Trapping metastable neon atoms

Citation for published version (APA):Tempelaars, J. G. C. (2001). Trapping metastable neon atoms. Eindhoven: Technische Universiteit Eindhoven.https://doi.org/10.6100/IR543676

DOI:10.6100/IR543676

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Trapping Metastable Neon Atoms

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN

DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG

VAN DE RECTOR MAGNIFICUS, PROF. DR. M. REM, VOOR

EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE

VOOR PROMOTIES IN HET OPENBAAR TE VERDEDIGEN

OP DONDERDAG 26 APRIL 2001 OM 16.00 UUR

DOOR

JEFFREY GODEFRIDUS CORNELIS TEMPELAARS

GEBOREN TE BERGEN OP ZOOM

DIT PROEFSCHRIFT IS GOEDGEKEURD

DOOR DE PROMOTOREN:PROF. DR. H.C.W. BEIJERINCK

EN

PROF. DR. B. J. VERHAAR

COPROMOTOR:DR. IR. E. J.D. VREDENBREGT

Druk: Universiteitsdrukkerij Technische Universiteit EindhovenOntwerp omslag: Astrid van den Hoek

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Tempelaars, Jeffrey Godefridus Cornelis

Trapping Metastable Neon Atoms / byJeffrey Godefridus Cornelis Tempelaars. -Eindhoven: Technische Universiteit Eindhoven, 2001. - Proefschrift. -ISBN 90-386-1759-3NUGI 812Trefw.: atomaire bundels / laserkoeling / atoombotsingen / atomen; wisselwerkingen /laserspectroscopie / neonSubject headings: atomic beams / laser cooling / optical cooling of atoms; trapping /interatomic potentials and forces / neon

aan mijn ouders

The work described in this thesis was carried out at the Physics Department of theEindhoven University of Technology and was part of the research program of the‘Stichting voor Fundamenteel Onderzoek der Materie’ (FOM), which is financially sup-ported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek’ (NWO).

Contents

1 Introduction 31 Cold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Bose-Einstein condensation of metastable neon atoms . . . . . . . . . . . 43 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Laser cooling and trapping 91 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Light forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Applications of laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Cold atomic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Intense beam of cold metastable Ne(3s) 3P2 atoms 251 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Source and collimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Magneto-optical compressor and sub-Doppler cooler . . . . . . . . . . . . 396 Beam characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Metastable neon atoms in a magneto-optical trap 531 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 Trap dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 Trap volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Loading rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 Trap population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 Trap decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Photoassociation spectroscopy of 85Rb2 0+u states 87

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872 Bound state calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

1

2 Contents

3 Application to the rubidium dimer . . . . . . . . . . . . . . . . . . . . . . . . 924 Photoassociation experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 Model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Summary 113

Samenvatting 115

Dankwoord 118

Curriculum Vitae 120

Chapter 1

Introduction

1 Cold atoms

Since the demonstration of the use of laser light to cool and confine neutral atomsin the late 1980s [1,2], the cooling and trapping field has made an enormous devel-opment. Several technological applications of cold atoms have been realized. Lasercooled atoms in atomic beams can be focused to a very small scale: because of thesmall de Broglie wavelength of the cold atoms, very high resolution can be obtainedin the fabrication of nanostructures [3]. The use of laser cooled atoms in atomicclocks gave a breakthrough in time-metrology: the application of fountains of coldatoms [4] increased the accuracy in the atomic frequency standard by two orders ofmagnitude [5]. Aside from these practical applications, cold and ultracold collisionsare of interest in themselves as fundamental quantum mechanical phenomena. Theincreasing use of atom traps as an experimental and technological tool increased theinterest in collisions at temperatures below 1 mK [6]. The questions how “bad” colli-sions lead to trap loss, and how “good” collisions could be used in cooling schemes,fascinated atom trappers constantly [7].

The insights obtained led to the experimental realization in 1995 of Bose-Einsteincondensation (BEC) in cold, dilute samples of alkali-metal atoms [8–10]. Bose-Einsteincondensation in atomic samples occurs when the mean inter-particle distance is ofthe same order as the thermal wavelength. In formula:

nΛ3dB = n(2π2/mkBT)3/2 ≥ 2.61, (1.1)

where n is the number density, ΛdB the thermally averaged de Broglie wavelengthat temperature T , m the atom’s mass, and kB Boltzmann’s constant. This conditionrequires densities of the order of 1014 atoms/cm3 and ultra low temperatures of theorder of 1 µK. It was a different cooling technique, called evaporative cooling [11,12], which made the final step to the quantum degenerate regime possible. Since1995, BEC’s of alkali-metal atoms were produced all over the world, and in 1998also atomic hydrogen was Bose-condensed [13,14]. Other atomic species which arecandidates for BEC are metastable rare gas atoms, with helium and neon the primepossibilities [15–18], as well as alkali-earth atoms [19].

3

4 Chapter 1

The attainment of BEC in weakly interacting gases opened a whole new areain physics. Quantum-statistical effects have been observed in BEC’s which are re-lated to superconductivity phenomena and superfluidity of liquid helium [20]. It hasbeen shown that the speed of light can be reduced to several meters per secondin BEC’s [21], and consequently, experiments have been suggested to create atomicsamples behaving like optical black holes [22]. Furthermore, experiments showedthat the coherence of the atoms can be conserved when atoms are coupled out of aBEC with optical or microwave pulses. Because of the analogy to the coherence oflaser light, this may be considered the birth of the atom laser [23].

2 Bose-Einstein condensation of metastable neon atoms

Experimentally, a Bose-Einstein condensate is created in roughly three steps. First,atoms are loaded into a magneto-optical trap (MOT) in which the combination ofa laser field with a magnetic quadrupole field confines the atoms in the trappingcenter. Second, the atoms are transferred from the MOT to a magneto-static trap.Finally, the atoms are cooled down in the magneto-static trap by a forced evaporativecooling process until BEC is reached. Until now, BEC has only been observed in dilutesamples of alkali-metal atoms and atomic hydrogen. In our group we are trying toproduce a BEC with metastable neon atoms [18]. A BEC of metastable rare gas atomsis attractive for a number of reasons. First of all, real time diagnostics can be usedsuch as Auger deexcitation of escaping atoms at a surface and UV photon detectionfollowing optical pumping to a non-metastable state. Second, tight magnetic trapsare easy to achieve for metastable atoms, due to their large magnetic moment. Third,different isotopes with integer and half-integer angular momentum are available,which potentially allows experiments with stable and unstable condensates as wellas Fermi gases. Furthermore, the high internal energy, e.g. 17 eV for Ne(3s)3P2,makes single atom detection possible, which opens new ways to look at the statisticsand formation of BEC’s. Finally, future experiments with atom lasers of electronicallyexcited atoms would be possible.

The experimental conditions are less favorable for metastable rare gas atoms.First of all, efficient beam-brightening techniques are necessary to overcome the lowefficiency of ∼ 10−4 with which rare gas atoms in the metastable state are producedin commonly used sources. Second, trap lifetimes and densities are limited by theprocess of Penning ionization. And finally, the finite lifetime of the metastable state,e.g. 24 s for the Ne(3s)3P2 state, limits the time scale of the experiments. Despite allthose experimental difficulties, the prospects for creating a BEC of metastable neonatoms are rather good [18]. Calculations done by Doery et al. [24] showed that, byspin-polarizing the atoms, which happens spontaneously in a magneto-static trap,the ionization of metastable neon atoms may be suppressed by four orders of mag-nitude. Consequently the trap lifetime due to ionizing losses can be increased tovalues much larger than the natural lifetime of the metastable state [18]. However,those calculations were based on modified sodium potentials, since accurate inter-action potentials of metastable neon atoms are not available at all. A full theoretical

Introduction 5

calculation based on new, ab initio short-range [25] and semiempirical long-rangepotentials [26], could therefore still alter these expectations.

3 Scope of this thesis

The road we are following to reach Bose-Einstein condensation with metastable neonatoms, is analogous to that of sodium [27], with the exception that we have to loadour magneto-optical trap with a brightened atomic beam, instead of from a vaporcell or slowed atomic beam. In this thesis the first successful steps towards BEC ofmetastable neon atoms are described.

Chapter 2 gives a general introduction into the physics of cooling and trappingneutral atoms. In chapter 3 we describe how we applied several laser cooling tech-niques to create an intense beam of cold metastable Ne(3s)3P2 atoms. The beam, witha flux of 5 × 1010 atoms/s, can be used for a variety of cold collision experiments.In chapter 4 we describe how we used the bright beam setup to create a magneto-optical trap containing almost 1010 metastable neon atoms. A large initial numberof atoms trapped in the MOT facilitates the evaporative cooling phase in a magneto-static trap, necessary to reach the quantum degenerate regime. Recently 109 atomswere transferred from the magneto-optical trap to a magneto-static trap [28], devel-oped in our lab [29,30].

Knowledge of the interaction parameters for cold colliding atoms was crucialfor understanding the properties of a BEC in the case of alkali-metal atoms. Thetechnique of photoassociative spectroscopy (PAS) has been used, with considerablesuccess, to determine collisional properties of the alkali atoms [31], and is equallyappropriate for studies of the rare gases [32,33]. This technique, explained in moredetail in chapter 2, is based on the formation of bound excited-state molecules byexciting two slowly colliding ground state atoms with a probe laser. In chapter 5of this thesis it is shown by an experiment with rubidium that PAS can be used toinvestigate the coupling between two molecular states.

Recently, photoassociation experiments on metastable helium atoms were re-ported by Herschbach et al. [33]. From PAS spectra they estimated long range excitedstate interaction parameters for metastable helium. In the future we plan to applythe PAS technique to metastable neon atoms to obtain experimental data on long-range ground-state potentials [32] as well as ionization properties [24]. The latterwill give information about the suppression of ionization of spin polarized samples.

References

[1] S. Chu, L. Hollberg, J.E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55,48 (1985).

[2] E.L. Raab, M. Prentiss, Alex Cable, Steven Chu, and D.E. Pritchard, Phys. Rev.Lett. 59, 2631 (1987).

6 Chapter 1

[3] J.J. Mcclelland, R.E. Scholten, E.C. Palm, and R.J. Celotta, Science 262, 877 (1993).

[4] M.A. Kasevich, E. Riis, S. Chu, and R.G. DeVoe, Phys. Rev. Lett. 63, 612 (1989).

[5] P. Lemonde, Phys. World, January, 39 (2001).

[6] H.J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-VerlagNew York, 1999.

[7] J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne, Rev. Mod. Phys. 71(1), 1, (1999).

[8] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Sci-ence 269, 198 (1995).

[9] K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn,and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).

[10] C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, Phys. Rev. Lett. 75, 1687(1995).

[11] H. F. Hess, Phys. Rev. B. 34, 3476 (1986).

[12] N. Masuhara, J.M. Doyle, J.C. Sandberg, D. Kleppner, T.J. Greytak, H.F. Hess, andG.P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).

[13] T.C. Killian, D.G. Fried, L. Willmann, D. Landhuis, S.C. Moss, T.J. Greytak, and D.Kleppner, Phys. Rev. Lett. 81, 3807 (1998).

[14] D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, andT.J. Greytak, Phys. Rev. Lett. 81, 3811 (1998).

[15] S. Nowak, A. Browaeys, J. Poupard, A. Robert, D. Boiron, C. Westbrook, and A.Aspect, Appl. Phys. B 70, 455 (2000).

[16] N. Herschbach, P.J.J. Tol, W. Hogervorst, and W. Vassen, Phys. Rev. A 61,050702(R) (2000).

[17] M. Zinner, C. Jentsch, G. Birkl, and W. Ertmer, private communication.

[18] H.C.W. Beijerinck, E.J.D. Vredenbregt, R.J.W. Stas, M.R. Doery, and J.G.C. Tem-pelaars, Phys. Rev. A 61, 023607 (2000).

[19] T. Ido, Y. Isoya, and H. Katori, Phys. Rev. A 61, 061403(R) (2000).

[20] F. Chevy, K.W. Madison, and J. Dalibard, Phys. Rev. Lett. 85, 2223 (2000).

[21] L. Vestergaard Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, Nature 397(6720),594 (1999).

[22] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000).

Introduction 7

[23] M.-O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, and W.Ketterle, Phys. Rev. Lett. 78, 582 (1997).

[24] M.R. Doery, E.J.D. Vredenbregt, S.S. Op de Beek, H.C.W. Beijerinck, and B.J. Ver-haar, Phys. Rev. A 58, 3673 (1998).

[25] S. Kotochigova, E. Tiesinga and I. Tupitsyn, Phys. Rev. A 61, 042712 (2000).

[26] A. Derevianko and A. Dalgarno, Phys. Rev. A 62, 062501 (2000).

[27] M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, and W.Ketterle, Phys. Rev. Lett. 77, 416 (1996).

[28] S.J.M. Kuppens, V.P. Mogendorff, J.G.C. Tempelaars, E.J.D. Vredenbregt, andH.C.W. Beijerinck, to be published.

[29] R. Stas, Laser cooling and trapping of metastable neon atoms, internal report,Eindhoven University of Technology (1999).

[30] V.P. Mogendorff, Towards BEC of metastable neon, internal report, EindhovenUniversity of Technology (2000).

[31] W.C. Stwalley and H. Wang, J. Mol. Spec. 195, 194 (1999).

[32] M.R. Doery, E.J.D. Vredenbregt, J.G.C. Tempelaars, H.C.W. Beijerinck, and B.J.Verhaar, Phys. Rev. A 57, 3603 (1998).

[33] N. Herschbach, P.J.J. Tol, W. Vassen, W. Hogervorst, G. Woestenenk, J.W. Thom-sen, P. van der Straten, and A. Niehaus, Phys. Rev. Lett. 84, 1874 (2000).

[34] R.A. Cline, J.D. Miller, and D.J. Heinzen, Phys. Rev. Lett. 73, 632 (1994).

8 Chapter 1

Chapter 2

Laser cooling and trapping

1 Introduction

The momentum carried by a beam of light can be used to manipulate atomic trajec-tories. This was already demonstrated in the early thirties by a small deflection ofa beam of sodium atoms illuminated by a resonant lamp [1]. Since the developmentof the laser, intense and highly directional light sources became available which en-hanced the efficiency of the momentum transfer from the light field to the atomsdrastically. In 1985, laser light was used to cool and trap neutral atoms for the firsttime [2]. In section 2 of this chapter we discuss the origin of optical forces and howthey are used in the laser cooling and trapping field. In section 3 we describe theprinciples of the neutral atom traps used in the experiments described in chapters 4and 5 of this thesis. Since in this thesis experiments with both metastable neonatoms and rubidium atoms are described, in section 4 we give a short descriptionof their atomic structure and discuss the physics of low temperature collisions forboth atoms.

2 Light forces

2.1 Principles

Let us consider a two-level atom at rest in a classical electro-magnetic field with elec-tric field component E(R, t) = E0(R)[cos(ωLt−kL · R)]. The optical force exerted onthe atom can be of two types: a dissipative, spontaneous force and a conservative,dipole force [3]. The spontaneous force arises from the absorption and spontaneousemission of a photon from the light field. Absorbing a photon from the light fieldresults in a transfer of a momentum kL to the atom in the direction of the lightpropagation. If the decay of the atom, after absorbing a photon, is spontaneous, theemission of the photon is in a random direction, so that over many events the cor-responding recoil averages to zero effect on the momentum of the atom. As a resultthe net force is in the direction of the light propagation, as shown schematically inFig. 2.1. The dipole force arises from the interaction of the dipole moment, induced

9

10 Chapter 2

Incoming light

Scattered light

Net force

Atom

Figure 2.1: Principle of the spontaneous radiation force. Photons are absorbed fromthe direction of the laser beam and reemitted spontaneously in random direction.The net force is in the direction of the light propagation.

by the oscillating electric field, with the gradient of the electric field amplitude. Thisbecomes clear when we look more quantitatively at the character of the forces, asdone in detail by Cohen-Tannoudji et al. [4].

The expression for the interaction Hamiltonian of an atom at rest in the electricfield E(R, t) is given by

H = −µeg · E(R, t), (2.1)

with µeg the transition dipole moment. The light force on the atom is then givenby [5]

F = −〈∇RH〉 = ∇R〈µeg · E(R, t)〉, (2.2)

where we neglected the spatial variation of the electric field over the size of the atom.The two types of forces arise from the gradient operator working on the electric field:the spontaneous force Fsp is related to the spatial dependence of the phase of theelectric field while the dipole force Fdip is related to the spatial dependence of theamplitude of the electric field. The expression for the two types of forces arise fromthe steady-state solutions of the optical Bloch equations [5]:

F = Fsp + Fdip

= ∇R(kL · R)Ω2

2

[Γ/2

(∆L)2 + (Γ/2)2 +Ω2/2

]

−∇R|E0|Ω2

2

[∆L

(∆L)2 + (Γ/2)2 +Ω2/2

], (2.3)

with ∆L = ωL − ω0 the detuning of the optical field from the atomic transitionfrequencyω0, Γ the natural width of the atomic transition; Ω reflects the strength ofthe coupling between field and atom and is called the Rabi frequency,

Ω = − µeg · E0

. (2.4)

The spontaneous force is a dissipative force and can be used to cool atoms. Thisis why it is sometimes called the cooling force. We can express this force as

Fsp = kLΓ2

[s

1+ s

], (2.5)

Laser cooling and trapping 11

with the saturation parameter s defined as

s = Ω2/2(∆L)2 + (Γ/2)2

. (2.6)

The cooling force is simply the momentum of a photon kL times the scatteringrate, as mentioned earlier. The maximum cooling force equals Fmax = kLΓ/2 andis reached for high saturation parameters. By defining the on-resonance saturationparameter s0 as

s0 = II0

= 2Ω2

Γ 2, (2.7)

with I the light intensity and I0 the saturation intensity depending on the atomictransition involved, we write for the cooling force

Fsp = kLΓ2

s01+ s0 + (2∆L/Γ)2

. (2.8)

The dipole force can be written in terms of the gradient of the light intensity ∇I:

Fdip = −∆L ∇I2I

[s

1+ s

]= −∆L ∇I

2Is0

1+ s0 + (2∆L/Γ)2. (2.9)

A red-detuned light beam, i.e. ∆L < 0, produces a force that attracts the atomsto the intensity maximum, while a blue-detuned light beam repels the atoms fromthe intensity maximum. The dipole force is sometimes called the trapping force,because, for example, a focused, red-detuned laser beam can be used for trappingneutral atoms (as described in section 3.2 of this chapter).

2.2 Velocity dependence

Cooling atoms with laser light requires a velocity dependent force. The dissipative,spontaneous force described above can be used for this purpose. Considering anatom moving in the direction of a slightly red-detuned laser beam, the effective de-tuning ∆eff from resonance contains not only the laser detuning ∆L, but also containsa contribution ∆D = −kL · v due to the Doppler effect. Therefore the force givenby Eq. (2.8) becomes velocity dependent by replacing ∆L by the effective detuning∆eff = ∆L +∆D. This is the basis for the standard laser cooling mechanism known as“Doppler cooling” or “optical molasses” [6].

Let us now consider a standing light wave produced by two counterpropagating,slightly red-detuned laser beams with the same frequency. An atom moving in thedirection of one of the laser beams experiences, due to the Doppler effect, a slightlyblue-shifted laser beam. As a consequence it will absorb more photons per unit oftime from the counterpropagating laser beam than from the copropagating beam.Hence the atom experiences a net force opposite to its own motion. Around v = 0the force is proportional to the velocity and can be seen as a damping force. Fors0 1 the force can be approximated by [7]

F = 8k2Ls0(∆L/Γ)v[1+ (2∆L/Γ)2]2

≡ −βv, (2.10)

12 Chapter 2

-xc xcvc-vc

ba

-2 -1 0 1 2-0.4

-0.2

0.0

0.2

0.4

kLv/Γ

F/F m

ax

-2 -1 0 1 2-0.4

-0.2

0.0

0.2

0.4

F/F m

axµ'Gx/hΓ

Figure 2.2: a: Velocity dependence of the cooling force for optimal conditions ∆L =−Γ/2, s0 = 1. The dashed lines indicate the capture velocity vc = −∆L/kL. b: Positiondependence of the cooling force on an atom at rest (v = 0) for optimal conditions∆L = −Γ/2, s0 = 1. The dashed lines indicate the capture range xc = −∆L/(µ′G).

with β the damping coefficient. Figure 2.2a shows the damping force for optimalconditions (∆L = −Γ/2 and s0 = 1) as a function of the atomic velocity. Atoms witha velocity below the capture velocity vc = −∆L/kL feel the damping force given byEq. (2.10). In chapter 3, we describe how we use standing light waves to capturemetastable neon atoms and collimate them into an parallel atomic beam.

Although the velocity-dependent force can be used to cool atoms, it can not beused to (spatially) trap atoms, since there is no position-dependent force to drivethe atoms to an equilibrium point in space. However, the spontaneous force can bemade position dependent by applying a spatially varying magnetic field.

2.3 Position dependence

In a magnetic field B the internal energy of an atom in a magnetic sublevel mi withrespect to B changes by an amount ∆E = −µ · B = µBBgmi, with µ the magneticmoment of the atom, µB the Bohr magneton, and g the Lande factor. The transitionfrequency for an atom with ground state g and excited state e is shifted by ∆B =µ′B/, with

µ′ ≡ (geme − ggmg)µB, (2.11)

the effective magnetic moment of the transition [5]. Figure 2.3 shows the one-dimensional situation of a hypothetical atom with atomic transition Jg = 0→ Je = 1,in an inhomogeneous magnetic field B = B(x) ≡ Gx. The atom is illuminated by twocounterpropagating laser beams, each detuned slightly to red of the zero magneticfield atomic transition. The laser beams are opposite circularly polarized, the beam

Laser cooling and trapping 13

0

B

xσ+ σ−

σ+σ− σ+σ− σ+σ−

me=

B=0B< 0 B> 0Force=0Force Force

Je=1

Jg=0

-1 0 +1

Figure 2.3: Principle of the position dependent radiation force for a Jg = 0 → Je = 1atomic transition. The inhomogeneous magnetic field splits the Zeeman levels ofthe exited state. As a consequence, an atom positioned at the left of the origin ismore resonant with the σ+-polarized beam and feels a resulting force directed tothe origin. An atom positioned at the right of the origin, is also pressed to the originbecause it is more resonant with the σ−-polarized beam.

running in the +x direction has σ+ polarization and the beam running in the −xdirection has σ− polarization. If the atom is located at a position left of the origin,the atom is, because of the Zeeman splitting of the Je = 1 state, more resonant withthe σ+ polarized beam and feels a force directed to the origin. An atom positionedon the right of the origin will be pushed back to the origin by the σ− polarized beam.

The position-dependent force on the atom is analogous to the damping force invelocity space, described in the previous section. Figure 2.2b shows the force on anatom in an inhomogeneous magnetic field B = B(x) ≡ Gx. Analogous to the velocity-dependent force, a capture radius can be defined by xc = −∆L/(µ′G). Position-dependent forces are used to trap atoms in magneto-optical traps, like described insection 3.1. In chapter 3 we describe two applications of position-dependent forcesin an atomic beam: Zeeman-compensated slowing to decelerate atoms in an atomicbeam and a two-dimensional version of a magneto-optical trap to funnel atoms intoa narrow beam.

2.4 Cooling limits

Cooling atoms with laser light also introduces heating effects caused by the random-ness of the momentum steps undergone by the atom with each emission or absorp-tion. The motion of the atom can be compared to a random walk in momentumspace caused by the randomness in direction of the spontaneous emitted photonsand the uncertainty in the number of absorbed photons from the light field. Theheating corresponding to this random walk process can be expressed in terms of a

14 Chapter 2

momentum diffusion coefficient D [5].In the laser cooling and trapping field it is convenient to define the temperature

of an atomic sample for each degree of freedom separately by kBTi/2 = m〈v2i 〉/2,

with kB Boltzmann’s constant and m the atomic mass [5]. The equilibrium temper-ature which can be reached by laser cooling can be found by comparing the heatingrate 〈Eheat〉 = D/m, introduced by the random walk process, with the cooling rate〈Ecool〉 = 〈Fspvi〉 = β〈v2

i 〉 of the damping force:

kBTi = Dβ

= −Γ4

(Γ

2∆L+ 2∆L

Γ

). (2.12)

The minimum temperature is reached for ∆L = −Γ/2: kBTD ≡ Γ/2, and is calledthe Doppler limit. For neon the Doppler temperature and corresponding Dopplervelocity equal TD = 196 µK, and vD =

√〈v2〉 = 0.29 m/s, respectively.

Most applications of laser cooling described in this thesis consider only Dopplercooling with corresponding temperatures given by Eq. (2.12). However, experimentshave shown that, by using polarization gradients or a homogeneous magnetic field,laser cooling below the Doppler limit is possible [7–11]. A new limit may be consid-ered caused by the exchange of the recoil energy ER = 2k2L/(2m) with the absorp-tion or emission of a single photon. The recoil velocity equals vR = 0.031 m/s formetastable neon, which corresponds to a recoil temperature defined by TR = 2ER/kBof TR = 2.3 µK. By eliminating the spontaneous emission process, cooling below therecoil limit is possible. This can be achieved in two ways. One uses optical pumpinginto a velocity-selective dark state (VSCPT) [12], while another scheme uses velocityselective Raman transitions within a Zeeman multiplet [13].

Even much lower temperatures can be reached with a different cooling technique,called evaporative cooling. This technique, first described by Hess [14], is based onthe removal of the hottest particles from an atomic sample and was first used onatomic hydrogen [15]. With this technique temperatures below 1 µK can be reached.At such low temperatures, and high enough atomic densities, Bose-Einstein conden-sation (BEC) can occur [16].

3 Applications of laser cooling

3.1 Magneto-optical trap (MOT)

The most common neutral atom trap makes use of both optical and magnetic fieldsto form a magneto-optical trap (MOT), which was first demonstrated by Raab et al.[17]. The principle of the MOT is shown schematically in Fig. 2.4. Three orthogonalpairs of counter propagating, circularly polarized laser beams intersect at the centerof a magnetic quadrupole field, generated by two anti-Helmholz coils. The force onthe atoms is velocity and position dependent and is for each direction ξ = x,y orz given by F(ξ,vξ) = F+(ξ, vξ) + F−(ξ, vξ), with F±(ξ, vξ), analogous to Eq. (2.8)written as

F±(ξ, vξ) = ±kLΓ2

s01+ s0 + (2∆±(ξ, vξ)/Γ)2

, (2.13)

Laser cooling and trapping 15

σ+

σ+

σ+

σ−

σ−

σ−

B

x y

z

Figure 2.4: Schematic view of a magneto-optical trap. Two anti-Helmholz coils gen-erate a spherical quadrupole magnetic field, indicated with the small arrows. Threepairs of counter propagating, opposite polarized, laser beams intersect at the originof the quadrupole field.

with the effective detuning for each beam given by ∆±(ξ, vξ) = ∆L∓kLvξ±µ′B/. Forsmall displacements and velocities the force can be expanded analogous to Eq. (2.10),resulting in

F = −βvξ − κξ, (2.14)

with β the damping coefficient defined in Eq. (2.10), and κ the spring constant givenby

κ = µ′GkL

β, (2.15)

with µ′ the effective magnetic moment given by Eq. (2.11) and G the magnetic fieldgradient. Note that the magnetic field gradient, produced at the center of the trap,has in the direction along the coil axis (z direction) twice the strength of that in theradial direction (x or y direction). The same holds for the spring constant for whichwe write κ ≡ κz = 2κx = 2κy .

The motion of the atoms in the MOT is characterized by that of a harmonic os-cillator with damping rate γ = β/m and oscillation frequency ω = √

κ/m [5]. Fortypical values of the magnetic field gradient G ≈ 0.1 T/m, the oscillation frequencyis a few kHz, much smaller than the damping rate which is a few hundred kHz.This leads to an overdamped atomic motion with a damping time 2γ/ω2 less than amillisecond.

The temperature of the atoms trapped in a MOT is the same as the temperatureof optical molasses [5]. When the atomic density is low enough, the spatial andmomentum distribution of the atoms is Gaussian and the cloud can be characterizedby three rms radii σx, σy , σz, and one temperature T [18]. The equipartition theoremgives then for each degree of freedom a relation between the temperature and the

16 Chapter 2

radius:12κiσ 2

i = 12kBT . (2.16)

Townsend et al. [18] call this regime the temperature limited regime, since the radiusis limited by the temperature. At higher atomic densities the interaction between theatoms has to be taken into account. Then the density is limited by repulsive forcesbetween the atoms caused by reabsorption of emitted photons. Densities whichcan be reached in MOT’s are limited to ∼ 1011 atoms/cm3 [18]. Ketterle et al. [19]proposed a scheme in which the photon-induced repulsion is reduced by pumpingthe atoms in the center of the trap into a dark ground state that is not sensitive tothe trapping light. Using this scheme, known as the dark spontaneous-force opticaltrap (dark SPOT), much higher densities could be reached. Because of the lack of adark state, this scheme could not be applied on metastable atom MOT’s. The atomicdensity in the metastable neon MOT described in chapter 4 of this thesis is lowenough that it operates in the temperature limited regime.

3.2 Far-off resonance trap (FORT)

Magneto-optical traps have the disadvantage that the frequency of the trappingbeams must be near resonance. This causes density limitations by trap loss dueto excited state collisions and the reabsorption of scattered photons. Dipole traps,such as a far-off resonance trap (FORT), make use of the dipole force instead of thespontaneous force. This has the advantage that it can be operated far from reso-nance, resulting in negligible excited state population [3]. The simplest form of aFORT consists of a single, linearly polarized, strongly focused Gaussian laser beam.Since the dipole force given by Eq. (2.9) is conservative, the trapping potential Udip

can be found by integrating the force:

Udip = −∫FdipdR = ∆L

2ln[1+ s0

1+ (2∆L/Γ)2

], (2.17)

which for large laser detunings and high intensities becomes

Udip s0Γ8∆L/Γ

. (2.18)

Because the trap depth is proportional to the light intensity, and inversely propor-tional to the laser detuning, the trap can be very deep even at large detunings whenhigh intensities are used. Consequently the fraction of atoms in the excited stateis very low since this fraction is proportional to s0Γ/(2∆L/Γ)2. The FORT used forthe photoassociation experiments described in chapter 5 of this thesis consists of afocused laser beam with a waist w 10 µm and a total power of P = 1 W, resultingin a trap depth Udip/kB 6 mK. Typical densities in a FORT are ∼ 1012 atoms/cm ata temperature of a few hundred µK [20]. Because FORT’s are rather difficult to load,the amount of trapped atoms is normally very low ∼ 104. However, experimentshave shown that it is even possible to load up to 107 atoms into a FORT [21].

