OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

Rudolf Grimm and Matthias WeidemullerMax-Planck-Institut fur Kernphysik, 69029 Heidelberg, Germany

Yurii B. OvchinnikovNational Institute of Standards and Technology, PHY B167, Gaithersburg, MD 20899, USA

Contents

I Introduction 1

II Optical dipole potential 2A Oscillator model . . . . . . . . . . . . . 2

1 Interaction of induced dipole withdriving field . . . . . . . . . . . . . 2

2 Atomic polarizability . . . . . . . . 33 Dipole potential and scattering rate 3

B Multi-level atoms . . . . . . . . . . . . 41 Ground-state light shifts and optical

potentials . . . . . . . . . . . . . . . 42 Alkali atoms . . . . . . . . . . . . . 5

III Experimental issues 7A Cooling and heating . . . . . . . . . . . 7

1 Cooling methods . . . . . . . . . . . 72 Heating mechanisms . . . . . . . . . 93 Heating rate . . . . . . . . . . . . . 10

B Experimental techniques . . . . . . . . 111 Trap loading . . . . . . . . . . . . . 112 Diagnostics . . . . . . . . . . . . . . 12

C Collisions . . . . . . . . . . . . . . . . . 13

IV Red-detuned dipole traps 14A Focused-beam traps . . . . . . . . . . . 14

1 Collisional studies . . . . . . . . . . 152 Spin relaxation . . . . . . . . . . . . 163 Polarization-gradient cooling . . . . 164 Bose-Einstein condensates . . . . . 175 Quasi-electrostatic traps . . . . . . 18

B Standing-wave traps . . . . . . . . . . . 181 Optical cooling to high phase-space

densities . . . . . . . . . . . . . . . 192 Quantum interference . . . . . . . . 193 Spin manipulation . . . . . . . . . . 204 Quasi-electrostatic lattices . . . . . 20

C Crossed-beam traps . . . . . . . . . . . 211 Evaporative cooling . . . . . . . . . 212 Interference effects . . . . . . . . . . 22

D Lattices . . . . . . . . . . . . . . . . . . 22

V Blue-detuned dipole traps 24A Light-sheet traps . . . . . . . . . . . . 25B Hollow-beam traps . . . . . . . . . . . 26

1 Plugged doughnut-beam trap . . . . 272 Single-beam traps . . . . . . . . . . 273 Conical atom trap . . . . . . . . . . 27

C Evanescent-wave traps . . . . . . . . . 281 Evanescent-wave atom mirror . . . 292 Gravito-optical cavities . . . . . . . 293 Gravito-optical surface trap . . . . . 30

VI Concluding remarks 32

I. INTRODUCTION

Methods for storage and trapping of charged and neu-tral particles have very often served as the experimentalkey to great scientific advances, covering physics in thevast energy range from elementary particles to ultracoldatomic quantum matter. The ultralow-energy region be-came experimentally accessible as a result of the dramaticdevelopments in the field of laser cooling and trapping,which have taken place over the last two decades (Sten-holm, 1986; Minogin and Letokhov, 1988; Arimondo etal., 1992; Metcalf and van der Straten, 1994; Chu, 1998;Cohen-Tannoudji, 1998; Phillips, 1998).

For charged particles, the strong Coulomb interactioncan be used for trapping in electric or electromagneticfields (Bergstrom et al., 1995; Ghosh, 1995). At the verylow temperatures reached by laser cooling, single ionsshow a variety of interesting quantum effects (Winelandet al., 1995), and ensembles form a crystalline orderedstate (Walther, 1993). Laser cooling in ion traps hasopened up completely new possibilities for ultrahigh pre-cision spectroscopy and related fundamental applications(Thompson, 1993). It is a very important feature of iontraps that the confining mechanism does not rely on theinternal structure of the ion, which is therefore accessiblefor all kinds of experiments.

For neutral atoms, it has become experimental rou-tine to produce ensembles in the mikrokelvin region, andmany experiments are being performed with such laser-cooled ultracold gases. It is thus possible to trap theatoms by much weaker mechanisms as compared to theCoulomb interaction. Traps for neutral atoms can berealized on the basis of three different interactions, eachclass having specific properties and offering particular ad-vantages:

• Radiation-pressure traps operating with near-resonant light (Pritchard et al., 1986; Raab et al.,1987) have a typical depth of a few Kelvin, and be-cause of very strong dissipation they allow to cap-ture and accumulate atoms even from a thermal

1

gas. In these traps, the atomic ensemble can becooled down to temperatures in the range of a few10µK. The trap performance, however, is limited inseveral ways by the strong optical excitation: Theattainable temperature is limited by the photon re-coil, the achievable density is limited by radiationtrapping and light-assisted inelastic collisions, andthe internal dynamics is strongly perturbed by res-onant processes on a time scale in the order of amicrosecond.

• Magnetic traps (Migdall et al., 1986; Bergeman etal., 1987) are based on the state-dependent forceon the magnetic dipole moment in an inhomoge-neous field. They represent ideal conservative trapswith typical depths in the order of 100mK, and areexcellent tools for evaporative cooling and Bose-Einstein condensation. For further applications,a fundamental restriction is imposed by the factthat the trapping mechanism relies on the internalatomic state. This means that experiments con-cerning the internal dynamics are limited to a fewspecial cases. Furthermore, possible trapping ge-ometries are restricted by the necessity to use ar-rangements of coils or permanent magnets.

• Optical dipole traps rely on the electric dipole in-teraction with far-detuned light, which is muchweaker than all mechanisms discussed above. Typ-ical trap depths are in the range below one mil-likelvin. The optical excitation can be kept ex-tremely low, so that such a trap is not limited bythe light-induced mechanisms present in radiation-pressure traps. Under appropriate conditions, thetrapping mechanism is independent of the partic-ular sub-level of the electronic ground state. Theinternal ground-state dynamics can thus be fullyexploited for experiments, which is possible on atime scale of many seconds. Moreover, a great vari-ety of different trapping geometries can be realizedas, e.g., highly anisotropic or multi-well potentials.

The subject of this review are atom traps of the lastdescribed class along with their unique features as storagedevices at ultralow energies.

