Bose 2.2- Manuals

Zusammenfassung Gegenstand dieser Arbeit ist die experimentelle Untersuchung eines dipolaren Bose-Einstein-Kondensates (BEK) mit Chromatomen in einem eindimensionalen optischenGitterpotential. Zunächst wird der Einfluss der periodischen Potentiallandschaft auf die wechselwirkungsabhängige Stabilität...

zur experimentellen Kontrolle: viele atomare Spezies verfügen über Feshbach-Resonanzen , in deren Nähe die s -Wellenstreulänge über ein externes Magnetfeld eingestellt werden kann [11, 12]. Aufgrund des großen magnetischen Dipolmoments der Chromatome spielt in den hier gezeigten Experimenten außerde...

die s -Wellenstreulänge jedoch reduziert und somit ein Quantengas mit starker dipolarer Wechselwirkung erzeugt werden [32]. Die Kalibration der Streulänge, d.h. die experimen- telle Bestimmung ihrer Abhängigkeit von der angelegten Magnetfeldstärke, spielt in denExperimenten eine wichtige Rolle und w...

Streulänge reduziert, bis der kritische Wert a crit erreicht ist, bei der ein plötzlicher Verlust der Atome im Kondensat eintritt. Für genügend kleine Gittertiefen werden po- sitive kritische Streulängen bis zu a crit = (12 ± 2) a 0 gemessen, mit a 0 ' 0 . 053 nm dem Bohrschen Radius. Aufgrund der h...

1 Introduction Since the realization of Bose-Einstein condensates (BECs) in ultracold dilute atomic vapours in 1995 [1–4], degenerate quantum gases have become an ideal environment to study many-body systems in the quantum regime. Essentially, BECs represent macrosopicquantum objects, with around 10...

the contact interactions in the system. Nevertheless, dipolar effects have been observede.g. by measuring the expansion velocity of the chromium BEC [31] or the frequency of acollective excitation mode [49]. An anisotropic sound velocity in the dipolar BEC (dBEC),depending on the orientation of the ...

This thesis In this thesis I report on the investigation of a 52 Cr BEC in a one-dimensional optical lattice, operating in a regime with dominant dipolar interactions. The experimental work presented here, relies greatly on the previous achievement of a 52 Cr BEC with the s -wave scattering length t...

spatially separated dipolar condensates is considered in the discussion. Chapter 3 describesthe production process of a 52 Cr BEC with tunable contact interactions. Special emphasis is put on the principle and the application of the Feshbach resonance technique as it isa key ingredient in our experi...

2 Dipolar Quantum Gases This chapter gives a brief introduction to the physics of bosonic dipolar quantum gases. It provides the basic formalisms to understand the experimental results presented in thisthesis. For more details, two recent reviews on dipolar quantum gases give an excellentoverview of...

a macroscopic phase coherence throughout the sample. A detailed treatment on the fundamental properties of BECs is given in Refs. [8, 89]. For an intuitive picture, let us consider the realistic case of a thermal sample of bosonic atoms confined in a harmonic trapping potential, with ω 0 the charact...

2.2 Two-Body Interactions The atomic density in a BEC is typically around n BEC = 10 14 − 10 15 cm − 3 , which is very low compared to solids ( n carbon ∼ 10 23 cm − 3 ) or even air ( n air ∼ 10 19 cm − 3 ). From such low density we might deduce that interactions between the particles do not play an...

with the van-der-Waals term of the molecular potential 7 , we obtain an interaction radius r 0 ∼ 100 a 0 of the molecular potential. This defines the maximal distance between two atoms at which the weakest bound-state can be formed. The interaction range r 0 is typically much smaller than the mean i...

r impact v/ 2 v /2 V(r) r a (a) (b) Fig. 2.1, Elastic two-body scattering: (a) Classical picture of two colliding atoms moving at the relative velocity v . The impact parameter r impact defines the minimum distance of the particles in the scattering process. (b) The two-body wave function ψ ( r ) (b...

using the reduced mass m red = m/ 2. We immediately see that the prefactors in the contact coupling strength g , defined in Eq. (2.4b), are chosen such that r = a is the intersection point of the wave function with the r -axis. Thus, the parameter a may be identified with the s -wave scattering leng...