Laser cooling and trapping 17

4 Cold atomic collisions

4.1 Principles

In the experiments described in this thesis, collisions between laser cooled atomsplay an important role. In this section we give a short overview of the physicsinvolved in atomic collisions at low temperatures. Collisions between two struc-tureless particles are generally described by the technique of potential scattering, inwhich the particles interact through a potential V(R) depending only on their rela-tive coordinate R [22]. The effect of the collision is expressed by the total scatteringcross section, which is found by solving the problem quantum mechanically. In theso-called partial-wave analysis the incoming wavefunction, describing the relativemotion of the particles, is expanded in spherical waves, each with an angular mo-mentum 1. The scattering of each partial wave is found by solving the radial partof the time-independent Schrodinger equation. For large internuclear distances thesolutions evolve to simple oscillatory functions, and the total elastic scattering crosssection can be written as [22]

σ(k) = 4πk2

∞∑1=0

(21 + 1) sin2 δ1(k), (2.19)

with k the wave number and δ1(k) the phase shift of the scattered wave relative tothe incident wave.

The number of partial waves that contribute to the collision depends on the col-lision energy E = 2k2/2µ, with µ the reduced mass. At low energy, the numberis reduced by the rotational barrier created by the rotational energy part Vrot of theinteraction potential:

U(R) = V(R)+ Vrot = V(R)+ 21(1 + 1)2µR2

. (2.20)

If the collision energy is lower than the 1 = 1 barrier, only central collisions (1 = 0)contribute to scattering. In this so-called s-wave scattering regime the phase shiftvaries as δ0(k) = −ka, with a the s-wave scattering length defined by

a = − limk→0

tanδ0(k)k

. (2.21)

For such ultracold collisions the total elastic scattering cross section for two identi-cal particles approaches

σ(k) k→0= 8πa2, (2.22)

where the factor of 8 instead of factor 4 occurs due to identical particle symmetry [3].The elastic scattering cross section determines the thermalization time of a dense

sample of atoms. This means that evaporative cooling of a cold sample of atoms witha large scattering length, e.g. sodium, is easier than cooling of a sample of atomswith a small scattering length, e.g. hydrogen [5]. The sign of the scattering length is

18 Chapter 2

important for the stability of a Bose-Einstein condensate. A positive scattering lengthprovides a stable condensate while for a negative scattering length a condensate isonly stable when it contains a small number of atoms. The scattering length can beestimated in several ways: one can measure the thermalization time of a cold sampleof atoms or precisely determine the ground state interaction potential. The latter canbe done by photoassociation spectroscopy, described later in this chapter. To findan expression for the potentials describing the interaction between two collidingparticles, i.e. Eq. (2.20), first the atomic structure must be analyzed. Since in thisthesis we describe experiments with rubidium atoms as well as metastable neonatoms, we describe briefly the atomic structure of alkali-metal atoms and rare-gasatoms below.

4.2 Atom-atom interactions

Alkali-metal atoms

The electron configuration of a ground state alkali-metal atom, consists of a num-ber of closed shells, called the core, with one valence electron. Since the orbitalmomentum of the core is zero, the orbital angular momentum of the atom is fullydetermined by the angular momentum of the valence electron. The latter consists ofan angular part l and a spin part s which are coupled to j = l+s. The electron angu-lar momentum is coupled to the nuclear spin i by the hyperfine interaction to formtotal angular momentum f = j+i. As an example of the fine and hyperfine structureof alkali-metal atoms we consider the isotope 85Rb with nuclear spin i = 5/2. The85Rb ground state, 2S1/2 in Russel-Saunders notation, is split by the hyperfine inter-action into f = 2 and f = 3 states. Exciting the valence electron to the 5p orbital,the 2P1/2 and 2P3/2 states are formed with f = 2, 3 and f = 1, 2, 3, 4 hyperfine states,respectively. Figure 2.5 shows a schematic diagram of the 85Rb energy levels. TheS1/2(f = 3) → P3/2(f = 4) transition, which is used for laser cooling, can be pumpedwith ordinary diode lasers.

The interaction between two ground state alkali-metal atoms is given by the cen-tral, electronic and hyperfine interaction. At long range the central electronic inter-action is written as V(R) = −Cn/Rn. The dispersion coefficient Cn is for groundstate collisions (S − S) given by the van der Waals interaction with n = 6. At verylong internuclear separation, the hyperfine-interaction dominates and three differ-ent atomic hyperfine ground state potentials can be distinguished. If one atom is inthe excited state, the central interaction is dominated by the ±C3/R3 dipole-dipoleinteraction, which for large internuclear separation is comparable to the spin-orbitcoupling. Neglecting the hyper-fine interaction sixteen S − P -potentials are distin-guished. In chapter 5 of this thesis we investigate the spin-orbit coupling betweentwo of those potentials for the 85Rb2-dimer. The spin-orbit coupling can cause fine-structure changing collisions.

Inelastic collisions such as fine-structure-changing and hyperfine-changing colli-sions are important contributions to heating and loss in alkali-metal traps. Trap lossmeasurements were done intensively for alkali-metal traps [3]. Typical loss rates,

Laser cooling and trapping 19

52S1/2

5 2P1/2

5 2P3/2

f43213

2

23

D1

D2

λ0= 780 nm

Figure 2.5: Schematic energy diagram of 85Rb with on the left the fine structurestates and on the right the hyperfine splitting. The D1 (795 nm, 5 S1/2 →5 P1/2), D2

line (780 nm, 5 S1/2 →5 P3/2), and the transition used for laser cooling are indicated.

usually expressed in cm3/s, are for hyperfine-changing collisions of the order of10−11 cm3/s, and for fine-structure-changing collisions of the order of 10−15 cm3/s.This means that a trap with an atomic density of 1011 atoms/cm3 will self-destructwithin one second when hyperfine-changing collisions are not suppressed.

Rare-gas atoms

The rare gases differ from the alkalis in that, as a general rule, the most abun-dant isotopes have total nuclear spin equal zero, so that hyperfine structure is notpresent [23]. On the other hand, singly excited rare-gas atoms are complicated byextensive fine structure, which we illustrate by considering neon. Figure 2.6 showsa part of the energy level scheme of neon. A ground state neon atom, 1S0 in Russel-Saunders notation, has an electronic configuration (1s)2(2s)2(2p)6. Excitation ofthe rare-gas atoms is not easily possible via optical transitions from the ground statesince this requires extreme-ultraviolet lasers. Fortunately, two of the first-excitedstates are metastable and can be used as effective ground states in laser coolingprocesses. When exciting one electron to a 3s orbital, four fine-structure states inthe (1s)2(2s)2(2p)5(3s) configuration are distinguished; two of them, the 3P0 and3P2 states, are metastable and have a calculated natural lifetime τ = 430 s andτ = 24.4 s, respectively [24]. The second set of ten excited states, numbered by αiin order of decreasing energy, has an electronic configuration (1s)2(2s)2(2p)5(3p)and are all short-lived (lifetime ≈ 20 ns). The lifetime of the α9 (3D3) state isτ = 19.4 ns [25] and the corresponding linewidth Γ = 8.2 (2π )MHz. For laser coolingand trapping of metastable neon the closed Ne(3s) 3P2 ↔(3p) 3D3 optical transition ata wavelength λ0 = 640.2 nm is used. This wavelength regime can be reached with

20 Chapter 2

1P1

3P2

3P1

3P0

1S0

3D3

λ0= 640.2 nm

J=0 J=1 J=2 J=30

16.5

16.9

18.2

18.6

19.0

E (

eV)

αi

1

3 2 45 67 8 9

10

Figure 2.6: Partial energy level scheme of neon. For the 3s fine-structure multipletwe use the Russel-Saunders notation; the 3p multiplet is numbered by αi in orderof decreasing energy. The (3s) 3P2 ↔(3p) 3D3 closed level transition, used for lasercooling, is indicated.

dye-lasers.The long-range interaction between two metastable rare-gas atoms is given by a

quadrupole-quadrupole term and the Van der Waals interaction: V(R) = −C5/R5 −C6/R6. The quadrupole-quadrupole term is caused by the quadrupole moment of theunfilled core, and is rather small compared to the Van der Waals interaction becausethe core is tightly bound [26]. The S−P interaction is again dominated by the C3/R3

dipole-dipole interaction; however, the quadrupole-quadrupole term for the heavierrare-gas atoms cannot be fully neglected [26].

The internal energy of the rare-gas metastable states is always large enough toallow for ionization during a collision. The internal energy of the neon 3P2 stateequals 16.6 eV which has to be compared to the neon ionization energy of 21.6 eV.The process of ionization for the neon 3P2 state is given by

Ne(3P2)+Ne(3P2) → Ne(1S0)+Ne+ + e− (PI)→ Ne+2 (v)+ e− (AI). (2.23)

The first reaction is called Penning ionization (PI) and results in an ion and a groundstate atom, both carrying an energy of 100 to 500 K. In the second reaction, calledassociative ionization (AI), a molecular ion is formed which contains the internalenergy in the form of vibrational energy (vibrational level labeled with v in Eq. (2.23)).Both reactions produce an electron with an energy of the order of 12 eV.

Laser cooling and trapping 21

The process of ionization is a major loss process in metastable atom traps.This loss process can be introduced in the scattering potential V(R) by addinga complex term: W(R) = V(R) − iΓ(R)/2, with Γ(R) the so-called autoionizationwidth. Doery et al. [26] used such potentials to calculate the ionization rate formetastable neon in an energy range of ≤ 1 mK, and found for an unpolarized sampleKi = 8×10−11 cm3/s. This means that a cold sample of metastable neon atoms, witha typical density of 1010 atoms/cm3, has an ionization lifetime τi of approximatelyone second. Fortunately, the ionization rate can be suppressed by spin-polarizingthe atoms. By spin-polarizing the atoms, the initial state in Eq. (2.23) has total elec-tron spin S = 2, while the final state can only have electron spin S = 0,1. Sincethe ionization process conserves electron spin, ionization is prohibited [26]. Cal-culations show that in the case of metastable neon the ionization rate may be sup-pressed by approximately four orders of magnitude [26]. In a magneto-static trapthe ionization lifetime is then approximately 200 s, much higher then the naturallifetime of the neon 3P2 state. The calculations done by Doery et al. [26] were basedon modified Na2 potentials, since no experimental data is available for metastableneon. An experimental technique to derive potential parameters is photoassociationspectroscopy, which is applied frequently on alkali-metal atoms [27–29] but can alsobe done in metastable atom traps [30,31].

4.3 Photoassociation spectroscopy

A powerful experimental technique to study cold atomic collisions is photoassoci-ation spectroscopy (PAS). The principle of PAS is shown schematically in Fig. 2.7.Atoms trapped in a MOT or FORT are illuminated by a probe-laser beam with a cer-tain frequency ν . During a collision of two ground-state atoms, one of them can beexcited at a certain internuclear distance and an excited molecular state is formed.The molecule dissociates again while the molecule decays back to the ground state.Normally the kinetic energy which is gained during this process is higher than thetrap depth which results in trap loss. This trap loss is measured as a function of thefrequency ν of the probe laser beam.

An excited diatomic molecule can only be formed when the frequency of theprobe laser is resonant with the transition from the ground state to a bound vibra-tional state v . Scanning the probe laser frequency results in a series of vibrationalenergies, which is indicated on the left in Fig. 2.7. Since the collision energy E is ofthe order of a few mK, the resolution of the PAS spectrum is of the order of 10 MHz.From such spectra a lot of information is gained of the collision dynamics.

The spacings between the vibrational levels is determined by the long-range partof the S − P -potential, i.e., the C3/R3 part of the central interaction. The relative in-tensity of the peaks in the spectrum gives information about the long range part ofthe S − S potential, i.e., the C6/R6 part of the central interaction, since the transitionrate is determined by the overlap between the ground and excited state wave func-tions. Knowing the long-range ground state potential the s-wave scattering lengthcan be determined with only limited knowledge of the short-range potentials [32].When the collision energy E is higher than the rotational energy barrier for 1 = 1,

22 Chapter 2

vv-1

v-2

Signal

E

ν2

S+S

S+Pν1

R2 R1R

V(R)

Figure 2.7: Principle of photoassociation spectroscopy. During a binary S+S collisionone of the atoms is excited to the P -state and a diatomic molecule is formed. Whenthe excited molecule decays, the atoms gain kinetic energy and can leave the trap.This trap loss is measured as a function of the frequency ν of the probe laser beam.

also the rotational energy spacing can be measured. This spacing gives informationabout the shape of the S − P potential, as well as possible couplings to other ex-cited state potentials. In Chapter 5 we describe a photoassociation experiment inwhich we measured the rovibrational spacings of two excited state potentials of the85Rb2 dimer. From the photoassociation data, a qualitative picture of the spin-orbitcoupling between the two states was developed.

References

[1] R. Frisch, Z. Phys. 86, 42 (1933).

[2] S. Chu, L. Hollberg, J.E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55,48 (1985).

[3] J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne, Rev. Mod. Phys. 71(1), 1, (1999).

[4] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-photon interactions:basic processes and applications, Wiley New York, 1992.

[5] H.J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-VerlagNew York, 1999.

Laser cooling and trapping 23

[6] See the special issue on laser cooling and trapping of atoms in J. Opt. Soc. Am.B 6, 2084 (1989).

[7] P.D. Lett, W.D. Phillips, S.L. Rolston, C.E. Tanner, R.N. Watts, and C.I. Westbrook,J. Opt. Soc. Am. B 6, 2084 (1989).

[8] P. Lett, R. Watts, C. Westbrook, W.D. Phillips, P. Gould, and H. Metcalf, Phys. Rev.Lett. 61, 169 (1988).

[9] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).

[10] M.D. Hoogerland, H.C.W. Beijerinck, K.A.H. van Leeuwen, P. van der Straten, andH.J. Metcalf, Europhys. Lett. 19, 669 (1992).

[11] B. Sheehy, S-Q. Shang, P. van der Straten, and H.J. Metcalf, Phys. Rev. Lett. 64,858 (1990).

[12] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji,Phys. Rev. Lett. 61, 826 (1988).

[13] M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).

[14] H. F. Hess, Phys. Rev. B. 34, 3476 (1986).

[15] N. Masuhara, J.M. Doyle, J.C. Sandberg, D. Kleppner, T.J. Greytak, H.F. Hess, andG.P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).

[16] M.H. Anderson et al., Science 269, 198 (1995); K.B. Davis et al., Phys. Rev. Lett.75, 3969 (1995); C.C. Bradley et al., Phys. Rev. Lett. 75, 1687 (1995); D.G. Friedet al., Phys. Rev. Lett. 81, 3811 (1998).

[17] E.L. Raab, M. Prentiss, A. Cable, S. Chu, and D.E. Pritchard, Phys. Rev. Lett. 59,2631 (1987).

[18] C.G. Townsend, N.H. Edwards, C.J. Cooper, K.P. Zetie, C.J. Foot, A.M. Steane, P.Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995).

[19] W. Ketterle, K.B. Davis, M.A. Joffe, A. Martin, and D.E. Pritchard, Phys. Rev. Lett.70, 2253 (1993).

[20] R.A. Cline, J.D. Miller and D.J. Heinzen, Phys. Rev. Lett. 73, 632 (1994).

[21] K.L. Corwin, S.J.M. Kuppens, D. Cho, and C.E. Wieman Phys. Rev. Lett. 83, 1311(1999).

[22] C.J. Joachain, Quantum Collision Theory, North-Holland, Amsterdam, 1975.

[23] Argon, Helium, and the Rare Gases, edited by G.A. Cook (Interscience Publish-ers, New York, 1961).

[24] N. Small-Warren and L. Chin, Phys. Rev. A 11, 1777 (1975).

24 Chapter 2

[25] S.A. Kandela and H. Schmoranzer, Phys. Lett. 86a, 101 (1981).

[26] M.R. Doery, E.J.D. Vredenbregt, S.S. Op de Beek, H.C.W. Beijerinck, and B.J. Ver-haar, Phys. Rev. A 58, 3673 (1998).

[27] L.P. Ratliff, M.E. Wagshul, P.D. Lett, S.L. Rolston, and W.D. Phillips, J. Chem. Phys.101, 1994.

[28] J.D. Miller, R.A. Cline, and D.J. Heinzen, Phys. Rev. Lett. 71, 2204 (1993).

[29] J.P. Burke, Jr., C.H. Greene, J.L. Bohn, H. Wang, P.L. Gould, and W.C. Stwalley,Phys. Rev. A 60, 4417 (1999).

[30] N. Herschbach, P.J.J. Tol, W. Vassen, W. Hogervorst, G. Woestenenk, J.W. Thom-sen, P. van der Straten, and A. Niehaus, Phys. Rev. Lett. 84, 1874 (2000).

[31] M.R. Doery, E.J.D. Vredenbregt, J.G.C. Tempelaars, H.C.W. Beijerinck, and B.J.Verhaar, Phys. Rev. A 57, 3603 (1998).

[32] F.A. van Abeelen and B.J. Verhaar, Phys. Rev. A 59, 578 (1999).

Chapter 3

Intense beam of cold metastableNe(3s) 3P2 atoms

1 Introduction

The proposal of using resonant laser light to cool and trap neutral atoms, stemsalready from 1975 [1,2]. Since then the cooling and trapping field has made a hugedevelopment starting with the first experimental data by Chu et al. [3] in 1985. Cool-ing and trapping of atoms made it possible to study collisions between cold atoms.As a consequence, a lot more is known about the collision dynamics of cold atoms. Arecent review of experiments and theory in cold and ultracold collisions is given byWeiner et al. [4]. This review includes advances in experiments with atomic beams,light traps, and purely magnetic traps. Most of the experiments are dealing withalkali-metal atoms, while collision experiments with metastable rare-gas atoms havebeen somewhat neglected [5–9]. This is due to the relative ease of building and oper-ating optical traps for alkali-metal atoms. In the case of rare-gas atoms it is not pos-sible to excite atoms from the ground state, since this requires extreme-ultravioletlasers. So no traps can be produced within a vapor cell, like in the alkali-metal case.

Fortunately, two of the first excited states of the rare-gas atoms, with the excep-tion of helium, are metastable and one of them can be used for laser cooling (heliumhas only one metastable first excited state). The production of these metastablestates, however, is very inefficient: at best, only about 1 in 103 atoms, excited in aplasma discharge source, are in the metastable state [10]. Furthermore, high-energy,background atoms, produced by such a plasma discharge, limit the trap density ifthe source is situated near the trapping region. That is why rather complicatedbeam-brightening techniques must be applied to be able to reach high trap densi-ties.

In this chapter we describe the experimental techniques to produce an intensebeam of cold metastable Ne(3s) 3P2 atoms. The beam, with a flux of 5×1010 atomsper second, has a transverse and longitudinal beam temperature of 285 µK and28 mK, respectively. This beam can be used for a variety of cold collision experi-ments, e.g., to study collision rate constants as a function of velocity, alignment and

25

26 Chapter 3

Collimator

TransverseDopplerCooler

Zeeman Slower

Magneto-OpticalCompressor

TransverseSub-Doppler

Cooler

Source Bright Beam

TransverseDopplerCooler

y

z

Figure 3.1: Schematic view of the neon beam line. Six laser cooling sections are in-dicated starting from the source with the collimator, the first transverse Dopplercooler, the Zeeman slower, the second transverse Doppler cooler, the magneto-optical compressor and the sub-Doppler cooler.

orientation of the colliding partners, as was done by Wang and Weiner [11] for ther-mal alkali-metal collisions and by Thorsheim et al. [12] for subthermal alkali-metalcollisions. In the last decade, in several other places metastable atom beams weredeveloped [13–17] and were used for various experiments [18–23]. We show in chap-ter 4 that our intense neon beam is ideal for loading a magneto-optical trap (MOT),as has been done by Woestenenk et al. [21] and Herschbach et al. [23] for metastablehelium.

Figure 3.1 shows a schematic overview of the neon beam line, with the six lasercooling sections necessary to create an intense beam of cold atoms. In this chapterwe describe the operation of the different laser cooling techniques which are appliedin the beam line, starting with an overview of the complete setup in section 2. In sec-tion 3 a description of the laser collimator and the first transverse Doppler coolingstage is given.

The collimator captures metastable Ne(3s) 3P2 atoms from a discharge-excitedsupersonic expansion operated with LN2 cooling and collimates them into a parallelbeam. An extra transverse Doppler-cooling section reduces the divergence of thebeam to a few times the Doppler limit. In the Zeeman slower, described in sec-tion 4, the atoms are axially slowed from 500 to 100 m/s. Again, an extra transverseDoppler-cooling section, positioned in between the two Zeeman solenoids, reducesthe divergence of the slowed atomic beam. In section 5, we describe the magneto-optical compressor (MOC), which captures the slowed atoms and funnels them intoa narrow beam. In addition sub-Doppler cooling is achieved in the last portion ofthe beam-line. The effects of the sub-Doppler cooler are also shown in section 5.Finally, in section 6 we give an overview of the beam characteristics and show thatthe intense beam of cold neon atoms is very useful for various experiments, e.g., forloading a magneto-optical trap (MOT).

Intense beam of cold metastable Ne(3s) 3P2 atoms 27

Table I: Characteristic quantities of the neon atom and the laser cooling transitionNe(3s) 3P2 ↔Ne(3p) 3D3.

Quantity Symbol Value

mass m 33.2×10−27 kginternal energy of 3P2 state Ei 16.6 eVwavelength λ0 640.225 nmwavevector k = 2π/λ0 9.81× 106 m−1

spontaneous decay rate Γ 8.20 (2π ) MHznatural lifetime of 3D3 state τ = 1/Γ 19.42 nssaturation intensity σ light I0,σ 4.08 mW/cm2

saturation intensity π light I0,π 7.24 mW/cm2

recoil velocity vrec = k/m 0.031 m/sDoppler limit, velocity vD = (Γ/2m)1/2 0.29 m/sDoppler limit, temperature TD = Γ/2kB 196 µK

2 Experimental setup

2.1 Laser setup

All laser-cooling sections use the Ne(3s) 3P2 ↔(3p) 3D3 optical transition, which wasalready indicated in Fig. 2.6 of section 4.2 in chapter 2. The most important pa-rameters for this transition are given in Table I. An overview of all the opticalcomponents, necessary to prepare the laser light for the different laser cooling sec-tions, is given in Fig. 3.2. All laser cooling sections are operated by using a singlecontinuous-wave single-frequency ring dye laser, (Coherent, type 899-21). The dyelaser typically has an output power of 700 mW while pumped with 7 W of light froma argon ion laser, type Coherent Innova 300. The dye laser is locked to a frequencythat is Zeeman shifted almost two line widths (∆L = ωL −ω0 = −1.8Γ ) to the redof the Ne (3s) 3P2 ↔(3p) 3D3 transition by using saturated absorption spectroscopy.The linewidth of the laser is about 1 MHz. Acousto optic modulators (AOMs) areused to shift the laser frequency for the collimator and the Zeeman slower. Opticaltelescopes are used to expand the laser beams to the required sizes. The laser beamcharacteristics of each laser cooling section are given in Table II. The average laserintensity 〈I〉, given in the table, is found by dividing the total laser power P by thearea 2wx,y × 2wz, with the waist radius taken to be the 1/e2 intensity drop of aGaussian laser beam.

2.2 Beam diagnostics

Figure 3.3 shows an artist’s impression of the beam line. The total length of thebeam line, taken from the nozzle of the discharge, is approximately 3 m. Besides

28 Chapter 3

Argon Ion LaserDye-Laser

Ne Cell*

CB

PD

PD

λ/4λ/2PCB

λ/2PCB

λ/2PCB

λ/2PCB

λ/2PCB

AOM400 MHz

AOM80 MHz λ/4 λ/4 λ/2

DopplerCooler

DopplerCooler

Collimator

Compressor

ZeemanSlower

Sub-DopplerCooler

700 mW

250 mW

130 mW 90 mW

40 mW

10 mW

10 mW

50 mW 80 mW

CT T

T

T

T

T

T

CT

CTCT

Figure 3.2: Schematic view of the optical components necessary for the differentlaser cooling sections of the neon beam line. All laser cooling sections are operatedwith a single dye-laser which is locked 1.8Γ to the red of the Ne (3s) 3P2 ↔(3p) 3D3

transition by using saturated absorption spectroscopy. T: optical telescope , CT:cylindrical optical telescope, CB: cubic beam splitter, PCB: polarizing cubic beamsplitter, PD: photo diode, AOM: acousto optic modulator, λ/2, λ/4: half and quarterwaveplates.

Table II: Laser beam characteristics for the different laser cooling sections.

Section P 2wx,y × 2wz or 2wr 〈s〉= 〈I〉/I01 ∆L/Γ(mW) (mm×mm or mm)

collimator2 4× 20 51× 17 0.3 +81st Doppler cooler 50 51 0.6 -1.8Zeeman slower 90 51 1.5 -502nd Doppler Cooler 40 85× 34 0.6 -1.8MOC3 250 30× 90 1 -1.8sub-Doppler cooler 10 5× 18 1.6 -1.8

1For the Zeeman slower and MOC circularly polarized light is used, I0,σ =4.08 mW/cm2, for the other sections linear polarized light is used, I0,π =7.24 mW/cm2 (Table I).2The collimator laser beam is split in two pairs of laser beams, one pairfor cooling in the x direction and one pair for cooling in the y direction,respectively.3For the MOC not the waist of the laser beam but the dimensions of theinteraction area are given.

Intense beam of cold metastable Ne(3s) 3P2 atoms 29

Source CollimatorZeeman Slower Compressor

Wire Scanners

0 1 2 3 z (m)

Figure 3.3: Artist’s impression of the neon beam line. The main laser cooling sec-tions and the wire scanners for beam diagnostics are indicated. Table III gives adetailed overview of the position and the length of the different parts of the setup.

the four main laser cooling sections, also four wire scanners, positioned along thebeam line for beam diagnostics, are indicated. Table III gives detailed informationabout the position and length of the different laser cooling sections and the positionof the wire scanners. We take the z axis along the beam axis; relative positions, e.g.,the length of the laser cooling sections, are indicated with z′.

A wire scanner consists of a stainless steel wire that can be moved transverselythrough the atomic beam by a stepping motor. When a metastable atom hits thewire it transfers its internal energy, 16.6 eV for the 3P2 state, to the metal, therebyemitting an electron from the wire. This Auger process has a quantum efficiency ofalmost unity [24] since the work function of an electron in the metal is about 5 eV. Byscanning the wire in transverse direction through the atomic beam and measuringthe number of emitted electrons from the wire, a one-dimensional beam profile isgenerated. The beam profile results from integrating the two-dimensional densitydistribution of the atomic beam along the length of the wire:

I(x) = ηAe∫ d/2

−d/2

∫ l/2

−l/2Φm(x,y)dydx ≈ ed

∫ l/2

−l/2Φm(x,y)dy, (3.1)

with ηA ≈ 1 the quantum efficiency of the Auger process, e the elementary charge, dand l the diameter and length of the wire, respectively, and Φm(x,y) the transversespatial distribution function of the beam of metastable atoms. Note that Eq. (3.1)is a lower limit for the beam signal since the exact value of the quantum efficiencyof the Auger process for Ne(3P2) atoms on stainless steel is not known; the reviewarticle of Hotop [24] reports values between ηA = 0.3− 0.91.

We use two types of wire scanners, one type consisting of a single wire with adiameter of 1 mm, and a second type, a so-called crossed-wire scanner, consistingof two perpendicular wires with a diameter of 0.1 mm. Behind the collimator weuse a set of two wire scanners of the first type, scanning in the x and y direction,respectively. A second set of such wire scanners is used 580 mm downstream from

30 Chapter 3

Scanner 4Scanner 3

x

yx'y'

φ=π/4

y-Scan

y'-Scan

y-Wire

x-Wire

y'-Wire

x'-Wire

Detectionplate

Figure 3.4: Crossed-wire scanners downstream of the sub-Doppler cooler used forbeam diagnostics and beam alignment. The x (x′) and y (y ′) wires measure inde-pendently the beam profile in two perpendicular directions while a scan in the y ′

(y) direction is made. Below the wires of scanner 4, a circularly shaped, opticallytransparent detection plate is situated.

Table III: Axial position and length of different components of the beam line.

Device axial position1 z length z′

(mm) (mm)

nozzle 0collimator 43 1501st wire scanner (x,y) 2101st Doppler cooler 230 50beam stop 3002nd wire scanner (x,y) 7901st Zeeman solenoid 1230 8502nd Doppler cooler 2260 402nd Zeeman solenoid 2370 150magneto-optical compressor 2670 90sub-Dopper cooler 2775 183rd wire scanner (x,y) 28004th wire scanner (x′,y ′) 2950Zeeman mirror 30001The axial position indicates the beginning of the differ-ent parts of the setup, measured from the nozzle position(z = 0).

Intense beam of cold metastable Ne(3s) 3P2 atoms 31

the first set (see Fig. 3.3 and Table III). Just behind the sub-Doppler cooler, and150 mm downstream of the sub-Doppler cooler, we use a crossed-wire scanner. Theadvantage of the crossed-wire scanners, shown schematically in Fig. 3.4, is that bymaking a scan in the y ′ (or y) direction, information in the x and y (or x′ andy ′) direction is obtained. The last crossed-wire scanner also contains an opticallytransparent detection area, consisting of a plate of glass covered with a conductingIndium-Tin-Oxide layer, with which the total atom flux can be measured.