Historically, the optical dipole force, acting as confin-ing mechanism in a dipole trap, was first considered byAskar’yan (1962) in connection with plasmas as well asneutral atoms. The possibility of trapping atoms withthis force was considered by Letokhov (1968) who sug-gested that atoms might be one-dimensionally confinedat the nodes or antinodes of a standing wave tuned farbelow or above the atomic transition frequency. Ashkin(1970) demonstrated the trapping of micron-sized parti-cles in laser light based on the combined action of ra-diation pressure and the dipole force. Later he sug-gested three-dimensional traps for neutral atoms (1978).The dipole force on neutral atoms was demontrated byBjorkholm et al. (1978) by focusing an atomic beam by

means of a focused laser beam. As a great breakthrough,Chu et al. (1986) exploited this force to realize the firstoptical trap for neutral atoms. After this demonstra-tion, enormous progress in laser cooling and trapping wasmade in many different directions, and much colder anddenser atomic samples became available for the efficientloading of shallow dipole traps. In the early ’90s opticaldipole forces began to attract rapidly increasing interestnot only for atom trapping, but also in the emerging fieldof atom optics (Adams et al., 1994).

In this review, we focus on dipole traps realized withfar-detuned light. In these traps an ultracold ensembleof atoms is confined in a nearly conservative potentialwell with very weak influence from spontaneous photonscattering. The basic physics of the dipole interactionis discussed in Sec. II. The experimental background ofdipole trapping experiments is then explained in Sec. III.Specific trapping schemes and experiments are presentedin Secs. IV and V, where we explore the wide range ofapplications of dipole traps considering particular exam-ples.

II. OPTICAL DIPOLE POTENTIAL

Here we introduce the basic concepts of atom trappingin optical dipole potentials that result from the interac-tion with far-detuned light. In this case of particular in-terest, the optical excitation is very low and the radiationforce due to photon scattering is negligible as comparedto the dipole force. In Sec. II A, we first consider theatom as a simple classical or quantum-mechanical oscil-lator to derive the main equations for the optical dipoleinteraction. We then discuss the case of real multi-levelatoms in Sec. II B, in particular of alkali atoms as usedin the great majority of experiments. discussed

A. Oscillator model

The optical dipole force arises from the dispersive in-teraction of the induced atomic dipole moment with theintensity gradient of the light field (Askar’yan, 1962;Kazantsev, 1973; Cook, 1979; Gordon and Ashkin, 1980).Because of its conservative character, the force can be de-rived from a potential, the minima of which can be usedfor atom trapping. The absorptive part of the dipoleinteraction in far-detuned light leads to residual photonscattering of the trapping light, which sets limits to theperformance of dipole traps. In the following, we derivethe basic equations for the dipole potential and the scat-tering rate by considering the atom as a simple oscillatorsubject to the classical radiation field.

2

1. Interaction of induced dipole with driving field

When an atom is placed into laser light, the electricfield E induces an atomic dipole moment p that oscil-lates at the driving frequency ω. In the usual complexnotation E(r, t) = e E(r) exp(−iωt) + c.c. and p(r, t) =e p(r) exp(−iωt) + c.c., where e is the unit polarizationvector, the amplitude p of the dipole moment is simplyrelated to the field amplitude E by

p = α E . (1)

Here α is the complex polarizability, which depends onthe driving frequency ω.

The interaction potential of the induced dipole momentp in the driving field E is given by

Udip = −12〈pE〉 = − 1

2ε0cRe(α) I , (2)

where the angular brackets denote the time averageover the rapid oscillating terms, the field intensity isI = 2ε0c|E|2, and the factor 1

2 takes into account thatthe dipole moment is an induced, not a permanent one.The potential energy of the atom in the field is thus pro-portional to the intensity I and to the real part of thepolarizability, which describes the in-phase componentof the dipole oscillation being responsible for the disper-sive properties of the interaction. The dipole force resultsfrom the gradient of the interaction potential

Fdip(r) = −∇Udip(r) =1

2ε0cRe(α)∇I(r) . (3)

It is thus a conservative force, proportional to the inten-sity gradient of the driving field.

The power absorbed by the oscillator from the drivingfield (and re-emitted as dipole radiation) is given by

Pabs = 〈pE〉 = 2ω Im(pE∗) =ω

ε0cIm(α) I . (4)

The absorption results from the imaginary part of thepolarizability, which describes the out-of-phase compo-nent of the dipole oscillation. Considering the light as astream of photons hω, the absorption can be interpretedin terms of photon scattering in cycles of absorption andsubsequent spontaneous reemisson processes. The corre-sponding scattering rate is

Γsc(r) =Pabs

hω=

1hε0c

Im(α) I(r) . (5)

We have now expressed the two main quantities ofinterest for dipole traps, the interaction potential andthe scattered radiation power, in terms of the position-dependent field intensity I(r) and the polarizability α(ω).We point out that these expressions are valid for any po-larizable neutral particle in an oscillating electric field.This can be an atom in a near-resonant or far off-resonantlaser field, or even a molecule in an optical or microwavefield.

2. Atomic polarizability

In order to calculate its polarizability α, we first con-sider the atom in Lorentz’s model of a classical oscillator.In this simple and very useful picture, an electron (massme, elementary charge e) is considered to be bound elas-tically to the core with an oscillation eigenfrequency ω0

corresponding to the optical transition frequency. Damp-ing results from the dipole radiation of the oscillatingelectron according to Larmor’s well-known formula (see,e.g., Jackson, 1962) for the power radiated by an accel-erated charge.

It is straightforward to calculate the polarizability byintegration of the equation of motion x + Γωx + ω2

0x =−eE(t)/me for the driven oscillation of the electron withthe result

α =e2

me

1ω2

0 − ω2 − iωΓω(6)

Here

Γω =e2ω2

6πε0mec3(7)

is the classical damping rate due to the radiative en-ergy loss. Substituting e2/me = 6πε0c

3Γω/ω2 and in-troducing the on-resonance damping rate Γ ≡ Γω0 =(ω0/ω)2Γω, we put Eq. 6 into the form

α = 6πε0c3 Γ/ω2

0

ω20 − ω2 − i (ω3/ω2

0) Γ. (8)

In a semiclassical approach, described in many text-books, the atomic polarizability can be calculated byconsidering the atom as a two-level quantum system in-teracting with the classical radiation field. One findsthat, when saturation effects can be neglected, the semi-classical calculation yields exactly the same result as theclassical calculation with only one modification: In gen-eral, the damping rate Γ (corresponding to the sponta-neous decay rate of the excited level) can no longer becalculated from Larmor’s formula, but it is determinedby the dipole matrix element between ground and excitedstate,

Γ =ω3

0

3πε0hc3|〈e|µ|g〉|2 . (9)

For many atoms with a strong dipole-allowed transitionstarting from its ground state, the classical formula Eq. 7nevertheless provides a good approximation to the spon-taneous decay rate. For the D lines of the alkali atomsNa, K, Rb, and Cs, the classical result agrees with thetrue decay rate to within a few percent.

