Loading a 2D Microtrap Lattice on a Magnetic Atom Chip

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Loading a 2D Microtrap Lattice on a

Magnetic Atom Chip

Master thesis in Physics

Track: Advanced Matter and Energy Physics

D.R.M. Pijn

Supervisor: Dr. R.J.C. Spreeuw

Daily supervisors: Dr. A. Tauschinsky, J.B. Naber,

Dr. L. Torralbo-Campo

Second reviewer: Dr. K. Dohnalov´

a

Master (track) coordinator: Prof. Dr. Mark Golden

November 24, 2014

Institute of Physics

Science Park 904

1098 XH Amsterdam

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Abstract

A scalable two-dimensional register of quantum bits with an arbitrary lattice symmetry would be an important step in the fields of quantum information and quantum simulation. Such a system is investigated in Amsterdam in the form of an atom chip experiment, in which ensembles of neutral87Rb atoms are trapped in a lattice of magnetic microtraps. The microtraps are based on a thin film of permanently magnetized FePt. Using optical lithography, the traps can be patterned in arbitrary lattice configurations and scaled down to nanometer sizes.

This thesis covers the loading of 105 _{atoms in a 10 μm spaced lattice, containing a hexagonal}

and square symmetry. The basic concepts of the magneto-optical trap and the magnetic Ioffe-Pritchard trap are described, as well as the techniques of polarization gradient cooling and radio-frequency evaporative cooling. Imaging is done in the form of absorption and fluorescence imaging. Temperatures are measured through time of flight imaging and by RF spectroscopy. The experimental setup around the atom chip is described, including a set of silver foil wires which are used to provide the trapping potentials for a u-wire magneto-optical trap and a z-wire magnetic trap in several of the loading stages. The set of external magnetic coils around the atom chip is characterized and the external RF coil is optimized. The imaging system is characterized with a resolution of 2.4(2) μm and a depth of focus of 5.40 μm.

The process of finding a working loading sequence is described in detail, as well as the most recent optimized version of the loading sequence and related measurements. Finally, magnetic field simulations are discussed as a guide in finding the parameters for a successful loading sequence.

The loading of this lattice is the first step geared towards a series of quantum information experiments with the current atom chip, and an important step in scaling down the lattice spacing for future atom chip experiments.

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Populaire samenvatting

Naarmate de elementaire onderdelen van computers steeds kleiner worden gaan quantummech-anische effecten een grotere rol spelen. Richard Feynman stelde al in 1982 voor om deze effecten juist te gebruiken: een computer die werkt volgens de wetten van de quantummechanica zou veel effici¨enter zijn in het simuleren van quantumsystemen, en zou bepaalde problemen op kunnen lossen die met klassieke computers niet binnen een beperkte tijd opgelost kunnen worden. De elementaire eenheden van zo’n quantumcomputer, de zogenaamde qubits, kunnen zich in elke superpositie van de waarden 0 en 1 bevinden. Elk quantumsysteem dat twee toestanden bevat is in principe geschikt om deze qubits in te coderen, maar er zijn bepaalde eigenschappen die een systeem gunstig of ongunstig maken.

In de Quantum Gases & Quantum Information onderzoeksgroep in Amsterdam worden de qubits gecodeerd in de toestand van koude rubidiumatomen die gevangen zitten in een rooster van magnetische microvallen boven een atoomchip. Elke val bevat een paar honderd atomen die samen één qubit vormen. Door de grondtoestanden van de atomen als qubit states te gebruiken bereiken we een hoge stabiliteit, terwijl tijdelijk geëxciteerde Rydbergtoestanden - toestanden met een hoog dipoolmoment - gebruikt worden voor interacties tussen de qubits. De magnetische microvallen zijn gebaseerd op een structuur van magnetisch materiaal op de chip, wat veel vrijheid biedt in de keuze van een roosterpatroon, en waardoor het systeem ook zeer schaalbaar is. De atomen worden gemanipuleerd en afgebeeld met behulp van lasers, waardoor een groot deel van het experiment uit optische elementen bestaat.

Mijn project bestond uit het laden van een vierkant en hexagonaal rooster, met een rooster-afstand van 10 μm. De ladingsprocedure begint met het loslaten van de atomen in een vac-uumkamer, waar ze gevangen worden in een magnetisch-optische val. In deze val worden de atomen met lasers afgekoeld tot een temperatuur van een paar honderd microkelvin en gevan-gen in de potentiaal van een magnetisch quadrupoolveld. Vervolgevan-gens worden de atomen in een volledig magnetische val geladen, die gebaseerd is op een z-vormige draad onder de chip. Hier worden de atomen verder afgekoeld door alleen de heetste atomen te verdampen met behulp van radiofrequente straling, tot ze een temperatuur bereiken van ∼10 μK. Door de magnetische velden aan te passen worden de atomen langzaam dichter bij de chip gebracht, tot ze condenseren in het rooster van microvallen.

Uit mijn onderzoek bleek dat de precieze instelling van o.a. de magnetische velden tijdens de ladingsprocedure zeer kritiek is. Twee belangrijke factoren zijn de temperatuur en de spinricht-ing van de atomen. Door de magnetische velden en de koelspinricht-ingsmechanismen te optimaliseren, experimenteel en met behulp van simulaties, hebben we een ladingsprocedure gevonden waarmee we een rooster van ongeveer 800 microvallen kunnen laden met gemiddeld 400 atomen per val, 6 μm boven de chip. Dit is de eerste stap op weg naar een quantumcomputer gebaseerd op rubidiumatomen in een rooster van magnetische microvallen boven een atoomchip.

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1 Introduction

As technology grows ever smaller we are approaching a regime where the laws of quantum mechanics start to play an important role. The study of quantum information uses the com-plexity of these laws to analyze problems that cannot easily be studied using classical computers (Feynman, 1982). Quantum systems are used to either simulate the behaviour of other quan-tum systems (Buluta and Nori, 2009) or to compute hard or unsolvable problems (problems for which the classical computing time increases exponentially or faster with the size of the problem) (Grover, 1996; Shor, 1997).

For quantum computation, quantum bits, or qubits, are defined as the smallest units in some physical quantum system. According to quantum mechanics they can either be in one of two separate states, or in a superposition of the two states. Quantum gates, operations that change the state of one or several qubits, are then performed in a specific algorithm such that the final state of the qubits holds part of the solution to the computational problem.

Although there is already much literature on the theoretical working and algorithms of a quan-tum computer (Nielsen and Chuang, 2010), the physical realization of such a quanquan-tum computer poses a challenge as it has to meet several strict criteria (DiVincenzo et al., 2000). Many quan-tum systems have been investigated extensively over the past decade; some of the most promising physical realizations include trapped ions (Monroe and Kim, 2013), superconductors (Devoret and Schoelkopf, 2013), spintronics in semiconductors (Awschalom et al., 2013), topological sys-tems (Stern and Lindner, 2013), and Rydberg atoms (Saffman et al., 2010).

1.1 Quantum information using Rydberg atoms on magnetic

atom chips

To be useful for quantum information processing, a physical system must be both weakly in-teracting with its environment to improve coherence, and well accessible to allow for easy ma-nipulation. Ultracold atoms have both properties, as they can be trapped in optical, electric or magnetic fields, while their internal states can be controlled and measured with lasers. Ions trapped by electric fields, and neutral atoms trapped by magnetic fields are both topics of inten-sive research, where the first approach is technically the most developed (Garcia-Ripoll et al., 2005). Neutral atoms, however, are interesting because of their weak interactions and their in-herent stability. Compared to ions they are less sensitive to their environment and their ground states have longer lifetimes. Interactions can be tuned using Rydberg atoms. These are ordinary atoms in a state with a high principal quantum number, which gives them the general behaviour of hydrogen and a strong dipole moment. By exciting only specific atoms to a Rydberg state, interactions can be made switchable. The atoms can then be used to probe atom or atom-surface interactions, or as a mechanism to create entanglement between multiple qubits (Saffman et al., 2010). Lasers are used to conveniently switch between Rydberg- and ground states, as the qubit states are still encoded in more stable hyperfine levels of the ground state.