Typical values for the measured normalized wire current I/d are in the range 1to 100 nA/mm, and can be translated into a 1D-intensity I by using the relation

1 nAmm−1 = 6.25× 109 atoms s−1mm−1. (3.2)

The total amount N of atoms in the beam can be found by integrating the beamprofile given by Eq. (3.1) over the scan direction x, or can be approximated by multi-plying the height of the beam profile I(0) by the beam diameter dbeam, for which wetake the Full Width Half Maximum (FWHM) of the beam profile:

N =∫I(x)dx ≈ I(0)dbeam. (3.3)

By comparing beam profiles taken with two successive wire scanners, informationabout the divergence of the atomic beam can be found. Experimentally the atomicbeam divergence is found by dividing the difference in FWHM beam diameters bythe distance between the two wire scanners, in formula Θ = (d2−d1)/(z2−z1). Thebeam divergence can also be expressed in terms of the transverse and longitudinalvelocity distribution of the atoms moving in the atomic beam. In these terms wedefine the divergence Θ of an atomic beam by the ratio between the FWHM of thetransverse velocity distribution and the mean longitudinal velocity distribution, informula Θ = 2

√2 ln(2)σv⊥/v||, with σv⊥ the rms value of the transverse velocity

distribution and v|| the mean of the longitudinal velocity distribution of the atoms.In section 6 we give an overview of the beam characteristics obtained from the beamscans made with the wire scanners. In the following sections each of the laser coolingstages will be discussed in detail, as well as the diagnostics and beam properties aftereach stage.

3 Source and collimator

3.1 Metastable atom source

The beam line starts with a discharge-excited supersonic expansion. The source,schematically shown in Fig. 3.5, produces a beam of metastable Ne(3s) 3P2 atoms ina DC discharge that runs through the nozzle of a supersonic expansion [25]. Thenozzle has a diameter of 150 µm, the source pressure is typically 5 mbar, and thedischarge current 7 mA at a voltage of ≈ 400 V. The axial velocity of the atoms isreduced from approximately 1000 m/s to 500 m/s by cooling the source with liquidnitrogen. The resulting center-line intensity of metastable 20Ne atoms is Im(0) =

32 Chapter 3

Ne

LN2

LN2

400 V

Nozzle Skimmer

Figure 3.5: The discharge excited metastable neon source. The boron-nitride nozzleplate is pressed to a reservoir filled with liquid nitrogen. The discharge is drawnfrom the cathode through the nozzle to the skimmer.

Table IV: Characteristics of the metastable neon source.

nozzle diameter 150 µmsource pressure 5 mbardischarge current 7 mAdischarge voltage 400 Vcenter-line metastable intensity Im(0) 2× 1014 s−1sr−1

mean velocity 480 m/svFWHM 100 m/sexcitation fraction 10−4-10−520Ne isotope fraction 0.913P2 fraction 0.83

2×1014 s−1sr−1. Other high-energy products emerging from the source are atoms inthe metastable 3P0 state, metastable atoms of the isotope 22Ne, and UV photons. InTable IV characteristics of the source are given.

The atomic beam, produced by the source, has a divergence ofΘsource = 350mrad,meaning that, at one meter distance from the source the beam has a diameterdbeam = 350 mm! This results in a very low beam intensity, which is proportionalto (dbeam)−2. This is the reason why immediately behind the source a collimationsection is placed, which cools the atoms in transverse direction into a nearly parallelbeam.

3.2 Collimating section

After leaving the source, the atoms enter the collimator in which they are capturedand collimated by a two-dimensional optical molasses. For the molasses beams weuse, analogous to Hoogerland et al. [26], curved wave fronts to achieve a large cap-ture angle while keeping the cooling time as short as possible. The curved wave

Intense beam of cold metastable Ne(3s) 3P2 atoms 33

β0

βn

α/2

150 mm

60 mmΘ

v⊥

v||c/2

Figure 3.6: Schematic view of the collimation section. One pair of mirrors and thepath of one laser beam is shown. The mirrors with a length of 150mm, are separated60 mm from each other and make an angle of α = 1.7 mrad. The laser beamsare injected at β0 = 130 mrad, corresponding to a capture angle Θc = 88 mrad.After each reflection the angle between the laser beam and the atomic beam axis isreduced with amount α, to keep the atoms in resonance with the laser light.

fronts are produced by using the zig-zag method, which is shown schematically inFig. 3.6. The laser light, with detuning ∆L = +65 (2π )MHz, is injected at an angleβ0 = 130 mrad with respect to the plane perpendicular to the atomic beam axis.With each reflection, the angle β is reduced by an amount α = 1.7 mrad, the anglebetween the mirrors. The Doppler shift ∆D seen by the atoms in the collimator isgiven by

∆D = −k · v = −kv|| sinβ+ kv⊥ cosβ ≈ −kv||β+ kv⊥, (3.4)

with v|| and v⊥ the longitudinal and transverse velocity of the atoms respectively,and β the angle with respect to the plane perpendicular to the atomic beam axis.With the condition ∆D + ∆L = 0 we find for the resonant transverse velocity of theatoms,

v⊥(z) = −∆L/k+ v||β(z). (3.5)

Because β(z) changes while the atoms move downstream through the collimator,also the transverse velocity v⊥(z) changes through the collimator. At the entranceof the collimator this velocity is, for β0 = 130 mrad and ∆L = +65 (2π)MHz, v⊥(z =0) = 21 m/s, resulting in a capture angle Θc = 2v⊥/v|| = 88 mrad. At the end ofthe collimator at z = 150 mm, i.e., after n = 12 reflections of the laser beam, theresonance velocity will be v⊥(z = 150 mm) = 11 m/s, corresponding to a beamdivergence Θcol = 46 mrad.

The advantage of the multiple laser-beam reflections, used in the zig-zag method,is the reduction in required laser power. The total laser power used for the collimatoris 80 mW, i.e., 20 mW per laser beam (Table II). Other methods, e.g., by using a broad

34 Chapter 3

-20 -10 0 10 2020

30

40

50

60

Detector Position (mm)

Cur

rent

(nA

/mm

)

77 mrad

Figure 3.7: Line-integrated profile of the atomic beam just behind the collimator,showing Θc = 77 mrad capture range. Full line: beam profile with collimation laseron; broken line: beam profile with collimation laser off.

converging laser beam [16], would require more than 300 mW of laser power forachieving the same capture angle.

Figure 3.7 shows experimental beam profiles taken in the x direction with thefirst wire scanner which is placed just behind the collimator. The effect of thecollimator is clearly visible. The atoms are captured and cooled in transverse di-rection, resulting in an atomic beam with a diameter of dcol = 9 mm containingNcol = 1.1 × 1012 atoms/s in the 3P2 state. Measured beam characteristics are listedin Table VI. From the beam profiles in Fig. 3.7 the capture angle Θc of the collimatorcan be estimated at Θc = 77 mrad. This is approximately the same as found withEq. (3.5).

The large effective detuning kv⊥ at the end of the collimator results in a trans-verse velocity of almost 11 m/s, as mentioned above. Measurements show a beamdivergence Θcol = 23 mrad, i.e., a transverse velocity of 5.5 m/s, which is still muchmore than the Doppler velocity (Table I). This is why an extra transverse Dopplercooling stage is positioned immediately behind the collimator. Extra transverseDoppler cooling reduces the divergence of the atomic beam. This can be seen inFig. 3.8, which shows beam profiles taken in the x direction with the second wirescanner, 580 mm downstream of the collimator. Measurements have shown thatwith extra transverse Doppler cooling the divergence of the beam reduces to ap-proximately 10 mrad, corresponding to seven times the Doppler limit.

Switching the collimation and Doppler cooling laser off, the beam flux decreasesdrastically, as shown by the dashed line in Fig. 3.8. This beam profile contains,besides atoms in the 3P2 state, also parasitic products of the source, i.e., atoms in themetastable 3P0 state, metastable atoms of the isotope 22Ne, and UV photons. Thoseproducts could disturb the collision experiments that the atomic beam is used for.

Intense beam of cold metastable Ne(3s) 3P2 atoms 35

-20 -10 0 10 200

5

10

Detector Position (mm)

Cur

rent

(nA

/mm

)

Figure 3.8: Influence of extra transverse Doppler cooling behind the collimator. Thebeam profiles are taken 580 mm downstream of the collimator; full line: collimationand Doppler cooling laser on; dash-dotted line: collimation laser only; dashed line:source only.

Therefore a beam stop, consisting of a disc with a diameter of 3 mm, is positionedin the center of the beam just behind the transverse Doppler cooler. The beam stopkeeps the parasitic products from reaching the end of the setup, while most of theatoms in the 3P2 state can pass due to parabolic character of the beam leaving thecollimator.

4 Zeeman slower

The collimated and transversely cooled atoms enter a midfield-zero Zeeman slowerin which they are decelerated by a counterpropagating laser beam. The changingDoppler shift ∆D is compensated by making use of the spatial varying Zeeman-effectin the slower [27]. For this slowing process circularly polarized σ+ light is used,which pumps the atoms to the 3P2|mg = +2〉 ↔3D3|me = +3〉 magnetic sublevelsystem. The laser beam is coupled into the Zeeman slower through a mirror witha 3 mm orifice in the center. The mirror is positioned behind the magneto-opticalcompressor (see Fig. 3.1) and transmits the atoms which are captured and moldedinto a narrow beam by the magneto-optical compressor (section 5).

Using σ+ polarized light, the resonance condition for an atom in the |mg = +2〉ground state, moving with longitudinal velocity v(z′), can be written as

kv(z′) = −∆L +∆B = −∆L + µB

B(z′), (3.6)

where kv(z′) is the Doppler shift, ∆L the laser detuning, ∆B = (µB/)B(z′) theZeeman shift due to the magnetic induction B(z′), and µB the Bohr magneton.

36 Chapter 3

Table V: Characteristics of the Zeeman slower.

Solenoid 1 Solenoid 2

length 850 mm 150 mmfield strength/current 7.39× 10−3 T/A −2.56× 10−3 T/Acurrent 3.7 A 6.8 A∆L −400 (2π) MHz −400 (2π) MHzmean capture velocity 480 m/smean final velocity 98 m/s

Slowing atoms with initial longitudinal velocity vi to final velocity vf , with uni-form deceleration a = ηamax = ηkΓ/(2m), η ≤ 1, requires a distance

∆z′ = z′f − z′

i =v2i − v2

f

2a, (3.7)

which gives for the longitudinal velocity v(z′)

v(z′) = vi

√√√√√1− z′ − z′i

z′f − z′

i

1−

v2f

v2i

. (3.8)

Combining this expression with the resonance condition, Eq. (3.6), the magnetic fieldB(z′) must have the following shape to get uniform deceleration

B(z′) = µB∆L + k

µBvi

√√√√√1− z′ − z′i

z′f − z′

i

1−

v2f

v2i

. (3.9)

The required field is produced by two solenoids producing fields in oppositedirection. In between the solenoids the magnetic field is zero. In the first solenoidthe atoms are slowed down from v(z′ = z′

i1) = vi1 to v(z′ = z′f1) = vf1, over

a distance ∆z′1 = z′

f1 − z′i1. In the second solenoid they are slowed further from

v(z′ = z′i2) = vi2 = vf1 to v(z′ = z′

f2) = vf2 over a distance ∆z′2 = z′

f2−z′i2. Knowing

from the resonance conditions in between the two solenoids that ∆L = −kvf1 we canwrite the magnetic field of the two coils as

B(z′) = B0

1− vi1

vf1

√√√√√1− z′ − z′i1

z′f1 − z′

i1

1−

v2f1

v2i1

z′

i1 ≤ z′ ≤ z′f1, (3.10)

B(z′) = B0

1−

√√√√√1− z′ − z′i2

z′f2 − z′

i2

1−

v2f2

v2f1

z′

i2 ≤ z′ ≤ z′f2, (3.11)

with B0 = ∆L/µB .

Intense beam of cold metastable Ne(3s) 3P2 atoms 37

kvi

kvf

kv(z)

∆B

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

z (m)

kv/2

π (G

Hz)

-0.4

-0.2

0.0

0.2

0.4

0.6

∆ B/2

π (G

Hz)

|∆ L|

Figure 3.9: Schematic representation of the Zeeman slowing process. Full line: Zee-man shift caused by the magnetic field of the Zeeman slower as a function of thedistance along the beam line (z = 0 indicates the beginning of first magnet of theZeeman slower); dashed line: resonance velocity kv(z) of an atom on its trajectorythrough the Zeeman slower, entering the Zeeman slower with velocity vi = 490 m/sand leaving it with velocity vf = 100 m/s. Here ∆L = −400(2π) MHz indicates thedetuning of the slowing laser.

To keep the atoms in resonance during the slowing process, the change in mag-netic field must not exceed the changing Doppler shift [28]

dBdz′ ≤

kµB

av(z′)

= kµB

ηkΓ2mv(z′)

. (3.12)

To fulfill to this expression we choose the total length of the solenoids such that theatoms can follow the magnetic field with a fraction of one tenth of the maximumradiation force, i.e., η = 0.1. In table V the characteristics of the Zeeman slower aregiven. The advantage of a midfield-zero Zeeman slower is that the longitudinal veloc-ity spread of the atoms leaving the slower is minimized because the high magneticfield at the end of the slower forces the atoms to get very abruptly out of resonance.A second advantage of a midfield-zero Zeeman slower is the possibility of putting avacuum pump in between the solenoids.

The Zeeman slowing process is schematically shown in Fig. 3.9, which depictsthe trajectory of an atom at resonance condition as given by Eq. (3.6). The longi-tudinal velocity of the atoms leaving the Zeeman slower is measured by a standardTime-of-Flight (TOF) technique. Figure 3.10 gives the velocity distribution of un-slowed, partially slowed (second Zeeman solenoid operated with reduced current),and fully slowed atoms. The fully slowed atoms have a final velocity of approxi-mately 100 m/s. Measurement show that approximately 50% of the atoms, enteringthe Zeeman slower are slowed to 100 m/s.

38 Chapter 3

0 200 400 600 800 1000

0

5

10

15

20

Velocity (m/s)

Sign

al (

arb.

uni

ts)

80 100 120

8 m/s

Figure 3.10: Measured longitudinal velocity distributions. Dashed line: initial ve-locity distribution, average velocity 500 m/s; open circles: only slowing in the firstmagnet of the Zeeman slower, average velocity 200 m/s; closed circles: slowing inthe full Zeeman slower, average velocity 100m/s. Inset: Close-up of 100m/s velocitydistribution showing 3.4 m/s rms width, i.e., 8 m/s FWHM.

-2 -1 0 1 20

2

4

6

8

10

Detector Position (mm)

Cur

rent

(nA

/mm

)

Figure 3.11: Line-integrated beam profile just behind the magneto-optical compres-sor showing the influence of extra transverse Doppler cooling between the twosolenoids of the Zeeman slower. Full line: beam profile with extra Doppler cool-ing; dashed line: beam profile without extra Doppler cooling.

Intense beam of cold metastable Ne(3s) 3P2 atoms 39

The atoms are not only slowed in the Zeeman slower but also cooled in the longi-tudinal direction. The temperature of the atoms can be derived from the width of thevelocity distributions, using the relation kBT|| = mσ 2

v|| , with σv|| the rms width of thelongitudinal velocity distribution. The 100 m/s velocity distribution has a 3.4 m/srms width, corresponding to a longitudinal beam temperature of T|| = 28 mK; thishas to be compared with the temperature 4.4 K of the 500 m/s atoms.

During the slowing process the transverse velocity of the atoms increases due tothe randomness in the direction of the spontaneously emitted photons. To counter-act this effect, an extra transverse Doppler-cooling stage is put in between the twosolenoids of the Zeeman slower. Transverse Doppler-cooling is possible because themagnetic field is almost zero between the two solenoids, as can be seen in Fig. 3.9.Reducing the divergence of the atomic beam increases the flux of atoms within thecapture range of the magneto-optical compressor. Extra transverse cooling increasesthe beam flux with approximately 50%, this can be seen in Fig. 3.11 which shows thebeam profile behind the magneto-optical compressor.

5 Magneto-optical compressor and sub-Doppler cooler

Due to the slowing process in the Zeeman slower, the diameter and divergence of theatomic beam has increased significantly. This results in a small phase space density(see section 6). A common way to compress an atomic beam in phase space, is touse a two-dimensional version of a magneto-optical trap [29], a so-called magneto-optical compressor (MOC). Below we give a description of the MOC, analogous to thedescription of the magneto-optical trap in section 3.1 of chapter 2.

5.1 Compression of atomic beam

The principle of the MOC is given in Fig. 3.12, showing two orthogonal pairs ofcircularly polarized laser beams, intersecting at the center of a magnetic quadrupolefield produced by four permanent magnets. To explain the operation of the MOC weconsider a hypothetical atom with a J = 0 → J = 1 transition. This simplification ischosen because for the case of the J = 2 → J = 3 transition for neon, the ongoingredistribution over the five magnetic sublevels of the lower state complicates theprocess [30].

The force on the atom is given by the sum of the spontaneous forces F± exertedby the σ+ and σ− beams [29]

F(x,vx) = F+(x,vx)+ F−(x,vx), (3.13)

with F±(x,vx) = ±kΓ2

s01+ s0 + (2∆±(x,vx)/Γ)2

, (3.14)

∆±(x,vx) = ∆L ∓ kvx ∓ µB

∂B∂x

x, (3.15)

where s0 is the saturation parameter for zero detuning, ∆±(x,vx) the effective de-tuning with ∆L the laser detuning, kvx the Doppler shift due to transverse velocity

40 Chapter 3

σ+

σ+

σ-

σ-

B

Figure 3.12: Principle of a two dimensional magneto-optical trap. The solid arrowsindicate the magnetic quadrupole field resulting from the four permanent magnets.The open arrows indicate the light field, consisting of two pair of counterpropagatingσ+ − σ− polarized laser beams. The atoms move perpendicular to the plane of thelaser beams, at the center of the quadrupole field.

-100 -75 -50 -25 0-15

-10

-5

0

5

10

15

x (m

m)

z' (mm)

Figure 3.13: Monte-Carlo simulation of some atomic trajectories in the MOC, showingcompression of the atomic beam.

Intense beam of cold metastable Ne(3s) 3P2 atoms 41

y

xσ+

σ-

BB

B B

Figure 3.14: Schematic view of the magneto-optical compressor. The mirror sectionprovides counterpropagating σ+-σ− laser beams in the x and y direction. Theshaded section indicates the area where the atoms, moving perpendicular to theplane of the laser beams, interact with the laser light.

vx, and (µB/)(∂B/∂x)x the Zeeman shift due to the local transverse magnetic field.The sign difference in the Zeeman term is caused by the excitation of the differentmagnetic subtransitions,mg = 0→ me = 1 for the σ+-beam andmg = 0→ me = −1for the σ−-beam.

For each position x there exists an equilibrium velocity vx,eq = −(µB/k)(∂B/∂x)xtowards the beam axis, for which the forces F± are balanced. As a consequence,the atoms are cooled to this equilibrium velocity and will approach the axis like anoverdamped harmonic oscillator; the cooling force can, for small displacements andvelocities, be written as [31]

F(x,vx) = −β(vx − vx,eq) = −βvx − κx, (3.16)

which is similar to Eq. (2.14) of chapter 2. In Fig. 3.13 some simulated atomic tra-jectories through the MOC are depicted. Atoms captured within the spatial captureradius xc ≈ 10 mm are dragged inwards to the atomic beam axis.

Figure 3.14 shows a schematic view of the MOC implemented in the beam-line.The two pairs of σ+ − σ− laser beams are produced by recycling a single σ+ laserbeam by a mirror section. The transverse size of the MOC equals an area of 2wx,y ×2wz = 30 × 90 mm2, the laser beam has an average saturation parameter of 〈s〉 =〈I〉/I0,σ = 1 and has a detuning of ∆L = −1.8Γ (see Table II).

The magnetic quadrupole field of the MOC is generated by four permanent mag-nets mounted outside of the vacuum system. The magnets, which are made of aNd-Fe-B alloy, have dimensions 60×60×30 mm3 and a magnetization of 1.15 T. Atthe position of the Sub-Doppler cooler four much smaller magnets are positioned(10×5×5 mm3, magnetization 1.15 T). These magnets, which can individually be

42 Chapter 3

-100 0 100 2000.0

0.2

0.4

0.6

z' (mm)

∂B/∂

x (T

/m)

MOC

SD

SG-effect

Figure 3.15: The gradient of the radial quadrupole magnetic field due to the MOC-magnets and the compensating sub-Doppler magnets. The arrows indicate the inter-action regions of the MOC and the sub-Doppler cooler, and the region from wherethe Stern-Gerlach force plays a role (Section 5.2). The position z′ = 0 corresponds tothe end of the interaction area, i.e., the center of the magnets of the magneto-opticalcompressor.

-15 -10 -5 0 5 10 15-0.6

-0.3

0.0

0.3

0.6

x (mm)

F/F m

ax

dB/dx=0.6 T/m

dB/dx=0.3 T/m

dB/dx=0.1 T/m

Figure 3.16: Position dependency of the transverse force on the atoms in the MOCfor vx = 0 m/s, ∆L = −1.8Γ , and s0 = 1. Full line: force at the end of the MOC(z′ = 0 mm), field gradient ∂B/∂x = 0.6 T/m; dashed line: force half way in theMOC (z′ = −50 mm), field gradient ∂B/∂x = 0.3 T/m; dash-dotted line: force at theentrance of the MOC (z′ = −90 mm), field gradient ∂B/∂x = 0.1 T/m.

Intense beam of cold metastable Ne(3s) 3P2 atoms 43

-20 -10 0 10 200

1

2

3

4

5

6

7

Detector Position (mm)

Cur

rent

(nA

/mm

)1 mm

Figure 3.17: Line-integrated profile of atomic beam, taken with the vertical wireof the 3rd wire scanner, positioned 40 mm behind the MOC. Full line: data withcompression laser on; dashed line: compression laser off. The dip, left of the peakin the beam profile with compression laser on, is caused by electrons which areemitted from the second wire of the crossed-wire scanner.

moved in radial direction, are situated much closer to the beam axis and compen-sate locally the magnetic field generated by the MOC magnets. Figure 3.15 showsthe magnetic field gradient of the MOC and sub-Doppler cooler. At the entranceof the MOC the quadrupole field gradient has a strength of 0.1 T/m and increaseslinearly to almost 0.6 T/m at the end of the MOC. The field gradient drops to ap-proximately zero at the position of the sub-Doppler cooler, due to the influence ofthe compensation magnets.

The transverse force on the atoms, for vx = 0 ms−1 and ∆L = −1.8Γ at threedifferent longitudinal positions in the MOC is given in Fig. 3.16. The curve depictingthe force at the entrance of the MOC shows a spatial capture radius of approximatelyxc = 10 mm, which was also shown in Fig. 3.13.

Figure 3.17 shows a beam profile measured with the vertical wire of the third wirescanner. The effect of the MOC is visible: the atoms are molded into a narrow beamwith a diameter of dMOC = 1 mm, containing NMOC = 5 × 1010 atoms/s. This resultsin a particle density of n = N/(πx2

⊥v||) = 6.4× 108 atoms/cm.Measurements show that a 0.25 fraction of the slowed atoms are captured by

the MOC. A beam profile measured with one of the wires of the fourth wire scanneris given in Fig. 3.18, showing a beam profile with a diameter of 2 mm. From thebeam profiles given in Fig. 3.17 and Fig. 3.18, the divergence of the atomic beam wasmeasured to be equal to 8 mrad which approximately corresponds to the transverseDoppler velocity: vD ≈ v⊥ = 0.35 m/s. This results in a transverse beam temperatureT⊥ = 285 µK. In Table VI the characteristics of the atomic beam are given.

44 Chapter 3

-5 0 50

1

2

3

4

5

Detector Position (mm)

Cur

rent

(nA

/mm

)2 mm

Figure 3.18: Line-integrated profile of atomic beam 190 mm behind the MOC. Fullline: data with compression laser on; dashed line: compression laser off.

5.2 Sub-Doppler cooling of the atomic beam

To decrease the divergence of the atomic beam to a value less than the Doppler limit,a transverse sub-Doppler cooling stage is situated 15 mm downstream of the MOC.Two orthogonal pairs of linearly polarized, counterpropagating laser beams give riseto a two-dimensional molasses with polarization gradients. This cooling technique,usually referred to as Sisyphus cooling [32], is expected to give a transverse beamvelocity of 0.11 m/s, i.e., three times less than the Doppler limit.

Steering

We systematically investigated the effects of sub-Doppler cooling on the atomicbeam [33]. Figure 3.19 shows beam profiles measured with the fourth wire scan-ner. Measuring the beam profile with this wire scanner gives a distribution in the x′

and y ′ direction. Each profile is measured with sub-Doppler cooling laser on and off.We found that the sub-Doppler cooling stage has only steering effects on the atomicbeam: no decrease of the atomic beam divergence was observed at all. The steeringeffects are clearly visible in Fig. 3.19; here, the sub-Doppler forces push the atomicbeam upwards by giving the atoms a transverse velocity vy = 3 m/s. We found thatthose steering effects can be explained very well in terms of velocity-selective reso-nances (VSR) [33–35]. The VSR-model says that a resonance velocity of vy = 3 m/s,for example, corresponds to a local magnetic field in the sub-Doppler cooler of 6 G,which is possible when the atomic beam enters the sub-Doppler cooler not exactlyin the middle, where the magnetic field is compensated.

We applied the steering effects for aligning the atomic beam through the orifice

Intense beam of cold metastable Ne(3s) 3P2 atoms 45

0 5 10 15 20 250

1

2

3

4

Detector Position (mm)

Cur

rent

(nA

/mm

)

x' y'

Figure 3.19: Influence of the sub-Doppler cooler on the atomic beam. Beam profilesin the x′ and y ′-direction are obtained by scanning crossed-wire scanner number 4in the y-direction. Full lines: sub-Doppler laser on; dashed line: sub-Doppler laseroff.

in the Zeeman mirror. This can be done by changing the radial positions of thecompensation magnets of the sub-Doppler, thereby changing the magnetic field atthe position of the sub-Doppler light field and, selecting different resonance velocityconditions.

Stern-Gerlach effect

The fact that no sub-Doppler velocities where observed can be explained in termsof Stern-Gerlach forces which the remaining magnetic field downstream of the sub-Doppler cooling stage exert on the atoms [33], as first reported by Koolen [36]. Withinhomogeneous magnetic fields it is possible to deflect atomic beams [37]. By creat-ing quadrupole or hexapole magnetic fields it is even possible to focus atomic beams,as described by Kaenders et al. [38]. The internal energy of an atom with magneticdipole moment µ in an external magnetic field B changes by

Vdip = −µ · B. (3.17)

Considering a moving atom that experiences only small variations in the externalfield, i.e., so-called adiabatic motion, the projection, mg, of the atom’s magneticdipole moment onto the external field remains constant. The corresponding Stern-Gerlach force is then given by

FSG = (µ · ∇)B = −µBggmg ∇|B|, (3.18)

with µB the Bohr magneton and gg (=1.5 in our case) the Lande factor.

46 Chapter 3

The Stern-Gerlach force can be ignored in the MOC and sub-Doppler cooler, be-cause there the optical forces dominate. In the region downstream of the sub-Doppler cooler, however, there is no light field but there is still a strong radialmagnetic field gradient from the MOC-magnets (Fig. 3.15).

Let us consider the influence of the Stern-Gerlach force on the atoms movingin the region indicated in Fig. 3.15, which shows the radial magnetic field gradient.Solving Newton’s second law, we find that an atom in the |mg〉 magnetic substate isaccelerated to

∆v⊥,SG = −µBggmg

mv||

∫|∂B/∂r |dz, (3.19)

with m the atomic mass, v|| the longitudinal velocity of the atoms, and ∂B/∂r theremaining radial magnetic field gradient as indicated in Fig. 3.15. Due to the depen-dence of the Stern-Gerlach force on the magnetic substate, the atoms are deflectedin 2J + 1 different directions. From the measured magnetic field gradient given inFig. 3.15 we estimate that ∆v⊥,SG = 0.2 m/s for the |mg = 2〉 magnetic substate.This leads to an additional beam divergence of 4 mrad. Presumably this contributesto the measured transverse velocity spread v⊥ = 0.35 m/s, which equals a beamdivergence of 8 mrad.

6 Beam characteristics

In the previous sections we showed that, by using several sets of wire scanners, itis possible to estimate the characteristics of the atomic beam such as the flux Nof atoms, the beam diameter and divergence. In atomic beam physics often thebrightness and brilliance are given to characterize a beam of atoms. Below we followLison et al. [39] to give explicit definitions for the brightness and brilliance of anatomic beam.

6.1 Brightness and brilliance

The flux density or intensity of a circularly shaped atomic beam is defined as N/πx2⊥,

where x⊥ is the beam radius. The geometrical solid angle occupied by the atoms ina beam is Ω = π(v⊥/v||)2, with v⊥ and v|| the transverse and average longitudinalvelocity of the atoms, respectively. Then the brightness is defined as the flux densityper solid angle, and is given by

R = Nπx2⊥Ω

. (3.20)

In analogy with the frequency spread of optical beams, i.e., coherence length,atomic beams can be characterized by their longitudinal velocity spread σv|| . Thespectral brightness or brilliance of an atomic beam, is then given by

B = R v||σv||

. (3.21)

Intense beam of cold metastable Ne(3s) 3P2 atoms 47

Note that both R and B have the same dimensions as flux density, usually given interms of s−1mm−2mrad−2, and that for thermal atomic beams R and B have similarvalues because v||/σv|| 1.

Furthermore, the brilliance is related to the Liouville phase-space density Λ,which is the number of particles N per unit of phase-space volume. The relationbetween B and Λ can easily be found by splitting the spatial and momentum co-ordinates in transverse and longitudinal components, which results for a circularlyshaped atomic beam in

Λ = 1π(x⊥p⊥)2

Nx||p||

. (3.22)

Writing, N = Nv||/x||, this can be can be expressed as

Λ = B πm3v4

||, (3.23)

with m the atomic mass. The quantum limit occurs when the quantity defined byΛ = Λh3, with h Planck’s constant, is unity.

6.2 Overall characteristics

From the beam characteristics, measured with the different sets of wire scanners,we calculated the brightness, brilliance and phase space density of our atomic beam,by using Eqs. (3.20), (3.21), and (3.23), respectively. In Table VI the measured beamcharacteristics, such as the beam flux N, the beam diameter d = 2x⊥, the longitudi-nal velocity and the rms value of the transverse velocity are given. With these valuesthe local values of the brightness, brilliance and phase space density were calculated,i.e., the values at the axial position given in the second column of the Table VI (the

Table VI: Characteristics of the atomic beam. All quantities are measured locally atthe positions z (second column) of the different sections mentioned in the first col-umn. S: source, C: collimator, D: Doppler cooler, Z: Zeeman slower, MOC: magneto-optical compressor, and SD: sub-Doppler cooler.