An important difference between the quantum-mechanical and the classical oscillator is the possible oc-curence of saturation. At too high intensities of the driv-ing field, the excited state gets strongly populated and

3

the above result (Eq. 8) is no longer valid. For dipoletrapping, however, we are essentially interested in thefar-detuned case with very low saturation and thus verylow scattering rates (Γsc � Γ). We can thus use ex-pression Eq. 8 also as an excellent approximation for thequantum-mechanical oscillator.

3. Dipole potential and scattering rate

With the above expression for the polarizability of theatomic oscillator the following explicit expressions are de-rived for the dipole potential and the scattering rate inthe relevant case of large detunings and negligible satu-ration:

Udip(r) = −3πc2

2ω30

(Γ

ω0 − ω+

Γω0 + ω

)I(r) , (10)

Γsc(r) =3πc2

2hω30

(ω

ω0

)3 ( Γω0 − ω

+Γ

ω0 + ω

)2

I(r) .

(11)

These general expressions are valid for any driving fre-quency ω and show two resonant contributions: Besidesthe usually considered resonance at ω = ω0, there is alsothe so-called counter-rotating term resonant at ω = −ω0.

In most experiments, the laser is tuned relatively closeto the resonance at ω0 such that the detuning ∆ ≡ ω −ω0 fulfills |∆| � ω0. In this case, the counter-rotatingterm can be neglected in the well-known rotating-waveapproximation (see, e.g., Allen and Eberly, 1972), andone can set ω/ω0 ≈ 1. This approximation will be madethroughout this article with a few exceptions discussedin Chapter IV.

In this case of main practical interest, the general ex-pressions for the dipole potential and the scattering ratesimplify to

Udip(r) =3πc2

2ω30

Γ∆

I(r) , (12)

Γsc(r) =3πc2

2hω30

(Γ∆

)2

I(r) . (13)

The basic physics of dipole trapping in far-detuned laserfields can be understood on the basis of these two equa-tions. Obviously, a simple relation exists between thescattering rate and the dipole potential,

hΓsc =Γ∆

Udip , (14)

which is a direct consequence of the fundamental rela-tion between the absorptive and dispersive response ofthe oscillator. Moreover, these equations show two veryessential points for dipole trapping:

• Sign of detuning: Below an atomic resonance(“red” detuning, ∆ < 0) the dipole potential isnegative and the interaction thus attracts atomsinto the light field. Potential minima are thereforefound at positions with maximum intensity. Aboveresonance (“blue” detuning, ∆ > 0) the dipole in-teraction repels atoms out of the field, and poten-tial minima correspond to minima of the intensity.According to this distinction, dipole traps can bedivided into two main classes, red-detuned traps(Sec. IV) and blue-detuned traps (Sec. V).

• Scaling with intensity and detuning: The dipole po-tential scales as I/∆, whereas the scattering ratescales as I/∆2. Therefore, optical dipole traps usu-ally use large detunings and high intensities to keepthe scattering rate as low as possible at a certainpotential depth.

B. Multi-level atoms

In real atoms used for dipole trapping experiments, theelectronic transition has a complex sub-structure. Themain consequence is that the dipole potential in generaldepends on the particular sub-state of the atom. Thiscan lead to some quantitative modifications and also in-teresting new effects. In terms of the oscillator modeldiscussed before, multi-level atoms can be described bystate-dependent atomic polarizabilities. Here we use analternative picture which provides very intuitive insightinto the motion of multi-level atoms in far-detuned laserfields: the concept of state-dependent ground-state po-tentials (Dalibard and Cohen-Tannoudji, 1985, 1989).Alkali atoms are discussed as a situation of great practicalimportance, in order to clarify the role of fine-structure,hyperfine-structure, and magnetic sub-structure.

1. Ground-state light shifts and optical potentials

The effect of far-detuned laser light on the atomic lev-els can be treated as a perturbation in second order of theelectric field, i.e. linear in terms of the field intensity. As ageneral result of second-order time-independent pertur-bation theory for non-degenerate states, an interaction(Hamiltonian H1) leads to an energy shift of the i-thstate (unperturbed energy Ei) that is given by

∆Ei =∑j 6=i

|〈j|H1|i〉|2Ei − Ej

. (15)

For an atom interacting with laser light, the interactionHamiltonian is H1 = −µE with µ = −er representingthe electric dipole operator. For the relevant energies Ei,one has to apply a ‘dressed state’ view (Cohen-Tannoudjiet al., 1992), considering the combined system ‘atom plus

4

field’. In its ground state the atom has zero internal en-ergy and the field energy is nhω according to the numbern of photons. This yields a total energy Ei = nhω forthe unperturbed state. When the atom is put into anexcited state by absorbing a photon, the sum of its inter-nal energy hω0 and the field energy (n − 1)hω becomesEj = hω0 + (n− 1)hω = −h∆ij + nhω. Thus the denom-inator in Eq. 15 becomes Ei − Ej = h∆ij

For a two-level atom, the interaction Hamiltonian isH1 = −µE and Eq. 15 simplifies to

∆E = ± |〈e|µ|g〉|2

∆|E|2 = ±3πc2

2ω30

Γ∆

I (16)

for the ground and excited state (upper and lower sign,respectively); we have used the relation I = 2ε0c|E|2,and Eq. 9 to substitute the dipole matrix element withthe decay rate Γ. This perturbative result obtained forthe energy shifts reveals a very interesting and importantfact: The optically induced shift (known as ‘light shift’ or‘ac Stark shift’) of the ground-state exactly correspondsto the dipole potential for the two-level atom (Eq. 12);the excited state shows the opposite shift. In the inter-esting case of low saturation, the atom resides most ofits time in the ground state and we can interpret thelight-shifted ground state as the relevant potential for themotion of atoms. This situation is illustrated in Fig. 1.

