period is twice the slit size) and the dimension ofthe DMD. Due to rounding errors in the routine that rounds the pixel value toeither black or white, the small offset, epsilon is required. The frame rate andthe buffer size then has to be given. The buffer size determines how many blackpadding frames to be added in the beginning and end of the patch video. Thereason for this is that the OMXP performs some irregular function calls to themodified function in the beginning and end of a video sequence that can activatethe uEye trigger. The last three entries give the paths to an already generatedlook up table, LUT, to a phasemap and the name of the output video. The LUTcontains the positions of all the sample patches. The phasemap is for corrected slits(see algorithm 2); alternatively the entry can be "none" if no correction is wanted.First, all the input parameters are saved as variables. The LUT is loaded as alist and the phasemap is loaded as a matrix. An object for writing the outputvideo is then initialized. To this object the output video name, the frame rate,frame size (DMD size) and the video CODEC are passed. The CODEC tells thevideo writing object how to encode and decode the data stream of generated video.Here the CODEC given by the FOURCC name (four-character code) MJPG is used.Other CODECs were tried out, but it was found that they produced artifacts inthe frames.The black padding frames are then added to the video.To save computational time, a template frame, containing nothing but the referencepatch, is then made.

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In each iteration of a for loop that runs over the LUT, a sample patch is generatedin the template frame. The frame is then written to the video and the samplepatch is deleted from the template frame again.The sample patch is generated by algorithm 3, where epsilon as well the parametershift have been added to the binary algorithm.

Algorithm 3 Corrected slit generation1: for pixel(x,y) in patch do2: if |mod(kx · (x+ epsilon + shift) + ky · (y+ epsilon) + ∆φ(x,y),2π)-π|≤ π/2 then

3: pattern(x,y) = 14: else5: pattern(x,y) = 0

The shift determines the phase shift of the grating of the sample patch and iscalculated as

shift = mod (i, n) · 2 · slitSizen

(5.5)

where i is the current iteration of the for loop over the LUT and n is the numberof times the grating should be shifted, which is calculated automatically from theLUT. If the thePhasemap is "none", then ∆φ(x,y) = 0.If a phasemap on the other hand is passed to the program, the phasemap is firstextrapolated in order to get a value for the relative phases of the mirrors in the outerpart of the ROI defined on the DMD. The extrapolation is done in a simple way asdepicted in figure 5.17: First the phasemap is extrapolated along the x-dimension(red arrows) from the calculated slope between the two outer pixels on each side ofthe 101 rows in the middle of the phasemap. The phasemap is then extrapolatedalong the y-dimension (green arrows) from the calculated slope between the twoouter pixels in each column.Finally, black padding frames are added to the video again, and the video is saved.It takes less than 10 minutes to produce a patch video, which is a speedup of morethat a factor of 9.

Phase extraction

The program that extracts the phases also takes a text file as input. The file hasto have the following form and order:

• dimX=1920• dimY=1080

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Figure 5.17.: Shows the idea behind extrapolating the phasemap. The circle repre-sents the phasemap. The red arrows represent extrapolation in the±x-direction, while the green arrows represent the extrapolation inthe ±y-direction.

• phasemapInputVideo=/path/phasemapVideo.avi• LUT=/path/LUTname.txt• outputPath=/path/• pathToROI=/path/ROI.txt• sizeOfROI=3• slitSize=10;

Here the two first entries specify the dimensions of the DMD. The third entryis the interference ‘phasemap’ video from which the phases has to be extracted.The LUT and output path then have to be specified. The sixth entry is a textfile containing the coordinates of the virtual pinhole (defined in section 5.5.2). Ifno pinhole is already chosen, "none" can be passed and the program will find theoptimal position for the pinhole, and write it to a text file named ROI.txt. Thetwo last entries are the size of the pinhole in pixels (given as size× size) and theslit size.First the input parameters are loaded into variables, the LUT is loaded into a listand the phasemap video is loaded into a video object.The frames containing the interference signal have to be separated from the blackpadding frames in the video. This is done by analyzing the video frame by framefirst from the beginning moving forward and then from the end moving backwards.For each frame the total signal is calculated, and when an increase in the signalabove a defined threshold is observed, the frame index is taken as the start/end ofthe video. The program calculates and compares the number of frames containingthe interference signal to the expected number of frames, which is found from theLUT. If they do not match, a warning is printed.

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If no ROI was given in the input file, a position of the virtual pinhole is found.This is done by calculating the linesum along the x- and y-dimension. The x- andy-coordinate are taken as the index in the respective sums with highest value.The mean intensities within the pinhole of the first five frames containing theinterference signal are calculated and the relative phase is extracted by fittingequation 5.3 to these intensities. The phase is attributed to the a DMD mirror.The mirror position is given by the LUT. The next five frames are analyzed in thesame way and so on, until all the phases are extracted and a phasemap is formed.For fitting, routines from the Gnu Scientific Library are imported.The phasemap is then unwrapped and interpolated as explained in section 5.5.4. Fi-nally the phasemap before unwrapping and interpolation (figure 5.10), the phasemapafter unwrapping but before interpolation and the phasemap after unwrapping andinterpolation (figure 5.12) are written as matrices to separate text files as well asthe position of the pinhole.For the program, it takes in the order of 1 to 2 minutes to generate a phasemap,which is more than a factor of 10 faster compared to the Matlab script.

5.5.9. StabilityThe new tools described in the previous sections open up for new types of measure-ments. In this section an investigation of stability of the DMD is presented.The stability determines for how long a time a phasemap actually describes therelative phases over the DMD, and previous work has shown that mechanicalinstability can be an issue for some DMD models [31].First, a phasemap was taken. The slit size of the uncorrected patches used is10Pixels and the wavelength of the light is 780 nm. Based on the phasemap, apatch video with phase corrected patches was generated. Over the span of 6days, ∼ 70 corrected phasemaps were taken. The first and the last phasemap arepresented in figure 5.18. It is noted, that the phasemaps are not identical but theoverall tendency seems to be the same.Since the phasemaps have been corrected, the validity of the phasemap is related tothe flatness. A clear slope is seen in figure 5.18, but since slopes and pistons do notaffect the quality of an imaging system, Zernike polynomials are fitted to the 70phasemaps and the terms corresponding to slopes and pistons are subtracted. Thepeak-vally value (PV) as well as the root-mean-square value (RMS) are calculated.The results are presented in figure 5.19, where the PV and RMS are measured inunits of the wavelengths, λ. The drift over the 6 days is in the order of 0.02λ forthe RMS and 0.09λ for the PV. The PV values are in general higher than the RMSvalues since they are more sensitive to potential bad fits, which can give rise to

63

(a) (b)

Figure 5.18.: Shows two phasemaps separated in time by 6 days. (a) is the firstphasemap in the stability test and (b) the last.

‘hot pixel’-like effects in the phasemap. A ‘hot pixel’ can be seen in figure 5.18baround coordinate (1300,700). The important thing to note from this test is thatthe flatness of the corrected phasemap does not exceed λ/4 which is the criteria fora diffraction limited system.

5.5.10. Amplitude mapIt is necessary to know the illumination profile over the DMD for using it to generateholograms, as it will be explained in detail in the next section. An amplitudemap(or intensitymap) is obtained by using a technique similar to the one described insection 5.5.2: A sample patch is shifted to different positions on the DMD. Noreference patch is present and the phase of the grating of the sample patch is notshifted. For each position, (x,y), of the patch, the total signal collected by thecamera is calculated and attributed to the mirror in the center of the patch, U(x,y).The oversampling is the same as for the phasemapping sequence and in order toavoid extrapolation, no ROI is defined on the DMD. To take the amplitudemap,a video is generated where each frame contains the sample patch. As previously,the frame rate is 30Hz, the patch size is 45Pixels, the slit size is 10Pixels and theoversampling is 15Pixels. The video is streamed via the hacked RPI, and the signalis captured by the uEye.After calculating the total intensity in all frames, the amplitude map is linearlyinterpolated. An amplitudemap is seen in figure 5.20.The reason for not extracting the amplitude map directly from the amplitude ofthe fit used for extracting the relative phases (equation 5.3) is the oversampling of

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Date

22/06-16 23/06-16 24/06-16 25/06-16 26/06-16 27/06-16

RM

S [λ

]

0

0.005

0.01

0.015

0.02

0.025

0.03

Root Mean Square corrected by Zernike polynomials

Date

22/06-16 23/06-16 24/06-16 25/06-16 26/06-16 27/06-16

PV

[λ

]

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Peak-Valley corrected by Zernike polynomials

Figure 5.19.: Shows the root mean square value and peak-valley value of thephasemaps as a function of the date, at which the phasemap wastaken. Zernike polynomials have been fitted to the phasemaps andthe orders representing slopes and pistons have been subtracted.

the sample patches during the phasemapping sequence. Due to the oversampling,the sample patch and the reference patch will spatially overlap in the middle of theROI at some point, which will result in less light being reflected in the directionof the camera. This will generate a dip in intensity in the middle of the intensitymap, which is difficult to compensate for.