Section z N d v|| σv⊥ R1 B1 Λh3

×1010 ×1020 ×1020 ×10−13(mm) (s−1) (mm) (m/s) (m/s)

S 0 3500 0.15 480 71 288 2733 1285C 210 110 9 480 4.8 0.55 5.3 2.5D 790 110 15 480 1.9 1.27 12.0 5.6Z 2800 19.2 45 98 1.9 1.02×10−3 2.5×10−2 6.7MOC 2800 4.8 1.0 98 0.35 15.0 370 1×105SD 2950 4.8 2.1 98 0.35 3.5 85.0 2.3×1041R and B given in terms of s−1m−2sr−1.

48 Chapter 3

positions of the different wire scanners). Those calculated values are given in thelast three columns of Table VI.

Looking at the measured beam flux, N, it is clear that, moving from the collimatorto the end of the setup, more than a factor of 20 in beam flux is lost. This is partiallycaused by the slowing process in the Zeeman slower. The Zeeman slower slows only50% of the atoms to 100 m/s, and a significant fraction of the slowed atoms do notreach the entrance of the MOC, because the divergence of the atomic beam increasesdue to the slowing process. Furthermore, only 25% of the atoms which reach theentrance of the MOC are captured by the MOC.

As mentioned in section 2.2, the beam divergence can be found by comparingbeam profiles taken with two successive wire scanners. From this it is found thatthe divergence of the atomic beam decreases from 350 mrad at the source to 8 mraddownstream of the magneto-optical compressor. This increases the brightness andbrilliance of the atomic beam drastically, which can be seen more clearly in Table VII,which gives the values of the brightness, brilliance and phase space density at theend of the setup for different operation modes of the atomic beam machine, goingfrom limited operation (only source on) to full operation (all laser cooling sectionson). All values are given relative to the full operation values, which are given in thelast row of Table VI. A graphic representation of this is given in Fig. 3.20 which showsa plot of the brightness and brilliance as a function of the phase-space density, fordifferent operation modes of the beam machine.

From Table VII and Fig. 3.20 it is clear that the brightness, brilliance and phase-space density increase a lot when going from less to full operation. Switching on theZeeman slower, decreases the brightness and brilliance, due to the increasing beamdiameter and divergence and the loss-processes mentioned earlier. The magneto-optical compressor compensates this loss in brightness and brilliance. Comparingthe operation mode with only the source on with the full operation mode, we see thatthe brightness and brilliance increase six orders of magnitude and the phase-space

Table VII: Brightness, brilliance and phase-space density at the end of the setup(z = 2950 mm) relative to the full operation values. RAll = 3.5 × 1020 s−1m−2sr−1,BAll = 8.5 × 1021 s−1m−2sr−1, and ΛAll = 2.3 × 10−9 (Table VI). S: only the sourceon, S+C: source and collimator on, S+C+D: source, collimator and Doppler cooleron, S+C+D+Z: source, collimator, Doppler cooler and Zeeman slower on, All: fulloperation: all laser cooling sections on.

Section R/RAll B/BAll Λ/ΛAll

S 1×10−6 7×10−7 1×10−9S+C 3×10−3 1×10−3 2×10−6S+C+D 8×10−2 3×10−2 5×10−5S+C+D+Z 6×10−5 6×10−5 6×10−5All 1 1 1

Intense beam of cold metastable Ne(3s) 3P2 atoms 49

10-18 10-16 10-14 10-12 10-10 10-81014

1016

1018

1020

1022

Λ h3

(s

-1m

-2sr

-1)

S

S+C

S+C+D

S+C+D+Z

AllBrightnessBrilliance

R B ,

Figure 3.20: Plot of the brightness R (open circles) and the brilliance B (solid cir-cles) vs phase-space density at the end of the setup (z = 2950 mm). S: source only;S+C: source and collimator on; S+C+D: source, collimator and Doppler cooler on;S+C+D+Z: source, collimator, Doppler cooler and Zeeman slower on; All: full opera-tion: all laser cooling sections on.

-2 -1 0 1 20

1

2

3

Detector Position (mm)

Cur

rent

(nA

/mm

)

-2 -1 0 1 20.0

0.1

0.2

0.3

0.4

0.5

Detector Position (mm)

Cur

rent

(nA

/mm

) Ne2220Ne

Figure 3.21: Beam profiles of a bright beam of the two bosonic neon isotopes 20Neand 22Ne.

50 Chapter 3

density nine orders of magnitude. The latter is still nine orders of magnitude fromthe quantum limit (BEC).

By locking the dye-laser on the other bosonic isotope of neon, 22Ne with naturalabundance 9%, we can also produce a cold and intense beam of this isotope. Thiscan be seen in Fig. 3.21, which shows atomic beam profiles of both isotopes. The22Ne atomic beam has the same width and divergence as the 20Ne atomic beam, andcontains, seven times less atoms than the 20Ne beam. The 22Ne beam contains moreatoms than would be expected from the natural abundance of 9%, which is probablycaused by more efficient cooling of the laser cooling sections.

7 Concluding remarks

We produced a cold and intense beam of metastable Ne(3s) 3P2 atoms. The coldbeam, with a diameter of 1 mm, has a flux of 5× 1010 atoms per second. The beamdivergence of 8 mrad corresponds to a transverse beam temperature of T⊥ = 285 µK,which is equal to the Doppler temperature. The longitudinal temperature of thebeam is T|| = 28 mK. The particle density immediately behind the magneto-opticalcompressor is n = 6× 108 atoms/cm3, and the phase-space density Λh3 = 1× 10−8.While the bright neon beam of Schiffer et al. [18], has a much larger phase-spacedensity Λh3 = 2 × 10−6 (due to the fact that they cool their beam in transversedirection to sub-Doppler temperatures), their beam contains 103 fewer atoms.

Our cold intense atomic beam can be used for a whole range of cold collisionexperiments, e.g., Penning electron energy spectroscopy [40] and photo-associationspectroscopy [41]. The beam can also be used for loading atom traps. In chapter 4 ofthis thesis we show that the bright neon beam is an excellent candidate for loadinga magneto-optical trap (MOT). Thanks to the high loading rate, created by the beam,we can easily make a MOT containing more than 109 metastable neon atoms.

References

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[7] M. Walhout, U. Sterr, C. Orzel, M. Hoogerland, and S.L. Rolston, Phys. Rev. Lett.74, 506 (1995).

Intense beam of cold metastable Ne(3s) 3P2 atoms 51

[8] K.A. Suominen, K. Burnett, P.S. Julienne, M. Walhout, U. Sterr, C. Orzel, M.Hoogerland, and S.L. Rolston, Phys. Rev. A 53, 1678 (1996).

[9] N. Herschbach, P.J.J. Tol, W. Vassen, W. Hogervorst, G. Woestenenk, J.W. Thom-sen, P. van der Straten, and A. Niehaus, Phys. Rev. Lett. 84, 1874 (2000).

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[12] H.R. Thorsheim, Y.W. Wang, and J. Weiner, Phys. Rev. A 41, 2873 (1990).

[13] A. Aspect, N. Vansteenkiste, R. Kaiser, H. Haberland, and M. Karrais, Chem.Phys. 145, 307 (1990).

[14] F. Shimizu, K. Shimizu, and H. Takuma, Chem. Phys. 145, 327 (1990).

[15] A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Optics Comm. 111, 155(1994).

[16] W. Rooijakkers, W. Hogervorst, and W. Vassen, Opt. Commun. 123, 321 (1996).

[17] G. Labeyrie, A. Browaeys, W. Rooijakkers, D. Voelker, J. Grosperrin, B. Wanner,C.I. Westbrook, and A. Aspect, Eur. Phys. J. D 7, 341 (1999).

[18] M. Schiffer, M. Christ, G. Wokurka, and W. Ertmer, Optics Comm. 134, 423(1997).

[19] H.C. Mastwijk, M. van Rijnbach, J.W. Thomsen, P. van der Straten, and A.Niehaus, Eur. J. Phys. D 4, 131 (1998).

[20] H.C. Mastwijk, J.W. Thomsen, P. van der Straten, and A. Niehaus, Phys. Rev. Let.80, 5516 (1998).

[21] G. Woestenenk, H.C. Mastwijk, J.W. Thomsen, P. van der Straten, M. Pieksma, M.van Rijnbach, and A. Niehaus, Nucl. Instr. and Meth. in Phys. Res. B 154, 194(1999).

[22] P. Engels, S. Salewski, H. Levsen, K. Sengstock, and W. Ertmer, Appl. Phys. B 69,407 (1999).

[23] N. Herschbach, P.J.J. Tol, W. Hogervorst, and W. Vassen, Phys. Rev. A. 61,050702(R) (2000).

[24] H. Hotop, Atomic, Molecular and Optical Physics: Atoms and Molecules, 29b,191, eds. F.B. Dunning and R.G. Hulet, (Academic Press, New York, 1996).

[25] M.J. Verheijen, H.C.W. Beijerinck, L.H.A.M. v. Moll, J. Driessen, and N.F. Verster,J. Phys. E: Sci. Instrum. 17, 904 (1984).

52 Chapter 3

[26] M.D. Hoogerland, J.P.J. Driessen, E.J.D. Vredenbregt, H.J.L. Megens, M.P.Schuwer, H.C.W. Beijerinck, and K.A.H. van Leeuwen, Appl. Phys. B 62, 323(1996).

[27] W.D. Philips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1992).

[28] V.S. Bagnato, C. Salomon, E., Jr. Marega, and S.C. Zilio, J. Opt. Soc. Am. B, 8(3),497 (1991).

[29] J. Nellessen, J. Werner, and W. Ertmer, Opt. Commun. 78, 300 (1990)

[30] M.D. Hoogerland, Laser manipulation of metastable neon atoms, Ph.D. thesisEindhoven University of Technology (1993).

[31] A.M. Steane, M. Chowdhury, and C.J. Foot, J. Opt. Soc. Am. B 9, 2142 (1992)

[32] Y. Castin, K. Berg-Sørensen, J. Dalibard, and K. Mølmer, Phys. Rev. A 50, 5092(1994).

[33] R. Stas, Laser cooling and trapping of metastable neon atoms, internal report,Eindhoven University of Technology (1999).

[34] S-Q. Shang, B. Sheehy, H. Metcalf, P. van der Straten, and G. Nienhuis, Phys. Rev.Lett. 67, 1094 (1991).

[35] M. Rauner, S. Kuppens, M. Schiffer, G. Birkl, K. Snegstock, and W. Ertmer, Phys.Rev. A 58, R42 (1998).

[36] A.E.A. Koolen, Dissipative Atom Optics with Cold Metastable Helium Atoms,Ph.D. thesis, Eindhoven University of Technology (2000).

[37] W.J. Rowlands, D.C. Lau, G.I. Opat, A.I. Sidrov, R.J. McLean, and P. Hannaford,Opt. Comm. 126, 55 (1996).

[38] W.G. Kaenders, F. Lison, I. Muller, A. Wynands, and D. Meschede, Phys. Rev. A54, 5067 (1996).

[39] F. Lison, P. Schuh, D. Haubrich, and D. Meschede, Phys. Rev. A 61, 013405(1999).

[40] J.L.W.P. Oerlemans, De cilindrische spiegelelektronenmeter, internal report (indutch), Eindhoven University of Technology (1998).

[41] M.R. Doery, E.J.D. Vredenbregt, J.G.C. Tempelaars, H.C.W. Beijerinck, and B.J.Verhaar, Phys. Rev. A 57, 3603 (1998).

Chapter 4

Metastable neon atoms in amagneto-optical trap

1 Introduction

The successful Bose-Einstein condensation (BEC) of alkali atoms has stimulated sev-eral groups [1–3] to extend the range to other species such as metastable rare gasatoms in particular, with helium and neon being the prime candidates [4]. Thesemetastable atoms distinguish themselves from the alkalis by their immense inter-nal energy of 4 to 20 eV, for Xe to He, respectively. This makes them particularlyinteresting new candidates for BEC. The internal energy can be used for efficientdetection schemes leading to real time diagnostics [4] of the phase transition andcollective deexcitation phenomena in the UV range. The internal energy also makesit a challenge because Penning ionization leads to high inelastic collisional loss ratesalready at relatively low densities as compared to alkali atoms.

In our group we aim at reaching BEC with metastable neon atoms. The firststep is to trap large samples of metastable neon atoms in a magneto-optical trap(MOT), which can then be transferred to a magnetic trap and evaporatively cooled.Evaporative cooling is greatly facilitated when the initial number of atoms is high,i.e., 109−1010 atoms. In this chapter we show how such large amounts of metastableneon atoms can be trapped in a MOT.

In contrast to alkali atoms a vapor cell MOT is not possible with metastableatoms. Our MOT is loaded with the bright and slowed atomic beam described inchapter 3. This allows for MOT loading rates of over 1010 atoms/s. The axial velocityof the atoms exiting from the beam machine is still 100 m/s, so that for efficienttrapping an additional Zeeman slower is necessary. This introduces an additionallaser beam to the MOT which cannot be neglected in the description of the MOT’sdynamics, i.e., a seven beam MOT is created. This seems awkward at first but nev-ertheless allows for a MOT containing 9 × 109 atoms, to our knowledge the largestMOT of metastable atoms reported. In this chapter we describe the characteristicsof such a MOT.

This chapter is organized as follows. We start in section 2 with a description ofthe dynamics of a metastable atom MOT. In particular we develop a model for a MOT

53

54 Chapter 4

disturbed by a seventh laser beam. The experimental setup is described in section 3.In the sections 4 to 8 we describe the experiments we did to investigate the behaviorof the MOT. We end in section 9 with concluding remarks.

2 Trap dynamics

In this section we describe the physics necessary to understand the behavior of themetastable neon MOT that is described in this chapter. The MOT is loaded withatoms from the bright atomic beam, which is described in chapter 3. The atomsare decelerated just before the MOT. This has the consequence that the laser beam,necessary for the slowing process, intersects with the trapping beams of the MOT.This results in a seven beam MOT instead of an ordinary six beam MOT, during theloading of the MOT. The influence of this seventh laser beam on the trap dynamicsis described in this section. We start with a description of the population of atomsin a metastable atom MOT.

2.1 Trap population

The evolution of the number of atoms N(t) in the MOT fulfills the equation

N(t) = R − N(t)τ

− β∫n2(r, t)d3r . (4.1)

Here, R is the loading rate, τ the time constant representing loss processes whichare independent of the number of trapped atoms (e.g., collisions with backgroundgas atoms), β the rate constant that describes the density dependent losses (whicharise from two-body collisions between trapped atoms) and n(r, t) is the densitydistribution.

The optical pumping process in the trap has the effect that a small fraction of thetrap population is in the excited P -state. The collision between two particles can bea collision of three types: a S − S, S − P or P − P collision. The two-body loss rate βis thus written as a function of the populations ΠS(P) in the S (P) state:

β = 2(Π2SKSS + 2ΠSΠPKSP +Π2

PKPP), (4.2)

with KSS(SP,PP) the rate coefficients for S − S (S − P, P − P) collisions, respectively.Experiments with metastable helium atoms [5–8] show that KSP is two orders ofmagnitude larger thanKSS and that theKPP can be neglected [5]. Mastwijk et al. [9],however, report a KSP which is only a factor of 20 larger then KSS , for metastablehelium. No experimental results are reported for metastable neon atoms but thereis no reason to expect a larger ratio between KSP and KSS for metastable neon.Moreover, our trap is operated with relatively low laser beam intensities, where thefraction of atoms in the excited state is less then one percent. In the detuning rangewhere we operate the trap, the excited state population can be neglected, so thatfrom now on we write β ≡ βSS = 2KSS .

Metastable neon atoms in a magneto-optical trap 55

Generally, the density in metastable atom MOT’s is low enough to assume thatthe spatial distribution is independent of the number of trapped atoms, i.e., n(r, t) =n(r)f (t) [8]. In this so-called temperature limited regime [10] it is convenient toassume a Gaussian density distribution with central density n0 and rms radius σρ inradial direction and σz in axial direction, the density distribution is given by

n(r) = n(ρ, z) = n0 exp

(− ρ2

2σ 2ρ− z2

2σ 2z

). (4.3)

Here the central density is related to the volume V via N = n0V , with the volume ofthe atom cloud given by

V = (2π)3/2σ 2ρσz. (4.4)

By defining an effective trap volume [11] given by

Veff = N2∫n2(r)d3r

= 23/2V, (4.5)

Eq. (4.1) can be written as

N(t) = R − N(t)τ

− βN2(t)Veff

. (4.6)

The evolution of the number of trapped atoms during the loading phase can befound by solving Eq. (4.6) with N(0) = 0 [8],

N(t) = NS1− exp(−t/τ0)

1+ (N2Sβ/VeffR) exp(−t/τ0)

, (4.7)

withτ0 = τ

(1+ 4βRτ2/Veff)1/2, (4.8)

and NS the steady state number of trapped atoms. An expression for the steadystate number of atoms can easily be found from Eq. (4.6) by substituting N = 0,resulting in

NS = Veff

2βτ

(1+ 4Rβτ2

Veff

)1/2

− 1

. (4.9)

For losses dominated by two-body trapped atom collisions, i.e., 2βτ/Veff 1, Eq. (4.9)simplifies to

NS (RVeff/β)1/2. (4.10)

In this case the steady state number of trapped atoms is proportional to the squareroot of the loading rate.

The trap decay can be found from Eq. (4.6) by substituting R = 0, resulting in

N(t′) = NS exp(−t′/τ)1+ (βNSτ/Veff)[1− exp(−t′/τ)], (4.11)

where NS = N(0) the initial number of trapped atoms. The dimensionless termβNSτ/Veff is a measure for the relative importance of the losses caused by two-body

56 Chapter 4

collisions [11]. Substituting β = 0 gives a purely exponential trap decay with lifetimeτ , while for βNSτ/Veff 1 Eq. (4.11) can be written as

N(t′) = NS

1+ (βNS/Veff)t′. (4.12)

The initial rate of decay is then proportional to the square of the number of atoms:N(0) = βN2

S/Veff. We define a decay time determined by two-body losses, i.e., ioniza-tion, by τi = Veff/(NSβ).

Now the expressions for the trap population are known, we can calculate them fordifferent trapping conditions, to find the best operating conditions. To do that wefirst need to know more about the distribution of the atoms in the MOT. Thereforethe trapping potential is needed. We express the trapping potential in a simpleDoppler model, described below.

2.2 Doppler model

The MOT is operated in the temperature limited regime, as mentioned earlier. Thethermal motion of the atoms in the trap causes them to spread out to an rms radiusσρ in radial direction and σz in axial direction as

12kBT = 1

2κρ〈σ 2

ρ 〉 =12κz〈σ 2

z 〉, (4.13)

with κρ and κz the spring constant of the trap in radial and axial direction, respec-tively (Eq. (2.15)) and T the temperature of the atoms determined by the balancebetween cooling and diffusion as described in chapter 2 (Eq. (2.12)). The volumeof the atom cloud, given by Eq. (4.4), can then be expressed in terms of the springconstants and the temperature:

V = (2πkBT)3/21√κ2ρκz

. (4.14)

Knowing that for large detunings of the trapping beams ∆M, the spring constantκi scales as κi ∼ (∆M)−3, and that the temperature is proportional to the detuning,T ∼ ∆M, the volume of the cloud should be proportional to the detuning to the powerof six, V ∼ (∆M)6.

The steady state number of trapped atoms can be calculated using Eq. (4.9) or,assuming that the internal trap loss process dominates over the external loss pro-cess, by using Eq. (4.10). Since the volume is proportional to the detuning to thepower six, the steady state number of atoms should be proportional to the detuningcubed, assuming that R and β are independent of the detuning. There is no reasonto assume that the loading rate depends strongly on the detuning of the trappingbeams since the slowing process is fully controlled by the seventh laser beam. Asmentioned earlier the loss rate β is only determined by S − S collisions and as aconsequence independent of the laser field. This model states that the number ofatoms is fully determined by the volume of the atom cloud, and because the volume

Metastable neon atoms in a magneto-optical trap 57

Table I: Slower and MOT characteristics.

slower field maximum Bmax 100 G

laser detuning ∆S −6.7 Γ1/e2 beam diameter ∼ 20 mmintensity sS ≡ IS/I0,σ 0.25

MOT field gradient G ≡ ∇zB = 2∇ρB 0.33 G/Acm1

laser detuning ∆M -1Γ to -8Γ 11/e2 beam diameter 35 mmtotal intensity sM ≡ 6IM/I0,σ 0.25 to 81

1In the experiments described in this chapter we variedthe detuning and intensity of the trapping beams andvaried the magnetic field gradient.

increases as a function of the laser detuning, the number of atoms increases as well.However, the increase of the volume cannot continue forever: for a certain detuningthe diameter of the atom cloud will be as large as the diameter of the trapping laserbeams! The volume, and therefore the number of trapped atoms, will not increaseanymore. Before this happens another process will occur, described in the next sec-tion, which will cause atoms to leak from the trap. The parameters used in the modeland, later on, in the experiments are given in Table I.

2.3 Trapping potential: geometrical trap loss

In our setup, described in detail in section 3, the laser beam for the extra Zeemanslower, crosses the trapping center along the x-axis, while the radial trapping beamsintersect with the x-axis under an angle of 45. The slower laser beam influences thedynamics of the MOT. This can be seen if we look at the potential of the MOT createdby combination of the MOT trapping beams and the slower beam. The potential UMOT

of the MOT is related to the force F = −∇UMOT on the atoms, with F the trappingforce described in section 3.1 of chapter 2. Considering only the direction of theslower laser beam, the force is given by the sum of the spontaneous forces causedby the radial MOT beams and the force caused by the slower beam.

The resulting potential is given in Fig. 4.1 for different detunings of the MOTbeams. The detuning ∆M of the MOT laser beams is expressed in units of the naturalline width Γ = 8.2(2π) MHz of the Ne 3D3 state. The intensities of the MOT laserbeams we express in terms of the total saturation intensity sM, which is defined bysix times the saturation intensity of one trapping beam sM = 6(IM)/(I0,σ ), with I0,σ =4.08 mW/cm2 the saturation intensity of the Ne(3s) 3P2(mJ = 2) ↔Ne(3p) 3D3(mJ =3) transition. We define the gradient G of the MOT magnetic field as G ≡ ∂B/∂z =2∂B/∂ρ, and we write G in units of G/cm (Table I). The intensity and detuning ofthe slower laser we keep fixed at sS = IS/I0,σ = 0.25 and ∆S = −6.7Γ . The force of

58 Chapter 4

ba

-50 -25 0 25 500.0

0.5

1.0

1.5

x (mm)

UM

OT (

K)

-10 -5 0 5 100

5

10

15

20

25

x (mm)U

MO

T (

mK

)

∆M=-Γ -2Γ -3Γ

Figure 4.1: a: Spatial profile of the confining potential of the MOT for three differentdetunings of the trapping beams. The minimum of the potentials is fixed to zeroKelvin. b: Detail of the deepest point of the potentials. The slower beam shiftsthe deepest point of the well in radial direction (−x direction). Trapping conditions:G = 10 G/cm, sM = 0.5.

ba

0 2 4 6 8 100

2

4

6

8

10

-∆M(Γ)

x 0 (

mm

)

0 2 4 60

2

4

6

8

10

-∆M(Γ)

x 0 (

mm

) x0+σx

x0 ∆C

sM=0.1 0.2 0.5 1 2 4

Figure 4.2: a: Shift of the trap minimum x0 as a function of the laser detuningfor different laser beam intensities, going from left to right: sM = 0.1 to sM = 4(G = 10 G/cm). b: Shift of the trap minimum x0 and the shift of the edge of the atomcloud x0 + σx as a function of the laser detuning (sM = 0.5 and G = 10 G/cm). Fora detuning of ∆M = ∆C = −4.3Γ the edge of the atom cloud reaches the edge of theaxial trapping beams dbeam = 9 mm.

Metastable neon atoms in a magneto-optical trap 59

the slower beam causes a tilt of the MOT potential, as can be seen in Fig. 4.1a, whichshows the MOT potential for three different laser detunings. This tilt has the effectthat the minimum of the potential shifts away from the center of the trap, which isshown in Fig. 4.1b. The depth of the potential is of the order of 1 K, as can be seenin Fig. 4.1a. This is much more than the expected cloud temperature of 1 mK. If weassume that the atom cloud is centered at the position of the potential minimum, weexpect also a shift of the atom cloud, when the laser detuning of the trapping beamsis changed.

We can calculate the shift of the deepest point of the potential for differenttrapping conditions. Figure 4.2a shows this shift for a MOT with field gradientG = 10 G/cm and trapping beam intensities corresponding to sM = 0.1 to sM = 4.For low trapping beam intensities, the trap minimum shifts very rapidly away fromthe trapping center while for high intensities this happens more slowly. Figure 4.2bshows the shift of the center of the atom cloud x0 together with the shift of the edgeof the atom cloud x0 + σx for a MOT with field gradient G = 10 G/cm and trappingbeam intensity sM = 0.5. The rms radius in radial direction is calculated using theexpression given in Eq. (4.13). At a certain critical detuning ∆M ≡ ∆C of the trappinglaser beams, the shift of the cloud center together with the increase in the volumeis so large that the edge of the atom cloud hits the boundary of the axial trappingbeams, which have a diameter of approximately 18 mm. When this happens thevolume cannot increase anymore while the shift of the cloud center continues, forincreasing laser detuning. This causes a cut-off of the atom cloud which results ina decrease of the volume and therefore a decrease of the number of trapped atoms.Furthermore the trap starts to leak atoms. We call this phenomenon geometric traploss.

Knowing the relation between the number of trapped atoms and the volume ofthe cloud (Eq. (4.10)), we can calculate the number of trapped atoms as a function ofthe detuning of the trapping beams. For the loading rate and two-body loss rate wesubstitute values which are independent of the laser intensity and detuning, namelyR = 1 × 1010 s−1 and β = 5 × 10−10 cm3/s. Figure 4.3 shows the calculated numberof atoms as a function of the detuning of the trapping beams for three differenttrapping conditions: (sM, G) = (0.2,10 G/cm), (0.5,10 G/cm), and (0.5,15 G/cm).Indicated in the figure is the critical detuning ∆C determined for the different trap-ping conditions in the same way as defined in Fig. 4.2b. We define the maximumnumber of trapped atoms as N(∆M) = N(∆C) ≡ Nmax. We see that Nmax is larger forhigher laser beam intensities and higher magnetic field gradients. From this calcula-tion we see that it is possible to trap more than 109 atoms in a trap with field gradientG 10 G/cm, laser detuning ∆M −5Γ and laser beam intensities corresponding tosM 0.5.

60 Chapter 4

Nmax

∆C

sM=0.2

G=10 10 15 G/cm

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

-∆M(Γ)

N (

109 )

0.5

Figure 4.3: Calculated number of trapped atoms as a function of the detuning ofthe trapping beams for three different trapping conditions. The critical detuning∆C , and the corresponding maximum number of trapped atoms Nmax, is larger forhigher laser beam intensities and higher magnetic field gradients. The number oftrapped atoms is calculated by using Eq. (4.10). For the loading rate and two-bodyloss rate we substitute R = 1× 1010 atoms/s and β = 5× 10−10 cm3/s, respectively.Both loading and loss rate are taken independent of the detuning and intensity ofthe trapping beams.

3 Experimental setup

3.1 Trapping chamber

Figure 4.4 shows an artist’s impression of the trapping chamber, while Fig. 4.5 showsa schematic side view of the chamber. On the left the last part of the beam-machinedescribed in chapter 3, is visible. The trapping chamber is connected to the vacuumsystem of the beammachine via a bellows, which makes it possible to align the centerof the trapping chamber with the atomic beam axis. A detector, consisting of a pieceof glass covered with a conducting Indium-Tin-Oxide layer, can be moved onto theatomic beam axis to measure the flux of atoms entering the trapping region. Behindthe detector a flow resistance is positioned to provide a pressure drop between thevacuum system of the beam-machine and an extra differential pumping chamber.The differential chamber is pumped with an ion-getter pump (type Varian StarCellVaclon, pumping speed 20 l/s), and a Ti-sublimation pump (pumps not visible inFig. 4.4). Again, a flow resistance is positioned between the differential pumpingchamber and the trapping chamber, and a valve separates the trapping chamberfrom the rest of the setup. The stainless steel trapping chamber has an octagonalshape and has seven anti-reflection coated windows. One port of the chamber is

Metastable neon atoms in a magneto-optical trap 61

Compensation

ValveIon-getter pump

Ti-sub.-pumpx

y

z

Detector

Beammachine

pumpingDiff.

chamber

coils

Figure 4.4: Artist’s impression of the trapping chamber. On the left the last partof the neon beam-machine is visible (chapter 3). The amount of atoms entering thetrapping chamber can be detected with a movable detection plate (detector). Notshown in this figure is a valve between the chamber of the beam machine and thedifferential chamber.

used for the connectors to a channeltron situated above the trapping center. Thetrapping chamber is also pumped with an ion-getter pump (type Ultek 202-2000,pumping speed 20 l/s), and a Ti-sublimation pump. Ideally the pressures dropsfrom 10−8 mbar in the vacuum system of the beam-machine via ∼ 10−9 mbar in thedifferential pumping chamber to ∼ 10−10 mbar in the trapping chamber1.

3.2 Magnetic field coils

The MOT quadrupole field is produced by anti-Helmholz coils made of 4 mm di-ameter hollow tubing. Each coil consist of 24 turns: three layers of eight turnseach. The magnetic field gradient of the coil configuration was measured to beG ≡ ∇zB = 2∇ρB = 0.33 G/Acm [12] (see Table I). The coils can carry a maximumcurrent of 200 A, and are cooled by flowing water through the copper tubing.

To compensate for disturbing magnetic fields (e.g., the Earth’s magnetic field)three pairs of Helmholz coils are positioned in the x, y and z-plane around thetrapping chamber (see Fig. 4.4). At the center of the trap, the coils produce a mag-netic field of B0 = 4.2 G/A in the x and z direction and B0 = 3.3 G/A in the ydirection, respectively [12]. The maximum current through the coils is 3 A, so mag-netic fields up to 10 G can be compensated. The compensation coils can also be usedfor positioning the atom cloud in the middle of the trapping region, i.e., at the centerof the MOT quadrupole field. This is important when the atoms are transferred fromthe MOT to a magneto-static trap: then the centers of both traps must overlap to

1We cannot measure the pressure in the differential pumping and trapping chamber, the ion-getter pump currents of both chambers indicate a pressure below 10−8 mbar.

