Bose 2.2 User Manual

Stability and Collapse Dynamics ofDipolar Bose-Einstein Condensatesin One-Dimensional Optical LatticesVon der Fakultät Mathematik und Physik der Unive

Streulänge reduziert, bis der kritische Wertacriterreicht ist, bei der ein plötzlicherVerlust der Atome im Kondensat eintritt. Für genügend kleine Git

6 Collapse of a Dipolar BEC in a 1D Optical LatticeThe subject of this chapter is the collapse dynamics of the dBEC in the 1D optical lattice,i.e. its

Due to the isotropic nature of the contact interactions, there is no preferential directionfor the atom bursts that are emitted from a purely contact

time (ms) 00 ms 0.1 ms 0.2 ms 0.4 ms0.3 ms000 0.5 1 1.5102010203012 30.5 ms(a) (c)(b)BFig. 6.1, Collapse dynamics of a dBEC in a single round trap:(a)

collapse process, they remain at their positions during the explosion of the cloud. Thisleads to the characteristic d-wave density pattern that is pre

above or below the stability threshold, as shown in Fig. 6.2(a). We then hold the systemin this final configuration for a variable timethold, before we

yzFig. 6.3, Collapse dynamics of a dBEC in a 1D lattice:Evolution of the systemfor increasing holding timetholdand different final lattice depthU, follo

In contrast, if the final lattice depth is chosen above the stability threshold, i.e. in thecasesU= 12.6ERandU= 8.2ER, there is no visible decay of the

for short holding timesthold≤0.6msfrom an exponential fit99to the remnant fraction,as shown in Fig. 6.5(a). We note that the choice of the exponential

6.2.3 Numerical Simulations of the Collapse DynamicsIn a collaboration with the theory group in Hannover, we have examined more closelythe collapse dy

wurde [36]. Dieses neuartige Kollapsszenario wird durch Echtzeit-Simulationen der Gruppeaus Hannover bestätigt.Die im Rahmen dieser Arbeit durchgeführ

0.45 ms0.85 ms 1.25 ms1.45 ms1.85 mst = 00.15 ms0.40 ms 0.60 ms1.25 ms1.45 mst = 0yztime t (ms)60.0 0.51.0 1.591215in-trapTOFU = 12.6U = 0U = 3.2U = 6

observe the collapse of the zero-momentum component in the images in Fig. 6.6(a)(upper row). The collapsed cloud then develops thed-wave pattern that

We emphasize that the inter-site coherence in the lattice plays a crucial role for theTOF-triggered collapse to occur. Considering the expansion of an

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

yzFig. 6.7, Delayed lattice switch-off:Atomic density distribution after TOF, for differ-ent delay times ∆tbetween ODT and lattice switch-off. The parame

a(t), shown in Fig. 6.8(a). We therefore cross the stability threshold, given by the criticalscattering lengthacrit, only after the end of the program

In the image taken forthold= 0.2ms, we observe an even stronger expansion of the twocentral density peaks in they-direction, while the two outer peaks

7 Summary and OutlookSummaryThe main subject of this thesis was the study of the static and dynamic properties of adipolar chromium Bose-Einstein cond

interaction strength fixed. Using this new technique, we found that the in-trap collapsedynamics of an unstable dBEC is slowed down for increasing latt

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

in Ref. [205], our new setup may be suited to directly image the various partial waves atselectable collision energies: making use of the tunable trap

A appendixA.1 Scattering Properties of Bosonic Dipolar GasesIn this section, we discuss the basic elastic and inelastic scattering properties of cold

different angular momenta, with the strength given by the potential matrix elementsV(m)ll0,V(m)ll0def= hlm|1 − 3(ˆr · ˆz)2|l0mi="1 −32l + 1 (l − m

Fig. A.1, Elastic dipolar scattering:Total elastic scattering cross sectionσtotal(redline) in dipole units (d.u.) in dependence of the energyEin units

species dipole moment D ED/kB87Rb 1 µB0.6 a016 K52Cr 6 µB23 a013 mK164Dy 10 µB200 a053 KK-Rb ∼ 0.5 Debye 60, 000 a01 nKTab. A.1, Dipole length D and d

The situation is different in atomic systems with permanent magnetic dipoles wherethe dipole lengthDis fixed: here, the dipolar inelastic scattering is

A.3 Calculations on the Ground State in a 1D Optical LatticeIn this part, we show the explicit calculations of the ground-state properties of a contac