ωh

E

0

ωhexcitedstate

stateground

0

FIG. 1. Light shifts for a two-level atom. Left-hand side,red-detuned light (∆ < 0) shifts the ground state down and theexcited state up by same amounts. Right-hand side, a spatiallyinhomogeneous field like a Gaussian laser beam produces aground-state potential well, in which an atom can be trapped.

For applying Eq. 15 to a multi-level atom with transi-tion substructure1, one has to know the dipole matrix el-ements µij = 〈ei|µ|gi〉 between specific electronic ground

1Perturbation theory for non-degenerate states can be ap-plied in the absence of any coupling between degenerateground states. This is the case for pure linear π or circularσ± polarization, but not for mixed polarizations where Ra-man couplings between different magnetic sub-states becomeimportant; see, e.g., Deutsch and Jessen (1997).

states |gi〉 and excited states |ej〉. It is well-known inatomic physics (see, e.g., Sobelman, 1979) that a specifictransition matrix element

µij = cij ‖µ‖ , (17)

can be written as a product of a reduced matrix element‖µ‖ and a real transition coefficient cij . The fully reducedmatrix element depends on the electronic orbital wave-functions only and is directly related to the spontaneousdecay rate Γ according to Eq. 9. The coefficients cij ,which take into account the coupling strength betweenspecific sub-levels i and j of the electronic ground andexcited state, depend on the laser polarization and theelectronic and nuclear angular momenta involved. Theycan be calculated in the formalim of irreducible tensoroperators or can be found in corresponding tables.

With this reduction of the matrix elements, we cannow write the energy shift of an electronic ground state|gi〉 in the form

∆Ei =3πc2 Γ2ω3

0

I ×∑

j

c2ij

∆ij, (18)

where the summation is carried out over all electron-ically excited states |ej〉 . This means, for a calcula-tion of the state-dependent ground-state dipole potentialUdip, i = ∆Ei, one has to sum up the contributions of allcoupled excited states, taking into account the relevantline strengths c2

ij and detunings ∆ij .

2. Alkali atoms

Most experiments in laser cooling and trapping areperformed with alkali atoms because of their closed opti-cal transitions lying in a convenient spectral range. Themain properties of alkali atoms of relevance for dipoletrapping are summarized in Table I. As an example, thefull level scheme of the relevant n s → n p transition isshown in Fig. 2(a) for a nuclear spin I = 3

2 , as in thecase of 7Li, 23Na, 39, 41K, and 87Rb. Spin-orbit couplingin the excited state (energy splitting h∆′

FS) leads to thewell-known D line doublet 2S1/2 →2 P1/2,

2 P3/2. Thecoupling to the nuclear spin then produces the hyperfinestructure of both ground and excited states with energiesh∆HFS and h∆′

HFS, respectively. The splitting energies,obeying ∆′

FS � ∆HFS � ∆′HFS, represent the three rele-

vant atomic energy scales.

5

∆,FS

21P

2

P2

21

21

23

21

0

L=0

L’=1

(b)

J’=

J’=(c)

J =

HFS∆

HFS∆,

F=2

F=1

F’=2

F’=1

(a) F’=3

23

2S

ω

FIG. 2. Level scheme of an alkali atom: (a) full sub-structure for a nuclear spin I = 3

2, (b) reduced scheme

for very large detunings exceeding the fine-structure splitting(|∆| � ∆′

FS), and (c) reduced scheme for large detunings inthe range ∆′

FS>∼ |∆| � ∆HFS, ∆′

HFS.

On the basis of Eq. 18, one can derive a general resultfor the potential of a ground state with total angularmomentum F and magnetic quantum number mF , whichis valid for both linear and circular polarization as longas all optical detunings stay large 2 compared with theexcited-state hyperfine splitting ∆′

HFS:

Udip(r) =πc2 Γ2ω3

0

(2 + PgF mF

∆2, F+

1− PgF mF

∆1, F

)I(r) .

(19)

Here gF is the well-known Lande factor and P charac-terizes the laser polarization (P = 0,±1 for linearly andcircularly σ± polarized light). The detunings ∆2, F and∆1, F refer to the energy splitting between the particularground state 2S1/2, F and the center of the hyperfine-split 2P3/2 and 2P1/2 excited states, respectively. Thetwo terms in brackets of Eq. 19 thus represent the con-tributions of the D2 and the D1 line to the total dipolepotential.

2The assumption of unresolved excited-state hyperfine struc-ture greatly simplifies the calculation according to Eq. 18 be-cause of existing sum rules for the line strength coefficientsc2ij ; see also Deutsch and Jessen (1997).

In order to discuss this result, let us first considerthe case of very large detunings greatly exceeding thefine-structure splitting (|∆F, 1|, |∆F, 2| � ∆′

FS), in whichwe can completely neglect the even much smaller hyper-fine splitting. Introducing a detuning ∆ with respect tothe center of the D-line dublett, we can linearly expandEq. 19 in terms of the small parameter ∆′

FS/∆:

Udip(r) =3πc2

2ω30

Γ∆

(1 +

13PgF mF

∆′FS

∆

)I(r) . (20)

While the first-order term describes a small residual de-pendence on the polarization P and the magnetic sub-state mF , the dominating zero-order term is just the re-sult obtained for a two-level atom (Eq. 12). The latterfact can be understood by a simple argument: If thefine-structure is not resolved, then the detuning repre-sents the leading term in the total Hamiltonian and theatomic sub-structure can be ignored in a first perturba-tive step by reducing the atom to a very simple s → ptransition; see Fig. 2(b). Such a transition behaves like atwo-level atom with the full line strength for any laser po-larization, and the ground-state light shift is thus equalto the one of a two-level atom. This single ground statecouples to the electronic and nuclear spin in exactly thesame way as it would do without the light. In this sim-ple case, all resulting hyperfine and magnetic substatesstates directly acquire the light shift of the initial atomics state.

