Department of Physics and Astronomy

Department of Physics and Astronomy University of Heidelberg Diploma thesis in Physics submitted by Ion Stroescu born in Bucharest Issued 2010 On ...
Author: Kerrie Dawson
1 downloads 2 Views 5MB Size
Department of Physics and Astronomy University of Heidelberg

Diploma thesis in Physics submitted by Ion Stroescu born in Bucharest

Issued 2010

On a cold beam of potassium atoms

This diploma thesis has been carried out by Ion Stroescu at the Kirchhoff Institute for Physics under the supervision of Prof. Dr. M. K. Oberthaler

On a cold beam of potassium atoms This thesis summarizes the endeavour to design and build an apparatus as well as the laser system for the creation of a Bose-Einstein condensate of potassium. Our approach uses a two-dimensional magneto-optical trap to produce a high flux cold beam of potassium atoms, which respresents a common starting point of experiments with ultra-cold atoms. This thesis introduces the basic theory of Bose-Einstein condensation, laser cooling, magneto-optical traps and Feshbach spectroscopy together with some important properties of potassium crucial for the success of the experiment. The optical setup is described in detail, including a short introduction to the laser types we use, how they are locked to the spectroscopy and the way the light is acousto-optically modulated in order to fulfil the requirements for laser cooling. Furthermore, the vacuum chamber is explained, in particular the source of the cold atomic beam with its unique feature, the double clip coil. The atom beam divergence is discussed, as well as the experimental chamber. Finally the first observation of 39 K in a magneto-optical trap is reported and a short outlook is presented on the future of the experiment.

¨ Uber einen kalten Strahl von Kaliumatomen Diese Diplomarbeit fasst das Unterfangen zusammen eine Apparatur sowie das Lasersystem f¨ ur die Erzeugung eines Bose-Einstein-Kondensats aus Kalium zu entwickeln und aufzubauen. Unsere Methode verwendet eine zweidimensionale magnetooptische Falle, um einen kalten Strahl von Kaliumatomen mit hohem Fluss herzustellen, welcher einen u ¨blichen Startpunkt f¨ ur Experimente mit ultrakalten Atomen darstellt. Diese Arbeit f¨ uhrt die grundlegende Theorie der Bose-Einstein Kondensation, Laserk¨ uhlung, magnetooptischer Fallen und Feshbach-Spektroskopie ein, zusammen mit einigen wichtigen Eigenschaften von Kalium, die ausschlaggebend f¨ ur den Erfolg des Experiments sind. Der optische Aufbau wird im Detail beschrieben, einschließlich einer kurzen Einf¨ uhrung in die verwendeten Lasertypen, wie diese an die Spektroskopie stabilisiert werden und die Art in der das Licht akustooptisch moduliert wird, um die Anforderungen der Laserk¨ uhlung zu erf¨ ullen. Außerdem wird die Vakuumkammer erl¨ autert, insbesondere die Quelle des kalten Atomstrahls mit ihrem einzigartigen Merkmal, der double clip coil. Die Atomstrahldivergenz wird diskutiert, ebenso wie die Experimentierkammer. Zum Schluss wird von der ersten Beobachtung von 39 K in einer magnetooptischen Falle berichtet und ein kurzer Ausblick u ¨ber die Zukunft des Experiments pr¨asentiert.

Contents

1

Introduction

1

2

Theoretical background 2.1 Laser cooling . . . . . . . . . . . 2.1.1 Magneto-optical trapping 2.2 Bose-Einstein condensation . . . 2.2.1 Scattering length . . . . . 2.3 Feshbach resonances . . . . . . . 2.4 Sympathetic cooling . . . . . . .

. . . . . .

3 3 5 6 7 9 10

3

Properties of potassium 3.1 Optical properties . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Scattering properties . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14

4

Optical setup 4.1 Laser system . . . . . . . . . . . . . . . . . . . . . . . 4.2 Light modulation . . . . . . . . . . . . . . . . . . . . . 4.2.1 Measuring the beam waist . . . . . . . . . . . . 4.2.2 The potassium spectroscopy cell . . . . . . . . 4.2.3 Doppler-free saturated absorption spectroscopy 4.2.4 Lock-in amplification . . . . . . . . . . . . . . . 4.2.5 Acousto-optical modulation . . . . . . . . . . . 4.3 Light transport . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Optical fibres . . . . . . . . . . . . . . . . . . . 4.3.2 Preparing the MOT beams . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

17 19 23 24 25 26 28 31 32 32 33

Vacuum chamber 5.1 2D-MOT . . . . . . . . . . . . 5.1.1 Double clip coil . . . . . 5.1.2 Atomic beam divergence 5.2 Pumps, gauges and valves . . . 5.3 Differential pumping system . . 5.4 Experimental chamber . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

35 37 39 40 41 44 45

5

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

i

Contents

5.4.1 5.4.2

Glass cell . . . . . . . . . . . . . . . . . . . . . . . . . . Dipole trap mirrors . . . . . . . . . . . . . . . . . . . . .

45 46

6

State of play 6.1 Observation of the MOT . . . . . . . . . . . . . . . . . . . . . . 6.2 Atom number and loading rate . . . . . . . . . . . . . . . . . . 6.3 Magnetic field dependence and temperature . . . . . . . . . . .

47 47 51 52

7

The 7.1 7.2 7.3 7.4

57 57 57 59 61

future of the experiment Feshbach coils . . . . . . . . . . . . Optical dipole trap . . . . . . . . . Sympathetic cooling with rubidium Experiments with potassium . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

A Acknowledgements

69

B Declaration

71

ii

List of Figures

2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19

Scheme of the energy levels in a magneto-optical trap . . . . . Scattered wave functions with and without potential . . . . . . Open and closed channel in Feshbach theory . . . . . . . . . . . Occurrence of a Ramsauer-Townsend minimum for negative background scattering lengths . . . . . . . . . . . . . . . . . . . Enhanced three-body recombination close to Fesbach resonance Feshbach resonance of 39 K at about 400 G . . . . . . . . . . . .

10 10 11

Vapour pressure of potassium in the temperature range from 0 ◦ C to 127 ◦ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level scheme of the bosonic potassium isotopes 39 K and 41 K .

14 15

Optical setup for laser cooling of potassium . . . . . . . . . . . Doppler-free spectrum of 39 K with Verdi V10 as pumping laser Doppler-free spectrum of 39 K with MonoDisk-515-MP as pumping laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inside of the Coherent MBR-110 Titanium:Sapphire laser . . . Ideal MBR-110 cavity beam position on the etalon . . . . . . . Scheme of the Coherent MBR-110 Titanium:Sapphire laser . . Oscilloscope signal of the MBR-110 reference cavity fringes . . Transmittance spectrum of O2 with minimum at 766.701 nm . . Measured beam waist and fitted error function . . . . . . . . . Doppler-free spectrum of 39 K with identified transitions . . . . Scheme of the setup for Doppler-free saturated absorption spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scheme of the origin of a cross-over signal . . . . . . . . . . . . Doppler-free saturated absorption spectroscopy with and without MBR-110 reference cavity enabled . . . . . . . . . . . . . . Procedure of frequency locking using a PI controller . . . . . . The three fundamental methods of signal modulation . . . . . . The three critical points of line shape modulation . . . . . . . . Scheme of an AOM path in the cat’s eye configuration . . . . . The components of a self-made AOM driver . . . . . . . . . . . Illustration of the numerical aperture of a lens . . . . . . . . .

5 8 9

18 20 21 22 22 23 23 24 25 25 26 26 27 28 29 29 31 32 32

iii

List of Figures

iv

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17

Labelled scheme of the vacuum apparatus . . . . . . . . . . . The Kimball spherical octagon . . . . . . . . . . . . . . . . . Scheme of the vacuum apparatus including item frame . . . . Photograph of the double clip coil prior to its installation . . Expected velocity distribution of the atomic beam . . . . . . The double clip coil built into the Kimball spherical octagon Measured radial magnetic field of the double clip coil . . . . . Direction of current flowing through double clip coil . . . . . Illustration of radial quadrupole field generated by four wires Cut through the schematics of the vacuum chamber . . . . . Scheme for the calculation of the atomic beam divergence . . Schematic top view of the vacuum apparatus . . . . . . . . . Picture of the Varian UHV-24p Ionization Gauge . . . . . . . Drawing of the Varian TSP cryopanel . . . . . . . . . . . . . Technical drawing of the differential pumping tubule . . . . . Photograph of the glass cell mounted to the 6-way cross . . . Technical drawing of the dipole trap mirrors . . . . . . . . . .

. . . . . . . . . . . . . . . . .

36 37 38 39 39 39 40 40 40 41 41 42 42 43 44 45 46

6.1 6.2 6.3 6.4

48 49 49

6.5 6.6 6.7 6.8

False colour images of the 39 K-MOT . . . . . . . . . . . . . . . Picture of 39 K atoms in a magneto-optical trap . . . . . . . . . Polarization of the six MOT beams . . . . . . . . . . . . . . . . Image of the 39 K-MOT taken with the Hamamatsu CCD camera ORCA-05G . . . . . . . . . . . . . . . . . . . . . . . . . . . Scheme of the detuning settings for the 3D-MOT . . . . . . . . Measurement of the 39 K MOT loading rate . . . . . . . . . . . Dependence of atom number on magnetic field coil current . . . Paired measurements for MOT temperature estimation . . . . .

7.1 7.2 7.3

Ideal alignment of separation posts between the Feshbach coils Suggestive technical drawing of the Feshbach coil holders . . . Optical setup for laser cooling of potassium and rubidium . . .

57 58 60

50 50 52 53 54

1 Introduction Bose-Einstein condensation was experimentally realized for the first time in 1995 [1, 2, 3], more than 70 years after being theoretically predicted [4, 5, 6]. Since then it has enabled a multitude of interesting experiments which would not have been possible otherwise. Among the most remarkable consequences of Bose-Einstein condensation are superfluidity [7, 8, 9], leading to quantized vortices [10], and coherence across a macroscopic region [11]. Out of the many alkali atoms that Bose-Einstein condensation has been achieved with, like 7 Li [3], 23 Na [2], 41 K [12], 85 Rb [13], 87 Rb [1] and 133 Cs [14], potassium deserves our special attention. Potassium is but one of two alkali atoms with both bosonic (39 K and 41 K) and fermionic (40 K) naturally occurring isotopes, the other being lithium. Furthermore, the existence of favourable Feshbach resonances [15] allows us to study a Bose-Einstein condensate of 39 K exhibiting a wide range of interaction strength, from an ideal to a strongly interacting Bose gas. In addition the interaction can be varied between the attractive and the repulsive regime. Like 7 Li the bosonic potassium isotope 39 K has a negative background scattering length [16] and thus intrinsically exhibits an attractive interaction. One of the requirements for creating a Bose-Einstein condensate is a source of cold atoms. Although it is possible to magneto-optically trap atoms from the background vapour inside the experimental chamber, two other methods have turned out to be much more efficient, both of which utilize a second chamber with much higher pressure, separated from the experimental chamber by a differential pumping system. One approach uses the Zeeman effect to continuously adjust the energy levels of the atoms coming from a hot oven according to the detuning of a laser beam that slows them down. This configuration is called a Zeeman slower. The other solution is a two-dimensional magneto-optical trap (2D-MOT), sometimes called funnel, that produces a cold beam of atoms trapped inside the experimental chamber. We have chosen the latter approach for several reasons. Using a 2D-MOT allows us to design a vacuum chamber that is much more compact, with a distance of less than 63 cm between the centre of the 2D-MOT and the place where condensation takes place. At this short distance, given a certain divergence, the spreading of the atomic beam is small enough to not contribute significantly to the background pressure inside the experimental chamber. Furthermore,

1

Chapter 1

Introduction

we aim for a high flux of the atomic beam in order to achieve a fast MOT loading time. This facilitates a high repetition rate of the experiment, thus improving the statistical measurements. The choice of 39 K is largely based on a broad Feshbach resonance at a moderate magnetic field of about 400 G. Its width of about 50 G allows fine tuning of the scattering length, and thus the interaction strength, about zero, as well as at high values close to the resonance. Alternatively, 133 Cs has favourable Feshbach resonances as well [17, 18], however we decided against this species due to its high three-body recombination loss rate [19]. For our aim to create a Bose-Einstein condensate of 39 K with a 2D-MOT as the source of a cold beam of potassium atoms we have designed a double clip coil located inside the high vacuum chamber that generates the required radial magnetic quadrupole field. The scope of this work is the design and setup of both the laser system, which is used for laser cooling the atoms, as well as the vacuum chamber. The thesis is structured in the following way. First we address the theoretical background in Chapter 2, introducing laser cooling and trapping of atoms, Bose-Einstein condensation, as well as changing the interaction properties via Feshbach resonances. The following Chapter 3 summarizes some of the properties of potassium which are relevant for our experiment. In Chapter 4 we discuss the optical setup, from the generation of light and its frequency modulation to shaping the beams according to our requirements. Here we devote a large part to the description of the employed commercial equipment, since a profound understanding thereof was very helpful in establishing a failure-free operation. We will also cover Doppler-free saturated absorption spectroscopy of potassium and how to lock the laser frequency to such a signal using lock-in amplification and a proportional-integral controller. The other important part of the experimental setup is the vacuum chamber. Chapter 5 is devoted to its two main parts: the two-dimensional magnetooptical trap as a source for cold potassium atoms and the experimental chamber. A special focus is put on the unique double clip coil we use to generate the radial quadrupole field required for the 2D-MOT. In Chapter 6 the first observation of 39 K atoms in a magneto-optical trap is documented. The thesis is concluded in Chapter 7 with an outlook on the future of the experiment, that involves using 87 Rb to sympathetically cool 39 K, since efficient evaporative cooling is not possible with 39 K alone due to the RamsauerTownsend effect.

2

2 Theoretical background The first section of this chapter presents an introduction to laser cooling, in particular magneto-optical trapping of atoms. We then discuss the basic theory of Bose-Einstein condensation, which serves as a motivation for a later stage of the experiment. The following part of the theory section deals with Feshbach resonances, the method of finding those resonances via three-body losses and how they can be used to tune interactions in a Bose-Einstein condensate. Finally, sympathetic cooling is discussed, a concept which is essential for the condensation of 39 K, since this potassium isotope does not allow efficient sub-Doppler cooling. We also introduce the Ramsauer-Townsend effect which impairs the efficiency of evaporative cooling.

2.1 Laser cooling The interaction of atoms with photons allows us to manipulate their speed and trap them for experimental purposes using laser light. The principles of laser cooling can be understood by investigating the deceleration of an atom by a counter-propagating laser beam. The radiation force of light, Frad = IA/c, depends on its intensity I and the illuminated area A, where c is the speed of light. For an atom, the corresponding area is given by the absorption cross-section σ, thus the force exerted by the laser beam is Frad = σI/c. In the simple theoretical model of a two-level atom, this force can be expressed in a different form as Fscatt = ~k ρ˙ 22 , where ~k is the photon momentum and ρ22 is the population of the excited state [20]. The lifetime of this state is given by τ = Γ−1 , with the natural linewidth Γ. Without any external driving, one obtains the differential equation ρ˙ 22 = −Γρ22 .

(2.1)

The scattering force is then given by Fscatt = ~kRscatt with the scattering rate Rscatt = Γρ22 . The steady-state solution of the optical Bloch equations for times t  τ gives Rscatt =

Γ Ω2 /2 , 2 δ 2 + Ω2 /2 + Γ2 /4

(2.2)

3

For this the Nobel Prize in Physics was awarded to Steven Chu, Claude Cohen-Tannoudji and William D. Phillips in 1997.

The minus sign in eqn 2.1 accounts for the fact that the population of the excited state decreases exponentially with time. When discussing the light force, however, we are only concerned with its absolute value.

Details about the optical Bloch equations may be found in [21].

Chapter 2

Theoretical background

where δ = ω − ω0 + kv is the detuning of the laser with frequency ω with respect to the atom’s resonance frequency ω0 , including the Doppler shift kv for an atom at speed v. The Rabi frequency Ω is related to the saturation intensity via I/Is = 2Ω2 /Γ2 [21]. Substituting Ω in eqn 2.2 yields the scattering rate as a function of intensity and results in a scattering force given by Fscatt = ~k

I/Is Γ . 2 1 + I/Is + 4δ 2 /Γ2

(2.3)

For large intensities, I  Is , this scattering force approaches Fmax = ~kΓ/2. In other words the scattering rate is given by Rscatt = Γ/2, since the populations of ground and excited state are equal in this limit. For an atom of mass m the maximum acceleration according to Newton’s law is amax =

This technique is called optical molasses and was proposed by Steven Chu and co-workers from Stanford University in 1985.