5.5.11. HologramsIn this section the working principle of the Fourier DMDs and how arbitrary wavefronts are generated based on concepts introduced in the previous chapters will bedescribedin in line with [31].As mentioned previously, the DMD has to model the Fourier transform (FT) ofthe wanted output signal, F . First the Fourier transform, H, is calculated fromF . H has to have the same dimensionality as the part of the DMD used and issplit into an absolute part, Ut(x,y), and a phase part, φ(x,y), where (x,y) is thecoordinate of the DMD pixels. The amplitude can only be attenuated by the DMD.The attenuation of a DMD pixels is given as the ratio between the intensity andthe absolute part of the FT:

a′(x,y) = U(x,y)Ut(x,y) . (5.6)

Limiting for the system is the minimum value of a′ which is denoted a′l. As presentedin section 5.5.1, the amplitude is modulated by changing the width, a(x,y), of the

65

Figure 5.20.: Shows the amplitudemap over the entire DMD

slits. The width of the slits is scaled as

a(x,y) = ala′(x,y) (5.7)

which trivially is ≤ 1. The width of the slits on the DMD, w(x,y), is taken to bein the interval [0, π], and can thus be calculated as

w(x,y) = πa(x,y) (5.8)

The binary algorithm that takes φ and the varying widths into account, is given byalgorithm 4, where pattern still is the binary image to be uploaded on the DMD.

Algorithm 4 Corrected slit generation1: for pixel(x,y) in patch do2: if |mod(kx · x+ ky · y + ∆φ(x,y) + φ(x,y),2π)-π| ≤ w(x,y)/2 then3: pattern(x,y) = 14: else5: pattern(x,y) = 0

A few examples of different artistic holograms are given in figure 5.21, where boththe actual DMD configuration and the corresponding hologram is showed. Eventhough the DMD is a binary device, a greyscale is seen in the Mona Lisa-hologram.Holograms of Gaussian dimple arrays of different size are presented in figure 5.22,where the input image, the DMD configuration and the corresponding holograms

66

Figure 5.21.: The left column contains the configuration of the DMD-mirros ascalculated by algorithm 4. Here the black pixels represent the mirrorsin the on-state and white pixels represent the mirrors in the off-state.The column to the right shows the corresponding holograms. It isclearly seen, especially in the Mona Lisa hologram, that the image isnot binary but has a gray scale. The colors have been inverted in thebottom hologram.

67

are shown in the three columns. From the column in the middle it is clearly seen,that the more complex the input image is, the fewer mirrors are actually used whichcorresponds to using a small fraction of the input power.

68

Figure 5.22.: Shows holograms of different dimple arrays. The rows show a 2× 2,a 5 × 5 and a 10 × 10 dimple array. In the column to the left theMatlab-produced input images are shown. The column in the middleshows the calculated configuration of the DMD mirrors. The columnto the right shows the corresponding holograms where the contrasthas been increased to make the dimples visible. The colors have beeninverted and from the column in the middle it is seen that the morecomplex the input image is, the fewer mirrors are in the off-state.

69

6 Alignment of the high resolution board

The high resolution board (HiRes board) contains all the optics and DMDs neededfor high resolution imaging, addressing and manipulating single atoms and thegeneration of arbitrary potentials.The board can be subdivided into the following modules:

• A 787.5 nm arm, which is responsible for addressing single atoms in line withthe theory presented in section 3.3. The arm contains a DMD in a direct-and in a Fourier configuration.

• A 940 nm arm, which is responsible for generating arbitrary potential land-scapes which are to be projected onto the optically trapped atoms. This armalso contains a DMD in a direct- and in a Fourier configuration.

• A fluorescence arm, which is used for high resolution imaging of the atoms,that are fluorescing at 780 nm.

• A molasses arm, which is used to induce optical molasses in the opticallytrapped atoms

• A pilot beam, which is used for aligning the entire HiRes board with respectto the science chamber.

A CAD drawing of the HiRes board is presented in figure 6.1. Due to aestheticconsiderations, only the rough position of the different DMD arms and the HiResobjective are indicated. The board consists of a top par and bottom part. The tiltand height of the top part relative to the bottom part are adjusted by micrometertranslation screws.In the following chapter, it will be explained how the different modules presentedabove were designed and aligned on the top part of the board.

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780nmoverviewmaging system

780nm high resolutionimaging system

pilot beam

787nm Direct DMD940nm Fourier DMD

940nm Direct DMD

787nm Fourier DMD Top Part

BottomPart

HiResObjective

Figure 6.1.: Shows the CAD drawing of the HiRes board. The rough position ofthe DMD arms are shown. The figure is obtained and modified from[37].

6.1. TowerFor the alignment process, reference beams at 787.5 nm, 780 nm and 940 nm wereneeded to define the optical axes of the setup. The beam diameters were chosen tobe 8mm, which compares to the 7mm aperture of the HiRes objective. This ensuresthat the system could be aligned without the risk of clipping on the aperture ofthe objective afterwards. For the reference beams a vertical Thorlabs cage system(tower) was built. The tower is presented in figure 6.2, where the green arrowmarks the outcoupler for the reference beams and the blue arrows mark the mirrorsused for aligning the beams onto the center of the tower by two irises (red arrows).An iris was mounted onto the optical axis of the objective by mounting it on thePiFoc (yellow arrow), which will be used to position the objective with nanometerprecision with respect to the atoms. By looking at the signal with a uEye camera(IDS) while closing the irises, the tower was aligned onto the optical axis of theobjective.

6.2. Cage systemsAfter the alignment of the reference beam, the foundations of the four arms werebuilt. A silver coated mirror (Thorlabs) was placed just under the PiFoc (denoted

71

Figure 6.2.: Shows the vertical tower. The outcoupler (green arrow) and the twomirrors (blue arrows) are seen in the top. The three irises (red arrows)are also seen.

72

shooting-up mirror), to translate the reference beam to the horizontal plane.After the shooting-up mirror, a 940 nm dichroic mirror (Custom made, Laseroptik)was placed in order to overlap the 940 nm arm with the other arms.After another mirror, a beam sampler (BS) was placed for overlapping the 787.5 nmarm with the fluorescence- and molasses arm. As a first choice, a non-wedged BSwas chosen (Custom made, Lens-optics). It was found that internal reflectionsinside the BS caused interference and a wedged BS (Thorlabs) was chosen instead.A wedge only introduces a shift in position of the final image and does thus notinfluence the imaging quality. After the BS, in the fluorescence/molasses path, amechanical flip mirror was installed.The molasses arm and fluorescence arms were also overlapped via a wedged BS. Thetwo beam samplers are placed so that they compensate for each other’s potentialshift of the beam due to internal reflection for the 780 nm imaging beam.The setup is shown schematically in 6.3, where the mirror closest to the sciencechamber is the shoot-up mirror. An explanation of the symbols is given in figure6.4. The drawback of using a BS for overlapping the 787.5 nm arm is, that only 4%of the 787.5 nm light is reflected towards the science chamber. Since no dichroicmirror for separating 787.5 nm and 780 nm exists, a beam sampler was the onlysolution.A cage system for the fluorescence arm was then installed. The reference beam wasaligned to the cage system with two centered irises and by adjusting the shoot-upmirror as well as the mirror after the 940 nm dichroic mirror. The fluorescence armis seen in figure 6.5. A f = 750 nm achromatic lens (AC508-750-B, Thorlabs) isused for focusing the beam onto a camera(Andor, iXon 897).After this, the cage system for the 787.5 nm arm was built in. It was first alignedonto fixed irises mounted in the cage system by making the clipping and diffractiongenerated by closing the two irises as equal as possible. Due to a shortage of spaceon the HiRes board, only one silver coated mirror (Thorlabs) was used for adjustingthe beam, as seen on figure 6.3. After the mirror, a flip mirror was placed, whichallows imaging of the intermediate plane onto a uEye camera. The flip mirror isfollowed by a quarter-wave plate, a half-wave plate and another quarter-wave plate,which enables complete control over the polarization of the light. This is neededin order to create perfectly circular polarized light to manipulate the mF -statesof atoms as explained in section 3.3. After the wave plates a non-polarizing beamsplitter (CCM1-BS014, Thorlabs) for separating the arm into two paths was placed;one path was used for a direct DMD and one path was used for a Fourier DMD. Anon-polarizing beam splitter was used, since circular polarized 787.5 nm light willbe used in the HiRes experiment.

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To the Science Chamber

FluorescenceArm

787.5nm Arm

940nm Arm

Molasses Arm

(a) (b)

Figure 6.3.: (a) Shows the part of the HiRes board immediately after the PiFoc. Thesymbols are explained in figure 6.4. The mirror closest to the sciencechamber is the shoot-up mirror. The arrows indicate the propagationdirection of the light, when the board is finished and in use. (b) Showsthe PiFoc with the mounted iris, the 940 nm dichroic mirror and thefirst silver coated mirror. The tower has not yet been placed.

DMD

Fiber Coupler

PD Photo Diode

Mirror

PBS

Lens

Beam Sampler

λ/4 Wave plate

λ/2 Wave plate

—

— Achromat

Beam Splitter

Flip Mirror

CAM Camera

Iris

Dichroic

Figure 6.4.: Explains the different symbols in the schematic drawings of the HiResboard.

CAM

—

—f = 750mm

From thescience chamber

Figure 6.5.: Shows the 780 nm fluorescence-arm of the HiRes board.

74

Table 6.1.: List of DMDs on the HiRes board

787.5 nm DirectFourier

940 nm DirectFourier

Reference

Beamz

x

Figure 6.6.: Shows a how the reference beam is hitting the DMD after the cagesystem and is reflected onto a uEye camera.

The same approach was used for installing the 940 nm arm: Fixed irises weremounted in the cage system and the 940 nm dichroic mirror was used to adjust thebeam. No wave plates were placed in this arm and a polarized beam splitter (PBS)was used to separate the arm into the direct- and Fourier arm. A PBS was used,since only linearly polarized 940 nm light will be used in experiments.

