Implementation of a lambda-enhanced gray molassesphoton.physnet.uni-hamburg.de/.../Masterthesis-Matthias-Tarnowski.pdf · Master Thesis Matthias Tarnowski ... In section3.3I discuss

FachbereichPhysik

Implementation and Characterization of a GrayMolasses and of Tunable Hexagonal Optical

Lattices for 40K

Implementierung und Charakterisierung einergrauen Melasse und eines variablen hexagonalen

optischen Gitters für 40K

Master Thesis

Matthias Tarnowski

Universität HamburgMIN-Fakultät, Department Physik

Institut für Laser-Physik

April 2015

Abstract

Sub-Doppler cooling is an important step to reach quantum degeneracy in atomic gases.It has recently been demonstrated that the Λ-enhanced gray-molasses scheme leads to lowtemperatures for alkali atoms. In the first part of this thesis, I present a detailed studyof a blue-detuned gray-molasses cooling scheme for fermionic 40K on the D1 transition.The Raman resonance condition between the cooling and repumping frequency in a Λ-typesystem has indeed to be fulfilled to observe the anticipated narrow feature of very lowtemperatures around the resonance. With optimal parameters we achieve a temperature of6 µK. In combination with repumping and optical pumping on the D1 transition we realizeefficient loading of a K-Rb mixture into a magnetic trap and fast subsequent evaporation.We significantly reduce the cycle time of our experimental sequence compared to coolingon the D2 transition. The results demonstrate ideal starting conditions for a sympatheticcooling experiment to produce a potassium degenerate gas.

The precise control of experimentally tunable parameters makes ultracold quantum gasesa perfect model system for simulating for example solid state physics. In the second partof this thesis the implementation of complete control over the polarization of the latticebeams provides a new tuning possibility for optical lattices. A set of quarter- and half-wave plates are used to create different lattice geometries, and especially AB-offset latticesare of special importance since they allow for the exploration of novel physics regardingthe impact on topological band structures. The band structure of a honeycomb lattice isprobed with amplitude modulation spectroscopy. This technique is introduced as a tool tomake the experimental setup of the optical lattice more reliable regarding beam imbalancesand polarization errors.

Zusammenfassung

Sub-Doppler Kühlung ist ein wichtiger Schritt zur Erzeugung eines quantenentarteten Fer-migases. Erst vor Kurzem wurde gezeigt, dass es bei Verwendung einer Λ-Konfigurationmöglich ist, sogar noch tiefere Temperaturen für Alkali-Atome zu erreichen als es mit ei-ner konventionellen grauen Melasse möglich ist. Im ersten Teil dieser Arbeit präsentiereich eine detaillierte Untersuchung der blau verstimmten grauen Melasse von 40K auf derD1 Linie. Dabei beobachten wir, dass die Raman Resonanzbedingung zwischen Kühl- undRückpumplaser in einem Λ-System erfüllt sein muss, um das erwartete schmale Merkmalvon besonders tiefen Temperaturen zu finden. Mit optimalen Parametern erreichen wir eineTemperatur von 6µK. In Kombination mit dem optischen Pumpen und Rückpumpen aufder D1 Linie erreichen wir effektives Laden einer K-Rb Mischung in eine Magnetfalle mitanschließender schneller Evaporation. Dabei reduzieren wir die experimentelle Zykluszeitsignifikant verglichen mit der vorher implementierten Kühlung auf der D2 Linie. Die Er-gebnisse zeigen ideale Startbedingungen für die Erzeugung von ultrakalten Kaliumgasen ineinem Experiment, das auf sympathetischer Kühlung basiert.

Die präzise Kontrolle über experimentell einstellbare Größen machen ultrakalte Quantenga-se zu einem perfekten Modellsystem zur Simulierung von Phänomenen beispielsweise aus derFestkörperphysik. Im zweiten Teil dieser Arbeit wird eine neue Möglichkeit zur Einstellungoptischer Gitter vorgestellt, indem volle Kontrolle über die Polarisation eingeführt wird.Durch Kombination von λ/2- und λ/4-Plättchen in den Gitterstrahlen werden unterschied-liche Gittergeometrien erreicht, wobei AB-Versatz Gitter von besonderem Interesse sind, dasie die Möglichkeit bieten neuartige Physik zu untersuchen, womit besonders der Einfluss auftopologische Bänder gemeint ist. Zusätzlich wird die Amplitudenmodulationsspektroskopiezur Untersuchung der Bandstruktur eines hexagonalen Gitters vorgestellt. Diese Technikwird als Werkzeug zur verlässlichen Justage des Gitters bezüglich des Strahlausgleichs ein-geführt.

Referenten

Referent: Prof. Dr. Klaus SengstockUniversität HamburgFakultät für Mathematik, Informatik und NaturwissenschaftenDepartment Physik – Institut für Laser-Physik„Quantengase und Spektroskopie“

Koreferent: Prof. Dr. Henning MoritzUniversität HamburgFakultät für Mathematik, Informatik und NaturwissenschaftenDepartment Physik – Institut für Laser-Physik„Quantenmaterie“

Erklärung zur Eigenständigkeit

Hiermit bestätige ich, dass die vorliegende Arbeit von mir selbstständig verfasst wurde undich keine anderen als die angegebenen Hilfsmittel - insbesondere keine im Quellenverzeichnisnicht benannten Internet-Quellen - benutzt habe und die Arbeit von mir vorher nicht ineinem anderen Prüfungsverfahren eingereicht wurde. Die eingereichte schriftliche Fassungentspricht der auf dem elektronischen Speichermedium.Ich bin damit einverstanden, dass die Masterarbeit veröffentlicht wird.

Hamburg, den 27. April 2015

Matthias Tarnowski

ix

Contents

1 Introduction 1

2 A Λ-Enhanced Gray Molasses 52.1 Theoretical introduction to a Λ-enhanced gray molasses . . . . . . . . . . . 5

2.1.1 Dark states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 The Sisyphus-effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Bright and gray molasses . . . . . . . . . . . . . . . . . . . . . . . . 82.1.4 The Λ-enhanced gray molasses . . . . . . . . . . . . . . . . . . . . . 10

2.2 Laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Setup of the optics system . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Frequency stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Experimental setup and conduction of measurements . . . . . . . . . 192.3.2 Results: Gray molasses . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Results: Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Tunable Hexagonal Optical Lattices 393.1 Theoretical introduction to tunable hexagonal lattices . . . . . . . . . . . . 39

3.1.1 Optical potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 Tuning of geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.3 Wave plates and polarization . . . . . . . . . . . . . . . . . . . . . . 453.1.4 Amplitude modulation spectroscopy . . . . . . . . . . . . . . . . . . 47

3.2 Experimental setup for the optical lattice . . . . . . . . . . . . . . . . . . . 513.2.1 Polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.2 Intensity stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2.3 Phase-lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Tuning of polarization: Calibration of the wave plates . . . . . . . . . . . . 563.4 Multiband spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.1 Amplitude modulation spectroscopy in a honeycomb lattice . . . . . 61

3.4.2 Calibration of the lattice . . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Conclusion and Outlook 71

List of Figures 73

List of Tables 74

Bibliography 80

1

1 Introduction

Ultracold atoms are widely used for quantum simulation because of their versatility andcontrollability [1]. Important examples include the exploration of the BCS-BEC crossoverby changing the interaction strength via Feshbach resonances, the observation of the Mott-insulator transition in optical lattices, band structure engineering in 3D and 2D opticallattices, and the study of low-dimensional systems [2]. By further decreasing the temper-ature even new exotic phases are predicted which is the reason why intense experimentaleffort is currently under way to get to new minima in temperature for ultracold fermionicgases in order to reach these novel regimes. Due to the existence of broad Feshbach reso-nances at low magnetic fields, potassium is an attractive alkali element [3].Most experiments on quantum degenerate gases begin with a laser cooling phase that is fol-lowed by evaporative cooling in a conservative trap. The final quantum degeneracy stronglydepends on the temperature at the end of the laser cooling phase and sub-Doppler cool-ing is often a key ingredient [4] for initiating efficient evaporation. In our experiment forfermionic 40K a cooling cycle on the D2 transition for a bright optical molasses has beenused. However, 40K features a narrow hyperfine structure in the excited state of the D2

transition that hinders efficient sub-Doppler cooling by cooling to the red of this transition[5]. The same is true for other isotopes of potassium and lithium.To overcome this limitation a gray molasses cooling scheme on the D1 transition at 770 nmcan be implemented to produce cold and dense atomic samples. To even further cool downthe sample a new technique using a Λ-scheme has been introduced in 2013 by the Salomongroup in Paris [6] and is now referred to as a Λ-enhanced gray molasses on the D1 transition.

The principal aim of this thesis was to implement and optimize such a Λ-enhanced graymolasses which would be a starting step for implementing an all-optical cooling setup. Anall-optical approach means loading a potassium gas directly from the gray molasses into adipole trap followed by fast evaporation via a Feshbach resonance without operating a mag-netic trap. Of course, very low temperature is needed and also a high phase space density.This yields much faster experimental cycle times and would therefore make the experimentvery efficient when acquiring enormous amounts of data. It was shown by Burchianti et al.

2 1 Introduction

[7] and Salomon et al. [8] that the gray molasses produces excellent starting conditions foran all optical production of degenerate gases.In this thesis I show that even without such an all-optical approach it is possible to re-duce the cycle time significantly when employing a sympathetic cooling experiment withfermionic 40K and bosonic 87Rb in a magnetic trap. Another advantage of this new schemeis that combined optical pumping and repumping is very easily implemented without signif-icant heating effects. This cooling scheme provides ideal starting conditions for sympatheticcooling of a potassium cloud in a bath of rubidium.

A nowadays very famous field of application for ultracold fermions opened up when the su-perposition of an ultracold gas with an optical lattice was suggested [9–11]. These systemsare well-suited for engineering solid-state simulations, as fermions resemble the electronsof a crystal and the optical potential its periodic lattice. These systems are described bythe Fermi-Hubbard model [12], though it is not possible to solve the Hubbard Hamilto-nian analytically which underlines that ultracold fermions are in fact quantum simulators.Optical lattices are widely tunable: crucial parameters like the depth of the lattice, thedimensionality and geometry of the atomic cloud in the lattice, the interaction strength canbe tuned freely allowing for very specific system realizations.In the second part of this thesis I describe the introduction of a new tuning knob into ourexperimental setup, which is the lattice geometry. The existing hexagonal lattice setup [13]is enhanced by exercising complete control over the polarization. A pair of wave platesfor each lattice beam allow for realizing a wide spectrum of different lattice geometries.This setup allows creating AB-offset lattices [14], where the two sites A and B are offset inenergy, which breaks inversion symmetry and allows to explore its impact on topologicalbands [15, 16].To fully rely on the lattice system, calibrations of the wave plates, of the lattice depthand other modifications regarding the intensity stabilization and phase-lock of the existingsetup had to be performed. In order to probe the lattice systems, we use the technique ofamplitude modulation spectroscopy, giving us a tool to examine the band structure [17, 18].

Bose-Fermi-Mixture project

The work for this thesis has been done at the Bose-Fermi-Mixture project (BFM), so theresults of this thesis are a result of a team effort. The BFM team besides me consists ofNick Fläschner, Dominik Vogel, Dr. Benno Rem, Dr. Christof Weitenberg and Prof. Dr.Klaus Sengstock.

3

Structure of this thesis

The second chapter of this thesis deals with the Λ-enhanced gray molasses and is dividedin three parts.In section 2.1 I present a theoretical introduction to sub-Doppler cooling. Concepts like darkstates and Sisyphus cooling and especially the Λ-enhanced gray molasses cooling mechanismfor 40K are explained. For better understanding a comparison of the previously used brightmolasses and the newly implemented gray molasses is added.In section 2.2 I present the setup of the optics system on the laser table used for the D1

gray molasses for 40K. Here, a short introduction to Doppler-free saturation spectroscopyis included, as well.In section 2.3 the experimental results of the Λ-enhanced gray molasses are presented andoptimal parameters for operating this cooling scheme are found. I also show the results ofD1 optical pumping and repumping before loading into the magnetic trap. Here, our resultsare compared with other experiments that also use a gray molasses scheme.

The third chapter of this thesis deals with tunable hexagonal optical lattices and is dividedin four parts.In section 3.1 I discuss the theoretical description of dipole potentials and how to createhexagonal lattices. I explain how we get complete control over the polarization of eachlattice beam via a set of quarter- and half-wave plates, which enables us to realize differentlattice geometries. The method of amplitude modulation spectroscopy to probe the bandstructure of such a system is introduced, as well.In section 3.2 the experimental realization of tunable hexagonal lattices is presented. I alsodescribe the implementation of a polarimeter into the experimental setup used for calibra-tion of the wave plates. The stability of our lattice setup regarding intensity stabilizationand phase-lock are discussed, as well. Here, heating rates in a honeycomb lattice and 1Dlattice without using a phase-lock scheme are measured.In section 3.3 I discuss our calibration method of the wave plates enabling us to tune toeach possible polarization. In section 3.4 I show first experimental results using amplitudemodulation spectroscopy to probe the band structure of a honeycomb lattice. An impor-tant step towards determination of the band structure with high precision is the calibrationof the lattice with regard to beam imbalances. The process of acquiring a reliable latticestructure is discussed in detail here.

4 1 Introduction

5

2 A Λ-Enhanced Gray Molasses

2.1 Theoretical introduction to a Λ-enhanced gray molasses

Until the end of the 1980s, it was expected that the limit for laser cooling would be theDoppler temperature. This limit would be attained by shining light on atoms from alldirections of space, frequency red-detuned by δ = −Γ/2 from an atomic resonance, where1/Γ is the lifetime of the excited state. In this case, each atom would perform a randomwalk in the momentum space due to the spontaneous emission, resulting an atomic ensembleat equilibrium temperature of

TD = ~Γ2kB

. (2.1)

For 40K this is a temperature of 145µK. However, experiments showed that temperatureslower than expected by one or two orders of magnitude could be reached. This effect wasthen explained as a consequence of another cooling mechanism: the Sisyphus or polarizationgradient cooling [4]. With this technique the temperature can decrease to the scale of therecoil energy

Trecoil = ~2k2L

2mkB. (2.2)

For 40K the recoil temperature is 0.41 µK. In case of the fermionic alkaline isotope 40K thehyperfine structure of the P3/2 excited state is too narrow to allow efficient Sisyphus cool-ing due to strong off-resonant coupling. The hyperfine-structure of the P1/2 state is moreresolved, thus one can use the D1 transition S1/2 → P1/2. In the 1990s many Sisyphus-likecooling mechanisms were reported, one of them is the gray molasses cooling scheme.Gray molasses cooling was proposed in [19, 20] and realized in a three dimensional configu-ration in the mid 90s on the D2 transition of cesium and rubidium by cooling these samplesdown to six times the single photon recoil energy [21–23]. In 2013 the Salomon group

6 2 A Λ-Enhanced Gray Molasses

observed an additional effect when implementing an additional Λ-system on resonance con-dition to even further cool down a sample of lithium [6]. In order to understand how thisΛ-enhanced gray molasses works one has to review three theoretical concepts. These arethe existence of dark states, the Sisyphus cooling mechanism, and the Λ-configuration itself.