Fmax ~k Γ vrec = = , m m2 2τ

(2.4)

with the recoil velocity vrec = ~k/m ≡ h/λm. This is the velocity exhibited by an atom that absorbs a single photon with momentum ~k. For the D2 line of 39 K the recoil velocity is about 1.3 cm/s, which corresponds to a maximum acceleration of amax = 3 × 105 m/s2 . In order to cool a gas in which the atoms move randomly in each spatial direction, multiple laser beams are required. In fact, for each dimension two counter-propagating beams are needed. If both have a frequency that is tuned several natural linewidths below the resonance (red-detuned), moving atoms will experience a repelling force from the laser beam that they are running into, since the light will be closer to resonance due to the Doppler effect, while the co-propagating laser beam is even further detuned and does not affect the atoms. For atoms at rest the mean influence of both beams is compensated. The lowest temperature expected with this technique is given by the Doppler cooling limit, kB TD = ~Γ/2, which arises from considerations regarding the minimal kinetic energy of the atoms. The fact that experiments have shown much lower temperatures in certain cases has led Jean Dalibard and Claude Cohen-Tannoudji to the description of Sisyphus cooling in 1989 [22]. They extended the two-level atom model by considering magnetic substates and optical pumping between those, such that a new dissipative process diminishes the minimal temperature. We will not elaborate on this effect, since it does not play a significant role in the case of potassium. Due to the narrow spacing in the hyperfine structure of the excited levels, as seen in Fig. 3.2, potassium atoms initially in the |F = 2i state will quickly be transferred into the |F = 1i state and vice versa (we do not have the case of a closed transition). This strong optical pumping prevents efficient sub-Doppler cooling.

4

2.1

mJ –1

B

E

mJ 1

B

0 J=1

0 1

Laser cooling

σ

+



σ

ω z

–1

J=0

Fig. 2.1 In this scheme of a two-level atom with states |J = 0i and |J = 1i the frequency ω of the circularly-polarized light is detuned below the resonance of the |J = 1, mJ = 0i state.

2.1.1 Magneto-optical trapping In addition to cooling the atoms we wish to trap them, i.e. achieve spatial confinement, in order to increase their density. This is accomplished by adding a magnetic field gradient to the existing laser beams from the optical molasses technique and ensuring that the light has a circular polarization. Two coils mounted in a distance that is equal to their radius will generate a uniform field gradient when the currents flow through them in opposite directions, while preserving B = 0 at the centre; this is called an anti-Helmholtz configuration. In the central region the magnetic substates of the atoms will experience a linear energy shift due to the Zeeman effect. Assuming, for simplicity, an atom with two states |J = 0i and |J = 1i and circularly-polarized light, as shown in Fig. 2.1, the following situation arises. An atom at position z will be resonant with the σ − -light, which induces a transition into the |mJ = −1i substate, while the atom is pushed towards the centre of the trap. A similar situation is given for the counter-propagating σ + -light, which is only resonant for atoms at position −z. The total force inside this magneto-optical trap (MOT) is given by +



σ σ FMOT = Fscatt (ω − kv − (ω0 + βz)) − Fscatt (ω + kv − (ω0 − βz)) ∂F ∂F ' −2 kv + 2 βz, ∂ω ∂ω0

(2.5)

where Fscatt is the scattering force, known from the optical molasses technique, as a function of the detuning. The terms ω0 + βz and ω0 − βz are the resonant absorption frequencies for a ∆mJ = +1 and a ∆mJ = −1 transition, respectively, both at position z. The approximation is only valid for small velocities kv  Γ and small Zeeman shifts βz  Γ. The Zeeman shift is given by

5

Here J is generic for an angular momentum, while later we will address the hyperfine states denoted by F .

It is convenient to use only one laser beam and retro-reflect it after passing a quarter-wave plate. This way the returning light always has the opposite polarization.

Chapter 2

Theoretical background

gJ µB dB z, (2.6) ~ dz with the Land´e factor gJ ' 1 and the Bohr magneton µB = e~/2me . Since Fscatt is a function of δ = ω − ω0 , we have ∂F/∂ω = −∂F/∂ω0 and the total force can be rewritten as βz =

∂F αβ (kv + βz) = −αv − z, (2.7) ∂ω k with a damping coefficient α = 2k∂F/∂ω and a spring constant αβ/k. Atoms which enter the region where all six laser beams cross undergo an overdamped oscillation. Thispbecomes manifest in a damping ratio ζ > 1. This ratio is given by ζ = α/ 4mαβ/k, where m is the mass of the atom. For potassium with a mass m ≈ 39 u and a photon wavelength of 767 nm, assuming a magnetic field gradient of 10 G/cm and a light intensity of 0.1 Is , we estimate ζ ' 5. Here we make use of the fact that for δ = −Γ/2 the damping coefficient can be expressed as α = ~k 2 I/Is [23]. As a result of the additional position-dependent force the MOT has a higher velocity capture range than the equivalent molasses (same laser light, but no magnetic field gradient). However, due to the increased density of the atomic cloud, the lowest achievable temperature is higher in a MOT. FMOT = −2

For a force written in the form F = −cv − kx the damping ratio is defined as c , ζ= √ 2 mk where m is the mass of the object.

2.2 Bose-Einstein condensation Contrary to the condensation of vapour into a fluid, caused by attractive forces, Bose-Einstein condensation is a purely statistical effect that does not rely on interaction. Historically, Bose investigated the behaviour of photons, non-interacting particles, whereupon Einstein carried the idea over to atoms. At a certain temperature T , an atom with mass m has a thermal de Broglie wavelength of λdB = √

The fact that composite particles, like atoms, made out of an even (odd) number of particles with halfinteger spin, obey Bose-Einstein (Fermi-Dirac) statistics is shown in [24].

h , 2πmkB T

(2.8)

where h is Planck’s constant and kB the Boltzmann constant. This length scale is a measure for the position uncertainty of the atom. When this wavelength grows to the order of the interparticle distance, the atoms become indistinguishable and their quantum nature emerges. The spin of the particle now plays a major role in determining their statistical behaviour. We will focus on bosons, particles with integer spin, and not deal with the quantum degeneracy of half-integer spin fermions. Bosons have a tendency to flock together, such that they preferably occupy states that already accommodate other bosons. The thermal de Broglie wavelength can be increased by lowering the temperature. At the same time the interparticle distance has to be decreased,

6

2.2

Bose-Einstein condensation

such that the number density n goes up. Quantum statistics become important when the phase-space density satisfies nλ3dB ∼ 1. Let us consider an ideal Bose gas in a box of volume V . The total number of particles can be written as N = N0 + NT , where N0 is the number of atoms in the state with the lowest energy 0 and NT is the number of thermal atoms. In general the condition for Bose-Einstein condensation, i.e. the macroscopic occupation of a single-particle state, is given by NT (Tc , µ = 0 ) = N,

(2.9)

where µ is the chemical potential. From this the critical temperature Tc can be obtained as a function of the (fixed) number density n = N/V , namely 2π~2 Tc = kB m



n g3/2 (1)

2/3 ,

(2.10)

with g3/2 (z) being a special case of the Bose function gp (z) depending on the fugacity z = exp(βµ), where β = 1/kB T [25]. In particular g3/2 (1) = 2.612 for µ = 0, i.e. z = 1. The number of thermal atoms is given by NT = V λ−3 dB g3/2 (z). For T < Tc we can set µ = 0 and obtain  NT =

T Tc

3/2 N,

(2.11)

and with this a condensate fraction of N0 =1− N



T Tc

3/2 .

(2.12)

For temperatures above Tc the total number of atoms corresponds to NT and the phase-space density is given by nλ3dB = g3/2 (z).

2.2.1 Scattering length So far we have only discussed non-interacting Bose gases. In the regime of dilute gases, however, alkalis exhibit a weak interaction and have to be treated accordingly. At ultracold temperatures, i.e. very low collisional energies, the interaction of atoms in a gas can be regarded as scattering of hard p spheres. ikz Let us consider an incident plane wave of the form e , where k = 2mE/~2 depends on the particle’s mass m and energy E. Assuming that this particle is scattered from a potential V (r), the resulting wave function is ψ(r) = eikz + f (θ) eikr /r. The entire information about the scattering potential V (r) is contained in the scattering amplitude f (θ). At low energies, the atoms have no orbital angular momentum, i.e. l = 0, and f ∝ Y0,0 , hence the name s-wave scattering. The scattered wave function then becomes ψ ' 1 + f /r, since

7

The angle θ is defined between the z-axis and the direction of r.

Chapter 2

Theoretical background

Fig. 2.2 In the presence of a molecular potential, the scattered wave function gathers an additional phase as compared with the wave function scattered from a point-like object (shown as a dotted line). In regions where the potential is negligible, the scattered wave function is proportional to sin(kr − δ). Figure from [27].

For more information about elastic scattering at low energies see [26].

kr and kz vary little over the range of interaction. The scattering length is defined as a = − lim f (θ), E→0

We can now define weak interaction or the condition of a dilute gas as na3  1.

(2.13)

such that ψ(r) = 0 on a spherical surface of r = a. Although this is an illustrative description of scattering of a hard sphere, note that quantum mechanics allows a < 0, which corresponds to an attractive interaction. For alkali atoms the electronic spin state is determined by the spin 1/2 of the outermost electron. When discussing collisions between two alkalis, one must differentiate between the singlet and triplet electron spin state. Their collisional behaviour may be very different, hence one considers a singlet scattering length and a triplet scattering length. At very low energies it is adequate to take into account only the lowest angular momentum term (l = 0) of the partial wave expansion. The scattering of a particle from a potential can then be described by a phase-shift δ of the scattered wave function in comparison to the unperturbed one, as shown in Fig. 2.2. Using this method an alternative definition of the scattering length is given by 1 lim k cot δ(k) = − , k→0 a where the phase-shift δ(k) depends on the wave number k.

8

(2.14)

2.3

Feshbach resonances

2.3 Feshbach resonances

 a = abg 1 −

∆B B − B0

 ,

(2.15)

where the width of the resonance is defined as ∆B = Bzc − B0 with Bzc being the magnetic field at which the scattering length vanishes. In the vicinity of the zero-crossing, i.e. B ≈ Bzc , eqn 2.15 can be linearized to result in a(B) ∼

abg (B − Bzc ), ∆B

(2.16)

which demonstrates that in order to precisely tune the scattering length, a small background scattering length is favourable, as well as a large width of the Feshbach resonance. Both attributes are present when working with a particular Feshbach resonance of 39 K at about 400 G (see Tables 3.1 and 3.2). Dissipation due to molecule formation is not possible for two particles, due to momentum conservation. However, when three particles interact, two of them may form a molecule, while the third one carries away the excess momentum. The binding energy of such a molecule is large enough for the particle to leave the trap. Close to a Feshbach resonance the absolute value of the scattering rate increases vastly, strongly enhancing three-body recombination. In this regime the three-body recombination rate scales according to

9

The cross-section is twice the spherical surface 4πa2 due to bosonic enhancement, the fact that the probability of occupying a state is increased by N + 1, if N bosons are already in that state.

closed channel Energy

We have seen that at low energies the interaction in a Bose-Einstein condensate is solely governed by the s-wave scattering length a. Experimental data for this parameter can be obtained by measuring the elastic cross-section σ = 8πa2 and the results range from large positive values of about 100 a0 for 87 Rb and 23 Na, where a0 ≈ 52.9 pm is the Bohr radius, to smaller values for spin-polarized hydrogen and even negative values of about −400 a0 for 85 Rb and about −30 a0 for 39 K [28, 29, 30]. The scattering length may be tuned by means of so-called Feshbach resonances. Two colliding particles in a particular spin state may form a bound state, associated with a certain molecular potential, which we shall call closed channel. Particles of a different spin state will scatter according to a potential that differs from the former. Such an open channel is shown in Fig. 2.3, together with the closed one. If the two spin states have different magnetic moments, the position of the bound state relative to the asymptotic energy of the collisional channel can be tuned via an external magnetic field. The point at which the energies coincide shall be given by B0 . If there is a small coupling between the two channels, the scattering length can be expressed as a ∼ 1/(B − B0 ) [25]. On the other hand, if the magnetic field is far away from the resonance, the scattering length assumes a constant value abg , called background scattering length. A suitable parametrization for the entire range of the magnetic field is given by

open channel

Interparticle distance

Fig. 2.3 The scattering length diverges when the energy of the bound state coincides with the asymptotic energy of the open channel.

Chapter 2

Theoretical background

(a)

(b)

Fig. 2.4 (a) The cross-section for elastic s-wave scattering of 85 Rb has a minimum at 375 µK. At 650 µK a g-wave resonance can be seen in the total 85 Rb cross-section. On the other hand, 87 Rb does not show this behaviour, due to its positive background scattering length. Figure from [32]. (b) The solid line marks the total cross-section for 87 Rb–40 K. The dashed and dotted curves represent s-wave and p-wave cross-sections, respectively. Zero-temperature values for scattering among the same species are also indicated. Figure from [33].

K3 ∝ a4 [31], and the resulting decline of the atom number can be used to experimentally obtain the position of a Feshbach resonance (see Fig. 2.5).

2.4 Sympathetic cooling

Fig. 2.5 The three-body recombination is enhanced close to a Feshbach resonance, shown here for the |F = 3, mF = 3i state of 133 Cs. Figure from [17].

The rate of inelastic collisions, which heat up the atomic cloud, has to be taken into account as well. 87 Rb has good scattering properties with respect to that as well.

In order to create a Bose-Einstein condensate, i.e. increase the phase-space density beyond unity, the technique of evaporative cooling has proven to be crucial. By removing the hottest atoms and allowing the gas to thermalize, the temperature can be lowered significantly, while the density remains comparatively high. Although the number of atoms may be reduced by a few orders of magnitude, the remaining atoms exhibit a higher phase-space density. The method of evaporative cooling relies on collisions between the atoms in a gas. One requirement for optimal scattering is a large, positive background scattering length. In the case of 87 Rb with a ∼ 100 a0 , where a0 is the Bohr radius, this is well established. In contrast, 39 K does not offer such a pleasant starting point. The background scattering length of about −33 a0 is both rather small and negative. This poses a problem, because of the suppression of s-wave collisions at relatively low energies for negative scattering lengths, known as the Ramsauer-Townsend effect. At the temperature where the swave scattering cross-section vanishes, the largest contribution to the total cross-section comes from p-wave scattering. At this point, however, the total cross-section is orders of magnitude lower than the zero-temperature value, as seen in Fig. 2.4(a). It can be shown that the cross-section corresponding to the lth partial wave obeys σl ∝ T 2l at low energies [34], such that at low temperatures, the p-wave threshold falls off like T 2 . This power law can be

10

2.4

Sympathetic cooling

Fig. 2.6 The scattering length of 39 K shows a broad resonance at about 400 G for collisions between |mF = 1i states. Figure from [38].

observed in the theoretical prediction of the p-wave partial scattering crosssections in Fig. 2.4(b). The elastic s-wave cross-section of 39 K has a Ramsauer-Townsend minimum at T ∼ 320 µK, where contributions from other partial waves are still small [35]. For this reason, efficient evaporative cooling requires the addition of a second species that acts as a coolant. A promising candidate for sympathetic cooling is 87 Rb, since an inter-species background scattering length of aKRb = 36 a0 assures a large total cross-section for all temperatures. There is even a 39 K–87 Rb Feshbach resonance that can be used to tune the inter-species scattering. The threshold at which the p-wave contribution to the total cross-section starts to decrease can be estimated by considering a long-range form of the collisional potential ~2 l(l + 1) C6 − 6 (2.17) 2µR2 R where R is the internuclear separation, µ is the reduced mass and C6 is the van der Waals coefficient. Setting the collision energy Ecoll = 3kB T /2 to the maximum of U gives the p-wave threshold. For collisions between potassium and rubidium C6 = 4274(13) a.u., where the value in parentheses states the theoretically estimated uncertainty [36], and the p-wave threshold is at about 110 µK. Homonuclear collisions of potassium atoms have a van der Waals coefficient of C6 = 3897(15) a.u. [37]. In summary, we expect that a rather difficult cooling scheme is required 39 for K. However, the ability to tune the interaction strength of the Bose gas very precisely about zero and even to negative values of the scattering length outweighs the disadvantages. U (R) '

11

Chapter 2

12

Theoretical background

3 Properties of potassium Potassium is an alkali metal with atomic number Z = 19 and a standard atomic weight of 39.0983(1) [39]. The single outer electron is responsible for the high chemical reactivity, in particular when immersed in water. The melting point is at 336.53 K or 63.38 ◦ C, the boiling point at 1032 K or 759 ◦ C [40]. Potassium has two stable isotopes, 39 K and 41 K, both bosonic, with a natural abundance (NA) of 93.26% and 6.73%, respectively [41]. The fermionic isotope 40 K has a half-life of 1.28 × 109 years and a NA of 0.012% [42]. Experiments with 40 K therefore use enriched dispensers of the fermionic fellow. An important property when dealing with cold atom sources is vapour pressure. At thermal equilibrium between a solid (or fluid) and its vapour the latter exerts a pressure on the former, called vapour pressure, which is a function of temperature. When the vapour pressure of a solid (or fluid) is higher than the pressure of the surrounding gas, it will sublimate (or vaporize). The temperature dependence of the vapour pressure is given by the ClausiusClapeyron law   L p(T ) = p0 exp − , (3.1) kB T where L is the latent heat, kB the Boltzmann constant and T the temperature. For the purpose of plotting it is useful to rewrite this equation as B , (3.2) T where A and B are constants that depend on the material. For potassium these constants were obtained from [43] and used to graph the vapour pressure in Fig. 3.1. The constants also depend on the aggregate state, but change only slightly when going from solid to liquid. We use the knowledge of the vapour pressure to adjust the potassium pressure inside the 2D-MOT by appropriately heating an oven that contains 1 g of potassium. log p = A −

3.1 Optical properties The ground state of potassium is 42 S1/2 , as shown in Fig. 3.2. From there the two most prominent transitions are D1 to 42 P1/2 and D2 to 42 P3/2 ,

13

Vapour pressure (mbar)

Chapter 3

10

-4

10

-5

Properties of potassium

-6

10 10

-7

10

-8

10

-9

solid liquid 280

300

320

340

Temperature (K)

360

380

400

Fig. 3.1 The vapour pressure of potassium is shown in the temperature range from 0 ◦ C to 127 ◦ C. At 336.53 K potassium changes from a solid into a liquid.