In the more general case of a resolved fine-structure, but unresolved hyperfine structure (∆′

FS>∼|∆F, 1|, |∆F, 2| � ∆HFS), one may first consider the atom

in spin-orbit coupling, neglecting the coupling to the nu-clear spin. The interaction with the laser field can thusbe considered in the electronic angular momentum con-figuration of the two D lines, J = 1

2 → J ′ = 12 , 3

2 . Inthis situation, illustrated in Fig. 2(c), one can first cal-culate the light shifts of the two electronic ground statesmJ = ± 1

2 , and in a later step consider their coupling tothe nuclear spin. In this situation, it is important to dis-tinguish between linearly and circularly polarized light:

For linear polarization, both electronic ground states(mJ = ± 1

2 ) are shifted by the same amount because ofsimple symmetry reasons. After coupling to the nuclearspin, the resulting F, mF states have to remain degen-erate like the two original mJ states. Consequently, allmagnetic sublevels show the same light shifts accordingto the line strength factors of 2/3 for the D2 line and 1/3for the D1 line.

For circular polarization (σ±), the light lifts the de-generacy of the two magnetic sublevels of the electronic2S 1

2ground state, and the situation gets more compli-

cated. The relevant line strength factors are then givenby 2

3 (1 ± mJ) for the D2 line and 13 (1 ∓ 2mJ) for the

D1 line. The lifted degeneracy of the two ground statescan be interpreted in terms of a ‘fictitous magnetic field’(Cohen-Tannoudji and Dupont-Roc, 1972; Cho, 1997;Zielonkowski et al., 1998a), which is very useful to un-

6

derstand how the lifted mJ degeneracy affects the F, mF

levels after coupling to the nuclear spin. According tothe usual theory of the linear Zeeman effect in weak mag-netic fields, coupling to the nuclear spin affects the mag-netic sub-structure such that one has to replace gJmJ bygF mF , with gJ and gF denoting the ground state Landefactors. Using this analogy and gJ = 2 for alkali atoms,we can substitute mJ by 1

2gF mF to calculate the relevantline strength factors 1

3 (2± gF mF ) and 13 (1∓ gF mF ) for

the D2 and the D1 line, respectively. These factors leadto the mF dependent shifts for circularly polarized lightin Eq. 19. One finds that this result stays valid as long asthe excited-state hyperfine splitting remains unresolved.

For the photon scattering rate Γsc of a multi-level atom,the same line strength factors are relevant as for thedipole potential, since absorption and light shifts are de-termined by the same transititon matrix elements. Forlinear polarization, in the most general case ∆ � ∆′

HFSconsidered here, one thus explicitely obtains

Γsc(r) =πc2 Γ2

2hω30

(2

∆ 22,F

+1

∆ 21,F

)I(r) (21)

This result is independent of mF , but in general dependson the hyperfine state F via the detunings. For lin-early polarized light, optical pumping tends to equallydistribute the population among the different mF states,and thus leads to complete depolarization. If the de-tuning is large compared to the ground-state hyperfinesplitting, then all sub-states F, mF are equally populatedby redistribution via photon scattering. For circular po-larization, Zeeman pumping effects become very impor-tant, which depend on the particular detuning regimeand which we do not want to discuss in more detail here.It is interesting to note that the general relation betweendipole potential and scattering rate takes the simple formof Eq. 14 either if the contribution of one of the two Dlines dominates for rather small detunings or if the de-tuning is large compared to the fine-structure splitting.

Our discussion on multi-level alkali atoms shows thatlinearly polarized light is usually the right choice for adipole trap3, because the magnetic sub-levels mF of acertain hyperfine ground-state F are shifted by sameamounts. This only requires a detuning large comparedto the excited-state hyperfine splitting, which is usuallyvery well fulfilled in dipole trapping experiments. If thedetuning also exceeds the ground-state hyperfine split-ting then all sub-states of the electronic ground state areequally shifted, and the dipole potential becomes com-pletely independent of mF and F . For circularly polar-ized light there is a mF -dependent contribution, whichleads to a splitting analogous to a magnetic field. This

3an interesting exception is the work by Corwin et al. (1997)on trapping in circularly polarized light; see also Sec. IVA2.

term vanishes only if the optical detuning greatly exceedsthe fine-structure splitting.

III. EXPERIMENTAL ISSUES

Here we discuss several issues of practical importancefor experiments on dipole trapping. Cooling and heatingin the trap is considered in Sec. III A, followed by a sum-mary of the typical experimental procedures in Sec. III B.Finally, the particular role of collisions is discussed inSec. III C.

A. Cooling and heating

Atom trapping in dipole potentials requires cooling toload the trap and eventually also to counteract heating inthe dipole trap. We briefly review the various availablecooling methods and their specific features with respectto dipole trapping. Then we discuss sources of heating,and we derive explicite expressions for the heating ratein the case of thermal equilibrium in a dipole trap. Thisallows for a direct comparison between dipole traps op-erating with red and blue detuning.

1. Cooling methods

Efficient cooling techniques are an essential require-ment to load atoms into a dipole trap since the attainabletrap depths are generally below 1mK. Once trapped, fur-ther cooling can be applied to achieve high phase-spacedensities and to compensate possible heating mechanisms(see Sec. III A 2) which would otherwise boil the atomsout of the trap. The development of cooling methodsfor neutral atoms has proceeded at breathtaking speedduring the last decade, and numerous excellent reviewshave been written illuminating these developments (Foot,1991; Arimondo et al., 1992; Metcalf and van der Straten,1994; Sengstock and Ertmer, 1995; Ketterle and vanDruten, 1996; Adams and Riis, 1997; Chu, 1998; Cohen-Tannoudji, 1998; Phillips, 1998). In this section, webriefly discuss methods which are of relevance for coolingatoms in dipole traps. Chapters IV and V describe theexperimental implementations of the cooling schemes toparticular trap configurations.

Doppler cooling. Doppler cooling is based on cyclesof near-resonant absorption of a photon and subsequentspontaneous emission resulting in a net atomic momen-tum change per cycle of one photon momentum hk withk = 2π/λ denoting the wavenumber of the absorbed pho-ton. Cooling is counteracted by heating due to the mo-mentum fluctuations by the recoil of the spontaneouslyemitted photons (Minogin and Letokhov, 1987). Equilib-rium between cooling and heating determines the lowestachievable temperature. For Doppler cooling of two-level

7

atoms in standing waves (“optical molasses”), the min-imum temperature is given by the Doppler temperaturekBTD = hΓ/2. Typical values of the Doppler tempera-ture are around 100 µK, which is just sufficiently low toload atoms into a deep dipole trap. The first demonstra-tion of dipole trapping (Chu et al., 1986) used Dopplercooling for loading the trap and keeping the atoms frombeing boiled out of the trap. With the discovery of meth-ods reaching much lower temperatures, Doppler coolinghas lost its importance for the direct application to dipoletraps.