0 r (a) (b) Fig. 2.2, Dipole-dipole interaction (DDI): (a) Two dipoles polarized by an external magnetic field B along the z -direction. The separation r = | r | and the angle ϑ = ∠ ( z , r ) enter the DDI potential given by Eq. (2.6). (b) The interaction between two dipoles is attractive in a head-...

After investigating the long-range behaviour of the DDI potential, we now consider its behaviour at small distances r . If the DDI potential acts on a two-body wave function that does not vanish at the origin, i.e. ψ ( r = 0) 6 = 0, the divergence of V dd at this point must be cured. In an according...

We stress here that the dipolar length a dd does not correspond to a finite interaction radius of the dipolar interactions. Such radius cannot be defined for long-range interactions. With the prefactors given in Eq. (2.9a), the value of a dd seems chosen arbitrarily at the moment. But we will see in...

operators 11 ˆa „ k and ˆa k , such that ˆΨ „ ( r ) = X k φ k ( r ) ˆa „ k and ˆΨ( r ) = X k φ k ( r ) ˆa k , (2.11) with φ k ( r ) a set of single-particle states. Assuming a macroscopic population N of the lowest lying single-particle state φ 0 (such that N + 1 ' N 1) we can replace the according ...

Inserting the ansatz ψ ( r , t ) = ψ ( r ) exp( − iµt/ ~ ) into Eq. (2.14), where µ is the chemical potential of the condensate, we obtain the stationary Gross-Pitaevskii equation µ ψ ( r ) = " − ~ 2 2 m ∇ 2 + V ext ( r ) + Φ contact ( r ) + Φ dip ( r ) # ψ ( r ) , (2.15a) with Φ contact ( r ) a...

(i) N 1: The macroscopic population of a single particle state allows for the replacement of the creation and annihilation operators by classical numbers [91], ˆa ≈ ˆa „ ≈ √ N . With typical BEC atom numbers N ∼ 10 4 the condition N ≈ N + 1 1 is well satisfied. (ii) r 0 λ dB : The de-Broglie wavelen...

Since in our experiments with a chromium BEC, all the validity criteria are fulfilled, we will make use of the mean-field description in the remaining part of this thesis. 2.4 Solutions of the Non-Local Gross-Pitaevskii Equation In this section, we discuss different solutions of the stationary Gross...

energy V ext ( R ) = mω 2 0 R 2 / 2. In the other limit of small and decreasing radii, Heisenberg’s uncertainty relation p = ~ /R leads to a divergence in the kinetic energy, given by E kin = p 2 / (2 m ) = ~ 2 / (2 mR 2 ). This energy divergence results in the so-called quantum pressure , which sta...

2.4.3 TF-Approximation with Contact and Dipolar Interactions The long-range character of the dipole-dipole interaction significantly complicates the description of a BEC in the TF-approximation. It has been found [110, 111] that in thisregime, the dipolar mean-field potential Φ dip only contains ter...

Hence, for a given external trapping potential, we obtain an exact solution for thestationary GPE, with the chemical potential given by µ = gn 0 [1 − dd f dip ( κ )]. 0.01 0.1 1 10 100 -2 -1 0 1 (a) (b) B Fig. 2.3, Dipolar anisotropic function and dipolar mean-field potential: (a) The dipolar anisot...

ratio [114]. It is given by N ( a − a dd ) /a ho 1 for trap ratios λ . 1, used e.g. in the measurements for the calibration of the scattering length, presented in section 3.2.3. Concluding this section, we have seen that the dipolar interactions influence the ground- state properties of a BEC. While...