2.1.1 Dark states

The concept that gives the gray molasses its name is the existence of so-called dark states.For an atomic ground state with angular momentum F , a gray molasses operates on theF → F = F ′ or F → F ′ = F − 1 optical transition. For any polarization of the localelectromagnetic field, the ground state manifold possesses dark states which are not coupledto the excited state by incident light. Figure 2.1 shows a simple example for σ+-light anda F → F = F ′ transition. The mF = 2 state is a dark state because another excitation isforbidden by the selection rules for electric dipole radiation. If an atom gets into this stateit will stay there forever because it is not coupled by light. For σ−-light the mF = −2 stateis dark and for π-light the mF = 0 state is dark. For other polarizations a superposition ofmF states is dark. The same way one always finds two dark states when a F → F ′ = F − 1transition is considered. The D1-line of alkali metals has this exact property. 40K featuresa transition from F = 9

2 to F ′ = 72 that will be discussed in the experimental part of this

chapter.

mF

dark statefor σ+ light

σ+

-2 -1 0 1 2

-2 -1 0 1 2

Figure 2.1: Dark states. mF = 2 is a dark state for σ+-polarized light for a F = 2→ F ′ = 2 transition.

2.1 Theoretical introduction to a Λ-enhanced gray molasses 7

2.1.2 The Sisyphus-effect

The second important model to describe sub-Doppler cooling is the Sisyphus-effect. Oneingredient for this way of cooling is the existence of polarization gradients i.e., a spatialmodulation of the polarization has to be present. The simple model in 1D is shown in alin ⊥ lin arrangement of laser beams in figure 2.2. If two counter-propagating laser beamswith linear polarization perpendicular to each other interfere to form a standing wave, aperiodic modulation of the polarization with period λ/2 is attained. The polarization variessmoothly from linear to σ− to linear to σ+ back to the initial linear polarization [4]. Notethat only the polarization and not the intensity is spatially modulated.

Figure 2.2: Polarization gradient. In a 1D model the polarization for a lin ⊥ lin configuration isspatially modulated with a period of λ/2. The polarization changes from linear to σ− to linear to σ+

back to linear. Figure taken from reference [4].

The second ingredient for sub-Doppler cooling is the so-called light shift originating from theatom-light-coupling. This effect causes a shift of the energy levels. In a polarization gradientfield the energy shift of the atomic levels is modulated in space, also depending on themagnetic quantum number mF. When talking about the Sisyphus effect one has a specificpicture in mind: constantly moving up a potential hill and falling down by means of opticalpumping processes, repetitively. The atom moves up a potential hill, thereby convertingkinetic energy in potential energy. On top of the hill an optical pumping process from ahigher to a lower energy level will happen with high probability, leading to a populationtransfer from one magnetic sublevel into another. Since the emitted light is generally ofhigher frequency than the laser light, energy is dissipated in the radiation field and the atomcan start again moving up the hill but now with lower kinetic energy. This will happen aslong as the atom has enough kinetic energy to climb the potential hill. Repeated cycles


lead to a decrease of velocity of the atom, thus the temperature sinks.As the kinetic energy decreases and atoms are not able to climb the potential hill anymore, a cooling mechanism based on the Sisyphus-effect becomes inefficient. An equilibriumtemperature is reached, scaling with kBT ∝ I/δ as long as the temperature is still well abovethe recoil temperature. The capture velocity, which describes the highest velocity an atomcan have to be captured in the molasses, is usually very low compared to standard MOTcooling, but it is proportional to the light intensity. So, with higher intensity more atomscan be caught. Notice that I have discussed 1D models, but these effects also occur in3D. In this case we get a polarization landscape by interference of three pairs of counter-propagating beams, meaning that there is always a polarization gradient. Then the coolingmechanism is the same as described in the 1D model.

2.1.3 Bright and gray molasses

The bright and gray molasses are different in a way, that is explained in the following.Figure 2.3 points out the differences. In a red-detuned bright molasses the light shift shiftsthe energy of the ground state manifold in negative direction. Consider the J = 1/2 →J ′ = 3/2 transition and a lin ⊥ lin standing wave with alternating polarization [4]. Atthe location where the light field is purely σ+, the ground state population is driven tothe mJ = 1/2 sublevel by means of optical pumping. This is because absorption produces∆m = +1 transitions, whereas the subsequent spontaneous emission produces ∆m = ±1, 0,so on average this corresponds to ∆m ≥ 0 for each scattering event. At a location of σ−

light it’s the other way around and atoms are pumped into the mJ = 1/2 state. Travelinghalf a wavelength in the light field , the atoms readjust their population completely frommJ = 1/2 to mJ = −1/2 and back again. The light shift depends on the Clebsch-Gordancoefficient Cge as ∆Eg ∝ C2

ge. Cge describes the coupling between atom and light with gdenoting the ground state and e the excited state of the atom. The coefficient depends onthe magnetic sublevel and on the polarization of the field, leading to different light shiftsfor different magnetic sublevels. The light shift for the mJ = 1/2 state is three times largerthan that of the mJ = −1/2 state when the light is completely σ+. For σ− polarization thelight shift for the mJ = −1/2 state is three times larger. This causes the population to bealways in the state with largest light shift [24]. Then the Sisyphus effect leads to repeatedcooling cycles as described before.

In contrast a gray molasses is blue-detuned and dark states are present. The bright statesare light-shifted into positive direction because of the blue detuning. The dark states are,of course, not coupled by light, so there is no light shift, their energy does not change with


Energy

Energy

z z

0 0

λ/8 λ/4 3λ/8 λ/2 5λ/8 λ/8 λ/4 3λ/8 λ/2 5λ/8

dark

bright

mJ=1/2

mJ=-1/2σ+ σ-σ-

Figure 2.3: Bright and gray molasses. Bright (left) and gray (right) molasses shown in comparison. Ina bright molasses the atoms climb the potential hill, on top, they are pumped back with high probabilityinto a state with lower potential or larger light shift, losing kinetic energy in the process. In a graymolasses atoms are lifted from a dark state to a bright state via motional coupling, from where they canclimb the potential hill. Near the top a high scattering probability leads to pumping back into the darkstate. Collecting very slow atoms in the dark state leads to lower temperatures.

position and therefore have a flat potential. The potential of the bright states is similar tothe bright molasses spatially modulated depending of the local polarization and intensityvariations of the laser field. The cooling mechanism can be described as follows. An atommoves with finite velocity. It couples to bright states due to an effect called motionalcoupling, as the Hamiltonian is not only described by light-atom interaction, but also thekinetic energy has to be included, resulting in dark and bright states that are not stationary[25]. The strength of this motional coupling is proportional to the velocity of the atom.Furthermore, this coupling is strongest, when the light shift of the bright states is lowestor in other words, when the energy difference of the specific states Ebright−Edark = Ebright

is minimal. That is the main difference to the bright molasses, the existence of motionalcoupling. The atom can climb the potential hill and because of a higher decay rate there itis pumped back into the dark state. Energy is again dissipated which leads to cooling.The dark state is only for velocity class 0 really a true dark state. Otherwise there is stillmotional coupling. During the cooling mechanism slow atoms are accumulated in darkstates, which leads to lower temperatures in gray molasses schemes.


The simplified discussion so far must be generalized to the case involving many hyperfinestates. In the case of 40K there are 10 + 8 hyperfine states involved. In reference [26]optical Bloch equations are solved numerically for the 40K system in the presence of bothcooling and repumping laser fields with the result that the essential picture remains valid.Main results are the existence of 8 bright states, 2 weakly coupled states and 8 dark statescombining both hyperfine manifolds. The optical pumping rate is low for weakly coupledstates and it practically vanishes for the dark states. Also a good correlation between thelight shift magnitude with the optical pumping rate is shown, since the departure rateof bright states increases with light shift, while the weakly-coupled states are long-lived.In other words, an atom that climbed the bright state potential hill is more likely to bedepumped than the one that is still at the bottom of the potential, which favors efficientSisyphus cooling. The calculations have been performed assuming a lin ⊥ lin configuration,however, in our experimental setup a σ+−σ− configuration is realized instead, because both,MOT (for which σ+ and σ− light is needed) and molasses beams, take the same optical path.In reference [4] it is shown that both configurations lead to the same minimum temperature.

2.1.4 The Λ-enhanced gray molasses

Figure 2.4 shows a three level Λ-system with two coupling fields. State |1〉 and state |2〉 arenot coupled by light, but both states are coupled to state |3〉 by two different light fieldsΩ1 and Ω2.

Ω1

Ω2

|1>

|2>

|3>

δ1δ2

Figure 2.4: The Λ-configuration. In a Λ-configuration state |1〉 and |2〉 are coupled by light to anexcited state |3〉, but |1〉 and |2〉 are not coupled. At Raman condition δ = 0 a dark state consisting of|1〉 and |2〉 occurs. This state is not coupled with state |3〉 anymore.

The previously described gray molasses scheme can be further improved by adding sucha Λ-configuration as shown in [6]. The enhanced cooling can be explained qualitatively


as follows. For ω1 ≈ ω2 = ω the Hamiltonian of the combined atom-light system can bewritten as

H = ~Ω1 cos kz(|1〉〈3|) + h.c.

+ ~Ω2 cos kz + Φ(|2〉〈3|) + h.c.

+ ~δ1|1〉〈1|+ ~δ2|2〉〈2|. (2.3)

In a simplified picture we set one of the intensities much larger than the other. We assumethat Ω1 Ω2, so the new dressed bright states are given by

|2′〉 = cos θ|2〉 − sin θ|3〉

|3′〉 = sin θ|2〉+ cos θ|3〉, (2.4)

where tan 2θ = Ω2/δ2.

When the Raman condition (two-photon-resonance) δ1 = δ2 is fulfilled, coherent populationtrapping occurs in a Λ-system. The Raman condition leads to the formation of a dark state,which does not couple to the excited state any more. This non-coupling state is given by

|NC〉 = Ω1|1〉 − Ω1|2〉Ω2

1 + Ω22

. (2.5)

The interaction of the two laser beams with the three level atoms thus results in threenew perturbed energy levels and eigenstates. Figure 2.5 shows a simple model, whichfeatures a ’normal’ dark state with respect to σ+ light that always occurs in a gray molassesscheme. New dark states are formed by quantum interference of hyperfine manifolds in aΛ-configuration. In a sense this is an additional gray molasses mechanism, which worksparallel to the conventional gray molasses scheme. Thus, even lower temperatures arepossible to achieve. For a more quantitatively analysis, see reference [27].


Ω1

Ω2

|1>

|2>

|3>

Raman resonance

dark state

new dark states

F'=2

F=2

F=1

Figure 2.5: Λ-enhancement. When the Raman condition is fulfilled, additional dark states appeardue to quantum interference. These new dark states participate additionally to the conventional graymolasses using the ’normal’ dark state. This leads to further cooling.

2.2 Laser setup

In the following the experimental setup of the optics system necessary for creating a Λ-enhanced gray molasses is described. The underlying cooling and repumping scheme isshown in figure 2.6.

coolingrepumping

F=9/2

F=7/2

F'=7/2

F'=9/2

δ

2S1/2

2P1/2

1286 MHz

155 MHz

D1770 nm

Figure 2.6: Cooling and repumpimg scheme. The gray molasses is realized on theD1 optical transitionat 770 nm. Cooling light works on the F=9/2 to F’=7/2 transition, repumping on the F=7/2 to F’=7/2transition. The ground state manifolds are seperated by 1.286GHz. Numerical values taken from [28].

2.2 Laser setup 13

The gray molasses for 40K works on the D1 transition at 770.1 nm (for reference: the D2

transition is at 766.7 nm) i.e., a transition from the ground state 2S1/2 to the 2P1/2 manifold.One reason is a better resolved excited state manifold compared to the D2 transition. Thesecond reason is by having a blue-detuned cooling beam with respect to the 2P1/2 manifold,both possible transitions result in a gray molasses cooling. For efficient cooling we needa cooling and a repumping beam, which also sets up a Λ-type configuration. The coolinglight works on the F = 9/2 → F ′ = 7/2 transition. As pointed out in the previoussection it has to be blue-detuned by an amount δc. The repumping light works on theF = 7/2 → F ′ = 7/2 transition with a detuning to the blue of δr in order to get atoms,which fall into the F = 7/2 state, back into the cooling cycle. On Raman condition wehave δ1 = δ2 or δ = 0, with δ = δr − δc being the detuning from the Raman condition. Theeffect of off-resonant excitations to the F ′ = 9/2 level should be small, since it is separatedfrom F ′ = 7/2 by 155.3MHz or 25.7 Γ.Actually, the repumping beam is no independent beam but is generated as a sideband ofthe cooling beam with a frequency of 1.286GHz via an electro-optic modulator (EOM).This configuration allows the beams to be phase-coherent, which is necessary for the Λ-configuration to work since it relies on quantum interference effects.

2.2.1 Setup of the optics system

During the course of this thesis, an optical setup for operating a gray molasses has beenbuilt at the BFM project. This setup is shown in figure 2.7. Note that this is a conceptualsketch and not to scale. In the following we walk briefly through the setup.

A Ti:Sapphire laser (Coherent) is used as laser source. The laser is pumped by a 10WVerdi (Coherent) at 532 nm. The Ti:Sapphire has a measured output power of 800mW at770.1 nm. The beam passes through a Faraday isolator to avoid reflections back into thelaser. Then the beam is split into two arms via a half-wave plate and a polarizing beamsplitter (PBS). Roughly 20mW of power are transmitted into the spectroscopy arm, whichis indicated by a green box in figure 2.7. The laser is locked on a spectroscopy of 39K on theD1 line. This Doppler-free saturation spectroscopy is explained in detail in the next section.Notice, that the detuning of the cooling beam can be set via the double-pass acousto-opticmodulator (AOM) (Crystal Technologies 3200-121,center frequency 200MHz) that is placedin the spectroscopy branch.The reflected branch at the PBS is focused into an AOM (Crystal Technologies 3110-120,center frequency of 110MHz). The AOM is adjusted in such a way that most of the poweris led into the plus first order. The coupling efficiency in the first order is about 85%. An


EOM 1

AO

M 3

spectroscopy cell EOM 2

AOM 1

AO

M 2

cavity

Ti:SapphVer

diphoto diode

photo diode

photo diode

fiber coupler

fiber coupler

edge filter

4%

optical diode

λ/4

λ/4

λ/2

λ/2

λ=770nm

λ=532nm

+1 0-1

-1

λ=767nm

D1 opical pumping

D1 coolingand repumping

D2 cooling

spectroscopy

Figure 2.7: Sketch of the setup for the gray molasses. The spectroscopy arm is framed by a greenbox. With the double-pass AOM3 it is possible to tune the frequency of the laser output. In the middlearm the cooling and repumping light is prepared from the +1 order of AOM1. The subsequent EOMgenerates the repumper as a sideband of the cooling beam. The ratio of cooling and repumping powercan be measured by using a Fabry-Pérot cavity. After superposing D1 and D2 light from the MOTbranch, the beams are coupled into a fiber and led to the experiment table. The optical pumping andrepumping light are created from the zeroth order of AOM1, subsequent EOM and finally AOM2, thenit is coupled into a fiber and led to another part of the laser table, where the beam is superimposed withD2 optical pumping light of the rubidium branch (not pictured here).

electro-optic modulator (Qubig E0-K40-K3, 1.13GHz-1.38GHz) is placed right after theAOM, such that both, the zeroth and first order of the AOM pass through this device. Viathis EOM we modulate the repumper on the cooling beam as sidebands. The zeroth order ispicked up and led through another AOM (Crystal Technologies 3080-122, center frequency80MHz) in a way that most of the laser power goes into the minus first order. The light isthen coupled into a fiber and brought to another part of the laser table in order to overlapthis beam with the previously used branch for optical pumping that also features rubidium

2.2 Laser setup 15

optical pumping light. This easy to implement D1 optical pumping and repumping, whichis used to capture most atoms in the magnetic trap, proves to be very valuable as will bediscussed in more detail in section 2.3.After being modulated by the EOM, the minus first order is led to a bandpass edge filter,at which the D1 (770 nm) and D2 (767 nm) light (that is still used for the MOT phase)is superposed and coupled into a fiber, which leads to the experiment table. The trans-mission efficiency is here about 90%. Before being overlapped a small portion of the beamis picked up (only 4%) and led into a Fabry-Pérot cavity in order to monitor the ratioof cooling and repumping power, that can be made visible on an oscilloscope. On theexperiment table the molasses beams take the same path as the MOT beams, giving riseto three counter-propagating pairs of beams in σ+−σ− configuration. The waist size is 1 cm.