6.3. Alignment of the DMDsThe next step was to place in the four DMDs listed in table 6.1. Two are placed ineach of the 787.5 nm- and 940 nm arm.First the direct 787.5 nm DMD was installed. The DMD was placed within in thebeam path of the reference beam. A uEye was placed in the direction of the zerothdiffraction order from the DMD and a small circle with r = 10DMD Pixels wasuploaded to the DMD. A diffraction pattern was seen as the circle clipped thebeam. Another diffraction pattern was seen, when one of the irises was closed. Byshifting the position of the entire DMD along the x-dimension, the two diffractionpatterns were overlapped as much as possible. This is shown in figure 6.6, where thereference beam comes from the cage system described above. The final overlappingwas done by shifting the position of the circle. The offset position of the circle wasnoted.

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The approach for aligning the Fourier DMD in the Fourier arm was the same asthat of the direct DMD.The 940 nm direct DMD and the Fourier DMD were installed in the same way.

6.4. Laser sourcesFor each DMD, individual laser sources are needed. First, the sources for the787.5 nm direct DMD was installed:An outcoupler for 787.5 nm light with a diameter of 14.76mm (denoted "787.5 nmbig") was placed next to the DMD. After the outcoupler a half-wave plate and apolarized beam splitter (PBS) were placed. The combination of the half-wave plateand the cube is for defining a clear polarization. Next, two mirrors were placedwith a beam sampler between them. The BS was placed in such a way that itdirects a part of the beam onto a photo diode (PDA36A-EC, Thorlabs), whichwill be used for monitoring beam intensity. This is seen in figure 6.7. The mirrorclosest to the DMD was placed roughly where the zeroth order was detected withthe uEye. The beam was aligned with respect to the reference beam by placinga uEye camera between the outcoupler and the first iris in the tower such that it‘faced’ the DMD. A circle with r = 50DMD Pixels was uploaded on the DMD, andthe resulting beam reflected towards the tower was aligned onto the two irises inthe tower, by looking at the diffraction pattern as they were closed.As earlier described, the mirrors of the DMD tilts ±12°. The above describedlaser source was aligned to the +12° state of the mirrors. Another outcoupler wasadded to the setup, which is denoted "the 787.5 nm small". This beam was alignedwith respect to the −12° state of the mirrors. Every working principle for thisconfiguration is the same as before, except that the "on/off"-state of the mirrors isreversed. The diameter of this beam is 2.5mm, and the idea is to use it for fewerdimples with higher power. The alignment procedure was the same as for the otheroutcoupler.The outcoupler providing a laser source for the 787.5 nm Fourier DMD was placedand aligned in almost the same way. The difference was, however, that instead ofusing a regular circle on the DMD, a circle containing slits, generated by algorithm1 with a slit size of 10DMD Pixels which gives kx = ky = 2π/20, was used. TheDMD was aligned to the first diffraction order instead of the zeroth order. The787.5 nm arm is shown in figure 6.8. The procedure for aligning lenses will beexplained in section 6.5.The 940 nm arm is shown in figure 6.9. The overall aligning procedure has beenthe same as that for the 787.5 nm arm.

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PD

z

x

Towardsthe tower

Figure 6.7.: Shows the placement of the outcoupler. The beam propagates towardsthe tower, where it is aligned onto irises and hence the reference beam.

PD

f = 750mm

787.5nm

f = 500mm—

—

1:1 telescope, f=200mm

CAM

787.5nmDirect

PD

787.5nm

787.5nmFourier

PD

787.5nm small

To thescience chamber

Figure 6.8.: Shows the complete 787.5 nm arm of the HiRes board.

940nm

PD

940nmDirect

f=750mm

940nm small

—

—

f = 500mm

940nm Fourier

PD

1:1 telescope, f=200mm

CAM

PD

To thescience chamber

Figure 6.9.: Shows the complete 940 nm arm of the HiRes board.

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6.5. Lenses

Each of the 787.5 nm- and 940 nm-arms, contain a 750mm lens (LA1978-B, Thor-labs), an achromatic lens (AC254-500-B, Thorlabs) with f = 500mm (denotedf500A) and a 1:1 telescope consisting of two f = 200mm achromatic lenses (f200A)(AC254-200-B,Thorlabs).The 750mm lens was built in first. It is placed after the outcoupler and thealignment was carried out by placing an iris before and after the lens, closing theirises and moving the lens in plane normal to the propagation direction of the beamuntil it was aligned with respect to the optical axis defined by the reference beams.The irises were then removed. The lens focusses the light towards the DMD andinto the f500A lens.This is needed for minimizing the effect of the 10 kHz flickering of the DMD whichswitches the state of all the mirrors simultaneously. When all mirrors are turnedon, the DMD reflects the entire wavefront which will interfere with the signalgenerated by the pattern uploaded to the DMD. Placing the 750mm lens will makethe flickering signal diverge through the arm after the passing the focal point. Thisresults in only a weak signal at the plane of the atoms in the final setup. At thesame time, the imaging quality is increased due to an effect called isoplanatism[36].When using the DMD to manipulate the wave front, a smaller region of mirrors areused, which can be considered light sources. The reflected beam will be collimatedby the f500A lens.The f500A lens was placed after the direct DMD separated by f and its focal lengthwas chosen such that the magnification of the system, M = 5mm/500mm = 0.01,projects a super pixel of 7× 7 DMD mirrors onto lattice site, in the plane of thetrapped atoms.The first approach to align the f500A lens in the 787.5 nm arm was to place ina shear plate (Thorlabs) after the beamsampler that overlaps the 787.5 nm- andfluorescence arm and use shear plate interferometry to check the collimation. Theshear plate was mounted with a tilt of 45° and due to internal reflections withinthe shear plate, a pattern of interference fringes should be formed, which wasmonitored by a uEye camera. By placing in the f500A lens and uploading a circle ofr = 25DMD pixels the interference signal in figure 6.10 was seen. The idea was totranslate the f500A in the z-dimension until the fringes became completely straight.This would indicate a perfect collimation of the beam reflected by the DMD.Unfortunately, the change in the interference signal as the f500A was translatedwas too small to indicate anything.

78

Figure 6.10.: Shows the interference signal from the shear plate, as a circle ofr = 25DMD pixels was displayed on the DMD. The colors are invertedand the contrast is increased.

The approach was then changed to mapping out the aberrations, defocus included,in the 787.5 nm arm. This was done by building the Twyman-Green inspiredinterferometer (TGI) presented in figure 6.11a on a separate optical bread board.The light comes from an outcoupler which has been calibrated by analyzing thebeam with a commercial Shack-Hartmann wavefront sensor (Tholabs). The beamis split by a PBS into a test arm and a reference arm, where a silver coated λ/10mirror (Thorlabs) back-reflects the light onto a uEye camera. In each arm, aquarter-wave plate has been placed. Since the light passes through the plate twice,it acts as a half-wave plate and rotates the polarization by 90°. The quarter-waveplates were rotated in such a way that they maximized the interference signal. Thetwo mirrors in the test arm are used for overlapping the beam with the 787.5 nmarm via the BS. The direct DMD is used in "parked mode" which back-reflects thelight through the arm to the PBS in the interferometer, which overlaps it withthe signal from the reference arm onto the uEye. Spherical fringes in the resultinginterference signal correspond to a defocus and straight fringes correspond to a tilt[34].As a quick test, an interference signal with no lenses in the test arm and the beamback-reflected by the DMD was obtained. The interference signal is presented infigure 6.12a. To get a feeling for the aberrations induced by the DMD, a silvercoated λ/10 mirror (Thorlabs) was used to reflect the beam instead of the DMD.The resulting signal is seen in 6.12b. Fewer fringes are clearly seen indicating thatthe DMD itself is a source of aberrations.

79

CAM

ReferenceArm

TestArm

(a)

PD

f = 750mm

787.5nm

f = 500mm—

—

787.5nmDirect

CAM

ReferenceArm

MeasurementArm

(b)

Figure 6.11.: Shows the external (a) and the built in (b) Twyman-Green inspiredinterferometer. For simplicity, only the relevant parts of the arm isshown in (b)

(a) (b)

Figure 6.12.: Shows the interference signals without a lens from the Twyman-Greeninterferometer where the beam has been reflected by the DMD (a)and by a mirror (b). The colors have been inverted.

80

Figure 6.13.: Shows the interference pattern for the optimal position of the f500A.The colors have been inverted.

The mirror in front of the DMD was removed. The f500A lens was mounted on anxy-translation stage and translated along the propagation direction of the beamuntil the optimal position was found. It was then moved in the plane normal to thepropagation direction until it was aligned onto the irises in the cage system. Theinterferometric signal for the optimal position of the f500A is seen in figure 6.13.For placing the 1:1 telescope, a second TGI was built around the beam splitterused for overlapping the direct- and Fourier DMD. This is shown in figure 6.11b,where light was reflected into the system via the DMD and a uEye was placed inthe Fourier path. A silver coated mirror (Thorlabs) was placed in the end of thearm for back-reflecting the beam. This constitutes the measurement arm. At theend of the reference arm another silver coated λ/10 mirror was placed.The 1:1 telescope was then built in. The distance between the f500A lens andthe first f200A lens does not matter. It was installed in the cage system andaligned onto the irises. The last f200A lens was placed in, aligned to the irises andinterferometrically aligned along the propagation direction of the beam. For thealignment, a small circle with r = 5DMD pixels was uploaded to the DMD whichacted as a point source. Figure 6.14 shows the interference signal for a randomposition and the optimal position of the last f200A lens. For the random positioncurved fringes are clearly seen, which are not present at the optimal position,indicating an optical path difference smaller than λ/2. Another interference patternconsisting of spherical rings is seen in each of the interference patterns. Thisprobably arises from diffraction effects of the beam clipping onto the small circleuploaded to the DMD.

81

(a) (b)

Figure 6.14.: Shows the interferometric signal for a random position (a) and theoptimal position (b) of the last f200A. No fringes are visible, hencethe optical path difference is smaller than λ/2.