This expression means that two infinite planes of dipoles do not interact [121]. In this limit, the system is translationally invariant in the xy -plane, and therefore the mean-field potential is zero everywhere. uniform disc Gaussian r B,z d lat (a) (b) (c) 0 1 2 3 -4 -2 0 0 1 -10 -5 0 5 Fig. 2.4, ...

Considering a real trapped dipolar gas with weak interactions, the density of the sample will rather be described by a Gaussian than a disc shape, as we have shown in section 2.4.1.We therefore write the 2D density distribution of the samples in the Gaussian form n Gauss 2D ( ρ ) = N πσ 2 ρ e − ρ 2 ...

Refs. [51, 118]), E (2) inter ( d lat ) = − g dd N 2 (2 π ) 3 / 2 σ 3 ρ 1 Z 0 d u (1 − 3 u 2 )(1 − u 2 ( η + L 2 )) (1 − ηu 2 ) 5 / 2 exp − L 2 u 2 2(1 − ηu 2 ) ! , (2.33) where L def = d lat /σ ρ is the normalized distance between the samples, and η def = 1 − κ − 2 . The remaining integral in Eq. (...

inter-site energy vanishes. Thus, we recover the result obtained in the Thomas-Fermiapproximation, where we found a vanishing dipolar contribution to the chemical potentialof a spherically symmetric cloud (see section 2.4.3). The pancake-shaped density distribution leads to an even more complex beha...

B,z 0 -50 -100 -150 -200 0 10 20 30 40 50 (a) (b) Fig. 2.6, Inter-site energy in a dipolar multi-site system with constant filling of the sites: (a) Sketch of the system. The N lat samples, containing N atoms each, with the dipoles oriented along the symmetry axis z , are equally separated by the di...

3 Producing a 52 Cr BEC with Tunable Interactions To investigate dipolar effects in an ultra-cold bosonic gas, we routinely create a 52 Cr condensate with tunable contact interactions. The complex production process of thecondensate has been developed in our group over several years, with detailed d...

3.1.2 Procedure We create a beam of chromium atoms by sublimation in an effusion cell 25 at a temperature T ∼ 1450 ◦ C . The atomic beam is first collimated by a set of apertures, and then radially cooled by a two-dimensional optical molasses [129, Ch.9.3]. For the radial cooling, weuse blue laser l...

value 28 , we minimize the Zeeman energy that is released in a dipolar relaxation process. Still, the heating is too strong to reach the critical temperature for condensation. To overcome the limitation in temperature, imposed by the dipolar relaxation, we stop the RF evaporation after reaching the ...

3.1.3 Laser Systems We now give a brief overview of the lasers that are involved in the production process of the BEC. 425 nm MOT laser system The laser light at wavelength λ = 425 . 6 nm, resonant with the 7 S 3 ↔ 7 P 4 transition, is used for several purposes on the experiment: radial cooling in o...

427 nm optical pumping laser system The setup of the optical pumping laser system ( λ = 427 . 6 nm) is similar to the MOT laser system, however, involving much lower laser powers. We produce around 60 mWof light at a wavelength λ = 855 . 2 nm from an external-cavity diode laser 38 , which is frequen...

3.2 Tuning of the Contact Interactions In a chromium BEC, the dipolar interactions are much stronger than e.g. in condensatesof alkali elements, due to the six times larger magnetic moment of the atoms. However, when compared to the contact interaction strength, the dipolar interactions are still we...

Let us consider more closely the tuning of the scattering length in the vicinity of a Feshbach resonance (see Fig. 3.3). If the spin projections of the two coupled states arenot the same, i.e. ∆ M S 6 = 0, also their associated magnetic moments show a difference ∆ µ m 6 = 0. Then, the relative poten...

3.2.2 Experimental Realization of the Feshbach System The required magnetic field strength B ∼ 600 G to reach the Feshbach resonance is conveniently produced in our setup by using the offset coils from the magnetic trap [74]. They are water-cooled, such that the heat is efficiently removed when oper...