For a cycling transition, i.e. one where the excited state can only decay into one ground state, the saturation intensity is given by Is =

πhc . 3λ3 τ

induced by light with a wavelength of 770.108 nm and 766.701 nm, respectively. The lifetime of the former is 26.72(5) ns, while the D2 line has a lifetime of 26.37(5) ns [44], corresponding to a natural linewidth of Γ/2π = 6.036(12) MHz. The Doppler temperature, i.e. the lowest temperature achievable by Doppler cooling, given by kB TD = ~Γ/2 is 144 µK. The momentum transfer of an emitted photon to the atom is mvrec = ~k, where k = 2π/λ is the wavenumber of the photon and m the atom’s mass. For potassium the recoil velocity is vrec ≈ 1.3 cm/s and the associated temperature, 2 , is about 400 nK. However, the recoil temperature given by kB Trec = 21 mvrec as described in Section 2.1, due to the narrow spacing of the excited 42 P3/2 levels, this limit of sub-Doppler cooling cannot be reached. The D1 and the D2 lines of potassium have a saturation intensity Is of 1.73 mW/cm2 and 1.75 mW/cm2 , respectively.

3.2 Scattering properties At low collisional energies, a requirement usually fulfilled in experiments with cold gases, only partial waves with the lowest angular momenta need to be taken into account. At ultracold temperatures the situation is well described solely by s-wave scattering. Table 3.1 summarizes the values of the s-wave scattering length a for collisions between the three potassium isotopes 39 K, 40 K and 41 K.

14

3.2

Scattering properties

Fig. 3.2 This level scheme of the bosonic potassium isotopes 39 K and 41 K, taken from [45], shows the hyperfine structure of the ground state 42 S1/2 and the excited state of the D2 transition 42 P3/2 . The cooling transition is denoted by ω2 , while ω1 is the repumping frequency. The detunings δ1 and δ2 are set to values of a few natural linewidths. Since the level spacing of the excited states |F 0 i is on the order of the detunings, the cooling light also excites states other than |F 0 = 3i, which may decay at a high rate into the |F = 1i ground state. Hence the power of the repumping light has to be almost as high as the one of the cooling light. With 93.3% the natural abundance of 39 K is much higher than that of 41 K.

15

Chapter 3

Properties of potassium

Isotopes

as (a0 )

at (a0 )

39+39 39+40 39+41 40+40 40+41 41+41

138.49(12) -2.84(10) 113.07(12) 104.41(9) -54.28(21) 85.53(6)

-33.48(18) -1985(69) 177.10(27) 169.67(24) 97.39(9) 60.54(6)

Table 3.1 The singlet and triplet s-wave scattering lengths for all combinations of the potassium isotopes 39 K, 40 K and 41 K were taken from [46].

As discussed in Section 2.3, in the vicinity of a Feshbach resonance the s-wave scattering length can be parametrized by   ∆B a(B) = abg 1 − , (3.3) B − B0 The width is defined as ∆B = Bzc − B0 , where Bzc is the field at which the scattering length vanishes, called zero-crossing.

where abg is the zero-energy background scattering length, B0 is the position of the resonance and ∆B its width. Table 3.2 lists the Feshbach resonances of 39 K for l = 0 that have been experimentally verified to date. For a complete list, including the theoretically predicted resonances, see [15]. The most promising resonance is between |mF = 1i states at 403 G. Due to its large width of 52 G it is possible to precisely tune the scattering length, i.e. the interaction strength, about zero.

mF , mF

B0 (G)

−∆B (G)

abg (a0 )

1, 1

25.85(10) 403.4(7) 752.3(1) 59.3(6) 66.0(9) 32.6(1.5) 162.8(9) 562.2(1.5)

0.47 52 0.4 9.6 7.9 -55 37 56

-33 -29 -35 -18 -18 -19 -19 -29

0, 0 −1, −1

Table 3.2 This summary of the measured Feshbach resonances of 39 K, together with the theoretically predicted widths and background scattering lengths, was taken from [15].

16

4 Optical setup The first step towards experiments with ultracold atoms is laser cooling. The potassium D2 transition between the 42 S1/2 ground state and the 42 P3/2 excited state requires light at 766.701 nm, in our case generated by the Titanium:Sapphire laser MBR-110 from Coherent, which is pumped by the thindisk laser MonoDisk-515-MP from ELS. At first we used a Coherent Verdi V10 to pump the MBR-110, but had to switch to the ELS. A small fraction of the infrared light from the Ti:Sa is sent to the spectroscopy cell, where it is used to lock the laser to the cross-over signal between the |F = 1i and |F = 2i hyperfine ground states and the excited states. Due to the narrow spacing of the potassium 42 P3/2 hyperfine structure, the repumping laser power has to be almost as high as the power of the cooling light, in contrast to rubidium, where the repumper is much weaker than the cooler. As depicted in Fig. 4.1, the main part of the 767 nm light passes two mirrors that can compensate any drift in the pointing of the Ti:Sa, followed by a telescope, which reduces the beam waist by a factor of two, to about 0.6 mm, while at the same time ensuring that the laser beam is collimated. The irises help to maintain a propagation of the light central with respect to the lenses. Without this telescope we experienced cases where the light coming directly out of the Ti:Sa doubled its waist after 2 m. The collimated laser beam is divided into two parts: one path for the 3D-MOT light and one for the 2D-MOT and the push beam. The frequency modulation is done via acousto-optical modulators (AOMs). For the 2D-MOT two frequencies (cooling and repumping light) are required, hence we use two AOMs here. In order to optimize the flux of the atomic beam we have the possibility to produce a push beam with two frequency components, in spite of reports that for 39 K one mainly needs repumping light for this beam [45]. Another two AOMs are used to generate the required frequencies for the 3DMOT. The cooling and repumping laser beams are superimposed, by means of a polarizing beamsplitter cube (PBS). Since they now travel in the same direction but have perpendicular polarization, we need a half-wave plate and another PBS to end up with two beams, both consisting of 50% cooling and 50% repumping light. Those two beams propagate in perpendicular directions, yet each of them is fully polarized, a requirement set by the polarization-

17

The spectroscopy signal as well as the locking procedure are explained in detail in Section 4.2.3 and 4.2.4.

The vacuum chamber is explained in detail in Chapter 5.

Chapter 4

Optical setup

125 mm

3D-MOT 125 mm

125 mm

push beam Quarter-wave plate Half-wave plate Polarizing beam splitter

50 mm 100 mm

Mirror Lens

2D-MOT

K

Photo diode Iris Fibre Beam dump

ELS

Ti:Sa

Shutter Acousto-optical modulator Fig. 4.1 This scheme shows the optical setup for laser cooling of potassium. Some of the light emitted by the Ti:Sa is branched off to the heated spectroscopy cell and retro-reflected onto a photo diode, while the main beam is forwarded to the AOMs for frequency modulation. Two frequencies are superimposed at a time and coupled into optical fibres.

18

4.1

Laser system

maintaining optical fibres we use to transport the light to the vacuum chamber. The unavoidable downside of this solution can be observed in the case of the push beam, where only one laser beam is needed and the other one has to be dumped, thus discarding half of the initial laser power. The same is true for one of the 3D-MOT beams, since only three are needed. Fortunately, the power required for the 3D-MOT is significantly lower than for the 2D-MOT, while the push beam requires even less power, therefore the loss is acceptable. Directly after outcoupling the 2D-MOT light from the fibre, we use telescopes to widen the diameter of the two beams from 1 mm to 5 mm, followed by cylindrical telescopes that enlarge the beam diameter in one direction to 25 mm, while leaving the other axis unchanged. Thus an elliptical beam with an aspect ratio of 5:1 is formed. These beams traverse the 2D-MOT chamber and are retro-reflected. For the 3D-MOT we use a cage-system, where the light is extracted from the fibre without an outcoupler. The divergent beam passes a quarter-wave plate and is collimated by a lens to a waist of 15 mm before being retro-reflected.

Retro-reflection of MOT beams, while resulting in a slight power imbalance due to reflections, has the advantage of using the available laser power twice.

4.1 Laser system In this section we will discuss the different lasers we employ or have employed in our experiment, in particular the difference between a high-end commercial pumping laser (Coherent Verdi V10) and a prototype pumping laser (ELS MonoDisk-515-MP). A deep understanding of the equipment was crucial for the successful operation of these lasers, especially in the case of the MonoDisk515-MP thin-disk laser. The light force felt by the atoms is proportional to the intensity of the beam. For the given area of illumination inside our 2D-MOT the power needs to be about 260 mW. Assuming another 10 mW for the push beam and 150 mW for the 3D-MOT (3×20 mW for repumping and 3×30 mW for cooling light), the total laser power after the optical fibres has to be 420 mW. Taking into account that about 40% of the power is inevitably lost when using fibres and the double-pass AOM paths have an efficiency of about 70%, this experiment requires at least 1 W of infrared light. This light is generated by the Titanium:Sapphire laser MBR-110, which is pumped by about 10 W of 515 nm light from an ELS thin-disk laser. In the early stage of the experiment, we used the Coherent Verdi V10 as a pumping laser. The first spectroscopy signals (see Fig. 4.2) were taken with this configuration. The Verdi V10 is a continuous wave diode-pumped solid state (DPSS) laser at 532 nm that uses a single core fibre-coupling scheme. This results in a high stability concerning beam profile and noise, even as the diode array ages. As a consequence the laser is fully sealed and does not require any maintenance. The diode array, consisting of 19 linearly arranged diodes, has a power output of 40 W and a typical lifetime of 20,000 hours. It is

19

This assumption is based on the values from [45], where in each 2DMOT beam they used 50 mW and 80 mW for repuming and cooling light, respectively.

Chapter 4

Optical setup

(a)

(b)

Fig. 4.2 The Doppler-free absorption signal taken with the Verdi V10 as the pumping laser is shown (a) as a whole, and (b) zoomed in. The bottom curve (channel 1) is the output of the photo diode, while the top curve (channel 2) shows the error signal from the lock-in amplifier.

Lithium Triborate (LiB3 O5 ) is a nonlinear optical crystal that allows non-critical phase matching (NCPM), i.e. one which is not critical to angular variation.

located in the power supply, allowing for a simple exchange without opening the laser head. The 19 separate diodes produce elliptical, astigmatic and highly divergent beams that are combined, through the fibres, into one beam with circular profile. This single beam, ideal for the TEM00 mode of the DPSS cavity, is fed into the active medium, in this case a Nd:YVO4 laser rod. In contrast to usual DPSS lasers where the light coming from the separate diodes is collected into a multi-core fibre, thus resulting in a beam profile that changes its shape when one or more of the diodes start to deteriorate, the Verdi uses a single-core fibre with a uniform output shape. The latter changes homogeneously with time, preventing the population of non-TEM00 cavity modes as well as a change of the gain profile of the Nd:YVO4 rod. Nd:YVO4 emits light at 914 nm, 1064 nm and 1342 nm. While the stimulated emission cross-section at 1064 nm is four times larger than for Nd:YAG, the optical conversion efficiency is larger than 60%. In combination with a frequency doubling NCPM LBO crystal, a high output power at 532 nm can be achieved. The Verdi V10 has a nominal output power of at least 10 W and a linewidth of less than 5 MHz. We used this device as the pump of a Ti:Sa laser to obtain the spectrum of 39 K that can be seen in Figure 4.2. The photo diode signal shows one of the ground states of the hyperfine structure as well as the cross-over signal, while the other hyperfine ground state is not clearly visible. Channel 2 of the oscilloscope is an impressive portrayal of the lockin amplifier’s capability, since both hyperfine ground states can be observed in the error signal. The principles of lock-in amplification are described in Section 4.2.4.

20

4.1

(a)

Laser system

(b)

Fig. 4.3 For this measurement of the Doppler-free spectrum the ELS MonoDisk-515-MP thin-disk laser was used to pump the Ti:Sa. Both (a) wide and (b) narrow intervals are shown. Here the top curve is the photo diode signal and the bottom curve shows the lock-in error signal.

After a few months with the Verdi, we had to switch to the ELS MonoDisk515-MP thin-disk laser as a pump. The laser body houses a thin crystal disk, the active laser material, that is attached to a heat sink and, pumped by diode lasers, injects 1030 nm YAG light into a cavity. This light passes two etalons, the first output mirror and then two curved mirrors, of which the second one acts as another outcoupler. Between the two curved mirrors a second harmonic generator (SHG) is placed, that doubles the frequency and thus halves the wavelength of the light to 515 nm. The SHG is a birefringent crystal, in which incident light that is polarized along one of the axes creates a second harmonic with polarization along the other axis. The temperature dependent refractive index is different for initial beam and second harmonic, thus by changing the temperature of the crystal one is able to find an optimal value where the two waves have the same phase and interfere constructively. This process is called phase matching. The ELS provides a TEM00 beam with M 2 < 1.1 and a beam diameter of 3 mm ± 10%. The beam divergence is supposed to be 0.5 mrad and the power instability < 2%. However, at times we have experienced power fluctuations of more than 10% over a short time. The measurement of the hyperfine states of 39 K is shown again in Figure 4.3, this time using the ELS as a pump. When comparing the lock-in amplifier’s error signal in both pictures, one can see that this signal is inverted in 4.3(a), i.e. mirrored with respect to the abscissa. Being mainly the first derivative of the photo diode’s signal, the error signal, e.g. in the case of the cross-over, should be negative for frequencies lower than the resonance and positive for higher frequencies, as can be seen in 4.3(b). The loop amplifier used for constant feedback of the error signal to the MBR-110 provides the

21

To avoid overheating the thin disk has to be constantly sprinkled with deionized water, that is supplied by an external water purification unit.

In this case the birefringent crystal is heated to 37 ◦ C.

The beam quality factor M 2 is a measure of how close a laser beam is to a diffraction-limited Gaussian beam, the later having per definition M 2 = 1.

MBR110 Chapter 4

Optical setup

Optical Schematic

Pum

3 2

4 6

1

6 5

Fig. 4.4 A monolithic block (1) inside the MBR-110 houses all the critical elements of the cavity, like the piezo-mounted mirror (2) and the optical diode (3). The etalon (4) together 1. Monolithic Resonator high stability in a compact and robust design. Features and with the reference cavity (5)Provides are required for frequency stabilization. The laser light passes Benefits two mirrors2. in the Brewster angle (6), which allows fasta unity scanning of inthe Piezo-mounted Mirror Designed for high-frequency response, and providing gain bandwidth excesslaser frequency 40 kHz, the piezo-mounted mirror ensures a tight lock corresponding to a very without any deviation of theof beam. (Copyright Coherent) narrow linewidth.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Fig. 4.5 The MBR cavity beam should permeate the etalon at position 12, 13 (looking towards the output coupler).

3. Miniature Optical Diode

This highly compact device ensures unidirectional operation over the entire tuning range.