After placing the lenses, a uEye camera for intermediate imaging of the wave planewas placed in the in the focal point of the first lens in the telescope via the flipmirror mentioned earlier.The same approach was used for aligning optics in the 940 nm arm.

6.6. Molasses armThe optical molasses arm provides the beam for optical molasses in the opticallytrapped atoms. A schematic drawing of the arm is presented in figure 6.15. Fromthe outcoupler, the laser passes a f = 1000mm lens (AC508-1000-B, Thorlabs)that focuses the light into the objective to create a beam with the largest possibleRayleigh range. The beam sampler in the figure overlaps the molasses arm withthe fluorescence arm.

6.7. Pilot BeamThe pilot beam was used for aligning the entire HiRes board with respect to thescience chamber. The setup was divided into the two parts seen in figure 6.16. Theleft part was placed outside the HiRes board and basically consists of a 780 nm laserdiode coupled into a fiber. The right side was placed on the Hires board, under thescience chamber. The beam comes out of the fiber, propagates through the mirrors

82

f=1000mm

780nmMolasses Beam

Figure 6.15.: Shows the 780 nm molasses arm of the HiRes board.

and shoots up on a half inch reference mirror glued with high temperature resistantglue on the flange of the high resolution view port. The beam is back-reflectedthrough the setup and is picked up by a power meter (Thorlabs).When installing the HiRes board, the vacuum was first broken and the sciencechamber was opened. After building the high resolution viewport as explained insection 4.2.2, the HiRes board was moved in place while looking through the sciencechamber from above. Tests were then carried out to find its optimal position andat this optimal position, the Pilot beam was aligned onto the reference mirror suchthat the back-reflected beam was picked up by the power meter. The board wasthen taken out, the chamber was closed and the system was baked out in order toreach ultra high vacuum. Afterwards, the board was repositioned in such a waythat the back-reflected pilot beam was picked up by the power meter. This ensureda correct tilt of the board with respect to the science chamber.The installation of the HiRes board will be explained in great detail in chapter 7.As a concluding remark for the alignment procedure, a schematic figure of thealigned HiRes board containing all the different modules is presented in figure 6.17.

83

780nm PilotTo thereference mirror

PM

PM

780nmlaser diode

Power Meter

Laser Diode

Figure 6.16.: Shows the pilot beam. The laser setup, which is placed outside theHiRes board, is presented to the left. The part of the setup placed onthe HiRes board is shown to the right.

PD

f = 750mm

787.5nm

f = 500mm

DMD

Fiber Coupler

PD Photo Diode

Mirror

PBS

Lens

Beam Sampler

λ/4 Wave plate

λ/2 Wave plate

—

— Achromat

—

—

Beam Splitter

1:1 telescope, f=200mm

Flip Mirror

CAM Camera

CAM

Iris

Dichroic

787.5nmDirect

PD

787.5nm small

940nm

PD

940nmDirect

f=750mm

940nm small

—

—

f = 500mm

940nm Fourier

PD

1:1 telescope, f=200mm

CAM

787.5nm

787.5nmFourier

CAM

Fluorescence arm

PD

—

—

f = 750mm

To the Science Chamber

PD

780nm Pilot

f=1000mm

780nmMolasses Beam

To the reference mirror

Figure 6.17.: Shows a schematic drawing of the aligned HiRes board.

84

7Initial tests and installation of theHiRes Board

In this chapter it is described how the aligned HiRes board was initially tested andhow it was installed into the HiRes experiment. The initial tests are described insection 7.1 and the installation is presented in section 7.2.

7.1. Initial testing of the HiRes boardA series of tests were carried out before the board was installed into the HiResexperiment to characterize the performance of the system. The tests are describedin the following:

Faraday Fourier DMDInitially we wanted to fit an extra outcoupler providing a 780 nm beam for the787.5 nm Fourier DMD to perform experiments with non-destructive Faraday imag-ing technique[28]. The setup would use the DMD in −12° "off"-state in the sameway as the "small beams" are used with the direct DMDs. It was, however, notpossible to fit the outcoupler onto the HiRes board. The workaround was to usethe 787.5 nm outcoupler and fiber for the 780nm Fourier beam and compensate forthe difference in wavelength by using a different k-vector in the underlying gratinguploaded to the DMD. The 780 nm beam was aligned by systematically changingthe k-vector until it was overlapped with the two irises in the tower. The k-vectorwas found to be:The values were found to be:

kx = 2π22.4 ky = 2π

18.4 (7.1)

compared to kx = ky = 2π/20 for the 787.5nm beam.

85

Table 7.1.: Efficiencies and potential depth for the direct DMDs. The efficienciesare listed for a dimple with a waist of 10 DMD pixels

Wavelength [ nm] DMD Efficiency (ηori) Udip

P0[Erec

W ] Udip

P0[µK

W ]787.5 Small 4 · 10−5 −715.6 −129.67

Big 3.6 · 10−6 −64.4 −11.67940 Small 10−3 −3229 −585.13

Big 2 · 10−4 −645.8 −117.0253

Efficiencies of the DMDs

The efficiencies of the DMDs are of high relevance, since the power of the beamgoing into the science chamber determines the depth of the trapping potentials.The efficiency of each DMD was found by taking the ratio between the measuredpower immediately after the corresponding fiber and immediately after the beamsplitting cube used for overlapping the direct- and Fourier path.It has to be kept in mind that the efficiency changes with the number of mirrorsused and hence the area of the used mirrors. For the measured efficiencies, ηori,of the direct DMDs, a dimple with a waist of 10DMD Pixels was uploaded. Thiscorresponds roughly to a waist in the focal plane of wori ≈ 500 nm. The efficienciesare given in table 7.1. Due to the beam sampler in the the 787.5 nm arm, whichonly transmits 4% of the power, the 787.5 nm efficiencies have been multiplied bya factor of 0.04.For a dimple of a given width in the focal plane, w, the efficiency can be calculatedas

η(w) = ηori

(w

wori

)2(7.2)

For the Fourier DMDs the efficiencies have been measured the same way, justwith a slit pattern uploaded to the DMD. The width of the slit pattern was1000DMD Pixels which roughly corresponds to the size of the ROI defined inchapter 5. The size of the width ensures that the entire aperture of the highresolution objective is used which in turn gives the smallest possible width, wori, inthe focal plane which can be calculated via [43]:

wori = κλ

2NA = κλf

D(7.3)

where κ = 0.9 is a constant that takes the illumination of the lens into account, λ isthe wavelength of the light, NA is the numerical aperture which has been rewritten

86

Table 7.2.: Efficiencies for the Fourier DMDs.

Wavelength [nm] Efficiency (ηori)787.5 12 · 10−5

940 7 · 10−3

in the last expression in terms of the focal length, f , and the aperture of the lens,D.The measured efficiencies are given in table 7.2, where the effect of the beamsampler again has been taken into account.The efficiency for an arbitrary waist, w, can for the Fourier DMDs be calculatedvia:

η(w) = ηori

(w

wori

)−2. (7.4)

where the negative power takes into account, that big structures in Fourier spacegives small structures in real space.For a given efficiency, the depth of the corresponding dimple can be calculated.Taking into account, that the laser can drive a transition from the groundstate tomultiple excited states, the potential is given by [26]

Udip(r) = πc2Γ2ω2

0

(2 + PgFmF

∆2,F+ 1− PgFmF

∆1,F

)I(r) (7.5)

where Γ is the damping rate, ω0 is the transition frequency, P = 0,±1 for linearand circular (σ±) polarization of the light, I(r) is the intensity, ∆2,F and ∆1,F arethe energy splitting between the ground state and the center of the hyperfine splitof the excited states 2P3/2 and 2P1/2. The intensity is given as power per area andif one is interested in the maximum depth of the dimple, Udip(r = 0), only themaximum intensity has to be considered, which is given by [35]

Imax = 2Pπw2 (7.6)

where P is the power in the laser beam and πw2 is the Gaussian beam spot size. Thepower after the DMDs is given as the input power,P0, multiplied by the efficiency,P = ηP0. Substituting this into 7.6, we see that

Imax/P0 = 2ηπw2 . (7.7)

From equation 7.2 it is seen that η ∝ w2 for the direct DMDs and hence Imax/P0 andthe dimple depth are independent of the dimple waist even though the transmittedpower changes with waist.

87

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Waist [µm]

-10 3

-10 2

-10 1

-10 0

-10 -1

Ud

ip(r

=0

) [E

rec

/W]

Dimple depth as a function of waist

787nm

940nm

(a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Waist [µm]

-10 2

-10 1

-10 0

-10 -1

-10 -2

Ud

ip(r

=0

) [µ

K/W

]

Dimple depth as a function of waist

787nm

940nm

(b)

Figure 7.1.: Shows the dimple depth measured in units of recoil energy per power(a) and temperature per power (b) as a function of dimple waist in thefocal plane.

The dimple depth per input power has been calculated for linear polarized light.The results are also given in table 7.1 in the units of µKW−1 and ErecW−1 whereErec is the so called recoil energy.The recoil energy is the maximum energy a photon can transfer to a trapped atomand is given by

Erec = ~2k2

2m (7.8)

where ~k is the momentum of the photon. The recoil energy can thus be interpretedas the amount of ‘kicks’ an atom can get from photons before leaving the trap.For the Fourier DMDs, it is seen that η ∝ w−2 and hence the Imax/P0 and thedimple depth depend on the waist. In figure 7.1, Udip(r = 0)/P0 has been plottedas a function of the dimple waist in the focal plane, w. It is seen, that the trapis deepest for smallest waists. Gravity has not been taken into account in thecalculations above since a 2D confinement has been assumed.A power of 500mW can be expected after the 940 nm fibers and a power of 50mWcan be expected after the 787.5 nm fibers. For the latter wavelength, this amountsto 2 µW after the 787.5 nm small DMD.The differential light shift of one optical lattice site compared to the neighbouringsite can be calculated via 7.5. It amounts to 0.18GHzmW−1 for 787.5 nm anda beam waist of 520 nm. The shift for the small DMD is then 360 kHz, whichcompares to the 70 kHz in the Munich experiment [49]. For the 787.5 nm big DMD,the shift is 32 kHz.