Eddy currents The current in the Feshbach coils is dynamically computer-controlled by changing the set-point of the PI-controller [74, Ch.4.2.2]. If, however, we change the desired current value faster than the inverse bandwidth of the servo-loop ( ∼ 1 kHz), oscillations of the Feshbach current occu...

0 0 20 40 60 80 100 10 20 30 (a) (b) dipolar expansion (TF) TF vs. simulations TF simulations 0 20 40 60 80 100 10 0 20 30 40 Fig. 3.4, Dipolar expansion: (a) Ratios R 5 y /N (red dots) and R 5 z /N (blue dots), calcu- lated for different scattering lengths using the TF approximation (parameters: ω ...

width and the center of the resonance is obtained by fitting the function a ( I FB ) = a bg · 1 − ∆ I FB I FB − I FB , 0 ! , (3.3) to the data, where the width and the center of the FR are expressed in terms of theFeshbach current. The goal of this two-step fitting procedure is to extract not onlyth...

The precise determination of the function a ( I FB ) is the main goal of the calibration procedure. However, to display the results in real physical parameters, we have to convertthe Feshbach current into a magnetic field strength. Therefore, we have performed spec-troscopy measurements on a thermal...

4 A BEC in a One-Dimensional Optical Lattice In this chapter, the basic properties of a BEC trapped in a one-dimensional (1D) opticallattice potential are described, neglecting the dipolar interactions. We focus only on thetopics that are relevant for this thesis. An overview over the broad physics ...

zoom x z Fig. 4.1, Interference of two coherent laser beams: Two crossing laser beams (propagation direction given by the arrows) produce a regular 1D array of intensity maxima (darker shading means higher intensity). The spacing d lat between the intensity maxima is defined by the wavelength of the...

of ultra-cold gases in optical lattices: k lat = π d lat the lattice wave number , (4.3a) E R = ~ 2 k 2 lat 2 m = ~ 2 π 2 2 md 2 lat the recoil energy, and (4.3b) s = U lat E R the dimensionless lattice depth. (4.3c) 4.1.2 Experimental Realization of the 1D Lattice The optical lattice along the z -d...

mirror lens l /2 wave plate beam block ODT1 shutter 1st fiber laser optical isolator AOM CCD probe beam Feshbach coils z x y g B q Fig. 4.2, Schematic drawing of the optical lattice setup (top view): The lattice laser beam (solid red line) is intensity controlled by an AOM and may be blocked by a sh...

with the parameter A def = E /E R − U lat / (2 E R ) and the lattice parameter Q def = U lat / (4 E R ) = s/ 4. The solutions of Eq. (4.5) are the so-called Mathieu functions M ( A, Q, ˜ z ), which can be computed numerically for known parameters A and Q . Recalling the periodicity of the lattice po...

initial (flat) state to the ground state, described by the Mathieu function with lowesteigenenergy E . We rather project the initial state ψ (˜ z , t = 0) into the basis of Mathieu functions, such that we can write ψ (˜ z , t = 0) = r max X r =0 | M ( A r , Q, ˜ z ) i h M ( A r , Q, ˜ z ) | ψ (˜ z, ...

We compare the measurements with the calculations by the following procedure. We integrate the single absorption images along the y -direction and fit an 1D Gaussian function to every diffraction order in a semi-automatic procedure [122, Ch.A.2.2]. In this way, we extract the relative populations P ...

we show the results of the evaluation of the absorption images from Fig. 4.3. We obtain independent fitting results of the lattice depth for each diffraction order. All of themmatch within a small intervall around the mean value U lat = (144 ± 2) E R (disregarding the weakly populated 6 th order). F...

2. deeper lattices: At increasing lattice depths, the movement of the wavepackets becomes gradually inhibited by the strong confinement at the positions of theindividual lattice sites. In this case, the delocalized Bloch waves are not well suitedfor an intuitive description of the system anymore. In...