4. Etalon

A single, thin etalon ensures single-mode operation using an innovative servo-lock to

possibility to invert this eliminate signal. mode-hops. The Coherent is high-finesse a single frequency laser, 5. Reference Cavity MBR-110 An Invar-spaced, (>25) interferometer, held in a Titanium:Sapphire temperature-controlled environment and sealed to resist pressure changes, provides good long-term stability which uses a one way bow-tie ring configuration in order to reduce spatial for the single-frequency laser. Locking to this cavity can achieve relative linewidths as low as 10 kHz. hole burning, a deformation of the spectral gain shape due to interference patGalvanometer-mounted Brewster plates enable scanning in excess of 40 GHz at 800 nm, without any terns of a 6.standing wave.Twin The pump Tilting Brewster Plates resultant beam deviation. laser is centred onto the crystal using two external mirrors, while the telescope at the entrance of the MBR-110 can be 7. Ti:Sapphire Selected, high figure-of-merit material provides a wide tuning range from 700 nm to 1000 nm. used to adjust the diameter of the incident pumping light. The Ti:Sa crystal 8. Birefringent Filter low-insertion-loss, three-plate birefringent filter provides smooth tuning over a broad emits fluorescence light Atuning into the ring cavity towards mirrors M1 and M2 (see range. Fig. 4.6). 9. The forward The fluorescence from M2 passes the optical isolator and Mirrors entire tuning range is covered using the minimum number of laser mirrors, thus reducingmirror the need for time-consuming mirror changes. All mirrors use a threaded insert reaches the piezo-mounted M3 (used to delicately adjust the cavity system, ensuring ease and reproducibility of mirror replacement. length), while the backward fluorescence passes the birefringent filter on its 10. Output Couplers Optimum output couplings are chosen to suite the varying gain profile of Ti:Sapphire, ensuring high output powerslight and overallpaths efficiency throughout way to the output coupler M4. Both have thetotuning berange. aligned throughout the entire cavity. The optical diode forces light to propagate in only one direction through the ring cavity, while a well calibrated etalon guarantees single mode operation. For optimal performance, the position where the cavity beam runs through the etalon should be adjusted according to Fig. 4.5. Together with the temperature-controlled, high finesse reference cavity, whose error signal drives the piezo-mounted cavity mirror M3, a linewidth of less than 100 kHz is achievable. The MBR-110 is designed in such a way that realignment of the cavity is rarely needed. Even if the pump laser is moved, it turned out to suffice to centre the pump light onto the Ti:Sa crystal using two external mirrors in order to achieve lasing. However, if critical parameters of the pump laser are changed, in particular the wavelength or the beam waist, the cavity might have to be adjusted. The MBR-110 manual provides a step by step instruction on how to perform the alignment from the very beginning, but is rather vague in describing the last step required for successful lasing. Once the fluorescence

22

Options and Accessories

Reduced Functionality Options

Additional Features

s

nge.

d

ert

Single-Frequency Ti:Sapphire Laser

4.2

Light modulation

Optical Schematic of the MBR-110 Ti:Sapphire Laser

9

Pump Lenses

M1

7

M2

Ti:S Crystal

Photodiodes

8

Optical Diode

Birefringent Filter M4

M3

Piezo-mounted Mirror

Etalon

Brewster Plates

Piezo

Laser Output

Output Coupler

10

Reference Cavity

Mirror

Beamsplitter

Mirror

Fig. 4.6 The Titanium:Sapphire crystal (7) emits fluorescence light in both directions of the ring cavity. Using the irreducible The amount of four mirrors (9) this light has to be superimposed Options and Frequency-doubling Option ultra-narrow linewidth performance of the MBR-110 Ti:Sapphire laser can be translated Accessories the (MBD-200) into the blue near-UV (350-500 nm) region of the spectrum by using the MBD-200 throughout entire cavity in order toandachieve lasing. The birefringent filter (8) coarsely Monolithic Block Frequency Doubler. sets the wavelength within a wide tuning range from 700 nm to 1000 nm. The output coupler (10) Reduced can be exchanged to suit desired range (SW, MW or LW). (Copyright MBR-110 PE (Passive Etalon) the MBR-110 ring cavitywavelength with intracavity etalon only. Functionality Coherent) Options

MBR-110 EL (Etalon Lock)

Addition of electronic control of the intracavity etalon to remove mode-hops.

MBR-110 Ps (Passive Scan)

Addition of intracavity Brewster plates to MBR-110 EL enables continuous scanning up

to 20 GHz. light from M1 is first positioned upon M4 and from there onto M3, it is actually possible to observe a secondNo external fluorescent spot outside of the cavity. It can be Additional Focusing Optics focusing optics are required when using any commercially available laser pump Features (see footnote 1, back page). useful to look for the second fluorescent spot not just outside of the cavity, but outside ofMBR the MBR-110 body. However, overlapping the two E-110entire Electronic Laser By custom-designing the electronic locking system simply to match the characteristics of the Servo-control Unit Ti:Sapphire ring laser, exceptionally narrow linewidths are achieved. This fully dedicated, spots using an infra-red viewer requires a considerable amount of patience. microprocessor-controlled unit is used to lock and control the single-frequency scan of the laser. An LCD display and menu-driven software enable fast and efficient control of the This process is somewhat more systematic if a photodiode is used to monitor laser’s locking and scanning functions. the signal of the two beams. Once the cavity is aligned, thus lasing has occurred, and the power has been optimized by walking the intra-cavity beams with all four mirrors, both vertically and horizontally, the etalon has to be set up, in order to enable the single frequency operating mode. With the etalon lock enabled the reference cavity alignment may reduce the laser linewidth to less than 100 kHz. The immediate attempt to tune the laser to 766.701 nm will fail due to the excitation of oxygen molecules that are present in the MBR-110 cavity [47]. This absorption of the cavity light is responsible for a severe loss of power. In order to reach the desired wavelength, the laser body has to be filled with nitrogen (or any other available gas, e.g. argon). Once enough oxygen molecules have evaded, the flux of nitrogen can be lowered. However each opening of the laser body demands a full refill with nitrogen. There is no noticeable O2 absorption outside of the cavity. Albeit the absorption is small, it is larger than the gain inside the cavity. The relevant part of the transmittance spectrum of oxygen is shown in Fig. 4.8.

4.2 Light modulation After the light has been generated it needs to be modulated and frequencylocked before it can be used for laser cooling atoms. The former is done

23

Fig. 4.7 The cavity fringes can be observed by monitoring the Reference Cavity Fringes output (yellow dot) from the electronics head using an oscilloscope that is triggered by the reference cavity sweep output from the MBR-E110. Note that the modulation switch has to be turned on (down position).

A reason for opening the laser body could be the need of a cavity realignment in case the power has decreased and optimizing the external mirrors is not sufficient.

Chapter 4

Optical setup

Fig. 4.8 The transmittance spectrum of oxygen shows a dip at 13043 cm−1 , corresponding to λ = 766.701 nm. (Copyright Physikalisch-Chemisches Institut of the Justus-LiebigUniversit¨ at Gießen)

by acousto-optical modulators (AOMs), whereas the frequency stabilization is achieved via a combination of lock-in amplification and a proportionalintegral controller (PI). The lock-in amplifier (LIA) receives its signal from the Doppler-free saturated absorption spectroscopy done with a potassium reference cell and forwards it to the feedback controller. Throughout our optical setup we wish to maintain a constant beam waist of 0.6 mm, to which the AOMs and the optical fibres are adjusted. Since, however, the waist of the beam inevitably becomes larger after propagating some distance, we are forced to use further corrective telescopes.

4.2.1 Measuring the beam waist Knowing the waist of the laser beam is very important when dealing with AOMs and fibre couplers. The crystal inside the AOM has a certain active area and the housing a given aperture. The fibre coupler’s lens is also tailored to a specific beam diameter. We have decided on a constant waist of 0.6 mm throughout the optical setup and considered this when choosing the AOMs as well as the couplers. A simple method of measuring the waist of a Gaussian beam is the use of a razor blade mounted onto a micrometer translation stage to cut off part of the laser beam and monitor the remaining intensity with a power meter. Figure 4.9 shows the data points taken for a measurement of the beam coming out of the Ti:Sa and the fit using an error function, given by 2a y=√ π

Z

bx−c

2

e−τ dτ + d,

(4.1)

0

where √ a, b, c and d are real parameters. The width can be extracted from b = ( 2σ)−1 , where σ 2 is the variance of the distribution. The waist, ob-

24

4.2

Light modulation

x 10−3 1.6

1.2

1.4

Power (W)

1

1.2

0.8

1

0.6

0.8

0.4

0.6 0.4

0.2

0.2

0 0

200 400 600 800 1000 1200 1400 1600 1800 Position (µm)

0

0

200

400 600 800 1000 1200 1400 1600 1800 Position (µm)

(a)

(b)

Fig. 4.9 (a) The error function is fitted to the dataset of the beam waist measurement using the razor blade method. (b) The first derivative of the error function yields the Gaussian beam profile from which a waist of 620(35) µm can be extracted.

tained directly from the first derivative of the error function, i.e. the Gaussian function that represents the transverse beam profile, is measured to be √ 2 = 620(35) µm. w = 2σ = b

(4.2)

The full width at half maximum (FWHM) is connected to the standard deviation via √ FWHM = 2σ 2 ln 2.

In case the output from the Ti:Sa changes its waist, which for a suboptimal cavity alignment may occur when the pumping laser is changed, we have implemented a telescope (see Fig. 4.1).

4.2.2 The potassium spectroscopy cell Locking the laser frequency to a desired value requires a spectroscopy cell, i.e. a glass cell filled only with the element to be investigated. We use the cylindrical Thorlabs CP25075-K Ref. Cell with flat windows and a diameter of 25 mm. Due to the low vapour pressure of potassium the spectroscopy cell needs to be heated to about 70 ◦ C. For this purpose we build an oven, i.e. a concentric tube about the cell, swathed in resistance wire, that we operate at about 32 V and 0.9 A. The end caps of this tube have holes allowing the light to pass through the spectroscopy cell, however the ambient air that enters the oven through these holes causes the end caps of the spectroscopy cell to be colder than the bulk, such that the potassium condenses there. To avoid this, we use a cold finger, a mesh of copper wound around the bulk of the spectroscopy cell, that dissipates the heat out of the oven to an aluminium block attached to the optical table, thus rendering the bulk colder than the end caps (a temperature difference of about 5 ◦ C is sufficient).

25

Fig. 4.10 The Doppler-free spectrum of 39 K taken after optimization of the Ti:Sa etalon and servo lock as well as the laser power of the spectroscopy beams. The crossover signal (X) and the transitions from both hyperfine ground states are indicated.

Chapter 4

Optical setup

/2

K /2

/4

Fig. 4.11 This scheme of our setup for Doppler-free saturated absorption spectroscopy shows the laser beam originating from the Ti:Sa (above), where a small fraction of it passes the potassium spectroscopy cell twice and falls onto a photodiode, while the main part is forwarded to the experiment.

In Fig. 4.10 the spectrum of 39 K taken with the heated spectroscopy cell is depicted. The cross-over signal is equidistant to the two hyperfine ground state levels. Identifying the two peaks with |F = 1i and |F = 2i from this signal is not possible at first sight, since it is not clear whether the frequency increases or decreases when moving from left to right in this plot. However, were able to assign the transitions based on their intensity. The different intensities can be understood by considering the degree of degeneracy of each state. The |F = 1i state has three magnetic substates, while |F = 2i has five, hence the latter should be more pronounced, in agreement with [48].

Fʹ ωL

ω1

ω2 F=2 F=1

Fig. 4.12 This scheme of a threelevel atom illustrates the occurrence of a cross-over signal when the laser frequency is ωL = (ω1 + ω2 )/2. The atoms in |F = 2i move with the laser beam, the ones in |F = 1i move towards the laser beam.

The probe beam should be weak enough to exclude power broadening.

4.2.3 Doppler-free saturated absorption spectroscopy The spectrum of the hyperfine ground states of 39 K is needed to set a lock point for the Ti:Sa. We are interested in the D2 transition from 42 S1/2 to 42 P3/2 that is induced by light with a wavelength of 766.701 nm. The splitting of the hyperfine ground states |F = 1i and |F = 2i is 462 MHz, so we need a method of tuning the frequency within this range. This is done via the internal etalon of the Ti:Sa. When the frequency of the light passing through the reference cell is swept across the resonance, a decreased transmission is observed in form of a lower photo diode signal. Due to the Doppler effect which is responsible for the atoms seeing the light at a different frequency ω 0 = ω − k · v, depending on the magnitude and direction of their velocity v, the linewidth of a transition is broadened, usually to the extent that multiple lines blur into each other and become irresolvable. This effect can be overcome by using two counterpropagating laser beams (originating from one source) to perform the spectroscopy. First the pump beam passes the reference cell in one direction and excites part of the atoms. Since the laser intensity is much higher than the saturation intensity of potassium Is = 1.75 mW/cm2 , almost half of the atoms are in the excited state, while the other half is in the ground state. Then the second beam, acting as a probe beam, is sent through the cell and detected on a photodiode. Since the two beams run in opposite directions, simultaneously

26

4.2

(a)

Light modulation

(b)

Fig. 4.13 For comparison the Doppler-free saturated absorption spectroscopy was acquired (a) using only the etalon lock and (b) with the additional servo-lock enabled. Note the noise reduction when both devices are activated.

they can only interact with atoms at rest (v = 0), thus disabling the Doppler broadening. We use a particular setup (see Fig. 4.11), where the pump beam is reflected after passing the potassium cell and guided once more through the cell, this time acting as the probe beam. The typical spectrum of the hyperfine structure of 39 K, as seen in Fig. 4.13, shows two transmission maxima, corresponding to the two hyperfine ground states |F = 1i and |F = 2i, as well as a transmission minimum. The latter is called cross-over signal and arises when the pump laser with ωL = (ω1 +ω2 )/2 is simultaneously resonant with atoms in |F = 2i having a speed of +v and atoms in |F = 1i having a speed of −v (see Fig. 4.12). In this case, the absorption of the probe beam is increased. Cross-over signals are usually more pronounced, since a larger class of atoms is involved. For a certain class of atoms with speed v the pump beam has the resonant frequency ω1 , whereas the probe beam’s frequency is ω2 . In this case the atoms are optically pumped from the |F = 1i into the |F = 2i state, increasing the population in |F = 2i and thus increasing the absorption of the probe beam. A similar situation arises for atoms with the same speed v, travelling in the opposite direction. Those atoms are transferred from the |F = 2i into the |F = 1i state, where they are absorbed by the probe beam. Both processes contribute to a reduced transmission signal on the photo diode at that frequency. Considering a three-level atom with one ground state and two excited states leads to a transmission maximum at the cross-over frequency. We estimate that the width of the cross-over signal is about 10% of the interval between the |F = 1i and |F = 2i signal, corresponding to a value of ∆νx ≈ 46 MHz or about eight times the natural linewidth. This is caused by the fact that all excited hyperfine states are involved in the transition. The |F 0 = 3i and |F 0 = 0i states are 34 MHz apart, giving rise to one broad signal.

27

At 70 ◦ C the potassium atoms have a most probable speed of about 380 m/s, which corresponds to a Doppler shift of kv ∼ 3 GHz.

The optical pumping occurs via the excited hyperfine states |F 0 i of the 42 P3/2 state.

For more information on Dopplerfree laser spectroscopy, especially cross-over resonances in saturation spectroscopy, see [21].

Chapter 4

Optical setup

(a)

(b)

(c)

Fig. 4.14 The three oscilloscope images show the typical locking procedure. (a) First, the spectrum is obtained (top curve), together with the signal from the lock-in amplifier (bottom curve). (b) Then the range is narrowed down close to the locking point. (c) Once the proportional (P) and integral (I) part of the loop amplifier are enabled, both signals should remain constant, indicating that the laser is frequency locked.

In fact, we were able to observe a substructure of the cross-over transition in the demodulated signal from the lock-in amplifier, which can be understood as two overlapping lines, one from the |F 0 = 3i and one from the |F 0 = 0, 1, 2i states, the difference between those being 21 MHz (see Fig. 3.2). Additionally, our spectroscopy signal is influenced by saturation power broadening, thus we end up with an overall linewidth of the cross-over transition of about 46 MHz. In Section 4.1 it was mentioned that the Ti:Sa has two methods of frequency stabilization. One being the etalon that is located inside the cavity. The other is a reference cavity, into which a small fraction of the Ti:Sa output is injected and used as a feed forward. Although it is possible to lock the laser using the etalon alone, it is favourable to use both devices, since otherwise a single mode operation cannot be guaranteed. The impact of the servo-lock, i.e. using the reference cavity in addition to the etalon, can be observed in Fig. 4.13. In Figure 4.10 a mode hop can be observed in the left-hand part of the signal. Where the Doppler-broadened absorption signal would normally rise and reach saturation, it decreases instead. In fact the signal left of the mode hop is the mirror image of the actual spectroscopy. Mode hops indicate that the laser is not operating in a single mode throughout the entire sweeping range. In our case this could be due to one of the many absorption lines of oxygen (see Fig. 4.8) or simply because of the limited free spectral range (FSR) of the Ti:Sa’s etalon. On the other hand, for a given finesse, a large FSR implies a large resonator bandwidth, which contrasts with single-frequency operation.