Optically, neutral atoms have been trapped in arrays of dipole traps (Dumke et al., 2002; Saffman and Walker, 2005). Magnetically, neutral atoms were initially trapped with macroscopic coils, while later experiments have made use of miniaturized current carrying wires in conjunction with an external homogeneous magnetic field to generate magnetic traps of almost arbitrary geometry. Miniaturization of the traps leads to much stronger confinement as the field gradients can be made very high. By integrating the microscopic wires with well known chip technology to create ‘atom chips’, many applications can be thought of; including model systems for

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three-and one-dimensional quantum gases, disordered systems, quantum information processing, in-tegrated atom optics, matter-wave interferometry, precision force sensing, and studies of the interaction between atoms and surfaces (Fort´agh and Zimmermann, 2007). Thinking of quan-tum information processing, atom chips provide a trapping environment that is easily scalable and by changing the geometry, very flexible in probing different types of interactions.

Permanent magnetic atom chips, for which the current carrying wires are replaced by a pattern of magnetic material, are a topic of research in only a few places (Jose et al., 2014; Llorente Garc´ıa et al., 2013). Using magnetic material, the patterns can be made even smaller without having problems due to high current densities. Using optical lithography to pattern the magnetic material provides more freedom in trap sizes and geometries: with current solid state technology patterns can be made in nanometer ranges, opening up possibilities of quantum simulations with single atoms. Trap geometry boundaries are also especially interesting to investigate with these techniques as they are much harder to create with optical lattices. These ideas have resulted in a setup in Amsterdam that traps ultracold atoms in a magnetic lattice on an atom chip (Gerritsma et al., 2007).

1.2 Overview of the experiment

A layer of permanently magnetized FePt is patterned on a Si chip using optical lithograpy, to create a lattice of microtraps with a lattice spacing of 10 μm. This is a good range to start probing Rydberg interactions between traps (Saffman et al., 2010), while still keeping individual traps optically resolvable. A hexagonal and a square lattice are patterned side by side to experiment with different geometries and study boundary effects. The chip is mounted on a layer of z-shaped silver wires that are used to initially trap and move a cloud of87_{Rb atoms}

before they are loaded into the microtrap lattice. The chip and mount are enclosed by a glass cell and kept at a vacuum pressure of 10-11 _{mbar to reduce background collisions and thereby}

improve trap lifetimes. Three pairs of external magnetic coils are mounted around the glass cell to create various fields necessary for trapping and moving the atoms, and to compensate for background fields. During the experimental sequence, 87Rb atoms are first trapped in a mirror-magneto optical trap (M-MOT) (Reichel et al., 1999) and laser-cooled to a few hundred microkelvin. The atoms are then taken through several loading stages that involve the use of the silver wires, optical pumping, magnetic trapping in Ioffe-Pritchard type of traps (Pritchard, 1983), and evaporative cooling before they are finally loaded into the microtrap lattice, resulting in ensembles of a few hundred atoms per trap.

Each ensemble forms a single qubit. The qubit state |0i is represented by all atoms in the hyper-fine ground state 5S1/2|F = 1, mF = −1i. The qubit state |1i is represented by a superposition

of one atom in the hyperfine ground state 5S1/2|F = 2, mF = 2i and all other atoms in the

hyperfine ground state 5S1/2|F = 1, mF = −1i. The qubit |1i state is thus encoded in the state

of one atom, which means that single atom loss could lead to a qubit decaying to the |0i state, but not to the loss of a qubit. The use of ensembles also provides a high Rabi frequency, which enables faster quantum gates. Rydberg states are used as a transitional state to ensure that only one atom ends up in an excited state, and to create interaction between traps by use of the Rydberg blockade mechanism (Tong et al., 2004). All information about the state of the atoms is probed by absorption imaging, which is based on a high-numerical aperture lens mounted in-vacuum to obtain high resolution images.

Except for the chip and the silver wires, the setup is largely based on a previous version of the experiment, where a 20 μm lattice spacing was investigated (Gerritsma, 2007). This setup was designed for relatively easier replacement of the chip, to gradually move towards smaller lattice spacings while optimizing the loading procedure for each chip generation.

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1.3 Outline of this thesis

This thesis covers the research I did starting from the focusing of the in-vacuum lens, the installment of the 10 μm chip, up to the loading of the microtrap lattice. While loading the lattice was in the previous version of the experiment just a matter of cooling and lowering the atoms onto the chip, much difficulty was encountered in the final loading stages and most work went into optimizing the parameters of these stages. In addition, evaporative cooling was not working as expected, and a new coil was devised. Finally, to aid in the optimization of the loading and explain the problems that were encountered, the magnetic fields in the last few stages were simulated in Mathematica and compared to experimental data.

Chapter 2 covers the theory for concepts used in the rest of the thesis. Experimental techniques for cooling and trapping atoms in each loading stage are briefly explained. Some theory for absorption imaging and the methods we used to compute physical quantities from absorption images are covered next. Finally, the equations and methods used for simulating the magnetic traps are discussed.

Chapter 3 describes the parts of the experimental setup as well as related measurements and calibrations. The calibration of the new RF coil is discussed and characterizing measurements for the magnetic coils are shown. The optics, and especially the focusing and measuring of an upper bound for the resolution of the in-vacuum lens are also described in this chapter. Chapter 4 covers the detailed experimental sequence for loading the microtrap lattice. It dis-cusses the problems and solutions that were found along the way, and compares the results to simulations.

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2 Theory

This chapter gives an explanation of the experimental techniques and concepts that are used in the experiment. Absorption imaging is discussed as a method to probe and calculate the various physical quantities that we used to optimize the loading process and the theoretical background for the simulations is given.

2.1 Loading stages

Starting with a cloud of 87Rb atoms in a vacuum-chamber, the experiment is divided into several stages that prepare the atoms for trapping in the microtrap lattice by gradually cooling to microkelvin temperatures and shifting their position closer and closer to the chip.

2.1.1 Magneto-optical trap

In the first stage, the atoms are trapped and cooled in a magneto-optical trap (MOT) (Foot, 2005). This type of cooling is achieved generally by six slightly detuned orthogonal laser beams that collectively form a closed loop system that cools the atoms to several hundreds of mi-crokelvins, as explained below and illustrated by Figure 2.1. Note that in this setup we use only four laser beams, reflected at a 45° angle on the chip to cover all six directions, to be able to create a MOT several millimeters above the chip and smoothly transition to the uMOT, which is based on silver wires below the chip (explained in section 2.1.2).

When an atom in the center of the beams moves in a certain direction it observes a slightly higher frequency of the opposing beam as a consequence of the Doppler effect, which brings it closer into resonance and increases the chance of absorbing a photon from that beam. When a photon is absorbed with momentum ~k, the atom receives a kick in the opposite direction of movement, slowing it down. Shortly afterwards the photon is spontaneously emitted, but with the slightly higher energy of the resonance frequency, thereby decreasing the kinetic (thermal) energy of the atom. Atoms are now slowed down, but can still slowly diffuse out of the laser beams. Coupled with a quadrupole magnetic field, the atoms can be trapped in the center of the beams: as illustrated in Figure 2.1, a pair of magnetic coils with currents in opposite direction produces a magnetic field with B = 0 in the center and increasing in every direction. Atoms moving away from the center experience a Zeeman shift that brings them closer to resonance with one of the laser beams, depending on the polarization, such that the atoms are always pushed towards the center of the trap.