U1is a constant that depends neither on the number of atoms nor on the site indexand is determined by the dimensionality of the system114. We now cons

function of the BEC in the formψG(ρ, z)def=1π3/4l2ρlzexp"−ρ22 l2ρ−z22 l2z#, (A.16)wherelρ(lz) is the radial (axial) width of the condensate. With

symmetric trap along the polarization directionz, are approximated by the GaussianformsΨ1(ρ, z) =1π3/4σρ√σzexp −ρ22σ2ρ−z22σ2z!, (A.18a)and Ψ2(ρ, z) =1

1 IntroductionSince the realization of Bose-Einstein condensates (BECs) in ultracold dilute atomicvapours in 1995 [1–4], degenerate quantum gases have

The choice of the Gaussian form for the density distributionsnj(r) allows for a simpleexpression of their Fourier transformsF{nj} =N(2π)3/2exp−14k2ρσ

A.6 Excitation Spectrum of a 2D Homogeneous Dipolar BECHere, we show a brief derivation of the excitation spectrum in a 2D homogeneous dipolargas, dis

A.7 Fitting Procedure in Calibration of the Scattering LengthIn this part, we describe the fitting procedure used in the calibration of the scatteringl

routine ’leasqr.m’ directly provides the correlation between the two fitting parameters.Replacing for clarity the variablesxdef=∆IFBandydef= IFB,0, wit

the programmed field value and its actual value measured by Zeeman spectroscopy. Wehave identified two reasons for the deviations: (i) due to the limite

strong oscillations of the Feshbach current which are damping out after around 2ms.These oscillations are reduced when using a linear ramp (with a ram

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 Ato

[13]M. Baranov: “Theoretical progress in many-body physics with ultracold dipolargases.” Physics Reports 464, 71–111 (2008)[14]T. Lahaye, C. Menotti,

[29]R. Wilson, and J. Bohn: “Emergent structure in a dipolar Bose gas in a one-dimensional lattice.” Physical Review A 83, 023623 (2011)[30]A. Griesma

the contact interactions in the system. Nevertheless, dipolar effects have been observede.g. by measuring the expansion velocity of the chromium BEC [3

[43]R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov: “Optical Dipole Traps for Neu-tral Atoms.” in: Advances In Atomic, Molecular, and Optical Physics

[56]R. V. Krems, G. C. Groenenboom, and A. Dalgarno: “Electronic InteractionAnisotropy between Atoms in Arbitrary Angular Momentum States.” The Journa

[69]P. Pedri, and L. Santos: “Two-Dimensional Bright Solitons in Dipolar Bose-EinsteinCondensates.” Physical Review Letters 95, 200404 (2005)[70]I. Ti

[85]F. London: “On the Bose-Einstein Condensation.” Physical Review54, 947–954(1938)[86]L. D. Landau, E. M. Lifshitz, and L. E. Reichl: “Statistical P

[102]S. Ronen, D. Bortolotti, D. Blume, and J. Bohn: “Dipolar Bose-Einstein condensateswith dipole-dependent scattering length.” Physical Review A 74,

[118]D. Peter: “Theoretical investigations of dipolar quantum gases in multi-well poten-tials.” (Diploma thesis, Stuttgart, 2011)[119]M. Rosenkranz, Y

[135]P. O. Schmidt, S. Hensler, J. Werner, T. Binhammer, A. Görlitz, and T. Pfau:“Continuous loading of cold atoms into a Ioffe Pritchard magnetic trap

[151]C. Wieman, and T. Hänsch: “Doppler-Free Laser Polarization Spectroscopy.” Phys-ical Review Letters 36, 1170–1173 (1976)[152]E. D. Black: “An intr

[165]K. Xu, T. Mukaiyama, J. R. Abo-Shaeer, J. K. Chin, D. E. Miller, and W. Ketterle:“Formation of Quantum-Degenerate Sodium Molecules.” Physical Rev

[181]A. Trombettoni, and A. Smerzi: “Discrete Solitons and Breathers with DiluteBose-Einstein Condensates.” Physical Review Letters 86, 2353–2356 (200

last dipolar system being presented here are homonuclear Rb2molecules, with one ofthe atoms excited to a high-lying Rydberg state [64]. Contrary to th

[196]J. H. Denschlag, J. E. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho,K. Helmerson, S. L. Rolston, and W. D. Phillips: “A Bose-Einstein c

[210]C. Ticknor, and S. T. Rittenhouse: “Three Body Recombination of Ultracold Dipolesto Weakly Bound Dimers.” Physical Review Letters 105, 013201 (20

DanksagungAbschließend möchte ich noch vielen lieben Leuten ganz herzlich danken, ohne die eineerfolgreiche Durchführung dieser Doktorarbeit unmöglich

wenn das Experiment in diesem ersten Jahr meiner Doktorarbeit nicht immer so wolltewie wir, haben wir nie den Mut verloren und viel Spaß zusammen geha

wissenschaftliche Bereicherung unseres Experiments.Auch bei den Theorie-Gruppen sind aller guten Dinge drei: ich möchte mich bei Prof.Wunner und Patri