4.2.4 Lock-in amplification The laser frequency is locked to the cross-over signal between |F = 1 − 2i and the excited states |F 0 i. Any deviation from this frequency value is captured by

28

Frequency

(a)

Light modulation

Amplitude

Amplitude

Amplitude

4.2

Frequency

Frequency

(b)

(c)

Fig. 4.15 The three fundamental methods of signal modulation are (a) amplitude modulation (collective vertical motion of all points), (b) inhomogeneous magnetic field modulation leading to a time dependent linewidth broadening due to different magnetic sensitive transitions and (c) frequency modulation (collective horizontal motion of all points). The top scheme shows the initial line shape (shaded) and the one obtained by the corresponding modulation, while the bottom curve depicts the difference of the two signals.

a lock-in amplifier and used to correct the MBR’s piezo cavity mirror position via a PI controller. We use a coil around the potassium spectroscopy cell to induce a magnetic field that shifts the energy levels via the Zeeman effect with a modulation frequency of about 80 kHz. The result of the sinusoidal modulation is then captured by a transimpedance amplifier, a circuit used to rapidly read out a photo diode. There are three kinds of basic modulation of a spectroscopy signal as shown in Fig. 4.15. One is a vertical modulation, where every point of the signal collectively oscillates up and down. This is achieved through modulation of the laser power, also called amplitude modulation. Fig. 4.15(a) shows that this modulation has the strongest impact on the extremum, yielding a demodulated signal (difference of modulated and initial signal) that resembles the original line shape. Another way to obtain a demodulated signal is through a broadening of the linewidth. This can be achieved with a magnetic field modulation, that affects the magnetic sub-levels of the atoms. Considering the |F = 1i state at B = 0, the mF = 0, ±1 states are degenerate and give rise to a single transition peak with its natural linewidth (ignoring any broadening processes). If a magnetic field is applied, the energy levels are shifted, depending on mF . In fact mF = 1 is shifted in the opposite direction of mF = −1, thus effectively broadening the linewidth. The result is shown in Fig. 4.15(b), where the extremum senses (almost) no deviation, while the edges are modulated the most, an effect clearly displayed by the lock-in amplifier signal. Then there is the possibility of a collective horizontal motion of the initial signal, corresponding to a frequency modulation of the laser. This method

29

Fig. 4.16 The three critical points of line shape modulation are illustrated in this scheme. While A and B are points on the edges, e.g. at half maximum of the signal, point C is at the extremum.

Chapter 4

The lock-in amplifier applies a combination of low-pass and high-pass filters. The resulting band-pass filter quality factor is Q = f0 /B = p 1/(R L/C), where f0 is the centre frequency and B the bandwidth.

Optical setup

produces an error signal, shown in Fig. 4.15(c), that resembles the derivative of the initial Lorentzian line shape. At the zero-crossing the curve can be well approximated by a straight line, which is used by the PI controller to drive the laser frequency back to its set value. The typical lock-in amplifier signal is a result of all three modulations, each weighted differently. Since the power fluctuation caused by the instability of the laser is highly unlikely to happen at the set modulation frequency, amplitude modulation will be suppressed by the lock-in amplifier’s band-pass. On the other hand, when choosing magnetic field modulation, one can expect to induce frequency modulation as well. If there is a slight asymmetry in the magnetic field, some mF states will contribute more to the signal than the others, effectively causing an offset of the entire line shape. The lock-in amplifier uses phase-sensitive detection (PSD) to retrieve information from an input with low signal-to-noise ratio (SNR). PSD can detect a signal at 10 kHz with a bandwidth of only 0.01 Hz. For this the lock-in amplifier multiplies the photo diode signal, which can be parametrized as Vsig sin(ωsig t + θsig ), with its reference Vref sin(ωref t + θref ), resulting in VPSD = Vsig Vref sin(ωsig t + θsig ) sin(ωref t + θref ) 1 = Vsig Vref cos((ωsig − ωref )t + θsig − θref ) 2 1 − Vsig Vref cos((ωsig + ωref )t + θsig + θref ). 2

(4.3)

The reference can be an internal reference, which is phase-locked to the TTL sync output of the function generator that drives the modulation. A low-pass filter removes the term with ωsig + ωref . If the signal frequency ωsig is equal to the reference frequency ωref , only a DC signal remains, given by

Only inputs with frequencies at the reference frequency result in an output and shot noise is typically not important, since the bandwidth is very small.

1 (4.4) VPSD = Vsig Vref cos(θsig − θref ). 2 By adjusting the phase of the reference signal close to θsig , one tries to achieve cos(θsig − θref ) ≈ 1, thus having a maximum value of VPSD . Another way is to use a second PSD, that would yield VPSD2 ∝ Vsig sin(θsig − θref ) and use the in-phase component X = Vsig cos(θsig √ − θref ) and the quadrature component Y = Vsig sin(θsig − θref ) to obtain R = X 2 + Y 2 = Vsig . This procedure is called dual-phase lock-in. We use the Single-Board Lock-In Amplifier LIA-BVD-150-H with single phase detection (in-phase component X) and a working frequency of 50 120 kHz. The settings are shown in table 4.1. The values for the time constant, sensitivity as well as coarse and fine phase have to be found empirically and may change with varying photo diode intensity. The PI controller has the option to invert the error signal, i.e. mirror it with respect to the abscissa. This results in a derivative of the signal dip

30

4.2

Dyn.-Res. Mode PLL Ref.-Thr.

L 2f S 2V

Light modulation

H 1f F 0V

Table 4.1 The switches of the LIA-BVD-150-H are set to (bold) Ultra Stable & Low Drift and 1-f Mode, which corresponds to frequencies larger than 60 kHz. In addition Slow PLLLocking is enabled and the Reference-Input-Threshold is set to 0 V, since we use a sine modulation. If the reference signal was TTL, a threshold of +2 V ought to be specified.

(like the cross-over signal) that is positive for frequencies smaller than the resonance and negative for higher frequencies, as can be observed in Fig. 4.3. The resonance frequency of the magnetic field modulation coil is given by √ f0 = (2π LC)−1 and its inductance is L=

µ0 N 2 A ≈ 6 × 10−5 H, l + 0.9r

(4.5)

The quality factor of an inductor is given by Q = ωL/R, where R is the resistance and ωL the inductive reactance at resonance.

assuming the length l = 5.5 cm, radius r = 1.25 cm and the number of windings to be N = 80. The cross-section area of the coil is A = πr2 and µ0 = 4π × 10−7 H/m the magnetic constant. Using the LCR Meter Escort ELC-3131D we have measured the inductance to be 57 µH with a quality factor of Q = 0.515 at 1 kHz. Choosing a capacity of 68 nF leads to a resonance frequency of f0 ≈ 80 kHz.

4.2.5 Acousto-optical modulation After the Ti:Sa has been locked to the cross-over signal between the two hyperfine ground states, we divide the light into several beams and use acoustooptical modulators (AOMs) to shift the frequency according to the requirements set by the cooling beam and the repumping beam, respectively. Since the cross-over is exactly between the |F = 1i and |F = 2i states, which are separated by 462 MHz, the situation is the following: the frequency of the cooling beam has to be reduced by 231 MHz, while the repumping beam’s frequency has to be increased by 231 MHz (see Fig. 3.2). Inside the AOM there is a tellurium dioxide (TeO2 ) crystal that exhibits an oscillation driven by a piezo crystal. The radio frequency (RF) modulation of the piezo crystal leads to a sound wave in the TeO2 crystal from which incident photons are scattered. When a photon is scattered it may absorb (emit) a phonon and increase (decrease) its frequency. Our experiment is comprised of a double-pass AOM setup, where the light passes each AOM twice, resulting in a frequency shift of twice the RF frequency. Since a change of the RF frequency also changes the scattering angle, we use a so-called cat’s eye configuration [49], consisting of a lens and a planar mirror (see Fig. 4.17), that ensures a constant beam path, independent on the

31

f f

Fig. 4.17 The cat’s eye configuration ensures that the beam deflected by the AOM is led back into the aperture, even if the diffraction angle is changed.

Chapter 4

The frequency shift is a direct consequence of momentum and energy conservation of both photons and phonons. On the other hand, if a standing wave was formed inside the crystal, no frequency shift would occur.

Ideally a diffraction efficiency of 83% can be achieved at 633 nm.

V1 VCO

V2 VVA

AMP

AOM

Fig. 4.18 This scheme of a selfmade AOM driver shows all the required components.

Optical setup

applied RF frequency. It is important that the lens is focused both on the crystal and on the mirror, since the lens does not only diffract the collimated laser beam, but also focuses its waist onto the mirror. We use a focal length of 125 mm, because at distances closer to the AOM, the first and zeroth order beam are not well separated. The Crystal Technology AOM 3110-120 we use has an active aperture (crystal size) of 2.5 mm × 0.6 mm, suitable for a beam waist of 0.6 mm. The centre frequency is 110 MHz, with a bandwidth of 24 MHz. Initially we used self-made drivers to control the AOMs. These drivers consist of a voltage controlled oscillator (VCO), a voltage variable attenuator (VVA) and an amplifier (AMP), connected as shown in Fig. 4.18. We used the Mini-Circuits VCO POS-150+ to generate an RF sine output, whose frequency can be adjusted by a control voltage V1 . This RF output was attenuated as required by the Mini-Circuits VVA ZX73-2500+, using another control voltage V2 . In the last step, the signal was amplified by the Mini-Circuits ZHL-1A. The VVA is needed, since the ZHL-1A has a fixed level of amplification. Eventually we switched to AOM drivers that were engineered by the Physics Institute in Heidelberg. At the core, they also use the POS-150+. With 3 W RF power, however, their amplifier is superior to the ZHL-1A, increasing the diffraction efficiency of our AOM paths.

4.3 Light transport After the light used for laser cooling has been frequency locked and adjusted to the D2 transition of potassium, it needs to be transported to the vacuum chamber via optical fibres and the outcoming beams have to be shaped according to our 2D-MOT symmetry.



Fig. 4.19 The numerical aperture of a lens is given by NA = n sin θ, where n is the index of refraction and θ the depicted half-angle. It characterizes the capability of the lens to focus light.

4.3.1 Optical fibres In general it is useful to decouple optical setup and vacuum chamber by placing them on separate optical tables. This way any vibrations or thermal variations are restricted to one part of the experiment and do not affect the other. The connection between both optical tables is then established by optical fibres, in our case the Thorlabs Patch Cable P3-630PM-FC-5, a polarizationmaintaining fibre with FC/APC connectors (ceramic 8◦ angled ferrules) at each end. We use the Sch¨after+Kirchhoff fibre collimators 60FC-4-A3.1-02 and 60FC-4-M5-10 for incoupling and outcoupling, respectively. The former has an aspheric lens with a focal length of f = 3.1 mm and a numerical aperture (NA) of 0.68, while the later is a monochromat with f = 5.1 mm and NA = 0.25. Both collimators are suited for FC/APC connection (8◦ -polish) and 600–1000 nm light.

32

4.3

Light transport

4.3.2 Preparing the MOT beams As depicted in Fig. 4.1 we use six fibres to transport the light: two for the 2DMOT, one for the push beam and three for the 3D-MOT. Behind the fibre the 2D-MOT beams are enlarged to a waist of 5 mm using a spherical telescope with f1 = 25 mm and f2 = 125 mm. Then the beams pass a quarter-wave plate and become circularly-polarized. In the last step, before reaching the 2D-MOT chamber, the beams pass a cylindrical telescope with f3 = -12.5 mm and f4 = 60 mm and are enlarged in the horizontal direction only, resulting in elliptical beams with half-axes of 25 mm and 5 mm. These beams are then send through the vacuum chamber, passing another quarter-wave plate, before being retro-reflected. The elliptical beam shape ensures a larger capture range along the axis given by the propagation of the atomic beam, while optimizing the light intensity in radial direction. The push beam is extracted from the fibre collimator and its waist of about 1 mm is left unchanged. It is oriented such that it passes the centre of the 2DMOT coils, runs through the differential pumping tubule and hits the centre of the glass cell. The light for the 3D-MOT is injected straight into a cage-system, in which fibre holder, quarter-wave plate and a lens are mounted collinearly by virtue of four rods. Since we do not use an outcoupler here, the light from the fibre is divergent. It passes a quarter-wave plate that ensures circular polarization and is collimated by a f = 75 mm lens to a waist of 15 mm. After it has traversed the glass cell, the light is retro-reflected, thus requiring only half the power compared to six independent beams.

33

These elements of the vacuum chamber, differential pumping tubule and glass cell, are addressed in Section 5.3 and 5.4.1, respectively.

Chapter 4

34

Optical setup

5 Vacuum chamber Experiments with ultracold quantum gases usually take place in an ultra-high vacuum environment, since a considerable amount of residual gas will limit the lifetime of the trapped atoms due to collisions. In the past, experiments have been done in a single chamber by loading a magneto-optical trap (MOT) with the atoms released from a dispenser and then proceeding with magnetic trapping and evaporative cooling. More recent experiments consist of two parts: the experimental chamber and a source of cold atoms that feeds the former. By separating the two chambers with a differential pumping tubule, one can obtain a pressure difference of 104 . The high pressure inside the atom source implies a high flux of atoms that enter the second chamber, where a much lower background pressure is present. We use a two-dimensional realization of a magneto-optical trap (MOT) to create a cold atomic beam that loads the experimental cell. The atoms are cooled by the 2D-MOT in radial, but not axial direction. They are guided by a push beam through a differential pumping tubule into the ultra high vacuum (UHV). There the atoms are trapped in a 3D-MOT inside a glass cell, where magnetic trapping and evaporative cooling will later take place. We have decided to use a horizontal setup, despite the gravitational deflection of the atomic beam, since it allows us to place the opto-mechanics close to the optical table and thus avoid vibration of the posts. In this respect a vertical setup is much harder to maintain and requires at least one additional breadboard for the 3D-MOT optics. It would also demand the imaging system to be placed on the breadboard, high above the stable optical table. We don’t expect the effect of gravity to be significant, but given the divergence the atomic beam intrinsically has, we have tried to keep the distance between 2D-MOT and 3D-MOT to a minimum. Before the assembly of the apparatus most components went through an elaborate cleaning procedure. It is important to have the vacuum parts thoroughly cleaned, when aiming for a pressure in the range of 10−11 mbar and below. For this, all parts are rinsed with deionized water and immersed in an ultrasonically agitated bath of acetone for approximately 30 minutes, before they are heated up to well over 100 ◦ C. Then the procedure of cleaning with water and ultrasound is repeated. In the course of building up the experiment we have tried to learn from

35

An extensive cleaning instruction is found in [50].

Chapter 5

Vacuum chamber

9

5

4 10

7 8 1 3 6

11

2 Fig. 5.1 The vacuum apparatus features the Kimball spherical octagon (1), with the two ovens for potassium (2) and rubidium (3). A 55 l/s ion pump (4) is connected to the Kimball, as well as a turbomolecular pump (5) for the first stage of the vacuum. The connectors of the double clip coil (6) are brazed into the feedthrough. The atomic beam leaving the Kimball passes a gate valve (7) and a 6-way reducing cross before reaching the glass cell (8). The UHV is maintained by a 150 l/s ion pump (9) and measured by a Bayard-Alpert gauge (10). Lateral viewports (11) enable access for dipole trap laser beams to the experimental chamber.

36

5.1

2D-MOT

other experiments and if possible improve on them. As for the gaskets we used silver plated OFHC copper for all regular flanges and annealed, silver plated copper for the viewports. Whenever possible we used non-magnetic screws (A4) or at least low-magnetic screws (A2) in order to connect the ConFlat (CF) vacuum parts. We only use CF parts, that are UHV suitable with a leak rate of less than 10−11 mbar l s−1 and are bakeable up to 450 ◦ C. The M8 screws were tightened diagonally with 10 Nm, 15 Nm and 20 Nm consecutively, the M6 screws only with 10 Nm followed by 15 Nm. After the vacuum parts had been put together and the ion pumps had reached a constant pressure, the apparatus was covered with one layer of aluminium foil and heating elements were attached to it. On top of that several layers of aluminium foil were applied for thermal insulation. With the turbomolecular pump still operating, the entire chamber was slowly heated up to 180–200 ◦ C. At 100 ◦ C, the residual water is evaporated from the internal chamber walls. At about 180 ◦ C there is another critical point, where the diffusion coefficient of hydrogen is large enough that it is purged from the metal within a few hours. The whole vacuum apparatus is supported by an item frame, as seen in Fig. 5.3. Aluminium bars with a special profile are fixed to each other by modular screw sets. This allows us to put together a frame that is intrinsically light and easy to modify. While adding little weight to the apparatus as a whole, the item frame is very sturdy and permits an uncomplicated transfer, e.g. when moving the apparatus to another optical table.

If the copper contains oxygen molecules, they will combine with residual H2 to form water molecules, which are larger and cannot permeate the crystal structure of copper. As a consequence little pores emerge in the cooper, which compromise the vacuum [51]. Therefore oxygen-free high thermal conductivity (OFHC) copper is desirable when working with ultra-high vacuum.

Fig. 5.2 The Kimball spherical octagon offers two CF100 and eight CF40 flanges. (Copyright Kimball Physics Inc.)

5.1 2D-MOT There are two ways of realizing a source of cold atoms, both utilizing the force that an atom experiences when absorbing light. One is a so-called Zeeman slower, consisting of a magnetic field and a laser beam which is propagating in the opposite direction of atoms coming from an oven. The atoms are slowed down by absorbing the light and the Doppler-shift is compensated by an inhomogeneous magnetic field configuration. The other type of cold atom source – the one we are using in this experiment – is a 2D-MOT, sometimes called a funnel. In this case the atoms are only cooled (and trapped, i.e. forced to the centre) in radial direction, while remaining thermal in axial direction. The 2D-MOT chamber needs to reflect the symmetry of the trap and at the same time present optical access for the MOT beams as well as connections to the vacuum pumps and the potassium and rubidium ovens. This is best realized by the Kimball spherical octagon seen in Fig. 5.2. With its eight CF40 flanges and two CF100 flanges it provides the required optical access. Unfortunately the Kimball spherical octagon is too large for the magnetic quadrupole coils to be mounted on the outside, so we decided to integrate the coils into the vacuum chamber.

