IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) R91–R134 doi:10.1088/0953-4075/40/14/R01

PhD TUTORIAL

Generation of a dipolar Bose–Einstein condensate

Axel Griesmaier

5. Physikalisches Institut, Universitat Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

E-mail: a.griesmaier@physik.uni-stuttgart.de

Received 21 April 2007, in final form 6 June 2007Published 4 July 2007Online at stacks.iop.org/JPhysB/40/R91

AbstractThis tutorial presents a detailed discussion of the generation of a Bose–Einsteincondensate (BEC) of chromium atoms. This constitutes the first realization of aBose–Einstein condensate of atoms with strong dipole–dipole interaction. Dueto the special electronic and magnetic properties of chromium atoms, standardmethods cannot be applied to generate a chromium BEC. The tutorial containsa detailed discussion of the techniques that we use to create the chromium BEC.Compared to all other BECs that have been created so far, chromium atomshave an extraordinarily large magnetic dipole moment and therefore underliestrong magnetic dipole–dipole interaction which is in contrast to the contactinteraction that stems from s-wave scattering of the atoms, long-range andanisotropic. A steadily growing number of theoretical publications show thatmany interesting new phenomena are expected to occur in a ‘dipolar’ BEC.An experimental investigation of the expansion dynamics of the chromiumBEC provides the first experimental proof of a mechanical effect of dipole–dipole interaction in a gas. We describe the condensate by a hydrodynamicmodel of a superfluid that takes dipole–dipole interaction into account. Ourexperimental results are in excellent agreement with this theory. We are able todetermine the relative strength parameter of the dipole–dipole interaction anddeduce the s-wave scattering length of chromium atoms. These two quantitiesallow us to completely describe the interaction of the atoms in the condensate.The experimental results presented here make the chromium BEC the mostpromising system for further investigations of dipolar effects in degeneratequantum gases.

(Some figures in this article are in colour only in the electronic version)

Contents

1. Introduction: interactions—the flavour of quantum gases R921.1. A bit of BEC history R921.2. Why dipolar BECs are interesting R941.3. The way to a dipolar quantum gas R94

0953-4075/07/140091+44$30.00 © 2007 IOP Publishing Ltd Printed in the UK R91

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2. Theoretical concepts R962.1. Interactions in a Bose gas R962.2. Magnetic dipole–dipole interaction in a Bose–Einstein condensate R101

3. Experimental apparatus and detection techniques R1053.1. The apparatus R1053.2. Absorption imaging R109

4. Trapping chromium atoms R1124.1. Chromium atoms in an optical dipole trap R1124.2. Loading the optical trap R1144.3. Transfer to the lowest Zeeman state R116

5. Evaporative cooling R1175.1. Plain evaporation R1185.2. Cooling in the crossed configuration R119

6. Bose–Einstein condensation of chromium atoms R1216.1. Reaching degeneracy R121

7. Analysing the BEC transition R1238. Expansion of the condensate R1239. Observation of dipole–dipole interaction in a degenerate quantum gas R125

9.1. Measurement of the dipole–dipole strength parameter R1279.2. Determination of the dipole–dipole strength parameter R128

10. Conclusion R130Acknowledgments R131References R131

1. Introduction: interactions—the flavour of quantum gases

1.1. A bit of BEC history

The first observation of Bose–Einstein condensation in a dilute atomic vapour—the emergenceof a macroscopic quantum state in a many-body system—marked the start of a new era inquantum and atomic physics. Within only a few months in 1995, three groups reported thesuccessful generation of Bose–Einstein condensates (BEC) in ultracold rubidium [1], sodium[2] and lithium [3] gases. Only 6 years later, Carl Wieman, Eric Cornell and WolfgangKetterle were awarded the 2001 Nobel prize in physics for their pioneering work on thephysics of this new state of matter. Since then, the field has gained the ever growing interest ofexperimentalists as well as theoreticians. In the following years BECs succeeded in vapoursof hydrogen [4] in 1998, potassium [5] and metastable helium [6] in 2001, and cesium [7] andytterbium [8] in 2003.

All of these elements have their own characteristics. Rubidium and sodium are the ‘workhorses’. They are very well understood systems which permit the routine production of verylarge condensates [9] with a comparably low technical effort. Lithium is outstanding due to itseffective attractive interaction, and the existence of bosonic and fermionic isotopes which havebeen brought to degeneracy at the same time [10]. Hydrogen is the simplest atomic systemwhich allows for exact calculations of interatomic potentials. Potassium is also interestingbecause it has bosonic and fermionic isotopes, and metastable helium is outstanding due to itshigh internal energy which offers distinguished diagnostic possibilities [11]. Cesium atomsshow a very broad Feshbach resonance [12–14] which makes them perfectly suited for controlof the scattering properties, besides being of technical interest since cesium atomic clocks

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define our time standard. Ytterbium—so far the only non-alkali, non-rare gas atom in theBEC family—stands out due to its vanishing magnetic moment and narrow optical transitionswhich makes it a promising candidate for precision measurements.

With chromium, the ninth element has joined the family of Bose–Einstein condensates[15, 16]. The peculiarities of chromium point in yet another direction. It is outstandingbecause it introduces a new kind of interaction into the field of degenerate quantum gases. Incontrast to all chemical elements that have been successfully brought to quantum degeneracy,chromium atoms have an extraordinarily large magnetic moment of 6 µB in their 7S3 groundstate. Since the long-range and anisotropic magnetic dipole–dipole interaction (MDDI)scales with the square of the magnetic moment, this comes along with a MDDI betweentwo chromium atoms that is a factor of 36 higher than in alkali BECs. The large magneticmoment stems from the unique electronic structure. While the alkalis, which form the largestgroup among the Bose–Einstein condensates, all have rather simple electronic configurationwith only one valence electron, chromium has six valence electrons with aligned spins(total spin quantum number S = 3). In contrast to a classical gas, interactions play animportant role in a BEC. In fact, although these interactions are very weak, all essentialproperties of Bose–Einstein condensates of dilute atomic gases are determined by the strength,range and symmetry of the interactions present. An additional interaction with differentsymmetry gives therefore rise to new physical phenomena that can be explored in a chromiumBEC.

In all other Bose–Einstein condensates, the short-range and isotropic contact interactionthat arises from s-wave scattering between the atoms is by far dominant. Many excitingphenomena based on this type of interaction have been studied (reviews are given in [17, 18]).Early experiments with the newly available state of matter aimed at studies of the influenceof interactions on the thermodynamic properties of the gas. Deviations of the specific heatand the transition temperature from ideal gas theory were studied [19] and the mean-fieldinteraction energy was measured [20]. Interactions also manifest in the excitation spectrum of aBose–Einstein condensate by a modification of the eigenfrequencies of elementary excitations[21–23]. With the generation of quantized vortices and vortex lattices [24–26] in Bose–Einsteincondensates, another spectacular type of collective excitations was observed, a phenomenoncharacteristic for superfluid systems.

The use of Feshbach resonances to tune the contact interaction has opened new possibilitiesof control over such systems. By applying an external magnetic field to tune the s-wavescattering length it is possible to explore extreme regimes from strongly repulsive to verysmall and strongly attractive interaction. One prominent example called ‘Bosenova’ is thecollapse and explosion of a BEC when the contact interaction is suddenly changed fromrepulsive to attractive [27]. Feshbach resonances have also been used to create BECs ofbosonic molecules formed by fermionic atoms [28–30] and to study the BEC-BCS crossoverin these systems [31–36].

The quantum-phase transition from a superfluid to a Mott insulator state [37, 38] observedin three- and one-dimensional lattices, respectively, is also a result of the sophisticated interplaybetween external potentials, quantum-statistics and inter-particle interaction. Optical latticeswere also used to enter the Tonks–Girardeau regime [39], where the so-called fermionizationof bosonic atoms due their repulsive interaction is observed when the dimensionality of theconfining potential is reduced. With these experiments, Bose–Einstein condensates and dilutegases entered the field of strongly correlated many-body systems—a regime that had previouslybeen a domain of solid-state physics. In this sense, an atomic gas can act as a model systemfor solid-state physics [40].

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1.2. Why dipolar BECs are interesting

It is evident that creating a Bose–Einstein condensate of atoms that are subject to an interactionwith different symmetry and range than the contact interaction will also introduce newcollective phenomena. It has been shown in [41–44] that the anisotropy of the MDDIintroduces an anisotropy in the density distribution of a trapped dipolar condensate. Theexperimental observation of a modification of the condensate expansion due to the MDDI thatwas proposed in [45] is a central point of this tutorial. The regime of dominant dipole–dipoleinteraction can be entered by using a Feshbach resonance to lower the contact interaction.Moreover, a continuous transition between both regimes of dominant dipole–dipole or contactinteraction is possible.

The properties of a BEC are modified even more when the dipole–dipole interactionis the dominant interaction. Due to the anisotropic character of the MDDI, most of theexpected effects depend strongly on the symmetry of the trap. Modifications of the ground-state wavefunction [41, 43, 46] and the stability criteria of dipolar condensates in differenttrap geometries were studied [41, 42, 47]. In pancake-shaped traps [48], modifications ofthe eigenmodes of elementary excitations are expected [47, 49], as well as the occurrenceof a roton-maxon in the excitation spectrum. The existence of new quantum phases wasproposed for the case of a dipolar condensate in a two-dimensional optical lattice. Dependingon the angle between the magnetic moments and the lattice plain and on the lattice depth,transitions between the superfluid, the supersolid (a superfluid with a periodic modulationof the macroscopic wavefunction) and the insulating checkerboard phase (similar to a Mottinsulator but where occupied lattice sites are neighboured by unoccupied ones), are predictedto occur [50]. Dipolar BECs are also discussed in the context of spinor condensates[51, 52]. The combination of large spin and magnetic moment leads to interesting neweffects like the conversion of spin into angular momentum [53]. Very recently, the influence ofMDDI on the formation of vortex lattices in rotating dipolar BECs has been studied [54–56].In these publications, a dramatic influence of the relative strength of the MDDI with respectto the s-wave interaction on the symmetry of the generated lattice structure was found. Ifthe dipole–dipole interaction is strong enough, the existence of 2D solitons is expected [57].Tuning of the MDDI is possible by spinning the quantization axis of the atomic dipoles [58]together with the Feshbach tuning of the contact interaction. This allows for the explorationof all these effects in regimes of almost arbitrary different ratio between the two types ofinteraction. The huge variety of physical phenomena that are predicted for dipolar BECs andthe steadily growing number of theoretical work on these effects make dipolar quantum gasesone of the most exciting fields of atom optics.

1.3. The way to a dipolar quantum gas

Finding and generating a degenerate gas where dipole–dipole interaction plays an importantrole was, however, not so easy. The Bose–Einstein condensates that existed before thechromium BEC were produced mainly from group-I species which all have only verysmall magnetic dipole moments around one Bohr magneton. The only exceptions areytterbium, which is diamagnetic in its ground state, and metastable helium, whose magneticdipole moment is also small. Promising candidates for the observation of dipole–dipoleinteraction in a quantum gas would be heteronuclear molecules that come up with largepermanent electric moments. But although a lot of effort is currently being made toproduce ultracold ensembles of molecules either by buffer gas cooling [59], selecting slowatoms from a thermal source [60], cooling them directly [61, 62], or by producing them

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from ultracold heteronuclear atomic mixtures using Feshbach resonances [63–66], long-lived degenerate ensembles of heteronuclear ground-state molecules do not exist. Recently,ultracold heteronuclear molecules have also been produced for the first time using an rftechnique to create the molecules in an optical lattice close to a Feshbach resonance [67]. Onemajor problem at the moment is that molecules produced by the use of these techniques arenot in their internal ground state but in short-lived highly excited ro-vibrational states, and, asin the latter case, that the production rates of molecules in lower internal states are very low.

The alternative is to use atoms with large magnetic moments. Although the dipole–dipoleinteraction between such atoms is much weaker than for electric dipoles, the magnetic momentof 6 µB of chromium atoms is large enough to lead to observable effects of the dipole–dipoleinteraction in a Bose–Einstein condensate which has been shown in the framework of my PhDproject and will be discussed in this tutorial. Moreover, if a Feshbach resonance is used totune the s-wave scattering length to smaller values, the relative contribution of the magneticdipole–dipole interaction can be made even larger, such that it should be possible to enter theregime were it represents the main contribution to the interaction potential of the atoms in thecondensate.

The path, however, that finally led to the generation of a chromium BEC was long andrequired a lot of experimental and theoretical work on all aspects of cooling and trapping.Two experimental setups and a yearly growing number of lasers involved in the experimentwere needed. In contrast to the alkalis, only little was known about the spectroscopic andscattering properties of atomic chromium when our chromium experiment was started. Thecomplex electronic structure made ab initio calculations of the molecular potentials difficultand important spectroscopic properties and trapping strategies were unknown. The s-wavescattering length—the most important parameter for evaporative cooling, of which neithersign nor modulus was known—had to be determined experimentally to find out whether thegeneration of a Cr-BEC would work at all. The successful operation of a chromium magneto-optical trap and repumping on the intercombination lines [68] as well as the development of acontinuous loading scheme [69–71] of chromium atoms into a magnetic trap were importantsteps towards the production of a chromium BEC.

Further progress was achieved with the first measurement [72] of the s-wave scatteringlength aCr = 170 ± 39 a0 and the development of a Doppler cooling technique that allowedus to cool the sample optically within the magnetic trap [73]. Later, the scattering length wasdetermined with much higher accuracy to be aCr = 102 ± 13 a0 at zero magnetic field from acomparison of experimentally observed Feshbach resonances to theoretical results [74].

With respect to magnetic trapping and evaporative cooling in a magnetic trap, anextraordinary large magnetic moment is not always of advantage. Besides the many promisingpossibilities of studying dipole–dipole interaction in a quantum gas with chromium, the dipole–dipole interaction also induces new loss mechanisms. The probability of inelastic collisionsdue to dipolar relaxation [75] scales with the third power of the total spin. This leads to verylarge loss rates of magnetically trapped chromium atoms of βdr = 2.5 × 10−12 cm3 s−1 atB = 1 G and T = 10 µK. This extreme dipolar relaxation rate causes standard condensationtechniques to fail in a magnetic trap and necessitates much more elaborate methods for thecreation of a chromium BEC. Also a different approach relying on cryogenic buffer gas loadingand evaporative cooling of chromium did not succeed due to large losses [76, 77].