As each photon carries only a small momentum, this process needs to happen many times before the atoms are cooled down to microkelvin temperatures. There is a small probability that atoms are excited off-resonantly to the wrong hyperfine excited state, from where they can decay to the wrong hyperfine ground state. This would take them out of resonance with the laser and out of the cooling cycle. A repumper laser is used to excite these atoms back into the cycle.

The minimum temperature is limited by the natural linewidth of the used transition, which causes even the very cold atoms to ocassionally absorb a photon, giving them a momentum kick of ~k, where ~ is Planck’s constant divided by 2π and k is the angular wavenumber of the photon. The Doppler cooling limit is given by

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where ~ is Planck’s constant divided by 2π, γ is the inverse excited-state lifetime τ−1, and kB

is Boltzmann’s constant. For Rb: TDoppler ' 145 μK. (Steck, 2010)

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Figure 2.1: (a) A pair of coils in anti-Helmholtz configuration produces a magnetic field that increases in every direction from the center of the trap, at the intersection of three orthogonal pairs of circularly polarized laser beams. (b) The Zeeman splitting of the atomic sub-levels depends on the position in the magnetic field gradient. The hyperfine states move into resonance with the polarized laser beam that is directed towards the center of the trap. Source: (Foot, 2005)

2.1.2 uMOT

After capturing and initial cooling atoms in the MOT, the cloud is transferred to a so-called u-wire MOT (uMOT) which brings the atoms closer to the surface of the chip (Gerritsma, 2007). The external magnetic coils are now used in a Helmholtz configuration to create a homogeneous magnetic bias field. Together with the field of a u-shaped silver wire below the chip, a quadrupole field is formed around the point where the bias field and the field from the wire cancel each other. A schematical overview is given in Figure 2.2 and a cross-section of the field is shown in Figure 2.3.

Iu

By

Figure 2.2: Schematical overview of the u-wire. The field produced by a current Iu running through a

u-shaped wire is combined with a homogeneous field By to create a quadrupole field.

2.1.3 Polarization gradient cooling & optical pumping

Using the polarization gradient of counterpropagating circularly polarized laser beams, the atoms are further cooled to temperatures below the Doppler limit. While the details of this cooling mechanism are described in (Dalibard and Cohen-Tannoudji, 1989), it is important to note that the frequency of the laser beams should be detuned below resonance, and that the magnetic fields should be reduced to weaken the field gradient. The new temperature limit is set by the

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2.1 Loading stages

(a) (b)

Figure 2.3: (a) Vector plot of an ideal quadrupole field, with the field axes indicated by a red and green line. (b) Vector plot of the field produced by a u-wire and a homogeneous field. This is a cross-section of the field above the center of the middle wire segment, perpendicular to the wire.

small momentum that an atom which was initially at rest after absorption gains by spontaneous emission of a photon. This process sets the lowest temperature that can be reached while the light is still on, known as the recoil temperature:

Trecoil= ~ 2_k2

2mkB

(2.2)

where ~ is Planck’s constant divided by 2π, k is the wavenumber of the light, m is the mass of an atom, and kB is Boltzmann’s constant. For Rb: Trecoil' 350 nK. (Steck, 2010)

To prepare for the next stage the atoms have to be in a magnetically trappable state, i.e. a state that has a positive Zeeman shift and therefore has a minimal potential energy in a magnetic field minimum (Foot, 2005). To realize this, an optical pumping beam transfers all the atoms to the |F = 2, mF = +2i state, as is explained in Figure 2.4.

Final state Decay

σ+

Excitation

Figure 2.4: Optical pumping: σ+ polarized light drives ∆mF = +1 transitions such that after

sev-eral excitations all atoms end up in the encircled |F = 2, mF = +2i state. Source:

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2.1.4 Magnetic trap

In this stage the atoms are transferred from the magneto-optical trap to the magnetic trap. The trapping force in this trap is no longer supplied by radiation-pressure, but by the shift in poten-tial energy due to the Zeeman effect. The atoms have to be very cold (hundreds of microkelvins), i.e. have a very low kinetic energy, to be contained by a magnetic trap, but the advantage of this kind of trap is that there is no limit to how cold the atoms can get. Magnetic traps are often used as a last trapping stage for Bose-Einstein condensation as this requires conditions of very high densities and very low temperatures. (Foot, 2005)

We use the field generated by a z-shaped wire in combination with a homogeneous external bias field in the y-direction to create a slightly elongated magnetic trap (Reichel et al., 1999), as is shown in Figure 2.5. A magnetic field minimum is created at the height where the bias field cancels the field from the wire, but this time the end segments of the wire both create a field in the positive x-direction which cannot be canceled by the bias field. The result is a magnetic field minimum that increases positively in every direction with a nonzero x-component at the minimum, known as a Ioffe-Pritchard trap. The field in the minimum is called the Ioffe field, and the direction sets the quantization axis for the atoms in the trap. By changing the combination of the current through the wire and the strength of the bias field we can adjust the location of the trap minimum and the field gradient around the minimum: a higher current (Iz) pushes the

trap minimum away from the chip and a higher field (By) pushes towards the chip. The position

can then be adjusted by changing either one and the compression can be adjusted by changing both while keeping the ratio Iz/By constant.

By Iz x y z (a) _(b)

Figure 2.5: (a) A current through a z-shaped wire is combined with an external homogenous bias field in the y-direction. (b) This graph shows a cut-through of the magnetic field at the height of the field minimum in the z-direction. The bias field (By) and the field from the middle

segment cancel each other above the center of the middle wire segment, resulting in a field minimum where atoms can be trapped. The height and steepness of the trap depend on the current and the bias field strength. A red dot indicates the position of the minimum, where the atoms will be trapped.

2.1.5 Evaporative cooling

Further cooling below the recoil limit is done in the magnetic trap using the technique of evap-orative cooling. The principle of this technique is to continually remove only the hot atoms from the trap, leaving the colder atoms and thereby lowering the average temperature in the following way. At any finite temperature the atoms are oscillating in the trap, with a maxi-mum radius that depends on their temperature. As the atoms are confined in a magnetic field gradient and the Zeeman shift depends on the field strength, the splitting between different

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2.1 Loading stages

spin states depends on the position of the atoms and so does the frequency of the RF radiation that is necessary to flip their spin. The energy levels of spin-flipped atoms shift in the opposite direction, turning the potential valley into a hill and causing them to leave the trap. By flipping the spin of atoms at a certain position(radius) in the trap with RF-radiation of the appropriate frequency only the atoms that are hot enough to reach that position are removed. This process effectively cuts off all atoms above a certain energy and can be referred to as an RF-knife. Then, by slowly ramping down the frequency while giving the atoms enough time to rethermalize only the hottest atoms in the tail of the velocity distribution are continually removed and the average temperature is lowered. As there is no fundamental limit to this type of cooling, the minimum detectable number of atoms is the only restriction. (Foot, 2005)

2.1.6 Magnetic microtraps

The 10 μm spaced lattice of microtraps uses the same trapping concept as for the Z-wire trap. However, the Z-wire is now replaced by a patterned layer of permanently magnetized material on a chip. As the magnetic material is magnetized out-of-plane, the magnetic field lines curl around the edges of the pattern as they would around a wire. The edges of the material can thus be treated as microscopic wires, and with an external bias field a lattice of Ioffe-Pritchard traps can be formed. The location of the traps depends on the bias field and the Ioffe field, with two options shown in Figure 2.6. Although the position can be chosen for optimal trapping parameters, the loading sequence has to be matched accordingly to prevent atom loss.

Figure 2.6: Vector plot of the field created by the magnetic material. The magnetic material (shown in purple) is magnetized out of the plane, with the magnetic field lines curling around the edges of the material. Solid lines show manifolds where traps can be formed with appropriate bias fields. Two optional trap locations are indicated by red dots and will be discussed later on.