2.2.2 Frequency stabilization

In order to work properly, the laser has to be stabilized to a certain frequency. FM-spectroscopy consists of a phase modulation of the laser light and a following absorptionspectroscopy that measures absorption signals of specific atomic transitions [29]. In fig-ure 2.7 the green box frames the spectroscopy branch in our experimental setup. Thedouble-pass AOM is not important for this technique, but is needed for freely tuning thefrequency of the locked laser light in a range of around 80MHz. The laser light is ledthrough a self-made EOM (25 MHz). In the frequency spectrum two sidebands are createdby this phase modulation. After that the light passes through a glass cell that contains a gasof 39K, thus a frequency dependent change in phase and amplitude occurs. Measuring thebeat signal on a fast photo diode we get the absorption and dispersion signals that generatethe error signal. Then the laser can be locked to a sharp slope of this signal. To lock thelaser on the right frequency we use a technique called Doppler-free saturation spectroscopy.Actually, we lock the laser on a D1 transition of the isotope 39K because this isotope hasthe highest abundance of all potassium isotopes. After the pump beam passes through theglass cell atoms of a specific velocity class are excited. The reflected probe beam excites adifferent velocity class because of the Doppler shift. The Doppler-shifted effective frequencyω′, which interacts with the atom, can be written as

ω′ = ωL − ~k~v + v2

2c2 + . . . , (2.6)

where ω0 denotes the non-shifted frequency of the transition, ~v is the velocity of the atoms


and ~k the wavevector. Higher orders can be neglected. On resonance frequency, the mediumbecomes transparent, since the atoms already got excited by the incoming pumping beam.The Doppler-broadened absorption spectrum now shows the characteristic Lamb-dip andadditional features corresponding to the crossover lines.By passing an EOM, sidebands of the pump beam are generated with a radio frequencyωm. The light field then reads

E(t) = E0 eiω0t eiM sin(ωm) +c.c., (2.7)

where M is the modulation index and E0 the amplitude of the electric field. For M 1follows

E(t) = E0[eiω0t−M2 ei(ω0−ωm)t +M

2 ei(ω0+ωm)t] + c.c., (2.8)

giving the strong carrier signal with frequency ω0 and two weaker sidebands at (ω0−ωm) and(ω0 + ωm), phase-shifted 180 with respect to each other. The transmitted signal throughthe glass cell reads [29]

ET(t) = E0T0 eiω0t−E0M

2 T−1 ei(ω0−ωm)t +E0M

2 T+1 ei(ω0+ωm)t +c.c., (2.9)

where Tj = e(δj(ω)−iΦj(ω)) is the frequency dependent transmission coefficient. Here, δ de-notes the absorption coefficient and Φ the phase shift. Measuring with a fast photo diodeprovides a time-averaged intensity signal according to

〈I(t)〉 = 〈|ET (t)|2〉. (2.10)

This light field ET(t) creates a signal on the photo diode with central frequency and abeating of the sidebands if T+1 6= T−1. Neglecting terms with M2 and terms with |δ0 −δ±1|, |φ0− φ±1| 1, and taking into account that the fast photo diode only averages oversignals with doubled frequency, the signal on the photo diode simplifies to

〈I(t)〉 ∝ e−2δ0((δ−1 − δ+1)M cos(ωmt) + (Φ+1 + Φ−1 − 2Φ0)M sin(ωmt)). (2.11)

2.2 Laser setup 17

The signal of the fast photo diode is demodulated via a mixer, which is basically a multi-plication of the photo diode signal and the EOM driver signal (25MHz). Subsequently, thesignal is low pass filtered giving us a possible error signal [30]

Uerror = M(−12∆δ cosφ+ 1

2∆Φ sinφ). (2.12)

In our case the cos term represents the derivation of the absorption signal. Manipulatingthe phase φ between the EOM signal and the photo diode signal via the EOM-driver,eliminates the sin term. Thus, the created error signal has a zero crossing, which can belocked electronically via a PID controller. The lock process is performed by a self-madelock-box.

F=9/2

F=7/2

F'=7/2

F'=9/2

2S1/2

2P1/2

2P1/2

2S1/2

D1

770nm

F=1

F=2

F'=2

F'=1

39K 40K

474 MHz

1286 MHz

56 MHz

Figure 2.8: Spectroscopy scheme. The Titan:sapphire laser is locked to a spectroscopy of 39K on theD1 transition. Instead of locking to the crossover F = 1 → F ′ = 1, 2 (black dashed line), a reliableerror signal is attained by changing the phase of the demodulation. We now lock the laser to a frequencyindicated by the green dashed line. To get the frequency for the cooling transition (blue), AOM3 in thespectroscopy branch and AOM1 (cf. figure 2.7) shift the frequency to the desired value. In this case474MHz are necessary for a resonant transition. Numerical values taken from [28].

In the scheme pictured in figure 2.8, there are three possible transitions to which the lasercan be locked. The transition from F = 1 to F ′ = 1 or F ′ = 2 or the crossover between


those two. The error signals of these transitions created by the described FM-spectroscopyhave been relatively weak and locking the laser was not possible. That is why we merged thethree slopes to one slope via shifting the phase φ of the EOM-driver. The resulting slope issharp and the signal strong enough that the laser can be locked reliably and remains lockedover a period of days.

170 180 190 200 210 2200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spectroscopy AOM frequency (MHz)

Nor

mal

ized

ato

m n

umbe

r

Figure 2.9: Fluorescence imaging. A Lorentzian curve is the result of fluorescence imaging with D1light to find the resonance position. In this case the resonance position is at 182MHz setting for thespectroscopy AOM.

Now, we have to find the setting of AOM3 for the resonant transition of the D1 cooling light.With a fluorescence image measurement the resonance frequency can be found. Therefore,atoms are loaded into a MOT and D1 light incidents on the atomic cloud with variablefrequency (via the double-pass AOM in the spectroscopy branch, AOM3). That way atomsare excited according to the D1 transition. When the excited states decay, fluorescencelight is emitted in all directions. For different frequencies images of the fluorescence signalare taken with the Andor camera. On resonance the signal should have its maximum andfigure 2.9 shows exactly that behavior. A Lorentzian curve is the result of such a resonancemeasurement. The AOM has to be set to 182MHz. Because we use a double-pass AOM,this corresponds to a frequency shift of 364MHz.

2.3 Experimental results 19

2.3 Experimental results

2.3.1 Experimental setup and conduction of measurements

All of the experiments presented in this thesis were performed at the ’Bose-Fermi mixture’experiment in the group of Klaus Sengstock at the University of Hamburg. As the namealready indicates, the experiment is designed to simultaneously trap bosonic 87Rb andfermionic 40K [31, 32]. A detailed description of experimental aspects can be found in thetheses of Silke and Christian Ospelkaus. The focus of this chapter is on the creation ofultracold fermionic gases, regarded as only a short summary of the most important aspects.First, a brief overview of the setup in general is given. A sketch of the apparatus used inthis experiment is shown in figure 2.10.

2D-MOT

3D-MOT

LA

D2

Pumping Stage &Vacuum System

Detection

Figure 2.10: Experimental setup. The two-species 3D-magneto-optical-trap (MOT) (D1,D2,LA) isloaded from a 2D-MOT. Two glass cells which are connected via a differential pumping stage seperatingthe upper vaccuum system from the lower. The molasses beams take the same path as the 3D-MOTbeams. Figure taken from reference [33].

In the upper glass cell, the dispensers continuously release 40K and 87Rb atoms, whichare captured in a 2D-MOT at a background pressure of approximately 10−9 mbar. Using


a blue-detuned pushing beam, the atoms are then transferred into the lower glass cell,the ’science chamber’, where they are trapped in a 3D-MOT at a background pressurebelow 10−11 mbar. For 40K, a Dark-Spot-MOT is used [34, 35]. After that there is aoptical molasses phase, which will be the main issue in this chapter. For further coolingthe atoms are optically pumped into the magnetically trappable low-field seeking states|F = 2,mF = 2〉 for rubidium and |F = 9/2,mF = 9/2〉 for potassium to then be loaded intothe magnetic trap, which is a hybrid cloverleaf-4D-trap [36]. This issue will be discussed inthe end of this chapter. In the magnetic trap, an exponentially decreasing radio frequencyis applied to continuously remove the hottest atoms. This process is known as evaporativecooling. For evaporative cooling to be effective, the atomic ensemble has to rethermalizequickly which means that high elastic collision rates are required. However, due to the Pauliprinciple [37], a spin-polarized fermionic sample cannot be directly cooled evaporatively.One approach is cooling the fermions sympathetically, using bosons as a reservoir. Thismethod is applied in this experiment and by varying the MOT loading times and thus themixing ratio of bosonic 87Rb and fermionic 40K, different particle numbers and temperaturesof the fermions can be reliably set.

A typical experimental routine used to optimize the gray molasses cooling method in thecontext of this thesis is depicted in figure 2.11. Here the previously used D2 molasses isreplaced by a molasses phase on the D1 transition.

MOT (K

)

D1 m

olass

es

TOFim

aging

time

10 s

10 m

s

18 ms

Figure 2.11: Experimental procedure. A typical experimental sequence for measurements regardingthe optimization of our gray molasses scheme. It replaces a previously used D2 molasses. Only 40K isloaded during the MOT stage. Timeframes are not to scale.

In order to characterize the D1 gray molasses, parameters like the detuning and power ofcooling and repumping beam has to be tuned. After a MOT phase of about 10 to 20 sfollowed by the gray molasses phase of typically about 10ms the atoms are released andafter a time-of-flight of about 18ms the atoms are detected by absorption imaging revealingthe momentum distribution of the atomic cloud. The long duration of the MOT phase isdue to a low dispenser currency and thus low 40K pressure in the glass cell. For imaging, a


high-resolution self-built objective is used that is described in reference [33]. In this regimethe width of the cloud is a measure for temperature but it is not possible to extract thetemperature from one image because the initial size of the cloud weighs in too much. Tocircumvent this problem a longer time-of-flight would be useful but it was not possible be-cause the cloud gets bigger than the CCD chip of the camera. Further reduction of themagnification of the imaging system did not work due to aberration effects.Most other publications operate a short compressed MOT (CMOT) after the MOT phasein order to reach a higher phase-space density. In our setup we refrained from implement-ing this stage, since rubidium shall be loaded simultaneously for later evaporation in themagnetic trap.The optimizing procedure of the molasses phase mostly consisted of reducing the widthand maximizing the atom number of the atomic cloud and increasing the atom numbercaught in the molasses after time-of-flight while tuning through relevant parameters. Notethat after 18ms time-of-flight all atoms, that are not captured in the molasses, have flownoff quickly due to their high energy and consequently do not disturb the measurement ofthe atom number. Indeed, without operating a gray molasses stage, there is no signal vis-ible after this expansion time. The atom number is measured after a short pulse of D2

repumping light (previously used in the D2 molasses) in order to pump the atoms into thedetection transition. In the following laser intensities are given in units of the saturationintensity Isat = 1.75 mW/cm2. Also note that in the following all measurements were takenwithout loading of rubidium. After optimizing the lone 40K molasses the temperature withsimultaneous loading of rubidium is determined as well.

2.3.2 Results: Gray molasses

Detuning from the Raman condition

As described in section 2.2 the repumping and the cooling beam are phase coherent as theformer is generated as a sideband of the ladder via an EOM. It is possible to tune thedetuning of the repumper with respect to the cooling beam i.e., the Raman detuning δ.As explained in sectionsec:theory, the temperature should reach a minimum at the Ramanresonance condition, which means δ = 0. If there is no detuning between the cooler andthe repumper with respect to the excited level, additional Λ-systems emerge for the groundhyperfine manifolds and the result would be a lower temperature due to additional darkstates on two photon resonance. Here the hyperfine manifolds mF = 7/2 and mF = 9/2


form those additional dark states, which act figuratively speaking in an additional graymolasses scheme. In figure 2.12 this is seen in the experiment. The resonance positionis exactly at the Raman resonance condition and has a very narrow width of 0.45MHz(FWHM) i.e., less than 0.1 Γ. For red Raman detuning δ < 0 and far from the resonancethe two mechanisms decouple. The conventional gray molasses scheme works outside ofthis resonance position where the temperature has a more or less constant value around20µK. Note that most other parameters used in this measurement are not optimal. Forblue Raman detuning δ > 0 we observe a sharp peak of heating, which is,however, notfully resolved in this measurement. This could be explained as a destructive effect of therepumping light (cf. [6] for a detailed calculation) in the conventional gray molasses. Whenthe repumper is detuned in a way that atoms in the F = 7/2 state are pumped to thetop of a hill of a bright state potential, the atoms increase their kinetic energy as they falldown the optical potential and this generates heating. However, the heating feature is notexamined any further, since we are mostly interested in the cooling process.

−15 −10 −5 0 5 10 15

10

15

20

25

30

35

40

45

50

55

Detuning from Raman condition δ (MHz)

Tem

pera

ture(μK)

−1.5 −1 −0.5 0 0.5 1 1.5

10

20

30

40

50

Figure 2.12: Detuning from the Raman condition-temperature. On resonance condition δ = 0 asharp minimum occurs. This is due to additional Λ-systems and the following appearance of new darkstates. The insert shows the resonance position magnified. Other parameters: δcool = 2 Γ, Icool = 17 Isat4ms molasses duration, Prepump

Pcool= 0.2.


Furthermore, figure 2.13 shows that a maximum number of atoms is caught at the Ramancondition.

Detuning from Raman condition (MHz)-15 -10 -5 0 5 10 15

Ato

m n

umbe

r (x

10

6)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 2.13: Detuning from the Raman condition-atom number. On resonance condition δ = 0most atoms are caught in the molasses.

All further measurements and optimizations have been conducted at the Raman resonancecondition.

Power of the repumping beam

The power of the repumper is another parameter that has been examined. Important tonote is that by adding repumping power, the cooling power will be less. That is whyfigure 2.14 shows the dependence of the temperature of the atomic cloud on the ratioof repumping and cooling intensity. The ratio is measured via use of the Fabry-Pérotcavity whose signal is displayed as carrier and sidebands of the laser beam on a connectedoscilloscope (see figure 2.15). At a ratio of Prepump

Pcool= 0.12 the temperature saturates.