For lattice depths U lat ∼ 10 E R , there is still particle exchange between the lattice sites. However, the lattice is sufficiently deep that the localized wave functions showalmost no overlap from one site to the next. This greatly simplifies the descriptionof an interacting condensate in a 1D opt...

where the ϕ j ( t ) are the phases of the separated on-site condensates. As long as there is tunneling in the system, all the phases of the sub-condensates are the same and aredetermined through the chemical potential µ , with ϕ j ( t ) = ϕ ( t ) = µ t/ ~ . In the TBA, the sinusoidal lattice potenti...

where we have neglected any contributions from the tunneling. The global chemical poten- tial µ is thus expressed as the sum of the on-site potential energy ε j def = mω 2 z ( d lat j ) 2 / 2 def = Ω j 2 arising from the underlying harmonic trapping in the z -direction, and the local chemical potent...

where N 0 def = ( µ/U 1 ) 2 is the atom number in the central lattice site and the “inversion point” j inv def = q µ/ Ω yields the number of populated lattice sites: N lat = 2 j inv + 1. With the total number of atoms defined by N = P j N j , we finally obtain the global chemical potential µ = 15 N ...

parameter ( a, U lat , ω ρ,z ), we also change the local values of e µ such that it differs from one lattice site to another. As we can calculate both j and µ j (see section 4.3.2), we obtain analytical expressions for the phases ϕ j ( t ) = e µ ( j d lat ) t/ ~ of the independent condensates in the...

experiment simulation time (ms) 0.78 1.56 3.9 7.8 11.7 15.6 2.6 5.2 13 26 39 52 U ( E ) lat R U lat time start TOF (a) (b) Fig. 4.6, Expansion of a BEC from a 1D lattice with increasing lattice depth: (a) Experimental sequence: we ramp up linearly the lattice depth at a rate of 10 / 3 E R ms − 1 bef...

5 Stability of a Dipolar BEC in a 1D Optical Lattice In this chapter, we address the question about the stability of a dipolar condensate in theone-dimensional lattice potential. Due to the anisotropy of the dipole-dipole interaction,a single dBEC was found to be more stable in an oblate trap (panca...

where α denotes the angle between the quasi-momentum of the excitations and the polarization direction of the dipoles (the contact coupling strength g is defined in Eq. (2.4b) and the dipolar coupling strength g dd is given by Eq. (2.9b)). Let us look at the limiting cases of the excitation spectrum...

the case α = 0, the sound waves create lines of maximum density, with the dipoles sitting side-by-side as shown in Fig. 5.1 (a) . In such configuration, the dipoles interact repulsively, leading to an increase in energy and hence a stable configuration of the system. This isdifferent in the case α =...

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.5 1.0 1.5 2.0 (-20) a 0 (-21) a 0 (-21.9) a 0 roton-maxon spectrum momentum dependence DI 0.01 0.1 1 10 -0.5 0.0 0.5 1.0 (a) (b) Fig. 5.2, Excitations in a 2D homogeneous dipolar BEC: (a) The function H 2 D q ⊥ l z / √ 2 , characterizing the momentum dependent dipolar i...

strong trapping in the z -direction. The system is then well inside the quasi-2D regime (see section 4.3.2), with its transverse size l z given by the harmonic oscillator length. It is shown in Ref. [16] that in such geometry the dipoles do not reach sufficiently strongattraction to produce a roton ...

Measurements The geometry dependent stability of a single trapped dipolar BEC has been experimen- tally investigated in our group [35]. In the experiment, a cylindrically symmetric trap isused, with the symmetry axis oriented along the polarization direction z of the dipoles. The trap geometry is th...

Calculation of the critical scattering length The stability threshold of the dipolar condensate can be computed for a given set of trap parameters. The general procedure is to solve the Gross-Pitaevskii equation (2.15a),searching for the critical scattering length below which there exists no physica...

the variational calculations match the measured critical scattering length in the regimeof prolate traps ( λ < 1). Thus, the instability mechanism in this regime is identified with the phonon instability. At larger trap ratios, however, the variational calculations predict a more stable situation...