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2.2 Absorption imaging

All information about the experiment is obtained through imaging. We use absorption images to determine important parameters like the temperature, the number of atoms, their collision rate, their position, their spatial distribution, the lifetime of the trap, and the phase space density. By looking at these parameters after each step of the loading sequence and optimizing them for each step we gradually find a loading sequence that produces enough atoms in the microtraps to start experimenting.

In absorption imaging the attenuation of a laser beam passing through a cloud of atoms is measured by looking at the shadow of the cloud, which is captured on a CCD camera. Laser light resonant with the |F = 2, mF = ±2i to |F0 = 3, m0F = ±3i transition is scattered by the

atoms and the wave of scattered light is focused on a CCD camera by two lenses. As discussed in more detail in (Mulder, 2013), for objects that are much larger than the wavelength of the light, the shadow is not formed by an interference pattern between the incident and scattered light and no lenses are necessary to focus the image. Therefore we have used three different imaging methods, conceptually shown in Figure 2.7.

Laser L1 L2 CCD atom (a) chip mirror image atoms CCD Fiber (b) chip Fiber CCD atoms L1 (c)

Figure 2.7: (a) Conventional absorption imaging setup. Resonant light scattered by an atom is focused on a CCD by two lenses. The measured shadow is formed by interference between the scattered (blue) and incident (red) light. (b) Side imaging setup to measure the distance between the chip and the atoms. Light is reflected off the chip-surface and the distance is determined as a function of the distance between the reflected and unreflected atom cloud image. (c) Direct side imaging setup to observe and optimize various aspects of the loading sequence. A lens is included to provide a magnification.

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2.2 Absorption imaging

2.2.1 Optical density

Lambert-Beer’s law states that the transmission of light (T ) through a column of atoms depends logarithmically on the product of the scattering cross-section (σ), the density of the atoms in the column (n), and the length of the column (l):

T =If I0

= e−σnl (2.3)

where If is the intensity of the light after attenuation and I0is the intensity of the light before

attenuation. The optical density is defined as the column density times the scattering cross-section. (Ketterle et al., 1999)

O.D. ≡ σ Z

n(z) dz (2.4)

This gives the transmission in terms of the optical density:

T =If I0 = e−O.D. (2.5) or equivalently, O.D. = −LnIf I0 .

In the experiment, three different images are taken to obtain I0 and If. First, an absorption

image is taken with the atoms still in place. Then the atoms are released and a light image is taken. Finally, the light is turned off and a dark image is taken to correct for background light. The optical density is then calculated for every pixel as

O.D. = −Ln absorption − dark light − dark

(2.6)

(a) Absorption (b) Light

(c) Dark (d) Optical Density

Figure 2.8: Examples of the three absorption images that are taken to calculate the optical density. A colormap is assigned relative to the maximal differences in photon counts per pixel, which means that the same colors indicate different amounts of light in different images. The count number indicates the total photon counts over the whole image.

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2.2.2 Atom number

By adding the optical density for every pixel in a certain region of interest (ROI), dividing over the scattering cross-section σ0 and correcting for the magnification M and the pixel size wpx,

we can estimate the total number of atoms in the selected region of interest:

Nsum= w_px M 2 1 σ0 X ODROI (2.7)

A more accurate way to determine the number of atoms is to use the assumption that the density distribution is Gaussian. By fitting the atomic cloud with a Gaussian function we obtain its 1/e2 width 4σ and the absorption amplitude A in the x- and y-direction. Taking the mean of the two amplitudes, the number of atoms is then given by

Nint= w_px M 2 1 σ0 Ax+ Ay 2 ∞ Z −∞ e− x2 2σ2_x _dx ∞ Z −∞ e− y2 2σ2_y _dy _(2.8)

Using the Gaussian integral

∞ Z −∞ e−(x−b)22c2 dx = c √ 2π leads to Nint= w_px M 2 1 σ0 Ax+ Ay 2 2πσxσy (2.9)

Note that this method can only be used for images that contain a fraction of the atomic cloud that is large enough to fit a Gaussian function. When the fit is not accurate, the first method to calculate the number of atoms Nsum is more reliable.

Fluorescence imaging

A third way to calculate the number of atoms is by using fluorescent light that is emitted during the MOT stage (Torralbo-Campo, 2012). We use a lens to focus some of the emitted light on a photodiode, as shown in Figure 2.9.

r d

Figure 2.9: Many photons are spontaneously emitted during the MOT stage. A fraction of this flu-orescent light is captured and focused on a photodiode. For a lens with radius r at a distance d, the cone over which the light is captured is defined by the solid angle Ω = 2π 1 − cos tan−1 r_d.

The number of atoms is given by

Nf l= 4π Ω Pf l 0.96n_P at (2.10)

where Ω is the angle over which the fluorescent light reaches the detector, n is the number of uncoated glass surfaces that are passed, Pf lis the fluorescence power detected on the photodiode,

and Pat is the amount of energy emitted per atom. For a two-level system, Pat is calculated as

Pat= ~ω Γ 2 I/Is 1 + I/Is+ 4δ2/γ2 (2.11)

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where ~ is Planck’s constant divided by 2π, ω is the atomic resonance frequency, Γ is the natural linewidth of the transition, δ is the detuning between the laser frequency and the atomic resonance frequency, I is the intensity of the light that excites the atoms, and Isis the saturation

intensity. I is given by

I = 2P

πw2 (2.12)

where P is the incoming power (in this case 6 times the power per beam, as each dimension is covered by two counterpropagating beams) and w is the width of the beam. The fluorescence power Pf l is calculated as

Pf l=

V

Rρ (2.13)

where V is the voltage measured on the photodiode, R is the resistance of the oscilloscope, and ρ is the responsivity (in A/W) of the photodiode. The voltage difference caused by connecting the photodiode directly to the 1 MΩ oscilloscope input is included in the calculation by taking the resistance into account here.

Note that the reabsorption of fluorescent photons makes this method of measuring the number of atoms less reliable. Also because we consider only a two-level system here, this method should only be used as a rough approximation or as an estimate of the order of magnitude.

2.2.3 Temperature

We use two methods to determine the temperature. The first is using time of flight, where the atoms are released from the trap. While falling under gravity and expanding, the average velocity of the atoms is measured. The second method uses RF spectroscopy to probe the spectral distribution of the number of atoms in the trap, and thereby the temperature. For the second method only a brief explanation is given, based on chapter 6 of (Gerritsma, 2007).

Time of flight

For time of flight (TOF) images the magnetic fields are turned off after the experimental se-quence and the cloud of atoms is allowed to expand for a period of several milliseconds, after which an image is taken. As the images are taken from below gravity does not play a role and the cloud is expected to expand isotropically in all directions. We measure the velocity distribution for individual components separately and thereby obtain temperatures Txand Tyby the method

described below (Spreeuw, private communication).

The one-dimensional Maxwell-Boltzmann distribution for velocity component vxis given by

f (vx) = N exp − mv 2 x 2kBTx = N exp − v 2 x 2σ2 vx (2.14)

where N is a normalization constant so that R f (vx)dvx = 1. The average of the squared

ve-locity is ¯v2

x=R v2xf (vx)dvx= kBTx/m, which is consistent with the average kinetic energy per

dimension in an ideal gas (1

2kBT ). The 1/e

2 _{or r.m.s. width of the velocity distribution is then}

defined as σ2 vx ≡ ¯v

2

x= kBTx/m.

The expanded cloud on the TOF image can be fitted by the Gaussian function g(i) ∼ exp−(i−i0)2

2σ2 i

, defined in pixel coordinates i. Correcting for the magnification M and the linear pixel size wpx

yields the physical r.m.s. width σx= σiwpx/M . The physical width varies in time as

σ_x2(t) = σ_x2(0) + σ2_v

xt

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Fitting the squared cloud width σ_x2 for a series of measurements with the function a + bt2, we obtain b = σ2_v_x= kBTx/m, and similarly Ty.