However, the temperature decrease is very small, so this parameter is not critical. Forfurther measurements a ratio of Prepump

Pcool= 0.2 has been chosen.


0

1

2

3

4

5

6

Ato

m n

umbe

r (x

10

6 )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.92

0.94

0.96

0.98

1

Tem

pera

ture

(a.

u.)

Repump power (Irepump

/Icool

)

Figure 2.14: Repumping power. The temperature saturates at ratio of PrepumpPcool

= 0.12, but the coolingeffect is rather small. Other parameters: δcool = 4.5 Γ,Icool = 17 Isat, 13ms molasses duration.

Figure 2.15: Typical signal of the cavity. The x-axis displays the cavity length while the y-axis displaysthe cavity transmission. The ratio Prepump

Pcoolis determined by dividing one side-peak by the main peak.

Figure taken from the data-sheet of the EOM [38].


Duration of the molasses phase

An important parameter is the duration of the molasses phase. This phase should not betoo long because of the diffusion of the cloud. In general, 40K and 87Rb shall be loadedtogether for sympathetic cooling in a magnetic trap. The optimal molasses duration for87Rb in our experimental setup is 10ms, so the duration for the D1 molasses must be inthe same area to guarantee a robust simultaneous operation of potassium and rubidium.Figure 2.16 shows the atom number in the cloud and the temperature depending on theduration of the molasses phase.

3.5

4

4.5

5

5.5

Ato

m n

umbe

r (x

10

6 )

0 2 4 6 8 10 120.5

0.6

0.7

0.8

0.9

1

Tem

pera

ture

(a.

u.)

Duration of molasses (ms)

Figure 2.16: Duration of molasses. On the left side the atom number (red) saturates at a durationof 4ms. On the right side the temperature (blue) saturates at a duration of 6 to 8ms. An exponentialfit (solid line) gives a time constant of 1.4ms for the decrease in temperature. Other parameters:δcool = 4.5 Γ, Icool = 17Isat, Prepump

Pcool= 0.2.

The atom number saturates at a molasses duration of about 4ms, but the temperaturesaturates at a duration of 6 to 8ms, an exponential fit even reveals a time constant of1.4ms. This result provides a compatibility with a simultaneous 87Rb molasses. Finally,we have increased the duration of the molasses up to 1 s in order to measure the diffusioncoefficient. The cloud size after time-of-flight can be used to characterize the in-situ sizewhen the time-of-flight is held constant, thus avoiding effects of a high in-situ density ofthe cloud. The spatial diffusion coefficient Dx is then determined from the cloud size aftera time-of-flight of 18ms for different molasses times tm and the relation


σ2 = σ20 + 2Dxtm. (2.13)

In reference [39] a theoretical minimum for the spatial diffusion coefficient was calculatedin a 1D-model as ≈ 26 ~

M with M being the mass of the respective atom. In their 85Rbexperiment they found a diffusion coefficient by a factor of three higher than the theoreticalminimum while operating a standard bright molasses. We observed (see figure 2.17 thespatial diffusion coefficient to be about 0.37 ± 0.01mm2/s, which is ten times higher thanthe 1D theoretical minimum. In another 40K experiment a diffusion coefficient of about1.2mm2/s was observed, which is three times higher compared to our experiment [40] andthirty times higher than the theoretical minimum. This corresponds well to a temperature,which is roughly three times lower in our experiment as will be shown later in this section.

0 200 400 600 800 10001.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

Duration of molasses tm

(ms)

σ2

(mm

2 )

Figure 2.17: Diffusion of the molasses. Cloud size after a fixed time-of-flight of 18ms as a functionof the molasses duration tm. The solid line is a linear fit to the data to obtain the diffusion coefficient.

Power of the cooling beam

The cooling power proves to be a very tricky and important parameter for the D1 molasses.Figure 2.18 shows the atom number and temperature in dependence of the cooling power.

The interesting observation to be made is that for high cooling power all atoms are caught,


0

2

4

6

Ato

m n

umbe

r (x

10

6 )

0 2 4 6 8 10 12 14 16

0.7

0.8

0.9

1

Tem

pera

ture

(a.

u.)

Cooling intensity Icool

(I/Isat

)

Figure 2.18: Cooling intensity. The temperature (blue) has a minimum at a cooling intensity, for whichnot all atoms (red) are caught in the molasses.

but the temperature is too high and far away from the minimum. However, at minimumtemperature a starting decrease of the atom number can be observed due to a low capturevelocity. Also a linear increase of temperature for higher cooling power is observed, accord-ing to T = I

δ . For too low cooling power the scattering rate is just not high enough anda molasses cannot operate effectively, explaining a rapid increase in temperature. In orderto guarantee a robust operation of the molasses scheme with respect to power decrease ofthe cooling beams (e.g. due to bad fiber coupling) a new cooling scheme has to be imple-mented. The aim is to capture most atoms while cooling them to minimum temperature.The implementation of such a scheme is described in the next section [26].

A two-stage cooling scheme

The results of the previous section imply that a two stage cooling scheme would be optimalfor capturing as much atoms as possible at lowest possible temperature. That means thata capture phase is needed with high and constant cooling power. The previous resultsprove that a duration of 4ms seems to be a perfect fit for this stage. To cool the sample acooling stage has to follow up when the power gets ramped down to a value that has to bedetermined. This stage has a duration of 9ms. A sketch of this two-stage cooling theme is


pictured in figure 2.19

17 Isat

3.4 Isat

cool

ing

pow

er

4 ms 13 mstime

Figure 2.19: Two-stage cooling scheme. A capture phase with high and constant cooling power at17 Isat with a duration of 4ms is followed by a cooling phase of 9ms in which the power is ramped downto 1/5 of the initial cooling intensity.

Figure 2.20 shows the results of the measurement of the final cooling power after the rampin the cooling phase.

3

4

5

6

Ato

m n

umbe

r (x

10

6 )

0 2 4 6 8 10 12 14 16

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Tem

pera

ture

(a.

u.)

Final cooling intensity Icool,final

(I/Isat

)

Figure 2.20: Final cooling intensity. Atom number (red) and temperature (blue) in dependence of thefinal cooling power after the ramp.

Most atoms are caught and the lowest temperature is reached at a final cooling power of


0.2 Icool, where Icool = 17 Isat. The duration of the molasses is now a combined 13ms, whichseems to be a bit long but even with simultaneous loading of rubidium this long durationconstitutes no problem regarding temperature and atom number although further and morespecific measurements regarding the two species molasses have to be performed to say forsure.

Detuning of the cooling beam

After implementing the two-stage cooling scheme the last parameter that has to be opti-mized is the detuning of the cooling beam. For a gray molasses to work, the beam has tobe blue-detuned. Figure 2.21 shows the results of this measurement.

0

2

4

6

Ato

m n

umbe

r (x

10

6 )

0 2 4 6 8 100.2

0.4

0.6

0.8

1

Tem

pera

ture

(a.

u.)

Cooling detuning δcool

(Γ)

Figure 2.21: Detuning of the cooling beam. The temperature is minimal at a detuning of about27MHz or 4.5 Γ to the blue. Here the two-stage cooling Other parameters: δcool = 4.5Γ, Icool = 17Isat,Prepump

Pcool= 0.2. The two-stage cooling scheme is also employed in this measurement.

The minimum temperature is reached for a detuning of 4.5 Γ or 27MHz for 40K. At highdetuning the temperature saturates in a wide regime, so the exact value of the detuningseems not to be essential. That result is actually very important if you want to impose anoptical lattice which would result in a light-shift of a few MHz depending on the location.Since in this regime the temperature is constant, there is no effect expected.


Table 2.1: Optimal parameters.Parameter Valuetcapture 4mstcool 9msIcool,initial 17 Isat

Icool,final 3.4 Isat

Irepump 0.2 Icool

δcool +4.5 Γδrepump +4.5 Γ

Temperature measurement

After iterations of the optimization process the temperature has to be determined by mea-suring a time-of-flight series. For different time-of-flights of duration t the e−1/2 widthof the atomic cloud is determined assuming an initial gaussian distribution of the density

ρ(r, 0) = e−r2

(2σ0)2 with an initial width σ0. The width σ after time-of-flight t can be writtenas [41]

σ =√σ2

0 + kBT

mt2. (2.14)

The temperature T can be extracted from a fit. The temperature has been measured foroptimal parameters as described in the previous sections. The parameters are summarizedin table 2.1.

Figure 2.22 shows the width of the cloud σ in dependence of the time-of-flight and thecorresponding fit function.

For best parameters the cloud has a temperature of 5.7±0.2 µK. For comparison, two groupsin France reached a temperature of 6 µK for bosonic 39K [42] and 20 µK for 40K [26]. Inreference [43] a temperature of 11µK is reached for 40K. Our measurements constitute thecoldest molasses of 40K and demonstrate that one can reach the same temperature as for39K. In terms of the recoil energy Erecoil = ~2k2

2m = kBTrecoil our temperature corresponds toT = 14Trecoil.Finally, we use the in-situ cloud size extracted from the time-of-flight measurement andcalculate the averaged density n ≈ 10101/cm3 assuming the atoms (N = 107) are caught ina sphere with radius r = σ0 ≈ 600 µm, which is also determined from the fit. This yields a


0 5 10 15 20 25

600

650

700

750

800

850

900

950

1000

Time-of-Flight (ms)

1/e1/

2ex

pans

ion

wid

th (μ

m)

Figure 2.22: Time-of-flight series. Size of the atomic cloud dependent of the time-of-flight. Solid lineshows a fit to determine the temperature T . Optimal parameters stated in table 2.1 are employed.

phase-space density in the order of 10−5, The phase-space density is calculated as φ = nλ3B

with λB being the thermal de Broglie wavelength. Although the density is a factor of tenlower compared to other experiments, the phase-space density is roughly in the same orderdue to the lower temperature in our experiment.

Comparison with other publications

In table 2.2 experimental results for gray molasses cooling in other groups are shown.

Table 2.2: Comparison of parameters for a gray molasses.author species T [µK] atoms δ

ΓIcoolIsat

IrepIcool

TTrecoil

BFM 40K 5.7± 0.2 107 4.5 17 0.2 14Fernandes [26] 40K 20 7 · 108 2.3 11-14 0.12 50Sievers [43] 40K 11 3.2 · 109 2.3 14 0.12 27Salomon [42] 39K 6 1.5 · 109 3.5-0.2 3.5 0.33 15Grier [6] 7Li 60 5 · 108 4.5 45 0.02 20Burchianti [7] 6Li 40 109 5.4 2.7 0.19 11


Compared to other 40K-molasses schemes we reach the lowest temperature for 40K. How-ever, it has to be mentioned that the atom number in our experiment is significantly lower(factor 10 to 100). Thus the density of the cloud is lower in our experiment and that canexplain the lower temperature, since density effects lead to heating due to light-inducedtwo-body collisions. It should be noted that in contrast to our experiment all others op-erated a short compressed MOT stage previous to the molasses stage to further increasethe density. Cooling and repumping power are in the same regime. Other experimentshave implemented a gray molasses scheme for 39K, 6Li and 7Li. In all cases the lowesttemperature reached is similar in terms of the recoil temperature Trecoil (10− 50Trecoil).

Simultaneous operation with rubidium

Finally, the simultaneous operation of the 87Rb and 40K molasses shall be discussed. Therubidium molasses works on the D2 transition. In a MOT collision losses of 40K atoms area huge factor (note that the 40K MOT works on the D2 line). However, there seems to beonly little influence during the combined D1 molasses for potassium and D2 molasses forrubidium. For optimal potassium parameters found in the previous section the atom numberin two-species operation does not differ from single species operation. The temperature ofthe potassium cloud rises to 8µK, which is a 25% increase but still far below the Dopplerlimit. Reason for this little influence is the short molasses duration and the minimizationof light-induced collision losses or heating as the atoms are accumulated in dark states.

2.3.3 Results: Optical pumping

Because we don’t have sufficiently deep dipole traps available for direct loading into anoptical dipole trap, the alternative approach to reach a degenerate quantum gas has to beimproved: sympathetic cooling with bosonic rubidium in a radio frequency evaporation.The experimental procedure is depicted in figure 2.23.

It has been proven useful to implement a short so-called optical pumping stage in theexperimental cycle just after the optical molasses before loading into the magnetic trap.While using a weak homogenous magnetic field to create a quantization axis the opticalpumping beam along the same axis pumps the atoms from not trappable hyperfine statesto trappable states. The potential of a magnetic field is given by the scalar product of themagnetic moment µ and the magnetic field B(r) [44]:


mag

netic

trap

evap

orat

ion

MOT (K

)

D1/D2

mola

sses

TOFim

aging

time

opt

. pum

ping

MOT (K

+Rb)

10 s 2 s

12 m

s

15 s

18 m

s

150 μ

s

1.5 s

Figure 2.23: Experimental procedure with sympathetic cooling. In simultaneous operation theMOT is loaded with 40K and 87Rb. Rubidium is only loaded in the last one or two seconds of the MOTstage. After a simultaneous gray molasses for 40K and bright molasses for 87Rb a short optical pumpingstage (150 µK) transfers the atoms into trappable states. Then the atoms are cooled sympathetically ina bath of rubidium while evaporating.

Vmag(r) = −µ ·B(r)

Vmag(r) = −mFgFµB|B|, (2.15)

with gF being the Landé-factor and µB the Bohr magneton. Since local maxima cannotexist for a magnetic field, only so-called low-field seeker can be trapped. Low-field seekerfulfill the condition

mFgF < 0, (2.16)

as these states have a minimum energy at locations where the magnetic field B(r) is mini-mal. In case of 40K the F = 9/2,mF = 9/2 is the best trappable state. Note that the opticalpumping process is stochastic due to the spontaneous decay of the excited states. To getall atoms into the mF = 9/2 many scattering processes must occur, since at the beginningeach hyperfine state is occupied equally. In the previous inefficient D2 cooling and opticalpumping scheme, the temperature (and atom number in a quantum degenerate gas) wasvery critical with respect to power of the optical pumping beam and duration of this stagebecause of heating effects for too much power and duration. By using optical pumping onthe D1 line heating should be suppressed because mF = 9/2 is a dark state. Figure 2.24shows the implemented optical pumping scheme in our experiment. We need an optical


pumping beam and a repumper to get the atoms out of the dark states. The repumper isagain modulated on the optical pumping beam via the same EOM in the cooling setup (cf.section 2.2). The optical pumping beam works on the F = 9/2 to F ′ = 9/2 transition, therepumper on the F = 7/2 to F ′ = 9/2 transition. A look at figure 2.25 shows why theoptical pumping on the D1 line is so efficient.

opticalpumping

repumping

F=9/2

F=7/2

F'=7/2

F'=9/2

2S1/2

2P1/2

1286 MHz

155 MHz

D1770 nm

Figure 2.24: Optical pumping scheme. Optical pumping and repumping on the D1 transition. Theoptical pumping light works on the F=9/2 to F’=9/2 transition, whereas the repumper works on theF = 7/2 to F ′ = 9/2 transition. The repumper is generated as a sideband of the optical pumping beamvia an EOM.