62 Chapter 4

y

x

Detector

Slower beam

Radial trapping beams

Slower-coil

Atoms

0 200 400 600 800 1000 mm

Bellows

Valve

Flowresistance

ValveMOT-coils

Slower-coil

Beammachine

Diff.-pumpingchamber chamber

Trapping

10 7100 100

Figure 4.5: Schematic side view of the trapping chamber.

minimize the loss in phase-space density. The compensation coils can also be usedfor reducing the influence of the slower laser beam on the trap dynamics.

3.3 Extra slower

The atoms entering the trapping chamber have an average velocity of 100 m/s (seechapter 3, section IV). Typical capture velocities of MOT’s are 50 m/s, so the atomsneed to be slowed down further to be captured by the MOT. This is done in an extraslower positioned close to the center of the trap. It uses two coils producing oppo-site magnetic fields. The first coil, positioned 120 mm downstream of the trappingcenter, consists of 100 turns and has a diameter of 80 mm. A second coil with 50turns and a diameter of 300 mm is positioned 190 mm upstream of the trappingcenter, see Fig. 4.5. The second coil is necessary to null the magnetic field producedby the first coil in the center of the MOT. The magnetic field produced by the firstcoil is at its maximum Bmax = 100 G, at 6 A current.

3.4 Laser setup

Figure 4.6 gives an overview of all the optical components necessary for preparingthe MOT trapping beams and the laser beam for trap diagnostics (section 3.5). Thelight for the trapping beams comes from a continuous-wave single-frequency ringdye laser (Spectra Physics, type 380D). The dye laser typically has an output powerof 120 mW while pumped with 4 W of light from a argon ion laser, type CoherentInnova 70. The dye laser is locked to a frequency that can be Zeeman shifted betweenone and twelve line widths to the red of the Ne (3s) 3P2 ↔(3p) 3D3 transition by using

Metastable neon atoms in a magneto-optical trap 63

T

Ne Cell* CB

PD

PDλ/4

Dye

T

λ/4

T

λ/4

λ/2

PCB

CB

λ/4

Tλ/2

PCB

S

AOM40 MHz

λ/2PCB

AOMs30-50 MHz

Fiber

CCD Camera

Probe-beam

Slower-beam

Radial trappingbeams

Axial trappingbeams

Ar +

L

λ/4

Figure 4.6: Top view of the trapping chamber with schematically a view of the opticalcomponents of the magneto-optical trap. The light for the six trapping beams allcome from a single dye laser which is locked to the Zeeman-tuned Ne(3s)3P2↔(3p)3D3

transition by saturated absorbtion. The light for the extra Zeeman slower and theprobe-beam come from the dye laser used for preparing the atomic beam (chapter3). L: single lens, T: optical telescope, CB: cubic beamsplitter, PCB: polarizing cubicbeam splitter, PD: photo diode, S: beam-shutter, AOM: acousto optical modulator,λ/2, λ/4: half and quarter wave plates.

saturated absorption spectroscopy. This Zeeman shifting is done by varying thecurrent through a coil producing a DC magnetic field in the saturated absorptioncell. The linewidth of the laser is about 1 MHz.

The total laser power to the MOT can be regulated by a half-wave plate and apolarizing beam splitter cube. The light can be switched on and off within 2 ms witha mechanical beam shutter (type Uniblitz VMM-T1). By using optical telescopes thelaser beam is expanded to a waist diameter of ∼ 35 mm (1/e2 intensity drop). Thenecessary three pairs of counterpropagating σ+−σ− polarized beams are created bysplitting the main trapping beam into three beams of equal intensity, and reflectingeach one after passing appropriate quarter wave plates and the trapping center, backto itself (see Fig. 4.5 and Fig. 4.6). The maximum available laser power results in anintensity of IM = 5.5 mW/cm2 for each trapping beam, corresponding to a totalsaturation parameter (i.e., sum of the six trapping beams) sM = 8. The trapping laserbeams are cut-off by a diaphragm resulting in a diameter of approximately 18 mm.

The laser light for the extra slower and for a probe-beam used for trap diag-nostics, are derived from the dye laser used for preparing the atomic beam (seechapter 3, section 2a). This dye laser is locked almost two line widths to the red of

64 Chapter 4

the Ne (3s) 3P2 ↔(3p) 3D3 transition; acousto optic modulators (AOMs) are used toshift the beams to the right frequencies.

For the slower beam, 10 mW of laser light is split off and is shifted by a 40 MHzAOM to a laser detuning of ∆S = −6.7Γ . The beam is expanded by two optical tele-scopes to a 1/e2 waist diameter of ∼ 20 mm and a quarter wave plate is used to makethe light σ+-polarized. The laser intensity is IS = 1 mW/cm2 which corresponds to asaturation intensity of sS = 0.25 (Table I).

For trap diagnostics using absorption imaging (see next section) a laser beamwith variable intensity and detuning is necessary. We prepare this laser beam byusing two AOMs to split off a variable amount of laser power from the dye laserused for preparing the atomic beam. This can be done because the probe beam isnormally only used when the loading of the trap is stopped, so that in principle allthe laser power otherwise needed to prepare the atomic beam can be used. We usetwo tunable 40 MHz AOMs, the first one shifting the beam to the blue and the secondone to the red. In this way a detuning of a few line widths to the red or blue can begenerated. The probe-beam is coupled into a single-mode polarization maintainingfiber and expanded by an optical telescope. The fiber provides spatial filtering ofthe wavefront. The beam then crosses the trapping center in axial direction and isimaged on a CCD-camera (Fig. 4.6).

3.5 Diagnostics

To investigate the dynamics of the atoms in the MOT we use a few techniques todiagnose the cloud of atoms. Figure 4.7 is an artist’s impression of the trap showingthe detectors to measure the cloud characteristics such as number of atoms, volumeand temperature. The number of trapped atoms is determined by measuring thepower of the fluorescence emitted by the atoms. The fluorescence, emitted withina solid angle of Ωdet = (7.6 ± 0.3) × 10−4, is focused on a photo-diode via a lens-system, in which two apertures prevent background light of reaching the photo-diode. The measured efficiency of the system of lenses and photo-diode equalsηdet · Tlenses = 0.125 ± 0.005 A/W. The loading and decay rate of the MOT can alsobe derived from the fluorescence power by studying time-dependent phenomenaresulting from switching the loading on and off.

The volume of the atom cloud can be measured by analyzing fluorescence im-ages taken with a CCD-camera positioned below the trapping chamber, see Fig. 4.7.By fitting the intensity profile of the CCD-frames with Gaussians, the volume can beestimated as described in section 4. The measured relation between camera-imagepixels and real sizes is for the x and z direction identical and equals 13.1± 0.5 pix-els/mm. Both the number of atoms and the volume of the atom cloud can be mea-sured by using an absorption imaging technique, which is described in detail insection 7.1. In this technique a probe laser beam passes the trapping area in ax-ial direction and gives information about the position (x and y) and the numberof atoms. The relation between CCD-camera pixels and real sizes was for the xdirection: 16.1 ± 0.5 pixels/mm and for the y direction: 9.1 ± 0.5 pixels/mm. Bytaking images of the cloud time-dependently also the temperature of the atoms can

Metastable neon atoms in a magneto-optical trap 65

Channeltron

CCD-camera

Photo-diode

LensesApertures

Atomic beam

Slower-coils

MOT-coils

Slower beam

Beamsplitter

y

xz

MOT beams

Figure 4.7: Artist’s impression of the detection setup. The probe laser beam usedfor absorption imaging, moves along the z-axis and is not shown in this figure.

be measured with the absorption imaging technique as described in section 7.1. Thetemperature of the cloud can also be measured by measuring a time-of-flight (TOF)metastable atom signal with the channeltron positioned 4 cm above the trappingcenter (see Fig. 4.7). In front of the entrance of the channeltron (type Galileo 4039)a grid is placed which can be put on a positive voltage to measure only metastableatoms, or on a negative voltage to measure ions and metastable atoms.

In the next sections we describe the measurements we did to systematically in-vestigate the behavior of our MOT, starting with the cloud volume in the next section.

4 Trap volume

4.1 Fluorescence technique: spatial distribution

The volume of the cloud of atoms in the MOT is measured by fitting CCD pictureframes containing a fluorescence image of the cloud in the x − z-plane. Figure 4.8ashows such an image with a plot of the horizontal and vertical intensity profilethrough the center of the cloud. These intensity profiles are fitted with Gaussiansresulting in rms widths in the x- and z-directions, as shown in Fig. 4.8b.

Assuming that the spatial distribution in the x and y direction are identical, wewrite σx = σy ≡ σρ, with ρ the radial coordinate. The volume V of the cloud isthen calculated using Eq. (4.4). The relative uncertainty in the measured volume is

66 Chapter 4

-8 -6 -4 -2 0 2 4 6 80

20

40

60

80

x (mm)

Pixe

l Con

tent

sx

z

ba

Figure 4.8: a: CCD image of the trapped cloud of atoms in the MOT. Horizontal andvertical intensity profiles through the center of the image are shown. b: Gaussian fitthrough the horizontal (x) intensity profile which is shown in a.

approximately 12%, and is determined by the uncertainty in the Gaussian fits andthe uncertainty in the conversion from pixels to real sizes.

4.2 Volume measurements

The volume of the atom cloud fully determines the maximum number of atoms thatcan be trapped. As mentioned in section 2.2 the volume itself is expected to dependvery strongly on the detuning of the MOT laser light, i.e., (∆M)6. Here we investigatehow V depends on the trap parameters sM, ∆M, and G.

The strong detuning dependence of V is illustrated by the data shown in Fig. 4.9a.Different sets of data are shown taken for different values of sM at a fixed G =23 G/cm. Varying ∆M from −2Γ to −5Γ the volume typically increases by a factorof 20 to 50, depending on sM. At higher intensities the volume doesn’t increase asmuch. Note that the volume can become as large as 0.5 cm3. It is at these largevolumes that the largest number of atoms is trapped. At the critical detuning thenumber of trapped atoms was Nmax = 6× 109, for the sM = 0.4 data set. The numberof atoms was measured with the fluorescence technique described in the next sec-tion. We checked that the volume is independent of the number of atoms in the trap.This was found by decreasing the loading rate and measuring the volume again.

Knowing the volume of the cloud and the number of atoms in the cloud we canestimate the central density n0 = N/V . With V increasing rapidly as a function of∆M the density remains low. This is illustrated by the data in Fig. 4.9b, where thecentral density n0 is plotted again as a function of detuning. At the detunings wherethe largest samples are trapped, i.e., at the critical ∆C , the density is only about

Metastable neon atoms in a magneto-optical trap 67

0 2 4 6 80.0

0.2

0.4

0.6

-∆M(Γ)

V (

cm3 )

∆C

ba

1.0

2.0

0 2 4 6 80

20

40

60

-∆M(Γ)n 0

(10

9 cm

-3)

∆CsM=0.4=0.4sM

Figure 4.9: a: Volume of the cloud of atoms in the MOT as a function of the MOTlaser detuning for different intensities of the trapping beams sM, the magnetic fieldgradient is fixed G = 23 G/cm. b: Central intensity n0 = N/V as a function of thedetuning of the trapping beam, for sM = 0.4 and G = 23 G/cm. In both graphs thecritical detuning is indicated for the sM = 0.4 trapping condition.

1× 1010 atoms/cm3.In order to gain a full understanding of the behavior of the MOT volume we

develop the Doppler model mentioned in section 2.2 in detail here. The basic as-sumption of the model is that the volume is given by the equipartition of kineticand potential energy, resulting in Eq. (4.14). The Doppler temperature is given byEq. (2.11) in Chapter 2. The spring constant is taken to be the derivative of thephoton scattering force (Eq. (2.12) of chapter 2). Inserting these into Eq. (4.14) onearrives at

V = V0

[(2∆M/Γ + 1

2(∆M/Γ))(1+ s0 + 4(∆2

M/Γ 2))2

(∆M/Γ) s0 (G/G0)

] 32

, (4.15)

with

V0 = π32

32

[Γ

kL µBG0

] 32

, (4.16)

where G0 = 10 G/cm is a reference value, kL is the wavenumber of the laser, ∆M isthe MOT laser detuning in units of the linewidth, and s0 = 1

6sM is the single beamon-resonance saturation parameter. The value of V0 = 2.5× 10−9 cm3. Note that allunits are included in V0 by the introduction of G0.

Comparing Eq. (4.15) to the data we find that the measured volumes are muchlarger than the model predicts. However, by using V0 as a single fit (scaling) param-eter we obtain the solid curves shown in Fig. 4.9a. Clearly, the model predicts thecorrect dependence of the volume on detuning. Moreover, data sets taken for dif-ferent intensities (sM = 0.4 to 2.0) and magnetic field gradients (10 to 23 G/cm) all

68 Chapter 4

0 2 4 6 8-2

0

2

4

6

8

10

-∆M(Γ)

x 0,z

0 (m

m)

∆C

x0z0

Figure 4.10: Shift of the center of the atom cloud in radial (x0) and axial (z0) directionas a function of the detuning of the trapping beams. The black dots indicate themeasured shift in radial direction and the open circles indicate the measured shiftin axial direction. The solid line shows the predicted shift by the Doppler model(see section 2.3). The first data point is fixed to the theoretical curve. Trappingconditions: sM = 0.4, G = 23 G/cm, with corresponding critical detuning ∆C = −4.3Γ .

result in the same value of Vfit0 = (1.1± 0.2)× 10−8 cm3. Here the uncertainty refers

to the spread of the fits.The discrepancy between V0 and Vfit

0 can be explained partly from the fact thatthe model is based on the Doppler force for a simple two level atom. In reality theforce becomes smaller because the Clebsch-Gordan coefficients of the different sub-levels have to be included. If we take care of this and substitute 0.5s0 for s0 thefit parameter Vfit

0 agrees with V0. This correction for the effective Clebsch-Gordancoefficient is larger then we expect, as described in section 5. However, in the modelcalculations we use s0 → 0.5s0. Radiation trapping can be neglected because of therather low density. This is confirmed by the fact that the volume is independent ofthe number of atoms in the trap.

From the CCD-camera images it is also possible to obtain the position of the atomcloud. Figure 4.10 shows the coordinate of the cloud center (x0, z0) as a function ofthe detuning of the trapping beams for the sM = 0.4 data set of Fig. 4.9a. We see that,going from ∆M = −2Γ to −5Γ , the cloud center moves 6 mm in radial (x) direction,while in axial direction (z) the cloud stays more or less fixed. The solid line in thefigure indicates the radial shift calculated with the Doppler model of section 2.2.We see that the Doppler model predicts a correct dependence of the radial shift onthe detuning however more rapidly than the measurements show. The shift, fullydetermined by the balance between the radial trapping beams and the slower beam,follows the measurements better when the intensity ratio between trapping beams

Metastable neon atoms in a magneto-optical trap 69

0 2 4 6 80.0

0.5

1.0

1.5

2.0

-∆M(Γ)

σ x/σ

z

∆C

212

Figure 4.11: Ratio between the radial and axial rms width of the atom cloud σx/σzas a function of the detuning of the trapping beams. Trapping conditions: sM = 0.4,G = 23 G/cm, with corresponding critical detuning ∆C = −4.3Γ .

and slower beam is decreased.This could also be the case in the experiments since the ratio between the rms

width in radial and axial direction σx/σz was higher then 1/√2, as predicted by

Eq. 4.13. This can be seen in Fig. 4.11 which shows the ratio σx/σz as a function ofthe detuning of the trapping beams, corresponding again with the sM = 0.4 data setof Fig. 4.9a. As can be seen the ratio is always larger then 1

2

√2, and increases as a

function of the detuning. This means that during the measurements the intensity ofthe radial trapping beams was larger than the intensity of the axial trapping beams,i.e., sM,x > sM,z. We cannot explain the increase of this ratio with the detuning.Apparently the atom cloud moves through a region where the balance between theintensity of the axial and radial trapping beams changes. The fact that the ratio isalways higher then 1

2

√2 means that the intensity of the radial trapping beams was

higher then one sixth of the total intensity IM which we measured. This explainswhy the calculated shift of the atom cloud increases more rapidly with ∆M than themeasured one.

In conclusion, we find that the volume of the cloud of atoms in the MOT canindeed be described qualitatively very well by a simple Doppler model. However,quantitative agreement is only obtained to within a factor of four. The shift of theatom cloud with the detuning is described very well by the geometrical loss model.Finally, Fig. 4.12 shows the CCD-camera images of the atom cloud for different laserdetunings. The radial shift and the increasing of the volume is clearly visible as wellas the changing aspect ratio of the cloud.

70 Chapter 4

0

4

8

-4

12

16

x (m

m)

∆M

-2.1Γ-2.7Γ

-3.3Γ

-4.0Γ

=-4.5Γ

Figure 4.12: CCD-images of the trapped cloud of atoms corresponding to some ofthe data points of Fig. 4.10.

5 Loading rate

5.1 Fluorescence technique: number of atoms

A standard way to determine the amount of atoms in a MOT is to measure the powerof the fluorescence emitted by the atoms in the trap [13]. The total power emittedby one atom can be written as

Pscat = ωLΓΠ(e), (4.17)

with ωL the laser frequency and Π(e) the population of the excited states and Γthe decay rate. For a two-level atom illuminated by a single travelling wave of Rabifrequency Ω and detuning ∆L, this can be written as

Pscat = ωLΓ2

Ω2/2∆2

L + Γ 2/4+Ω2/2. (4.18)

In a MOT, however, one must calculate an average over all the transitions betweenthe various Zeeman sublevels between the ground and excited states. In this casethe total scattering rate per atom can be approximated [13]

Pscat ωLΓ2

CΩ2tot/2

∆2L + Γ 2/4+ CΩ2

tot/2, (4.19)

where Ω2tot is determined by six times the average light intensity of one trapping

beam, and C is a phenomenological parameter containing an average of Clebsch-Gordan coefficients. Townsend et al. [10] found an experimental value of C = 0.7 ±0.2 for a Cs MOT, much larger than the average squared Clebsch-Gordan coefficientswhich yields C = 0.4. In the case of the Ne 3P2→3D3 transition the average over the

Metastable neon atoms in a magneto-optical trap 71

Clebsch-Gordan coefficients equals 0.46; following Townsend et al. [10] we take inour experiments C = 0.7± 0.2.

In our setup the atoms are slowed by an extra Zeeman slower in order to becaptured by the MOT (section 3.3). The extra slower laser beam influences the pop-ulation of the excited states. To correct for this seventh laser beam in the MOT wewrite for the population of the excited states

Π(e)tot = Π(e)

M +Π(e)S

= 12

[CsM

1+ CsM + 4∆2M/Γ 2

+ CsS1+ CsS + 4∆2

S/Γ 2

], (4.20)

with ∆M and ∆S the detuning of the MOT beams and slower beam, respectively, andsM = IM/I0,σ with IM six times the intensity of one trapping beam, I0,σ = 4.1 mW/cm2

the saturation intensity for the used transition and sS = IS/I0,σ with IS the intensityof the slower beam.

The fluorescence power emitted by the atoms is measured with a photo-diode (seesection 3.5). The current Idet from the photo-diode is proportional to the amount ofatoms N trapped in the MOT:

Idet = ηdet · Tlenses ·Ωdet · Pscat ·N. (4.21)

Substituting Tdet = ηdet · Tlenses we write for the number of atoms

N = IdetTdet ·Ωdet · Pscat

. (4.22)

When the uncertainty in the averaged phenomenological parameter C is neglectedthe relative uncertainty in the number of trapped atoms is about 10%. The largestcontribution to this uncertainty is produced by the 2 MHz, i.e., 0.25Γ , uncertainty inthe frequency of both trapping and slower laser. The phenomenological parameter Cis not known for our trap, but since the averaged squared Clebsch-Gordan coefficientfor the 3P2→3D3 transition is much smaller than C = 0.7, Eq. (4.20) gives a upper limitto the population of the excited states.

5.2 Measurements

Figure 4.13 shows an example of a loading and decay curve for a trap with sM = 0.4and G = 23 G/cm. At t = 0 the loading of the MOT is started by switching on theatomic beam. This is done by unblocking the laser beams which go to the lasercooling section of the beam-machine, as well as the extra Zeeman slower. As canbe seen in the figure, within one second the MOT reaches its steady-state number ofatoms NS . At t = 2.1 s the loading of the trap is stopped by blocking the laser beamsagain. Again also the extra slower laser beam is switched off.

Figure 4.14a shows a detail of the loading process. To determine the loadingrate from this curve, we fit the curve with a straight line at t = 0, as indicatedin the figure. In this measurement, where ∆M = −4.4Γ , the loading rate is R =

72 Chapter 4

0 1 2 3

0

1

2

3

4

5

t (s)

N (

109 )

Figure 4.13: Example of a loading and decay curve. At t=0 the loading of the trapis started, i.e., the atomic beam is switched on, at t = 2.1 s the loading is stopped.Trapping conditions: ∆M = −4.4Γ , sM = 0.4 and G = 23 G/cm.

(1.2 ± 0.2) × 1010 atoms/s. We can measure the atom flux to the trapping chamberwith the detection plate at the entrance of of the trapping chamber, see Fig. 4.5.The atom flux will be lower at the trapping center since the atomic beam has adiameter of more than 6 mm at the entrance of the trapping chamber and has topass two flow resistance with a diameter of only 7 mm, see Fig. 4.5. The atomicbeam will be cut off by the flow resistances. This was also measured by taking aCCD-camera image of the fluorescence of the atomic beam at the trapping center. Inthis example the atom flux was measured to be 2.8 × 1010 atoms/s. Thus a loadingrate of R = (1.2± 0.2)× 1010 atoms/s means a loading efficiency of 43%.

Figure 4.14b shows the loading rate as a function of the detuning of the trappingbeams ∆M. As can be seen, the loading rate increases as a function of the laser de-tuning, until the critical detuning is reached, after which the loading rate decreasesagain. The uncertainty in the loading rate is estimated by the uncertainty in thenumber of atoms and the uncertainty in the fit. The maximum loading efficiency wasmeasured at the critical detuning and equals 54%. We cannot explain the increase ofthe loading rate with the detuning; perhaps the shift in radial direction of the cloudcauses a change in the loading rate. The decrease of the loading rate at the criticaldetuning can be explained by the shifting of the cloud center. Because the cloud isshifted to the edge of the trapping beams probably less atoms can be captured. Thisdetuning dependence of the loading rate was not so clear at other trapping condi-tions, i.e., higher laser beam intensities and lower field gradients, then the loadingrate stays more or less constant and decreases at the critical detuning.

In conclusion, we obtain loading rates of up to 1.5× 1010 atoms/s which is morethan 50% of the atomic flux of the bright atomic beam, described in chapter 3.

Metastable neon atoms in a magneto-optical trap 73

0.00 0.25 0.50

0

1

2

3

4

t (s)

N (

109 )

ba

0 2 4 6 80

5

10

15

20

-∆M(Γ)R

(10

9 s-1

)

∆C

Figure 4.14: a: Detail of the loading process shown in Fig. 4.13. The loading rateR is estimated by fitting the loading curve at t = 0 with a straight line. b: Loadingrate of the MOT as a function of the laser detuning of the trapping beams. Trappingconditions: sM = 0.4, G = 23 G/cm, with corresponding critical detuning ∆C = −4.3Γ .

6 Trap population

We measured the amount of atoms in the MOT with the fluorescence-detectionmethod described in section 5.1. Figure 4.15a shows the number of trapped atomsas a function of the laser detuning for different laser beam intensities. In each graphthe number of trapped atoms increases until a certain critical detuning ∆C , beyondwhich the number of atoms suddenly decreases. For the measured critical detuningwe take the detuning at which the measured amount of trapped atoms is maximum.We see that this critical detuning shifts to larger values for higher laser beam inten-sities.

In Fig. 4.15b the measured critical detuning ∆C as a function of the intensity ofthe trapping beams is shown together with the predicted critical detuning, calculatedwith the geometric loss model of section 2.3. We see that the measured criticaldetunings agree very well with the geometric loss model.

As shown in section 2.3 and Fig. 4.3, the model predicts an increase of Nmax withthe intensity of the trapping beams. This is not what we see in the measurementsshown in Fig. 4.15a. Here the maximum number of atoms Nmax, measured at thecritical detuning, decreases from Nmax = 6.0 × 109 to Nmax = 2.5 × 109 when goingfrom sM = 0.25 to sM = 4. We do not have an explanation for this decrease. Thesteady state number of trapped atoms is determined by the loading rate, the volumeand the loss rate, as given by Eq. (4.10). Assuming a constant loading rate andvolume at ∆C, the decrease of Nmax with the intensity is determined by the two-bodyloss rate β. For the highest laser beam intensity sM = 4 the fraction of atoms in

74 Chapter 4

0 2 4 6 80

2

4

6

8

-∆M(Γ)

N (

109 )

0.13

0.20

0.25

0.5

1

2 4

ba

sM=

0 1 2 3 4 50

2

4

6

8

sM-∆

C (

Γ)

Figure 4.15: a: Measured number of trapped atoms N as a function of the laserdetuning ∆M, for different intensities of the trapping beams sM. The magnetic fieldgradient is fixed to G =10 G/cm. b: Critical detuning ∆C as a function of the intensityof the trapping beams sM. Black dots: measurements, solid line: predictions of theDoppler model.

the excited state is only 1.2% at the critical detuning. If we assume that KSP is notmore than two orders of magnitude larger thanKSS , as mentioned in section 2.1, thecontribution of S − P -collisions to the two-body loss rate β is of the same order, orless than the contribution of S −S-collisions to the two-body loss rate. Further on inthis section we show some experimental evidence for the assumption that β = 2KSS .

For higher laser beam intensities the drop in the amount of atoms beyond thecritical detuning is more gradual, as can be seen in Fig. 4.15a. This is also predictedby the geometrical loss model as can be seen in Fig. 4.2a: the lines predicting the shiftof the center of the cloud are more gradual at the critical detuning for higher laserintensities. This means that the leaking process for higher laser beam intensitiesincreases more slowly with ∆M, this is also what we see in the measurements.

We also changed the gradient of the magnetic field and measured again the num-ber of atoms as a function of the detuning of the trapping beams, the intensity of thetrapping laser beams was fixed to sM = 0.4. The result of this measurement is shownin Fig. 4.16a. Again the same behavior is visible. The critical detuning shifts to largervalues for higher field gradients. This is also what the geometrical loss model pre-dicts as can be seen in Fig. 4.16b which shows the measured and predicted criticaldetuning. Again the model predicts the values of the critical detuning very well.

From these measurements it is clear that there is an optimum magnetic field gra-dient for which the number of trapped atoms is maximum. This optimum gradientis about G = 20 G/cm; the number of trapped atoms then equals Nmax = 9 × 109,almost 1010 atoms! The atom cloud has a volume of more than 2 cm3 at this opti-

Metastable neon atoms in a magneto-optical trap 75

0 2 4 6 80

2

4

6

8

10

- ∆Μ ( Γ)

N (

109 ) G=10

26 G/cm13

17 20

0 10 20 300

2

4

6

8

G (G/cm)-∆

C (

Γ)

ba

Figure 4.16: a: Measured number of atoms N as a function of the laser detuning ∆M,for different magnetic field gradients G. The intensity of the trapping beams is fixedsM = 0.4. b: Critical detuning ∆C as a function of the magnetic field gradient G. Themeasured critical detuning corresponding to each trapping condition is indicated bythe black dots, solid line: predictions of the Doppler model.

mum trapping condition! The existence of an optimum magnetic field gradient is,however, not predicted by the Doppler model.

We checked the fluorescence method by measuring the number of atoms with theabsorption imaging method described in section 7.1. Both methods gave the samevalues within 20%.

We can calculate the two body-loss rate β from the measured steady state numberof atoms, the volume, and the loading rate by rewriting Eq. (4.9) to

β = R −NS/τN2

SVeff. (4.23)

The result of this is shown in Fig. 4.17, which shows the calculated two-body lossrate. The trapping conditions are analogous to those corresponding to Fig. 4.14b, i.e.,sM = 0.4 and G = 23 G/cm. For the lifetime corresponding to background collisionswe substitute τ = 3.75 s, which was found from a decay measurement discussed insection 8. With the exclusion of the last two data points the two-body loss rate isfound to be constant as a function of the laser detuning. The average loss rate isfound to be β = (5.1 ± 1.4) × 10−10 cm3/s. The much larger values found for thelast two data points in Fig. 4.14b are due to the uncertainty in the measured volumeand the number of trapped atoms caused by the leaking process of the trap. Goingfrom ∆M = −2Γ to the critical detuning ∆C , the fraction of atoms in the excited statedecreases from 1.4% to 0.3%. Since the two-body loss stays more or less constant asa function of the laser detuning, we can conclude that the measured loss rate couldbe ascribed purely to S − S collisions. Doery et al. [14] found a theoretical value

76 Chapter 4

0 2 4 6 8-0.5

0.0

0.5

1.0

1.5

2.0

-∆M(Γ)

β(10

-9 c

m3 /s

)0.1

∆C

Figure 4.17: Two-body loss rate calculated from the measured number of atoms, thevolume of the atom cloud and the loading rate. The solid line indicates the averageof the data points without the last two points and is found to be β = (5.1 ± 1.4) ×10−10 cm3/s.

of β = 2KSS = 1.6 × 10−10 cm3/s, which is a factor three lower than the value wemeasured. In section 8 we show direct measurements of β from decay curves.

By locking both the laser for the beam machine and the laser for the MOT beamsto the 22Ne isotope, we trapped N = 1× 109 atoms of this isotope. For the trappingconditions used this was three time less atoms than for the 20Ne isotope. So byoptimizing the trap it should also be possible to trap at least 3 × 109 atoms of the22Ne isotope.

In conclusion, we trapped almost 1010 20Ne atoms. This was achieved underrather unconventional low intensity of the trapping beams, i.e., sM 0.25. Theoptimum trapping field is around G = 20 G/cm. At the optimum trapping conditionsthe cloud has a extraordinarily large volume of more than 2 cm3. The simple Dopplermodel and the picture of the seven beam MOT discussed in section 2 explains thetrap behavior quite well.