Given the trap frequency, we can also obtain an approximation for the temperature from a single time of flight image:

By assuming that the hottest atoms will travel much further during expansion than in the trap, σ2

vxt

2_σ2

x(0), Eq. (2.15) can be reduced to σ2x(t) ≈ σ2vxt

2₌ kBTx m t 2_{, or} Tx≈ σ2 x(t) m kBt2 (2.16)

Having a first approximation of Tx we can now estimate the initial width of the thermal cloud.

For an ideal gas trapped in a harmonic potential U = 1₂mωx2x2, the density at point x is given

by

n(x) = n0 e−U/kBTx= n0 e−mω

2

xx2/2kBTx _(2.17)

For the thermal radius R, the distance from the trap center at which the density has dropped to 1/e of its maximum value, we find R =q2kBTx

mω2 x

.

Noting that 2σ2

x = R2 and combining this with Eq. (2.16) leads to an estimate for the initial

width of the thermal cloud:

2σ2_x(0) = R2(0) = 2kBTx mω2 x ≈2σ 2 x(t) ω2 xt2 (2.18)

We substitute this into Eq. (2.15). Using σ2 vxt

2₌ kBTx

m t

2 _{for the second term, we obtain a first}

approximation for the temperature as a function of the time of flight, the trap frequency, and the width of the expanded cloud on a single image:

Tx= σx2(t) 1 − _ω21 xt2 m kBt2 (2.19)

which we use as an indication for the temperature when we are not performing a series of time of flight measurements.

RF spectroscopy

Following the approach of (Gerritsma, 2007), a radio frequency pulse is used to couple atoms at a certain trap radius to an untrapped state, such that all atoms hot enough to reach this radius are expelled from the trap. The number of remaining atoms is then probed by absorption imaging, and using multiple pulses the atomic loss rate is measured as a function of RF frequency. A model is fitted to the data, giving the temperature as one of its fitting parameters. The model is based on a Landau-Zener picture and considers a two level system where the two levels are coupled during the rf pulse, with a Rabi frequency proportional to the rf amplitude: ~ΩRabi= µBgFBrf.

The atomic loss rate is calculated as the number of resonant atoms times the Landau-Zener spin-flip probability and results in the following expression:

1 N dN dt = 4√π(mF~)3/2Ω2Rabi (kBT )3/2 √ ωrfe− mF ~ωrf kB T _(2.20)

where we use the RF frequency above the trap bottom frequency: ωrf = 2π(f − f0).

The data is fitted using a formula of the more general form

F (ωrf, α, T ) = H(ωrf) ωrfαA e

−mF ~ωrf_{kB T}

(2.21) where H(ωrf) is the Heaviside step function, and α and A are fit parameters.

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2.2.4 Collision rate

The collision rate is an interesting quantity to calculate, as it determines the rate at which evaporative cooling can be performed: a high collision rate leads to a fast thermal equilibrium, which is necessary to ensure that only the atoms in the tail of the velocity distribution are continually evaporated during the cooling phase. As evaporative cooling times are in the same order of magnitude as the lifetime of the trapped atoms (seconds), it is useful to optimize this quantity. The elastic collision rate in the center of the trap is defined in (Walraven, 2010) as the inverse of the collision time τc:

1 τc

= √1

2n0σ ¯v (2.22)

where n0 is the density in the center of the cloud, σ is the collisional cross-section, and ¯v is the

average atomic velocity. For bosons in the same state σ = 8πa2_{, where a is the s-wave scattering}

length, which is 95 times the Bohr radius for Rubidium-87. The average speed of the atoms is given by ¯v =p8kBT /π m.

The density in the center of the cloud n0 is derived using our definition of the thermal radius in

section 2.2.3. When we define the number of atoms in a Gaussian shaped atomic cloud as

N = ∞ Z −∞ ∞ Z −∞ ∞ Z −∞ n0 e −x2 R2_x _e −y2 R2_y _e −z2 R2_z _{dx dy dz} _(2.23) then, using R = q 2kBT mω2 we obtain n0= N π3/2_R xRyRz = N _m 2πkBT 3/2 ¯ ω3 (2.24)

where ¯ω is the average trap frequency, ¯ω3= ωxωyωz.

2.2.5 Phase space density

The phase space density is an important parameter when aiming for Bose-Einstein Condensation or in general for a very cold and dense cloud of atoms. While the temperature can be lowered by turning down the strength of the trap, or the density can be increased by compressing, the phase space density captures both a high density and a low temperature into one parameter. It also determines when quantum statistics become important and Bose-Einstein Condensation occurs, at φ ≈ 1. When the collision rate is high enough, evaporative cooling can lead to runaway evaporation, where the density increases despite the loss of atoms, accelerating the evaporation rate and increasing the phase space density by several orders of magnitude.

The phase space density is defined in (Townsend et al., 1995) as

φ = n0Λ3= N ~¯ω kBT 3 (2.25)

where we used the definition of the thermal de Broglie wavelength Λ ≡p2π~2_/(mk

BT ), and

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2.2.6 Designing the imaging system

The imaging system is subject to certain restrictions and the components have to be chosen accordingly. For the conventional imaging setup shown in Figure 2.7a, both the atoms and the CCD have to be put in the focal point of the lens to obtain a sharp image. When the distance between the two lenses is exactly the sum of their focal distances f1+f2, the incoming collimated

beam is transformed back into a collimated beam, with a magnification given by M = f2

f1. Note

that the collimation can be corrected by additional lenses if the distance between the two lenses is not exactly the sum of their focal distances. The width of the lenses has to be larger than that of the laser beam to prevent diffraction effects from interfering with the images. However, the laser should be wide enough to achieve a uniform illumination of the atoms that are to be imaged, even though the intensity profile of the laser is Gaussian. The power of the laser should be high enough to produce enough counts to distinguish different features on the images but low enough to prevent saturation of the atoms or the camera. Finally, the laser should be locked properly as detuning leads to less absorption and an underestimation of the number of atoms. To achieve a high resolution, a lens with a numerical aperture of NA = 0.4 is mounted inside the vacuum cell, directly above the chip. While giving a high resolution, this was a technical challenge as the lens had to be focused with micrometer precision and the remaining optics and alignment had to be adapted to the high divergence of the outcoming beam.

The chip is used as a mirror for the probe beam, which means that a mirror image of the atoms will be formed by light that is first reflected and then absorbed. Whether we need to focus on one of the two images depends on the depth of focus, which we calculate below using the point spread function.

Point spread function

Due to wavelike nature of light, any beam coming from a finite size aperture (such as a lens) will have a certain divergence. For an imaging system, this limits the resolution, even when the system is perfectly in focus. The image formed by a point source in a focused optical system is described by the point spread function (PSF). For a lens with a circular aperture, the PSF has the shape of an Airy pattern, and is defined in (Saleh and Teich, 1991) as

PSF(r, λ, N ) = 2J1 πr λN πr λN ! (2.26)

where J1 is the Bessel function of the first kind of order one, r is the radial distance from the

optical axis in the image plane, λ is the wavelength, and N is the f-number of the system. The intensity of light in the image plane is given by the square of the PSF.

The f-number is given by N = f /D, where f is the focal distance and D the diameter of the lens. For the in-vacuum lens, f = 18.75 mm, D = 15 mm, and N = 1.25. The PSF can be approximated by a Gaussian function, which will be useful for calculating the depth of focus and the resolution later on. If we fit the Gaussian to the central peak of the PSF and equate the amplitudes, we find σ ≈ 0.42λN . The fitted Gaussian is shown next to the theoretical PSF for the chosen optics and a wavelength of λ = 780 nm in Figure 2.10.

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2.2 Absorption imaging -3 -2 -1 0 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 r HΜmL I Harb .units L e -r 2 2 Σ2 PSF2_H_rL

Figure 2.10: Comparison of a Gaussian function with a width of σ = 0.42λN to the square of the PSF.