The preparation technique that is finally used to generate a chromium Bose–Einsteincondensate combines magneto-optical, magnetic and optical cooling, pumping, and trappingtechniques [16]. It requires novel strategies, which are adapted to the special electronicstructure of chromium and the need to circumvent relaxation processes that originate from thedipolar character of the atoms.

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2. Theoretical concepts

2.1. Interactions in a Bose gas

2.1.1. Bose–Einstein condensation in a trap. Experiments with Bose–Einstein condensatesof dilute gases are performed in atom traps formed by magnetic or optical potentials thatcan usually be approximated by a three-dimensional harmonic oscillator model Uext =m/

2(ω2

xx2 + ω2

yy2 + ω2

zz2). The energy of a quantum-mechanical state in this potential

characterized by the quantum numbers �n = (nx, ny, nz) is E�n = h(ωx(nx + 1/2) + ωy(ny +1/2) + ωz(nz + 1/2)). In such a trap, the phase-space distribution of non-interacting thermalbosons is

f (�r, �p) = 1

(2πh)3

1

exp(

Uext(�r)+�p2/2m

kBT

) (1)

and the spatial distribution is given by the integral over momentum space

nT (�r) =∫

f (�r, �p) d3p = 1

λ3dB

g3/2(e− Uext(�r)

kB T ). (2)

Here g(3/2) is the polylogarithm or Bose–Einstein integral [78] gα(z) =∑∞j=1

zj

jα .In a similar way, one obtains the momentum distribution

nT (�p) =∫

f (�r, �p) d3r = 1

(λdBmωho)3g3/2(e

− p2

2mkB T ), (3)

where ωho = (ωxωyωz)1/3 is the geometric mean of the trap frequencies.

The total number of atoms is given by the sum of the occupation numbers of all statesNtot = ∑i ni . The set of occupation numbers {ni} of the states |i〉 for indistinguishable,non-interacting bosons is given by the Bose–Einstein distribution function

〈ni〉 = 1

�i

∞∑ni=0

nizni e−βEini = z

∂

∂zln �i = 1

z−1 eβEi − 1= 1

eβ(Ei−µ) − 1, (4)

where �i are the single-particle partition functions

�i =∞∑

ni=0

zni e−βniEi = 1

1 − z e−βEi, (5)

and z = eµ/

kB T = eβµ is the fugacity given by the chemical potential µ that represents theenergy needed to add a particle to the system while all other variables of the system (such astemperature and pressure etc) are kept constant.

For the case of a harmonic potential, the total number of atoms therefore has to followthe equation

N =∑

nx,ny ,nz

1

eβ(Enx ,ny ,nz −µ) − 1. (6)

To provide that (a) 〈ni〉 stays positive for all states and (b) the occupation number 〈n0〉 does notdiverge, which would happen if E0 = µ, follows that µ be smaller than the smallest single-particle energy E0. This leads to the dilemma that for T → 0, all occupation numbers ni andtherefore also Ntot would be 0. If one demands a fixed non-zero number of particles Ntot inthe system, the only way to circumvent this problem is that for T → 0, the chemical potentialµ tends towards E0 in such a way that the lowest single-particle state has a macroscopic butnot infinite occupation. The occurrence of such a macroscopic population of a quantum state

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is called Bose–Einstein condensation. The ground-state population N0 becomes macroscopicwhen the chemical potential approaches the ground-state energy

µ → µC = 3

2hω, (7)

where ω = (ωx +ωy +ωz)/3 is the arithmetic mean of the trapping frequencies. The populationof the excited states is

N − N0 =∑

nx,ny ,nz �=0

1

eβ(ωxnx+ωyny+ωznz) − 1=∫ ∞

0

dnx dny dnz

eβ(ωxnx+ωyny+ωznz) − 1, (8)

where the sum on the left may be replaced by the integral on the right if the level spacinghω is much smaller than the thermal energy kBT , such that these states can be treated like acontinuum. Solving the integral leads to

N − N0 = ζ(3)

(kBT

hωho

)3

. (9)

For a fixed number of indistinguishable bosonic particles, the population of the excitedstates saturates when the temperature approaches the critical temperature for Bose–Einsteincondensation T → Tc which leads to the transition temperature Tc [79]:

T 0c = hωho

kB

(N

ζ(3)

)1/3

≈ 0.94hωho

kB

N1/3. (10)

Below this temperature, the ground-state population grows as

N0

N= 1 −

(T

T 0c

)3

. (11)

The ground state of N non-interacting particles is a simple product state1 ( �r1, �r2, . . . , �rN) =∏j φ0(�rj ), where φ0(�r) is the lowest single-particle state

φ0(�r) =(mωho

πh

)3/4e− m

2h (ωxx2+ωyy

2+ωzz2). (12)

The density distribution

n(�r) = N |φ0(�r)|2. (13)

has a Gaussian shape whose width given by the oscillator lengths σi = 1/√

(2)aho,i =(h

2mωi

)1/2, i = x, y, z and independent of the number of particles. Thus a condensate is always

much smaller than a thermal distribution which has a width of σT = aho,i(kBT /hω)1/2, i =x, y, z. The appearance of a sharp peak in the density distribution is therefore one indicationof the presence of a Bose–Einstein condensate in a trapped gas.

The momentum distribution in the condensate is obtained by integrating (13) in space.The thermal distribution is always isotropic, whereas the condensed particles are localized withan uncertainty of h3 in the phase space. Hence their momentum distribution in an anisotropicpotential is anisotropic, too. The variance σp,i of their momentum distribution in direction i isinversely proportional to the harmonic oscillator length aho,i = √

h/(mωi) in that direction:

σp,i = h√2aho

=√

hmωi/2. (14)

If a gas is released from a non-spherical trap and allowed us to expand freely by a suddenswitch-off of the confining potential, the way it expands is a clear indication of whether it

1 Writing the state of the system as a product of identical single-particle states implies that the state of a particle beindependent of the states of all other particles.

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is a thermal gas or in a Bose-condensed state. After long expansion times, a thermal cloudwill always obtain a spherical shape due to its isotropic momentum distribution whereas acondensate expands anisotropically with a larger momentum in the direction where it initiallyobeyed the stronger confinement.

2.1.2. Critical temperature of a trapped interacting gas. In the case of repulsive interaction,the peak density is reduced and one can expect the transition of the critical pase space densitynpeakλ

3dB > ζ(3/2) to be reached at lower temperatures compared to a non-interacting gas.

Interaction among atoms in the thermal cloud and between thermal atoms and the condensatefraction leads also to a suppression of the thermal density in the centre and to the occupationof higher energy states than without interaction. To estimate the influence of interaction onthe critical temperature, one can use an approach similar to that for trapped ideal gases wherethe atoms additionally interact with the self-consistent mean field such that they are movingin an effective potential [80]:

H = −h2∇2

2m+ Ueff(�r) = −h2∇2

2m+ Uext(�r) + 2gn(�r), (15)

where n(�r) is the total density n = nc +nT . The thermal density can be calculated by replacingUext(�r) in equation (2) by Ueff(�r) − µ. Right at the critical temperature Tc, the total numberof atoms fulfils

N =∫

nT (�r, µc, Tc) d3r (16)

where the critical chemical potential µc is the lowest energy eigenvalue of theHamiltonian (15). The leading contribution to the total energy in a large system stems fromthe interactions; thus it can be approximated by µ0

c = 2gn(0). In [81, 82] Giorgini et al havecalculated the shift δTc = Tc − T 0

c of the critical temperature in the presence of interaction byan expansion of the right-hand side of equation (16) around µc = mu0

c and Tc = T 0c . Including

also corrections that account for the limited size of the sample they calculate the relative shiftof the critical temperature of an interacting trapped gas:

δTc

T 0c

≈ −0.728ωho

ωN−1/3 − 1.33

a

aho

N1/6. (17)

For a simpler representation of the condensate fraction in the presence of interactions, we firstdefine the following two parameters: the first one is the ratio between the chemical potential(equation (27)) of the interacting system calculated with the Thomas–Fermi approximation atT = 0 and the critical temperature T 0

c of the non-interacting model:

ϑ = µ

kBT 0c

= α

(N1/6 a

aho

)2/5

, (18)

where additionally the numerical coefficient α = 152/5ζ(3)1/3/2 1.57 has been introduced.The second parameter is the reduced temperature t = T

T 0c

, i.e. the ratio between the temperatureof the system and the critical temperature of the ideal gas. The temperature dependence of thecondensed number of atoms can be calculated by integrating the distribution function (1) overthe whole phase space:

NT = 1

(2πh)3

∫1

exp[(p2/2m + Ueff(�r) − µ)/kBT ] − 1d3r d3p, (19)

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where one neglects the kinetic term and takes the Thomas–Fermi approximation for theeffective mean-field potential Ueff(�r) − µ ≡ |Uext(�r) − µ|. With the parameters ϑ and t thatwe have introduced above, the condensate fraction of an interacting gas is given by [82]

N0

N= 1 − t3 − ζ(2)

ζ(3)ϑt2(1 − t3)2/5. (20)

2.1.3. Mean-field description of an interacting BEC. In the second quantization, theHamiltonian that describes a system of N interacting bosons trapped in an external potentialis given by

H =∫

d�r �†(�r)[− h2

2m∇2 + Uext(�r)

]�(�r) +

1

2

∫d�r d�r ′�†(�r)�†(�r ′)V (�r − �r ′)�(�r)�(�r ′),

(21)

where �†(�r) and �(�r) are the bosonic creation and annihilation operators at position �r ,respectively. V (�r − �r ′) is the two-body interaction potential and Uext(�r) is the externaltrapping potential.

The equation of motion for the condensate wavefunction is derived by writing down theHeisenberg equation ih ∂

∂t�(�r, t) = [�, H ] with the Hamiltonian from (21). The problem

can be simplified using a mean-field approach (described in detail, e.g., in [82]) by replacingthe field operator � with its expectation value φ(�r) = 〈�(�r, t)〉. To justify this, most of theparticles have to be in the ground state, which means that this approximation can be good onlyin systems far below the transition temperature. In such a fully condensed state, all atoms are inthe same single-particle ground state where φ(�r) is given by the product of the single-particlestates. The classical field φ(�r) is related to the condensate density by n0(�r) = |φ(�r)|2 andoften referred to as the condensate wavefunction.

The relevant interactions in most ultracold gases are collisions which can be described bya single parameter, the s-wave scattering length [83–85], independent of the exact details ofthe interaction potentials. Therefore the interaction term V (�r − �r ′) can be replaced by a δ-likecontact potential

V (�r ′ − �r) = gδ(�r ′ − �r) = 4πh2a

mδ(�r ′ − �r) (22)

with a coupling constant g that is only related to the s-wave scattering length a and mass m.The scattering length a characterizes the range of the interaction. If this range is much smallerthan the average inter-particle distance (n|a|3 1), the total N-particle interaction can berepresented by the sum of all pair interactions and the above treatment is justified. If we usethis contact potential and replace the field operator � by the expectation value, we get thetime-dependent Gross–Pitaevskii equation (GPE)

ih∂

∂tφ(�r, t) =

(− h2

2m∇2 + Uext(�r)gφ2(�r, t)

)φ(�r, t) (23)

which describes the atomic motion in the external field and the molecular field generated by allthe other atoms in the condensate. In the stationary case where the only time dependence is inthe global phase, the time dependence can be separated out and one gets the time independentGPE (

−h2∇2

2m+ Uext(�r) + g|φ(�r)|2

)φ(r) = µφ(�r). (24)

Although in the experiments discussed in this paper, the gases are always dilute and weaklyinteracting systems (n|a|3 1), interactions can contribute significantly to the GPE. The

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average total interaction energy in (24) with an average density n on the order of N/a3

ho isEint ≈ N2ga−3

ho ∝ N2a/a3

ho and the average kinetic energy is on the order of the ground-stateenergy of the harmonic oscillator. Thus Ekin ∝ Na−2

ho . A measure for the ratio of interactionenergy to kinetic energy is then given by Eint

Ekin≈ N a

aho.

2.1.4. The Thomas–Fermi approximation. In typical experiments, a/aho is on the order of1/1000 which means that for atom numbers of 103 and more, interactions play an importantrole. Typical BEC experiments realize condensates containing 100–107 atoms and thechromium condensates that will be discussed in this work contained up to 105 atoms. Hencein most experimental situations, and particularly our chromium condensates, interactions arenot only important but are the dominating contribution to the GPE.

In this case, the contribution of the quantum pressure (kinetic energy) term h2∇2/2m ·√n(�r) in the GPE only plays a role in the very outer regions of the condensate and can be

neglected. This is the so-called Thomas–Fermi (TF) approximation where the GPE simplifiesto

(Uext(�r) + g|φ(�r)2)φ(�r) = µφ(�r). (25)

One finds a direct solution for the condensate wavefunction and density distribution:

nT F (�r) = φT F (�r)2 = max

(1

g(µ − Uext(�r), 0)

). (26)

The relation between the chemical potential, number of particles and trapping frequencies isfixed by the normalization of the density nT F (�r) which yields

µ = hωho

2

(15Na

aho

)2/5

. (27)

This implies that the energy that is needed to add a particle to the system is the same everywhereand is equal to the chemical potential, given solely by the sum of the external potential andinteraction energy µ = Uext(�r)+Eint = Uext(�r)+gn(�r). As a consequence, the condensate hasa sharp boundary where the density vanishes (n = 0). This boundary is given by the conditionUext(�r) = µ that defines the Thomas–Fermi radii of the cloud in terms of trap frequencies andthe chemical potential (27):

Ri =√

2µ

mω2i

, i = x, y, z. (28)

Solution (26) implies that the density profile in the Thomas–Fermi approximation recoversthe inverse shape of the trapping potential with aspect ratios of

Ri

Rj

= ωj

ωi

. (29)

The density n0 in the centre of the cloud, where Uext ≡ 0, is given by the interaction energy

n0 = µ

g. (30)

The TF solution is a good approximation if N |a|aho

� 1 or Eint � Ekin.