37

The idea of using a Zeeman slower was soon discarded for it does not offer cooling in transverse direction, thus the highly divergent atomic beam would produce a considerable background pressure.

Chapter 5

Vacuum chamber

Fig. 5.3 Supporting the apparatus with an item frame offers both stability and modularity. Since the 150 l/s ion pump, being the heaviest component, is located so far above the plane of the optical table, a strong support is mandatory. On the other hand, the ion pump is kept as far away from the glass cell as possible, since a residual magnetic field will distort measurements. The ion pump is suspended from two steel U-profiles, which are attached to the frame. Small pillars underneath the Kimball spherical octagon allow us to align the 2D-MOT with respect to the experimental glass cell via the bellow.

38

5.1

2D-MOT

Fig. 5.4 A hollow copper wire was used to wind the double clip coil that generates the radial quadrupole field for the 2D-MOT. In this configuration the axial magnetic field should be zero within the coil volume.

The ovens are separated from the Kimball by a 126 mm CF40 straight connector and a double sided blank flange with a hole of 3 mm in diameter. By heating the oven that contains 1 g of potassium to about 70 ◦ C we produce a rather diffuse beam of potassium coming out of the aperture. The straight connector has to be heated as well in order to avoid condensation of the potassium to the connector wall. During the bake out we used the Peltier element TEC1-12709 for thermoelectric cooling of the ovens, in order to conserve as much potassium and rubidium as possible. During normal operation, the Peltiers can be used to heat the ovens, which is particularly important in the case of potassium. Based on [45] we expect a speed distribution for the atomic beam that peaks at v ≈ 30 m/s, corresponding to a common capture velocity of a 3DMOT. This speed distribution is not the Maxwell-Boltzmann distribution, but rather a Gaussian, as seen in Fig. 5.5, since the atoms are cooled in two dimensions, while remaining thermal in the third.

Fig. 5.5 The velocity distribution measured in [45] peaks at 32 m/s and has a FWHM of 4.5 m/s.

5.1.1 Double clip coil Using the Kimball spherical octagon as a 2D-MOT chamber requires a custom coil design, since a low heat emission is desired. This lead us to the development of the double clip coil made out of a single winding, as seen in Fig. 5.4. We used a hollow copper wire with a 3.6 mm × 3.6 mm cross-section to wind a coil that produces a radial quadrupole field, i.e. B(r) ∝ r, while providing virtually no gradient in the axial direction, as can be understood from the

39

Fig. 5.6 The double clip coil built into the Kimball spherical octagon, including electrical insulation.

Chapter 5

Vacuum chamber

Magnetic field (G) 20 10

- 0.03

- 0.02

- 0.01

0.01

0.02

0.03

Position (m)

-10 -20

Fig. 5.7 The blue data points show the measurement of the radial magnetic field generated by the double clip coil. The solid line is a theoretical curve that assumes the dimensions of the coil to be 8 cm × 4 cm × 4 cm, a current of 80 A and a thickness of the wire of 2 mm. From this data a gradient of 15 G/cm can be inferred. Additionally, the measurement of the magnetic field in axial direction at 60 A is shown in red.

x

y z

Fig. 5.8 Direction of the current flowing through the double clip coil.

y x

Fig. 5.9 In this cross-section of the double clip coil the direction of the current in the copper wire is indicated, as well as the direction of the generated magnetic field (blue).

scheme in Fig. 5.8. One of the CF40 flanges of the Kimball spherical octagon is used for the coil’s ceramic feed-through, into which the copper connectors are brazed. The cross-section of the copper wire has a hole with a diameter of 2.6 mm, allowing the coil to be water cooled. Applying 120 A, which result in magnetic field gradients of ∼ 20 G/cm, does not measurably increase the temperature of the coil, since the heat is instantly carried away by the water. For good measure the feed-through has to be able to withstand a high temperature, therefore is was attached with the high-temperature soft solder Pb96 Sn2 Ag2 that has a melting point of 304–310 ◦ C. The interlock is set to deactivate the power supply as soon as the thermoelements attached to the coil connectors register a value above room temperature. However, cautiousness is still advisable, since a melting of the solder would instantly compromise the vacuum and have devastating consequences for the entire apparatus.

5.1.2 Atomic beam divergence Our goal is to produce a cold beam of potassium atoms, where cold means having a narrow velocity distribution that peaks at values suitable for the capture range of the 3D-MOT. It also means that the atomic beam should have a low divergence, which is crucial for a high flux. To achieve this, we took several measures. First, we have the possibility to increase the magnetic field gradient in the 2D-MOT to about 20 G/cm. This in turn increases the radial force that is felt by the atoms, resulting in a smaller beam diameter to start with. Second, with less than 63 cm, the distance between the centre of the 2D-MOT and the experimental glass cell is kept as small as possible (see Fig. 5.12). Third, we have the differential pumping tubule that acts as a

40

5.2

Pumps, gauges and valves

1

y z

3

x

2

4

Fig. 5.10 A cut through the schematics of the vacuum chamber illustrates the path of the potassium atoms that are guided from the oven (1), through the 2D-MOT (2), into the differential pumping tubule (3) and further towards the glass cell (4). If the beam divergence is too high, those atoms that do not make it into the differential pumping tubule are lost. An atom that goes through the 5 mm nozzle of the tubule, however, is very likely to stay within the capture range of the 3D-MOT.

divergence filter. Atoms with low enough divergence pass through the tubule and are sure to reach the (geometric) capture range of the 3D-MOT. Those atoms that leave the 2D-MOT at an angle that is too high, will not reach the UHV chamber and contribute to the background pressure. They may however return and traverse the 2D-MOT region again, with the chance to get to the glass cell after multiple iterations. Assuming a transverse mean speed of the atoms of about 0.3 m/s and a longitudinal speed of roughly 30 m/s, when starting with a beam diameter of 1 cm in the 2D-MOT, this beam will have increased its diameter to 3 cm after travelling for 1 m (see Fig. 5.11). From this we can assume a beam divergence of 30 mrad. Indeed, a consistent value of (34 ± 6) mrad has been measured with a similar setup in [45]. If the mean axial speed of the atoms is indeed about 30 m/s, they will reach the centre of the glass cell in 20 ms. The gravitational deflection at this point is about 2 mm, which is only a fifth of the beam’s intrinsic widening.

5.2 Pumps, gauges and valves The fore-vacuum is established by a combination of a scroll pump and a turbomolecular pump (TMP). The Leybold SC-5-D Oilfree Scroll Vacuum Pump sets the pressure below 0.1 mbar, after which the TMP can be turned on. We use the Pfeiffer Vacuum TurboDrag Pump TMU 071 Y P with Electronic Drive Unit TC 600. This is a CF-F DN63 turbomolecular pump that can

41

0.3 m/s 1 cm

30 m/s

1m Fig. 5.11 The radial capture range of the 2D-MOT is roughly 1 cm, from which the potassium atoms travel to the 3D-MOT at a speed of about 30 m/s. The actual distance between 2D-MOT and 3D-MOT is in fact less than 63 cm.

Chapter 5

Vacuum chamber

624

176 267

8,2



285

Fig. 5.12 This schematic top view of the vacuum apparatus shows the beam path of optional laser beams shone in through the lateral CF100 viewports. Those beams may cross inside the glass cell at angles from 4◦ to 10◦ (here the situation is depicted for 8.26◦ ). From the centre of the 2D-MOT the atoms travel 63 cm to the experimental region, 2 cm away from the walls of the glass cell.

reach a pressure lower than 5 × 10−10 mbar (when baked) and a maximum rotation speed of 1500 Hz or 90000 min−1 . The Y indicates that the pump can be mounted in any orientation. Pumps with a P in the product name have a sealing gas connector that prevents aggressive gases from entering the motor and bearing region. The nominal pumping speed is 59 l/s for N2 and 50 l/s for He and the maximum acceptable magnetic field is 3 mT. The fore-vacuum has to be lower than 10 mbar, which is easy to achieve using a scroll pump. Unlike the TMH (aluminium) model, this TMU (stainless steel) pump can be baked up to 120 ◦ C. Connected to the Kimball spherical octagon via a 4-way cross is the Varian VacIon Plus 55 Diode, an ion pump with 55 l/s nominal pumping speed for N2 and an ultimate pressure below 10−11 mbar. The VacIon Plus 55 is bakeable up to 350 ◦ C and has an operating lifetime of 50,000 hours at 1 × 10−6 mbar.

Fig. 5.13 Picture of the Varian UHV-24p Ionization Gauge. (Copyright Varian)

On the ultra-high vacuum (UHV) side we have the Varian VacIon Plus 150 Diode in combination with a titanium sublimation pump (TSP). As the name implies, the VacIon Plus 150 has a nominal pumping speed for nitrogen of 150 l/s and the same operating lifetime as the VacIon Plus 55. Baking temperature and ultimate pressure are also identical, however in combination with the TSP, the VacIon Plus 150 can achieve a net pumping speed of 610 l/s for N2 and 1,380 l/s for H2 . The TSP is extremely efficient for pumping getterable gases, while ion pumps are suited for non-getterable gases, like argon and methane. When the TSP is flashed, i.e. a current of 55 A is run through one of the three titanium filaments at about 5 V, the cryopanel is covered

42

5.2

Pumps, gauges and valves

Fig. 5.14 To achieve higher pumping speeds, the cryopanel, which houses the titanium filaments, may be cooled with water or even liquid nitrogen. (Copyright Varian)

with titanium and absorbs any getterable gas that comes in contact with it. Our TSP cryopanel has an inner pumping surface of 826 cm2 and a reservoir volume of 1.8 l. Based on technical information we assume that the stray magnetic field generated by the VacIon Plus 150 is less than 0.4 G at the centre of the 6-way cross and less than 0.7 G above the glass cell (same height as the ion pump). Given that the distance between ion pump and experimental region is 48 cm, the residual magnetic field at this position is negligible. In order to measure the pressure in our 2D-MOT, i.e. high vacuum (HV) region, we use the BALZERS UHV ionization gauge IMR 132 with a pressure range of 10−3 to 10−11 mbar. This Bayard-Alpert vacuum gauge is built up as follows. A positively charged acceleration grid (anode, platinum-iridium) is arranged about the central thin ion collector. Two heating filaments (cathodes, tungsten) are placed outside of the grid. The hot cathode (only one is used at any moment, but it is always possible to switch to the other) emits electrons that are accelerated by the anode to about 100 eV. On their way to the anode the electrons ionize the residual gas, the resulting number of positive ions being proportional to electron current and pressure. The lowest measurable pressure depends on the x-ray limit. The electrons emitted by the cathode create soft x-ray radiation at the positive grid, which sets free photo-electrons from the ion collector. Those photo-electrons reach the grid and feign an ion current. In the experimental chamber, i.e. UHV region, we use the Varian UHV-24p Ionization Gauge (see Fig. 5.13) that allows a reliable pressure measurement from 1.3 × 10−3 mbar down to 6.7 × 10−11 mbar, with a reduced performance for an even lower pressure. Due to an extremely thin collector the x-ray limit is lowered to 6.7 × 10−12 mbar (compared to 2.7 × 10−11 mbar for the UHV-24 model). The UHV-24p is bakeable up to 450 ◦ C and the dual thoriated iridium filaments are easily replaceable. Since a maximum exposure to the vacuum

43

Because with any ion pump it is possible to also measure the pressure, every ion gauge is itself an ion pump. The pumping speed for a nude ion gauge is typically around 0.5 l/s. The higher the emission current the higher is the pumping speed.

Chapter 5

Vacuum chamber

70

13

10

8

127

5 150

Fig. 5.15 The differential pumping tubule is edged into a CF40 double-sided blank flange. Its total length of 150 mm and inner diameter of 5 mm result in a conductance small enough to allow for a pressure ratio of 104 between high and ultra-high vacuum region.

gives the highest possible accuracy, we have mounted the vacuum gauge such that it protrudes into the straight connector between 6-way cross and 150 l/s ion pump (see Fig. 5.1). We employ two valves in our experiment. First there is an angle valve located between the Kimball spherical octagon and the turbomolecular pump that is closed after the bake out and only opened in case we need to aerate the chamber, e.g. to replenish the ovens. This device is a stainless steel CF40 angle valve from novotek with a conductance of 37.7 l/s and a leak rate of less than 1 × 10−9 mbar l s−1 . It is bakeable up to 150 ◦ C. The second valve is a gate valve, which connects the Kimball spherical octagon and the 6-way cross. Together with the differential pumping tubule that is attached directly to it, the gate valve separates the HV from the UHV region.

5.3 Differential pumping system

Assuming molecular flow, the conductance of a tube with radius r and length l is given by r 8r3 πkB T L= , 3l 2m where m is the molecular mass, T the temperature and kB the Boltzmann constant.

Having a rather high pressure in the 2D-MOT of about 10−8 mbar, while maintaining an ultra-high vacuum in the experimental chamber, is possible with differential pumping. A small tubule of 5 mm inner diameter and 150 mm length reduces the conductance to L = 0.087 l/s. The total amount of gas passing through the tubule is given by Q = (pHV − pUHV )L = pUHV S. This leads to a pressure ratio of pHV S S =1+ ≈ , pUHV L L

(5.1)

where pHV is the pressure in the 2D-MOT chamber and pUHV the pressure in the glass cell. Given a suction capacity of the titanium sublimation pump of S ≈ 1000 l/s, it is possible to achieve a pressure ratio of ∼ 10−4 . We know that the pressure inside the experimental chamber is less than 1.7 × 10−11 mbar, which is the lowest value that can be read out by the ion gauge controller.

44

5.4

Experimental chamber

Fig. 5.16 The experiments take place in a 80 mm × 38 mm × 38 mm glass cell connected via a glass-metal transition to a CF40 flange. The glass cell flange is mounted to the 6-way cross via a zero-length CF63-CF40 reducing flange.

Inside the Kimball spherical octagon we have measured 3.2 × 10−7 mbar, however this measurement was performed with a self-made filament which was not calibrated and therefore probably yields a value which is too high.

5.4 Experimental chamber The last part of this chapter concerns the creation of the 3D-MOT in the experimental chamber, i.e. the glass cell, as well as the idea behind the dipole trap mirrors built into the vacuum. In order to produce a 3D-MOT of 39 K and witness the influence of the atomic beam properties on loading time and total number of atoms in the MOT, we have built temporary coils, with 18 windings each, out of a 4 mm × 1 mm copper wire (see Fig. 5.16). The coils are quadratic with an edge length of about 14 cm and mounted 7 cm apart, hence representing the Helmholtz configuration. Since the current in the two coils runs in opposite directions, we have the situation of a uniform-gradient field close to the centre of the trap, i.e. we have an anti-Helmholtz coil.

5.4.1 Glass cell All later experiments will take place in the glass cell mentioned above, which is located on the UHV side. The main part is a rectangular volume of 80 mm × 38 mm × 38 mm connected to the glass-metal-transition of a CF40 flange via

45

Chapter 5

Vacuum chamber

5

16,907

5

8,422

60

30

5

Fig. 5.17 The dipole trap mirrors are attached to a CF100 blank flange via a pedestal. Each mirror encloses an angle of 45◦ with the differential pumping tube that protrudes between the two mirrors. In the top view the atomic beam emerges from the right.

The same glass cell design is used in another experiment in the same work group, hence we only need one replacement cell for both setups.

a glass tube of 38 mm diameter (see Fig. 5.16). The walls of the rectangular volume are 4 mm thick, easily withstanding a pressure difference of 1 atm. This type of cell grants very good optical access, while keeping the volume that needs to be evacuated very small.

5.4.2 Dipole trap mirrors The 6-way cross we use to connect 2D-MOT chamber and glass cell exhibits several interesting features. In addition to the bellow near the CF40 flange that vibrationally decouples the two main parts of the vacuum apparatus, it offers large radial access via four CF100 flanges (see Fig. 5.1). The top one is used to connect a large 150 l/s ion pump, thus not compromising the conductance in any way. The bottom flange is used to support two mirrors, mounted at an angle of about 45 ◦ with respect to the atomic beam axis (see Fig. 5.17). Through the large CF100 viewports mounted on the two side flanges we are now able to send in laser beams, that are reflected from the mirrors and guided into the glass cell (see Fig. 5.12). We have built in two halves of a 50.8 mm × 50.8 mm protected silver mirror from Thorlabs. This N-BK7 mirror has a thickness of 6 mm and a front surface flatness of λ/8 (at λ = 633 nm). Silver-coated mirrors can be used for light of 400 nm to 2 µm. As a protection against oxidation, the silvered surface has an overcoat of SiO.

46

6 State of play Over a period of twelve months we have set up the Titanium:Sapphire laser to produce the light required for laser cooling of potassium, obtained a spectrum of the D2 line of 39 K, locked the Ti:Sa to this signal, designed and set up the paths for acousto-optical modulation and optical fibres, established a way of shaping the laser beam for the magneto-optical trap (MOT), designed and assembled a vacuum chamber, baked the apparatus and achieved a pressure of less than 10−11 mbar. For testing purposes we have observed a 3D-MOT of 39 K atoms inside the Kimball spherical octagon with the use of external coils in addition to the two-dimensional confinement posed by the double clip coil inside the chamber. Recently we were able to reproducibly acquire signals of a 3D-MOT in the ultra-high vacuum region of the glass cell and witnessed the impact of the atomic beam on it.