Depth of focus

The depth of focus is defined in (Siegman, 1986) as twice the Rayleigh range; the distance from the focal point over which the beam radius is increased by a factor√2, which is given by ZR =

πw2₀

λ (see Fig. 2.11). To calculate the depth of focus, we approximate the waist of the

beam in the focal point (w0) by looking at the width of the PSF. The 1/e2-radius of the Gauss

function that we used to approximate the PSF is given by 2σ = 0.84 λN = 0.82 μm. The depth of focus is then

2 ZR=

2πw02

λ = 5.40 μm. (2.27)

With a spacing of 2 × 10 = 20 μm between the atoms and their mirror images, this means that we cannot have both in focus simultaneously.

Figure 2.11: Overview of common units around the focal point. w0 is the waist, ZR is the Rayleigh

range, and b is the confocal parameter or depth of focus. Source: http://en.wikipedia.org

Resolution

The angular or spatial resolution of a lens describes the ability of the lens to resolve detail, or more quantitatively: the smallest distance over which one can distinguish different elements in an image. If we consider an image to be made up of points, and each point is represented in the image by a PSF, the Rayleigh criterion states that two points are just resolved if the separation of the two points is equal to the radius of their Airy disks (the central lobe of the PSF) (Lord Rayleigh, 1879). Therefore we take the radius of the Airy disk as the resolution, given by

∆` = 1.220λf

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2.3 Magnetic trapping simulated

For the last several loading stages it becomes increasingly difficult to find an optimal path as there are many parameters to optimize and only a loading path that is very close to optimal will prevent the loss of all atoms. To get a better picture of the problems we had and quickly see the results of other approaches, a simulation was made in Mathematica. The simulation was used to find the position of the magnetic field minimum as a function of several parameters and to plot the field around the minimum. The goal was to find a set of parameters that cause the magnetic field minimum to smoothly transfer from a position close to that of the MOT to the microtraps, 10 μm above the surface of the chip. This section covers the definitions for the simulations used in sec. 4.3.

The magnetization of the chip is defined by a set of Fourier coefficients coming from an opti-mization algorithm for the trap shape which is discussed in (Tauschinsky, 2013). The simulation uses the first 69 of these Fourier coefficients to define an infinite periodic lattice. A cross-section of the simulated magnetization on the chip is shown in Figure 2.12, based on the function

~ M (x, y) =X n=1 An,3cos 2π a (An,1x + An,2y) + An,4sin 2π a (An,1x + An,2y) (2.29)

where An,3 and An,4 are Fourier coefficients, An,1 and An,2 are lattice wave vectors. a is the

lattice spacing (10 μm). -10 -5 5 10 y HΜmL 0.2 0.4 0.6 0.8 1.0 1.2 M Harb.unitsL

Figure 2.12: Cross-section of the magnetization on the chip along the line x = 0. The magnetization is normalized and between 0 (no material) or 1 (material). The odd periodicity is caused by a small rotation of the lattice’s symmetry axes relative to the lab coordinate system. A two-dimensional density plot of the magnetization is shown in the background of Figure 2.6.

The magnetic field can be written as the gradient of a vector potential, ~Bµ = −∇φ, with as

vector potential φ =1 2µ0hM0 X n=1 An,3cos 2π a (An,1x + An,2y) −An,4sin 2π a (An,1x + An,2y) e−2πaz √ A2 n,1+A2n,2 (2.30)

where the parameters h and M0define the overall strength of the lattice. h is the film thickness

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2.3 Magnetic trapping simulated

compared to the field of an infinite straight wire B = µ0I

2πz with a current I = hM0. The field

gradient and curvature tensors are defined as

~ ~ g ≡X ij gij êi· ê|j = − X ij ∂2_φ ∂xi∂xj ˆ ei· ê|j (2.31) ~ ~ ~c ≡X ijk cijk eî· ê|j · ê|k = − X ijk ∂3_φ ∂xi∂xj∂xk ˆ ei· ê|j· ê|k (2.32)

where gij and cijk are matrix element coefficients, ˆei is a unit vector, and ˆe|j is a transposed

unit vector.

For the simulation of the z-wire on a much larger scale, we will assume that the end segments stretch out to infinity and that the wire is infinitely thin. The middle part of the z-wire is placed along the x-axis, with a length dx. The field can be derived from the Biot-Savart law and is given by ~ Bxwire(Iz, x, y, z, dx) = µ0Iz 4π x + dx/2 p(x + dx/2)2_{+ y}2_{+ z}2 − x − dx/2 p(x − dx/2)2_{+ y}2_{+ z}2 ! 1 y2_{+ z}2   0 −z y   (2.33)

The y-segments of the z-wire stretch out to plus or minus infinity on one side and are modeled by ~ Bywire,+∞(Iz, x, y, z) = µ0Iz 4π y p y2_{+ x}2_{+ z}2 + 1 ! 1 x2_{+ z}2   z 0 −x   (2.34) ~ Bywire,−∞(Iz, x, y, z) = µ0Iz 4π 1 − y p y2_{+ x}2_{+ z}2 ! 1 x2_{+ z}2   z 0 −x   (2.35)

Combining the equations gives for the complete z-wire:

~

Bzwire(Iz, x, y, z) = ~Bxwire(Iz, x, y, z, dx) + ~Bywire,−∞(Iz, x + dx/2, y, z)

+ ~Bywire,+∞(Iz, x − dx/2, y, z)

(2.36)

Two additional silver wires are used to pinch the long x-axis direction of the trap. They are modeled by two infinite wires in the y-direction, with x-offset ± xp. Currents through the pinch

wires are running in the same direction.

~ Bywire,∞(Ip, x, y, z) = µ0Ip 2π 1 x2_{+ z}2   z 0 −x   (2.37) ~ Bpinch(Ip, x, y, z) = ~Bywire,∞(Ip, x − xp, y, z) + ~Bywire,∞(Ip, x + xp, y, z) (2.38)

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The field from the external coils can be regarded as a homogeneous field close to the atoms and is modeled by adding a constant term Bext= (Bx, By, Bz). The complete magnetic field is then

given by

~

Btotal(Iz, Ip, x, y, z, ~Bext) = ~Bµ(x, y, z) + ~Bzwire(Iz, x, y, z + zsub)

+ ~Bpinch(Ip, x, y, z + zsub) + ~Bext

(2.39)

where we have included an offset for the z-wire and the pinch wires, which lie approximately 500 μm below the surface of the chip. This final equation will be used for the simulations in chapter 4.

0 2 mm

0 20 μm

Figure 2.13: Separate simulations of ~Bext, ~Bzwire, and ~Bµ, from top to bottom. B~ext is shown as a

vectorplot, and for ~Bzwireand ~Bµthe magnitude of the magnetic field | ~B| is shown at the

trap height.

Trap frequency

Using a model explained in chapter 2 of (Gerritsma, 2007) we can compute the trap frequency. We define a tensor~~~t as a function of the gradient and the curvature of the magnetic field. The trap frequency ω is then a function of the eigenvalues Tn of this tensor.

~ ~ ~t ≡~~g · ~~g +~~_~_{c · ~}_B _(2.40) ω = s mFgFµBTn mRb| ~B| (2.41)

We can use this formula for the trap frequency to calculate the temperature, collision rate, and phase space density discussed in the previous sections.

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3 Experimental setup

This chapter gives an overview of the experimental setup and the related measurements & calibrations. In particular, the calibration of the RF coils is discussed in section 3.2.1, and the resolution measurements are discussed in section 3.2.2.

3.1 Overview

The following sections give a short overview of all the experimental parts. For a more detailed specification, see (Tauschinsky, 2013) & (Leung et al., 2014). For a detailed report on the construction of the chip mount, see (Pijn, 2013).