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ezey

r

B

0

µ

ex

θ

ϕ

Figure 1. Two dipoles �µ aligned in an external field �B.

2.2. Magnetic dipole–dipole interaction in a Bose–Einstein condensate

The interactions that have been discussed in the previous sections are isotropic and shortrange which permitted us to treat them like delta-like contact potentials in the mean-fieldapproach. The magnetic dipole–dipole interaction that plays an important role in a chromiumBose–Einstein condensate, in contrast, is anisotropic and cannot be treated as a short-rangeinteraction.

Due to these two differences, the presence of significant dipole–dipole interaction changesthe properties of a BEC and also the mathematical treatment dramatically. Due to thelong-range character, not only the local density but the whole density distribution of atomsdetermines the interaction potential of an atom in the cloud.

2.2.1. Interaction potential. Consider two atoms (figure 1) with a magnetic moment�µm = µm�eµ oriented along the direction of an external magnetic field �B (�µm‖ �B). Here�eµ is a unit vector defining the orientation of the magnetic moment. The interaction energy ofthe two dipoles separated by the vector �r is given by

Udd(�r) = µ0µ2m

4πr3

(1 − 3(�eµ�r)2

r2

). (31)

The strength of the dipole–dipole interaction is given by the pre-factor of (31). This strengthcan be compared to the coupling constant g of the s-wave interaction (22) and measured bythe dimensionless dipole–dipole strength parameter

εdd = µ0µ2mm

12πh2a. (32)

It is chosen such that a homogeneous condensate is unstable2 against collapse if εdd > 1 in astatic magnetic field [41]. With the scattering length aCr = 103 a0 of 52Cr [74] and its magneticmoment of µ = 6 µB in the ground state3, the dipole–dipole strength parameter of chromiumis εCr

dd = 0.148. This is much larger than for any other element that has been cooled to quantumdegeneracy so far. The alkalis which form the largest family of Bose–Einstein condensateshave a magnetic moment which is six times smaller than that of chromium; thus the magneticdipole–dipole interaction among them is a factor of 36 smaller than in a chromium BEC. Thecorresponding strength parameters are εdd = 0.0064 for 87Rb and εdd = 0.0035 for 23Na.

2 If the effective interaction is attractive, the condensate collapses [27].3 The value of 103 a0 was obtained from Feshbach resonance measurements in thermal chromium clouds [74].

R102 PhD Tutorial

As has been shown in [58], in general the dipole–dipole interaction Udd can be tuned byusing a time-dependent field to align the dipoles

B(t) = B(cos ϕ �ez + sin ϕ(cos(�t) �ex + sin(�t) �ey)), (33)

where �ex, �ey and �ez are unit vectors defining the x-, y- and z-axis, respectively. The rotationfrequency � of the magnetic field around the z-axis has to be chosen such that the magneticmoments can adiabatically follow the rotation but fast enough for the atomic motion on thetime scale of one rotation to be negligible. In this case, the atoms will feel a time-averageddipole–dipole interaction. This is provided if ωLarmor � � � ωtrap and puts a constrainton the minimum field to be used since the Larmor frequency of the atomic precession isωLarmor = µmB/h. The effective long-range interaction between atoms oriented in such a fieldand separated by the vector �r is then given by

Udd(�r, ϕ) = −µ0µ2m

4π

(3 cos2 ϕ − 1

2

)(3 cos2 � − 1

r3

), (34)

where ϕ is the angle between the magnetic field direction and the z-axis and � is the anglebetween the separation vector r and the z-axis.

Using this method to tune the dipole–dipole interaction strength and additionally utilizinga Feshbach resonance [74] to tune the s-wave scattering length a close to zero will permit usto explore different regimes of dominating short-range or long-range interaction and effectiveattractive or repulsive dipole–dipole interaction.

2.2.2. Trapped dipolar condensates. Due to the long-range character of the dipole–dipoleinteraction, the mean-field potential depends on the whole density distribution of a condensateinstead of only the local density. For a density distribution n(�r), the mean-field potential is[41, 49]

dd =∫

Udd(�r − �r ′)n(�r ′) d3r ′ =∫

Udd(�r − �r ′)|φ(�r ′)|2 d3r ′. (35)

With this additional interaction, the Gross–Pitaevskii equation (23) gets the form

ih∂

∂tφ(�r, t) =

(h2

2m∇2 + Uext(�r) + g|φ(�r, t)|2 +

∫Udd(�r − �r ′)|φ(�r ′)|2 d3r ′

)φ(�r, t). (36)

O’Dell et al have shown in [49] that even under the influence of the dipole–dipole mean-fieldpotential dd(�r), the wavefunction has the shape of an inverted parabola in the Thomas–Fermilimit (compare section 2.1.4). As in the case of pure contact interaction, a wavefunction ofthe form

|φ(�r)|2 = nc(�r) = nc,0

(1 − x2

R2x

− y2

R2y

− z2

R2z

)(37)

is a self-consistent solution of the superfluid hydrodynamic equations [82] derived from theGross–Pitaevskii equation (36), also in the presence of the dipole–dipole interaction [49, 86].Rx,Ry , and Rz are the Thomas–Fermi radii similar to the discussion in section 2.1.4. Theanisotropy of the dipole–dipole interaction manifests itself in a modification of the aspect ratioof the trapped condensate. In the case of an axially symmetric trap, one derives the followingequations for the aspect ratio κ = R⊥/Rz [58, 86]:

λ2 = κ2 1 − εdd

(f (κ) + κ

∂f

∂κ(κ))([3 cos2 ϕ − 1]/2)

1 − εdd

(f (κ) − 1

2κ∂f

∂κ(κ))([3 cos2 ϕ − 1]/2)

(38)

PhD Tutorial R103

Figure 2. (a) Inverted parabolic profile of a BEC in the Thomas–Fermi limit without dipole–dipoleinteraction. The distribution corresponds to a spherically symmetric trap. (b) Saddle-shaped dipolepotential generated by dipolar atoms of a BEC in a spherical trap. The atomic dipoles whichare illustrated as small magnets in the figure are aligned by an external magnetic field B. Note theorientation of the saddle potential relative to the magnetic field direction.

where λ = ωz

ω⊥and f (κ) = 1+2κ2

1−κ2 − 3κ2 tanh−1 √1−κ2

(1−κ2)3/2 . Setting the dipole–dipole interactionto zero (εdd = 0), i.e. notionally ‘switching off’ the dipole–dipole interaction, (38) revealsthe well-known TF relation (29) for the aspect ratio. Given a trap anisotropy λ, one canfind a solution for the condensate anisotropy κ by solving (38) numerically. To understandthe anisotropy of the condensate, we look for the dipole potential (35) that is generated bythe aligned dipoles of the BEC. Let us, for simplicity, assume a spherically symmetric trap(ωx = ωy = ωz ≡ ω0). The more general case of an anisotropic distribution does not changethe qualitative considerations made in the following.

As already discussed, the density distribution of the Bose–Einstein condensate has theshape of an inverted paraboloid as depicted in figure 2(a). Inside the condensate (r < RT F ),the dipole–dipole mean-field potential at a position �r with distance r from the centre of massis given by [49, 58, 86]

insidedd (�r) = εddmω2

0

5

[1 − 3

(�emu · �r|r|)2]

r2 r � RT F . (39)

This equation shows that the potential is harmonic in r but has an angular dependence. Notethat the term in the square brackets varies depending on the angle between the orientation ofthe dipoles �eµ and the position vector �r from −2 to +4 for parallel or antiparallel positionand polarization vectors, respectively. The potential therefore has the form of a saddle witha negative curvature along the direction of magnetization and a positive curvature in thetransverse direction. Outside the condensate, the potential looks like the field generated by Ndipoles that are located in the centre of the distribution and is given by

outsidedd (�r) = εddmω2

0

5

[1 − 3

( �emu · �r|r|)2]

R5T F

|r|3 r > RT F . (40)

The total potential dd(�r) = insidedd + outside

dd (�r) in the y–z-plane is depicted in figure 2(b)where z is the direction of alignment of the dipoles.

The presence of such an anisotropic potential in addition to the isotropic trap has theconsequence that the total energy

Etot = Etrap + Econtact + Edd (41)

can be minimized if the atoms redistribute from regions where dd is positive to regionswhere it is negative. The redistribution leads to a decreasing contribution of the dipole–dipole

R104 PhD Tutorial

energy Edd and a repulsive contact interaction among the atoms prevents the BEC fromcollapsing. Even if the redistribution happens on the cost of increased potential energy andcontact-interaction energy, it will be favoured as long as the total energy is diminished in doingso. Hence the redistribution leads to an elongation along the direction of magnetization and acontraction perpendicular to it.

2.2.3. Expansion dynamics of a dipolar condensate. Now let us consider a dipolar BECreleased from a trap. After switching off the trapping potential, only the contact interaction andthe dipole–dipole interaction contribute to the potential terms in the GPS. Since the interactionenergy will be converted into kinetic energy during expansion, the quantum pressure termhas to be considered, too. The contact interaction part of the mean-field potential revealsthe same parabola shape as with pure contact interaction and the dipole potential dd(�r)still has its (harmonic) saddle shape (see figure 2(b)). Caused by the negative curvatureof this energy term in the direction of magnetization, the gradient of the total mean-fieldenergy (Umf = g|φ(�r)|2 + dd(�r)) in this direction will be larger than without dipole–dipole interaction. Therefore the atoms obey a larger acceleration along the direction ofmagnetization. In the directions perpendicular to the magnetization, the curvature of the dipolepotential is positive. Thus the repulsive contact interaction is weakened by the dipole–dipoleinteraction in the transversal direction and the acceleration perpendicular to the magnetizationis smaller. The general trend of deforming the condensate with an elongation along themagnetization and a contraction in the transversal directions is kept also during the expansionof the condensate.

This behaviour of the expanding condensate is somewhat counter-intuitive because onemight have in mind the simple picture of two parallel bar magnets that repel each other in thedirection perpendicular to the bars. This naive picture would suggest that the cloud expandsfaster in the direction perpendicular to the polarization of the atoms. But since the atoms ina gas can move almost freely, i.e only on the cost of higher potential energy in the trap andrestricted by the repulsive contact interaction, the total energy is minimized by redistributingatoms from the transversal to the longitudinal direction as long as the gain in potential energyand contact interaction energy is small compared to the loss of dipole–dipole interactionenergy. Compared to a non-dipolar condensate, expansion is therefore always faster in thedirection of polarization and the condensate becomes more elongated in this direction duringtime of flight.

To analyse this quantitatively, the expansion has to be treated numerically but with a muchsimpler equation (47) than a three-dimensional nonlinear Schrodinger equation (36) [43, 44,87, 45]. In the Thomas–Fermi limit (section 2.1.4), solutions for the density distribution√

n(�r, t) = φ(�r, t) eiϕ(�r,t) and the velocity field �v that is related to the phase ϕ(�r, t) by�v(�r, t) = (h/m) �∇ϕ(�r, t) have the form

n(�r, t) = |(�r, t)|2 = 15N

8πRxRyRz

(1 − x2

R2x

− y2

R2y

− z2

R2z

)(42)

�v(�r, t) = 1

2�∇(αxx

2 + αyy2 + αzz

2). (43)

According to the hydrodynamic theory of Bose–Einstein condensates [82, 88, 89], thedensity and the superfluid velocity fulfil the continuity equation and the Euler equation

∂n

∂t= −�∇(nv) (44)

PhD Tutorial R105

∂�v∂t

= −�∇(

v2

2+ Uext +

h2∇2√n

2m2√

n+

Umf (�r)m

), (45)

respectively, where the external potential Uext has to be set to 0 to study the expansion, andwhere Umf is the mean-field potential including s-wave and dipole–dipole interactions. Thedensity and velocity are time dependent through the radii Ri(t) and the coefficients αi(t).Because αi = vi

ri, αi(t) are given by

αi(t) = ∂

∂t(log(Ri(t)). (46)

The radii follow the equations of motion

md2Ri

dt2= −7

∂

∂Ri

Htot(Rx, Ry, Rz)

N, (47)

Htot/N being the total energy per particle. Htot = Hkin + Hext + Hmf consists of the classicalkinetic energy (neglecting ground-state motion)

Hkin = 1

14Nm(Rx

2+ Ry

2+ Rz

2), (48)

the energy in the external trapping potential

Hext = 1

14Nm(ω2

xR2x + ω2

yR2y + ω2

zR2z

), (49)

and the mean-field energy which consists of the s-wave scattering and dipole–dipole interactionenergies

Hmf = Hcont + Hdd = 15

7

(N2h2a

m

)1

RxRyRz

(1 − f (Rx, Ry, Rz)εdd). (50)

Here f (Rx, Ry, Rz) is a function that depends on the trap anisotropy and the orientation of theatomic dipoles. Defining the aspect ratios κyx = Ry/Rx and κzx = Ry/Rx and consideringthe case of polarization along x, the function f (κyx, κzx) is given by

f (κyx, κzx) = 1 + 3κyxκzx

E(ϕ\α) − F(ϕ\α)(1 − κ2

zx

)√1 − κ2

yx

, (51)

with

sin ϕ =√

1 − κ2yx, (52)

and

sin2 α = 1 − κ2zx

1 − κ2yx

, (53)

and where F(ϕ\α) and E(ϕ\α) are incomplete elliptic integrals of the first and second kinds,respectively [90]. The equations of motion (47) can now be solved numerically, using (50),(51), and (48) to calculate the derivatives of Htot.