We want to pump all atoms into the mF = 9/2 state to reach a maximum number ofatoms that can be trapped in a magnetic trap [32]. Fortunately, on the D1 line this stateis a dark state with respect to σ+ light. On the D2 line this state is no dark state, so anyadditional scattering processes would just lead to heating because at these low temperaturesany scattering process would lead to a considerable momentum transfer, which leads tosignificant heating.Optical pumping on the D1 line should consequently lead to a saturation of the atomnumber in the magnetic trap after evaporation because no atoms are lost due to heatingmechanisms. Figure 2.26 shows the atom number in the magnetic trap after evaporationfor variable optical pumping power in case of a D1 molasses and D2 molasses (recorded inOctober 2013). In this case the atom number is also a measure for temperature becauseheating processes would directly lead to atom loss due to the mechanism of the evaporation


F=9/2

F=9/2

F'=9/2

F'=11/2

mF

mF

dark state

σ+

σ+

heating

D1

D2

Figure 2.25: Optical pumping for D1 and D2. Top: Optical pumping on the D1 line features a darkstate mF = 9/2. Bottom: Optical pumping on the D2 leads to heating when mF = 9/2 is occupied forfurther scattering processes.

in general. In this measurement the frequency of the D1 optical pumper and repumper is1.5 Γ red-detuned to the transition depicted in figure 2.24, the power of the repumper istwo thirds of the optical pumping power.

On theD2 transition a decrease of atom number for higher pumping power is observed, whileinstead on the D1 line we observe saturation for even very high optical pumping power andalso much higher atom number cooled to the same temperature of roughly T = 0.1TFermi

This example shows that the cooling mechanism on the D1 line is very efficient and usefulto have in our experiment. Note that here the loading time of the MOT was actuallylonger for D2 optical pumping (10 s D1, 13 s D2). Optimizing an experiment really meansoptimizing the experimental cycle time, so to get the same number of atoms with the sametemperature, we are able to reduce the experimental cycle time by about 30% with the newD1 optical gray molasses in combination with D1 optical pumping compared to the old D2

bright molasses scheme. The reduction of cycle time is mostly due to cutting the evaporationduration in half (from 20 s to 10 s) thanks to a lower starting frequency of the rf knife (from40MHz to 30MHz). In the end we get a quantum degenerate gas in the magnetic trap at


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9

Optical pumping intensity (I/Isat

)

atom

num

ber

afte

r ev

apor

atio

n (x

10

5 )

D2

D1

Figure 2.26: Atom number after optical pumping and evaporation to a fixed temperature. Theatom number after evaporation saturates for high optical pumping power on the D1 line (blue) even forhigh pumping power. On the D2 line (red) instead, there is a distinct optimum with lower atom number.

a temperature of 0.1TFermi. The results prove that a lower starting temperature for 40Kprovides indeed ideal starting conditions in a sympathetic cooling experiment.

2.4 Summary

In this section the Λ-enhanced gray molasses for 40K has been characterized and optimizedby tuning of different parameters. We have implemented a two-stage cooling scheme witha capture phase of 4ms and and cooling phase of 9ms. The cooling beam is 4.5Γ blue-detuned. The Raman resonance condition between repumper and cooling beam is fulfilled,i.e. δ = 0. The power ratio of repumper and cooling beam is Prepump

Pcool= 0.2. The cooling

power is 17 Isat during the capture stage and is then ramped down to 3.4 Isat in the coolingstage.With those parameters it is possible to cool down an atomic sample of 40K from about0.5mK in the MOT to 5.7± 0.2 µK. Depending on the duration of the MOT it is possibleto capture 107 atoms, which is about 90% from the MOT.To increase the atom number after evaporation in the magnetic trap we have implemented acombined optical pumping and repumping scheme after the molasses phase with a duration

2.4 Summary 37

of 150 µs. This scheme proves to be very efficient, heating effects are minimized. Thanks tothat the evaporation duration can be reduced significantly. To reach a quantum degenerateFermi gas with the same temperature and atom number as in a previously used D2 molassesthe time of the experimental cycle is reduced by about 30%. Of course, faster cycle timesare desirable especially when high amounts of repetitions are required.

In future the Ti:Sapph laser is to be replaced by a diode laser plus a tapered amplifier. Infact, this setup has already been build and is ready to be implemented. This would freeup the more costly Ti:Sapph for more demanding tasks e.g., potential shaping. Anothernot yet systematically examined issue is the influence of a simultaneously used rubidiummolasses and how it effects the potassium molasses in a gray molasses scheme compared tothe previously used bright molasses. In a simultaneous MOT, losses due to light-assistedcollisions have been observed [45, 46]. However, we found little influence on temperatureand atom number while operating a simultaneous molasses stage.Loading into the existing dipole trap has not been successful due to insufficient power.However, an ’all-optical’ approach i.e., directly loading a dipole trap from the molasses andfurther evaporation in the dipole trap via use of a Feshbach resonance, can be desirable todecrease experimental cycle times even further.


39

3 Tunable Hexagonal Optical Lattices

3.1 Theoretical introduction to tunable hexagonal lattices

In this section a brief theoretical introduction to optical potentials in general and tunable2D-lattices specifically is presented. One possibility to tune the geometry of optical latticesis presented here by using an array of wave plates. Another method of tuning the geometryis described in reference [47]. This section is concluded by the description of one method toprobe these systems or specifically the band structure, which is the amplitude modulationspectroscopy.

3.1.1 Optical potentials

There are two forces concerned when imposing laser light onto an atomic sample, namelythe spontaneous and the dipole force. The dissipative or spontaneous force can changethe momentum of an atom by a sequence of absorption and emission and has dissipativecharacter, thus providing the basis of laser cooling [24]. The dipole force is conservativeand can be used to create optical potentials for trapping atoms. The dipole force can beunderstood semi-classically as the interaction between the light-induced dipole moment µ ofthe atom and the gradient of the electric field of the laser light. In the dressed atom picture[4] it can be computed quantum mechanically. The dipole potential and the scattering rateare given by [48]:

Udip(r) = −3πc2

2ω30

( Γω0 − ωL

+ Γω0 + ωL

)I(r)

Γsc(r) = −3πc2

2ω30

(ωLω0

)3( Γω0 − ωL

+ Γω0 + ωL

)2I(r) (3.1)

40 3 Tunable Hexagonal Optical Lattices

where Γ = ω30

3πε0~c3 |〈e|µ|g〉|2 is the spontaneous decay rate of the excited state, ω0 is theatomic resonance frequency between ground state and excited state, ωL the frequency ofthe laser and |〈e|µ|g〉|2 the expectation value of the atomic dipole moment.For a red-detuned laser the dipole force is attractive and thus the atoms are attracted tothe maxima of the electric field, whereas for blue-detuned light the force is repulsive andthe atoms are attracted to the minima of the electric field.Note that the expression given in equation 3.1 is a simplified one. In general, for multi-levelatoms other dependencies like coupling to different hyperfine states have to be taken intoaccount, too [48]. However, in our setup, the detuning is much larger than any hyperfineor fine splitting so that these structures cannot be resolved.As shown beams can create optical potentials for atoms. These potentials can be shapedinto different forms depending on their intended experimental use. Periodic structures,so-called optical lattices, can be created via the interference of multiple laser beams. Thesimplest kind of optical lattice is the sinusoidal potential of a standing wave created by oneretro-reflected laser beam:

VL = −V0 cos2(kLx) (3.2)

where the lattice spacing a = λL/2 = π/kL is given by the laser wave length and the latticedepth V0 = 4Udip(0) is defined by the depth of the dipole potential. However, in this thesisI want to turn the focus onto 2D optical lattices created with a three beam setup wherethe angle between each two beams is exactly 120. The optics setup is described in detailin reference [13], although there have been made some changes during the course of thisthesis.

Hexagonal lattices

Hexagonal lattices can be created by interfering three traveling laser beams with an angleof 120 between each pair of beams. For the following calculations, the wave vectors of thethree laser beams have been chosen as follows:

k1 = kL

0

1

0

, k2 = kL2

√

3

−1

0

, k3 = kL2

−√

3

−1

0

(3.3)

3.1 Theoretical introduction to tunable hexagonal lattices 41

By changing the polarization of the laser beams with regard to the plane spanned by thebeams of the lattice, the lattice can now be tuned from a triangular to a honeycomb lattice.Here, the overall lattice geometry stays the same, however, potential minima and maximaexchange their role so that the effective potential for the atoms is reversed.When the polarization of the laser beams is perpendicular to the plane, a triangular latticestructure is created. The laser beams at λ = 1064 nm are red-detuned with respect to theD1 and D2 lines of 40K and the polarization vectors read:

e1 = e2 = e2 =

0

0

1

(3.4)

Then the potential for the triangular lattice case is [49]:

Vtri(r) = V034 + 1

2[cos((b1 − b2)r − Φ2) + cos(b1r − Φ3) + cos(b2r + Φ2 − Φ3)], (3.5)

where b1 and b2 constitute the reciprocal lattice vectors, defined by b1 = k1 − k3 andb2 = k2 − k3. The phase of the first laser beam has been chosen as Φ1 = 0. The beamconfiguration and the corresponding potential is plotted in figure 3.1.

When the polarization of the laser beams is instead set parallel to the plane of the latticei.e.,

e1 =

1

0

0

e2 =

−1

2

−√

32

0

e3 =

−1

2√3

2

0

(3.6)

a honeycomb lattice structure is created. The potential then reads [49]:

Vhc(r) = V034 −

14[cos((b1 − b2)r − Φ2) + cos(b1r − Φ3) + cos(b2r + Φ2 − Φ3)]. (3.7)

The hexagonal lattice has only tiny lattice wells whose relative depth is only 1/9Vhc. Thusfor achieving the same lattice depth in the honeycomb lattice as in the triangular lattice, a


120°

ε ε

ε

120°

120°Figure 3.1: Triangular lattice. Three interfering beams in out-of-plane polarization configurationintersecting at 120 angles form a triangular lattice with a triangular configuration of the potentialminima (blue).

much higher laser power is needed. A sketch of the beams and the corresponding potentialis depicted in figure 3.2.

In contrast to the triangular lattice, the unit cell of a honeycomb lattice comprises two sites(A and B), corresponding to a two-atomic basis in solid state physics.

3.1.2 Tuning of geometry

So far, we were able to tune between a triangular and a honeycomb lattice. By introducingan additional phase shift between out-of-plane (s) and in-plane (p) polarization it is possibleto realize an arbitrary addition of these two potentials allowing for realizing many differentlattice geometries. We engineer a static lattice V (r) by exercising independent control overthe two polarization components of each of the three laser fields [14]

Ei = E(cos(θ) eiαi z + sin(θ)ki × z) ei(kir−ωt−φi) . (3.8)

Here, each phase αi denotes a retardation between s- and p-polarization of the fields and θ


120°

ε ε

ε

120°

120°Figure 3.2: Honeycomb lattice. Three interfering beams in in-plane polarization configuration inter-secting at 120 angles form a honeycomb lattice with a hexagonal configuration of potential minima(blue).

sets the ratio between the amplitudes of p and s components of Ei. Our hexagonal lattice isformed by three red-detuned traveling waves with wavevectors k1, k2 and k3 as describedbefore. The three beams interfere with each other to produce a lattice potential

V (r) = Vhc(r) cos2(θ) + Vtri(r − r′) sin2(θ) (3.9)

with Vtri and Vhc given in equations 3.5 and 3.7, respectively. We are now able to createdifferent lattice structures by setting the polarization orientation θ to tune the intensities,and using the retardation αi to tune the displacements of the two basis lattices, honeycomb(Vhc) and triangular (Vtri), relative to one another. Although the lattice geometry is alwayshexagonal, the effective lattice structure is determined by the relative strength of the tun-neling bonds. By manipulating θ and α we are able to change tunneling bonds and hencerealize effectively different lattice structures. In our setup possible lattice structures includedimers, square and 1D-chains and different kinds of AB-offset lattices, where the two sitesA and B are offset in energy ~δ. A phase diagram of lattice geometries that can be reachedis shown in figure 3.3.

When shifting the potential of a triangular lattice with respect to the honeycomb lattice


xzy

AB

ABA

B

Figure 3.3: Phase diagram of lattice geometries. While shifting the triangular lattice with respectto the honeycomb lattice in principal directions x, y and z as depicted on the top left, different latticegeometries show up. Red lines in the phase diagram mark a transition point of lattice geometries.Effective geometries include dimer and square lattices and 1D chains. Different realizations of AB offsetsare also possible.

potential in principal directions x,y and z as depicted in the figure (this corresponds totuning αi), different kinds of lattice geometries are obtained depending on the populationangle θ as well. For tuning sin2(θ) from 0 to 1 there is a transition from a honeycomb toa triangular lattice with both potentials compensating each other at sin2(θ) = 0.33. Thisis the regime of the polarization lattice with no intensity modulation of the light field, butwith modulation of the polarization [50]. In the far-detuned regime this results in a flatpotential. Shifting the triangular potential in x-direction and low values of θ creates dimerlattices featuring staggered sets of double wells. For high values of θ square lattices can becreated since four potential wells are connected by equal tunneling bonds. When shiftingthe triangular potential in y-direction and low θ, coupled AB-1D chains can be realized upto the point, where we reach an AB-honeycomb lattice, which means that the two sites (A


and B) contained in the unit cell are offset in energy, also shown in figure 3.4. This givesrise to interesting physics regarding manipulation of Dirac cones [47] and topological bands[15]. Shifting in z-direction also produces offset lattices, in this case in form of dimers.

Figure 3.4: AB-offset lattice. An hexagonal lattice with sites A and B, which are offset in energy, iscreated for θi = 0.05π and α1 = 0, α2 = 2π/2 and α3 = 4π/2. These kind of lattices are one ingredienttowards creation of topological bands as inversion symmetry is broken [14].

In reference [47] another method of tuning the lattice geometry is presented. Three retro-reflected beams create a two-dimensional lattice potential. Two of those beams are perpen-dicular to each other, which creates a checkerboard pattern, while the third one is parallelto one of the other beams and creates an independent standing wave. Different latticegeometries are created by an arbitrary detuning between this third beam and the othersand tuning of the intensities of each beam. Dimers, honeycomb, triangular, square and 1Dchains lattice structures can thus be created. Compared with their setup our setup hasthe same tunability, but features a true hexagonal symmetry and not just ’effectively’ (theBrillouin zone is a hexagon in our case instead of a square) and should be more stable asit is a static setup. However, tuning of the geometry is performed faster in their setup.Considerations about the technical difficulties for implementing a dynamical tuning of thelattice structure via motorized wave plates are subject in reference [51]. Other ideas includethe implementation of EOMs or liquid crystal retarders for fast tuning of the geometry.