6.1 Observation of the MOT The first signal of 39 K atoms in a three-dimensional magneto-optical trap was acquired in the Kimball spherical octagon, our 2D-MOT chamber. This was done in a preliminary setup, in which we installed an additional pair of coils outside of the vacuum chamber in order to ensure a confinement in all three directions. These coils with 15 windings and a rectangular circumference of 24 cm × 16 cm generated a magnetic field gradient of about 5 G/cm, enough to trap the atoms in axial direction. After optimizing the fluorescence signal by readjusting the laser beams, the current through both coils and especially the detuning of cooling and repumping light, an image of the atoms was recorded using a webcam without its infrared filter. This image is shown in Fig. 6.2. In order to improve the image quality we switched to the EXi Aqua frontilluminated interline CCD camera from QImaging. The CCD sensor has 1392 × 1040 pixels, each with a size of 6.45 µm × 6.45 µm, and its quantum efficiency is greater than 40% at 700 nm. The first false colour images can be seen in Fig. 6.1. Three shots were taken: one with both magnetic field and laser beams turned on, one with the magnetic field turned off and one with neither field nor light. There is a multitude of parameters that need to be set correctly for a successful magneto-optical trapping of atoms. Predominantly they concern

47

Chapter 6

State of play

50

50

100

100

150

150

200

200

250

250

300

300

50 100 150 200 250

80

80

60

60

40

40

20

20

0

0

100

200

300

0

50 100 150 200 250

0

(a)

50

100

100

150

150

200

200

250

250 300

50 100 150 200 250

300

50 100 150 200 250

3.1

16

3

14

2.9

12

2.8

10 8

200

(b)

50

300

100

2.7 0

100

(c)

200

300

2.6

0

100

200

300

(d)

Fig. 6.1 The uppermost row shows false colour images of the 39 K-MOT (a) before and (b) after postprocessing. The graph below each image represents the signal integrated along the abscissa. (c) In order to obtain the corrected picture, an image was taken in absence of the magnetic field and subtracted from the raw image. (d) The same image was also aquired without laser beams.

48

6.1

Observation of the MOT

Fig. 6.2 One of the first signals of 39 K atoms in a magneto-optical trap can be observed in the very centre of this picture. We used a pair of rectangular coils in addition to the double clip coil, together with counter-propagating laser beams in axial direction, to enable trapping of the atoms in all three dimensions.

the alignment, polarization, detuning and power balancing of the laser beams, as well as the magnetic field gradient and its position of zero-crossing. The potassium pressure inside the chamber needs to be sufficiently high, which can be adjusted by heating the oven. The alignment of the MOT beams was ensured initially, without any atoms involved, by placing stencils onto the viewports and guiding the light in and out of the chamber through their centre. The polarization was then adjusted to the magnetic field gradient as shown in Fig. 6.3. Power balancing was done individually for each beam using the ratios from [52] as a guide. By placing the coils in the appropriate position, the zero-crossing of the magnetic field has to be aligned with the centre of the region where all six MOT beams overlap. If the magnetic field gradient is too low, the atoms are not confined strongly enough to yield a fluorescence signal that is distinguishable from stray light. On the other hand, if the gradient is too high, the capture range of the MOT is reduced to a point where most atoms pass the beams without being trapped. Hence, the ideal gradient for the given laser power has to be determined by maximizing the fluorescence (see Fig. 6.7). The last and most delicate step is the choice of the detuning. Here we were not able to rely on previous works, but had to empirically determine the right values. For this we used function generators to sweep the AOM frequencies for both cooling and repumping light about the respective resonance. Recently we have observed a 39 K MOT inside the glass cell as well. The

49

σ– σ+ σ– B

σ+ z y

σ– σ+

x

Fig. 6.3 The polarization of the six MOT beams (red) has to be set according to the direction of the magnetic field (blue).

Chapter 6

State of play

3

x 10

4

50 2.5

100 150

2

200 1.5

250 300

1

350 400

0.5

450 100

200

(a)

300

400

500

0

0

100

200

300

400

500

(b)

Fig. 6.4 (a) This image of a 39 K MOT inside the glass cell was taken at a laser power of about 8 mW in each beam. (b) The line profile shows that the background signal, i.e. an image taken without the magnetic field, was subtracted from the raw image.

δ2

δ1 ω1

ωL

ωr



ω2 ωc

F=2 F=1

Fig. 6.5 The initial laser frequency ωL is reduced by ωc to yield the cooling light and increased by ωr to yield the repumping light. The difference to resonance is given by the detuning δ1 and δ2 for repumping and cooling light, respectively.

image shown in Fig. 6.4 was taken with the Hamamatsu Digital CCD Camera ORCA-05G. This is a progressive scan interline CCD camera with 1344 × 1024 pixels and a 6.45 µm × 6.45 µm chip size. It has a 12 bit analogue/digitalconverter and a typical readout noise of 10 electrons r.m.s. at a full well capacity of 15,000 electrons. During this measurement the external coils about the glass cell were operated at a current of 24 A, while the current through the double clip coil was 60 A, corresponding to a magnetic field gradient of 11 G/cm. Table 6.1 shows the values for the laser power in each beam and the AOM frequencies we used for this measurement. These settings are preliminary and can certainly be optimized. The detuning can be deduced from the AOM frequencies in the following way. The cooling light was prepared by reducing the initial laser frequency ωL by ωc = 226 MHz (twice the AOM frequency), as seen in Fig. 6.5. Similarly, for the repumping light the laser frequency was increased by ωr = 230 MHz. Given the particular shape of the error signal, the Ti:Sa is not necessarily locked exactly to the cross-over signal, but possibly a few natural linewidths aside. Scanning the cooling light we found the resonance at an AOM frequency of 122 MHz, corresponding to a total shift of 244 MHz, implying an offset of the locking point with respect to the cross-over signal of 13 MHz. From this we estimate a detuning of the cooler and repumper of δ2 ∼ 3Γ and δ1 ∼ 2Γ, respectively. At first the potassium oven was heated to well above 150 ◦ C, corresponding to a vapour pressure of about 4 × 10−4 mbar inside the oven, such that we could load the 3D-MOT from the increased background gas inside the glass

50

6.2

2D-MOT power 3D-MOT power Push beam power AOM frequency

Atom number and loading rate

Cooler

Repumper

10 mW 5 mW – 113 MHz

10 mW 3 mW 250 µW 115 MHz

Table 6.1 These preliminary values for 2D-MOT and 3D-MOT indicate the power in each beam, so the total power was 40 mW in the 2D-MOT and 24 mW in the 3D-MOT. The push beam was only comprised of repumping light. The conversion of the AOM frequencies into actual detuning values is discussed in the text.

cell. With the beams properly aligned, we then switched on the 2D-MOT and observed a strong influence on the 3D-MOT, when the oven temperature was turned down to about 70 ◦ C, i.e. the pressure inside the oven was lowered to about 2 × 10−6 mbar and with that the pressure in the glass cell decreased as well. In this case it was possible to load the 3D-MOT with the atomic beam turned on, but not when the beam was blocked.

6.2 Atom number and loading rate An estimate of the atom number in the 3D-MOT can be obtained from the image acquired with the CCD camera. We have measured a 3D-MOT beam intensity of 4.53 mW/cm2 , resulting in a ratio I/Is of about 2.6, where the saturation intensity of the D2 line of 39 K is given by Is = 1.75 mW/cm2 . Assuming a detuning of δ = 3Γ, eqn 2.2 yields a scattering rate of Rscatt = 1.97 × 105 Hz, at which photons are isotropically emitted. This fluorescence is collected by a lens with a radius of r = 22.5 mm in a distance of about R = 90 mm from the MOT and measured by the CCD camera. The correction factor for the reduced solid angle is given by πr2 = 1.56%. (6.1) 4πR2 The fluorescence light passes the glass cell wall and an imaging system comprised of two lenses, such that a transmission correction factor of T = 0.966 has to be considered, due to a power loss of 4% for each transition between glass and air. We have measured a quantum efficiency of the Hamamatsu ORCA-05G of QE = 35% and set the gain to 1 and the exposure time to τ = 20 ms for all the following measurements. With these parameters we can calculate the number of atoms in the MOT Ω=

Nc (6.2) Rscatt T Ω τ QE given the total number of counts Nc in an image taken by the CCD camera. For the measurement shown in Fig. 6.4(a) we estimate a total of Na = 2.9×105 Na =

51

The atomic beam can be blocked by switching off the double clip coil or blocking one of the appropriate AOMs.

Chapter 6

State of play

x 104 12

Atom number

10 8 6 4 2 0 0

50

100

150 200 Time (s)

250

300

350

Fig. 6.6 Every 10 seconds an image of the 39 K MOT was taken, from which the atom number was deduced. The measured data was fitted with f (x) = a(1 − exp(−bx)).

atoms in the MOT. This number is orders of magnitude less than we would expect and could be increased by optimizing the laser power and especially the 2D-MOT, i.e. ensuring that the atomic beam has a low divergence and is aligned with the 3D-MOT capture region. The temporal change in the number of atoms in a MOT is given by dN = γ − βN, (6.3) dt where γ is the loading rate and β considers the loss of atoms due to collisions. This differential equation is solved by N (t) = N0 (1 − e−t/τ ),

(6.4)

with a maximum number of atoms N0 and a characteristic time τ = β −1 . The loading rate is then given by γ = βN0 = N0 /τ . We have fitted eqn 6.4 to the measured data shown in Fig. 6.6 and obtained N0 = 1.29(3) × 105 and β = 0.021(2) s−1 . This corresponds to τ = 48(5) s and a loading rate of γ = 2700(300) s−1 . Again this is much smaller than expected and clearly shows that the atomic beam needs to be optimized.

6.3 Magnetic field dependence and temperature We have measured the dependence of the atom number on the magnetic field gradient of the 3D-MOT coils. This measurement was done with a total laser power of 7.7 mW (4.4 mW and 3.3 mW for cooling and repumping light, respectively) in each of the three beams and its results are shown in Fig. 6.7. A

52

6.3

2.5

Magnetic field dependence and temperature

x 105

Atom number

2 1.5 1 0.5 0

6

8

10 12 14 16 18 20 Magnetic field coil current (A)

22

24

Fig. 6.7 The atom number was measured for different currents through the magnetic field coil. For currents lower than 6 A the signal could not be distinguished from the background, while currents higher than 24 A severely heated up the coil.

maximum current is reached at 24 A, beyond which the coils would generate too much heat and the current would be turned off by the interlock. From this measurement we see that the optimal gradient is given for a current of 12–14 A. We can estimate the temperature of the MOT, if we assume that it is a sphere with constant number density and that all p atoms have the same velocity given by the most probable velocity vˆ(T ) = 2kB T /m obtained from the Maxwell-Boltzmann distribution. We release the atomic cloud with initial density n0 = N0 /V0 by turning the laser off for a short time ∆t = 400 µs and then recapture the atoms. The occupied volume after the expansion is 4π (r0 + vˆ∆t)3 , (6.5) 3 where r0 ∼ 1 mm is the initial radius. Since ∆t is very small, we can write r V1 3∆t 2kB T =1+ + O(∆t2 ) (6.6) V0 r0 m V1 =

and neglect terms of the order of ∆t2 and higher. Assuming the number density to remain the same after expansion, such that N0 /N1 = V0 /V1 , we obtain   2 m r0 N1 T = −1 . 2kB 3∆t N0

(6.7)

Figure 6.8 shows five paired measurements of the atom number in the MOT with a release time of 400 µs in each case. From this we obtain an

53

Chapter 6

State of play

14

x 104

12

Atom number

10 8 6 4 2 0

1

2

3

4

5 6 7 Measurement

8

9

10

Fig. 6.8 Each two measurements belonging together are connected with a red line. The release time between two measurements was 400 µs in each case.

See [23] for a derivation of this expression.

atom number ratio of N1 /N0 = 0.52 ± 0.08 and consequently a temperature of T = 378 ± 115 µK. By considering a balance between the damping force and the heating that occurs when photons are constantly absorbed and emitted the temperature can be expressed as T =−

~Γ 1 + I/Is + 4δ 2 /Γ2 , 8kB δ/Γ

(6.8)

which yields T = 476 µK or about three times the Doppler temperature. Although our measurement agrees with this value within the uncertainty, the simplified model of a sphere with constant number density n0 could be improved by considering instead a Gaussian density distribution   1 z2 n(z, σ) = √ (6.9) exp − 2 , 2σ 2πσ 2 which is the solution to a one-dimensional harmonic trap. In this case the standard deviation σ increases according to q σ(T ) = σ02 + σv2 (T )∆t2 , (6.10) p where σ0 is the initial Gaussian radius and σv = kB T /m is the Gaussian radius of the velocity distribution [53]. Measurements of the atom number dependence on the magnetic field gradient of the double clip coil or potassium oven temperature have not yielded any insights, since the loading rate is so small. In order to improve the loading rate and with that the total number of atoms in the MOT, the atomic

54

6.3

Magnetic field dependence and temperature

beam needs to be optimized by increasing and balancing the power in the 2DMOT beams and geometrically aligning the differential pumping tubule with respect to the 3D-MOT. Since we have not observed a significant influence of the push beam on the loading rate yet, this is another subject that needs to be investigated. While we have qualitatively seen that the atomic beam is crucial for the loading of the 3D-MOT at low pressures of 10−11 mbar in the glass cell, a detailed characterization of the 2D-MOT, especially its influence on the 3DMOT loading time, remains to be conducted.

55

Chapter 6

56

State of play

7 The future of the experiment There are several tasks that need to be approached short to long-term to further develop the experiment. The next step towards Bose-Einstein condensation, after the 2D-MOT and 3D-MOT have been characterized and the atomic flux has been optimized, is loading the 39 K MOT into a magnetic trap, to lower the temperature and increase the phase-space density. A proposal for the coils that would generate the required magnetic field is presented in this chapter. Furthermore, the addition of an optical dipole trap is favoured for evaporative cooling. Because of the poor collisional properties of 39 K, both magnetic trapping and evaporative cooling require the addition of rubidium, which sympathetically cools the potassium. A suggestion for an optical setup that includes rubidium is presented in Section 7.3. Finally there is an outlook on the long-term future of this experiment.

7.1 Feshbach coils To generate the field required for the magnetic trap and later tune it about Feshbach resonances, coils are needed that can sustain ∼ 400 A, giving rise to fields of up to ∼ 1000 G. For this purpose we have come up with a pair of Feshbach coils, each consisting of 28 windings (7 in radial and 4 in axial direction) of a copper wire 4.08 mm in diameter. They are designated to be housed in a frame with a wall thickness of 4 mm in radial and 1.5 mm in axial direction, while being mounted a distance of 45 mm apart from each other. This way, they will fit well around the glass cell and at the same time be as close together as possible, in order to minimize the current needed for the high magnetic fields. The position of the separation posts is designed such that the optical access is conserved as much as possible, by aligning two of the posts with the edges of the glass cell and keeping the other two close to the flange. Fig. 7.2 shows a draft of the frame with the relevant dimensioning.

7.2 Optical dipole trap The two CF100 viewports of the 6-way cross allow us to shine in laser beams at a wide range of angles. Inside the vacuum there are two mirrors, as discussed

57

39

K

Fig. 7.1 Since the separation posts between the two frames are aligned with the edges of the glass cell, little optical access is lost.

17,820

The future of the experiment

4 116,220

11

7,5



173,340

40,553

42

Chapter 7

6

35

°

Fig. 7.2 The separation post holders serve merely as an indication of the optimal angles. The final frame would be more stable in order to withstand the strong forces that occur when switching the current.

58

7.3

Sympathetic cooling with rubidium

in Section 5.4.2, that will reflect the incident light and guide it through the glass cell. Such a laser beam can be used to create an optical dipole trap in which evaporative cooling would take place. Since an optical dipole trap requires a large amount of power, the current laser system has to be extended. One option is the use of a tapered amplifier (TA) to generate the light required for laser cooling of potassium and possibly devote the Ti:Sa to the optical dipole trap. The TA however requires a diode laser to be seeded with. We have purchased the TA diode EYP-TPA-0765-01500-3006-CMT03-0000 with a centre wavelength of 765 mm and a recommended output power of 1.5 W, as well as the GaAs semiconductor laser diode EYP-RWE-0790-04000-0750-SOT01-0000 for use in an external cavity diode laser (ECDL) with a centre wavelength of 770 nm and a typical (extracavity) output power of 100 mW, both from eagleyard Photonics GmbH. Other than for the optical dipole trap, the two mirrors inside the vacuum could be used to reflect laser beams that would form an optical lattice. Two laser beams, one coming from each viewport, would overlap and generate a one-dimensional lattice. If two laser beams are shone in through each viewport, a two-dimensional lattice may be achieved without losing any optical access to the glass cell.