Polarization-gradient cooling. The Doppler tempera-ture is a somewhat artificial limit since it is based onthe simplifying assumption of a two-level atom. It wassoon discovered that atoms with a more complex levelstructure can be cooled below TD in standing waves withspatially varying polarizations (Lett et al., 1988). Thecooling mechanisms are based on optical pumping be-tween ground state Zeeman sublevels. The friction forceproviding cooling results either from unbalanced radia-tion pressures through motion-induced atomic orienta-tion, or from a redistribution among the photons in thestanding wave (Dalibard and Cohen-Tannoudji, 1989).In the latter case, the cooling force can be illustrativelyexplained by the so-called Sisyphus effect for a movingatom. The atom looses kinetic energy by climbing upthe dipole potential induced by the standing wave ofthe trapping light. When reaching the top of this po-tential hill, the atom is optically pumped back into thebottom of the next potential valley from where again itstarts to climb (Dalibard and Cohen-Tannoudji, 1985).Polarization-gradient cooling can be achieved in stand-ing waves at frequencies below an atomic resonance (red-detuned molasses) (Lett et al., 1988; Salomon et al., 1990)as well as above an atomic resonance (blue-detuned mo-lasses) (Boiron et al., 1995; Hemmerich et al., 1995). Inblue-detuned molasses, atoms primarily populate stateswhich are decoupled from the light field resulting in a re-duction of photon scattering, but also in a smaller coolingrate as compared to red-detuned molasses (Boiron et al.,1996).

With polarization-gradient molasses one can preparefree-space atomic samples at temperatures on the orderof ∼ 10Trec with the recoil temperature

Trec =h2k2

m(22)

being defined as the temperature associated with the ki-netic energy gain by emission of one photon. For thealkali atoms, recoil temperatures are given in Table I.The achievable temperatures are much below the typi-cal depth of a dipole trap. Polarization-gradient coolingtherefore allows efficient loading of dipole traps (see Sec.III B 1), either by cooling inside a magneto-optical trap(Steane and Foot, 1991) or by cooling in a short molassesphase before transfer into the dipole trap.

Besides enhancing the loading efficiency, polarization-gradient cooling was directly applied to atoms trapped

in a dipole potential by subjecting them to near-resonantstanding waves with polarization gradients (Boiron et al.,1998; Winoto et al., 1998). Since the cooling mechanismrelies on the modification of the ground-state sublevels bythe cooling light, a necessary condition for efficient cool-ing is the independency of the trapping potential from theZeeman substate which, in contrast to magnetic traps,can easily be fulfilled in dipole traps as explained in Sec.II B 2.

Raman cooling. The recoil temperature marks thelimit for cooling methods based on the repeated absorp-tion and emission of photons such as Doppler cooling andpolarization-gradient cooling. To overcome this limit,different routes were explored to decouple the cold atomsfrom the resonant laser excitation once they have reachedsmall velocities. All approaches are based upon transi-tions with a narrow line width which are thus extremelyvelocity-selective such as dark-state resonances (Aspectet al., 1988) or Raman transition between ground statesublevels (Kasevich and Chu, 1992). With free atomicsamples, the interaction time, and thus the spectral reso-lution determining the final temperature, was limited bythe time the thermally expanding atomic cloud spent inthe laser field (Davidson et al., 1994). Taking advantageof the long storage times in dipole traps, Raman coolingwas shown to efficiently work for trapped atomic samples(Lee et al., 1994; Lee et al., 1996; Kuhn et al., 1996).

The basic principle of Raman cooling is the follow-ing. Raman pulses from two counter-propagating laserbeams transfer atoms from one ground state |1〉 to an-other ground state |2〉 transferring 2hk momentum 4. Us-ing sequences of Raman pulses with varying frequencywidth, detuning, and propagation direction, pulses canbe tailored which excite all atoms except those with avelocity near v = 0 (Kasevich et al., 1992; Reichel etal., 1995; Kuhn et al., 1996). The cooling cycle is com-pleted by a light pulse resonantly exciting atoms in state|2〉 in order to optically pump the atoms back to the |1〉state through spontaneous emission. Each spontaneousemission randomizes the velocity distribution so that afraction of atoms will acquire a velocity v ≈ 0. By re-peating many sequences of Raman pulses followed by op-tical pumping pulses, atoms are accumulated in a smallvelocity interval around v = 0. The final width of thevelocity distribution, i.e. the temperature, is determinedby the spectral resolution of the Raman pulses. In dipoletraps, high resolution can be achieved by the long storagetimes. In addition, motional coupling of the degrees offreedom through the trap potential allowed to cool theatomic motion in all three dimensions with Raman pulsesapplied along only one spatial direction (Lee et al., 1996;Kuhn et al., 1996).

4The two states are, e.g., the two hyperfine ground states ofan alkali atom.

8

Resolved-sideband Raman cooling. Cooling with well-resolved motional sidebands, a technique well knownfrom laser cooling in ion traps (Ghosh, 1995), was veryrecently also applied to optical dipole traps (Hamann etal., 1998; Perrin et al., 1998; Vuletic et al., 1998; Bou-choule et al.; 1998). For resolved-sideband cooling, atomsmust be tightly confined along (at least) one spatial di-mension with oscillation frequencies ωosc large enough tobe resolved by Raman transitions between two groundstate levels. In contrast to Raman cooling explained inthe preceeding paragraph, the confining potential of thetrap is therefore a necessary prerequisite for the applica-tion of sideband cooling.

Atomic motion in the tightly confining potential is de-scribed by a wavepacket formed by the superpositionof vibrational states |nosc〉. In the Lamb-Dicke regime,where the rms size of the wavepacket is small compared tothe wavelength of the cooling transition, an absorption-spontaneous emission cycle almost exclusively returns tothe same vibrational state it started from (∆nosc = 0).The Lamb-Dicke regime is reached by trapping atoms indipole potentials formed in the interference pattern of far-detuned laser beams (see Secs. IVB and IVC). Sidebandcooling consists of repeated cycles of Raman pulses whichare tuned to excite transitions with ∆nosc = −1 followedby an optical pumping pulse involving spontaneous emis-sion back to the initial state with ∆nosc = 0. In this way,the motional ground state nosc = 0 is selectively popu-lated since it is the only state not interacting with theRaman pulses. Achievable temperatures are only limitedby the separation and the width of the Raman sidebandsdetermining the suppression of off-resonant excitation ofthe nosc = 0 state.