This thesisIn this thesis I report on the investigation of a52Cr BEC in a one-dimensional opticallattice, operating in a regime with dominant dipolar

spatially separated dipolar condensates is considered in the discussion. Chapter 3 describesthe production process of a52Cr BEC with tunable contact i

2 Dipolar Quantum GasesThis chapter gives a brief introduction to the physics of bosonic dipolar quantum gases.It provides the basic formalisms to und

a macroscopic phase coherence throughout the sample. A detailed treatment on thefundamental properties of BECs is given in Refs. [8, 89].For an intuit

2.2 Two-Body InteractionsThe atomic density in a BEC is typically aroundnBEC= 1014−1015cm−3, which is verylow compared to solids (ncarbon∼1023cm−3) or

with the van-der-Waals term of the molecular potential7, we obtain an interaction radiusr0∼100a0of the molecular potential. This defines the maximal di

rimpactv/2v/2V(r)ra(a) (b)Fig. 2.1, Elastic two-body scattering:(a) Classical picture of two colliding atomsmoving at the relative velocityv. The impa

using the reduced massmred=m/2. We immediately see that the prefactors in thecontact coupling strengthg, defined in Eq.(2.4b), are chosen such thatr=ai

0r(a) (b)Fig. 2.2, Dipole-dipole interaction (DDI):(a) Two dipoles polarized by an externalmagnetic fieldBalong thez-direction. The separationr=|r|and

After investigating the long-range behaviour of the DDI potential, we now consider itsbehaviour at small distancesr. If the DDI potential acts on a tw

We stress here that the dipolar lengthadddoes not correspond to a finite interaction radiusof the dipolar interactions. Such radius cannot be defined fo

operators11ˆakand ˆak, such thatˆΨ (r) =Xkφk(r) ˆakandˆΨ(r) =Xkφk(r) ˆak, (2.11)withφk(r) a set of single-particle states. Assuming a macroscopic popu

Inserting the ansatzψ(r, t) =ψ(r)exp(−iµt/~) into Eq. (2.14), whereµis the chemicalpotential of the condensate, we obtain the stationary Gross-Pitaevs

AbstractThe subject of this thesis is the investigation of the stability and the collapse dynamics ofa dipolar52Cr Bose-Einstein condensate (BEC) in a

(i) N 1: The macroscopic population of a single particle state allows for thereplacement of the creation and annihilation operators by classical numb

Since in our experiments with a chromium BEC, all the validity criteria are fulfilled, wewill make use of the mean-field description in the remaining pa

energyVext(R) =mω20R2/2. In the other limit of small and decreasing radii, Heisenberg’suncertainty relationp=~/Rleads to a divergence in the kinetic e

2.4.3 TF-Approximation with Contact and Dipolar InteractionsThe long-range character of the dipole-dipole interaction significantly complicates thedesc

Hence, for a given external trapping potential, we obtain an exact solution for thestationary GPE, with the chemical potential given by µ = gn0[1 − d

ratio [114]. It is given byN(a − add)/aho1 for trap ratiosλ .1, used e.g. in themeasurements for the calibration of the scattering length, presented

to different layers readsVdiscdd(rin, dlat) =µ0µ2m4π1 − 3 d2lat/(r2in+ d2lat)(r2in+ d2lat)3/2=µ0µ2m4πr2in− 2 d2lat(r2in+ d2lat)5/2.(2.27)Since we have

This expression means that two infinite planes of dipoles do not interact [121]. Inthis limit, the system is translationally invariant in thexy-plane,

Considering a real trapped dipolar gas with weak interactions, the density of the samplewill rather be described by a Gaussian than a disc shape, as w

Refs. [51, 118]),E(2)inter(dlat) = −gddN2(2π)3/2σ3ρ1Z0du(1 − 3u2)(1 − u2(η + L2))(1 − ηu2)5/2exp −L2u22(1 − ηu2)!, (2.33)whereLdef= dlat/σρis the norm

inter-site energy vanishes. Thus, we recover the result obtained in the Thomas-Fermiapproximation, where we found a vanishing dipolar contribution to

B,z0-50-100-150-2000 10 20 304050(a) (b)Fig. 2.6, Inter-site energy in a dipolar multi-site system with constant filling ofthe sites:(a) Sketch of the

separated dipolar BECs. The calculations have shown that the inter-site interactionenergy is strongly enhanced when considering more than two layers o