5.3.1 Measurement procedure Before describing the details of the experimental procedure, let us consider the principleof the stability measurement. We first have to find an observable that is suited to identifythe critical scattering length. From the discussion of the stability of a single trappedco...

evaporation by continuously lowering the power of the ODT laser beams 83 . Before reaching degeneracy, we switch on the strong magnetic field to a strength B evap = 602 G, well above the Feshbach resonance (FR) located at B 0 = 589 . 1 G. At the scattering length a ( B evap ) = 90 a 0 , we finish th...

ramp of the form U (˜ t ) = U lat h ( k + 1) ˜ t k + k ˜ t k +1 i , (5.4) with ˜ t = t/T ramp the time in units of the ramp duration T ramp = 20 ms and k = 11 the ramping parameter. The ramp, shown in Fig. 5.3 (a) , is globally slow enough to ensure adi- abaticity. In addition it is made slow at its...

the images, taken after the expansion from the lattice, we observe the interference patternof the condensate and, in addition, a thermal cloud in the background (see Fig. 5.8 (a) ). To extract the atom number in the condensate, we first integrate the recorded 2D 0 2 4 6 8 10 0 10 20 30 40 50 y z abs...

atom number N BEC even when we load the condensate into very deep lattices: here, the absorption images show complicated multi-peak patterns in the z -direction, as shown in section 4.4.2. This effect arises due to the dephasing of the on-site condensates and theirsubsequent interference during the ...

In the following we discuss the mechanisms that define the stability of the dBEC in the 1D lattice. To do so we divide the stability diagram into three regimes: (i) For shallow lattices with U lat . 3 E R , the lattice has no significant effect on the stability of the system. Hence, the relevant tra...

a weak dependence of the stability threshold on the BEC atom number 89 , as shown in Fig. 5.10. For lattice depths U lat > 15 E R the variation of a crit with the BEC atom number is very small and well below our experimental uncertainties. Dealing with very deeplattices we note that, in absence o...

5.4.1 Analysis of Inter-site Effects in the Lattice In our experiment, both on-site and inter-site interactions are always present and cannotbe “switched off” selectively to test their relative influence on the stability of the system.However, the effect of the inter-site interactions can be reveale...

In a second approach, we analyse the inter-site effects by performing variational calcu- lations with a Gaussian ansatz for the on-site wave functions. For an efficient calculation,all these wave functions are assumed to be the same and radially symmetric. We definethe on-site trap geometry by the r...

show the energy spectrum of the collective excitations for N lat = 15 dipolar condensates, using parameters close to the experimental ones. The lowest lying mode of the band- 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 50 40 30 20 10 -22.75 -18.20 -13.75 (a) (b) homogeneous layers lattice + trap Fig. 5.12, In...

states emerge mainly, if only few lattice sites 91 are populated [28], as shown in Fig. 5.12 (b) . Loading many lattice sites, the roton instability may not be distinguished anymore fromthe long-wavelength phonon instability, as the characteristic roton wavelength exceeds theradial size of the syste...

6 Collapse of a Dipolar BEC in a 1D Optical Lattice The subject of this chapter is the collapse dynamics of the dBEC in the 1D optical lattice, i.e. its time evolution in the unstable regime after the stability threshold is crossedrapidly. Usually in such studies, the system is driven into instabili...

Due to the isotropic nature of the contact interactions, there is no preferential directionfor the atom bursts that are emitted from a purely contact interacting BEC after thecollapse 92 . In contrast, the collapse dynamics of a dipolar BEC, even when confined in a spherically symmetric trap, shows ...

time (ms) 0 0 ms 0.1 ms 0.2 ms 0.4 ms 0.3 ms 0 0 0 0.5 1 1.5 10 20 10 20 30 1 2 3 0.5 ms (a) (c) (b) B Fig. 6.1, Collapse dynamics of a dBEC in a single round trap: (a) Experimental sequence to induce the collapse. Both the programmed ramp (red line) of the scattering length a ( t ) and the real ram...