3.1.3 Wave plates and polarization

We decided to realize the tuning of geometry and setting of the population θ and retardationα via an array of wave plates. Since the initial polarization state is linear because of apolarizing beam splitter in the lattice telescope (see figure 3.9) we only need a combination


of a λ/4 and λ/2 wave plates to reach each polarization possible, which are represented bythe Poincaré sphere [52]. The Poincaré sphere is basically a Bloch sphere for polarizationstates. The north and south pole is represented by right and left circular polarized light,respectively. The equator represents purely linear polarized light, while every other state onthe surface of the sphere represents an elliptical polarization. The beauty of the Poincarésphere is the simple representation of effects of a polarizing or birefringent medium likeour wave plates. In case of wave plates the output polarization can be obtained from theinput polarization by rotating it on the sphere about the axis FS, representing the medium,clockwise while looking from F to S by an angle δ. δ is the phase difference introducedbetween the fast and the slow polarization states and for a half-wave plate δ = π, while fora quarter-wave plate δ = π/2. In order to show that we are able to cover the whole Poincarésphere when starting with an arbitrary linear polarization state, we show the effect of thequarter-wave plate on a linear state in figure 3.5.

H F

VQ

S

R

L

H

R

L

Figure 3.5: Poincaré sphere with quarter-wave plate. Left: Beginning with linear polarization (here:horizontal), a quarter-wave plate transforms this state to the elliptical polarization state Q by rotationof π/2 for a given direction of the fast axis . Right: When rotating the fast axis the output polarizationstate will describe an 8 on the sphere. From north to south pole each longitude can be reached. Here,for simplicity the view is along the original horizontal state.

The horizontally polarized state is transformed to an elliptical polarization state. By tuningthe fast/slow axis, i.e. turning the wave plate, we can see that we cover each point betweennorth and south pole. However, with one wave plate we do not cover the whole surface ofthe sphere. A half-wave plate can transform this elliptical state into another elliptical state(onto the other hemisphere) depending on the direction of the fast/slow axis. In figure 3.6the evolution of the state with rotating fast axis is illustrated. It is represented by a circle


parallel to the equator. We can now clearly see that a combination of both wave plates leadsto covering the whole sphere, meaning that we are able to tune the polarization arbitrarily.The question how the wave plates have to be set is answered in section 3.3.

H

R

L

F

S

Q

P

R

L

H

S

F

Figure 3.6: Poincaré sphere with half-wave plate. Left: Starting with an elliptical polarization stateQ (created by the quarter-wave plate) the half-wave plate transforms the state into another ellipticalpolarization state P by rotation of π depending on the direction of the fast axis . Right: When rotatingthe fast axis each state on a circle (red) parallel to the equator is obtained.

3.1.4 Amplitude modulation spectroscopy

In this section the single-particle Hamiltonian describing our lattice setup is introduced.The eigenvalues forming the dispersion relation or the band structure are presented forthe simple case of a honeycomb lattice. Excitations within the band-insulating groundstate are produced by a periodic modulation of the lattice amplitude. This time-dependentperturbation of the Hamiltonian is described afterwards.

Hamiltonian

The Hamilton operator that describes the quantum system according to Schrödinger’s equa-tion is introduced here. Since in our case only fermionic and spin-polarized 40K is presentthe Hamiltonian can be written as the sum of two terms. One describes the kinetic energyof the particle, while the other characterizes the optical potential. No term for particleinteraction is needed, as spin-polarized fermionic particles do not interact via short rangepotential at low temperature. The Hamiltonian has the form [53]


H0 = p2

2m + VLattice(r), (3.10)

where p is the momentum operator and r the location of the particle in a periodic potentialV . The lattice potential can for example be of the form in equation 3.7 in case of ahoneycomb lattice. The eigenstates of a system with a periodic potential are the well-known Bloch states, which have the same periodicity as the underlying potential:

〈z|n, q〉 = ψ(n)q (z) = eiqz

∑K

c(n)K,q eiKz (3.11)

with

H0|n, q〉 = E(n)q (z)|n, q〉. (3.12)

Here, the indexK in the sum has to be considered for all available reciprocal lattice vectors, qis the quasi-momentum constrained to the first Brillouin zone, cK,q are the Bloch coefficientsand n the respective band index. The eigenvalues E(q)(n) denote the band structure ordispersion relation.

Band structure in optical lattices

When an atom is trapped in a periodic potential, its energy spectrum is no longer contin-uous. Instead, its dispersion relation is given by particular energy bands and band gapswhich depend on the lattice parameters such as lattice geometry or depth. This effect is ofcourse well known in solid state physics.Solving the single-particle Hamiltonian within a lattice produces the eigenvalues for spe-cific quasimomenta q, which describe the energy of a particle in this potential with such amomentum. Here, the range of eigenvalues represents the different excited states, or bands,which each particle of the same momentum can occupy. The resulting function E(q)(n) iscalled dispersion relation and denotes the band structure of the system.A two-dimensional picture is often too complex, that is why specific paths in the reciprocalspace are chosen to represent the band structure in one dimension. In case of a hexagonalBrillouin zone a path over symmetry points is usually taken i.e., from the center of the zoneΓ to a point in the zone’s corner K over to the middle of one edge M and back to Γ (seefigure 3.7.


MK

Γ

Figure 3.7: High-symmetry path in a hexagonal Brillouin zone. In a hexagonal lattice the firstBrillouin zone also has hexagonal symmetry. The high-symmetry path starts at Γ to M, over K back toΓ.

This way multiple bands can be made visible next to each other. Figure 3.8 shows such acase for a band structure of a triangular lattice where a path MΓKM in momentum spaceis chosen.

M Γ K M

Ene

rgy

(Ere

c)

-34

-32

-30

-28

-26

-24

Figure 3.8: Bandstructure along the high-symmetry path. Calculation of the band structure betweensymmetry points of the Brillouin zone covering the first seven excited bands plus the zeroth band for alattice depth of 35Erec for a honeycomb lattice. Note the Dirac cone at the K point.

Amplitude modulation spectroscopy

The band structure of an optical lattice can be probed with a technique called amplitudemodulation spectroscopy. Excitations in the lattice can be induced by periodically modu-lating the lattice depth or the amplitude of the intensities of the beams respectively. The


modulation imprints energy on the system, thus particle-hole pairs are created as atoms areexcited from the ground state to higher bands depending on the frequency of the modula-tion. Because of the different curvature of the bands, the imprinted energy is only resonantfor one specific quasimomentum, making this method momentum-resolved when we startwith a band insulator i.e., the Fermi-energy lies in the band gap meaning that only thezeroth band is completely filled. The quasimomentum is conserved in this process as can beshown in case of a one-dimensional optical lattice, which is explained in detail in reference[17] and shall briefly be reviewed here.

Introducing a time-dependent oscillating perturbation to the system produces the newHamiltonian:

H(z, t) = p2

2m + V0 cos2(kzz) · [1 + ε cos(ωt)] (3.13)

where ε denotes the strength of the amplitude modulation and ω its angular frequency,while kz represents the lattice spacing in momentum space:

kz = 2πλLattice

. (3.14)

The eigenstates of the non-perturbed system correspond to Bloch states (see equation 3.11).Under the assumption of a band insulator as the initial state, any perturbation can to firstorder only create excitations to higher bands with n > 0 Inserting the time-dependent partof equation 3.13 into Fermi’s Golden Rule [54], the transition probability Wn

qq′ of two Blochstates described by their momenta q, q′ and band indices n and m = 0 is estimated by:

Wnqq′ ∝ |〈n, q|V (z)|0, q′〉|2f(T, ω0 − ω), (3.15)

where f(T, ω0 − ω) depicts the response of state coupling to a modulation time T and amodulation frequency ω, while ω0 is the state-specific resonance frequency on the transition.Modulating sufficiently long, the response function can be reduced to a delta functionδ(ω0−ω). Using equation 3.13, the time-dependent part of the Hamiltonian can be writtenas

V (z)εV0

= cos2(kz) =H0 − p2

2mV0

. (3.16)

3.2 Experimental setup for the optical lattice 51

Assuming a resonant excitation frequency ~ω = E(n)q′ − E(0)

q and simplifying equation 3.15leads to

Wnqq′ ∝ | 〈n, q|H0|0, q′〉︸︷︷︸

=0

−〈n, q| p2

2m |0, q′〉|2. (3.17)

Here, the part of the stationary Hamiltonian vanishes, as the Bloch states are eigenstatesof the Hamiltonian. Using p2 ∝ ∂2

z and inserting the definition of the Bloch states, equa-tion 3.17 becomes

Wnqq′ ∝ |

∑K,K′

c(n)K,qc

(0)K′,q′

∫eiqz e−iKz ∂2

z eq′z eiK′z dz|2. (3.18)

Further simplifying yields a representation of δ-functions

Wnqq′ ∝ |

∑K,K′

c(n)K,qc

(0)K′,q′(q′ +K ′)2

∫e−i(q−q′)z︸︷︷︸δ(q−q′)

e−i(K−K′)z︸︷︷︸δ(K−K′)

dz|2. (3.19)

It follows, that

Wnq ∝ |

∑K

c(n)K,qc

(0)K,q(q +K)|2, (3.20)

proving that indeed only states with the same quasimomentum q are coupled and no mo-mentum transfer other than integral multiple of the reciprocal lattice vector is allowed. Itshould be noted that the excitation frequency’s width grows with shorter duration of theamplitude modulation as described by Fourier transformation. This is important becausethe modulation duration should be long enough to get frequency resolution, on the otherhand a long duration gives dephasing mechanisms due to effects of the confining harmonicpotential too much time, in particular when holes are concerned [55].

3.2 Experimental setup for the optical lattice

The existing optical lattice setup required some changes during the course of this thesis.These changes include restructuring of the intensity stabilization by implementing polarime-ters, removing the phase-lock and implementing a set of quarter- and half-wave plates for


each lattice beam to tune the geometry. These changes are explained and characterized inthe following sections.

3.2.1 Polarimeter

In order to realize different lattice geometries as described in section 3.1 for two of the threebeams (’ADWin’ and ’Wand’) a combination of a quarter- and half-wave plate has beenimplemented directly behind the telescopes so that the beam first passes the quarter wave-plate, then the half-wave plate before it hits the atoms in the glass cell. Special care hasto be taken in case of the lattice beam from the bottom. Firstly, there is no quarter-waveplate involved, secondly, it first passes a half-wave plate, then is reflected off a mirror beforeit is led through another half-wave plate and passes the glass cell through the differentialpumping stage. If we would just use a half-wave plate after the telescope in front of themirror, the polarization is not purely s- or p-polarization but a mixture depending on theactual setting of the wave-plate. By reflecting off a mirror a phase-shift between s- and p-polarization occurs and that effect shall be avoided, since we do not want additional phase-shifts between s- and p-polarization that would have additional influence on the latticegeometry. But the first half-wave plate is still necessary, because the polarization out ofthe fiber is not purely s- or p-polarization as well. So the first half-wave plate tunes thepolarization to honeycomb configuration, i.e. purely p-polarization. Now, the mirror hasno effect on the phase between s and p and the second half-wave plate can be used to tunethe geometry of the lattice (switching between honeycomb and triangular configuration).In order to reliably tune the lattice geometry it is mandatory to control that the wave platesdo what they are supposed to do. That is why we have implemented a polarimeter for eachof the upper two beams. The bottom beam will later be calibrated with respect to theother two. A sketch of the lattice setup on the experiment table involving the polarimeterfor the ’Wand’ and ’ADWin’ axis can be found in figure 3.9.

After passing the glass cell the beam is led through a polarizing beam splitter which sep-arates p- and s-polarization. Both beams are picked up by a 4% mirror and led onto aphoto diode. This way the the power in s- and p-polarization can be made visible on anoscilloscope and be determined. We also examined the phase shift due to the coating ofthe glass cell (only outside walls are coated) and found it to be negligible. For completecontrol over the phase shift an additional branch can be introduced. A small portion ofthe beam must be picked up, then led through a quarter-wave plate and the same setupfor the original polarimeter follows. This way, the phase shift would be mapped onto thepopulation of s- and p-polarization. Due to space issues and material expenses we refrained


Lattice 'ADWin' Lattice 'Wand'

Lattice 'Unten'

telescope

glass cell

p-polarization

s-polarization

λ/4

λ/2

p-polarization

s-polarization

λ/2

λ/4

λ/2

λ/2

Figure 3.9: Setup of the polarimeter. After passing through rotatable quarter- and half-wave-platesthe laser light is separated for s- and p-polarization and each detected by a photo diode. The latticebeam ’Unten’ has two half-wave plates, the first sets p-polarization, that there is no phase-shift due tothe following mirror introduced. The second wave-plate actually sets the polarization for the lattice. Thisbeam has no quarter-wave plate because we don’t need a phase-shift in this direction. The phase shiftdue to the coating of the glass cell can be neglected.

from implementing it. Since mounting of the wave plates hasn’t been done precisely enoughleading to an unknown position of the fast axis, the polarimeters have to be calibrated.This will be discussed in section 3.3.

3.2.2 Intensity stabilization

The polarimeters that are described in the previous section are not only used to calibratethe lattice geometry. Previously, the lattice depth was stabilized electronically by using theretro-reflected beam at the plane end of the optical fiber. So the beam passes two times theoptical fiber which proved to be problematic, since the intensity that is stabilized is not theone that the atoms actually feel, which may be due to insufficiently precise coupling into thepolarization axis of the polarization-maintaining fibers, that are used for our lattice beamsfor transportation from the laser table to the experiment table. That in turn can lead tovarying lattice depth at the location of the atoms, which results in heating. To circumventthis problem one photo diode of the polarimeter setup is used to measure the laser intensity


and create the error signal for the intensity stabilization. It is important to note that inthis case not the total intensity is stabilized, but only the intensity that is in p-polarizationand special care has to be taken when calibrating the lattice depth. A simpler way wouldhave been to stabilize the total intensity just by picking up the lattice beam after the glasscell and leading to a photo diode. However, to avoid additional phase-shifts by the pick-upmirror, which would compromise the wave plate calibration (see section 3.3), we refrainedfrom realizing this way of intensity stabilization for the two diagonal beams. The pick-up mirror also provides different reflectivity values depending on the incident polarizationmaking this scheme even less useful. However, the bottom beam is actually stabilized thatway, because we do not use a polarimeter for this lattice axis.

3.2.3 Phase-lock

Reference [49] states that a phase-lock is necessary, since otherwise the lattice can translateor accelerate in space, which would lead to heating. However, even with operating a phase-lock mechanism [13] lifetimes of atoms in the lattice are very short resulting in a heatingrate of 0.9 Hz

Erecin a triangular lattice and 0.12 Hz

Erecin the respective 1D lattice in triangular

configuration. A high heating rate from spontaneous emission is not expected, since thelaser beams are very far detuned (1064 nm compared to the 780 nm transition for 87Rb ).However, beat signals of two lattice beams show phase noise in the kHz range even whenthe phase lock is operated (cf. figure 3.10). This is comparable to the band transitionfrequencies and is most likely responsible for the short observed lifetimes.

For that reason the phase-lock in that form was removed. Lifetimes of the three beams arenow considerably longer. Figure 3.11 shows the lifetime of rubidium atoms in dependenceof the lattice depth in a honeycomb configuration and figure 3.12 shows the lifetime in therespective 1D lattice in honeycomb configuration (p-polarization).