3 Producing a52Cr BEC with Tunable InteractionsTo investigate dipolar effects in an ultra-cold bosonic gas, we routinely create a52Crcondensate with tu

Zeeman-Slower beamCCDimaging systemcloverleafcoilscoolingbeamsODTbeamsopticalpumpingZeemanslowerCr effusion cell1450°C2D molassesbeamsprobebeamyxyzIII

3.1.2 ProcedureWe create a beam of chromium atoms by sublimation in an effusion cell25at a temperatureT ∼1450◦C. The atomic beam is first collimated by

From the exit of the ZS, the atoms travel only a short distance until they reach thecenter of the trapping chamber, where they are captured in a magne

value28, we minimize the Zeeman energy that is released in a dipolar relaxation process.Still, the heating is too strong to reach the critical tempera

3.1.3 Laser SystemsWe now give a brief overview of the lasers that are involved in the production process ofthe BEC.425 nm MOT laser systemThe laser l

427 nm optical pumping laser systemThe setup of the optical pumping laser system (λ= 427.6nm) is similar to the MOTlaser system, however, involving mu

ContentsZusammenfassung 71 Introduction 132 Dipolar Quantum Gases 192.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . 1

3.2 Tuning of the Contact InteractionsIn a chromium BEC, the dipolar interactions are much stronger than e.g. in condensatesof alkali elements, due to

scattering properties in the system can be tuned by changing the potential difference∆Ebetween the open and the closed channel, which is the basic idea

Let us consider more closely the tuning of the scattering length in the vicinity of aFeshbach resonance (see Fig. 3.3). If the spin projections of the

3.2.2 Experimental Realization of the Feshbach SystemThe required magnetic field strengthB ∼600Gto reach the Feshbach resonance isconveniently produced

Eddy currentsThe current in the Feshbach coils is dynamically computer-controlled by changing theset-point of the PI-controller [74, Ch.4.2.2]. If, ho

actly within the TF approximation51[169]. We are thus able to numerically calculatethe valuesR5y/NandR5z/Nafter TOF as a function of the scattering le

00 204060 80 100102030(a) (b)dipolar expansion (TF) TF vs. simulationsTFsimulations0 20 40 60 80 100100203040Fig. 3.4, Dipolar expansion:(a) RatiosR5y

width and the center of the resonance is obtained by fitting the functiona(IFB) = abg· 1 −∆IFBIFB− IFB,0!, (3.3)to the data, where the width and the ce

The precise determination of the functiona(IFB) is the main goal of the calibrationprocedure. However, to display the results in real physical paramet

4 A BEC in a One-Dimensional Optical LatticeIn this chapter, the basic properties of a BEC trapped in a one-dimensional (1D) opticallattice potential

4.3.2 Ground-State of a BEC in a 1D Lattice . . . . . . . . . . . . . . . 704.3.3 Phase Evolution of Decoupled BECs . . . . . . . . . . . . . . . . 72

zoomxzFig. 4.1, Interference of two coherent laser beams:Two crossing laser beams(propagation direction given by the arrows) produce a regular 1D arra

of ultra-cold gases in optical lattices:klat=πdlatthe lattice wave number, (4.3a)ER=~2k2lat2m=~2π22md2latthe recoil energy, and (4.3b)s =UlatERthe dim

mirrorlensl/2 wave platebeam blockODT1shutter1stfiber laseroptical isolatorAOMCCDprobe beamFeshbach coilszxygBqFig. 4.2, Schematic drawing of the opti

with the parameterAdef= E/ER−Ulat/(2ER) and the lattice parameterQdef= Ulat/(4ER) =s/4. The solutions of Eq.(4.5)are the so-called Mathieu functionsM(

initial (flat) state to the ground state, described by the Mathieu function with lowesteigenenergyE. We rather project the initial stateψ(˜z, t= 0) int

and depends only on the timetpulseand the lattice depthUlatvia the lattice param-eterQ=Ulat/(4ER). The normalization of the relative populationsPnis g

We compare the measurements with the calculations by the following procedure. Weintegrate the single absorption images along they-direction and fit an

we show the results of the evaluation of the absorption images from Fig. 4.3. We obtainindependent fitting results of the lattice depth for each diffrac

2.deeper lattices: At increasing lattice depths, the movement of the wavepacketsbecomes gradually inhibited by the strong confinement at the positions

For lattice depthsUlat∼10ER, there is still particle exchange between the latticesites. However, the lattice is sufficiently deep that the localized wav