collapse process, they remain at their positions during the explosion of the cloud. Thisleads to the characteristic d -wave density pattern that is preserved during the TOF. We see that the simulations reveal the real-time collapse dynamics of the dBEC, which is not accessible in the experiment. Due...

above or below the stability threshold, as shown in Fig. 6.2 (a) . We then hold the system in this final configuration for a variable time t hold , before we switch off all the trapping lasers and perform an 8 ms TOF (see Fig. 6.2 (b) for an illustration of the sequence). Our (a) (b) lattice depth t...

y z Fig. 6.3, Collapse dynamics of a dBEC in a 1D lattice: Evolution of the system for increasing holding time t hold and different final lattice depth U , following the experimental sequence described in section 6.2.1. Each image is an average of five absorption pictures with the thermal cloud remo...

for short holding times t hold ≤ 0 . 6 ms from an exponential fit 99 to the remnant fraction, as shown in Fig. 6.5 (a) . We note that the choice of the exponential fitting function is not related to any physical model, but describes well the data in the considered time intervaland thus enables to qu...

6.2.3 Numerical Simulations of the Collapse Dynamics In a collaboration with the theory group in Hannover, we have examined more closelythe collapse dynamics of the dipolar condensate in the lattice by means of numericalsimulations. The simulations are based on the time-dependent Gross-Pitaevskii eq...

0.45 ms 0.85 ms 1.25 ms 1.45 ms 1.85 ms t = 0 0.15 ms 0.40 ms 0.60 ms 1.25 ms 1.45 ms t = 0 y z time t (ms) 6 0.0 0.5 1.0 1.5 9 12 15 in-trap TOF U = 12.6 U = 0 U = 3.2 U = 6.3 U = 8.2 k y k z (a) (b) (c) Fig. 6.6, Real-time simulations of the collapsing dBEC: (a) Time evolution of the system for a ...

observe the collapse of the zero-momentum component in the images in Fig. 6.6 (a) (upper row). The collapsed cloud then develops the d -wave pattern that is also recorded after TOF in the experiment. We explain this novel collapse scenario inmore detail in the text below. (iii) Holding the dBEC at a...

We emphasize that the inter-site coherence in the lattice plays a crucial role for the TOF-triggered collapse to occur. Considering the expansion of an incoherent array of condensates, their interference pattern will not show well-separated density peaks. Thedensity distribution is instead given by ...

oblate dBEC, if the system forms a structured ground-state before it is driven into theunstable regime. We therefore aim to cross the narrow parameter range 101 where the structured ground-states are predicted to appear [28], while the system is still trapped inthe lattice. The principle of this mea...

a ( t ), shown in Fig. 6.8 (a) . We therefore cross the stability threshold, given by the critical scattering length a crit , only after the end of the programmed ramp. Then, we hold the condensate for a variable time t hold in the trap, before we switch off the ODT. After a delay ∆ t = 0 . 2 ms, we...

In the image taken for t hold = 0 . 2 ms, we observe an even stronger expansion of the two central density peaks in the y -direction, while the two outer peaks are hardly visible. There is no clear structure in the atomic clouds anymore, the observation of small density modulations might, however, b...

7 Summary and Outlook Summary The main subject of this thesis was the study of the static and dynamic properties of a dipolar chromium Bose-Einstein condensate (BEC) in a one-dimensional optical latticepotential. The stability of the system was investigated by measuring the critical s -wave scatteri...

interaction strength fixed. Using this new technique, we found that the in-trap collapsedynamics of an unstable dBEC is slowed down for increasing lattice depths U lat . Choosing U lat above the stability threshold, the system showed almost no in-trap time evolution. However, after the release from ...

as well as e.g. dipolar condensates in toroidal traps (created by “painting” a circle withthe laser beam of the dipole trap). The latter case has been subject to a theoretical study,showing that the dBEC may form a self-induced Josephson-junction [201]: with the dipoles aligned in the plane of the t...