3. Experimental apparatus and detection techniques

3.1. The apparatus

3.1.1. The UHV chamber. All experiments that are discussed in this paper have been carriedout in an ultra-high vacuum (UHV) steel chamber. An outline drawing of the main parts is

R106 PhD Tutorial

Zeeman-Slowerbeam

CCD

imaging system

cloverleafcoils

coolingbeams

ODTbeams

opticalpumping

Zeemanslower

Cr effusion cell1600°C

probebeam

y

x

y

z

I

II

opticalpumping

y

90°

(a)

(b)

Figure 3. Schematic setup of our experiment: (a) whole apparatus, (b) upper chamber seen in thedirection of the probe beam. The horizontal beam I of the dipole trap propagates in the z-directionand carries a power of up to 9.8 W. The vertical beam II defines our y-direction and has a maximumpower of 4.8 W.

depicted in figure 3. Large parts of the experimental setup have been discussed in detail inseveral previous works [91–93]. The vacuum apparatus consists of two chambers: the loweroven chamber and the upper trapping chamber. A beam of chromium atoms is generatedin the lower chamber by a resistively heated high temperature effusion cell operating at1600 ◦C. The pressure in this chamber is held on a 10−9 mbar level by a 75 l/s Ion getterpump during operation of the effusion cell. An 80 cm long Zeeman slower connects the ovento the upper science chamber which contains the magneto-optical, magnetic and optical trapsand where all experiments are performed. This chamber is also pumped by a 75 l/s Ion getterpump and additionally by a thin layer of titanium as a getter material on the inner walls of thechamber which can be refreshed by a titanium sublimator from time to time. The pressure inthe upper chamber stays beyond the range of our ion gauge of <10−11 mbar all the time duringoperation of the apparatus. This low pressure in the region of the trap is a very importantprerequisite to keep the probability of collisions between trapped atoms and molecules fromthe background gas low which allows for long storage time of the trapped atoms.

3.1.2. The magnetic trap. The magnetic trapping potential is generated by a set of water-cooled coils in a cloverleaf configuration [20]. The currents through the coils can be ramped

PhD Tutorial R107

Figure 4. Schematic setup of the laser system used to produce light with a wavelength of425.6 nm for magneto-optical cooling and detection.

up to 300 A to generate a maximum axial curvature of 324 G cm−2 and a maximum radialgradient of 201 G cm−1 resulting in typical trap frequencies of 73 Hz in axial and 800 Hz in theradial direction at 1 G offset field. All currents can be set and switched by our Labview basedcomputer control system [91, 94]. The symmetry axis of the magnetic trap is in the horizontaldirection and defines the z-axis of the experimental setup. The vertical axis is referred to asthe y-axis and the third one as the x-axis. Three extra sets of coils are wound around the bodyof the chamber to produce additional homogeneous magnetic fields (∼2 G A−1 per pair ofcoils) in all three directions. They are separately controllable to compensate for external fieldsand the offset field produced by the magnetic trap as well as to apply the quantization fieldsnecessary for imaging and optical pumping. Fast switching of all magnetic fields is providedby the use of MOSFETs and—particularly for large currents—IGBTs (insulated gate bipolartransistors)4.

3.1.3. Cooling and repumping lasers. The main laser system of the apparatus produces theblue light at 425.6 nm that is needed for optical cooling and imaging on the 7S3 ↔ 7P4

transition. Figure 4 shows the schematic setup. Infrared light with a wavelength of851.1 nm from a Ti:sapphire laser5 pumped by an argon ion laser6 is frequency-doubledby a brewster cut lithium triborate (LBO) crystal in a monolithic ring cavity. The systemis stabilized by Doppler-free polarization spectroscopy of the 7S3 ↔ 7P4 transition in achromium hollow cathode lamp with a lock-in technique to reduce noise. With a pumpinglaser power of 17 W and an infrared power of 2 W, we are able to produce 800 mW of bluelight. In the normal experimental operation, we limit the blue power to ∼500 mW becausehigher intensities lead to thermal lens effects in the crystals of the acousto-optical modulators(AOM) used to adjust frequencies and intensities.

Atoms can decay from the excited state 7P4 of the cooling transition to the metastable 5D4

and 5D3 states in which they are decoupled from the cooling light. To transfer the atoms in 5D4

back to the ground state via the 7P3 state, we use a 5 mW external cavity7 diode laser system

4 Semikron.5 Coherent, MBR110.6 Coherent, Sabre R 25 TSM.7 Littrow configuration [95].

R108 PhD Tutorial

l/2l/2

fibrefibre

retretardationardationplateplate

polarizingpolarizingbeam splitterbeam splitter

lenslens

mirrormirror

beam dumpbeam dump

AOM IIAOM II AOM IAOM I

beam IIbeam II

beam Ibeam I1st order1st order 1st order1st order

z-axiscooling beamz-axiscooling beam

opticalpumpingopticalpumping

yy

zz

λ/2

Figure 5. Outline of the optical dipole trap setup. Beam I is in the horizontal (z-)direction, BeamII is in the vertical (y) direction.

[96] resonant with the 5D4 ↔ 7P3 transition at 663.2 nm. A second laser system with identicalsetup is available for the 5D3 ↔ 7P3 transition at 654 nm but is usually not used because thegain in atom number is small compared to the previous one and not worth maintaining thissystem. These lasers are locked to the modes of a Fabry–Perot resonator close to the transitionwavelength using the Pound–Drever–Hall sideband modulation scheme [97]. The cavity ismade of Zerodur and Invar—which have very low thermal expansion coefficients—and it hasa free spectral range of 75 MHz. A small part of the light is used for locking and can beshifted by double-pass AOMs, such that the output of the laser systems is on resonance withthe atomic transitions. The thermal drift of the cavity is about 2 MHz h−1.

All lasers can be switched on and off by mechanical shutters and the cooling and imaginglight can additionally be dimmed and tuned in a range of ±10� around resonance by AOMs.

3.1.4. Optical dipole trap. The laser that is used to generate the light for the optical trap isan Yb-fibre laser8 which operates at a wavelength of 1064 nm and which provides a maximumpower of 20 W. This laser has a linewidth of �λ = 1.65 nm, corresponding to a coherencelength of λ2/�λ = 0.7 mm which prevents the formation of a standing wave in the opticaltrap. The optical setup that is used to prepare two TEM00 mode beams and to focus them tothe centre of the experimental chamber is depicted in figure 5. The light coming from thelaser is collimated to a diameter of 4.7 mm by the head of the fibre. The linear polarizationof the light allows one to split it up into two beams with adjustable intensity ratios using apolarizing beam splitter and a λ/2-plate. Acousto-optical modulators in both optical beampaths allow for a precise, independent control of both laser intensities and the possibilityof a rapid switch-off for time-of-flight imaging with a suppression of the laser intensity of∼45 dB. Two telescopes in each of the beam paths are used to expand the beams before theyare focussed by two f = 500 nm lenses with 5 cm diameter. We focus the main beam to awaist of 29.5 µm and shine it in horizontally through the centre of the cloverleaf coils of the

8 IPG PYL-20M-LP.

PhD Tutorial R109

Slave DiodeSlave Diode855nm855nmSDL120mWSDL120mW

MasterMasterDiodeDiode855nm855nmSDLSDL120mW120mW

Cavity FSR=75MHzCavity FSR=75MHz

ExperimentExperiment

SHGSHGRing CavityRing Cavity

KNBO crystKNBO crystalal

AOMAOM

double pdouble passassAOMAOMDiodeDiode

current modulationcurrent modulation25MHz25MHz

PZTPZT

--PIPI

PIPI

Hänsch-CoulliaudHänsch-Coulliaudlockinglocking

l/2l/2

l/2l/2

l/4l/4

fibrefibre

fibrefibre

photodiodephotodiode

Pound-Drever-Hall lockingPound-Drever-Hall locking

xx

70mW70mW

6mW6mW427.6nm427.6nm

PZTPZT

seedseed

Wavemeter

opt.opt.isolatorisolator

anamorphicanamorphicprism pprism pairair

retretardationardationplateplate

polarizingpolarizingbeam splitterbeam splitter

lenslens

mirrormirror

Figure 6. Schematic setup of the frequency-doubled master-slave diode laser system used togenerate the 427.6 nm light for optical pumping.

magnetic trap. It usually carries optical powers of 9 W to 12 W. The second beam is shone inthrough a viewport from below the chamber in the vertical direction. It is focused to 50 µmand is used to form a dimple in the potential of the horizontal beam (see section 4.1). Usualoperation of this beam is at 4-5 W. In the centre of the chamber, the two beams cross underan angle of 90◦. The two final mirrors before the chamber are equipped with coatings that arehighly reflective for the 1064 nm trap light and highly transmissive at 425 nm. In this way,the horizontal MOT beam and the vertical pumping beam on the 7S3 → 7P3 transition can beshone in from behind these two mirrors.

3.1.5. Laser system for optical pumping. Optical pumping (see section 4.3) of the atomsfrom mJ = +3 to mJ = −3 on the 7S3 ↔ 7P3 transition requires light with a wavelength of427.6 nm. To generate this light, an additional laser system was set up in the framework ofthis thesis. It is based on an injection-locked master-slave [98] diode laser system operatingat twice the blue wavelength (855.2 nm). This infrared light is frequency doubled by apotassium-niobate (KNbO3) crystal [99, 100] in a home-made monolithic ring cavity. Aschematic representation of the laser setup is found in figure 6.

Once the laser is on resonance (indicated by a decrease of the number of atoms whenwe expose the MOT to the pumping light), the grating of the master diode is locked to thenearest lock point of the Pound–Drever error signal of the same reference cavity on which alsothe repumping lasers for the metastable states are locked. Due to the free spectral range of75 MHz, this lock point is at most 37.5 MHz (in the infrared light) away from resonance. Theslave laser is shifted back on resonance by a double-pass acousto-optical modulator (AOM)between the master and slave diodes. With about 70 mW of infrared power coming from theslave diode, the system produces up to 6 mW of blue light at 427.6 nm. However, for thepumping only a few hundred microwatt are needed.

3.2. Absorption imaging

Throughout the experiments presented here, the method used to image the atomic cloudis absorption imaging. In this technique, the cloud is illuminated with a probe beam of

R110 PhD Tutorial

/4/4

L1L1

BB

A2CCDCCD A1A1

FCFC

L2L2

MM

/2/2

PBSPBSyy

xx

λ

λ

Figure 7. Outline of the imaging system. FC: Fibre coupler; PBC: polarizing beam splitter;λ/2, λ/4: wave plates; M: mirror; L1, L2: lenses used to expand the probe beam; A1, A2achromats f = 300 mm; CCD: camera.

resonant light in the direction of the imaging system (see figure 7) and casts a shadow on thephotosensitive chip of a CCD (charge coupled device) camera.

While passing in the x-direction through a cloud of atoms with density distributionn(�r), the intensity I (�r) of the probe beam with frequency ωp is reduced by dI (�r) =−σ(ωp)n(�r)I (�r) dx, where σ(ωp) is the photon scattering cross section. Integration along xyields:

I (�r) = I (x, y, z) = I0 e−σ(ωp)∫ x

−∞ n(x ′,y,z)dx ′(54)

which is equal to Beer’s law I (x) = I0 e−ρOx for constant density n where ρO is the opticaldensity. As a result, the intensity profile I (x, y, z) of the probe beam after crossing the cloudcontains information on the column density

n(y, z) =∫ ∞

−∞n(x ′, y, z) dx ′ = 1

σ(ωp)ln

(I0(x, y)

I (x, y)

)(55)

3.2.1. Imaging system. The setup of our imaging system is outlined in figure 7. Imagingis performed in horizontal (x-)direction. Two identical 2 inch achromatic lenses with a focallength of 300 mm and a measured magnification of 1.0 are used to map the density distributiononto the photosensitive chip a CCD camera9. Fast switching is provided by a double-passAOM in front of the fibre coupler. A λ/4-wave is used to generate σ+ or σ -light, dependingon which Zeeman state has to be detected. The spatial resolution of the imaging systemdetermined using a standard USAF-1951 test target10 is ∼8 µm—just above the diffractionlimit.

A small homogeneous magnetic field is applied along the direction of the probe beam toalign the atomic dipole moments for the detection of both, mJ = +3 or mJ = −3 atomsin the ground state. Its magnitude corresponds to a detuning of ±2.5 � from themJ = ±3 → mJ = ±4 transition, respectively. In both cases, the σ -polarized light pumpsthe atoms to the extreme Zeeman state mJ = ±3 from where they can only be excited tomJ = ±4. We therefore treat them as two-level atoms.

3.2.2. Extracting data from images. Because the extension of the trapped cloud is close toor even below the resolution of our imaging system, all images are taken after some time of

9 PCO Pixelfly QE.10 Newport RES-1.

PhD Tutorial R111

flight. Such an image not only monitors the initial distribution of atoms in space, but, as thetime of flight gets longer, it is more and more determined by the momentum distribution or, inthe Thomas–Fermi limit, by the interactions.

The total number of atoms is determined by fitting a distribution function to the densityprofile on the image and integrating this function over the whole image plane. The temperatureof the cloud is determined from the width of the respective distribution. Depending on whetherthe cloud is purely thermal, partly condensed or a pure condensate, the proper distributionfunctions have to be used for the fits.

Thermal clouds far above Tc. If interaction between the particles can be neglected, which isusually the case in thermal clouds, the velocity distribution is kept constant also after releasingthe cloud from the trap. The density distribution of the cloud after a time t of free flight istherefore given by the convolution n(�r, t) = ∫ n(�r − �vt, t = 0)nv(|�v|) d�v, where (n(�r) is thespatial distribution and nv(|�v|) is the (isotropic) velocity distribution. The resulting Gaussianhas a standard deviation [101] of

σi(t) =√

σi(0)2 +kBT

mt2 =

√kBT

mω2i

+kBT

mt2, i = x, y, z. (56)

and the column density distribution (55) recorded on the camera chip is given by

nT (y, z) = nT

√2πσx exp

(− (y − y)2

2σ 2y

− (z − z)2

2σ 2z

)(57)

which is used to fit the image. The fit delivers the number of atoms NT = (2π)3/2nT σxσyσz

and the temperature T = mkB

σi (t)2

1ω2i

+t2 , where i denotes one of the visible directions x, y or z.