ZusammenfassungGegenstand dieser Arbeit ist die experimentelle Untersuchung eines dipolaren Bose-Einstein-Kondensates (BEK) mit Chromatomen in einem e

where theϕj(t) are the phases of the separated on-site condensates. As long as thereis tunneling in the system, all the phases of the sub-condensates

where we have neglected any contributions from the tunneling. The global chemical poten-tialµis thus expressed as the sum of the on-site potential ene

whereN0def= (µ/U1)2is the atom number in the central lattice site and the “inversion point”jinvdef=qµ/Ωyields the number of populated lattice sites:Nl

parameter (a, Ulat, ωρ,z), we also change the local values ofeµsuch that it differs fromone lattice site to another. As we can calculate bothjandµj(se

Again, we may use the analogy with the diffraction of laser light on a grating. In contrastwith our previous considerations in section 4.2.2, the optic

In very deep lattices, the width of the Gaussian on-site wave functions is directly given bythe harmonic oscillator lengthalat. In this case, the expr

experiment simulationtime (ms)0.78 1.563.97.811.715.62.65.213263952U (E )lat RUlattimestart TOF(a)(b)Fig. 4.6, Expansion of a BEC from a 1D lattice w

In the regime 0< Ulat< Ufix, we calculate the equilibrium atom number distribution inthe lattice, given by Eq.(4.20), for the different lattice de

interactions crucially change the properties of the system, as we show in the next chapterpresenting our measurements on the stability of a dipolar BE

5 Stability of a Dipolar BEC in a 1D Optical LatticeIn this chapter, we address the question about the stability of a dipolar condensate in theone-dim

zur experimentellen Kontrolle: viele atomare Spezies verfügen über Feshbach-Resonanzen,in deren Nähe dies-Wellenstreulänge über ein externes Magnetfel

whereαdenotes the angle between the quasi-momentum of the excitations and thepolarization direction of the dipoles (the contact coupling strengthgis d

the caseα= 0, the sound waves create lines of maximum density, with the dipoles sittingside-by-side as shown in Fig. 5.1(a). In such configuration, the

0.0 0.1 0.2 0.3 0.4 0.50.00.51.01.52.0(-20) a0(-21) a0(-21.9) a0roton-maxon spectrummomentum dependence DI0.01 0.1 1 10-0.50.00.51.0(a) (b)Fig. 5.2, E

strong trapping in thez-direction. The system is then well inside the quasi-2D regime(see section 4.3.2), with its transverse sizelzgiven by the harmo

(a) (b)Fig. 5.3, Rosensweig instability in a classical ferrofluid:(a) A fluid consisting offerromagnetic particles (ferrofluid) in a Petri dish develops

MeasurementsThe geometry dependent stability of a single trapped dipolar BEC has been experimen-tally investigated in our group [35]. In the experimen

Calculation of the critical scattering lengthThe stability threshold of the dipolar condensate can be computed for a given set oftrap parameters. The

the variational calculations match the measured critical scattering length in the regimeof prolate traps (λ <1). Thus, the instability mechanism in

5.3.1 Measurement procedureBefore describing the details of the experimental procedure, let us consider the principleof the stability measurement. We

evaporation by continuously lowering the power of the ODT laser beams83. Before reachingdegeneracy, we switch on the strong magnetic field to a strengt

dies-Wellenstreulänge jedoch reduziert und somit ein Quantengas mit starker dipolarerWechselwirkung erzeugt werden [32]. Die Kalibration der Streuläng

ramp of the formU(˜t) = Ulath(k + 1)˜tk+ k˜tk+1i, (5.4)with˜t=t/Trampthe time in units of the ramp durationTramp= 20msandk= 11 theramping parameter. T

the images, taken after the expansion from the lattice, we observe the interference patternof the condensate and, in addition, a thermal cloud in the

atom numberNBECeven when we load the condensate into very deep lattices: here, theabsorption images show complicated multi-peak patterns in thez-direc

To extract the critical scattering lengthacritfrom the measurements, we perform a fitwith the empirically chosen functionNBEC(a) = maxhN0(a − acrit)β,

In the following we discuss the mechanisms that define the stability of the dBEC in the1D lattice. To do so we divide the stability diagram into three

a weak dependence of the stability threshold on the BEC atom number89, as shown inFig. 5.10. For lattice depthsUlat>15ERthe variation ofacritwith t

5.4.1 Analysis of Inter-site Effects in the LatticeIn our experiment, both on-site and inter-site interactions are always present and cannotbe “switche

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.

show the energy spectrum of the collective excitations forNlat= 15 dipolar condensates,using parameters close to the experimental ones. The lowest lyi

states emerge mainly, if only few lattice sites91are populated [28], as shown in Fig. 5.12(b).Loading many lattice sites, the roton instability may no