In order to measure heating in the lattice, good care has to be taken with choosing the dipoletrap depth, because we need to work in a regime where heating does not mean particle loss,but BEC (Bose-Einstein condensate) fraction loss. In deep lattices there is also strongerharmonic confinement from the lattice beams leading to higher trap frequencies. In thiscase particles may not only be lost, but also BEC fraction loss is present, while at lowlattice depths (and dipole trap depth) only atom number loss can be observed. That iswhy we only want to quantify heating by BEC fraction loss. Therefore, a optimal dipoletrap depth has to be found, so that particle loss is avoided. On the other hand the dipoletrap depth may not be too high, since it would limit the already low lifetime in the lattice


Figure 3.10: Beat signal of two beams. Shown is the beat signal of two beams in an unlocked andlocked state. Features in the kHz range with both arms being locked are likely to cause heating. Thefigure is taken from reference [13]

.

considerably.After the dipole trap is loaded, the 2D-lattice is ramped up and at the same time the dipoletrap power is ramped to the found optimum power. Then the wait time in the lattice isvaried, before ramping the lattice down again. The lifetime is then determined from theevolution of the condensate fraction of the rubidium BEC depending on the time of thelattice stage. This is done for several lattice depths and the heating rate in terms of thelattice depth is extracted from a fit, so that the influence of the dipole trap (latticedepth = 0)does not appear in the value of heating rate. We find a heating rate of 0.14± 0.01 Hz

Erecin a

two-dimensional lattice, which is a major improvement, so removing the phase-lock providessignificantly less heating. In the 1D lattice we found a heating rate of 0.055 ± 0.004 Hz

Erec,

which is also lower. Note that in honeycomb configuration the laser power at the atoms istwice as high compared to the triangular configuration for the same lattice depth. Withthis argument we would usually expect a higher heating rate in honeycomb configurationdue to larger spontaneous emission. Since it is instead even lower, we can assume thatour phase-lock setup was insufficiently suited for our experiment. However, it has to bementioned that the heating rates with phase-lock were recorded by measuring just one datapoint at a specific lattice depth and subtracting the heating rate of the dipole trap, socomparability might be compromised a bit. The question remains if an effective phase-lockwould be useful to reduce heating rates even further. Currently work on an DDS based


0 2 4 6 8 10 12 14 160.2

0.7

1.2

1.7

2.2

2.7

lifet

ime(

s)

Lattice depth (Erec

)

0

1

2

3H

eatin

g ra

te (

Hz)

0 2 4 6 8 10 12 14 160.2

0.7

1.2

1.7

2.2

2.7

lifet

ime

(s)

Lattice depth (Erec

)

Figure 3.11: Lifetime in a honeycomb lattice. Lifetime (blue) in dependence of the lattice depth ina honeycomb lattice. This measurement was performed with bosonic 87Rb. A fit (solid) provides theheating rate. The red dots show the reciprocal lifetime for a better view at the heating rate.

phase-lock scheme is in progress.

3.3 Tuning of polarization: Calibration of the wave plates

The implemented polarimeters have to be calibrated in order to guarantee reliable tuningof lattice geometries.As described in the previous section it is possible to measure the intensity in p- and s-polarization of the lattice beam by use of the polarimeters. The polarization output of eachfiber is p-polarization (0,1), regulated by an polarizing beam splitter built in the latticetelescope. In order to tune the polarization reliably, we have to know the position of thefast axis of each quarter- and half-wave plate. In figure 3.13 a measurement of the peak-to-peak intensity is shown for the ’ADWin’ axis. For a range of 180 on the scale of thequarter-wave plate the half-wave plate has to be turned at least 90. On the oscilloscope twophoto diode signals are visible that have a sinusoidal form while turning the half-wave plate.We measure the peak-to-peak distance for each angle on the quarter-wave plate scale. If theinput polarization is parallel to the fast axis of the quarter-wave plate maximum contrast

3.3 Tuning of polarization: Calibration of the wave plates 57

0

0.5

1

1.5

2

heat

ing

rate

(1/

s)

0 2 4 6 8 10 12 14 160.2

0.7

1.2

1.7

2.2

2.7

lifet

ime(

s)

Lattice depth (Erec

)

Figure 3.12: Lifetime in a 1D lattice in honeycomb configuration Lifetime (blue) in dependenceof the lattice depth in a 1D lattice. This measurement was performed with bosonic 87Rb. A fit (solid)provides the heating rate. The red dots show the reciprocal lifetime for a better view at the heating rate.

will be measured because there won’t be any circular polarization parts present any more.Thus, the maximum value provides the angle of the fast axis.

The position of the half-wave plate’s fast axis is determined in similar fashion. We set thequarter-wave plate, so that fast axis and polarization are parallel to each other. Then thephoto diode voltage is measured in dependence of the half-wave plate angle. The positionof the fast axis is represented by maximum voltage in p-polarization. Figure 3.14 shows theresult of this measurement for the ’ADWin’ axis.

Actually the fast and slow axis can’t be distinguished with this type of measurement, sothe found values for the fast axis can be wrong by exactly 90. For this reason we used theexperiment as an additional cross-check as the measured band structure (see section 3.4)will be significantly different from the expected one. By changing values of the fast axesby 90 this problem can be solved. In table 3.1 the position of the fast axis of each opticalelement is shown.

After characterizing the wave plates we turn the focus to the actual setting of these platesin order to tune to arbitrary polarizations and lattice geometries. We introduce a represen-tation of arbitrary polarization given by


0 20 40 60 80 100 1200

100

200

300

400

500

600

700

800

900

Angle on scale (°)

Pho

to d

iode

vol

tage

(m

V)

Figure 3.13: Calibration of the quarter-wave plates. The peak-to-peak value for different settings ofthe quarter-wave plate of the ’ADWin’ axis (blue: p-polarization, red: s-polarization). The position ofthe fast axis of the quarter-wave plate is determined as 92.

p =

sp

=

sin(θ)

cos(θ) eiα

, (3.21)

where p is the so-called Jones vector. Since population θ (i.e. the ’ratio’ of triangular andhoneycomb) and retardation α are providing a better grasp of what we intend to do withthe optical lattice, we don’t want to use the (s,p) representation of polarization states, butinstead (θ,α). For an arbitrary state of polarization i.e., s and p are both complex, thepopulation angle can be retrieved as

θ = arcsin(|s|), (3.22)

Table 3.1: Fast axis.Lattice beam λ/4 λ/2Axis 1 ’Unten’ 0

Axis 2 ’Wand’ 178 358

Axis 3 ’ADWin’ 92 186

3.3 Tuning of polarization: Calibration of the wave plates 59

190 200 210 220 2300

500

1000

1500

2000

2500

3000

3500

4000

Angle on scale (°)

Pho

to d

iode

vol

tage

(m

V)

Figure 3.14: Calibration of the half-wave plates. The voltage of the photo diodes for different settingsof the half-wave plate (blue: p-polarization, red: s-polarization) of the ’ADWin’ axis. The position ofthe fast axis of the half-wave plate is determined as 186.

and the relative phase α is the phase of the complex number

p

s= |p

s| eiα (3.23)

Assuming an initial state (s,p), the output state of polarization is obtained by usage of Jonesmatrices for the quarter- and half-wave plates. In the following positive angles are definedclockwise with respect to the vertical (0,1) axis as viewed along the beam propagation.Then the Jones matrix for a quarter-wave plate with arbitrary angle β between fast andvertical axis reads

Jλ/4(β) = R(β) eiπ4

1 0

0 −i

R(−β), (3.24)

where R(β) is the rotation matrix


R(β) =

cos(β) sin(β)

− sin(β) cos(β)

. (3.25)

Actually, the prefactor for the quarter-wave plate matrix can be omitted since an overallphase does not alter the polarization state. Respectively, for the half-wave plate we have

Jλ/2(β) = R(β)

1 0

0 −1

R(−β). (3.26)

Considering an initial state (θ, α) = (0, 0), which is basically the polarization output of theoptical fiber, the polarization after running through the optical elements is obtained with

sp

= Jλ/4(βλ/4)Jλ/2(βλ/2)

sin(θ)

cos(θ) eiα

= Jλ/4(βλ/4)Jλ/2(βλ/2)

0

1

. (3.27)

This vector can now be transformed into the (θ,α) representation with equation 3.22 and3.23. The process to determine the angles βλ/4 and βλ/2 for a given initial state anddesired α and θ is automated via a Mathematica least square routine. The calibrated valuesfor the position of the fast axis is then used to determine the angle, that has to be seton the respective wave-plate holder. To cross-check if the program code produces correctvalues for the wave-plates we used Kapitza-Dirac effect with bosonic 87Rb in the 1D latticesformed by two beams [56]. In order to perform this check, we have to load rubidium intoa dipole trap and flash the lattice on for just a few microseconds. Momentum transferproduces characteristic peaks. In a 1D lattice one polarization can be turned in honeycombconfiguration (p-polarization), the other in triangular configuration (s-polarization) andminimizing the Kapitza-Dirac peaks determines an exact perpendicular configuration. In apolarization lattice (θ = 18) there only is a flat potential for a non-spin-dependent lattice,which means, that the lattice depth is zero. In this case no Kapitza-Dirac should be visible.Indeed, there were minimum Kapitza Dirac peaks observed when adjusting the wave platesto a polarization lattice. Comparison with the values the program code produces, thedeviation was less than 1, which is good enough for our purposes.

3.4 Multiband spectroscopy 61

3.4 Multiband spectroscopy

To probe the band structure of our two-dimensional lattice system, we use the technique ofamplitude modulation spectroscopy. For these and future measurements a symmetric latticeregarding beam power balance and knowledge of polarization is very important. During thecourse of this thesis the calibration of the lattice regarding beam imbalances and errors inpolarizations had to be reviewed to make analyzing data more reliable. With full controlover the geometry different systems can be probed e.g., topological bands with or withoutinteraction.

3.4.1 Amplitude modulation spectroscopy in a honeycomb lattice

After the fermions are cooled sympathetically in a magnetic trap and loaded into a crosseddipole trap, we get a spin-polarized quantum-degenerate gas. The next step is to imposean optical lattice on the atoms. The experimental sequence for creating the lattice andsubsequent amplitude modulation is shown in figure 3.15.

latti

ce d

epth

time

adia

batic

ram

p

latticemodulation

21 ms TOF

10 m

s

500 μs

Figure 3.15: Experimental sequence for amplitude modulation spectroscopy. The atoms areadiabatically loaded into a optical lattice via linear ramping up the intensity of the three beams in 10ms.The intensity (lattice depth) is then modulated typically 10% of its mean value in the next 500 µs. Thenthe all optical potentials are switched off, and after a time-of-flight of 21ms the momentum distributionin the optical lattice is revealed.


Previously, the technique has been used with band mapping before time-of-flight [55]. Whenband mapping is employed, the optical lattice potential is ramped down exponentiallyrevealing a fully momentum-resolved spectrum, since even the occupied bands can be madevisible. However, the band mapping phase suffers from being too slow, so that dynamicsin higher bands make analyzing data, i.e. the excitations, very difficult. The dynamics areinduced by the harmonic confinement. By omitting the band mapping stage and insteadreleasing the atoms instantly for a time-of-flight measurement, we lose information aboutthe specific band, into which the atoms are excited depending on the modulation frequency.On the other hand the analysis of the holes in the first Brillouin zone is now made possible.Although holes are closed very fast, by not band mapping this effect can be circumventedto still get a momentum-resolved band structure, at least at these low lattice depths thatwe use, since the Bloch coefficients of the bands are sufficiently different in shallow lattices.For data evaluation we trace the high-symmetry paths in the Brillouin zone as depictedin figure 3.7 and average over the six identical paths. The amplitude of the modulation iskept at εi = 0.1 for each of the three beams (i = 1, 2, 3) in order to remain in the linearresponse regime (cf. section 3.1). The excitation spectrum is shown in figure 3.16. For everyexcitation frequency ω (or energy ~ω, as used in the plots), the optical density is plotted foreach pixel, which corresponds to a momentum k = q, along the announced path. Thus, theband structure is revealed. Note that it is just an effective band structure, since the zerothband is always subtracted. Blue lines correspond to strong absorption and we achieve verygood agreement with the calculated band structure. Obviously, not all bands are coupledby the modulation operator, when all three beams are modulated at once.

In order to couple other (or more) bands we modulate only two of the three beams, so incase of ε2 = 0 we get a different excitation spectrum, that is depicted in figure 3.17. Anexplanation would be that we change the symmetry of the perturbation operator. In caseof three beams modulated one would assume that the perturbation operator has the samesymmetry as the lattice. By modulating two beams, the symmetry changes, thus maybecoupling bands with different parity. However, parity is only a good quantum number in adeep lattice. In a shallow lattice that we use, parity is not well-defined. That is why somebands are coupled in both modulation schemes.

When we average over the three possible modulation pairs (1&2, 1&3, 2&3) and also includethe case ε1 = ε2 = ε3 = 0.3 to couple the upper s-band (first excited band), we obtain aband structure that is depicted in figure 3.18. With higher modulation amplitude we leavethe linear response regime, which is necessary because the matrix element describing thecoupling of the first two bands is too small due to parity reasons. Only at the position ofthe Dirac cone strong deviations from the theoretical band structure are observed. In this


minnorm. OD

max norm. OD

Figure 3.16: Amplitude modulation spectroscopy with symmetrical modulation. All three beamare modulated at εi = 0.1, while not every band is coupled. Solid lines show the calculated band structure.

case the frequency resolution of the amplitude modulation technique is not high enough,because of the finite modulation time. In section 3.4.2 it is demonstrated that the excitedbands are very sensitive to beam imbalances and polarization errors, even more sensitivethan the Dirac cones of the lowest two bands. So, when the experimentally observed higherbands match the calculated bands, the Dirac cones of the lowest bands are expected to sitat the K point.

With this technique the band structure can be determined with high precision as can beseen in figure 3.19, where a MATLAB routine only finds the local minima corresponding toexcitations without using a fitting routine and still providing a very good resemblance ofthe theoretical band structure, proving that this technique is indeed a reliable tool to probethe band structure. Excitations between the first and second excited band correspond totwo-photon excitations into the third band due to a high amplitude of the modulation. Inthe future we hope to see effects of interactions on the band structure induced by using aspin mixture and further enhanced tuning by Feshbach resonances.


minnorm. OD

max norm. OD

Figure 3.17: Amplitude modulation spectroscopy with non symmetrical modulation. Only twobeams are modulated at at ε1,3 = 0.1 and at ε2 = 0. Here different bands are coupled compared to thesymmetric modulation. Some bands are equally coupled in both schemes. Solid lines show the calculatedband structure.

3.4.2 Calibration of the lattice

In order to probe the band structure reliably, good care has to be taken regarding beam im-balances and polarization errors, as the band structure is effected considerably. In previouswork in a hexagonal lattice the lattice depth was calibrated by calibrating three differentone-dimensional lattices created by only two beams of the hexagonal lattice. The hexago-nal optical potential will only be as symmetric as possible if the three measured 1D latticedepths are all equal [49]. This method has been proven not effective enough as the bandstructure obtained after lattice amplitude modulation did not fit to the calculated bandstructure for the lattice depth that was assumed in the experiment. An illustration of howthe 1D lattice calibration works is given in figure 3.20.