A particularly elegant realization of sideband coolingwas accomplished by using the trapping light itself todrive the Raman transition instead of applying additionallaser fields (Hamann et al, 1998; Vuletic et al., 1998). Forthis purpose, a small magnetic field was applied shiftingthe energy of two adjacent ground-state Zeeman sublevelsrelative to each other by exactly one vibrational quan-tum hωosc. In this way, the bound states |mF ; nosc〉 and|mF −1; nosc−1〉 became degenerate, and Raman transi-tions between the two states could be excited with single-frequency light provided by the trapping field (degeneratesideband cooling) (Deutsch and Jessen, 1998). The greatadvantage of degenerate sideband cooling is that workswith only the two lowest-energy atomic ground states be-ing involved, resulting in the suppression of heating andtrap losses caused by inelastic binary collisions (see Sec.III C).

Evaporative cooling. Evaporative cooling, orginallydemonstated with magnetically trapped hydrogen (Hesset al., 1987), has been the key technique to achieve Bose-Einstein condensation in magnetic traps (Ketterle andvan Druten, 1996). It relies on the selective removalof high-energetic particles from a trap and subsequentthermalization of the remaining particles through elasticcollisions. Evaporative cooling requires high densities to

assure fast thermalization rates, and large initial parti-cle numbers since a large fraction of trapped atoms isremoved from the trap by evaporation. To become effec-tive, the ratio between inelastic collisions causing lossesand heating, and elastic collisions providing thermaliza-tion and evaporation, has to be large.

In dipole traps, inelastic processes can be greatly sup-pressed when the particles are prepared in their energet-ically lowest state. However, the requirement of largeparticle numbers and high density poses a dilemma forthe application of evaporative cooling to dipole traps. Intightly confining dipole traps such as a crossed dipoletrap, high peak densities can be reached, but the trap-ping volume, and thus the number of particles transferredinto the trap, is small. On the other hand, large trap-ping volumes yielding large numbers of trapped particlesprovide only weak confinement yielding small elastic col-lision rates. This is why, until now, only one experimentis reported on evaporative cooling in dipole traps start-ing with a small sample of atoms (Adams et al., 1995).By precooling large ensembles in dipole traps with op-tical methods explained in the preceeding paragraphs,much better starting conditions for evaporative coolingare achievable (Engler et al., 1998; Vuletic et al., 1998;Winoto et al., 1998) making evaporative cooling a stillinteresting option for future applications.

Adiabatic expansion. When adiabatically expandinga potential without changing its shape, the temperatureof the confined atoms is decreased without increasing thephase-space density 5. In dipole potentials, cooling byadiabatic expansion was realized by slowly ramping downthe trapping light intensity (Chen et al., 1992; Kastberget al., 1995). In far-detuned traps consisting of micropo-tentials induced by interference, adiabatic cooling is par-ticularly interesting when the modulation on the scale ofthe optical wavelength is slowly reduced without modi-fying the large-scale trapping potential, by, e.g. changingthe polarization of the interfering laser beams. Atomsare initially strongly localized in the micropotentials re-sulting in high peak densities. After adiabatic expansion,the temperature of the sample is reduced, as is the peakdensity. However, the density averaged over one periodof the interference structure is not modified.

2. Heating mechanisms

Heating by the trap light is an issue of particularimportance for optical dipole trapping. A fundamen-tal source of heating is the spontaneous scattering oftrap photons, which due to its random nature causes

5Adiabatic changes of the potential shape leading to an in-crease of the phase-space density are demonstrated in (Pinkseet al., 1997) and (Stamper-Kurn et al., 1998b).

9

fluctuations of the radiation force6. In the far-detunedcase considered here, the scattering is completely elastic(or quasi-elastic if a Raman process changes the atomicground state). This means that the energy of the scat-tered photon is determined by the frequency of the laserand not of the optical transition.

Both absorption and spontaneous re-emission pro-cesses show fluctuations and thus both contribute to thetotal heating (Minogin and Letokhov, 1987). At large de-tunings, where scattering processes follow Poisson statis-tics, the heating due to fluctuations in absorption corre-sponds to an increase of the thermal energy by exactlythe recoil energy Erec = kBTrec/2 per scattering event.This first contribution occurs in the propagation direc-tion of the light field and is thus anisotropic (so-calleddirectional diffusion). The second contribution is due tothe random direction of the photon recoil in spontaneousemission. This heating also increases the thermal energyby one recoil energy Erec per scattering event, but dis-tributed over all three dimensions. Neglecting the gen-eral dependence on the polarization of the spontaneouslyemitted photons, one can assume an isotropic distribu-tion for this heating mechanism. Taking into accountboth contributions, the longitudinal motion is heated onan average by 4Erec/3 per scattering process, whereasthe two transverse dimensions are each heated by Erec/3.The overall heating thus corresponds to an increase of thetotal thermal energy by 2Erec in a time Γ−1

sc .For simplicity, we do not consider the generally

anisotropic character of heating here; in most cases ofinterest the trap anyway mixes the motional degrees ona time scale faster than or comparable to the heating.We can thus use a simple global three-dimensional heat-ing power Pheat = ˙E corresponding to the increase of themean thermal energy E of the atomic motion with time.This heating power is directly related to the average pho-ton scattering rate Γsc by

Pheat = 2Erec Γsc = kBTrec Γsc . (23)

For intense light fields close to resonance, in partic-ular in standing-wave configurations, it is well knownthat the induced redistribution of photons between dif-ferent traveling-wave components can lead to dramaticheating (Gordon and Ashkin, 1980; Dalibard and Cohen-Tannoudji, 1985). In the far off-resonant case, however,this induced heating falls off very rapidly with the detun-ing and is usually completely negligible as compared tospontaneous heating.