More accurate measurements of the temperature can be performed by fitting (56) to thewidths obtained from a series of pictures of clouds (prepared in the same way and releasedfrom the same trapping potential but after different times of ballistic expansion). For timesof flight larger than the inverse trapping frequency ttof > 1/ω, the cloud expands linearly andhence the temperature can be obtained from only the expansion velocities.

Systems close to Tc and clouds containing a condensate. In clouds still above but close tothe critical temperature, the difference between the Maxwell–Boltzmann and Bose–Einsteindistributions starts to be important. At temperatures close to Tc, the enhanced occupationprobability of lower lying states in the harmonic trapping potential has to be regarded whenfitting the density distribution. It has been shown by Bagnato et al in [79] that in the ideal gaslimit, at temperatures much higher than the level spacing kBT � hω, a semiclassical approachcan be used to approximate the spatial distribution resulting in

n(�r) = 1

λ3dB

∞∑j=1

ej (µ−U(�r))

kB T

j 3/2= 1

λ3dB

g3/2

(e

(µ−U(�r))kB T

). (58)

Integration along the x-direction yields

nT (y, z) = nT

√πx0g2

(e− y2

y20− z2

z20

). (59)

To evaluate images by fitting the above formula to the data, we approximate the polylogarithmgα(z) = ∑∞

j=1zj

jα by carrying out the summation up to j = 20. The g2 function that isused in this equation assumes a chemical potential of µ = 0 (i.e. a fugacity z of 1). Besidesthe fact that this is not true for partly condensed clouds, this simplification neither includes

R112 PhD Tutorial

interaction of the thermal cloud with itself nor with the much denser condensate fraction inthe centre of the trap. Hence it is not suited to fit the temperature of the thermal cloud [102].We therefore use (57) to measure the temperature, restricting the fit to the far outer wings ofthe thermal cloud where the chemical potential is much smaller than the kinetic energy andBose enhancement is low [19]. The number of atoms however is obtained from the fit of (59).It still contains a systematic error due to the above-mentioned neglect of interaction.

The resulting best fit is subsequently subtracted from the absorption image and we fit thecolumn density of the Thomas–Fermi distribution (26),

nc(y, z) = max

4

3ncRx

(1 − y2

R2y

− z2

R2z

)3/2

, 0

, (60)

to the remaining part of the image to determine the size of the condensate. Here we assume thatconsidering the condensate as being in the Thomas–Fermi limit is always a good approximationsince the cloud usually contains a sufficiently large number of atoms in all experimentsdiscussed here.

4. Trapping chromium atoms

4.1. Chromium atoms in an optical dipole trap

One of the crucial factors that made Bose–Einstein condensation of chromium possible is thestep from a magnetic to a purely optical trapping potential. This technique allows one totrap atoms in the energetically lowest Zeeman state where dipolar relaxation is suppressed.The optical trap is loaded by superimposing the optical trap potential with the magnetic trapand cooling the trapped sample in this hybrid optical/magnetic trap to temperatures belowthe depth of the optical trap using radio frequency (rf) evaporation. A large part of thesample is transferred to the pure optical trap by ramping down the magnetic trapping potentialadiabatically subsequent to rf cooling. In this way, the atoms that were magnetically trappedbefore, slowly expand into the optical trapping potential.

The trap is formed by two crossing beams produced by a Yb doped fibre laser that delivers20 W of infrared light with a wavelength of 1064 nm (see section 3.1.4). The light is detunedfar to the red from any transition from the ground state 7S3 of chromium. In this regime, wherethe scattering rate of photons is much smaller than the spontaneous decay rate �sc �,a classical treatment of the dipole force describes the trapping mechanisms well. A nicediscussion of such traps can be found in [103, 104] by Grimm, Weidemuller and Ovchinnikovwhere they present a detailed review of theoretical and experimental aspects of optical traps.

Real atoms possess many electronic transitions between energy levels which can have acomplex fine and hyperfine sub-structure. In such multilevel systems, the shift of a groundstate |gj 〉 in the laser field depends on all dipole matrix elements between the ground state

and the excited states |ei〉�Ej = |〈ei |µ|gj 〉|2δij

I2πε0c

. However, for the realization of optical traps,usually far detuned lasers are used to prevent heating by keeping the photon scattering ratelow. In this very far detuned case, where the detuning of the trap laser from any transitionis much larger than the fine structure splitting (δ � �FS), can the fine, hyperfine, andthe magnetic sub-structure be ignored [104]. The multilevel problem reduces to summingthe contributions of the different electronic states that couple to the ground state withoutconsidering the substructure. Each of the possible transitions can be treated like a two-level

PhD Tutorial R113

a7S3

z7P

z5P

y7P

y5P

x7P

w7P

2

3

2

1

2

2

3

2

3

2

3

3

4

1

4

3

4

4

excitedstate

transitionwavelength λj

line width

210nm

236nm

336nm

360nm

373nm

427nm

1.1x10 6 1/s

5.5x10 6 1/s

0.12x10 6 1/s

150x10 6 1/s

0.16x10 6 1/s

31.5x10 6 1/s

Γ

Figure 8. Transitions contributing to the ground state 7S3 trap potential of chromium atoms in theoptical trap. Most important are the a7S ↔ y7P transition at 359 nm and the cooling transitiona7S ↔ z7P at 427 nm (numbers taken from [105])

system with the same transition strength, independent of the laser polarization.The resultingdipole potential Udip of the ground state and the total photon scattering rate �sc are

Udip =∑

j

3πc2�j

2ω3j

(1

ωj − ω+

1

ωj + ω

)I (�r, t) (61)

and

�sc =∑

j

3πc2�j

2hω3j

(ω

ωj

)3 ( 1

ωj − ωi

+1

ωj + ω

)I (�r, t). (62)

Here ωj is the frequency of a certain transition from the ground state, ω is the frequency ofthe laser field, and �0 is the average linewidth of the fine structure levels of the respectivetransition. These equations give good approximations under most realistic conditions of dipoletraps where the detuning is large and saturation effects can be neglected. The decay rates �0

and transition frequencies ω0 of the transitions that have to be considered are best obtainedfrom spectroscopic data and can for example be found in the NIST database [105].

The relation between scattering rate and dipole potential in far off resonant traps is verysimple,

�sc = �

hδUdip, (63)

showing that in a trap with the same depth, the scattering rate can be reduced by increasingthe detuning. Therefore, optical dipole traps are preferably operated rather at large intensitiesthan small detunings.

Figure 8 shows the properties of the transitions that have to be taken into accountto calculate the trap parameters for ground state 52Cr atoms [105]. All transitions are atwavelengths below 427 nm. The transitions, that contribute significantly to the trap potentialof chromium are the a7S ↔ y7P transition at 359 nm and the cooling transition a7S ↔ z7Pat 427 nm. All other transitions are at least two orders of magnitude weaker and even farther

R114 PhD Tutorial

detuned from resonance in the 1064 nm laser field. The resulting dipole potential and photonscattering rate are

Udip(I ) = −0.273 × 10−36Jm2

W 2· I, �sc(I ) = 0.533 × 10−11 1

s

m2

W 2· I (64)

The heating rate due to photon scattering is

T (I ) = 1.79 × 10−18 K

s

m2

W 2· I. (65)

4.1.1. Trap geometry. In a Gaussian beam, the dipole trap potential is given by

Udip(r, z) = U01

1 +(

zzR

)2 exp

(−2r2

w20

1

1 + (z/zR)2

), (66)

where r and z are the radial and longitudinal positions with respect to the focus of the beam,zR is the Rayleigh range and w0 is the minimum beam waist of the laser field. The trap depthU0 is the potential energy in the focus of the beam where the intensity of the light field is I0.If the thermal energies kBT are much smaller than the potential depth U0, most of the atomsreside in the central region of the potential. The extension of the cloud is much smaller thanthe waist in the radial direction and the Rayleigh range in the axial direction. In this region,the optical potential in (66) is best approximated by a three-dimensional harmonic oscillatorwith an axial symmetry along the optical axis:

Utrap(r, z) = −U0 +m

2

(ω2

r r2 + ω2

zz2). (67)

where ωr =√

4U0

mw20

and ωz =√

2U0

mz2R

are the harmonic oscillation frequencies in the radial

and axial directions, respectively. Typically, the radial trap frequencies of optical traps arerelatively high compared to magnetic traps but the depth is much smaller. Another differenceto magnetic traps is that trap depth and frequencies cannot be varied independently11.

4.2. Loading the optical trap

4.2.1. Preparation of a cold cloud in the magnetic trap. The high magnetic moment in theground state as well as in the metastable states of chromium allows for a continuous loadingscheme (CLIP-trap) [71, 69] of magneto-optically cooled atoms directly into a magneticIoffe–Pritchard (cloverleaf) trap. The loading procedure is illustrated in figure 9. After beingslowed down by a Zeeman slower, the atoms are laser cooled and trapped in a two-dimensionalmagneto-optical trap (MOT) in the radial direction of the Ioffe-Pritchard trap. The curvedfield along the trap axis does not allow a usual MOT operation. Therefore the atoms are onlycooled in this direction using a σ +–σ + optical molasses. For both the MOT and the molasses,the 7P4 ↔ 7S3 is used. Due to the branching ratio of �P→S/�P→D = 250 000:1 betweentransitions from the excited 7P4 to the 7S3 ground state or the long lived 5D4 and 5D3 statesan atom undergoes on average 250 000 cycles on the cooling transition before it eventuallydecays to one of the metastable states. These states have a magnetic moment of 6 µB and4 µB , respectively. Without the repumping laser, the metastable states are decoupled from thecooling cycle and atoms which end up in a low-field seeking sub-state12 of these states stay

11 This is effectively the case in magnetic traps, since it is possible to limit the trap depth with a radio frequency shieldbut keep the trap frequencies constant [20].12 Low-field seeking states or ‘low field seekers’ are Zeeman states that have lower Zeeman energy EZ = gF mF µB B

in lower magnetic fields, i.e. gF mF > 0.

PhD Tutorial R115

MOTMOT425nm425nm

|g> == SS7733 |g> == SS77

33 |g> == SS7733

|e> == PP7744 77

33PP

|d> == DD5544

|d> == DD5544

magnetically trappedmagnetically trapped magnetically trappedmagnetically trapped

magnetically trappedmagnetically trappedin ground statein ground state

repumperrepumper663nm663nm

ΓΓegeg ΓΓeded

250000250000::11

44 BBµµ

66 BBµµ

(a) (b) (c)

Figure 9. The continuous loading scheme consists of three stages: (a) Loading: accumulationof 1.3 × 108 magnetically trapped cold atoms in the metastable D-state due to a branching ratioof 250 000 : 1 between transitions to the ground state and the metastable state. (b) Repumping:the MOT light is switched off and the atoms are pumped to the ground state via the 5D4 → 7P3transition. (c) Ground-state trap: all lasers off, atoms are trapped in the ground state.

trapped in the shallow magnetic potential (radial gradient13 B ′ = 9.5 G cm−1, axial curvatureB ′′ = 7.5 G cm−2) used during the loading stage. With this technique we accumulate about1.3 × 108 magnetically trapped atoms at roughly the Doppler temperature [106] of 124 µKand a phase-space density (PSD) of a few times 10−9 within a loading time of 10 s. Thenumber of magnetically trapped atoms in the metastable states is limited due to light inducedcollisions [70]. After the steady-state number of atoms in the magnetic trap is reached, thecooling lasers are switched off and the atoms in the 5D4 state are pumped back to the groundstate within 20 ms using laser light resonant with the 5D4 ↔ 7P3 transition at 663.2 nm (seesection 3.1.3). The magnetic trap is subsequently being fully compressed by ramping up thecurrents through the trap coils to 300 A. In doing so the cloud is heated up again to ∼1 mKand Doppler cooling of the optically dense cloud [73] is performed within the trap at an offsetfield of 14 G using the cooling beam propagating in z-direction. This Doppler cooling stepincreases the phase-space density by two orders of magnitude to ∼10−7 without losing atoms.Subsequently, the currents through the coils are lowered to form a trap with an aspect ratio asclose as possible to the later shape of the optical dipole trap to provide for efficient transferbetween the traps.

4.2.2. Radio-frequency evaporation. In this trap, we perform radio-frequency (rf)evaporation [107, 108] where high energetic atoms are selectively removed from the trapby transferring them to untrapped Zeeman states when they come into resonance with a radio-frequency field while moving to the outer high-field regions of the magnetic trap. To reducethe temperature gradually, we sweep the rf frequency in 3 linear ramps from 45 MHz downto 1.25 MHz. After this step, the cloud contains 4.5 × 106 to 5 × 106 atoms at a phase-spacedensity of 10−5 and a temperature of 22 µK. The overall gain in phase-space density duringthe magnetic trapping phase is thus about four orders of magnitude whereas the number ofatoms is reduced by 1.5 orders of magnitude.

4.2.3. Transfer into the optical trap. Beginning from the first rf-ramp, the magnetic trap issuperimposed by a single-beam ODT in the horizontal direction with the same symmetry axisas the magnetic trap. This trap is formed by beam I in figure 5 and figure 3(b) with a maximumpower of 9.8 W. The energy cutoff in this hybrid trap is still given by the resonance position ofthe rf field. In contrast to the radial direction, where the optical trap is much steeper than themagnetic trap, the axial confinement is still dominated by the magnetic potential. Thus atoms

13 In addition, the trap produces a magnetic offset field B0 �= 0 in the centre of the trap that leads to a harmonicpotential in the radial direction, in a certain range around the trap centre, too [20].