3.1.1 Vacuum system

The chip and atoms are enclosed by a vacuum cell which is kept at a pressure of ∼ 10−11 mbar by a Vacion Plus 75 Starcell ion pump, which is occasionally backed up by a Varian 916-0061 titanium sublimator. The system is based around a 6-port CF-63 stainless steel cube, which connects to a 45° angled arm (holding the cuvette, chip mount, and chip), a thermocouple, a Varian UHV-24p ion gauge to monitor the pressure, a vacuum feedthrough for connections to the silver wires, and the ion pump. The vacuum feedthrough supports a maximum of 10 A per pin, which is why some of the connections are doubled to allow a current of 20 A. After assembly and focusing of the in-vacuum lens, and using an additional turbo pump for initial pumping, the system was baked out at 140 to remove water molecules. As a supply for 87_{Rb, we use}

enriched Rubidium dispensers (Alvatec, s-type) mounted to the side of the chip mount with typical pulse times of 6-8 seconds at 5.8 A and 1.5 V.

3.1.2 Magnetic chip

The potential for the lattice of magnetic microtraps is provided by a patterned and magnetized 200 nm thin FePt film, deposited on a 330 μm thick, 15x20 mm2 _{wide Si substrate. The pattern}

is designed using an optimization algorithm for creating traps in arbitrary lattice configurations, that optimizes for maximum trap stiffness. Two lattices were designed to study, for example, a different number of nearest-neighbors and boundary effects at the border of the two lattices; one with a hexagonal and one with a square symmetry, shown in Figure 3.1. A duplicate pattern was made, resulting in two regions of magnetic patterns shown in Figure 3.2b; each consisting of a square and hexagonal part. The version with the least amount of imperfections was chosen after production and the chip was placed accordingly.

The film is patterned using optical lithography at the Ben-Gurion University in Israel, as ex-plained in (Leung et al., 2014). As the chip is also used as a mirror for the MOT and imaging beams, the surface is planarized with a 1 μm-thick layer of polymer SU8 and coated with a 90 nm thick layer of gold. To reduce the adsorption of Rubidium, an extra 25 nm of quartz (SiO2)

is added on top of the gold layer. The chip is magnetized by applying an external magnetic field of 5 Tesla in the out-of-plane direction.

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(a) (b)

Figure 3.1: Patterns on the chip for the (a) square and (b) hexagonal lattice after the etching process. The alternating strips are etched and non-etched regions. The patterns are aligned such that the Ioffe axes of the microtraps coincide with the Ioffe axis of the z-wire magnetic trap, to ensure a similar quantization axis for the atomic spin direction. Source: (Leung et al., 2014)

3.1.3 Silver wires

The chip is clamped to a set of silver wires cut out of 250 μm thick silver foil, which are used in the uMOT and magnetic trap (MT) stages. The h-shaped wires shown in Figure 3.2 are used both as u-wires for the uMOT and as z-wires to create a Ioffe-Pritchard type of magnetic trap centered above the magnetic pattern on the chip. The pinch wires on the sides are used to increase the trap strength in the x-direction (along the long axis of the magnetic trap). The wires are glued to a copper base which serves both as an electronic ground and as an outlet for heat produced by the silver wires. A series of copper wires connects the silver wires to a plug on the other end of the vacuum feedthrough. Typical currents through the z-wire are 0-20 A, limited by the vacuum feedthrough.

(a) (b)

Figure 3.2: (a) Silver foil containing the u-, z-, and pinch-wires. Note that we used part of the h-shaped wires as u-wires for a better positioning. (b) Silver wire design with an overlay (red) of the two regions of permanently magnetized FePt. The red patterns on the edges of the chip are for alignment purposes.

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3.1 Overview

Cu base Ti lens support

Ti spring lens (L1) with Ti lens holder Cu support tube

Rb dispenser

cross section

Si chip (330 μm)

Silver trapping wire (250 μm)

epoxy H77S

FePt magnetic structure (200 nm) Ti clamps

Figure 3.3: Overview of the chip mount without the cuvette and the external magnetic coils, see sections 3.1.1 to 3.1.4. The lens surface closest to the chip is electrically connected by a copper wire to subject the atoms to an electrical field between the lens and the grounded chip for EIT experiments, see (Rutjes, 2012).

3.1.4 Imaging setups

Two types of imaging setups were used: bottom imaging, i.e. aimed at and reflected by the chip using the high resolution in-vacuum lens, and side imaging to determine the location of the atoms in the z-direction. For bottom imaging we initially used a magnification of 2.4 to image the atoms in the MOT, and later a magnification of 13.3 to image the microtraps, see Figures 3.4a and 3.4b. For both bottom imaging setups, linearly polarized light from the probe laser comes in through a single-mode polarization maintaining fiber. The light is directed towards the chip by a polarizing beam splitter cube (PBS) and converted to circular polarized light by a λ/4 waveplate before entering the vacuum cell. After reflection, the circular polarization is converted back to a 90° shifted linear polarization by the λ/4 waveplate. The PBS then directs the beam towards a final lens and an Andor iKon-M 934 back-illuminated deep-depletion CCD camera, containing 1024x1024 pixels of 13x13 μm with a readout time of 300 ms. The in-vacuum lens is of the type Edmund Optics no. NT47-727, with a focal distance of 18.75 mm, and a diameter of 15 mm. Beam sizes and focal distances for the other lenses are indicated in Figure 3.4. For the microtrap imaging, the 1/e2_{-beam radius at the chip is σ = 0.35 mm. For a Gaussian beam}

profile, an area of 0.081 mm2_{, containing ∼ 800 traps, will then be illuminated with an intensity}

variation of less than 10%. With a magnification of 13.3 and a pixel size of 13 μm, individual microtraps fall within a pixel and the trap spacing is ∼ 10 pixels. Using pulse times of 50 μs at a power of ∼ 0.5 mW for the probe beam, we obtain on average a highest per-pixel-count of 5k for light images.

For initial side imaging, the probe beam is reflected by the chip at a very small angle θ, and produces two shadow images of the atom cloud, as in Figure 2.7b. In the limit θ → 0, the distance between the images is twice the atom-chip distance. For final side imaging the beam is aligned parallel to the chip and a lens is included to produce an image with a magnification of 2.26. See Figure 3.5 for examples of images.

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(a) (b)

Figure 3.4: Optical setup for (a) MOT imaging, and (b) microtrap imaging. See text for explanation and Figure 3.5a-c for example images. Sizes are in millimeters.

(a) (b) (c)

(d) (e)

Figure 3.5: Examples of images obtained with each setup: (a) Light image in the MOT imaging setup showing the outlines of the magnetic structure on the chip. (b) MOT image (magnification 2.4). (c) Magnetic trap (magnification 13.3). The previous images are all bottom images. (d) Side imaging showing the atoms and their mirror image, to determine the distance to the chip. (e) Side imaging showing a cloud of trapped atoms (upper part) and a cloud of atoms falling from the magnetic trap under gravity (lower part).

3.1.5 Laser setups

Three laser setups are used to produce the MOT & OP, probe, and repumper laser beams. All lasers are Extended Cavity Diode Lasers, frequency stabilized to one of the transitions shown in Figure 3.8. The reference spectra are obtained by saturated absorption spectroscopy through a small vacuum cell containing a mixture of rubidium gas isotopes. The optical setups are briefly discussed and schematically shown in Figure 3.7. For more specific technical details, see (Tauschinsky, 2013).