88

f500A

f200A f200A

HiResObjective

Re-imagingObjective

CAM

M1 M2

Figure 7.2.: Shows the test setup with the re-imaging system. M1 and M2 indicatethe magnification of HiRes board and re-imaging system, respectively.

Test of generated spot sizeAn important test that was carried out was the investigation of how small a spotcould be generated with the different beams coming from the HiRes board. Toobig a spot would limit the possibility of addressing only a single lattice site.For this test, the tower was removed and a "re-imaging system" was built. Acrude schematic sketch is presented in figure 7.2, where an arm of the HiResboard is depicted from the direct DMD to the HiRes objective. After this, there-imaging objective (UPLSAPO20X, Olympus) was mounted on another PiFoc(P-721) followed by a lens (300mm) that focused the light onto a uEye camera.The overall method was to step the PiFoc of the HiRes objective through the focusof the HiRes objective for the different beams and analyze the resulting spots.In order to be able to analyze the spots afterwards, the magnification of there-imaging system, M2, was calculated from

Mtot = M2

M1(7.9)

whereMtot is the magnification the entire system andM1 = 100 is the magnificationof the HiRes arm. The total magnification was found by uploading a checkerboardwith a known pattern size to DMD and compare it to the size of the correspondingimage on the uEye. It was found that Mtot = 0.43 yielding M2 = 43.The test was carried out by uploading a small structure on the DMD, whichacted as a point source. For the direct DMDs, the point source was a small circlewith r = 6DMD Pixels and for the Fourier DMDs a hologram of a double phasecorrected Gaussian dimple with a waist of

√8 DMD Pixels constituted the point

source.For each beam coming from the HiRes board, the HiRes objective was translatedin steps of 100 nm, similar to the preliminary test of the objective described insection 4.3.

89

Figure 7.3.: Shows the spot at the optimal position of the HiRes objective for the787.5 nm Fourier DMD.

For each position of the objective, the "point source" was imaged with the uEye. AGaussian profile was fitted to a linecut through the center of the spot in the x- andy-dimension. From the fit, the waist of the spot was extracted.The spot at the optimal position for the objective for the 787.5 nm Fourier DMD isseen in figure 7.3 and the corresponding linecuts are presented in figure 7.4. It isnoted that the data points, the fit and the simulated point spread function coincide.The optimal position objective was chosen as the position that yielded the smallestwidth in both dimensions.The waists for the different beams are given in table 7.3. For comparison, thetheoretically smallest waists are given by equation 7.3 and amounts to 506 nm for787.5 nm light and to 604 nm for 940 nm light. Too faint a signal could be thereason for the 787.5 nm Fourier-waist and the 940 nm Direct-waist to be smallerthan the theoretical limit.

7.2. Installation of the board

Before installation of the HiRes board into the HiRes experiment, the HiResviewport had to be placed in the science chamber. For this, the vacuum of theexperiment had to be broken. The installation can be subdivided into two parts: Aplacement part and an alignment part. In the following sections, the two parts arepresented in detail.

90

1 2 3 4 5 6

Distance [ µm]

0

10

20

30

40

50

60

70

80

90

100

Inte

nsity [A

rb. valu

es]

Cutout through the center in the x-dim.

xLine

xFit

PSF

(a)

1 2 3 4 5

Distance [ µm]

0

10

20

30

40

50

60

70

80

90

100

Inte

nsity [A

rb. valu

es]

Cutout through the center in the y-dim.

yLine

yFit

PSF

(b)

Figure 7.4.: Shows the linecut through the center of the spot (figure 7.3), the fit andthe point spread function in the x-dimension (a) and the y-dimension(b).

Table 7.3.: Shows values of the waist in the x- and y-direction at the optimal positionof the objective for the different DMDs. The standard deviations of thefits have been taken as an estimate of the uncertainty.

wx [µm] wy [µm]787.5 nm Fourier 0.489± 0.010 0.489± 0.018

Direct 0.571± 0.015 0.602± 0.012Small 0.598± 0.012 0.605± 0.014

940 nm Fouier 0.622± 0.016 0.606± 0.012Direct 0.509± 0.032 0.513± 0.030Small 0.648± 0.018 0.654± 0.013

91

Figure 7.5.: Shows the two screws blocking the way (red arrows) for sliding in theHiRes board. The blue arrow marks the HiRes objective.

Placement of the HiRes board

While the vacuum was broken, the top of the HiRes board was carefully slid underthe science chamber. It was noticed, that two of the screws in the flange of thehigh resolution viewport were blocking the way, as can be seen in figure 7.5. Theheads of the screws were reduced in size giving barely enough space for the boardto be slid in place under the science chamber. The screws holding the viewportwere tightened in the same way as presented in section 4.2.2.The top part of HiRes board was then raised, so that the bottom part could be slidunderneath it. For raising the the board, four arms had been attached to the sidesof the board. A crude sketch of the board seen from above is presented in figure7.6a, where the arms are labeled A-D and the micrometer screws for controllingthe tilt and height of the top part of the board with respect to the bottom part aredenoted 1− 3. Each arm has a lifting mechanism as seen in figure 7.6b.The board was (very) carefully raised since we only had 1mm of free space aroundthe objective as shown in figure 7.7a. The board was lifted by rotating the liftingmechanism of the four arms simultaneously while monitoring the position of theobjective with respect to the view port through the top of the science chamber, asseen in figure 7.7b.After reaching a height of roughly 41mm the bottom part of the HiRes board wasslid underneath the top part. The arms were then removed and the micrometer

92

x

y

A

BC

D

1 2

3

α

βγδ

(a) (b)

Figure 7.6.: (a) Shows a crude sketch of the HiRes Board seen from above. Thered circles labeled 1 − 3 indicate the micrometer screws translatingin the z-direction. The arms (see text) are labeled A − D. α − δ

denotes micrometer screws which will be placed after the high precisionalignment of the board. The orange circle is the objective. (b) Showsone of the arms (red arrow) and the lifting mechanism (blue arrow).

93

41mm

39mm

Viewport

Objective

z

(a) (b)

Figure 7.7.: (a) Is a sketch of the objective as it is being lifted towards the viewport. (b) Shows view through the top of the science chamber (whitearrow). The position of the objective (orange arrow) is seen throughthe high resolution view port (blue arrow).

94

Viewport

Objective

z

Target plate

(a) (b)

Figure 7.8.: Shows a schematic version (a) and an image of the test setup (b).

translation screws denoted α− δ in figure 7.6a were placed in such a way that theycould translate the board in the x- or y-direction.The final positioning of the board is described in the next section.

AlignmentFor high precision alignment, a test similar to the "Position and tilt"-test describedin section 4.3 was carried out. A test setup for generating points sources was built.It is shown in figure 7.8 and consisted of an outcoupler providing 780 nm lightplaced in a cage system. The collimated light from the outcoupler passes a lens thatfocusses the light onto the target plate (introduced in figure 4.11) which generateda point source. The lens was aligned by maximizing the power coming through theholes in the plate. The cage system was placed roughly in the focal plane of theHiRes objective inside the chamber. Teflon spacers were placed between the cagesystem and the view port to avoid damaging the port.The point sources were imaged in the fluorescence arm, where a uEye was placedas the camera (see figure 6.5).By moving the cage system around we were able to image the part of the targetplate marked by the gray rectangle in figure 4.11. The obtained signal is presentedin figure 7.9a. A clear tilt is seen from the asymmetric rings. It was inferred that

95

(a) (b)

Figure 7.9.: Shows the imaged part of the target plate with tilt (a) and without atilt (b). Both images are highly saturated to make the rings as visibleas possible.

the HiRes board was tilted around the x-axis defined in figure 7.6a.The tilt was minimized by turning the micrometer screw labeled 3 in figure 7.6aand moving the test target into the focal plane of the HiRes objective. While doingthis, the cage system touched the Teflon spacer which indicated that the focusof the objective was too low. The entire board was then raised by turning thethree micrometer screws (1,2,3) 360° simultaneously in steps of 36°. The screwclosest to the objective was again turned in order to minimize the tilt of the board.Minimizing the tilt and raising the board was repeated until the tilt was as smallas possible. The corresponding signal is seen in figure 7.9b. The image is highlysaturated to make the rings as visible as possible.A non-saturated image was then taken and the magnification of the system wascalculated to M = 156.87 in the same way as described for the preliminary test ofthe objective (section 4.3).Knowing the magnification, the high precision alignment could be carried out,which corresponded to making the still visible rings in figure 7.9b as concentricand symmetric as possible: First, a small hole in the target plate was chosen. Theprocedure was then to systematically change the micrometer screw closest to theobjective. Only the setting of this screw was changed, since the coma effects wereonly visible in one dimension (figure 7.9a). For each each position of the screw, thePiFoc was swept through the focus while images of the spot was taken. The spotat the optimal position of the PiFoc and micrometer screw is shown in figure 7.10a.A linecut through the center of the profile, a Gaussian fit and the simulated pointspread function is shown in figure 7.11 for each dimension. The waist is extracted

96

(a) (b)

Figure 7.10.: (a) Shows the spot for the optimal setting of the micrometer screwand the optimal position of the PiFoc. Notice the concentric ring. (b)Shows a profile for a non-optimal setting of the micrometer screw.The ring is clearly non-concentric.