7 Temperature

In the previous sections we described how we measured most of the trapping char-acteristics necessary to understand the working of the metastable neon MOT. Theremaining quantity is the temperature of the atom cloud. In this section we describetwo experimental methods to estimate the temperature of the atom cloud, namelyvia absorption imaging and from metastable atom time of flight (TOF) signals.

Metastable neon atoms in a magneto-optical trap 77

1.0 2.0

1.0

2.0

0.0O

D

OD

x (mm)5 10 15 20

10

5

15y

(mm

)

Figure 4.18: CCD-camera image of the optical density distribution of the atom cloud.In this example the central optical density equals 1.7.

7.1 Absorption imaging

Spatial distribution

With the absorption imaging technique described below it is not only possible tomeasure the spatial distribution of the atom cloud but also the velocity distributionof the atoms. For both we use a probe laser beam which is tunable in intensity anddetuning (see section 3.4). The probe beam, with a 1/e2 waist diameter of ∼4 cm,passes the trapping chamber in axial direction as can be seen in Fig. 4.6. The po-larization of the laser light is chosen linear. After passing through the trappingchamber the probe beam is focused in such way on a CCD-camera, that an image ofthe intensity distribution of the probe beam, at the position of the trapping center, ismade with a camera. The atoms in the MOT absorb laser light from the probe beam,which gives information about the number of atoms and the position of the atoms.

While passing through the trapping chamber the probe beam is attenuated by theatom cloud. Satisfying Beer’s law we can write for the intensity distribution of theprobe-beam

I(ρ, z) = I(ρ,−∞) exp(−σa

∫ z

−∞n(ρ, z′)dz′

), (4.24)

with σa the optical absorption cross section and n(ρ, z) the spatial density distribu-tion of the atom cloud. The optical density OD is defined as

OD(ρ) ≡ − ln

(I(ρ,∞)I(ρ,−∞)

). (4.25)

Assuming a Gaussian density distribution given by Eq. (4.3), the optical density ofthe atom cloud is given by

OD(ρ) =√2πσaσzn(ρ,0). (4.26)

78 Chapter 4

-2 -1 0 1 20.0

0.5

1.0

1.5

2.0

∆probe(Γ)

OD

Figure 4.19: The optical density of the atom cloud as a function of the detuning ofthe probe laser beam.

Figure 4.18 shows an example of a CCD-image of the part of the intensity dis-tribution of the probe beam which contains the atom cloud. By fitting the intensitydistribution with Gaussians, the radial density distribution of the atom cloud can beobtained.

Number of atoms

The number of trapped atoms can be found by integrating the optical density givenby Eq. (4.26) over the radial coordinate ρ

N = 4√πσρ

σa

∫OD(ρ)dρ, (4.27)

where the absorption cross section σa can be calculated knowing the detuning of theprobe beam [15]. Figure 4.19 shows the central optical density as a function of thedetuning of the probe laser beam. The MOT, containing 3×108 atoms, was operatedwith a high intensity sM = 8 of the trapping beams, the detuning was ∆M = −4Γand the magnetic field gradient G = 10 G/cm. The intensity of the probe beam wassprobe = 0.03 and the exposure time 30 µs, small enough not to influence the trappopulation. The characteristic Lorentzian profile, with a FWHM-width of approxi-mately one linewidth, is clearly visible.

Velocity distribution

Switching off the MOT beams, the atom cloud expands ballistically. The radial veloc-ity distribution of the atom cloud can be measured by comparing absorption images

Metastable neon atoms in a magneto-optical trap 79

taken at different delay times after switching off the MOT beams. After a delay timet the rms radius in radial direction is given by

σ 2ρ (t) = σ 2

ρ (0)+kBTm

t2, (4.28)

with σρ(0) the rms radius on t = 0, and T the temperature of the atoms.

7.2 Metastable atom TOF

With the channeltron in the trapping chamber (see subsection 3.5) we can measuremetastable atoms escaping from the cloud. Switching off the MOT beams and mea-suring the time dependence of the metastable atom signal, the velocity distributionof the trapped cloud can also be obtained. Assuming that the velocity distribution ofthe trapped atoms satisfies a Maxwell-Boltzman distribution we write for the velocitydistribution

P(vx, vy, vz) = P0 exp(− v2x

2σ 2x− v2

y

2σ 2y− v2

z

2σ 2z). (4.29)

After switching off the trapping beams at t = 0 the metastable atom signal on thedetector can be written as

N(t) ∝ (vy,0 − gt)t4

exp(−v2y,0

2σ 2y). (4.30)

Here, vy,0 is the initial velocity of the atoms hitting the detector at time t

vy,0 =yd + 1

2gt2

t, (4.31)

with yd = 4 cm the distance from the channeltron to the trapping center, and g theacceleration caused by gravity.

7.3 Measurements

We measured the temperature of the atom cloud by using the absorption imagingtechnique as described above. The trapping conditions were sM = 1.1, ∆M = −2.7Γand G = 7.6 G/cm at which the MOT contains N = 6.5 × 108 atoms. After the MOTwas loaded the MOT beams were switched off and the atom cloud was exposed bythe probe beam after a delay time t. A CCD-camera image of the absorption profilewas taken after a exposure time of 30 µs. The detuning of the probe beam was taken∆probe = −Γ/2, and the intensity sprobe = 0.03. This measurement cycle was repeatedfor increasing delay times from 0 to 7 ms in steps of 1 ms.

The result of the measurements are given in Fig. 4.20, which shows absorptionimages of the atom cloud as a function of the delay time. The corresponding rmswidths in radial (x and y) direction are plotted as a function of the delay time. Afit with Eq. (4.28) through the data points is shown, resulting in a horizontal and

80 Chapter 4

0 2 4 6 80

1

2

3

4

5

t (ms)

σ x, σ

y (m

m)

σxσy

Figure 4.20: Absorption images of the atoms cloud for different delay times afterswitching of the MOT beams. Black dots: horizontal distribution, open circles verticaldistributions. Corresponding temperatures Tx = 0.78 ± 0.15 mK and Ty = 0.85 ±0.13 mK.

ba

0 25 50 75 100 125

0.0

0.4

0.8

t (ms)

NT

OF (

arb.

uni

ts)

0 25 50 75 100 125

0.0

0.4

0.8

t (ms)

NT

OF (

arb.

uni

ts)

=−3.9Γ∆M

−2.7Γ

0.8sM= 0.4sM=

=−4.6Γ∆M

−3.3Γ

Figure 4.21: a: Metastable atom TOF-spectra for a MOT operated with laser beamintensity sM = 0.8. b: TOF-spectra for a MOT operated with laser beam intensitysM = 0.4. Both graphs show spectra for a small and large detuning of the trappingbeams. The temperature is larger for smaller detunings, as can be seen by the shiftof the curves to smaller flight times. The spectrum for large detuning showed inFig. b shows a fast peak around 20 ms.

Metastable neon atoms in a magneto-optical trap 81

0 2 4 6 80.0

0.5

1.0

1.5

2.0

-∆M(Γ)

T (

mK

)

∆C

Figure 4.22: Temperature as a function of the laser detuning for a MOT operatedwith laser beam intensity sM = 0.8. The solid line shows the corresponding Dopplertemperature. The data point indicated with the open circle shows the result of thetemperature measurement done with the absorption imaging technique, showed inFig. 4.20. Trapping conditions for that measurement were different: sM = 1.1 andG = 7.6 G/cm, indicated is the temperature T = Tx = Ty = 0.8 mK.

vertical temperature of Tx = 0.78± 0.15 mK and Ty = 0.85± 0.13 mK, respectively.This is both slightly higher then the expected Doppler temperature at these trappingcondition which equals T = 0.55 mK.

Amuch easier way to determine the temperature of the atom cloud is the metastableatom TOF technique, described above. Not a series of measurements need to bedone to determine the temperature, like in the case of the absorption imaging tech-nique, but a single measurement directly gives the temperature. We systematicallymeasured the temperature of the atom cloud for different trapping condition, i.e.,different detuning of the trapping beams.

Figure 4.21 shows, for two trapping conditions, the time-of flight signals, both fora small laser detuning and for a large laser detuning (indicated in the figure). Fromthose TOF-signals it is clear that, for both trapping conditions, the temperature ofthe cloud increases for larger laser detunings. For large laser detunings the TOF-signal of the trap operated with low laser beam intensities (sM = 0.4) shows anadditional fast peak (Fig. 4.21b). This makes it impossible to fit the TOF-signals withthe expression given by Eq. (4.30). This fast peak is probably caused by the influenceof the slower beam on the trap dynamics.

The TOF-signals of the trap using large laser beam intensities (sM = 0.8) werefitted with Eq. (4.30). The fits are shown in Fig. 4.21a by the dashed lines. From thosefits the temperature of the atom cloud was estimated for different laser detunings.The result of this is given in Fig. 4.22, which shows an increase of temperature while

82 Chapter 4

using larger laser detunings. The temperature of the cloud is close to the Dopplertemperature, which is indicated in Fig. 4.22 by the solid line. Around the criticaldetuning the temperature starts to increase probably due to the geometrical lossprocess or through a larger influence of the slower laser beam.

In conclusion, we found that the temperature of the atom cloud satisfies theDoppler temperature.

8 Trap decay

In section 6 we showed how we estimated the two-body loss rate from the loadingrate and the steady state number of trapped atoms. In this section we show how weestimated the two-body loss rate directly form trap decay curves.

An example of the load and decay process of the MOT was already shown inFig. 4.13. At t = 0 the MOT is loaded by switching on the atomic beam, i.e., switchingon the laser light to the cooling sections of the beam machine. After a time t = 2.1 sthe loading of the MOT is stopped by switching off again the laser light of the beammachine. By switching off this laser light also the light to the extra slower is switchedoff.

Figure 4.23a shows a detail of the decay process of the MOT. At t = 2.1 s thenumber of atoms first seems to drop rapidly from 4 × 109 to 2.5 × 109, after whichthe number of atoms decay gradually, as can be seen in the figure. This sudden dropis caused by the fact that also the slower beam is switched off at t = 2.1 s. Thishas two effects; first, the fluorescense caused by absorption from the slower beamimmediately falls away, and this is not corrected for in this figure. Correcting for thisartifact results in 20% more atoms after t = 2.1 s. A second effect is that switchingoff the slower beam results in a new situation: a trap with six trapping beams insteadof six trapping beams and a slower beam. Since the forces in a six beam trap havea different equilibrium from the forces in a seven beam trap, the cloud in the trapsuddenly jumps to a different position while switching off the slower beam, whichcan cause losses of trapped atoms.

We fitted the decay curve with Eq. (4.11), for τ we substitute τ = 3.75 s, whichwas found from a measurement where the decay of the MOT was measured duringa longer time. At the steady state situation the volume of the atom cloud was mea-sured by taking CCD-camera images, as described in section 4. Figure 4.23b showsthe two-body loss rate β as a function of the laser detuning. The trapping conditionsare the same as used at the measurement to estimate the loading rate, as describedin section 5, i.e., sM = 0.4 and G = 23 G/cm. From this it is clear that β increasesdrastically near the critical laser detuning. This is not to be expected, as described insection 2 the two-body loss rate should be independent of the laser detuning whileusing low laser beam intensities. Moreover, the estimated values are much higherthan the theoretical value of 1.6× 10−10 cm3/s, which Doery et al. [14] calculated.

The extraordinarily high decay rate we measured is due to the kick to the atomcloud caused by switching off the slower laser beam. This has two effects. First ofall, because of the kick atoms will be lost from the trap. And secondly, the volume

Metastable neon atoms in a magneto-optical trap 83

ba

2.0 2.2 2.4 2.6 2.8 3.0

0

1

2

3

4

5

t (s)

N (

109 )

0 2 4 6 80

5

10

15

20

25

30

-∆M(Γ)β(

10-9

cm

3 /s)

∆C

Figure 4.23: a: Detail of the decay process shown in Fig. 4.13. The decay curve isfitted with the expression given by Eq. (4.11). b: Decay rate of the MOT as a functionof the laser detuning. Again the critical detuning is indicated.

and temperature of the atom cloud will change because of the sudden changing ofthe trapping parameters. The volume was measured during a steady state conditionof the trap, i.e., a seven beam MOT, while the decay of the MOT was measured for aunbalanced six beam MOT. This will cause a misinterpretation of the volume of thecloud, and consequently a discrepancy in the estimated decay rate.

We also measured a load and decay curve for trapping conditions at which theinfluence of the slower beam is less important. This has the effect that the kick tothe atom cloud was negligible when the slower beam was switched off. We chosehigher laser beam intensities of the trapping beams and a slightly lower intensityof the slower laser beam. Furthermore we set the detuning of the trapping laserbeams below the critical detuning. The trapping conditions we chosen were: sM =0.9, sS = 0.16, ∆M = −4Γ and G = 13 G/cm. The number of trapped atoms wasN = 7.6× 108 atoms and the volume of the cloud was V = 0.29 cm3.

Under these conditions we measured the load and decay curve shown in Fig. 4.24.The loading curve, showed in detail in Fig. 4.24a, was fitted with both the expressionof Eq. (4.7) and the straight line at t = 0 s. Both result in a loading rate of R =(1.2 ± 0.1) × 109 atoms/s. The decay curve, showed in detail in Fig. 4.24b, wasfitted with the expression given in Eq. (4.11). From that a two-body loss rate ofβ = (6.8 ± 1.0) × 10−10 cm3/s and a background loss rate of τ = 3.75 ± 0.10 s wasfound. This direct measurement of the two-body loss rate corresponds with theindirect measurements discussed in section 6. Calculating again the two-body lossrate from the loading rate and the steady state number of trapped atoms by usingthe expression given by Eq. (4.23), a value of β = (6.1± 1.0)× 10−10 cm3/s is found.

In conclusion, the trap decay measurements we did were spoiled by the influence

84 Chapter 4

ba

0.0 0.5 1.0 1.5 2.00

2

4

6

8

t (s)

N (

108 )

0 5 10 15 200

2

4

6

8

10

N (

108 )

5 7 9 11t (s)

N (

108 )

10

1

0.1

Figure 4.24: a: Loading and b: decay behavior of the MOT. On t = 0 s the loadingstarts, i.e., the atomic beam is switched on, at t = 5 s the loading is stopped. Theinset in Fig. a shows the complete loading-decay curve. The loading process fittedwith both the expression given in Eq. (4.7) and the straight line indicated in thegraph. The decay of the MOT, shown on a log-scale, is fitted with Eq. (4.11).

of the slower beam on the trap dynamics. This effect was counteracted by choosingtrap conditions were the influence of the slower beam can be neglected. Again atwo-body loss rate of β = (6± 1)× 10−10 cm3/s is found.

9 Concluding remarks

We trapped almost 1010 metastable neon atoms in a MOT. To our knowledge this isthe largest MOT of metastable atoms reported. So far this large number of atoms isreached because of the high loading rates we can obtain with the bright atomic beamdescribed in chapter 3. Furthermore, this optimum number is obtained under ratherunconventional conditions, e.g., very low intensity of the trapping beams. Underthis conditions the volume becomes very large, thereby reducing the density andconsequently reducing the two-body loss rate.

Another remarkable phenomenon of our MOT is the influence of the slower beamon the trap dynamics. As a consequence we do not have an ordinary six beam MOTbut a seven beam MOT. However, this seven beam MOT satisfies a simple Dopplermodel, developed in this chapter, very well. The influence of this seventh MOT beamcan be regulated easily by choosing different trapping parameters, such as higherintensities of the trapping beams or different compensating magnetic fields.

The temperature of the atom cloud was measured and found to be in the Dopplerlimited regime, i.e., temperatures around 1 mK. For the highest measured density,i.e., 4× 1010 atoms/cm3, this corresponds to a phase space density of nΛ3 10−7.

Metastable neon atoms in a magneto-optical trap 85

Furthermore, the two-body loss rate was measured to be β = 2KSS = (5 ± 1) ×10−10 cm3/s, which is three times higher than calculated by Doery et al. [14].

A MOT containing 1010 atoms is a good starting point on the road to BEC, asdescribed by Beijerinck et al. [4]. Apart from a MOT of 20Ne we can also make a MOTcontaining more than 109 atoms of the 22Ne isotope.

References

[1] S. Nowak, A. Browaeys, J. Poupard, A. Robert, D. Boiron, C. Westbrook, and A.Aspect, Appl. Phys. B 70, 455 (2000).

[2] N. Herschbach, P.J.J. Tol, W. Hogervorst, and W. Vassen, Phys. Rev. A 61,050702(R) (2000).

[3] M. Zinner, C. Jentsch, G. Birkl, and W. Ertmer, private communication.

[4] H.C.W. Beijerinck, E.J.D. Vredenbregt, R.J.W. Stas, M.R. Doery, and J.G.C. Tem-pelaars, Phys. Rev. A 61, 023607 (2000).

[5] F. Bardou, O. Emile, J.-M. Courty, C.I. Westbrook, and A. Aspect, Europhys. Lett.20, 681 (1992).

[6] M. Kumakura and N. Morita, Phys. Rev. Lett. 82, 2848 (1999).

[7] P.J.J. Tol, N. Herschbach, E.A. Hessels, W. Hogervorst, and W. Vassen, Phys. Rev.A. 60, R761 (1999).

[8] A. Browaeys, J. Poupard, A. Robert, S. Nowak, W. Rooijakkers, E. Arimondo, L.Marcassa, D. Boiron, C.I. Westbrook, and A. Aspect, Eur. Phys. J. D 8, 199 (2000).

[9] H.C. Mastwijk, J.W. Thomsen, P. van der Straten, and A. Niehaus, Phys. Rev. A80, 5516 (1998).

[10] C.G. Townsend, N.H. Edwards, C.J. Cooper, K.P. Zetie, C.J. Foot, A.M. Steane, P.Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995).

[11] J. Arlt, P. Bance, S. Hopkins, J. Martin, S. Webster, A. Wilson, K. Zetie, and C.J.Foot, J. Phys. B: At. Mol. Opt. Phys. 31, L321 (1998).

[12] V.P. Mogendorff, Towards BEC of metastable neon, internal report, EindhovenUniversity of Technology (2000).

[13] P.D. Lett, W.D. Philips, S.L. Rolston, C.E. Tanner, R.N. Watts, and C.I. Westbrook,J. Opt. Soc. Am. B 6, 2084 (1989).

[14] M.R. Doery, E.J.D. Vredenbregt, S.S. Op de Beek, H.C.W. Beijerinck, and B.J. Ver-haar, Phys. Rev. A 58, 3673 (1998).

[15] H.J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-VerlagNew York 1999.

86 Chapter 4

Chapter 5

Photoassociation spectroscopy of 85Rb20+u states

The work described in this chapter was a collaboration of the Ultracold Atomic PhysicsGroup of the University of Texas at Austin, the Atomic Physics Division of the NationalInstitute of Standards and Technology at Gaithersburg, and the Atomic Physics andQuantum Electronics group at the Physics Department of the Eindhoven University ofTechnology. The experimental part of the work has been done at the University ofTexas at Austin.

1 Introduction

Photoassociation spectroscopy has been shown to be a very powerful tool in coldatom physics. Thanks to this technique, the long-range behavior of ground andexcited state cold collisions is now well-understood. From photoassociation data in-teraction parameters such as ground state scattering lengths could be derived [1,2].By using the photoassociation technique it was also possible to predict [3, 4] andmeasure [5] Feshbach resonances in collisions of ultracold Rb atoms. Recently, bycombining photoassociation data with Feshbach resonance field data, the sodiumscattering length could be estimated with improved accuracy [6]. By calculating pho-toassociation spectra and comparing them with measured spectra, it is even possibleto obtain an estimate of the long range behavior of the hyperfine interaction [7,8].

Photoassociation spectroscopy is also very useful for studying excited state po-tentials. Cline et al. [9] measured the rotationally resolved bound levels of the purelylong-range Rb2 0−g state and estimated a number of excited state parameters such asthe C3 coefficient. From their spectra they also found an indication of predissocia-tion of the Rb2 0+u states, and estimated the probability of the Landau-Zener transi-tion to the potential that connects asymptotically to the 5 2S1/2+5 2P1/2 limit. Dueto this kind of measurement, much was learnt about the Rb excited state potentials.Far less, however, is known about the couplings between them.

In this chapter we use photoassociation spectroscopy to study the coupling be-tween the 85Rb A 1Σ+u and b 3Πu states. We compare coupled channel bound state

87

88 Chapter 5

calculations with experimental photoassociation spectroscopy data, and develop aqualitative picture of the spin-orbit coupling between the A 1Σ+u and b 3Πu states. Insection 2 we describe the Fourier Grid Hamiltonian method we used for our coupledchannel bound state calculations. Section 3 gives the results of those calculationsapplied to the coupled 85Rb A 1Σ+u and b 3Πu states. A description of the photoas-sociation experiment and the experimental results are given in section 4. In section5 we compare the results of the coupled channel bound state calculations with theexperimental photoassociation data. As a reference we also describe the couplingbetween the A 1Σ+u and b 3Πu states with a simple Landau-Zener formula.

2 Bound state calculations

With the photoassociation spectroscopy technique it is possible to measure rovi-brational states over a wide range of energies. To compare experimental photoas-sociation spectra with theory, it is necessary to calculate bound state eigenvaluesover an equally wide energy range. Standard methods using iterative procedures,e.g., the Numerov integration method, often involve time consuming calculationsand are therefore only useful for a relatively small energy range. Moreover, be-cause of perturbations, e.g., spin-orbit coupling, bound states can be very closetogether. This implies that small iterative steps are necessary as well as complexbasis-transformations to represent the coupling schemes over the whole internu-clear range. Dulieu and Julienne [10] showed that a noniterative Fourier grid methodis a very powerful tool to calculate bound states of coupled systems. They applied aFourier Grid Hamiltonian (FGH) code to calculate the 150 lowest rovibrational statesof the coupled A 1Σ+u and b 3Πu potentials for two extreme cases: the light Na2 dimerand the heavy Cs2 dimer. In this chapter we extend their approach to the case of Rb2.First we briefly discuss the main steps of the FGH method analogous to Monnervilleand Robbe [11].

2.1 Fourier grid method

The radial part of the eigenvalue problem of a diatomic molecule is written as

HΨ(R) = [T + V(R)]Ψ(R) = EΨ(R), (5.1)

with T and V the kinetic energy and potential operator, respectively. To make adiscrete variable representation (DVR) approach we need to discretize the Hilbertspace. We do this, analogous to Colbert and Miller [12], by considering the coordinateR restricted to the interval (R0, RN) and apply, in a first approach, a grid Ri withuniform spacing ∆R = (RN − R0)/N:

Ri = R0 + (RN − R0)i/N, i = 1 to N − 1. (5.2)

The Hamiltonian, Eq. (5.1), is in the coordinate representation given by

Hii′ = 〈Ri|H|Ri′〉 = 〈Ri|T|Ri′〉 + 〈Ri|V|Ri′〉. (5.3)

Photoassociation spectroscopy of 85Rb2 0+u states 89

Here the basis functions |Ri〉 satisfy the orthonormal and completeness relation-ships,

〈Ri|Ri′〉 = δii′ , (5.4)

IR =N−1∑i=1

|Ri〉〈Ri|. (5.5)

The discrete scalar product between two functions φn and φn′ in the discrete spaceis defined by

〈φn|φn′〉 =N−1∑m=1

φn(Rm)φn′(Rm)∆R. (5.6)

Assuming orthonormal wave functions φn(R) satisfying the completeness relation-ship,

Iφ =N−1∑n=1

|φn〉〈φn|, (5.7)

the Hamiltonian becomes [13],

Hii′ = 〈Ri|H|Ri′〉 = 〈Ri|T|Ri′〉 + 〈Ri|V|Ri′〉= 〈Ri|T|Iφ|Ri′〉 + 〈Ri|V|IR|Ri′〉

= 〈Ri|T|N−1∑n=1

|φn〉〈φn|Ri′〉 + 〈Ri|V|N−1∑n=1

|Rn〉〈Rn|Ri′〉

= −2

2µ

N−1∑n=1

〈Ri|φ′′n〉〈φn|Ri′〉 +

N−1∑n=1

V(Rn)〈Ri|Rn〉〈Rn|Ri′〉

= −2

2µ∆R

N−1∑n=1

φ′′n(Ri)φ∗

n(Ri′)+ V(Ri)δii′ (5.8)

with, in the last step, the scalar products 〈Ri|φ′′n〉 and 〈φn|Ri′〉 like in Eq. (5.6),

written as,

〈Ri|φn〉 =N−1∑m=1

Ri(Rm)φn(Rm)∆R =N−1∑m=1

δim√∆R

φn(Rm)∆R = φn(Ri)√∆R. (5.9)

We take wave functions that vanish at the endpoints R0 and RN , i.e., particle-in-a-box eigenfunctions,

φn(R) =(

2RN − R0

)1/2sin

[nπ(R − R0)RN − R0

], n = 1 to N − 1. (5.10)

With these functions the Hamiltonian Eq. (5.8) becomes,

Hii′ = −2

2µ

(π2

RN − R0

)22N

N−1∑n=1

n2 sin(nπiN

)sin

(nπi′

N

)+ V(Ri)δii′ , (5.11)

90 Chapter 5

and working out the summation we obtain

i ≠ i′ : Hii′ = π22

4µ(RN − R0)2(−1)i−i′

1

sin2[π(i− i′)/2N]− 1

sin2[π(i+ i′)/2N]

,

(5.12)

i = i′ : Hii = π22

4µ(RN − R0)2

2N2 + 1

3− 1

sin2(πi/N)

+ V(Ri).

For an arbitrary number of spin channels j, the Hamiltonian for our eigenvalue prob-lem is represented by

Hij,i′j′ = Tii′δjj′ + Vjj′(Ri)δii′ . (5.13)

The advantage of the FGH approach follows from Eq. (5.12): No numerical matrixelement calculations have to be performed, since an analytical formulation of thekinetic energy part of the Hamiltonian is available. Numerical accuracy demands thatthe number of grid points be large enough to reproduce the maximum momentumrange. In other words, the stepsize S = (RN −R0)/N, with R0 and RN the position ofthe inner and outer turning points, must be small enough to represent correctly theoscillations in the wavefunction at the internuclear separation where the potentialV is deepest (at that point the radial wave function oscillates most rapidly) [14]. Informula:

S ≤ λmin/2 = π√2µ2 [E − Vmin]

, (5.14)

with λmin the wavelength of the radial wave function at the position where the po-tential is deepest, E the binding energy (the energy of the bound state) and Vmin

the minimum of the potential. Rovibrational states extending to a large internuclearseparation involve a large gridlength RN−R0, and therefore a large number N of gridpoints. For example, calculating all vibrational states of the Rb2 A 1Σ+u(0+u) potentialup to the 52S1/2 + 52P1/2 dissociation limit requires N ≈ 104 grid points to fulfill thecriterion given by Eq. (5.14). This results in huge matrices with a huge demand oncomputer memory, especially when several spin states are involved.

For large internuclear separations, however, the local wavelength of the radialwave function is much larger than at the point where the potential is deepest. In thisregion it makes sense to use a larger stepsize to reduce N [14,15]. This gives a locallimitation of the allowed stepsize S(R) as given by:

S(R) ≤ λ(R;E)/2 = π√2µ2 [E − V(R)]

. (5.15)

However, changing the gridsize also changes the expression for the kinetic energy,as we discuss below.

2.2 Variable gridsize

To calculate the Rb2 bound states up to the 5 2S1/2 + 5 2P1/2 dissociation limit itis necessary to make the grid spacing S(R) variable in such way that the grid size

Photoassociation spectroscopy of 85Rb2 0+u states 91

matches the local wavelength fairly closely. Tiesinga et al. [14] did this by applyinga non-linear coordinate transformation given by ρ = f(R), where f is a monotonic,invertible function of R with a derivative that decreases with increasing R. There-fore, by defining a homogeneous grid in the transformed ρ coordinate, the distancebetween two grid points in the original coordinate space grows with increasing R.The relation between the variable grid spacing S(R) and the constant grid spacing∆ρ is given by

S(R) = ∆ρf ′(R)

. (5.16)

Comparing this with Eq. (5.15), we adopt a transformation function given by [15]:

ρ = f(R) = 2∫∞

R

drλ(r ;E)

= 1π

∫∞

R

(2µ2

[E − V(R)]) 1

2

dr (5.17)

with λ(r ;E) given by Eq. (5.15). An analytic expression for the transformation func-tion f(R) is based on the long-range behavior V(R) = −Cn/Rn of the potential.Using this long-range behavior and the approximation E = 0, Eq. (5.17) leads to

f(R) = 1π

∫∞

R

(2µ2

Cn

rn

)1/2dr = 1

π2

n− 2

(2µCn

2

)1/2 (1R

)n2−1

. (5.18)

Deviations from a pure Cn/Rn behavior at small internuclear distance and the posi-tion of the deepest point of the potential can make it necessary to add a term to thetransformation to provide a better mapping.

Transforming the coordinates also modifies the kinetic energy operator of theSchrodinger equation according to [14]

T = −2

2µd2

dR2= −2

2µ

p(ρ)

d2

dρ2+ q(ρ)

ddρ

(5.19)

with

p(ρ) = 1(F ′(ρ))2

and q(ρ) = − F ′′(ρ)(F ′(ρ))3

, (5.20)

where F(ρ) is the inverse of f(R). We introduce a wavefunction φ(ρ) which isnormalized with respect to ρ if Ψ is normalized with respect to R

Ψ(R) = φ(ρ)√F ′(ρ)

. (5.21)

With this substitution and V (ρ) = V(F(ρ)) the Schrodinger equation becomes[2

2µ

− 1F ′(ρ)

d2

dρ2

1F ′(ρ)

+ V (ρ)

]φ(ρ) = Eφ(ρ), (5.22)

where the potential term

V (ρ) = V (ρ)+ 2

2µ

−12

F(3)(ρ)(F ′(ρ))3

+ 34(F ′′(ρ))2

(F ′(ρ))4

(5.23)

in which only the first part, given by V (ρ), is spin dependent. Equation (5.12) can beused to discretize the second derivative term d2/dρ2 in Eq. (5.22).