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3.1 Overview

MOT & optical pumping laser

Light from a Toptica TA-100 Master Oscillator/Power Amplifier unit running at 92 mA is sent through a circular aperture. At this point the power is ∼ 400 mW. A small percentage of the light is diverted to a rubidium vacuum cell for locking through saturated absorption spectroscopy (Mulder, 2011). The laser is locked to the F = 2 → (1, 3) crossover line, 212 MHz below the F = 2 → 3 transition. The rest of the light is run through a double pass Acousto-Optical Modulator (AOM), giving a total detuning of _2πδ = −13 MHz for the MOT cooling beam. During polarization gradient cooling the AOM is used to increase the detuning to

δ

2π = −55 MHz. An Electro-Optical Modulator (EOM) in combination with a polarizing beam

splitter (PBS) is used for fast switching of the light output between two single-mode polarization maintaining fibers leading to the MOT and the optical pumping beam. For the optical pumping beam, another AOM is included, to shift the frequency to the F = 2 → 2 transition. Using this setup we obtain a power of 60 mW for the MOT beam and 10 mW for the optical pumping beam directly after the fiber leading to the experiment. The MOT beam is then run through some additional optics, resulting in ∼ 6 mW per beam at the experiment.

Probe laser

The probe laser is based on a Toptica DL-100 laser system running at 162 mA. After diverting some light for locking purposes, the laser is sent through a fiber to the setup shown in Figure 3.6b. The frequency of the light is swept across the F = 2 → 3 transition using two double-pass AOMs, with a total detuning of −60 < _2πδ < +40 MHz. Scanning across the resonance frequency is used to find the frequency for which the most atoms are resonant in each stage of the loading sequence, as each stage brings a different magnetic field and a different Zeeman shift with it. A single-mode polarization maintaining fiber is used to transport the light to the experiment, with a power of ∼ 0.5 mW and a beam radius of 1.4(2) mm directly after the fiber. Using the AOM’s as a fast on/off switch, typical pulse times for absorption imaging are 50 μs.

(a) (b)

Figure 3.6: (a) MOT & OP laser setup. The enclosed part is used for saturated absorption spectroscopy, where PD denotes photodiode. λ/2 waveplates shift the (linear) polarization for optimal coupling into a fiber and to control the distribution of light by a polarizing beam splitter cube. Lenses are included to focus light in the AOM and apertures are included to select the appropriate order of frequency-shifted light (denoted by ±1). (b) Probe laser setup starting after the first fiber.

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Repump laser

The repump laser is based on another Toptica DL-100 laser system, running at 92 mA, and locked to the F = 1 → 2 transition. An EOM is used as a fast shutter. Directly after the fiber, a power of ∼ 10 mW is obtained. An optical isolator is included to prevent damage by reflection of light into the laser cavity.

(a)

Figure 3.7: Repump laser setup. See text for

expla-nation. Figure 3.8: Level scheme of the hyperfine structure of 87Rb with indi-cated laser frequencies. Source: (Tauschinsky, 2013)

3.1.6 Magnetic & RF coils

Three pairs of water-cooled coils, shown in Figure 3.9a, provide the magnetic fields necessary to trap and move the atoms. The coils are mounted around the vacuum cell in a Helmholtz configuration, centered on the atom trap. Using parallel currents for each pair of coils we can provide homogeneous magnetic fields in all three dimensions, for the Magnetic Trap (MT) and to cancel background fields. Antiparallel currents are used to create a quadrupole magnetic field in the MOT stage (commonly referred to as an anti-Helmholtz configuration). A FET-switch is used for fast current switching.

An external RF coil is included, and is used for evaporative cooling during the loading sequence. The coil, schematically shown in Figure 3.9b, consists of two circular windings of stranded copper wire, and is positioned to create an RF field perpendicular to the (Ioffe) quantization axis of the magnetic field in the Magnetic Trap (MT), while still allowing for two of the MOT beams to pass through the center of the coil. The coil is designed to create a field of 50 mG over a range of 0.2 to 35 MHz near the atoms, see section 3.2.1.

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3.1 Overview +x (a) +z +y (b)

Figure 3.9: (a) Three pairs of coils mounted around the vacuum cell: (1) MOT coils. (2) Big coils, used for the MT. (3) Small coils, used mainly to cancel background fields. MOT lasers are shown coming in from the top and the sides. (b) Schematic representation & position of the RF coil. The coordinate axes define the x, y, and z direction for the magnetic fields and for the simulations in chapter 4.

3.1.7 Computer control, sequence & software

The experiment is controlled by an Amplicon Ventrix 4020 PC, equipped with two National Instruments 6713 PCI cards that each control 8 12-bit analog channels with a maximum of±10 V output, and a Viewpoint DIO64 digital I/O card containing 64 digital lines divided over 4 banks (Tauschinsky, 2013). Using these connections, all shutters, EOMs, coils, cameras, and most AOMs are controlled by a Python based Labalyzer User Interface, shown in Figure 3.10. The program can be run in two modes; in direct control of all switches and parameters, mainly for alignment purposes, or preprogrammed by a user-defined timeframe. Using the second method, all steps of the loading sequence up to a certain point are preprogrammed and executed sequentially, after which an image is taken. The timeframe is run in a loop while its variables can be scanned to optimize for certain parameters, such as the number of atoms in the trap or the temperature of the atoms. The typical running time for a timeframe, which sets the repetition rate for the experiment and the update frequency for the absorption images, is ∼ 30 seconds. The program performs a fit of a Gaussian function (y = Ae−(x−µ)2/2σ2

) on a cut-through in both directions at a chosen position in the absorption image. The average signal in both directions is also computed, as well as the average optical density in a chosen box. All other parameters are derived from this information. During a scan, the absorption images as well as the fit parameters are automatically saved for further analysis.

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(a)

(b)

Figure 3.10: (a) Example of the Labalyzer UI running a timeframe up to the MT stage, showing the cloud of magnetically trapped atoms from the side. The intersection point of the two fitted cut-throughs is the point with the highest optical density close to the user-placed red cross, indicated by a red circle. The fit (white), cut-through at the cross (blue), and average signal (red) are shown above and to the side of the absorption image. (b) Part of the timeframe, showing the transfer to the uMOT stage. Variables are first defined and then used in a series of steps formatted as: number of milliseconds to wait before executing this step - channel name - new value - unit - ramp or step - ramp duration in milliseconds.

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3.2 Initial Measurements

3.2.1 Coil calibration

Magnetic coils

Using an N2774A Agilent current probe, an FLC-100 fluxgate sensor, and a KSY14 Siemens Hall sensor, the magnetic field in the center of the coils was measured as a function of the current running through the coils in a Helmholtz configuration. With a sensitivity of 2 V/A, the fluxgate was most sensitive, but could only be used for a current range of -0.1-0.1 A. The Hall sensor was used to cover the complete current range of -10-10 A. As the sensitivity of the Hall sensor was not known, a fit of the fluxgate data was used to calibrate the measurements of the Hall sensor, shown in Figure 3.11a. A fit of the Hall sensor measurements in Figure 3.11b shows a field-current relation of 11 G/A and a background field of ∼ 0.1 G. Note that the coils were close to, but not exactly in their final position, which may result in a slightly different background field. Field gradient measurements for the MOT coils in anti-Helmholtz configuration are shown in Figure 3.11c. The measurements indicate a field gradient of 11.28-12.72 G/cm for a current of 9-10 A. -0.15 -0.1 -0.05 0 0.05 0.1 0.15 -1.0 -0.5 0.0 0.5 1.0 1.5 IHAL B HG L Fit: B = 0.46 + 10.45 I Flux sensor measurements

(a) -10 -5 0 5 10 -100 -50 0 50 100 IHAL B HG L Fit: B = 0.11 + 11 I Small coils Big coils (b) -10 -5 0 5 10 -5 0 5 10 15 20 xHmmL B HG L 9.0 A: 11.28 Gcm 9.5 A: 11.76 Gcm 10.0 A: 12.72 Gcm (c)

Figure 3.11: (a) Sensitive fluxgate sensor measurements using the big coils. (b) Hall sensor measurements over a large range of currents using the big and small coils, calibrated with the fit data of (a). (c) Hall sensor measurements of the field gradient for the MOT coils in anti-Helmholtz configuration. The axis is chosen along the central coil axis.

Loading a 2D Microtrap Lattice on a Magnetic Atom Chip