Table 7.4.: Shows the gap measured at the micrometer screws

Micrometer screw Gap [mm]1 4.702 4.653 1.30

from the fit and amounts to wx = 0.429(15) µm and wy = 0.434(13) µm, where thenumbers in the parenthesis are the standard deviation of the fit. From the linecutsit is seen that the profile is not completely symmetric. It is also seen that theline cuts are more narrow than the point spread function, which indicates thatthe imaging system performs diffraction limited through the high resolution viewport! For comparison a profile for a non-optimal setting of the micrometer screw ispresented in figure 7.10b. The ring around the central spot is clearly not concentric.

At this optimal position of the board, micrometer screw α− δ were tightened sothat they marked the exact position of the top part of the HiRes board. Thecombination of the settings of the micrometer screws (1− 3 and α− δ) uniquelydefined the optimal position and tilt of the HiRes board, and were hence noted.The gap between the bottom- and top part of the HiRes board was measured atthe position of screw 1− 3, see table 7.4.

97

0 0.5 1 1.5 2 2.5 3

Distance [ µm]

20

40

60

80

100

120

140

160

180

200

220

Inte

nsity [A

rb. valu

es]

Cutout through the center in the x-dim.

xLine

xFit

PSF

(a)

0 0.5 1 1.5 2 2.5 3

Distance [ µm]

20

40

60

80

100

120

140

160

180

200

220

Inte

nsity [A

rb. valu

es]

Cutout through the center in the y-dim.

yLine

yFit

PSF

(b)

Figure 7.11.: Shows the linecuts through the center of the profile presented in figure7.10a. A Gaussian function has been fitted to the profiles which arealso compared to the point spread function.

7.3. Pilot beam and removing of the HiRes boardAt the optimal tilt of the HiRes board, the pilot beam was aligned onto themirror glued on the flange of the high resolution viewport in such a way that theback-reflected beam was picked up by the power meter.The arms with the lifting mechanisms were then reattached to the top part of theboard and the bottom part was removed. The top part was carefully lowered viathe lifting mechanisms and slid out from under the science chamber.The Science chamber was closed by mounting the top view port again and theHiRes experiment was baked out for roughly three weeks in order to reach the ultrahigh vacuum regime (∼ 10−11mbar).After the bakeout, the HiRes board (top- and bottom part) was installed under thescience chamber in the same way as described above and aligned to the optimalposition defined by the positions of the micrometer screws. At this position, it wasseen that the back-reflected pilot beam was still picked up by the power meter,which indicated that the optimal position of the HiRes board was reached.

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8 Action with atoms

This chapter introduces some of the physical concepts needed for further testesand use of the HiRes board. In section 8.1 the concept of fluorescence signals isexplained. When driving an atomic transition with light close to resonance, thereis a probability that the atoms are excited to different states which gives rise todifferent decay channels. This is elaborated in section 8.2. Section 8.3 presentsan analysis of a fluorescence signal obtained from atoms trapped in the sciencechamber. The concept of Two-body light induced collisions, which can be used asa method for achieving unitary filling of optical potentials, is explained in section8.4. Finally the current status of the HiRes experiment is given in section 8.5.

8.1. Fluorescence signalsA laser beam, which is detuned close to or on resonance, drives an atomic transition.Photons will be reemitted isotropically when the atom deexcites and these photonscan be detected by cameras or photodiodes. In our case, we use cameras to detectthe fluorescence signal from atoms confined in optical traps.In the following, the expected fluorescence signal is calculated for 87Rb.In a proper description of 87Rb, the atom has to be treated as a multilevel atom.In 87Rb the D1-line, corresponding to the 52S1/2 → 52P1/2 transition, has awavelength of 794.98 nm and the D2-line, corresponding to the 52S1/2 → 52P3/2

transition, has a wavelength of 780.24 nm. In the Hires-experiment, the optimaldetuning for optical molasses has not been identified yet, but inspired by Munich,it will probably in the order of 80MHz from the D2-line [24], making it far detunedfrom the D1-line. In this case, the description of the atom can be simplified to thatof a two-level system. It is shown in [45] that the total photon scattering rate in

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

I/Isat

10 2

10 3

10 4

10 5

10 6

Rsc [s

-1]

Scattering Rate for Rb-87

∆ = 0MHz

∆ = -40MHz

∆ = -50MHz

∆ = -60MHz

∆ = -70MHz

∆ = -80MHz

∆ = -90MHz

Figure 8.1.: Shows the scattering rate for a Rubidium atom as a function of I/Isatfor different (red)detunings.

the steady state is given by

Γsc = Γ2

I/Isat

1 + 4 (∆/Γ)2 + I/Isat(8.1)

which has been integrated over all directions and frequencies (of the differenthyperfine levels). I/Isat, where Isat = 1.669(2000)mWcm−2 is the saturationintensity in the steady state[45] for circular polarized light and Γ = 2π · 6.065MHzis the natural line width.Equation 8.1 has been plotted in figure 8.1 for different detunings. It is seen that adecrease in detuning gives higher scattering rate. This makes perfect sense, since itcorresponds to getting closer to the resonance. It should be noted that if one hastrapped a cloud of atoms, the graphs in figure 8.1 should be multiplied by the totalatom number in order to get an estimate for the expected isotropic fluorescencesignal.

8.2. Decay channelsAs previously stated, the transition of interest for the Hires-experiment is mainlythe 52S1/2 → 52P3/2-transition which is also known as the D2-line. The amount of

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hyperfine splittings for each state corresponds to the amount of values the totalatomic angular momentum, F, can take. These values are given by the inequality|J − I| ≤ F ≤ |J + I|, where J is the size of the total electron angular momentumJ and I is the size of the total nuclear momentum I. In the ground state, 52S1/2,J = 1/2 and I = 3/2 which gives the two hyperfine states F = 1 or F = 2. In the52P3/2-state, J = 3/2 and I = 3/2, so F ′ can take the values 0,1,2 or 3. Note thatthe hyperfine states in the excited state are referred to with an apostrophe to avoidpotential confusion. This splitting is shown in figure 8.2. The molasses laser is reddetuned 80 MHz from the F = 2→ F ′ = 3 transition.According to the dipole approximation, the hyperfine transition selection rulerequires that ∆F = 0,±1 with the exception that F = 0→ F ′ = 0 is not allowed[14]. This means that when an atom is excited to the F ′ = 3 state, it can onlydecay to the F = 2 state, where it will interact with the molasses laser again.There is, however, a finite chance that the laser drives a transition from F = 2 toF ′ = 2, which can decay to the F = 1 state. Here it is improbable that the atomwill interact with the laser, hence the atom is lost from the cooling cycle. In orderto recapture these atoms a so called repump laser is used. This laser is tuned insuch a way that it addresses the F = 1→ F ′ = 2-transition. When the atoms arereexcited to the F ′ = 2, they will either decay into F = 1 and interact with therepump laser again or they will decay into F = 2 and become a part of the coolingscheme again.The probability that an atoms ends in the F = 1 state can be calculated. First,the probability for ending in the F ′ = 2 state is given by the ratio between thescattering rates (equation 8.1) for the transitions to the F ′ = 2- and F ′ = 3-statefrom the F = 2-state:

PF=2→F ′=2 = Γsc,F=2→F ′=2

Γsc,F=2→F ′=3≈ 18% (8.2)

where a detuning from the F ′ = 3-state of ∆F ′=3 = 80MHz has been used,corresponding to a detuning from the F ′ = 2-state of ∆F ′=2 = 187MHz. I/Isat = 1has been used to get a rough estimate of the probability.The probability that atoms in the F ′ = 2-state fall into the F = 1-state canbe found by calculating the relative coupling strength from the F ′ = 2-state toF = 2- and F = 1-state, which is given as the ratio between the summed squaredClebsch-Gordan coefficients to each of the final states. The probability is found toPF ′=2→F=1 = 50% and hence the probability of an atom starting in the F = 2-stateand ending in the F = 1-state is amounts to

PF=2→F=1 = PF=2→F ′=2 · PF ′=2→F=1 ≈ 9% (8.3)

which establishes the need for the repump laser.

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Figure 8.2.: Shows the hyperfine structure for the D2-line in 87Rb. The figure isadapted from [45].

102

100 200 300 400 500

50

100

150

200

250

300

350

400

450

500

Figure 8.3.: Shows a fluorescence image from the crossed dipole trap in the finalchamber.

8.3. Detecting fluorescence signalsAn early test of the HiRes experiment after the bake out was to look for a fluorescencesignal in the final chamber.After the cooling sequence, atoms were loaded into a crossed dipole trap in thefinal chamber where they were illuminated in order to scatter fluorescence photons.To maximize the scattering rate, described by equation 8.1, resonant light (∆ = 0)was used. The total intensity of the molasses beams was estimated to roughlyI/Isat = 12. The fluorescence signal was collected via the top viewport of the finalchamber through an imaging system focusing the signal onto an EMCCD camera(Andor, iXon 897) which had a quantum efficiency of QE = 0.81, an effectiveexposure time of T = 50ms2 and an electron multiplying gain of EM = 1.The acquired signal is presented in figure 8.3 and the total fluorescence signal onthe camera, C, was found by subtracting the background and summing all pixelvalues.Based on the theory presented in section 8.1, an estimate of the atom number canbe found from the fluorescence signal: The expected number of photons impingingon the camera, Np, from NA atoms during the exposure time is given by

Np = NAΩΓscT (8.4)1The quantum efficiency is the ratio between the number photo electrons and the number ofphoton impinging on the camera.

2The atoms are lost due to heating on a timescale shorter that the exposure time of the camera.The time scale of 50 milliseconds is taken as a rough estimate, since it the typical lifetime ofthe trap.