92 Chapter 5

52S1/2

5 2P1/2

5 2P3/2

f43213

2

23

3.04 GHz

362 MHz

29.3 MHz63.4 MHz121 MHz

D1

D2

795 nm

780 nm

Figure 5.1: Schematic level diagram for 85Rb with the first three fine structure states.Indicated are, on the left, the D1 (795 nm), 5 S1/2 →5 P1/2 and D2 line (780 nm),5 S1/2 →5 P3/2 and on the right the hyperfine intervals.

3 Application to the rubidium dimer

3.1 Coupled potentials

The formation of molecules by photoassociation takes place at rather large internu-clear distances. The outer vibrational turning points of such molecules lie between20 and 400 a0 for Rb. At these internuclear distances the overlap of the electronclouds is negligible so that we can describe these molecules as two separate atomswho interact via electrostatic interactions [14]. The atomic ground state Hamilto-nian, which describes a single ground state atom, only involves a hyperfine inter-action term which couples the nuclear spin i to the electron spin s to form totalangular momentum f = s + i. The isotope 85Rb has a nuclear spin i = 5/2, so theground state atom in a 2S1/2 fine structure state can have total angular momentumf = 2 or f = 3. The atomic Hamiltonian for the excited 2P states involves also thestrong spin-orbit interaction, which couples the electronic orbital angular momen-tum la to the electron spin s to form the electron angular momentum ja = la + s.The hyperfine interaction then couples again the electron angular momentum to thenuclear spin; for the 2P1/2 state this gives total angular momentum f = 2, 3 and forthe 2P3/2 state total angular momentum f = 1, 2, 3, 4. In Fig. 5.1 a schematic leveldiagram for 85Rb is shown.

For two atoms in the ground state the molecular Hamiltonian consists of theatomic hyperfine interaction and the central, electrostatic interaction. The latter is,for atoms at intermediate and large internuclear separation, given by the dispersionand exchange interactions, forming together the singlet and triplet potentials. Forvery large internuclear separation the central interaction becomes negligible com-pared to the atomic hyperfine interaction [16].

The molecular Hamiltonian for the interaction between a ground state atom andan excited state atom consists, for large internuclear separation, of the atomic hy-

Photoassociation spectroscopy of 85Rb2 0+u states 93

Table I: Potential parameters for the 1Σ+u and 3Πu potentials for 85Rb2 dimer, as usedfor the calculations of the vibrational states.

1Σ+u 3Πu

C31(a.u.) 17.872 8.936

C62 (103 a.u.) 12.050 8.047

C82 (105 a.u.) 28.050 4.203

D13 (cm−1) 12578.906

D23 (cm−1) 12816.469

∆FS (cm−1) 237.5631Value extracted from experimental long-range 0−g analysis [23]. Valuesgiven in Reference [22]: C3(1Σ+u) = 18.400 a.u. and C3(3Πu) = 9.202 a.u.2Reference [22]3Reference [20]

perfine interaction, the spin-orbit interaction and the resonant electric dipole inter-action. For the internuclear separations we are interested in, i.e., the internuclearrange between 20 a0 and 400 a0, the hyperfine interaction is negligible for excitedstates. The resonant electric dipole interaction has a 1/R3 radial dependence, andfor large internuclear separations it is comparable to the spin-orbit interaction. Boththe resonant electric dipole interaction and the spin-orbit interaction conserve theprojection of the total electronic angular momentum j = ja + jb on the molecularaxis, therefore the so called Hund’s case (c) forms a convenient basis for this molec-ular Hamiltonian [17]. Following Movre and Pichler [18], the states are labeled byΩ±

σ , with Ω = |M| the absolute value of the projection of the total electron angular

momentum j on the molecular axis and σ = g or u the electronic parity. The su-perscript ±, which is only present for Ω = 0, indicates the symmetry with respectto reflection in a plane containing the internuclear axis. Considering this, the totalHamiltonian for all combinations of the 2S ground state atom and the 2P excitedstate atom, is given by a 24×24 matrix which, because of the conservation of theprojection of total angular momentum, consists of three submatrices correspondingto Ω =0, 1 and 2 [19]. Diagonalisation of those submatrices results in the 16 long-range adiabatic Movre-Pichler potential curves which for Rb2 are given in Fig. 5.2. Forinfinite internuclear separation the curves go to the S1/2+P1/2 and S1/2+P3/2 limits,from now on called the D1 and D2 dissociation limits. For 85Rb2 these limits are atD1 = 12578 cm−1 and D2 = 12816 cm−1 relative to the S1/2+S1/2 limit [20] (Table I).

For intermediate internuclear separation, the resonant electric dipole interactionbecomes, because of its 1/R3 radial dependence, much larger than the spin-orbitinteraction. In this region, called the uncoupled region by Movre and Pichler [18],also the total spin S = sa + sb and Λ, the projection of total orbital angular mo-mentum L = la + lb on the internuclear axis are conserved. Therefore the Hund’scase (a) labels 2S+1|Λ|±σ are used, resulting in four non-degenerate potential curves

94 Chapter 5

0+u

0+u

D1

D2

2g1g1u0-

g

1u 0+g0-

u

0+g

2u1g

1u 0-u

1g0-

g

20 40 60 8012.4

12.6

12.8

13.0

R (units of a0)

V (

103 c

m-1

)

Figure 5.2: The adiabatic long-range potential curves for the 2S1/2+2P1/2,3/2 first ex-cited states of 85Rb2 labeled in Hund’s case (c). The two 0+u curves, belonging to the2P1/2 dissociation limit D1, and to the 2P3/2 dissociation limit D2, respectively, areindicated by arrows.

5 10 15 205

10

15

R (units of a0)

V (

103 c

m-1

)

3Σu3Πg1Πg

1Πu 1Σg3Σg

3Πu

1Σu

Figure 5.3: The eight adiabatic Born-Oppenheimer (ABO) potential curves for Rb2. Atlong, range all eight curves result in the same limit because spin-orbit interaction isnot included. The term adiabatic refers here to the electron motion.

Photoassociation spectroscopy of 85Rb2 0+u states 95

0 10 20 30 40

6

8

10

12

14

R (units of a0)

V

(10

3 cm

-1)

b3ΠuA1Σ+

uD1

D2

8 9 10 11

0+u(D1)

0+u(D2)

Figure 5.4: Potential curves for the A 1Σ+u and b 3Πu states of Rb2. The inset showsthe curves in their crossing region. Broken lines: diabatic potentials without spin-orbit interaction; full lines: adiabatic 0+u potentials with spin-orbit coupling included,resulting for infinite internuclear separation in the two dissociation limits D1 and D2

(Table I).

corresponding to the Λ = 1 (Π) states with a ±C3/R3 radial dependence and theΛ = 0 (Σ) states with ±2C3/R3 radial dependence. So for intermediate internucleardistance the 16 adiabatic potential curves (Fig. 5.2) change over to four distinct elec-tric dipole potentials [18]. At small internuclear separation, however, the exchangeinteraction splits each of the four potentials into two separate branches, resulting inthe eight adiabatic Born-Oppenheimer (ABO) potentials shown in Fig. 5.3 [14]. TheBorn-Oppenheimer approximation states that the electrons move adiabatically withrespect to the slow nucleus. The term adiabatic refers to the inclusion of 〈Ψel|T |Ψel〉in the potential. For large internuclear separation the eight ABO potentials go to tothe same limit.

We are interested in the A 1Σ+u(0+u) and b 3Πu(0+u) excited states and the couplingbetween them. We use the terms diabatic and adiabatic with respect to the spin-orbit interaction. In this context diabatic means that the spin-orbit interaction is notincluded, and adiabatic means that spin-orbit interaction is included. In Fig. 5.4 thepotential curves for the A 1Σ+u and b 3Πu states of Rb2 are shown over the completeinternuclear range. The adiabatic 0+u potentials follow from the diabatic A 1Σ+u andb 3Πu potentials by including the spin-orbit interaction. More details of the poten-tials are given in the next section, in which we calculate the rovibrational states ofthe coupled A 1Σ+u(0+u)− b 3Πu(0+u) system.

96 Chapter 5

6 8 10 12 140

5

10

15

20

25

(103 cm-1)

Bv

(10-3

cm

-1)

b3Πu

A1Σ+u

0+u(D2)

0+u(D1)

D1 D2

vE

Figure 5.5: Rotational constant Bv (J=0) in Rb2. Broken lines: vibrational levels ofthe A 1Σ+u and b 3Πu diabatic potentials, disregarding the spin-orbit interaction; fulllines: vibrational levels of the 0+u adiabatic potentials. The data points (+) indicatethe results of the coupled bound state calculation, using the A 1Σ+u and b 3Πu diabaticpotentials as basis and the spin-orbit interaction as the coupling.

3.2 Coupled A 1Σ+u(0+u)− b 3Πu(0+u) system

We will use the FGH method described in section 2 to calculate the rovibrationalenergies of the coupled 85Rb A 1Σ+u(0+u) and b 3Πu(0+u) states. By implementing thecoordinate transformation described in section 2.2 in the FGH computer code usedby Dulieu and Julienne [10] it was possible to calculate the Rb2 rovibrational statesup to the D1 dissociation limit. In our case we are dealing with two spin channels,the 1Σ+u and 3Πu states, so the total Hamiltonian given by Eq. (5.13) can be expressedin terms of square matrices of order 2(N − 1),

(HAA HABHBA HBB

)=(

T 00 T

)+(

VA VABVAB VB

), (5.24)

where VA, VB , and VAB are diagonal (N−1)×(N−1)matrices and T is a nondiagonal(N − 1)× (N − 1) matrix.

The A 1Σ+u and b 3Πu potentials we used are ab initio potentials calculated byFoucrault et al. [21]. For the long-range part of the potentials we used the dispersioncoefficients given by Marinescu and Dalgarno [22], except for the C3 coefficient. Forthat we used a value extracted from an experimental long-range 0−g analysis [23].Table I gives the most important potential parameters of the A 1Σ+u and b 3Πu po-tentials. We assume the spin-orbit coupling between the A 1Σ+u and b 3Πu potentialsto be R-independent. This means that we take a constant value for the couplingterm VAB in Eq. (5.24), namely the value of the spin-orbit interaction VSO at infinite

Photoassociation spectroscopy of 85Rb2 0+u states 97

12.54 12.56 12.580

2

4

6

Bv

(10-3

cm

-1)

v=437

v=443

0+u(D2)

0+u(D1)

D1

(103 cm-1)vE

Figure 5.6: Calculated rotational constants Bv for the vibrational states close to theD1 dissociation limit. The data points are for the case of coupled potentials. Thearrows indicate a strongly perturbed vibrational state v = 437 and a vibrationalstate, v = 443, whose Bv is close to the unperturbed adiabatic value. The solid linescorrespond to Bv for the case of the uncoupled adiabatic potentials.

internuclear separation:

VAB(R) = VSO(∞) =√2 ·∆FS/3, (5.25)

in which ∆FS is the fine structure splitting (Table I).By taking N = 1000 grid points and a physical grid extending from 3.5 a0 to

550 a0, it is possible to calculate all rovibrational states of the coupled A 1Σ+u andb 3Πu potentials up to the D1 dissociation limit. Knowing that the coupled A 1Σ+u andb 3Πu potentials contain 542 rovibrational states, labeled from v = 0 to v = 541,it is remarkable that with only N = 1000 grid points, so less than 2 grid points pernode for the highest rovibrational states, we can reach an accuracy in the vibrationalenergy better than 10−4 cm−1 [15]!

Analogous to Dulieu and Julienne [10], we calculate both the rovibrational ener-

gies and the rotational constant Bv = 2

2µ〈Ψv(R)|1/R2|Ψv(R)〉 for rotational quantumnumber J = 0. Because of its 1/R2 dependence, the rotational constant is very sensi-tive to the effective excursions of the vibrating molecule in the radial direction. Dueto the different radial range over which the two diabatic (or adiabatic) potentialsextend, the rotational constant is very sensitive to the coupling between the two po-tentials. The vibrational levels are sensitive to the shape of the potentials involved.Moreover, the rotational constant is a parameter which can be readily measured byhigh resolution photoassociation spectroscopy (see section 4). By comparing themeasured rotational constants to the calculated values, detailed insight is obtainedin the coupling of the A 1Σ+u and b 3Πu potentials.

In Fig. 5.5 we show the calculated rotational constant as a function of the bindingenergy Ev , for three cases: (1) for the eigenstates of the individual A 1Σ+u and b 3Πu

98 Chapter 5

20 40 60 80 10012.50

12.52

12.54

12.56

12.58

R (units of a0)

(10

3 cm

-1)

20 40 60 80 10012.50

12.52

12.54

12.56

12.58

R (units of a0)

cm

-1)

20 40 60 80 10012.50

12.52

12.54

12.56

12.58

R (units of a0)

cm

-1)

20 40 60 80 10012.50

12.52

12.54

12.56

12.58

R (units of a0)

cm

-1) 0+

u(D1)

0+u(D2) D1

Bv

vE

Figure 5.7: Vibrational levels of the adiabatic 0+u(D1) and 0+u(D2) potentials, togetherwith the corresponding rotational constants. The maxima in Bv(Ev) coincide approx-imately with the vibrational levels of the 0+u(D2) potential, with its wider vibrationalspacing and its smaller internuclear separation.

potentials, diabatic regarding to spin-orbit interaction; (2) for the eigenstates of theadiabatic 0+u potentials, and (3) for the coupled A 1Σ+u and b 3Πu channels. In thisfigure we see that the results of the coupled channel calculations follow neither thediabatic (broken) lines nor the adiabatic (solid) lines but oscillate. This oscillatorybehavior of the rotational constant is also found by Kokoouline et al. [15], who didthe same kind of calculations as described in this chapter. The oscillations in therotational constant can be seen more clearly when we look at the levels close tothe D1 dissociation limit, as shown in Fig. 5.6. The arrows indicate the vibrationalstates v = 437 and v = 443 with a maximum and a minimum rotational constant,respectively.

The oscillations in the rotational constant are due to perturbations caused bythe interleaving of the relatively sparse 3Πu vibrational levels in the much denserspectrum of the 1Σ+u vibrational levels dissociating to the D1 limit. In Fig. 5.7 wesee schematically that the position of the maxima of the peaks corresponds approx-imately to the values of the adiabatic 0+u(D2) vibrational energies.

The perturbations of the 0+u(D2) levels in the spectrum of the 0+u(D1) levels arealso visible in the vibrational wave functions. Figure 5.8 shows the vibrational wavefunction of the v = 437 and v = 443 states. We see that the wave function cor-responding to the vibrational state with the large rotational constant (v = 437) isstrongly perturbed around R = 20 a0, the position of the outer turning point of the0+u(D2) potential.

Photoassociation spectroscopy of 85Rb2 0+u states 99

0 20 40 600.00

0.02

0.04

R (units of a0)

| Ψ(R

)|2

v=437

0 20 40 600.00

0.04

0.08

R (units of a0)

v=443

v| Ψ

(R)|

2v

Figure 5.8: Vibrational wavefunctions |Ψv(R)|2 of the coupled potentials for thestrongly perturbed v = 437 vibrational state (top panel) and the only slightly per-turbed v = 443 vibrational state (lower panel) (Fig. 5.6).

4 Photoassociation experiment

4.1 Experimental setup

We study photoassociation of 85Rb atoms by measuring trap loss from a far-off reso-nance trap (FORT). The FORT we use consists of a single, linearly polarized, focusedGaussian laser beam. Like described in chapter 2, the trapping force is the opticaldipole force, a conservative force based on the interaction between the gradient ofthe electric field of the laser light and the induced atomic dipole moment. This inter-action has been described by Dalibard and Cohen-Tannoudji [24] in a dressed statepicture, and in terms of optical Bloch equations by Gordon and Ashkin [25]. Assum-ing that the laser detuning ∆L = ωL −ω0, with ωL the laser frequency and ω0 theatomic transition, is much larger than the Rabi frequency Ω and the spontaneous-emission rate of the atom Γ , the trap depth U0 due to the dipole force is, analog toEq. (2.18), given by [24,25]

U0 = Ω20/4∆L, (5.26)

with Ω0 the Rabi frequency in the waist of the FORT laser. The advantage of a FORTover a magneto-optical trap (MOT) is the absence of near-resonance light, whichdrastically reduces the spontaneous-emission rate γs = ΓΩ2

0/4∆2L and thereby the

100 Chapter 5

Molasses beamsRepumper

Dipole

PMT

PA Probe

Figure 5.9: Schematic view of the laser beams in the FORT setup.

heating by photon recoils [26]. The wavelength of the FORT laser beam, producedby a Coherent Model 899-01 Ti:sapphire laser, can be tuned between 4 and 67 nm tothe red of the 85Rb 5 2S1/2-5 2P1/2 transition. Typical values for the trap depth are1-10 mK.

In Fig. 5.9 a schematic view of the laser beams for the photoassociation spec-troscopy (PAS) experiment are given. The FORT is loaded from a vapor cell MOTconsisting of three pairs of counterpropagating σ+ − σ− beams, which intersect atright angles inside a Rb vapor cell at the zero-field point of a magnetic sphericalquadrupole field [26]. When the FORT is loaded the photoassociation (PA) beam isturned on and after a while the amount of atoms in the FORT is measured with laserinduced fluorescense.

Figure 5.10 gives a complete overview of the optical components necessary torun the experiment. The laser beams for the MOT are derived from a high-powerdiode laser producing 40 mW of laser light which is injection-locked to a grating-tuned DBR diode laser. The grating-tuned DBR diode laser is locked to the 85Rb 52S1/2 (f=3) ↔ 5 2P3/2 (f′=4) cross-over saturated absorption peak. One beam fromthe laser (26 mW) is detuned with a 80 MHz acousto optic modulator (AOM) to 10MHz (1.7 Γ ) to the red of the 85Rb 52S1/2 (f=3) ↔ 52P3/2 (f′=4) transition and is usedfor all the MOT beams. A second beam from this laser is detuned with a 100 MHzAOM and is used as a probe beam to measure the amount of atoms in the FORT withlaser-induced fluorescence. A second grating-tuned diode laser is locked with a 130MHz offset to the 85Rb 5 2S1/2 (f=2) ↔ 5 2P3/2 (f′=3) cross-over saturated absorptionpeak to be used as a repumper beam, preventing optical pumping into the 5 2S1/2(f=2) state. Figure 5.11 schematically shows the different laser transitions in the 85Rblevel diagram.

For the PA laser we use a narrowband Ti:Sapphire laser (Coherent 899-21, 1 MHz).The wavelength of this laser can be measured by a wavemeter with a resolution of30 MHz. The stabilized HeNe laser of this wavemeter was calibrated by locking theTi:Sapphire laser to a Rb saturation absorption peak of known frequency [20]. Dur-ing a PAS experiment, as described in the next section, the wavelength of the PAlaser is scanned and the amount of atoms in the FORT is measured. At the begin-ning and end of each scan, usually over 20 GHz, the wavelength is measured within

Photoassociation spectroscopy of 85Rb2 0+u states 101

Argon Ion Laser CoherentTi:Sapphire

899-21

Ti:SapphireCoherent 899-01

Wavemeter

AOM

AOM

AOM AOMAOM

AOM

300 MHz etalonPA

FORT

Rb Cell Rb Cell

MOT Probe

DBRDL

DLRepumper

Figure 5.10: Schematic view of the optical components used for the photoassociationexperiment.

52S1/2

52P1/2

52P3/2

f4

321

23

795 nm

780 nm

ad

c

b

e

Figure 5.11: Energy level diagram for 85Rb showing the different laser frequenciesused in the photoassociation experiment: (a) FORT laser beam, (b) PA laser beam, (c)MOT beams, (d) repumper, and (e) probe laser beam.

102 Chapter 5

12.544 12.545 12.546Wavelength (10 3 cm-1)

PA r

ate

(arb

. uni

ts) 0+

u 0-g

1g

J=012 3 4 5

0+u

Figure 5.12: Part of the experimental photoassociation spectrum, showing a rovibra-tional transition of each of the 0+u, 0

−g and 1g molecular states. The inset shows the

rotational structure of the vibrational level of the 0+u state in more detail, includingthe assignment of the rotational quantum number J.

0.001 cm−1. During a scan, frequency markers are derived from the transmissionpeaks of a stable 300 MHz (0.01 cm−1) etalon. By fitting this etalon signal, and usingthe begin and end readings from the wavemeter, it is possible to assign the PA laserfrequency within 0.002 cm−1.

4.2 Measuring routine

The experiments we did are similar to those described by Cline et al. [9]. Fully loaded,the MOT contains 106 atoms at a density of 1010 cm−3. From the MOT we loadedapproximately 104 85Rb atoms with a temperature of several hundred µK into theFORT. The density in the FORT was about 1012 cm−3. The FORT laser, with a waistof 11(1) µm and a power of 1.6 W, is tuned to 12288 cm−1, which is between twowell-resolved photoassociation resonances. To avoid power broadening and a shiftof the photoassociation resonances by the FORT laser, the FORT and PA beams werechopped out of phase with respect to each other at 200 kHz. By chopping the MOTbeams in phase with the FORT beam the atoms were kept in the f=2 ground state.The time-averaged well depth of the FORT is U0 = 5(1) mK. After a photoassoci-ation time of 500 ms, the remaining atoms were probed using laser-induced fluo-rescence. This measurement cycle was repeated for a succession of PA frequencies,during which the intensity of the PA laser is changed in several discrete steps from2 kW/cm2 for low PA frequencies to 1 W/cm2 for high frequencies close to the dis-sociation limit. We measured from the D1 dissociation limit (12579 cm−1) down to12500 cm−1, to obtain an overlap with earlier measured spectra by Miller et al. [26]and Cline et al. [27].

Photoassociation spectroscopy of 85Rb2 0+u states 103

12.51 12.53 12.55 12.570

1

2

3

4

5

6

(103 cm-1)

D1

Bv

(10-3

cm

-1)

vE

Figure 5.13: Experimental values of the rotational constants Bv close to the D1 dis-sociation limit. The open circles on the horizontal axis indicate vibrational states forwhich it was not possible to determine the rotational constant. Due to overlap oflines some vibrational states are missing.

4.3 Experimental results

Figure 5.12 shows a small part of the spectra we obtained. In this figure we clearlysee the separate peaks from the 1g, 0−g and 0+u rovibrational series. We separatedthe 0+u rovibrational series from the 1g and 0−g series by comparing the spectra with1g and 0−g spectra that were measured and assigned earlier [26, 27]. The rotationalstructure of these 0+u rovibrational states (see inset Fig. 5.12) has been analyzed ingreat detail. We found more than a hundred 0+u rovibrational states in the regionfrom 12579 cm−1 to 12500 cm−1. From 63 of these states the rotational structurecould be resolved. From the rotationally resolved 0+u spectra, we determined therotational constant Bv of the rovibrational states by analyzing the position of therotational lines with the relation Ev(J) = BvJ(J + 1).

We also looked for 0+u states in spectra measured and assigned earlier by Milleret al. [26] and Cline et al. [27]. Miller et al. obtained a well-resolved 85Rb2 photoasso-ciation spectrum from 11600 cm−1 to 12528 cm−1, so their data overlap with ours.Unfortunately, the resolution of their spectrum is not always high enough to observerotational structure. Moreover, they were more interested in the rotational structureof the 0−g and 1g rovibrational states so they only did high-resolution measurementsaround areas where they expected those states. Nevertheless, it was still possible toassign 16 0+u vibrational states and determine their rotational constant in the regionfrom 11900 cm−1 to 12500 cm−1.

In Fig. 5.13 we display the measured rotational constants for the highest rovibra-tional states as a function of the binding energy. The uncertainty in the measuredvibrational energy and estimated rotational constants is smaller than the size of the

104 Chapter 5

open circles, since the experimental resolution of the binding energy is better than0.002 cm−1. The uncertainty in the rotational constants follows from the accuracyof the fits, and is less than 10%. The open circles on the energy axis in the figure in-dicate the vibrational states for which it was not possible to determine the rotationalconstant. For some vibrational states it was not possible to determine the rotationalconstant, either because the spectrum was not clear enough or because the 0+u stateoverlaped with a 0−g or 1g vibrational state. For the 0+u vibrational states close to thedissociation limit, the rotational constant could not be determined because it wastoo small.

5 Model calculations

5.1 Coupled bound state calculations

Figure 5.14 shows the measured rotational constants for the highest rovibrationalstates as a function of the binding energy, together with the calculated values. We seethat both the calculated and measured rotational constants of the 0+u rovibrationalstates oscillate as a function of their binding energy. We also see that the maxima ofthe calculated oscillations are shifted in energy with respect to measurements. Also,the period of the oscillations in the theoretical case is larger than in the experimentalcase. From Fig. 5.7 and the description of section 3.2, it is clear that the period ofthe oscillations is determined by the spacing of the vibrational levels of the 0+u(D2)potential. Because of the shift in energy and the larger spacing in the theoreticalcase, we can conclude that the 3Πu potential used is not fully correct.

Because the spacing between the vibrational levels of the 0+u(D2) potential in thetheoretical case is too large, we can conclude that the 0+u(D2) potential is too steepin the internuclear range corresponding to the energy range we are looking at. Therovibrational levels of the 0+u(D2) potential with a binding energy between 12500cm−1 and the D1 dissociation limit have outer turning points around 20 a0. This isroughly the range where the ab initio 3Πu potential is connected to the long-rangepart by a spline-function. The vibrational level spacing close to dissociation is deter-mined mostly by the long-range part of the potential, i.e., the dispersion coefficientsC3, C6 and C8. The C3 coefficient of the 0+u(D2) potential can not be changed indepen-dently from the C3 coefficient of the 0+u(D1) potential, since they both follow fromthe dipole-dipole interaction theory and differ by a factor 2 (Table I). The higherorder dispersion coefficient, C6 and C8, however, are different for both potentials,but for large internuclear separation the influence of these higher order terms issmall. Therefore probably the semi-empirical connection between the ab initio 3Πupotential and the long-range part is not fully correct.

The width and the height of the peaks is determined by the strength of the spin-orbit coupling at the position of the crossing of the two potentials. This can beseen when we vary the coupling between the 1Σ+u and 3Πu potentials by changingthe spin-orbit interaction VSO at the position Rx of the crossing. We introduce amodified coupling at the crossing Rx = 9.5 a0 of the potentials by subtracting a

Photoassociation spectroscopy of 85Rb2 0+u states 105

12.51 12.53 12.55 12.570

1

2

3

4

5

6

(103 cm-1)

Bv

(10-3

cm

-1)

12.51 12.53 12.55 12.570

1

2

3

4

5

6

(103 cm-1)B

v (1

0-3 c

m-1

)

D1D1

ba

vE vE

Figure 5.14: Rotational constant for the highest 0+u vibrational states. Dots/full lines:coupled channel bound state calculations; open circles/dashed lines: measurements.a: Calculated with R-independent spin-orbit coupling (Eq. (5.25)); b: Calculated withmodified spin-orbit coupling (Eq. (5.27)).

Gaussian function with a width σ = 1 a0 and a height such that the coupling at thecrossing point Rx is 60% of its asymptotic value:

VSO(R) = VSO(∞)[1− 0.40exp(−(R − Rx)2

2σ 2

)]. (5.27)

In this way we obtained a smaller value of the coupling at the crossing point of thetwo potentials, in accordance with the calculations done by Dulieu and Julienne [10]for the Cs2 dimer. Because the Gaussian function is broader than the width of theavoided crossing, we are sure that the coupling is decreased by the same factor overthe full range of interest. Figure 5.14b shows the results of calculations done withthe smaller spin-orbit coupling defined by Eq. (5.27).

The influence of this smaller coupling is clear: the peaks in Fig. 5.14b are higherand narrower than those in Fig. 5.14a, and resemble the experimental data moreclosely. Due to the smaller spin-orbit coupling the modulation depth of the rota-tional constant as a function of the binding energy becomes larger. In other words,the rotational constant of a vibrational state of one potential is less disturbed by theother potential when the coupling strength is decreased. As expected, the position ofthe peaks and the spacing between the maxima of the oscillations is not influencedby the modified coupling, since they are determined solely by the spacing betweenthe vibrational levels of the 0+u(D2) potential (Fig. 5.7).

As mentioned in section 4.3 we also looked for 0+u vibrational states in spectrameasured and assigned earlier by Miller et al. [26] and Cline et al. [27]. We foundin the energy range from 11900 cm−1 to 12500 cm−1 16 0+u vibrational states andwe determined their rotational constant. Figure 5.15 shows the measured and cal-

106 Chapter 5

11.9 12.1 12.3 12.50

2

4

6

8

10

12

14

Bv

(10-3

cm

-1)

b3Πu

A1Σ+u

(103 cm-1)

0+u(D2)

0+u(D1)

D1vE

Figure 5.15: Rotational constant for the vibrational 0+u states with binding energy Evbetween 11900 cm−1 and the D1 dissociation limit. Dots/full lines: coupled channelbound state calculations, done with the modified spin-orbit coupling. Broken lines:vibrational levels of the A 1Σ+u and b 3Πu diabatic potentials, disregarding the spin-orbit interaction; full lines: vibrational levels of the 0+u adiabatic potentials. Opencircles: measurements.

culated rotational constant as a function of the binding energy in the region from11900 cm−1 to the D1 dissociation limit. We see that, over the complete energyrange, the measured rotational constants fall in the region enclosed by the coupledchannel calculations and that they neither follow the diabatic values nor follow theadiabatic values, but oscillate. The coupled channel calculations were done with themodified spin-orbit coupling.

In Fig. 5.15 we also see that the values of the rotational constant coming fromthe coupled bound state calculations for some vibrational levels with binding energybelow 12500 cm−1 fall below the values of the rotational constant for the diabatic1Σ+u potential and the adiabtic 0+u(D1) potential. This means that the wave functionsof those vibrational states in the coupled case are disturbed in such a way that moreof probability lies at larger internuclear separation than in the purely diabatic oradiabatic case.

Another thing which can be seen in Fig. 5.15 is that the lines indicating the ro-tational constants for the diabatic 3Πu potential and the adiabtic 0+u(D2) potentialshows a strange hump around 12300 cm−1. This could also be an indication that thesemi-empirical connection between the ab initio 3Πu potential and the long-rangepart is not fully correct, as mentioned earlier.