R116 PhD Tutorial

(a)

00 55 1010 151500

0.50.5

11

1.51.5

22

2.52.5

33

3.53.5

44

atom

num

berN

(t)

timet [s]

BB00~0.8 G~0.8 GBB00~0 G~0 G

x10x1055

(b)

Figure 10. (a) Image of the hybrid magnetic/optical trap. The beam of the optical trap wasintentionally misaligned for this image to show both traps. (b) Dipolar relaxation of 52Cr atomsmeasured in an optical dipole trap. Two lifetime measurements were performed at differenthomogeneous magnetic fields (B0 ≈ 0.8 G and B0 ≈ 0 G). The lifetime of the trapped sampleclearly depends on the magnetic field.

need a larger energy to reach the region where the rf-field is resonant in radial than in theaxial direction. If the temperature of the atoms is comparable or smaller than the depth of theoptical trap, evaporation can only happen in z-direction and the cooling efficiency becomesweak.

We adjust the magnetic offset field during the hybrid trapping phase to a few tens ofMilligauss by applying some extra current in the bias coils. A low offset field in the centreleads to a radial compression of the magnetic trap and therefore improves the mode matchingof the magnetic and the optical traps. The final frequency of the rf-sequence is chosen suchthat the highest number of atoms is transferred adiabatically to the single-beam optical trap byramping down the magnetic trapping potential within 100 ms. We achieve a transfer efficiencyof 40% with 2 × 106 atoms in the ODT. The phase-space density at this stage is 5 × 10−5, afactor of five higher than the phase-space density that is found in the pure magnetic trap afterthe final rf-ramp without an optical trap present.

4.3. Transfer to the lowest Zeeman state

Even at a very low magnetic offset field, dipolar relaxation leads to a redistribution of theatoms among the Zeeman sub-states and to heating. A heating rate of 3 µK s−1 at an offsetfield of 0.7 G, and a relaxation rate constant on the order of βdr = 1 ×10−12 cm−3 at 0.15 Ghave been measured in an optical dipole trap [92]. Figure 10(b) shows the field dependenceof dipolar relaxation in the results of two lifetime measurements in the optical trap at themagnetic offset fields of 0.8 G and ∼0 G, respectively.

The only way to suppress this relaxation process completely is to bring all atoms to theenergetically lowest Zeeman state in a magnetic offset field B0 � kBT /µ. Relaxation fromthis state to higher Zeeman states is energetically forbidden if the atoms do not have enoughkinetic energy to bring up the difference in Zeeman energy that is needed to occupy sucha level. We use optical pumping to polarize the atoms in the lowest Zeeman state. Whenthe trapped atoms are exposed to pure σ− light resonant with the 7S3 ↔ 7P3 transition ata wavelength of 427.6 nm , the energetically lowest mJ = −3 state is a dark state. Theσ− light drives �mJ = −1 transitions from |g,mJ 〉 to |e,mJ − 1〉 from where the decayhappens to |g,mJ − 1〉, |g,mJ 〉 or |g,mJ + 1〉 with a probability given by the correspondingClebsch–Gordan coefficients as depicted in figure 11(a). Atoms that end up in |g,−3〉 remain

PhD Tutorial R117

73S73S gJ=2.0018

73P73P

gJ=1.91 7

--

mmJJ

9/129/12

4/124/121/121/12

00

1/121/12

4/124/12

9/129/12

3/123/12

5/125/12

6/126/12

6/126/12

5/125/12

3/123/12

00-1-1-2-2-3-3 +1+1 +2+2 +3+3

3/123/12

5/125/12

6/126/12

6/126/12

5/125/12

3/123/12

(a)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time [s]

rela

tive

num

ber

of a

tom

s

mJ=+3 experimental

mJ= 3 theory–

states mJ=[0,±1,±2]mJ=+ 3 theory

mJ= 3 experimental–

(b)

σ

Figure 11. (a) Magnetic substructure of the 7S3 ↔ 7P3 transition. The σ− polarized light drivesthe atoms which are initially mainly in the energetically highest state mJ = +3 to the lowest statemJ = −3. The numbers close to the transitions are the squares of the Clebsch–Gordan coefficients.(b) Time-dependent occupation numbers of the Zeeman sub-states of 7S3 during optical pumping.

unaffected by the light field because there is no �mJ = −1 transition. A magnetic field of11.5 G along the axis of the pumping beam requires a detuning of 32 MHz to the red withrespect to the resonance at the zero magnetic field. The relative detunings for �mJ = 0and �mJ = +1 transitions induced by this field are 2.3� and 6.2� if the laser is tuned onresonance with the �mJ = −1 transition. This helps to prevent transitions other than thedesired �mJ = −1 which are not fully suppressed if the polarization of the beam is not purelycircular. We use 0.5 mW of σ−-polarized light for 1 ms in the optical dipole trap to pump theatoms to mJ = −3 immediately after the magnetic trap is switched off.

Figure 11(b) shows numerical solutions of the occupation numbers of the different mJ -states during optical pumping together with the experimental data. To be able to resolve thetime dependence in the experiment, the optical power in the beam was reduced to only a fewµW such that 300 ms were needed to transfer the atoms to mJ = −3. To distinguish atoms inmJ = +3 from those in mJ = −3 in absorption images, the probe beam was blue (red) detunedand σ+ (σ -) polarized to detect mJ = +3 (mJ = −3) atoms, respectively. The numericalsimulation also includes the rate of the very fast decay that is observed in the beginning ofthe optical trapping phase. When we use 0.5 mW of laser light, the efficiency of the transferis close to 100% which is reflected in a dramatic increase of the lifetime of the trapped cloudfrom ∼6 s in mJ = +3 to > 140 s in mJ = −3 as shown in figure 12. This long lifetime isa very important precondition for efficient evaporative cooling in the optical dipole trap. Toprevent later thermal redistribution of the spins, the magnetic offset field of 11.5 G used forpumping is kept on during all further preparation steps.

5. Evaporative cooling

The phase-space density (PSD) of the chromium sample after the transfer into the optical dipoletrap is ∼5 × 10−5. To increase the PSD, we use evaporative cooling in the optical dipole trap.Radio frequency (rf) induced evaporative cooling of chromium atoms in a magnetic trap hasalready been demonstrated in earlier works [92]. The principle of evaporative cooling isthe withdrawal of atoms from a sample which carry more than the average energy. Thusthe average energy per atom in the remaining sample is lower than before. Evaporativecooling has been discussed already 20 years ago by Hess [109] and was demonstrated with

R118 PhD Tutorial

0 20 40 60 80 100120140160 300

105

106

time [s]

num

ber

of a

tom

s

=6.3smJ=+3

JJ

N(N( =10=105300s)300s)=142s=142s

mm=- 3=- 3

τ

τ

Figure 12. Comparison of trap lifetimes in the single-beam ODT before (crosses) and after (circles)pumping the atoms to the lowest Zeeman substate. Solid lines are fits of an exponential decayN(t) = N0 exp(−γ t) to the data where γ is the decay rate.

0 20 40 60 80 100 120 140 160 300

105

106

num

ber

of a

tom

s

=142s=142smJ =- 3=- 3

0 50 100 1500.50.60.70.80.91.01.11.21.4

1/1/ -width-width

time [s]

zz[mm

][m

m]

N(N( =10=105

300s)300s)

time [s]

e

(a)

105 106

10-4

10-3

number of atoms

phas

e sp

ace

dens

ity

0.010.01 0.10.1 11 1010 100100

1010-4-4

1010-3-3

holding time [s]holding time [s]

phas

esp

phas

esp

ace

dens

ityac

ede

nsity

(b)

σ

τ

Figure 13. (a) Lifetime measurement in the single-beam ODT after pumping the atoms to thelowest Zeeman substate. The inset shows the change of the axial size of the expanded cloud intime-of-flight images depending on the time it lived in trap. (b) Double logarithmic plots of thephase-space density versus number of remaining atoms and versus holding time in the single-beamtrap (inset). Arrows mark the chronology in the plots. An open circle marks the optimum point intime to start forced evaporation in both plots.

magnetically trapped hydrogen in 1988 [110]. In 1994, rf induced evaporative cooling wasdemonstrated with sodium and rubidium atoms by the groups of Cornell and Ketterle [107,108] who managed to generate the first BECs in dilute gases one year later using exactly thistechnique. In our experiment, the evaporative cooling technique permits us to cool the samplein the dipole trap from an initial temperature of 50 µK to temperatures around 100 nK. Thegain in phase-space density is about four orders of magnitude, while one order of magnitudein the number of atoms is lost during evaporation.

5.1. Plain evaporation

During the first seconds in the ODT, a very fast decay of the number of atoms is observedwhich becomes slower after about half of the atoms are lost (see figure 13(a)). Along withthis decrease of the number comes a very large increase in the phase-space density which was

PhD Tutorial R119

1mm

1mm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

optic

al d

ensi

ty

Figure 14. False colour image of atoms 1.2 ms after release from the trap where the vertical beamhas been ramped up during plain evaporation. White bars in the figure indicate the length scalesof the two directions.

determined from time-of-flight pictures taken during the lifetime of the trap. As two-body lossprocesses are suppressed in the lowest Zeeman state and there is also no hint for three-bodyloss in the regime of relatively low density at this stage of optical trapping, the observedloss can be attributed to essentially pure plain evaporation. Figure 13(b) shows the evolutionof the phase-space density during the first 120 s in the ODT plotted versus the number ofremaining atoms and versus time. The straight line in the main graph has a slope of 3.6 ordersof magnitude gain in phase-space density per lost order of magnitude in the number of atomsand is a linear fit to the data. This illustrates the very high efficiency of the plain evaporationstage. After 5 s it gets less efficient which is identifiable by the data points in figure 13(b)snapping off from the straight line. During plain evaporation the temperature of the trappedcloud drops from an initial value of 60 µK to 14 µK at the end. After about 10 s in the dipoletrap, the change of the temperature is so small that it can no longer be measured. Even after300 s in the trap, when still more than 100 000 atoms are trapped, we find the same temperaturewithin error bars. It corresponds to a cutoff parameter of η = U0/kBT = 9.6.

5.2. Cooling in the crossed configuration

The collision rates in the optical trap formed by only one beam are not high enough to supportevaporative cooling under the starting conditions that we have. To increase the density andthe collision rate, we increase the power of the vertical trapping beam (figure 3(b)) during theplain evaporation stage linearly from 0 W to ∼4.5 W. This creates a dimple in the centre ofthe trapping potential. Such an adiabatic deformation can be viewed as changing the powerlaw of the trapping potential and thus it leads to an increased density and phase-space densityas suggested and experimentally demonstrated by Pinkse et al [111]. The principle has sincethen been used successfully for the reversible generation of Bose–Einstein condensates in ahybrid magnetic/optical trap [112] and for Bose–Einstein condensation in optical traps [7].Stamper-Kurn explains this effect by a two-box model in [102]. For a crossed dipole trap withchromium atoms, an increase of the phase-space density by a factor of 50 when the dimple isramped up has been already shown [92]. Figure 14 shows an image of atoms released froma trap at this stage. About 20% of the 7.5 × 105 atoms are trapped in the crossed region at apeak density that is one order of magnitude larger than in the wings.

The increased density in the centre also leads to a higher collision rate and allows us toforce evaporation in the crossed configuration by gradually reducing the laser intensity in the

R120 PhD Tutorial

(a) (b)

Figure 15. (a) Optimized evaporation ramp of trap of the beam intensity. The intensity of thevertical beam is ramped up to 100% during 5 s of plain evaporation in the horizontal beam.Subsequently the intensity of the horizontal beam is lowered in six steps. (b) Evolution of phase-space density versus number of atoms during preparation. After CLIP-loading (I). Doppler-cooling(II). RF cooling (III). Transfer to the ODT (IV). Plain evaporation (V). Forced evaporation (VI)to (XI). From step (VI) on, • represents the total number of atoms. ◦ represents only the atomstrapped in the crossed region. For steps (IV) to (XI), the numbers close to the data points are thepower in the horizontal beam in per cent of the maximum power. The inset shows a typical imageof a cloud at 70% power in the horizontal beam. Starting from step (VI), the phase-space densityalways refers to the measured peak value in the crossed region.

horizontal trapping beam. To keep the density and collision rate in the dimple at a high level,the intensity of the vertical beam is held at 100% throughout the further evaporation steps.

The dynamics during evaporative cooling in this configuration is however not asstraightforward as in the single-beam trap because one has to take into account the atomsthat are trapped in the crossed region as well as those located in the wings of the horizontaltrap. The main feature of these traps is the persistent loading of atoms from the wings into thecrossed region [113, 114] by elastic collisions. This effect clearly shows up in our experiment.

With the final optimized evaporation ramp, degeneracy was reached with 100 000 atomsat a temperature around 450 nK .

Six evaporation steps were needed to increase the phase-space density to values largerthan one. The final roadmap to reach quantum degeneracy is depicted in figure 15(a): Weramp the horizontal trapping beam within 5 s to 70%, 1.8 s to 50%, 1.1 s to 30%, 1.2 s to 20%,and 1.6 s to 10% of its initial intensity. In a final ramp, we finally pass the critical PSD of ∼1.2and Bose–Einstein condensation occurs. This last ramp takes 250 ms and we additionally holdthe trap at constant power for another 250 ms to let the system equilibrate before the atomsare released from the trap to take an absorption image.

Figure 15(b) illustrates the evolution of the phase-space density during the wholepreparation cycle where the phase space density is plotted versus the number of atoms inthe trap. Step (I) represents the magnetic trap after repumping the atoms to the ground statewhere we find 1.3 × 108 atoms at a PSD of ∼10−9. After Doppler cooling (step (II)), thesame number of atoms is found but the PSD is increased by two orders of magnitude to10−7. During rf cooling, 2.5 orders of magnitude in the number of atoms are lost and weend up at step (III) with ∼5 × 106 atoms and a PSD of 10−5. The transfer to the opticaltrap (step (IV)) increases the phase-space density by a factor of 5 after the magnetic trap hasbeen ramped down while we lose about 60% of the atoms. Right after step (IV), the atomsare transferred to the lowest Zeeman state and plain evaporation increases the PSD to 10−3 atstep (V). Between steps (IV) and (V), we ramp up the second beam of the crossed trap. The

PhD Tutorial R121

number of atoms at this point is >7.5 × 105. From there on, evaporation is forced by rampingdown the horizontal beam in 6 steps. The phase-space density and number of atoms aftereach of these evaporation steps are represented by (VI)–(XI). Starting from step (VI), opencircles in the figure represent the number of atoms trapped in the crossed region of the trap.Particularly during the first evaporation steps, a larger number of atoms are still trapped inthe outer regions of the first beam shown in the inset of figure 15(b). During the evaporationprocess, atoms are permanently loaded from the wings into the crossed region, causing thenumber of atoms in this region to stay almost constant during steps (VI)–(IX). After stage (IX),finally all remaining atoms are in the crossed trap and the wings of the horizontal beam areempty. From there on we lose less than half of the atoms in steps (X) and (XI) until quantumdegeneracy is reached at a remaining ∼6% of the initial power corresponding to ∼600 mW.