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where Ω is the solid angle spanned by the imaging system. Due to the quantumefficiency, the number of photo-electrons,Ne is

Ne = Np ·QE (8.5)

The corresponding expected count on the camera, C, is given by

C = Ne · EMg

(8.6)

where g = 1.35 is the is the number of electrons per pixel count. In our systemwe had Ω = 7.62× 10−4 sr and Γsc = 2.799× 106 s−1, which was calculated fromequation 8.1. From equations 8.4 to 8.6, and by using the values presented above,the atom number was found to NA ≈ 1.24× 105.

8.4. Two-body light induced collisionsIn section 3.3 the scheme for addressing single atoms was explained. A trivialprerequisite for addressing single atoms is to have a trap with unitary filling. Acommon used method for achieving this is to induce the so-called two-body lightinduced collisions (TBLIC). The Gallagher-Pritchard model (GP) establishes thephysical picture of the TBLIC and will in the following be explained in line with[41] and [48].The basic principle of the GP model is shown in 8.4. A laser, detuned ∆ fromresonance, excites one of the atoms in an atom pair separated by R0. The resultinginteraction potential between the atoms is attractive and will accelerate the atomstowards each other. The potential is assumed to be

V = −C3/R3, C3 = 71 eVÅ3 (8.7)

for R’s where the coulomb repulsion is weak. Due to an assumption about theradiative broadening being small compared to the detuning, only atom-pairs withseparations fulfilling the relation h∆ = −C3/R

30 will be excited. As usual, the

detuning is given by ∆ = νL − ν0 with ν0 being the frequency of the atomictransition and νL being the frequency of the laser.It can be shown that the atoms will be excited at a rate

R ∝ R20

∣∣∣∣∣ d∆dR0

∣∣∣∣∣−1

∝ ∆−2 (8.8)

As previously stated, the excited atom pair will accelerate towards each other. Isthe interatomic distance smaller than RT when the atoms radiate, their kinetic

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Figure 8.4.: The Gallagher-Pritchard model. Shows the interaction potential, V ,between excited atom-pairs as a function of interatomic distance, R. ∆is the detuning of the laser that, in this case, excites atoms separatedby R0. If the atoms reemit a photon while being separated by R < RT ,the atoms will have enough kinetic energy to leave the trap. Figureborrowed from [41].

energy will be be greater than trapping energy and they will leave the trap. RT isthe interatomic distance that fulfills the following criteria:

h∆− V (RT ) = 2h∆T (8.9)

with h∆T being the trap depth. It can be showed that the total probability of traploss is

P = sinh (Γt1)sinh (Γ(t0 + t1)) . (8.10)

Here t0 and t1 are the times it classically takes for the atoms to go from R0 toRT and from RT to R = 0, respectively. Γ is the spontaneous emission rate. Theproduct between equation 8.8 and 8.10 gives the trap-loss rate:

β ∝

∆2 sinh(−∆T

∆

)5/6−1

(8.11)

To arrive at the last equation, it has been assumed that t1 is independent of thedetuning and t0 + t1 ≈ 0.746

õR5

0/2C3 ≡ (−∆T/∆)5/6/Γ, where µ is the reducedmass of the system. The GP model has been the most successful model for trap-losscollisions, when it has been extended to include the hyperfine-structure of theatoms[29]. In order to include hyperfine structure in the GP model, one has to take

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excitation to hyperfine potential curves into account as well as the atom’s movementalong those curves [48]. This is done by summing over the different curves in equation8.11, with each curve having a unique ∆T . Extending the model is quite complex.Instead the trap-loss rate for 87Rb has been measured in [20] for the 5S1/2 → 5P3/2

transition. For a red detuning of 80 MHz the coefficient is β ≈ 1 · 10−12 cm3s−1.This result has been referenced to an intensity of Iref = 10 mW/cm2, where it hasbeen assumed that the trap loss coefficient is linear in intensity.Another research group has measured a trap-loss coefficient that amounts to β =2.4(11)× 10−9 cm3 s−1 for a laser intensity of I/Isat = 0.5, detuning of ∆ = 20MHza trap depth of U/~ = 36MHz, a temperature of T = 300 µK and trap volume ofV = 0.7 µm3 [22].Knowing the trap-loss coefficient, the rate at which the atoms leave the trap canbe calculated. In general the rate is given by

d

dtN(t) = R− N(t)

τ− β

∫n(r,t)2d3r (8.12)

where R is the rate at which atoms are loaded into the trap, N(t) is the number ofatoms in the trap to the time t, τ is the lifetime of the trap due to collisions withbackground gas and n(r,t) is the spatial density of the trapped atoms in the trap.If the spatial distribution of atoms is independent of the number of atoms in thetrap, n(r,t) can be written as a product of two functions; one describing the timedependency and one describing the spatial distribution of the trapped atoms. Inthis case, and assuming the density distribution is Gaussian (as it is for a thermalcloud), the last term in equation 8.12 can be rewritten as [15]

β∫n(r,t)2d3r = βN(t)2

Veff(8.13)

where Veff is the effective volume of the trap which is Veff = π3/2(2σx)(2σy)(2σz) inthis Gaussian case. Ignoring the two first terms in 8.12, one can write the followingexpression for the change in the number of trapped atoms due to two body losses:

d

dtN(t) = −βN(t)2

Veff(8.14)

By rearranging the terms and do the following integration∫ 1

N

dN(t)N(t)2 = − β

Veff

∫ t

0dt′, (8.15)

where the limits are taken such that we find the time it takes to end up with asingle atom. The following relation is found:

t = Veffβ

(1− 1

N

)(8.16)

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10 1 10 2 10 3 10 4 10 5

N

10 -2

10 -1

10 0

10 1t/I

ref [s

(m

w/c

m2)-1

]

Time as a function of cloud size

waist = 0.5 µm

waist = 1 µm

waist = 2 µm

waist = 3 µm

waist = 4 µm

Figure 8.5.: Shows the time it takes to reach a single trapped atom as a functionogthe number of initially trapped atoms for different trap waists, as givenby equation 8.16.

which gives the time, t, it takes to end up with only 1 atom trapped, when initiallyN was trapped, if the only loss rate is the two-body light induced collisions. Infigure 8.5 t/Iref has been plotted as a function of cloud size for different waists, w,with w =

√2σ where wx = wy and wz = πw2

x/λ according to the Rayleigh lengthand a trap-loss coefficient of β = 1 · 10−12 cm3s−1.It is seen that the time it takes to reach a single trapped atom is constant forN > 50. It is also seen that the time it takes increases with increasing waist. Thismakes sense, since an increasing waist will result in a decreasing density hencemaking it less likely to have two-body light induced collisions.

8.5. Status of the HiRes experimentDue to the finite extent of this project and the fact that work in the laboratoryin general takes a longer time than first anticipated, the final testing of the HiResboard is not included in this project. In this section, the test currently carried outin the laboratory is explained.At the time of writing, the trapped atoms in the final chamber and the focus ofthe high resolution objective are being overlapped. The atoms are trapped in the

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Fluorescence signal on iXon

100 200 300 400 500

Width [px]

50

100

150

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250

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450

500

Heig

ht [p

x]

600

800

1000

1200

1400

1600

1800

2000

2200

2400

Counts

(a)

Position of lens (mm)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

FW

HM

of

pe

ak (

px)

6

7

8

9

10

11

12

13

14Width of fluorescence

(b)

Figure 8.6.: Shows the fluorescence signal from the atoms trapped in the longitudinaldipole trap for a random position of the last lens in the imaging system(a) and the FWHM-values as a function of the lens position (b).

final chamber in the the so-called longitudinal dipole trap (mentioned in section3.1.4). The atoms are illuminated in order to scatter fluorescence photons, whichare collected through the top viewport of the final chamber via a 1:1 imagingsystem consisting of two 150mm lenses that focuses the light onto a CMOS camera(Andor, ion 897). An example of the resulting fluorescence image is presented in8.6a. The imaging lens closest to the camera was then moved in order to shiftthe focus on the camera. The translation of the lens was measured with a caliperfrom a fixed reference point. For each position of the lens a fluorescence signalwas obtained. A Gaussian function was fitted to a linecut along the height andthe full-width-half-maximum (FWHM) was extracted. The FWHM-values as afunction of the lens position are presented in figure 8.6b.For different positions of the lens (measured from the same reference point), adimple was created with the 787.5 nm direct DMD. A Gaussian function was fittedto the dimple and the FWHM was distracted. The region of interest on the iXoncontaining the dimple and the FWHM-values as a function of lens position arepresented in figure 8.7a and 8.7b respectively.By comparing figure 8.6b and 8.7b it is seen, that an offset of roughly 2mm ispresent between the focal plane of the fluorescence signal and the dimple. Simulatingthe imaging system in a program called GaussianBeam yielded that the focal planeof the dimple was 2mm above the atoms. As a result, the entire board was lowered1mm 3 and the dipole trap was raised by 1mm.

3Limited to 1mm due to gap size at micrometer screw 3 (see table 7.4)

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Dimple - 787.5 nm

5 10 15 20 25 30 35

Width [px]

5

10

15

20

25

30

35

40

45

Heig

ht [p

x]

1000

2000

3000

4000

5000

6000

7000

8000

Counts

(a) (b)

Figure 8.7.: Shows the 787.5 nm dimple signal for a random position of the lastlens in the imaging system (a) and the FWHM-values as a function ofthe lens position (b).