A appendix A.1 Scattering Properties of Bosonic Dipolar Gases In this section, we discuss the basic elastic and inelastic scattering properties of cold andultra-cold bosonic dipolar gases, mainly summarizing the theoretical results from Refs. [57,58, 101, 108, 109, 207–210]. The studies presented he...

different angular momenta, with the strength given by the potential matrix elements V ( m ) ll 0 , V ( m ) ll 0 def = h lm | 1 − 3(ˆ r · ˆ z ) 2 | l 0 m i = " 1 − 3 2 l + 1 ( l − m )( l + m ) 2 l − 1 + ( l − m + 1)( l + m + 1) 2 l + 3 !# δ l,l 0 − 3 2 l + 3 vuut (( l + 1) 2 − m 2 )(( l + 2) 2 − ...

Fig. A.1, Elastic dipolar scattering: Total elastic scattering cross section σ total (red line) in dipole units (d.u.) in dependence of the energy E in units of the dipolar energy E D [58]. Furthermore, the contributions σ lm,l 0 m of the most important scattering processes are shown, with the quant...

species dipole moment D E D /k B 87 Rb 1 µ B 0 . 6 a 0 16 K 52 Cr 6 µ B 23 a 0 13 mK 164 Dy 10 µ B 200 a 0 53 µ K K-Rb ∼ 0 . 5 Debye 60 , 000 a 0 1 nK Tab. A.1, Dipole length D and dipole energy E D for different dipolar systems: Except from chromium, the table displays the parameters for rubidium (...

A.3 Calculations on the Ground State in a 1D Optical Lattice In this part, we show the explicit calculations of the ground-state properties of a contactinteracting BEC in a 1D lattice, discussed in section 4.3. We obtain analytical results fore.g. the local chemical potential µ j and the atom number...

U 1 is a constant that depends neither on the number of atoms nor on the site index and is determined by the dimensionality of the system 114 . We now consider the global properties of the lattice system. As shown in section 4.3.2, we can write the globalchemical potential µ as the sum of the local ...

function of the BEC in the form ψ G ( ρ, z ) def = 1 π 3 / 4 l 2 ρ l z exp " − ρ 2 2 l 2 ρ − z 2 2 l 2 z # , (A.16) where l ρ ( l z ) is the radial (axial) width of the condensate. With this ansatz, we can calculate the energy per particle via Eq. (2.17). This yields the following terms for thek...

A.6 Excitation Spectrum of a 2D Homogeneous Dipolar BEC Here, we show a brief derivation of the excitation spectrum in a 2D homogeneous dipolargas, discussed in section 5.1.2. A more detailed treatment can be found in Refs. [16, 26, 27]. We start from the excitation spectrum in the 3D homogeneous sy...

A.7 Fitting Procedure in Calibration of the Scattering Length In this part, we describe the fitting procedure used in the calibration of the scatteringlength (see section 3.2.3). The calibration procedure relies on the linear scaling relationbetween the scattering length a and the quantity R 5 y /N ...

routine ’leasqr.m’ directly provides the correlation between the two fitting parameters.Replacing for clarity the variables x def = ∆ I FB and y def = I FB , 0 , with their fitting uncertainties ∆ x and ∆ y and their correlation s xy , we can compute the uncertainty of the scattering length [212] (∆...

the programmed field value and its actual value measured by Zeeman spectroscopy. Wehave identified two reasons for the deviations: (i) due to the limited bandwidth of the active current stabilization, the PI controller may introduce delays and oscillations in theFeshbach current when its set-point i...

References ∗ [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. Davis, M. Mewes, M. Andrews, N. Druten, D. Durfee, D. Kurn, and W. Ketterle: “ Bose-Einstein Con...

Danksagung Abschließend möchte ich noch vielen lieben Leuten ganz herzlich danken, ohne die eine erfolgreiche Durchführung dieser Doktorarbeit unmöglich gewesen wäre. Allen voran danke ich meiner Freundin Anna. Du hast mich die ganzen Jahre während der Doktorarbeit unterstützt, auch wenn es “mal” wi...