The overall gain in phase-space density during the optical trapping period, starting at aPSD of 5×10−5 with 1.5×106 atoms and ending at a PSD on the order of 1 with 105 atoms, ismore than four orders of magnitude, whereas we lose only 1.2 orders in the number of atoms.Thus the overall quotient of orders of PSD gain versus orders of atom loss is >3.6—almosttwice as large as in the magnetic trap. This suggests the use of higher laser power for theoptical trap to form a deeper potential and therefore be able to reduce the time of rf-evaporationin the magnetic trap and transfer the atoms into the ODT already at an earlier stage.

6. Bose–Einstein condensation of chromium atoms

6.1. Reaching degeneracy

When the trap depth is lowered in one final evaporation ramp to a value below 6% of themaximum, Bose–Einstein condensation of chromium atoms is observed at a temperature of450 nK by the appearance of a narrow peak in the centre of the density distribution onimages taken after 4 ms time of flight. Shortly before the peak appears, the closeness todegeneracy announces itself by a deviation of the momentum distribution from the ideal gaslimit. Close to Tc, the Gaussian and Bose fits of the density distribution start to differ. Thiseffect is illustrated in figure 16. The first column shows three-dimensional representations ofthe density distributions obtained from images taken after different final powers of the lastevaporation ramp. In these 3D diagrams, dense regions are represented by large values onthe vertical axis. The x- and y-directions correspond to the position on the CCD chip. Cutsthrough the density profiles through the centre of mass of the distribution are displayed in thesecond column together with the Gaussian-, Bose-, and Thomas–Fermi-fits to the distributions.The third column contains some important data obtained from these fits. The profile in thefirst row is well represented by a Gaussian (57), whereas that in the second row is clearlynon-Gaussian although it is still not in the degenerate regime. To describe the distributionproperly, we have to use (59) (represented by the dashed blue line). In the centre, wherethe distribution is affected most by the Bose enhancement, the Gaussian fit represented bythe black line deviates significantly from the measured density profile. Beginning from thethird row, corresponding to 5.8% of the maximum trap depth, the density profile shows aBose–Einstein condensate in the centre. When the trap depth is reduced further, this centralpeak grows on cost of a diminishing number of atoms in the surrounding thermal cloud. Thelast row shows the profile of a nearly pure Bose–Einstein condensate at a very low final powerof 2.5% of the maximum corresponding to 230 mW. Gravity adds a potential gradient in thevertical direction and ‘tilts’ the trapping potential. Therefore the trap depth in this situation isless than 2 µK. The temperature of the remaining thermal atoms has been measured 120 nKwhich is the lowest temperature ever observed in a sample of chromium atoms.

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position [µm]position [µm]

optic

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nsity

optic

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nsity

total number of atoms: 147.000total number of atoms: 147.000

final power in thefinal power in thehorizontal beam: 10%horizontal beam: 10%

purely thermal cloudpurely thermal cloud

temperature: 1.0µKtemperature: 1.0µK

calculated phase-space density: 0.9calculated phase-space density: 0.9

-200-200 -100-100 00 100100 200200

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total number of atoms: 60.000total number of atoms: 60.000

final power in thefinal power in thehorizontal beam: 6.2%horizontal beam: 6.2%

purely thermal cloudpurely thermal cloud

temperature: 460nKtemperature: 460nK

calculated phase-space density: 1calculated phase-space density: 1

Bose-distribution needed toBose-distribution needed todescribe the profiledescribe the profile

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total number of atoms: 60.000total number of atoms: 60.000

final power in thefinal power in thehorizontal beam: 5.8%horizontal beam: 5.8%

partly condensed cloudpartly condensed cloud

temperature: 415nKtemperature: 415nK

calculated phase-space density: 1.5calculated phase-space density: 1.5

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total number of atoms: 55.000total number of atoms: 55.000

final power in thefinal power in thehorizontal beam: 5.2%horizontal beam: 5.2%

partly condensed cloudpartly condensed cloud

temperature: 410nKtemperature: 410nK

calculated phase-space density: 1.5calculated phase-space density: 1.5

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number of atoms innumber of atoms inthe condensate: 20.000the condensate: 20.000

final power in thefinal power in thehorizontal beam: 2.5%horizontal beam: 2.5%

almost pure condensatealmost pure condensate

temperature: 120nKtemperature: 120nK

Figure 16. Density profiles of clouds obtained from absorption images that have been taken after4 ms of free expansion. The lines in the central column are fits to the density profiles: · · · · · ·(black): results of a Gaussian fit to the wings of the thermal cloud in the grey shaded area; - - - -(blue): fits of the g2-function to the wings; —— (red): fits of a g2 plus Thomas–Fermi distributionto the complete profile.

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7. Analysing the BEC transition

To analyse the dependence of the condensate fraction on the system temperature, the trapdepth was lowered from 10% to 2.1% in a series of consecutive experiments. Absorptionimages were taken after a relaxation time of 250 ms in the final trap and a time of flightof 4 ms. The temperature was determined from the width of a two-dimensional Gaussianto the outer wings of the thermal cloud (see section 3.2.2) and the number of atomsin the thermal and the condensed part was determined from two-dimensional fits of (60)and (59) to the density profiles. In an optical dipole trap, the trap parameters vary togetherwith the depth of the trap and lead to a dependence of Tc on the final trap depth. To eliminatethis dependence, the measured temperature was rescaled to the corresponding prediction[79] for the transition temperature T 0

c = hωho

kB

(N

ζ(3)

)1/3 ≈ 0.94 hωho

kBN1/3 using the measured

number of atoms and the trap parameters. This rescaled temperature was used to calculatethe expected condensate fraction under consideration of the finite size of the system andcontact interaction among the atoms [115] for every data point. A comparison of the T/Tc

dependence of the measured condensate fraction with this prediction showed a systematic∼5% shift to higher temperatures. Because the trapping parameters have been determinedwith high precision, the calibration of the imaging system was cross-checked to exclude errorsin the measurement of sizes, and timing errors are also expected to be orders of magnitudebelow the 4 ms of time of flight, the determination of the atom number was supposed tobe the most likely source for this systematic deviation. Right at the critical temperature,the relation N0

N= 1 − t3 − ζ(2)

ζ(3)ϑt2(1 − t3)2/5 for condensates of interacting atoms [82] can

be used to calculate the total number of atoms by setting N0 to zero and inserting the measuredtemperature in the equation Ntot = NT = T 3ζ(3)kB

3

ωho3h3 . To calibrate the number determination

with this method, we used an image of a cloud just above Tc where we had to a apply acorrection factor of 1.16 to the atom number to fulfil the above equation. This correctionfactor can be explained by a detuning of the probe light from resonance by 0.20� = 1 MHz,which corresponds to a shift of the lock point by only 1/40 of the signal amplitude of ourspectroscopy signal. Additionally, the spectroscopy signal is subject to a long term drift withrespect to the resonance in the chamber due to changes of the magnetic fields at the positionof the chamber and the position of the spectroscopy cell. With this calibration of the atomnumber determination, the measured dependence of the condensate fraction on the scaledtemperature agrees with the prediction for an interacting Bose-gas within errors as depictedin figure 17. However, the experimental data seem to lie systematically slightly below thetheoretical prediction. Since this kind of measurement is obviously very sensitive to a correctdetermination of the number of atoms, this deviation could still be due to an error in the atomnumber.

8. Expansion of the condensate

A further clear signature of Bose–Einstein condensation is the anisotropy of the expansionin time of flight14 when the sample is released from an anisotropic trapping potential. Tostudy this effect, we prepared condensates in traps with different anisotropy. Figure 18 showstypical series of time-of-flight images taken after 1 ms to 9 ms of free expansion. The upperrow (i) shows images of a thermal cloud just above condensation and (ii) shows a condensate

14 An anisotropy in the expansion of thermal clouds has also been predicted by Kagan et al [116] and observedexperimentally by Shvarchuck et al [117] in a thermal gas of 87Rb. However, this only happens in the hydrodynamicregime, where the mean-free path is much smaller than the extension of the cloud which is not the case in theexperiments presented here.

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Figure 17. Condensate fraction (N0/N) dependence on scaled temperature (T /T 0c ). Black circles

represent the measured data. The red dotted line represents the predicted fraction, approximated

by NN0

= 1 − ( TTc

)3

where Tc = T 0c + δT int

c + δTf s

c . δTf s

c is a shift in the critical temperature

due to the finite number of atoms and δT intc takes into account the contact interaction. T is the

temperature of the thermal cloud. - - - - is the theoretical expectation for an ideal gas and takesinto account the finite number of atoms. —— is for an infinite number of atoms in an ideal gas.The error bars are representative for all data points.

Figure 18. Three series of absorption images of expanding clouds taken after 1..9 ms time offlight. (i) Thermal cloud just above Tc. (ii) Condensate released from an almost spherical trap.(iii) Clearly anisotropic expansion after release from a trap with an aspect ratio of 1 : 4.8.

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(a)

(b)

Figure 19. Experimental cycle for measuring the condensate expansion (a) for y- and (b) forz-polarization. Left figures: in-trap polarization of the atoms relative to the chamber. Rightfigures: schematic cycle. During preparation the field is in either case along the (vertical) y-axis.To measure with z-polarization, the field has to be turned slowly (within 40 ms) before releasingthe atoms. After 1 ms of expansion, we switch suddenly to the x-direction in both cases.

which was released from a trap with a power of 240 mW in the horizontal beam. The trapfrequencies on the visible axes are fz = 144 Hz and fy = 222 Hz, resulting in a relativelysmall anisotropy of fy/fz = 1.5, reflected in an almost isotropic expansion of the condensate.In (iii), the condensate was prepared in the same trap, but before releasing the atoms, thelaser power was adiabatically increased to 2.3 W within 250 ms. The frequencies are fz =144 Hz and fy = 689 Hz corresponding to an aspect ratio of fy/fz = 4.8. As expected, thecondensate expands faster on the vertical where the confinement was stronger before the trapwas switched off. This leads to the typical change of the aspect ratio.

9. Observation of dipole–dipole interaction in a degenerate quantum gas

To measure the effect of the magnetic dipole–dipole interaction on the expansion dynamicsof the condensate, we prepare an almost pure BEC by decreasing the power in the horizontaltrapping beam to 280 mW. A schematic illustration of the subsequent experimental cycle isdepicted in figure 19. After 250 ms equilibration, the intensity of the horizontal beam isincreased adiabatically to 2.3 W to form an anisotropic trap (trap parameters fx = 942 Hz,fy = 712 Hz, and fz = 128 Hz). The homogeneous offset field of ∼11.5 G along the y-axisthat is used for optical pumping (section 4.3) is always kept on until the trap has been rampedup. After this change of the trap parameters, we either keep the field aligned along y (situation(a)) in figure 19) or we rotate the field adiabatically from the y- to the z-direction (situation(b)) in figure 19). This is done by increasing the field in z-direction linearly within 40 ms to∼11.5 G while reducing the field in y-direction during the same time to 0 G. After the field hasreached the steady state, we keep the atoms for another 7 ms in the trap to give them enoughtime to redistribute. The total storage time in both cases of longitudinal (‖�ez) or transversal(‖�ey) magnetization is equal. Subsequently, the atoms are released by a sudden switch-off of

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1

1.5

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2.5

3

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4

time of flight [ms]

aspe

ctra

tioR

y(t)/

Rz(t

)

(a)

9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11

2.3

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2.8

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3

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time of flight [ms]

aspe

ctra

tioR

y(t)/

Rz(t

)

3.22(b)

Figure 20. (a) Aspect ratio of the expanding dipolar condensate with an 11-point moving averageapplied to the data. Error bars represent errors from the fits to the density distribution. Upper, reddata: field aligned in the vertical y-direction. Lower, black data: field in the horizontal z-direction.The solid lines represent the corresponding theoretical predictions. The blue dotted line representsthe theoretical prediction for pure contact interaction. (b) shows a detail of the grey shaded area in(a). •(�) represent the mean values of 42 (32) measurements with y (z)-polarization. Solid errorbars represent one standard deviation of the statistical errors in both directions. The dashed/dottederror bars (displaced laterally for clarity) represent systematic errors.

the trapping beams. The polarization field is kept constant for 1 ms after release from the trapand then rotated suddenly to the transversal x-axis in either case by switching on the field inthe x-direction and switching off the z- or y-fields. This 1 ms of free expansion is long enoughfor the mean-field energy to drop to such a small part of its initial value that changing thealignment of the dipoles after this time does not influence the expansion anymore. In otherwords, after this time the gas is already so dilute that any kind of interaction among the atomscan be neglected compared to the kinetic energy. After an additional time of flight of up to 18ms plus 1 ms for the detection field to settle (total time of flight 2 ms to 20 ms), an absorptionimage of the cloud is taken.

The most convenient quantity to analyse the expansion is the aspect ratio κ = Ry/Rz

since it is insensitive to fluctuations of the number of atoms. The only quantities that have tobe known exactly are the trap parameters and the ratio εdd between magnetic dipole–dipoleinteraction and contact interaction (32). The trap frequencies of ωx/2π = ( 942 ± 6) Hz,ωy/2π = ( 712± 4) Hz, and ωz/2π = ( 128± 7) Hz have been determined using resonantparametric heating of a thermal cloud by modulating the intensities of the trapping beams[103].