The above mentioned procedure was carried out again. The FWHM-values as afunction of lens position for the fluorescence and the dimple are presented in 8.8aand 8.8b. It is seen that the smallest widths are achieved for the same position ofthe lens, which indicates that the atoms and the focal plane of the 787.5 nm-armare overlapped. The alignment was carried out for the 787.5 nm-arm and not the940 nm-arm, since the potential chromatic shift for the latter arm in the imagingsystem has not yet been investigated.In the future, loading atoms into a dimple created by the 940 nm-arm and achievingunitary filling by inducing two-body collisions have to be investigated.

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-0.5 0 0.5 1 1.5 2 2.5 3

Position of lens (mm)

2.5

3

3.5

4

4.5

5

5.5

6

6.5

FW

HM

of

pe

ak (

px)

Width of fluorescence

(a)

0 0.5 1 1.5 2 2.5 3

Position (mm)

2

2.5

3

3.5

4

4.5F

WH

M (

px)

Width 787.5 nm dimple, x-direction

(b)

Figure 8.8.: Shows the FWHM-values as a function of position of the lens for thefluorescence signal (a) and the 787.5 nm dimple after lowering the HiResboard and raising the dipole trap.

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9 Conclusion and outlook

In this project a high resolution imaging system has been built for imaging singlelattice sites in optical lattices with a spacing of 532 nm, addressing and manipulatingsingle to a few atoms and generating arbitrary conservative potential landscapes.A key component implemented in the imaging system is the DLP Lightcrafter6500 Evaluation Modules. These digital mircomirror devices (DMD) are used forspatially modulating the light projected onto the atoms. Two DMDs have beenaligned in a direct configuration at a wavelength of 787.5nm and 940 nm and twohave been aligned in Fourier configuration at the same wavelengths. The directconfiguration is good for generating big geometries with sharp edges, while theFourier configuration allows one to cancel out aberrations of the imaging system,which is well suited for small diffraction limited geometries.In the Fourier configuration, the relative phase of the illumination source overthe micromirrors has to be known. In this project, a new and fast technique formapping the phases has been developed based on a Raspberry Pi.In the heart of the new technique is a hardware optimized video-player calledOMXplayer. Despite a lack of documentation about the OMXplayer, the sourcecode was modified for the new phasemapping method to work.With the new technique, it was possible to bring the time for mapping the phasesdown to 10 minutes, which corresponds to speeding up the process by a factor of48 compared to the earlier method presented in [31]. This speed-up opened up fornew types of tests that was very cumbersome if not impossible to carry out before.For generating the so-called patch videos and extracting the phases, two C++-programs were written that were a factor 10 faster that previously used programs.Before building the imaging system, preliminary tests of high resolution objectiveand the high resolution viewport were carried out. The objective was mountedon piezo stepper and the focus was shifted in steps of 100 nm while systematically

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changing the tilt of the objective with respect to the imaged object. The testconfirmed that the objective performed diffraction limited at the optimal positionand gave an insight about the tolerance of the position and tilt of the objective.The flatness of the viewport was measured by mounting the viewport in test flangeand systematically tightening the screws holding it in place. It was shown that aflatness of 0.2λ could be achieved.The imaging system was aligned and the stability of the system was tested bytaking and analyzing ∼ 70 phasemaps over 6 days. A drift of 0.09λ was measuredfor the peak-valley value of the phasemaps.Finally, the high resolution system was placed in High Resolution experiment. Itwas tested in the final chamber that the imaging system still performed diffractionlimited through the high resolution viewport. It was also tested, that it was possibleto use the DMDs to create dimples with waists close to the theoretical limits.The current status of the HiRes experiment is alignment of the trapped atoms inthe final chamber with respect to the high resolution imaging system. When this isaligned, investigation of how to achieve unitary filling of optical traps via two-bodylight induced collisions in line with [42] [22] [32] [46] should be carried out.Future work includes implementing the three dimensional optical lattices, and sincethe HiRes experiment is controlled via a labVIEW program, it is also desirable inthe near future to develop a labVIEW program for controlling the DMDs and theRaspberry Pi, to streamline the control of the entire experiment.Beyond this point, there will be many interesting ways to utilize the flexibilityof the HiRes experiment to manipulate atoms. By generating and dynamicallychanging a potential landscape, so-called optical tweezers can be formed and usedto shuttle single trapped atoms around similar to [11], which forms the basis of oneproposed quantum computation architecture [51], where local collision interactionsbetween two atoms form the so-called two-qubit gates.A possibility is also to simulate condensed-matter systems, where the optical latticepotentials play the role of the atoms in the periodic lattice of the matter and thetrapped atoms mimic the electrons [13]. In general, adding arbitrary potentialsto the regular optical lattice enhances the range of Hamiltonians which can besimulated in the HiRes experiment.Conservative arbitrary potentials also allow us to investigate atomtronics. Heresystems analog to electric circuits are generated in ultra cold neutral atoms inoptical lattices. A current running through copper wire could be simulated by aperiodic optical potential with a fine-tuned tunneling coupling between adjacentsites [40].All in all, the increased experimental flexibility due to the novel high resolution

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imaging system developed and presented in this master’s project opens up newdoors for quantum simulation and computation experiments.

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A Raspberry Pi - Software hack

This is a step-by-step guide to hacking the OMXplayer to generate triggers via theGPIOs. Since no documentation is available and only a few comments are left inthe source code, it is difficult to get an overview of the structure of the program.

Step 1

Prepare an SD-card with the newest version of Raspbian

Step 2

Put the SD-card into the RPI and boot up on it.

Step 3

In the command line type "sudo raspi-config" and choose "Expand filesystem". Goto "Advanced Options" and choose "Memory split" and give the GPU 256Mb or512Mb.

Step 4

Remove all parts of the already installed OMXplayer and FFMPEG. This can bedone by the commands "sudo apt-get remove omxplayer" and "sudo apt-get removeffmpeg"

114

Step 5Download omxplayer from https://github.com/popcornmix/omxplayer, i.e. write"wget https://github.com/popcornmix/omxplayer" in the terminal.

Step 6Unpack the downloaded version of the OMXplayer. Read the README.mdfile, which says you have to run the following commands in the following order:"./prepare-native-raspbian.sh", "make ffmpeg", "make" and "sudo make install".

Step 7Download the C++ library called WiringPi, which provides functions for con-trolling the GPIOs. Download-instructions and further info can be found athttp://wiringpi.com/download-and-install/. For examples on how to use thelibrary, check out http://wiringpi.com/examples/blink/

Compiling programs with WiringPi requires that one adds "-lwiringPi" to thecompile line or in the Make-file.

Step 8This is the step for hacking the OMXplayer. In the folder, where program has beenunpacked in Step 6, there is a file called OMXCore.cpp. Open the file and include theWiringPi-library by adding the line "#include <wiringPi.h>" in the top. Definea global variable in the top "int bufferCallIndex = 0;". Find the function de-clared as "OMX_ERRORTYPE COMXCoreComponent::DecoderEmptyBufferDone" (line∼1625). In this function add something like:

1 // I n i t i a l i z i n g the GPIO−c on t r o l (Remember to d e f i n e ←bu f f e rCa l l I ndex in the top )

2

3 i f ( bufferCallIndex>4) // 5 = s i z e o f bu f f e r4 5 // Generating the t r i g g e r pu l s e6 pinMode (8 , OUTPUT ) ;7 digitalWrite (8 , LOW ) ; // On8 delay (20) ;9 digitalWrite (8 , HIGH ) ; // Off

10 printf ( " Buf f IndexAfter5 = %d \n" , bufferCallIndex−5) ;

115

11 12 bufferCallIndex++;

In this example the WiringPi Pin 8 is used (http://wiringpi.com/pins/). Alsothe delay(20) (pause of 20ms) defines the length of the pulse. This should be longenough for the camera to register the trigger and shorter than the exposure timeof each frame. The function that is modified in the OMXCore.cpp file is probablycalled when a frame is deleted from the buffer. Due to a lack of documentationabout the source code, it cannot be determined with certainty exactly what thefunction does. The buffer size seems to be 5, which is why the trigger pulse isnot generated for the first 5 times the function is called; these calls are related tofilling/emptying the buffer and not showing the frames. Remember to add 5 blackframes in the end of the file that has to be played.

Step 9In the file "omxplayer.cpp" include the WiringPi-library by adding "#include<wiringPi.h>" in the top and add the line "wiringPiSetup();" before the while-loop defined around line 1173 in omxplayer.cpp, as shown below:

1 .2 .3 .4 .5 // Se t t i ng up the wir ingPi ' connect ion '6 wiringPiSetup ( ) ;7

8 whi le ( ! m_stop )9

10

11 i f ( g_abort )12 goto do_exit ;13

14 double now = m_av_clock−>GetAbsoluteClock ( ) ;15 bool update = f a l s e ;16 .17 .18 .

Step 10Run the command "make" in the terminal.

116

Step 11A movie, for example named "test.avi", is played by typing "sudo ./omxplayertest.avi" in the command line. By default, the output should be sent via theHDMI-connection to the screen you have connected to the RPI.

117

Bibliography

[1] User manual - uEye Cameras. Tech. rep., IDS Imaging Development SystemsGmbH, March 2009. Driver Version 3.32.

[2] Dlp6500fye 0.65 1080p mvsp s600 dmd. Tech. Rep. DLPS053A, Texas Instru-ments Incorporated, Oct. 2014. Revised Feb. 2016.

[3] Dlpc900 programmer’s guide. Tech. Rep. DLPU018B, Texas InstrumentsIncorporated, Oct. 2014. Revised July 2015.

[4] Dlp® lightcrafter™6500 and 9000 evaluation module (evm) user’s guide. Tech.Rep. DLPU028C, Texas Instruments Incorporated, Oct. 2014. Revised Nov2016.

[5] Getting started with ti dlp® display technology - application report. Tech.Rep. DLPA059A, Texas Instruments Incorporated, Januar 2015. Revised Feb.2017.

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