7.3 Sympathetic cooling with rubidium Since efficient sub-Doppler cooling is not possible for 39 K, due to the narrow spacing of the 42 P3/2 excited states in the D2 transition, a coolant is needed. It seems natural to consider 87 Rb for the purpose of sympathetic cooling, since laser cooling of this element is both efficient and well established. Furthermore, the collisional properties of 39 K–87 Rb are rather favourable, having a interspecies s-wave scattering length aKRb = 36 a0 , as opposed to the negative background scattering length aK = −33 a0 of 39 K. In addition to that there is an inter-species Feshbach resonance between 39 K and 87 Rb that can be used to further improve the scattering cross-section. With this in mind, we have included a rubidium oven in our apparatus, connected to the Kimball spherical octagon in the same way as the potassium oven (see Fig. 5.1). However, a major upgrade of the laser system is necessary, since both cooling and repumping light is now needed for 87 Rb as well. Figure 7.3 shows the revised optical setup that features two tapered amplifiers (TA), one for potassium and one for rubidium, as well as an external cavity diode laser (ECDL). The rubidium TA is used to produce the cooling light, whereas the ECDL is solely responsible for the repumper. Superimposing all four frequencies and coupling them into optical fibers becomes even more intricate than in the case of potassium alone and requires dichroic half-wave plates that change the polarization of one wavelength, but not the other.

59

In Section 2.4 we have discussed that a negative background scattering length leads to a RamsauerTownsend minimum of the scattering cross-section at temperatures that are too high for magnetic trapping.

Since the Ti:Sa laser is no longer required in this setup, it can be used to generate the light for the optical dipole trap.

Chapter 7

The future of the experiment

Quarter-wave plate

Mirror

Fibre

Half-wave plate

Lens

Beam dump

Dichroic half-wave plate

Photo diode

Shutter

Polarizing beam splitter

Iris

Acousto-optical modulator 3D-MOT

125 mm

125 mm

ECDL

125 mm

push beam

125 mm 125 mm

125 mm

100 mm 50 mm

50 mm 100 mm

2D-MOT Rb

TA

K

TA

Fig. 7.3 This scheme of the optical setup for laser cooling of both 39 K and 87 Rb shows one possibility to superimpose all the required frequencies and direct them to 2D-MOT, push beam and 3D-MOT. The light is generated by tapered amplifiers (TA) and an external cavity diode laser (ECDL). The wavelengths are shown red for potassium, blue for rubidium and green for both. While the potassium AOMs have a centre frequency of 110 MHz, for rubidium this is 80 MHz.

60

7.4

Experiments with potassium

In the current setup we mainly use zero-order wave-plates, which are composed of two regular wave-plates that differ in thickness by a small amount (λ/2 and λ/4 for half-wave and quarter-wave plate, respectively) and are attached to each other such that the slow axis of one coincides with the fast axis of the other. The result is an increase in bandwidth, thus they can be used for both the potassium and the rubidium wavelength.

7.4 Experiments with potassium After 39 K has been condensated, the first new experiments will be conducted. They will exploit the possibility of creating a weakly interacting quantum gas by tuning the scattering length close to zero. This enhances the study of momentum distributions in a BEC, since no information is lost due to collisional processes during the time-of-flight imaging. Regarding this, a proposal to study turbulence phenomena in dilute ultracold gases is made in [54]. Another approach would be to study attractive interactions in a doublewell potential. Experiments with two weakly linked 87 Rb BECs have been conducted, showing tunneling and nonlinear self-trapping in this bosonic Josephson junction [55], and new interesting phenomena could be observed in case of a new species like 39 K. In the long run one could think about condensating 41 K with the same setup. Since the frequency difference between the 42 S1/2 ground state of the two isotopes is only 236 MHz and the spacing between the two hyperfine states |F = 1i and |F = 2i is 254 MHz, the existing AOMs could be used. In the case of 40 K, the ground state hyperfine spacing is 1.29 GHz, which is too far apart for our AOMs, hence the setup would have to be modified. Moreover, due to its very small natural abundance of 0.012% experiments with 40 K usually require an enriched dispenser.

61

Chapter 7

62

The future of the experiment

Bibliography [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science, 269:198–201, 1995. [2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle. Bose-Einstein Condensation in a Gas of Sodium Atoms. Phys. Rev. Lett., 75:3969–3973, 1995. [3] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions. Phys. Rev. Lett., 75:1687–1690, 1995. [4] S. N. Bose. Plancks Gesetz und Lichtquantenhypothese. Zeitschrift f¨ ur Physik, 26:178, 1924. [5] A. Einstein. Quantentheorie des einatomigen idealen Gases. Sitzungsber. Preuss. Akad. Wiss., page 261, 1924. [6] A. Einstein. Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung. Sitzungsber. Preuss. Akad. Wiss., page 3, 1925. [7] J. F. Allen and A. D. Misener. Flow of Liquid Helium II. Nature, 141:75, 1938. [8] P. Kapitza. Viscosity of Liquid Helium below the λ-Point. Nature, 141:74, 1938. [9] F. London. The λ-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy. Nature, 141:643–644, 1938. [10] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle. Observation of Vortex Lattices in Bose-Einstein Condensates. Science, 292:476–479, 2001. [11] T. Esslinger, I. Bloch, and T. W. H¨ ansch. Probing first-order spatial coherence of a Bose-Einstein condensate. Journal of Modern Optics, 47:2725–2732, 2000.

63

Bibliography

[12] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Inguscio. Bose-Einstein Condensation of Potassium Atoms by Sympathetic Cooling. Science, 294:1320–1322, 2001. [13] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman. Stable 85 Rb Bose-Einstein Condensates with Widely Tunable Interactions. Phys. Rev. Lett., 85:1795–1798, 2000. [14] T. Weber, J. Herbig, M. Mark, H.-C. N¨agerl, and R. Grimm. BoseEinstein Condensation of Cesium. Science, 299:232–235, 2003. [15] C. D’Errico, M. Zaccanti, M. Fattori, G. Roati, M. Inguscio, G. Modugno, and A. Simoni. Feshbach resonances in ultracold 39 K. New Journal of Physics, 9:223, 2007. [16] J. L. Bohn, J. P. Burke, Jr., C. H. Greene, H. Wang, P. L. Gould, and W. C. Stwalley. Collisional properties of ultracold potassium: Consequences for degenerate Bose and Fermi gases. Phys. Rev. A, 59:3660–3664, 1999. [17] C. Chin, V. Vuleti´c, A. J. Kerman, and S. Chu. High Resolution Feshbach Spectroscopy of Cesium. Phys. Rev. Lett., 85(13):2717–2720, 2000. [18] P. J. Leo, C. J. Williams, and P. S. Julienne. Collision Properties of Ultracold 133 Cs Atoms. Phys. Rev. Lett., 85(13):2721–2724, 2000. [19] E. Braaten. Three-Body Recombination in Atoms with Large Scattering Length. Nuclear Physics A, 790:713c–717c, 2007. [20] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg. Atom-Photon Interactions. WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004. [21] C. J. Foot. Atomic Physics. Oxford University Press Inc., New York, 2005. [22] J. Dalibard and C. Cohen-Tannoudji. Laser cooling below the Doppler limit by polarization gradients: simple theoretical models. J. Opt. Soc. Am. B, 6(11):2023–2045, 1989. [23] M. Fox. Quantum Optics: An Introduction. Oxford University Press Inc., New York, 2006. [24] P. Ehrenfest and J. R. Oppenheimer. Note on the Statistics of Nuclei. Phys. Rev., 37(4):333–338, 1931. [25] L. Pitaevskii and S. Stringari. Bose-Einstein Condensation. Oxford University Press Inc., New York, 2003.

64

Bibliography

[26] L. D. Landau and E. M. Lifshitz. Quantum Mechanics. Pergamon, Oxford, 1987. [27] L. S. Butcher, D. N. Stacey, C. J. Foot, and K. Burnett. Ultracold collisions for Bose-Einstein condensation. Phil. Trans. R. Soc. Lond. A, 357:1421–1439, 1999. [28] Ch. Buggle, J. L´eonard, W. von Klitzing, and J. T. M. Walraven. Interferometric Determination of the s and d-Wave Scattering Amplitudes in 87 Rb. Phys. Rev. Lett., 93(17):173202, 2004. [29] A. Crubellier, O. Dulieu, F. Masnou-Seeuws, M. Elbs, H. Kn¨ockel, and E. Tiemann. Simple determination of Na2 scattering lengths using observed bound levels at the ground state asymptote. Eur. Phys. J. D, 6:211–220, 1999. [30] E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, and B. J. Verhaar. Interisotope Determination of Ultracold Rubidium Interactions from Three High-Precision Experiments. Phys. Rev. Lett., 88(9):093201, 2002. [31] B. D. Esry, C. H. Greene, and J. P. Burke, Jr. Recombination of Three Atoms in the Ultracold Limit. Phys. Rev. Lett., 83(9):1751–1754, 1999. [32] J. P. Burke, Jr. Theoretical Investigation of Cold Alkali Atom Collisions. PhD thesis, University of Colorado, 1999. [33] J. M. Goldwin. Quantum Degeneracy and Interactions in the 87 Rb–40 K Bose-Fermi Mixture. PhD thesis, University of Colorado, 2005. [34] J. J. Sakurai. Modern Quantum Mechanics. Addison-Wesley, 1985. [35] L. De Sarlo, P. Maioli, G. Barontini, J. Catani, F. Minardi, and M. Inguscio. Collisional properties of sympathetically cooled 39 K. Phys. Rev. A, 75(2):022715, 2007. [36] A. Derevianko, J. F. Babb, and A. Dalgarno. High-precision calculations of van der Waals coefficients for heteronuclear alkali-metal dimers. Phys. Rev. A, 63(5):052704, 2001. [37] A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb. HighPrecision Calculations of Dispersion Coefficients, Static Dipole Polarizabilities, and Atom-Wall Interaction Constants for Alkali-Metal Atoms. Phys. Rev. Lett., 82(18):3589–3592, 1999. [38] M. Zaccanti. Tuning of the interactions in ultracold K-Rb quantum gases. PhD thesis, Univerti` a degli Studi di Firenze, Dipartimento di Fisica, 2007.

65

Bibliography

[39] M. E. Wieser and M. Berglund. Atomic weights of the elements 2007. Pure Appl. Chem., 81(11):2131–2156, 2009. [40] J. A. Dean. Lange’s Handbook of Chemistry. McGraw-Hill, Inc., 1952. [41] J. K. B¨ ohlke, J. R. de Laeter, P. De Bi`evre, H. Hidaka, H. S. Peiser, K. J. R. Rosman, and P. D. P. Taylor. Isotopic Compositions of the Elements, 2001. J. Phys. Chem. Ref. Data, 34(1):57–67, 2005. [42] R. S. Williamson III. Magneto-optical trapping of potassium isotopes. PhD thesis, University of Wisconsin, 1997. [43] C. B. Alcock, V. P. Itkin, and M. K. Horrigan. Vapor Pressure of the Metallic Elements. Canadian Metallurgical Quarterly, 23:309–313, 1984. [44] H. Wang, P. L. Gould, and W. C. Stwalley. Long-range interaction of the 39 K(4s)+39 K(4p) asymptote by photoassociative spectroscopy. I. The 0− g pure long-range state and the long-range potential constants. J. Chem. Phys., 106(19):7899–7912, 1997. [45] J. Catani, P. Maioli, L. De Sarlo, F. Minardi, and M. Inguscio. Intense slow beams of bosonic potassium isotopes. Phys. Rev. A, 73(3):033415, 2006. [46] S. Falke, H. Kn¨ockel, J. Friebe, M. Riedmann, E. Tiemann, and C. Lisdat. Potassium ground-state scattering parameters and Born-Oppenheimer potentials from molecular spectroscopy. Phys. Rev. A, 78(1):012503, 2008. [47] D. E. Burch and D. A. Gryvnak. Strengths, Widths, and Shapes of the Oxygen Lines near 13,100 cm−1 (7620 ˚ A). Applied Optics, 8(7):1493–1499, 1969. [48] N. Bendali, H. T. Duong, and J. L. Vialle. High-resolution laser spectroscopy on the D1 and D2 lines of 39,40,41 K using RF modulated laser light. J. Phys. B: At. Mol. Phys., 14:4231–4240, 1981. [49] J. J. Snyder. Paraxial ray analysis of a cat’s-eye retroreflector. Applied Optics, 14(8):1825–1828, 1975. [50] A. Sitnikov. Vacuum Component Cleaning Technical Procedure. Canadian Light Source Inc., 101 Perimeter Road, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 0X4 Canada, 8.7.33.1, rev. 2 edition, 2007. [51] H. Franke. Vakuumpraxis f¨ ur Fortgeschrittene. Varian Associates, Inc., 121 Hartwell Avenue, Lexington, MA 02173, 1996. [52] J. Catani. A New Apparatus for Ultracold K-Rb Bose-Bose Atomic Mixtures. PhD thesis, Univerti`a degli Studi di Firenze, Dipartimento di Fisica, 2006.

66

Bibliography

[53] T. M. Brzozowski, M. M¸aczy´ nska, M. Zawada, J. Zachorowski, and W. Gawlik. Time-of-flight measurement of the temperature of cold atoms for short trap-probe beam distances. J. Opt. B: Quantum Semiclass. Opt., 4:62–66, 2002. [54] C. Scheppach, J. Berges, and T. Gasenzer. Matter-wave turbulence: Beyond kinetic scaling. Phys. Rev. A, 81(3):033611, 2010. [55] M. Albiez, R. Gati, J. F¨ olling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler. Direct Observation of Tunneling and Nonlinear Self-Trapping in a single Bosonic Josephson Junction. Phys. Rev. Lett., 95:010402, 2005.

67

Bibliography

68

A Acknowledgements First of all I would like to thank Markus Oberthaler for providing me with the opportunity to work in this fascinating field of ultracold quantum gases and accepting me into the Synthetic Quantum System workgroup. His incessant motivation and trust in me to help build up a new experiment have been invaluable. I also thank Selim Jochim for being my second examiner and taking the time to talk to me about the experiment. Christian Groß has always been a great help and inspiration, both in and outside of the lab. I enjoyed rock climbing and the occasional barbecue as much as tweaking the Ti:Sa and sweeping AOM frequencies in search for the MOT. Without my fellow diploma student Wolfgang M¨ ussel and the student assistant Rostislav Doganov all this would not have been possible. Together we have worked hard to impel the experiment and had quite a bit of fun at that. My thanks to Rosti for his contribution to the design of the Feshbach coils. Tilman Zibold, Eike Nicklas and Helmut Strobel of the BEC team have helped me a lot when I was new to the experiment and I have been able to seek their advice ever since. The fondest memories include bets on distance estimation, playing air guitar at the KIP Christmas party and clubbing in Heidelberg. The NaLi team, consisting of Steven Knoop, Tobias Schuster, Raphael Scelle and Arno Trautmann, has been very helpful, especially in lending lab equipment like tools, mirrors, posts, etc. This is an official apology for not always returning those items! I would also like to mention Jens Appmeier, who finished his PhD during the course of my work, and wish him the best of luck for the future. Many thanks as well to the rest of the Matterwavers Joachim Welte, Jiˇr´ı Tomkoviˇc, Florian Ritterbusch, Christoph Kaup, Fabienne Haupert, Philippe Br¨aunig and Hanno Filter. Furthermore I would like to thank our team assistants Dagmar Hufnagel and Christiane J¨ ager. The electronics division, especially J¨ urgen Sch¨olles and Alexander Leonhardt, has been very helpful with the electrical connection of the double clip coil and the power supply of our AOM drivers. The vital contributions of the workshop, especially Werner Lamad´e, Sieg-

69

Chapter A

Acknowledgements

fried Spiegel, Michael Lutz and Morris Weißer, have enabled us to connect the double clip coil to the water cooling system and clean the vacuum parts with ultrasound. They have also kindly manufactured the differential pumping tubule and the dipole trap mirrors according to our drawings. The KIP administration, especially Corina M¨ uller, Sandra Kiefer and Felicitas Kleveta, has ensured a smooth procedure of ordering and deliveries, on which we heavily relied during the building phase. Besides all the work, I would not want to have missed the company of Martin Fast for dinner at Caf´e Botanik, rock climbing in Kirchheim, playing table tennis, having lunch at Marstall every Saturday, and much more. I am grateful to Hendrik-Marten Meyer for valuable physical insight, vivid impressions of Stefan Raab and Stromberg, hiking in Innsbruck, as well as the many things still to come. Thank you, Johanna Bohn, for being the best friend one can have after such a short time and helping me out in every situation, including the little things like proofreading. Finally, I would like to express my deep gratitude to Sam Cooke, Christopher Nolan, Jack Daniel, Bob Seger, Billy Joel, John Fogerty, Peter Fox, Bruce Dickinson, Steve Walsh, Luke Kelly, Matt Lucas, David Williams, and last but not least Jennifer Aniston, Courteney Cox Arquette, Lisa Kudrow, Matt LeBlanc, Matthew Perry and David Schwimmer.

70

B Declaration I hereby confirm that I wrote this thesis on my own and that I did not use other sources or means than stated.

Heidelberg, the Signature

71

Suggest Documents