Figure 20(a) shows the aspect ratio of the BEC for different times of ballistic expansionin sequential experiments where the total time of flight was varied between 2 ms and 14 ms.Due to the long preparation time of ∼1 min 20 s for a BEC, the total measuring time forthis graph was more than 4 h. To reduce the influence of systematic drifts during that time,the time of flight of subsequent pictures was chosen randomly. For the same reason, we alsochanged between y- and z-polarization every 10 shots. An 11-point linear moving average(corresponding to averaging over 2.2 ms time of flight in the graph) has been applied to bothsets of data to average out fluctuations in the condensate widths. The expansion does not showfeatures on this time scale that could be concealed by doing so and an average over 2.2 ms istherefore save.

We compare the experimental data [118] to the results of theoretical calculations [119]carried out as discussed in section 2.2.3. The theory does not contain any adjustable parameters.

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Instead it only relies on known or measured quantities, namely the number of atoms, thetrap frequencies, the magnetic moment, and the s-wave scattering length that characterizesthe contact interaction [74]. The blue dotted line represents the theoretical behaviour of agas interacting solely via s-wave scattering. Compared to this non-dipolar behaviour, theexpansion of the chromium condensate shows a dependence on the polarization of the atomsthat is in agreement with the theory: With transversal polarization (field along the y-axis),the condensate is elongated in the transversal direction and the aspect ratio is increased; whenpolarized along the longitudinal direction (field along the z-axis), the condensate is contractedin the vertical direction and the aspect ratio is decreased.

The error bars in figure 20(a) include only errors that stem from the fit of the condensatesize. Systematic errors, e.g. uncertainty of the magnification, are not contained. Thesesystematic errors can be found in figure 20(a) where the mean values of the results of 42 and32 measurements with y- and z-polarization are represented by a red circle and a black square,respectively15. All measurements were performed after the same 10 ms time of flight.

The solid error bars are the statistical errors of all measurements and represent ±1 standarddeviation with respect to the mean value. Systematic errors that stem from a systematic 2%uncertainty in the size of the cloud affect all measurements in the same way, i.e. they shiftthe measured aspect ratios of both longitudinal and transversal polarizations by the sameamount but do not change the relative difference between the two sets of expansion data. Theshift of the measured aspect ratios due to systematic errors is indicated by the dashed/dottedrepresentation of the error-bars (displaced laterally for clarity) in the lower figure. Taking thesystematic error into account, also the upper data point for y-polarization (which deviates alittle from the theoretical expectation) agrees with the theory within error bars.

A t-test analysis [120] of the two sets of data at 10 ms yields a 99% confidence interval of0.35–0.53 for the difference κ⊥ − κ‖ between the aspect ratios in the two cases of transversaland longitudinal polarization. Although the theoretical prediction of κ⊥ − κ‖ = 0.31 doesnot lie within this interval, the large confidence level confirms that the two mean values ofthe measurements are clearly separated and strongly correlated with the polarization of thesample. In the series of measurements at different times of flight however (upper graph), theagreement between theoretical prediction and measurement seems to be much better.

9.1. Measurement of the dipole–dipole strength parameter

The starting point for the determination of the dipole–dipole strength parameter εdd is toanalyse the asymptotic behaviour of the condensate radii Ri(t) for long times of flight. In thisasymptotic limit, which is governed by a collisionless and potential free (except for gravity)ballistic flight, the radii of the cloud can be parameterized as

Ri(t) = R∗i + v∗

i t . (68)

These radii, as well as the asymptotic velocities v∗i = Ri(t) of the expansion scale with the

square root of the total energy (27). Hence, they are proportional to (Na)1/5:

v∗i ∝ (Na)1/5 (69)

15 In all, 50 measurements have been performed at 10 ms with both polarizations. Some of the measurements had tobe withdrawn due to an obvious instability of the system which on the one hand lead to a number of shots where thenumber of atoms was substantially smaller than the average 4.0 ± 0.6 × 104 of the remaining measurements. On theother hand, in the measurements with z-polarization, some of the condensates did not fall down vertically but movedsignificantly to one or the other side during their flight, which we considered a signature that the condensate waskicked (probably by mechanical noise on the optical table) prior to or during switching off the trap. These imageswere also withdrawn.

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and

Ri(t) = R∗i + v∗

i t ∝ (Na)1/5, i = [x, y, z]. (70)

Note that the initial values Ri(0) = R∗i are not the Thomas–Fermi radii Ri but are actually

smaller than these because the initial acceleration due to the conversion of mean-field energyto kinetic energy is neglected if the parametrization (68) is used. To prove the proportionality,we consider the equations of motion (47) in the steady state at t = 0, where the accelerationsRi(0) have to vanish. The radii at t = 0 are given by the solution

Ri ≡ Ri(0) (71)

of the Thomas–Fermi equation including the dipole–dipole interaction

Ri(0) = − 2

m

∂

∂Ri

(mω2

i

14Ri(t)

2 +15

7

Nh2a

m

1

RxRyRz

(1 − εddf ({Ri})))

= 0. (72)

Up to a small correction caused by the dipole–dipole interaction, this solution is similar to

the solution obtained for the pure contact interaction Ri(0) ≈ Ri =√

2µ

mω2i

. Now we define

dimensionless radii R(t) that are rescaled to the Thomas–Fermi radii in the form

Ri(t) = Ri(t)

Ri

, (73)

with Ri(0) = 1. With these rescaled radii, (72) gets the form

¨Ri(0) = − 2

m

∂

∂Ri

(mω2

i

14R2

i +15

7

(Nh2a

m

)1

R2i

∏j Rj

1∏k Rk

(1 − εddf ({Ri})))

= 0. (74)

Since this condition must hold for any number of atoms N and any trap geometry defined bythe set of trap parameters {ωx, ωy, ωz}, all radii fulfil the condition

Ri(0) = Ri ∝ (aN)1/5, (75)

which was what we wanted to prove.

9.2. Determination of the dipole–dipole strength parameter

To extract the dipole–dipole strength parameter εdd , we analyse the ratio of the two asymptoticvelocities [121]. In particular, this ratio depends only on the asymmetry introduced by thedipole–dipole interaction since the contribution of the s-wave scattering to the total energy isindependent of the polarization. We use the two rescaled asymptotic velocities

v∗y = v∗

y

〈N1/5〉 (76)

v∗y(

�B‖�ey) and v∗y(

�B‖�ez) for polarization along �ey and �ez, respectively to determine εdd byanalysing their ratio. To first order in εdd (in the expected range of εdd , higher orders arenegligible), the ratio has the form

v∗y(

�B‖�ey)

v∗y(

�B‖�ez)= 1 + Dεdd, (77)

where D = 0.6523 is a real constant that can be obtained from a numerical calculation ofv∗

y(�B‖�ey)/v

∗y(

�B‖�ez) for the measured trap parameters ωx, ωy, ωz,m. It is now possible to

calculate εdd for the measured ratio v∗y(

�B‖�ey)/v∗y(

�B‖�ez) by inverting relation (77). Thisrelation has been tested numerically and is a very good approximation for a wide range of

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2 4 6 8 10 12 14 160.2

0.4

0.6

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1.2

1.4

1.6

1.8x 10-4

vvyy =9.56=9.56 0.24 100.24 10-3-3 m/sm/s±±**

time of flight [s]

Ry(t

)[m

]

x 10-3

(a)

2 4 6 8 10 12 14 16x 10x -3

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time of flight [s]

Ry(t

)[m

]

vvyy** 1010-3-3 m/sm/s

x 10-4(b)

Figure 21. Measured dependence of the condensate radius Ry(t) on the time of flight. (a) Dipolesaligned along the y-axis, (b) dipoles aligned along the z-axis, ◦: measured radii; +: measuredradii rescaled using (78); ——: linear fit to the rescaled radii.

scattering lengths. It delivers the correct value for εdd with less than 5% error if the scatteringlength is larger than 37a0. For a > 60 a0 (i.e. εdd < 0.25), the error is smaller than 1% andfor a > 85 a0, it is smaller than 0.2%, i.e. in the expected range of εdd , higher order terms arenegligible.

To determine the asymptotic velocity, we use the condensate radii R(t) measured in atime-of-flight series. Since the number of camera pixels that are covered by the condensate ismuch larger in the direction of fast expansion, one can expect the most accurate results fromthe radius Ry(t) in y-direction. We first consider the case of polarization along the y-axis.Figure 21 shows the dependence of the condensate radius y-axis with polarization along y onthe time of flight for 67 different expansion times. Since the number of atoms is fluctuatingduring such a series of experiments, a linear fit directly to the measured data would containthese fluctuations as a statistical error. To get rid of this dependence, we make use of thescaling behaviour (70). By dividing all measured radii Ry by the number of atoms in thecondensate and multiplying them with the mean value of the fifth root of all measured atomnumbers 〈N1/5〉, we get a series of time-dependent radii which are now independent of theatom number

Ry = Ry

N1/5〈N1/5〉. (78)

The average number of atoms in this measurement was 〈N〉 = 29 100. The red circles infigure 21 represent the measured Ry(t), the black crosses with error bars mark the rescaledR(t) which show much less fluctuations. A linear fit to the rescaled data for times largerthan 3 ms to focus only on the asymptotic behaviour yields v∗

y(�B‖�ey) = (9.56 ± 0.24) m s−1

and R∗y = (10.1 ± 1.0) µm for y-polarization. For z-polarization, we get v∗

y(�B‖�ez) =

(8.78 ± 0.12) m s−1 and R∗y = (10.3 ± 0.9)µ m. By using the above re-scaling, the errors

�v∗y = ±0.24 m s−1 and �v∗

y = ±0.12 m s−1 in the fitted slope v∗y for y- and z-polarization,

respectively, do not contain fluctuations of the atom-number anymore.From the measured asymptotic velocities, we obtain

εdd =(

v∗y(

�B‖�ey)

v∗y(

�B‖�ez)− 1

)/D. (79)

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Including the error

�εdd =

√√√√( ∂εdd

∂v∗y(

�B‖�ey)

)2

(�v∗y(

�B‖�ey))2 +

(∂εdd

∂v∗y(

�B‖�ez)

)2

(�v∗y(

�B‖�ez))2

from the velocity fits, this yields a measured value of

εdd = 0.159 ± 0.034. (80)

This is in a very good agreement with the strength parameter of εdd = 0.148 that is expectedfor a scattering length of a = 103 a0.

9.2.1. Deducing the s-wave scattering length of 52Cr. Since the dipole–dipole interactionstrength is exactly known from (32), this result can be used to calculate the scattering length:

a = µ0µ2mm

12πh2εdd

= (5.08 ± 1.06) × 10−9 m = (96 ± 20)a0. (81)

This result is in excellent agreement with the value of (103 ± 13)a0 that has been obtainedby comparing the measured positions of Feshbach resonances in chromium collisions withmultichannel calculations [74]. Furthermore, the relative error of ±20% makes it a comparablyprecise method. Many different techniques have been used to determine the s-wave scatteringlengths of ultracold atoms but most of them come along with large error bars, often becausethe number of atoms enters the measurement.

10. Conclusion

In conclusion, the experiments presented in this tutorial document the generation of a dipolarchromium Bose–Einstein condensate and the direct observation of magnetic dipole–dipoleinteraction in a quantum gas. This is the first mechanical manifestation of dipole–dipoleinteraction in a gas. Among all existing BECs, a chromium BEC is the only one in whichatoms interact significantly via magnetic dipole–dipole forces.

Bose–Einstein condensation of chromium is achieved by forced evaporative cooling ina crossed optical dipole trap that is loaded from an rf cooled magnetic trap. We identifytransition from a classical gas to a quantum degenerate state by the typical appearance of abimodal momentum distribution and an anisotropic expansion when releasing the cloud froman asymmetric trap.

When a homogeneous magnetic field is applied to polarize the chromium BEC, theatoms redistribute to minimize the total energy which manifests in a change of the expansiondynamics. We find an agreement of the measured dynamics with a parameter-free theorythat proves that the theory of dipolar superfluids which I summarized in the theoreticalsection of this tutorial is well suited to describe the expansion dynamics of a dipolar BEC.The experimental results show that the magnetic dipole–dipole interaction among the atomsin a chromium BEC is strong enough to influence the properties of the condensate andconstitute a proof of the dipolar character of a BEC of chromium atoms. The possibility ofgenerating chromium BECs in series of experiments under stable conditions and of storingthem for a sufficiently long time are important prerequisites for further experiments. Althoughheteronuclear molecular BECs are expected to have much larger (electric) dipole moments,such condensates are not available at the moment. It is also questionable whether thesecondensates will have long enough lifetimes to perform experiments, once they are available.Long lifetimes and large dipole moments are only expected for heteronuclear fermion–fermion

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molecules [122–124] that are generated in a low vibrational state which is not easy to achieve(for a review on ultracold molecules see [125]).

Since it is in fact favourable for most of the expected phenomena to have a largerrelative strength of the MDDI, gaining control over the s-wave scattering length was themost important next step to enter the regime of dominant dipole–dipole interaction. In a veryrecent experiment, our group has demonstrated that it is possible to approach this regime,making the chromium BEC the most promising system for further studies of strong dipolareffects in quantum gases.

Acknowledgments

I would like to thank Professor Tilman Pfau for his support, his counsel and for supervisingmy PhD thesis, as well as all members of the atom optics group in Stuttgart, in particularthe chromium team, for their encouragement and practical help. I especially acknowledgefruitful discussions and collaboration with Stefano Giovanazzi, Luis Santos, Paolo Pedri andAndrea Simoni who were working enthusiastically on the theoretical description of our BEC.The work presented in this tutorial was financially supported by the SPP1116 and SFB/TR21programmes of the German Science Foundation (DFG).

References

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Rev. Lett. 75 3969[3] Bradley C C, Sackett C A, Tollett J J and Hulet R G 1995 Phys. Rev. Lett. 75 1687[4] Fried D G, Killian T C, Willmann L, Landhuis D, Moss S C, Kleppner D and Greytak T J 1998 Phys. Rev.

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