The ultracold cloud is loaded into a lattice consisting of each two beams, so for threecombinations the lattice depth has to be determined. After lattice amplitude modulationand band mapping [57] the band structure can be obtained. The lattice depth of a 1Dlattice is extracted from a fit to a calculated band structure as described in [33]. The threepolarization unit vectors of each lattice beam read:


minnorm. OD

max norm. OD

Figure 3.18: Amplitude modulation spectroscopy averaged. The dispersion relation with an averageof three modulation pairs and also included the case ε1 = ε2 = ε3 = 0.3. Here, also the upper s-band iscoupled. Solid lines show the calculated band structure.

e1 =

cos(θ)

0

sin(θ)

, e2 =

−1

2 cos(θ)√

32 cos(θ)

sin(θ) eiα2

, e3 =

−1

2 cos(θ)

−√

32 cos(θ)

sin(θ) eiα3

. (3.28)

The measured lattice depth VLat of each combination provides information of the intensityof the two beams because of the interference factor

√IiIj. The intensity at the atoms is

determined from the fields:

I(r) = |ei√Ii eikir +ej

√Ij eikjr |2 (3.29)

This leads to a dipole potential considering D1 and D2 transition [48]

Udip = −I(r)πc2

2Γ1ω3

1( 1ω1 − ωg

+ 1ω1 + ωg

) + 2Γ2ω3

2( 1ω2 − ωg

+ 1ω2 + ωg

) (3.30)

For the lattice depth the interference terms√IiIj are the ones to be considered. From


Figure 3.19: Momentum resolved band structure. The band structure can be determined with highprecision. Solid lines show the calculated band structure.

Figure 3.20: Calibration of the lattice depth. For each two beams lattice depth calibration is performedvia amplitude modulation spectroscopy and excitation into the third band. For each combination the 1Dlattice depth is determined.

measuring three different pairs it is now possible to determine the intensity of each of thethree beams at the position of the atoms, which is done numerically. With these values we


are able to calibrate the lattice in such a way that we should create symmetric lattices interms of beam balance.

Unfortunately, we found that calibrating the lattice with this method suffers from system-atic errors leading to imbalanced two-dimensional lattices. This may happen because of anon-circular dipole trap beam profile leading to different trap frequencies in gravitationaland horizontal direction. The 1D calibration method relies on center-of-mass momentumdetermination, which produces incorrect values for the momenta due to the dynamics ofthe trap [17]. Because of different trap frequencies in different directions, the strength ofthe dynamics varies for these directions resulting in a non-symmetric determination of thelattice depth for the three combinations. However, this method still proves to be a goodstarting point for further fine tuning.We found, that the second and third excited band are sensitive to beam imbalances andhence are ideally suited to compensate these. In a perfectly balanced situation the respec-tive bands touch each other at the Γ-point. If the beams are not balanced, Dirac conesemerge at different momenta. The generation of a Dirac cone can thus directly be relatedto the responsible beam: when a beam is too weak in power, Dirac cones exist along thelattice beam direction as illustrated in figure 3.21. If the beam is too strong instead, theDirac cone will be generated perpendicular to the beam’s wave vector. The bigger the beamimbalance, the larger the distance of the Dirac cones from the Γ-point. By analyzing theband structure for six different paths corresponding to the direction of each lattice beam’swavevector, we are now able to identify beam imbalances of one lattice beam using twopaths in momentum space.

L2

L3

L1

M

K Γ

L1'

L3' L2'

Figure 3.21: Lattice calibration using paths in first Brillouin zone. The three lattice beams(L1=’Unten’, L2=’Wand’, L3=’ADWin’, cf. figure 3.9) and the respective perpendicular direction (L’)and the first Brillouin zone as seen on our Andor detection camera. Right: one path in momentum space.Other paths used for analysis are represented by rotating the Brillouin zone 60 clockwise. This yieldssix different paths. Two paths at a time are used to reveal errors caused by each one beam.


For errors regarding lattice ’Unten’ the path shown in figure 3.21 and the path resultingfrom a rotation of the Brillouin zone by 180 have to be considered. In other words: whenthere is a crossing between the M and Γ-point the respective beam is too strong, if it isbetween the Γ and K-point it is too weak. Note that no crossing at all means that all threebeams are too different to each other. Figure 3.22 shows the calculated first three excitedbands. The strength of one beam (L3) is varied in intensity with respect to the other two.The touching point between second and third excited band moves with increasing beamstrength of L3 from the M point over Γ towards the K point. Remarkably, the Dirac coneemerging from the zeroth and first excited bands also moves, but in the opposite direction.

M Γ K M

Streng

th of beam L3

Figure 3.22: Calculated band structure for an imbalanced beam. The first (green), second (blue)and third (red) excited bands are shown in a series of varying strength of the ’ADWin’ axis beam. Ifthe beam is weaker than the other two the Dirac cone is shifted between Γ and K point. If the beam isstronger, the cone is shifted between Γ and M point. When all three beams are balanced, the touchingpoint is at the Γ point.

Apparently, when the touching point of second and third band is at the Γ point i.e., thebalanced case, the Dirac cone sits at the K point, indicating a symmetric lattice calibra-


Streng

th of beam L3 ('A

DW

in')

M Γ K M

Figure 3.23: Lattice calibration with ADWin axis. The second and third excited bands are shown ina series of varying strength of the ’ADWin’ axis beam. Solid lines show the desired band structure. Ifthe beam is too weak the Dirac cone is shifted between Γ and K point. If the beam is too strong, thecone is shifted between Γ and M point.

tion. Figure 3.23 shows experimental data of the movement of the Dirac cone of the secondand third excited band with increasing strength of the ’ADWin’ lattice beam. Here, thebehavior shown in figure 3.22 is resembled well.Now, we are able to deduce which beams are responsible for the imbalance and correct theintensities in the experiment control. The lattice is calibrated by a calculation of the bandstructure with a specific lattice depth and changing the intensity of each beam iterativelyuntil the experimentally obtained band structure fits to the calculated one. Applying this


method we found that deviations from the 1D lattice calibration method are indeed consid-erable. Although the iterative calibration process is time consuming, the results make thisapproach worthwhile, as all figures shown in the previous section have been created afteremploying this method. In the previous section we saw that the Dirac cone between zerothand first band is not spectroscopically accessible, but the results from this section let usassume that the desired lattice is created when the excited bands resemble the expectedstructure.

3.5 Summary

The existing hexagonal lattice setup was changed and enhanced during the course of thisthesis. We now exercise full control over the geometry of hexagonal lattices with arbitrarypolarization. The control over polarization is introduced into the system with a pair ofquarter- and half-wave plates for two laser beams, one only needs a half-wave plate. Thissetup provides a versatile tool to create AB-offset lattices and other structures. To probe theband structure lattice amplitude modulation has been established as an excellent tool. Thelattice calibration method, making use of the observation of moving Dirac points in higherbands with respect to beam imbalances, is an important step towards more interestingphysics regarding dressing of bands and topological bands with or without interactions.

In the future dressed bands and interactions in these bands are a subject to explore. Dressingcan be realized via lattice shaking [58, 59] , a technique that is now also established in ourexperiment. Furthermore via quench dynamics it should be possible to measure the Berrycurvature momentum-resolved, that provides a good tool to examine topological bands [16].

71

4 Conclusion and Outlook

In the context of this thesis a Λ-enhanced D1 gray molasses cooling system has been im-plemented into the experimental setup. After optimization of different parameters andimplementing a two-stage cooling scheme with capture and cooling phase, a temperature of5.7± 0.2 µK was reached which constitutes to my best knowledge the coldest 40K molassesreported so far. Compared to other experiments our setup lacks the atom number by afactor of 100, but the phase-space density at ≈ 10−5 is in the same order. We observe thesharp resonance feature of very deep cooling at zero Raman detuning δ. The effect canbe explained assuming a Λ system and the consequential existence of further dark statesthat work in an additional gray molasses scheme. After the molasses stage the atoms areloaded into a magnetic trap. To guarantee a maximum number of trapped atoms, the atomshave to be brought in the right state. Therefore, we implemented an optical pumping andrepumping beam, that also work on the D1 transition and we show efficient loading ofthe magnetic trap. In order to sympathetically cool the potassium cloud, we have to loadbosonic rubidium, too. Simultaneous operation of potassium and rubidium molasses hadmerely little effect on temperature and atom number. We show that a much lower startingtemperature for potassium proves to be meaningful for sympathetic cooling in a thermalbath of rubidium, as we were able to reduce the experimental cycle time by roughly 30%,mostly by cutting the evaporation duration in half.Although we are satisfied with the performance of the experiment, one can think about im-plementing an all-optical scheme using direct loading of a dipole trap from the molasses andsubsequent evaporation using a Feshbach resonance to reduce experimental cycle time evenmore. In the current setup it is just not possible due to too low power that the Ti:Sapphlaser produces, which is used for the dipole trap. Work in progress is the replacement ofthe Ti:Sapph used for D1 cooling with a diode laser and tapered amplifier, to free up themore costly laser, which can then be used for potential shaping, for example.

Having implemented this new cooling scheme the focus turned to simulating solid statephysics in optical lattices. We exercise full control of the lattice beam’s polarization byimplementing a quarter-wave/ half-wave plate array for two of the three lattice beams, the

72 4 Conclusion and Outlook

third just needs a half-wave plate. This provides the possibility to create interesting latticegeometries like dimers, and AB-offset lattices.First experimental data in two different kinds of lattices are presented. Calibration of thewave plates, which is necessary for reliable tuning, is performed using a polarimeter, whichallows to observe both polarization directions. Another important result during remod-eling the lattice setup is a lower heating rate in the lattice without using a phase-lock.After completing the lattice setup, first measurements are performed using the amplitudemodulation technique to probe the band structure of a honeycomb lattice and we get ridof systematic errors when calibrating the lattice by making use of specific alterations inthe band structure when the beam power is imbalanced, enabling us to measure the bandstructure with very high precision.The next steps are to also probe the band structure of other lattice types, but more impor-tantly observe, for example, topological phase transitions in AB-offset lattices by measuringa momentum-resolved Berry curvature. In order to create these special systems we havealready implemented the lattice shaking technique into the setup, so lots of possibilitiespresent themselves at the BFM-project in the near future.

List of Figures 73

List of Figures

2.1 Dark states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Polarization gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Bright and gray molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Lambda system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Lamba-enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Cooling and repumping scheme . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Setup of the optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Spectroscopy scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 Fluorescence signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.11 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.12 Raman detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.13 Raman detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.14 Repumping power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.15 Typical signal of the cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.16 Duration of molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.17 Diffusion of the molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.18 Cooling intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.19 Two-stage cooling scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.20 Final cooling intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.21 Detuning of the cooling beam . . . . . . . . . . . . . . . . . . . . . . . . . . 292.22 Time-of-flight series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.23 Experimental procedure with sympathetic cooling . . . . . . . . . . . . . . . 332.24 Optical pumping scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.25 Optical pumping on D1 and D2 line . . . . . . . . . . . . . . . . . . . . . . 352.26 Atom number after evaporation and optical pumping . . . . . . . . . . . . . 36

3.1 Triangular lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Honeycomb lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Phase diagram of lattice geometries . . . . . . . . . . . . . . . . . . . . . . . 443.4 AB-offset lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5 Poincare sphere with quarter-wave plate . . . . . . . . . . . . . . . . . . . . 463.6 Poincare sphere with half-wave plate . . . . . . . . . . . . . . . . . . . . . . 473.7 High-symmetry path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.8 Bandstructure along the high-symmetry path . . . . . . . . . . . . . . . . . 49

74 4 Conclusion and Outlook

3.9 Setup of the polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.10 Beat signals of two beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.11 Lifetime in a honeycomb lattice . . . . . . . . . . . . . . . . . . . . . . . . . 563.12 Lifetime in a 1D lattice in honeycomb configuration . . . . . . . . . . . . . 573.13 Calibration of the quarter-wave plates . . . . . . . . . . . . . . . . . . . . . 583.14 Calibration of the half-wave plates . . . . . . . . . . . . . . . . . . . . . . . 593.15 Sequence for amplitude modulation spectroscopy . . . . . . . . . . . . . . . 613.16 Amplitude modulation spectroscopy with symmetrical modulation . . . . . 633.17 Amplitude modulation spectroscopy with non-symmetrical modulation . . . 643.18 Amplitude modulation spectroscopy averaged . . . . . . . . . . . . . . . . . 653.19 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.20 Calibration of the lattice depth . . . . . . . . . . . . . . . . . . . . . . . . . 663.21 Lattice calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.22 Calculated band structure for an imbalanced beam . . . . . . . . . . . . . . 683.23 Lattice calibration with ADWin axis . . . . . . . . . . . . . . . . . . . . . . 69

List of Tables

2.1 Optimal parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Comparison of parameters for a gray molasses. . . . . . . . . . . . . 31

3.1 Fast axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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Danksagung

An dieser Stelle möchte ich mich bei allen Menschen bedanken, die es mir ermöglicht habendiese Arbeit erfolgreich zu meistern.An erster Stelle möchte ich mich bei Herrn Prof. Dr. Klaus Sengstock für die Betreuung mei-ner Masterarbeit bedanken. Ich habe ein lehrreiches, spannendes und motivierendes Jahr inseiner Arbeitsgruppe erleben dürfen und kann behaupten in diesem einen Jahr mehr als imgesamten Studium vorher gelernt zu haben. Außerdem möchte ich mich für die Möglichkeitbedanken, bei der DPG teilzunehmen und einen Vortrag dort zu halten.Bedanken möchte ich mich weiter bei Herrn Prof. Dr. Henning Moritz für die freundlicheÜbernahme des Zweitgutachtens.Ganz ausdrücklicher Dank gilt natürlich dem BFM-Team, namentlich Dr. Christof Weiten-berg, Dr. Benno Rem, Nick Fläschner und Dominik Vogel, sowie Frieder Fröbel und HaukeSchmidt. Ohne euch wäre die Zeit nur halb so schön gewesen. Mein Dank gilt besondersNick, mit dem ich die meiste Zeit endlose Stunden im Labor verbracht habe und der miralles über das Experiment beigebracht hat. Von ihm durfte ich nicht nur viel über Physiklernen, sondern auch über viele Dinge in anderen Lebensbereichen. Dominik war immereine große Hilfe für mich, wenn es um Programmierung ging. Danke an Christof und Bennofür eine hervorragende Betreuung des gesamten Projektes. Für meine Fragen gab es im-mer ein offenes Ohr und die Atmosphäre sowohl im Labor als auch außerhalb war einfachphänomenal. Auch in schwierigen Phasen gab es immer Leute, die einen aufgebaut haben,sodass ich schlussendlich feststellen kann: in ein besseres Team hätte ich gar nicht kommenkönnen. Für das Korrekturlesen meiner Arbeit möchte ich mich noch einmal ausdrücklichbei Christof und Nick bedanken.Weiterhin möchte ich mich bei allen Mitgliedern der Gruppe Sengstock und den Mitar-beitern des Instituts für Laserphysik für das freundliche und motivierende Arbeitsklimadanken.Danke an meine Freunde, die mir am Wochenende die nötige Abwechslung vom Laboralltaggeben. Großer Dank gilt auch meiner Familie. Ohne euch hätte ich es nie soweit geschafft.Danke Nicole für deine Unterstützung. Ich bin froh, dass es dich gibt und ich hoffe, dassjetzt wieder weniger stressige Tage auf uns zu kommen.

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Implementation of a lambda-enhanced gray molassesphoton.physnet.uni-hamburg.de/.../Masterthesis-Matthias-Tarnowski.pdf · Master Thesis Matthias Tarnowski ... In section3.3I discuss