We tried to assign the measured rovibrational levels by comparing the measuredbinding energies with the calculated ones in two different ways, first we labeled themeasured vibrational levels such that the energy difference between the experimen-tal and calculated binding energy is minimum, i.e., we labeled a measured vibrationallevel with the vibrational wave number v of the calculated vibrational level which

Photoassociation spectroscopy of 85Rb2 0+u states 107

400 420 440 460 480 500 520-2

-1

0

1

v

exp -

calc

(cm

-1)

D112.51 12.53 12.55 12.57

(103 cm-1)expvE

vE

vE

Figure 5.16: Energy difference between the experimental vibrational energy Eexpv

and the calculated vibrational energy Ecalcv as a function of the vibrational number v

(lower horizontal axis) and the experimental vibrational energy Eexpv (top horizontal

axis). The line with the dots indicates a vibrational number assignment where aminimum energy difference is chosen, the line with the open circles indicates a morephysical vibrational number assignment (see text).

is closest to the measured one. The result of this labeling routine can be seen inFig. 5.16, which shows the difference between the measured vibrational energy Eexp

v

and the calculated vibrational energy Ecalcv as a function of the assigned vibrational

number v and the experimental binding energy. From this we found that the highestmeasured vibrational state has vibrational number v = 515. From the coupled chan-nel calculations we found that the vibrational level closest to the D1 dissociationlimit has vibrational number v = 541. We see in the figure that from v = 515 downto v = 470 the energy difference shows a smooth behavior, but starts to oscillatefrom v = 470. This oscillation comes from the fact that for some vibrational levelsthe energy difference from an experimental level with energy Eexp

v to a calculatedlevel with energy Ecalc

v−1 is smaller than the energy difference to a calculated levelwith energy Ecalc

v . In this way the labeling routine skips a vibrational level. A morephysical way to assign the levels is also shown in Fig. 5.16: here we demand that theenergy difference increases with decreasing energy. This comes from the uncertaintyin the potentials used: from the dissociation limit to the bottom of the potential theuncertainty accumulates. The jumps in the energy difference are probably an effectof the coupling between the potentials or come from the fact that it is not alwaysclear which experimental level belongs to which calculated level. We conclude thatthe coupling between the potentials, the uncertainties in the potentials, and the factthat not all vibrational levels were measured, makes it impossible to assign all the

108 Chapter 5

vibrational levels.

5.2 Landau-Zener description

We can also describe the coupling between the potentials with the simple Landau-Zener formula [28]. In the adiabatic representation there is a finite chance to crossfrom one adiabatic curve to the other, due to the finite value of the radial velocity ofthe atoms. The probability of a single diabatic crossing is given by

Px = exp (−vref/vx) , (5.28)

with vx the velocity of the atoms at the crossing radius Rx. The characteristic veloc-ity vref is given by

vref = V2AB

|d∆VdR |, (5.29)

with d∆V/dR the derivative of the difference potential ∆V = VA − VB at Rx, withVA and VB the diabatic potentials. From Eq. (5.28) we see that the probability tojump from one adiabatic curve to the other is small for a large value of vref andis large for a small value of vref. Decreasing the coupling potential VAB implies adecrease of the crossing parameter vref, i.e., an increase of the probability to jumpfrom one adiabatic curve to the other. This is also what we see in Fig. 5.14: witha large spin-orbit coupling the oscillations in the rotational constant are large, thepeaks are broad and the value of the rotational constant goes to the adiabatic case.In conclusion, a large spin-orbit coupling, i.e., large coupling potential VAB , resultsin a small probability to jump from one adiabatic potential curve to the other, theso-called adiabatic case. Conversely, a small spin-orbit coupling results in a largeprobability to jump from one adiabatic potential curve to the other, the so-calleddiabatic case.

5.3 Alkali dimer systems

With the Landau-Zener transition picture we can also easily explain differences be-tween Na2, Rb2 and Cs2. Dulieu and Julienne [10] found that, due to the small spin-orbit coupling for the Na2 dimer, the A 1Σ+u and b 3Πu states act purely diabatical;for the Cs2 dimer, due to a large spin-orbit coupling, the behavior is purely adia-batic. We found that in the case of the Rb2 dimer the A 1Σ+u and b 3Πu act neitherdiabatic nor adiabatic. We can determine the transition probability Px for the threealkali dimers by calculating the velocity vx and the characteristic velocity vref at thecrossing Rx of the two potentials.

In Table II the potential parameters and the transition probabilties for the threealkalis are given. We see that this estimate corresponds with the results of thebound state calculations described in this chapter and those done by Dulieu andJulienne [10]: a large transition probability for Na2, a small one for Cs2 and a value

Photoassociation spectroscopy of 85Rb2 0+u states 109

Table II: Potential parameters and Landau-Zener probabilities for Na2, Rb2 and Cs2for an unmodified coupling VAB = VSO(∞), for Rb2 also for a modified couplingVAB = 0.6 · VSO(∞).

Coupling parameter Na2 Rb2 Cs2M 23 85 133|V(Rx)| ( cm−1) 8255 5808 5431∆FS(cm−1) 17 238 351

d∆VdR (cm−1a−1

0 ) 850(250)1 620(160)1 590(70)1

VAB = VSO(∞) vref (m/s) 0.6(+0.3−0.2

)200

(+70−40

)450

(+60−50

)vx (m/s) 4100 1800 1400

Px 1 0.89(+0.02−0.03

)0.72

(+0.02−0.03

)VAB = 0.6 · VSO(∞) Px - 0.96

(+0.01−0.01

)-

1The derivative of ∆V is determined by hand from a potential graph, this causesthe given uncertainties in d∆V

dR and therefore the uncertainties in vref and Px.

in between for Rb2. In Table II also the transition probability for the modified spin-orbit interaction in the Rb2 case is given: a 40% smaller spin-orbit coupling resultsin an 8% higher probability to jump to the other adiabatic potential curve.

Cline et al. [9] estimated from broadened 0+u vibrational levels, belonging to the0+u(D2) potential, the probability of the Landau-Zener transition, and found P =2Px(1− Px) = 0.14± 0.02, with Px the probability of a single diabatic crossing givenby Eq. (5.28). Using this expression we find from Table II, P = 0.19+0.05−0.03 using theunmodified coupling, and P = 0.08+0.02−0.02 using the modified coupling.

6 Concluding remarks

In summary, we have shown experimentally and theoretically that in the case of theRb2 dimer the A 1Σ+u and b 3Πu states neither act purely diabatically nor act purelyadiabatically. To obtain a better quantitative agreement, the shape of the A 1Σ+u andb 3Πu potentials has to be modified. From a qualitative comparison between theexperimental and calculated results, we conclude that the spin-orbit coupling at thecoupling radius is smaller than its asymptotic value, a trend that is in agreementwith the results for Cs2 found by Dulieu et al. [10].

Improved potentials are being developed by Stevens et al.: they are calculatingnew Rb2 A 1Σ+u and b 3Πu with the same techniques they used to calculate the3Σ+u and 3Σ+g potentials for Cs2 and other alkali dimers [29]. In these calculations,the spin-orbit interaction between the potentials is also rigorously taken into ac-count [30]. In this way hopefully a more quantitative picture of the coupling betweenthe potentials can be formed.

A limitation of the model we used is that we only took the coupling between the

110 Chapter 5

A 1Σ+u and b 3Πu states into account and not the coupling to other states. Muchlarger eigenvalue problems would have to be solved in that case [14].

References

[1] C.C. Tsai, R.S. Freeland, J.M. Vogels, H.M.J.M. Boesten, B.J. Verhaar, and D.J.Heinzen, Phys. Rev. Lett. 79, 1245 (1997).

[2] J.R. Gardner, R.A. Cline, J.D. Miller, D.J. Heinzen, H.M.J.M. Boesten, and B.J. Ver-haar, Phys. Rev. Lett. 74, 3764 (1995).

[3] J.M. Vogels, C.C. Tsai, S.J.J.M.F. Kokkelmans, B.J. Verhaar, and D.J. Heinzen,Phys. Rev. A 56, R1067 (1997).

[4] F.A. Van Abeelen, D.J. Heinzen, and B.J. Verhaar, Phys. Rev. A 57, R4102 (1998).

[5] Ph. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, and B.J. Verhaar,Phys. Rev. Lett. 81, 69 (1998).

[6] F.A. Van Abeelen and B.J. Verhaar, Phys. Rev. A. 59, 578 (1999).

[7] C.J. Williams, E. Tiesinga, and P.S. Julienne, Phys. Rev. A. 53, R1939 (1996).

[8] X. Wang, H. Wang, P.L. Gould, W.C Stwalley, E. Tiesinga, and P.S. Julienne, Phys.Rev. A. 57, 4600 (1998).

[9] R.A. Cline, J.D. Miller, and D.J. Heinzen, Phys. Rev. Lett. 73, 632 (1994).

[10] O. Dulieu and P.S. Julienne, J. Chem. Phys. 103, 60 (1995).

[11] M. Monnerville and J.M. Robbe, J. Chem. Phys. 101, 7580 (1994).

[12] D.T. Colbert and W.H. Miller, J. Chem. Phys, 96, 1982 (1992).

[13] The author likes to thank F.A. Van Abeelen for his help with the derivation ofEq. (5.8).

[14] E. Tiesinga, C.J. Williams, and P.S. Julienne, Phys. Rev. A 57, 4257 (1998).

[15] V. Kokoouline, O. Dulieu, R. Kosloff, and F. Masnou-Seeuws, J. Chem. Phys.,110(20), 9865 (1999).

[16] A.J. Moerdijk, B.J. Verhaar, and A. Axelsson, Phys. Rev. A, 51, 4852 (1995).

[17] G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of DiatomicMolecules, 2nd ed. (Van Nostrand Reinhold Co., New York 1950).

[18] M. Movre and G. Pichler, J. Phys. B: Atom. Molec. Phys. 10 (13) (1977).

[19] M. Marinescu, A. Dalgarno, Z. Phys. D., 36, 239 (1996).

Photoassociation spectroscopy of 85Rb2 0+u states 111

[20] G.P. Barwood, P. Gill, and W.R.C. Rowley, Appl. Phys. B 53, 142 (1991).

[21] M. Foucrault, Ph. Millie and J.P. Daudey, J. Chem. Phys. 96 (1992).

[22] M. Marinescu, A. Dalgarno, Phys. Rev. A, 52, 311 (1995).

[23] C.J. Williams and D.J. Heinzen (private communication).

[24] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).

[25] J.P. Gordon and A. Ashkin, Phys. Rev. A, 21, 1606 (1980).

[26] J.D. Miller, R.A. Cline, and D.J. Heinzen, Phys. Rev. Lett. 71, 2204 (1993).

[27] R.A. Cline, J.D. Miller, and D.J. Heinzen, (unpublished).

[28] E.E. Nikitin and S.Ya. Umanskii, Theory of Slow Atomic Collisions (Springer-Verlag, Berlin, 1984).

[29] M. Kraus and W.J. Stevens, J. Chem. Phys., 93, 4236 (1990).

[30] W.J. Stevens (private communication).

112 Chapter 5

Summary 113

Summary

Since the early eighties a variety of laser cooling techniques have been applied tocool and trap neutral atoms. The increasing knowledge of trap loss mechanismshas led to the confinement of larger and colder samples of atoms. The trapping ofatoms in purely magnetic traps and the technique of forced evaporative cooling wereresponsible for the attainment of Bose-Einstein condensation (BEC) in cold, dilutegases. In this quantum regime the atomic gases have typically a number density ofmore than 1014 atoms/cm3 and a temperature of the order of 1 µK. Till now thisregime has only been reached with alkali-metal atoms and atomic hydrogen. In thisthesis we describe the first experimental steps towards a BEC of metastable neonatoms. This research was carried out in the Atomic Physics and Quantum Electronicsgroup at the Physics Department of the Eindhoven University of Technology.

The usual scheme to create a BEC starts with the trapping of a large sample ofcold atoms in a magneto-optical trap (MOT). This first step is already complicatedfor metastable rare gas atoms, since the efficiency of producing rare gas atoms inthe metastable state is very low, i.e., ∼ 10−4. However, by applying several lasercooling techniques we succeed in creating a cold and intense beam of metastableNe(3s) 3P2 atoms. The atoms leaving the source are collimated, slowed, and finallycompressed to a beam with a diameter of 1 mm and a flux of 5× 1010 atoms/s. Thetransverse and longitudinal temperature of the bright beam are T⊥ = 285 µK andT|| = 28 mK, respectively, while the maximum brilliance of the beam equals 4× 1022

s−1m−2sr−1. Apart from a bright beam of the 20Ne isotope, it is easy to create abeam of the 22Ne isotope, with the only difference that the flux of the latter is seventimes smaller. Our cold intense atomic beam can be used for a whole range of coldcollision experiments, e.g., photoassociation spectroscopy. In this thesis it is shownthat the bright neon beam is also an excellent source for loading a magneto-opticaltrap.

The beam flux of the atomic beam provides a loading rate of up to 1.5 × 1010

atoms/s of the MOT developed in our lab. Because of this high loading rate we wereable to trap almost 1010 metastable neon atoms in the MOT; to our knowledge thisis the largest number of trapped metastable atoms reported. This optimum numberis obtained under rather unconventional conditions. First of all, we do not have anordinary MOT consisting of six trapping beams but a seven-beam MOT. The seventhlaser beam is introduced by the extra slower necessary for efficient capturing theatoms from the bright beam. Furthermore, the optimum number of trapped atomsis obtained using a very low intensity of the trapping beams. Under these conditionsthe trap volume becomes extraordinarily large, thereby reducing the number densityand consequently reducing the trap losses caused by ionization.

We developed a simple Doppler model which describes the trapping conditions ofour MOT very well. Even the influence of the seventh MOT beam satisfies the Dopplermodel. Our MOT is operated in the Doppler limited regime, with corresponding tem-peratures around 1 mK. For the highest measured density, i.e., 4× 1010 atoms/cm3,

114 Summary

this corresponds to a phase space density of 10−7. The lifetime of our MOT (∼ 0.2 s)is dominated by the two-body loss rate mostly caused by ionization, which we foundto be β = 2KSS = (5 ± 1) × 10−10 cm3/s. In the regime we operate our MOT in, theionization losses are fully determined by ground state collisions.

A MOT containing 1010 atoms is a good starting point on the road to BEC. Apartfrom a MOT of 20Ne we can also make a MOT containing more than 109 atoms ofthe 22Ne isotope. This opens perspectives when the 20Ne fails to meet the require-ments for easily reaching BEC, e.g. a large and positive scattering length. Besidesthe Penning ionization loss rate, almost no experimental collision data is availablefor metastable neon. That is why in our group photoassociation spectroscopy exper-iments are planned to obtain those collision parameters.

In this thesis we describe how we used this technique to obtain the coupling be-tween two excited states of the 85Rb2 dimer. That work, described in chapter 5 of thisthesis, was a collaboration of the Ultracold Atomic Physics Group at the Universityof Texas at Austin, the Atomic Physics Division of the National Institute of Standardsand Technology at Gaithersburg, and the Atomic Physics and Quantum Electronicsgroup at the Physics Department of the Eindhoven University of Technology. By us-ing photoassociation spectroscopy of laser-cooled 85Rb atoms, we studied the 85Rb20+u states that lie within 80 cm−1 of the 52S1/2 + 52P1/2 dissociation limit, with a res-olution better than 0.002 cm−1. These levels arise from both the A 1Σ+u and b 3Πuelectronic states. We found that the rotational constants of the 0+u levels oscillate asa function of their vibrational number. We compared our measurements with close-coupled bound state calculations, from which we gained a qualitatively understand-ing of the mechanism causing the oscillations. We conclude that nuclear motion onthe A 1Σ+u and b 3Πu potentials is strongly coupled so that the Born-Oppenheimerapproximation does not provide a valid zero-order description of these states. Thecoupling between the A 1Σ+u and b 3Πu potentials in the case of Rb2 is neither adi-abatic, as in the case of Cs2, nor diabatic, as in the case of Na2. To obtain morequantitative results, more has to be known about the 1Σ+u and 3Πu potentials and theradial dependence of the spin-orbit coupling between them.

We showed in this thesis that in our group a setup was developed to study coldcollisions of metastable neon atoms. In the future, both the bright atomic beam andthe magneto-optical trap can be used for a variety of collision experiments. Alreadysome experience with photoassociation spectroscopy was obtained by studying the85Rb2 dimer. In the near future the road to Bose-Einstein condensation of metastableneon will be continued by optimizing the loading of a magneto-static trap.

Samenvatting 115

Samenvatting

Sinds het begin van de jaren tachtig zijn tal van laserkoelingstechnieken toegepastom neutrale atomen te koelen en op te sluiten. Een beter begrip van de verliesme-chanismen die in atoomvallen optreden, heeft geleid tot het opsluiten van grotere enkoudere samples van atomen. Het opsluiten van atomen in puur magnetische vallenen de techniek van gedwongen afdampkoelen waren verantwoordelijk voor het berei-ken van Bose-Einstein condensatie (BEC) in koude, ijle gassen. In dit quantum regimehebben de atomaire gassen typisch een dichtheid van meer dan 1014 atomen/cm3 eneen temperatuur in de orde van 1 µK. Tot nu toe is dit regime alleen bereikt metalkali-metaal atomen en atomair waterstof. In dit proefschrift beschrijven we de eer-ste experimentele stappen in de richting van BEC van metastabiele neon atomen. Ditonderzoek is uitgevoerd in de capaciteitsgroep Atoomfysica en Quantumelektronicavan de faculteit Technische Natuurkunde aan de Technische Universiteit Eindhoven.

Het gebruikelijke schema om een BEC te creeren, begint met het opsluiten vaneen grote hoeveelheid koude atomen in een magneto-optische val (MOT). Deze eerstestap is reeds gecompliceerd voor metastabiele edelgas atomen omdat de efficientiewaarmee edelgas atomen in de metastabiele toestand worden geproduceerd erg laagis, ongeveer van de orde 10−4. Door het toepassen van verschillende laserkoelings-technieken zijn we er echter in geslaagd een koude, intense bundel van metastabieleNe(3s) 3P2 atomen te creeren. De atomen die de bron verlaten, worden gecollimeerd,afgeremd en uiteindelijk gecomprimeerd tot een bundel met een diameter van 1 mmen een flux van 5 × 1010 atomen/s. De transversale en longitudinale temperatuurvan de bundel zijn respectievelijk T⊥ = 285 µK en T|| = 28 mK, terwijl de maximalespectrale helderheid van de bundel 4× 1022 s−1m−2sr−1 bedraagt. Naast een helderebundel van de 20Ne isotoop is het eenvoudig een bundel van de 22Ne isotoop te ma-ken, met als enige verschil dat de flux van deze laatste zeven keer kleiner is. Onzekoude, intense atoombundel kan voor een skala van koude-botsings experimentengebruikt worden zoals bijvoorbeeld voor fotoassociatie spectroscopie. In dit proef-schrift laten we zien dat de heldere neon bundel ook gebruikt kan worden als bronvoor het laden van een magneto-optische val.

De bundelflux van de atoombundel maakt het mogelijk laadsnelheden van de inons lab ontwikkelde MOT, van 1.5 × 1010 atomen/s te bereiken. Dankzij deze hogelaadsnelheid waren we in staat bijna 1010 metastabiele neon atomen op te sluiten inde MOT, wat naar ons inziens het grootste aantal opgesloten metastabiele atomenis wat is gerapporteerd. Dit optimale aantal wordt bereikt onder enigszins oncon-ventionele omstandigheden. Ten eerste beschikken wij niet over een gebruikelijkeMOT bestaande uit zes opsluitlaserbundels, maar over een zeven bundel MOT. De ze-vende bundel is afkomstig van de extra slower, nodig voor het efficient invangen vanatomen uit de heldere bundel. Verder wordt het optimum aantal gevangen atomenbereikt door gebruik te maken van opsluitbundels met een erg lage intensiteit. On-der deze omstandigheden wordt het volume van de atoomwolk uitzonderlijk groot

116 Samenvatting

waardoor de atomaire dichtheid wordt gereduceerd en derhalve ook de verliezenveroorzaakt door ionisatie.

We hebben een simpel Doppler model ontwikkeld dat de condities waaronderde atomen opgesloten zitten in onze MOT goed beschrijft. Zelfs de invloed vande zevende MOT bundel voldoet aan het Doppler model. Onze MOT is werkzaamin het Doppler-gelimiteerde regime, welke correspondeert met temperaturen rond1 mK. Voor de hoogst gemeten atomaire dichtheid, 4 × 1010 atomen/cm3, komtdit overeen met een faseruimte dichtheid van 10−7. De levensduur van onze MOT(∼ 0.2 s) wordt gedomineerd door twee-deeltjes verliezen welke voor het groot-ste deel bepaald worden door ionisatie; de gevonden verliezen waren gelijk aanβ = 2KSS = (5± 1)× 10−10 cm3/s. In het regime waarin onze MOT werkt worden deionisatieverliezen volledig bepaald door grondtoestands botsingen.

Een MOT die 1010 atomen bevat is een goed startpunt op de weg naar BEC. Naasteen MOT van 20Ne kunnen we ook een MOT maken die meer dan 109 atomen van het22Ne isotoop bevat. Dit biedt perspectieven wanneer 20Ne niet voldoet aan de eisenom gemakkelijk BEC te bereiken, zoals bijvoorbeeld het hebben van een grote enpositieve verstrooiingslengte. Naast de Penningionisatie verlies-snelheid is er bijnageen experimentele botsingsdata beschikbaar over metastabiel neon. Dat is de re-den waarom er in onze groep fotoassociatie experimenten zijn gepland om dezebotsingsparameters te bepalen.

In dit proefschrift beschrijven we hoe we deze techniek gebruikt hebben om kop-pelingsparameters tussen twee toestanden van de 85Rb2 dimeer te bepalen. Dit werk,beschreven in hoofdstuk 5 van dit proefschrift was een samenwerking tussen de Ul-tracold Atomic Physics Group van de University of Texas in Austin, de Atomic Phy-sics Division van het National Institute of Standards and Technology in Gaithersburgen de Atoomfysica en Quantumelektronica groep van de faculteit Technische Natuur-kunde aan de Technische Universiteit Eindhoven. We hebben de 85Rb2 0+u toestandenbestudeerd die binnen 80 cm−1 van de 52S1/2 + 52P1/2 dissociatie limiet liggen. Dezetoestanden werden bepaald met een resolutie beter dan 0.002 cm−1 door gebruik temaken van fotoassociatie spectroscopie van laser-gekoelde 85Rb atomen. Deze ener-gieniveau’s behoren tot de elektronische A 1Σ+u en b 3Πu toestanden. We vonden datde rotatiekonstanten van de 0+u toestanden oscilleren als een functie van het nummervan hun vibratieniveau. We hebben de metingen vergeleken met een gekoppelde-gebondentoestanden berekening, waaruit we een kwalitatief beeld hebben gevormdvan wat de oscillaties veroorzaakt. We concluderen dat de kernbeweging van deA 1Σ+u en b 3Πu potentialen sterk gekoppeld is zodat de Born-Oppenheimer bena-dering geen goede nulde orde beschrijving oplevert. De koppeling tussen de A 1Σ+uen b 3Πu potentialen is voor Rb2 noch adiabatisch zoals bij Cs2 het geval is, nochdiabatisch zoals bij Na2 het geval is. Meer kennis van de 1Σ+u en 3Πu potentialen isnodig om meer kwantitatieve resultaten te verkrijgen.

In dit proefschrift hebben we laten zien dat in onze groep een opstelling is ont-wikkeld om koude botsingen tussen metastabiele neon atomen mee te bestuderen.In de toekomst kan zowel de heldere atoombundel als de magneto-optische val ge-bruikt worden voor een verscheidenheid aan botsingsexperimenten. Met het bestu-deren van de 85Rb2 dimeer is al enige ervaring opgedaan met fotoassociatie spectro-

Samenvatting 117

scopie. In de nabije toekomst wordt de weg naar Bose-Einstein condensatie verdervervolgd door het optimaliseren van het laden van een magneto-statische val.

118 Dankwoord

Dankwoord

Een promotieonderzoek doe je zeker niet in je eentje. Dit proefschrift was dan ookniet tot stand gekomen zonder de hulp van vele anderen. Daarom wil ik hierbijiedereen bedanken die bijgedragen heeft aan de totstandkoming van dit proefschriften voor de plezierige tijd die ik de afgelopen vier en een half jaar heb gehad.De meeste dank ben ik verschuldigd aan mijn copromoter Edgar Vredenbregt. Edgarheeft mij alle facetten van het laserkoelvak bijgebracht. Dat je drukregelaars subtielmoet behandelen, werd me de eerste dag in het lab al door hem duidelijk gemaakt.Geen enkel probleem kon ik tegen het lijf lopen of Edgar had er een oplossing voor.Zonder zijn hulp was het GEMINI-project zeker niet zo ver gekomen. Edgar, bedanktvoor alles.Mijn promotor Herman Beijerinck dank ik, naast alle hulp op het gebied van de fy-sica, ook voor zijn persoonlijke adviezen en begeleiding. Zijn onuitputtelijke stroomnieuwe ideeen werkte zeer stimulerend.Mijn tweede promotor Boudewijn Verhaar, die mij tijdens mijn afstuderen al enthou-siast maakte voor de atoomfysica, dank ik voor alle heldere discussies op het gebiedvan de botsingsfysica.Veel dank ben ik ook verschuldigd aan de technische staf van de AQT-groep. Ikben de tel kwijt geraakt van het aantal modificaties aan de GEMINI-opstelling ende vele malen dat we samen met Rien de Koning de verschillende vacuum kamersopengehaald hebben (en dankzij Rien ook weer netjes dichtgekregen hebben). Naastalle ontwerpen, tekeningen en constructies ben ik Rien ook dankbaar voor het steedsweer oppeppen van de laser. Voor Louis van Moll kan een klusje (en bijbehorendepromovendus) niet gek genoeg zijn of hij bedenkt er de juiste oplossing voor. Altijdstond hij voor het GEMINI-team klaar. Hiervoor ben ik hem zeer dankbaar. Jolandavan de Ven heeft de taken van Rien vrij snel overgenomen. Haar dank ik o.a. voor demooie 3D-tekeningen.Het enthousiasme en doorzettingsvermogen van Simon Kuppens hebben er voor ge-zorgd dat we in korte tijd een neon MOT tot stand gebracht hebben. De discussiesmet Simon over de MOT en de rest van mijn proefschrift waren bijzonder nuttig.Veel dank hiervoor.In het lab hebben we ook veel kunnen lachen dankzij de afstudeerders John Oerle-mans, Roland Stas, Veronique Mogendorff, Bert Claessens en Eric van Kempen. Naastde leuke tijd dank ik hen ook voor het vele werk dat zij verricht hebben. Hetzelfdegeldt voor de stagiairs Wilbert Mestrom en wederom Bert Claessens aka d’n Bertral-lius.Ton van Leeuwen dank ik voor al zijn waardevolle opmerkingen tijdens werkbespre-kingen en zijn hulp bij computerproblemen. Tevens dank ik Ton voor de leuke tijdtijdens de colleges en instructies N3 voor T.Onze secretaresses Rina Boom en Marianne van den Elshout dank ik voor de hulp bijalle administratieve klusjes.

Dankwoord 119

De leden van het AQT “AIO-overleg” orgaan, te weten, De Dikke, De Kale, De Lulloen Esther (Roel Knops, Armand Koolen, Roel Bosch en Patrick Sebel) bedank ik vooralle (vaak iets te) gezellige bijeenkomsten en de leuke tijd binnen de groep.Herman Batelaan dank ik voor de vele waardevolle discussies op de gang en in hetlab. De gesprekken met Jo Hermans leverden altijd weer een nieuwe, heldere kijk opvele fysicaproblemen. Peter van der Straten bedank ik voor de gezellige donderdagenen de discussies over de inhoud van dit proefschrift.De theoreten Frank van Abeelen, Servaas Kokkelmans en Johnny Vogels ben ik dank-baar voor de dicussies en leuke tijd op conferenties.I would like to thank Misha Kurzanov for all the work he did on the MOT and forbeing such a nice colleague. I thank Marek Synowiec for all the laughing in the labduring his visit at our group.I am very thankful to Dan Heinzen, Riley Freeland, Philippe Courteille, Jean-BriceCombebias and Kanny Ly for the great time during my visit at the University ofTexas. I also like to thank Paul Julienne, Carl Williams, and Eite Tiesinga for thehelpful discussions about the Rb2 problem. I like to thank Marya Doery for the nicetime during my visit at NIST Gaithersburg.

Naast bovengenoemde mensen ben ik ook veel dank verschuldigd aan vrienden, ke-nissen en familie.Louis Selen dank ik voor de gezellige tijd in huis en voor de tien kilo die ik, mededankzij zijn bijzondere kookkwaliteiten, de afgelopen jaren ben aangekomen.Quint Videler dank ik voor de avonturen tijdens het zeilen en de andijviestamp metspekavonden.Hugo de Jong, net zo verslaafd aan het goede leven als ik, dank ik voor alle ongeintijdens het uitgaan en op party’s.De bijzondere leden van Fysisch Genootschap Nwyvre dank ik voor alle, per definitie,veel te gezellige tijden.De Vastenavendvrienden JdeJ & LdeP, Bart & Miss Thole, Henk & Miranda, JJ en Stof-fels dank ik voor alle leut tijdens het dweile.DJ Stijn en DJ Robob ben ik erg dankbaar voor het aanleveren van al het moois opvinyl.Mijn ouders en de rest van de familie dank ik voor alle steun en liefde die ze mij alheel mijn leven geven.Tenslotte dank ik mijn lieve Aaf voor alle liefde en het zo gelukkig maken van me.

120 Curriculum Vitae

Curriculum Vitae

9 december 1971 Geboren te Bergen op Zoom

1984-1990 HAVO,Roncalli Scholengemeenschap te Bergen op Zoom

1990-1991 Propadeuse Technische Natuurkunde,Hoge School Eindhoven

1991-1996 Studie Technische Natuurkunde,Technische Universiteit Eindhoven,doctoraal examen augustus 1996

1996-2001 Onderzoeker-In-Opleiding bij Stichting FOM,werkgroep AQ-E-a,Experimentele Atoomfysica en Quantumelektronica,Faculteit Technische Natuurkunde,Technische Universiteit Eindhoven

april-augustus 1998 Werkbezoek Ultracold Atomic Physics Group,University of Texas, Austin, TX, USA