In addition to the discussed fundamental heating indipole traps, it was recently pointed out by Savard et

6Under typical conditions of a dipole trap, the scatter-ing force that results in traveling-wave configurations staysvery weak as compared to the dipole force and can thus beneglected.

al. (1997), that technical heating can occur due to inten-sity fluctuations and pointing instabilities in the trappingfields. In the first case, fluctuations occuring at twice thecharacteristic trap frequencies are relevant, as they canparametrically drive the oscillatory atomic motion. Inthe second case, a shaking of the potential at the trapfrequencies increases the motional amplitude. Experi-mentally, these issues will strongly depend on the par-ticular laser source and its technical noise spectrum, buthave not been studied in detail yet. Several experimentshave indeed shown indications for heating in dipole trapsby unidentified mechanisms (Adams et al., 1995; Lee etal., 1996; Zielonkowski et al., 1998b; Vuletic et al., 1998),which may have to do with fluctuations of the trappinglight.

3. Heating rate

For an ultracold atomic gas in a dipole trap it is oftena good assumption to consider a thermal equilibrium sit-uation, in which the energy distribution is related to atemperature T . The further assumption of a power-lawpotential then allows one to derive very useful expressionsfor the mean photon scattering rate and the correspond-ing heating rate of the ensemble, which also illustrateimportant differences between traps operating with redand blue detuned light.

In thermal equilibrium, the mean kinetic energy peratom in a three-dimensional trap is Ekin = 3kBT/2. In-troducing the parameter κ ≡ Epot/Ekin as the ratio ofpotential and kinetic energy, we can express the meantotal energy E as

E =32(1 + κ) kBT . (24)

For many real dipole traps as described in Secs. IVand V, it is a good approximation to assume a separablepower-law potential with a constant offset U0 of the form

U(x, y, z) = U0 + a1xn1 + a2y

n2 + a3zn3 . (25)

In such a case, the virial theorem can be used to calculatethe ratio between potential and kinetic energy

κ =23

(1n1

+1n2

+1n3

). (26)

For a 3D harmonic trap this gives κ = 1, for an ideal 3Dbox potential κ = 0.

The relation between mean energy and temperature(Eq. 24) allows us to reexpress the heating power result-ing from photon scattering (Eq. 23) as a heating rate

T =2/3

1 + κTrec Γsc , (27)

describing the corresponding increase of temperaturewith time.

10

The mean scattering rate Γsc can, in turn, be calcu-lated from the temperature of the sample, according tothe following arguments: Eq. 14 relates the average scat-tering rate to the mean dipole potential Udip experiencedby the atoms. In a pure dipole trap7 described by Eq. 25,the mean optical potential is related to the mean poten-tial energy Epot, the mean kinetic energy Ekin, and thetemperature T by

Udip = U0 + Epot = U0 + κEkin = U0 +3κ

2T . (28)

This relation allows us to express the mean scatteringrate as

Γsc =Γ

h∆(U0 +

3κ

2kBT ) . (29)

Based on this result, let us now discuss two specificsituations which are typical for real experiments as de-scribed in Secs. IV and V; see illustrations in Fig. 3. In ared-detuned dipole trap (∆ < 0), the atoms are trappedin an intensity maximum with U0 < 0, and the trap depthU = |U0| is usually large compared to the thermal energykBT . In a blue-detuned trap (∆ > 0), a potential mini-mum corresponds to an intensity minimum, which in anideal case means zero intensity. In this case, U0 = 0 andthe potential depth U is determined by the height of therepulsive walls surrounding the center of the trap.

For red and blue-detuned traps with U � kBT , Eqs. 27and 29 yield the following heating rates:

Tred =2/3

1 + κTrec

Γh|∆| U , (30a)

Tblue =κ

1 + κTrec

Γh∆

kBT . (30b)

Obviously, a red-detuned trap shows linear heating(which decreases when kBT approaches U), whereasheating behaves exponentially in a blue-detuned trap.Note that in blue-detuned traps a fundamental lowerlimit to heating is set by the zero-point energy of theatomic motion, which we have neglected in our classicalconsideration.

7this excludes hybrid potentials in which other fields (grav-ity, magnetic or electric fields) are important for the trapping.

Ured

k T

blue

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B

FIG. 3. Illustration of dipole traps with red and blue de-tuning. In the first case, a simple Gaussian laser beam isassumed. In the second case, a Laguerre-Gaussian LG01

“doughnut” mode is chosen which provides the same potentialdepth and the same curvature in the trap center (note thatthe latter case requires e2 times more laser power or smallerdetuning).

Eqs. 30 allow for a very illustrative direct comparisonbetween a blue and a red-detuned trap: The ratio ofheating at the same magnitude of detuning |∆| is givenby

Tblue

Tred

=3κ

2kBT

U. (31)

This comparison shows that blue detuning offers substan-tial advantages in two experimental situations:

• U � kBT , very deep potentials for tight confine-ment,

• κ � 1, box-like potentials with hard repulsivewalls.

When, in other words, a harmonic potential of moderatedepth is to be realized for a certain experiment, the ad-vantage of blue detuning is not substantial. The choice ofred detuning may be even more appropriate as the betterconcentration of the available laser power in such a trapallows one to use larger detunings to create the requiredpotential depth.

B. Experimental techniques

1. Trap loading

The standard way to load a dipole trap is to startfrom a magneto-optical trap (MOT). This well-knownradiation-pressure trap operating with near-resonantlight was first demonstrated by Raab et al. in 1987 andhas now become the standard source of ultracold atoms inmany laboratories all over the world. A MOT can providetemperatures down to a few 10 Trec, when its operationis optimized for sub-Doppler cooling (see Sec. III A 1).This sets a natural scale for the minimum depth of adipole trap as required for efficient loading. Due to theirlower recoil temperatures (Table I), heavier alkali atomsrequire less trap depth as the lighter ones and thus allowfor larger detunings. For the heavy Cs atoms, for exam-ple, dipole traps with depths as low as ∼ 10µK can be

11

directly loaded from a MOT (Zielonkowski et al., 1998b).Trap loading at much lower depths can be reached withBose-Einstein condensates (Stamper-Kurn et al., 1998).

For dipole-trap loading, a MOT is typically operated intwo stages. First, its frequency detuning is set quite closeto resonance (detuning of a few natural linewidths) to op-timize capture by the resonant scattering force. Then, af-ter the loading phase, the MOT parameters are changedto optimize sub-Doppler cooling (Drewsen et al., 1994;Townsend et al., 1995). Most importantly, the detun-ing is switched to much higher values (typically 10 – 20linewidths), and eventually also the laser intensity is low-ered. For the heavier alkali atoms8, this procedure pro-vides maximum phase-space densities for trap loading.Another option is to ramp up the magnetic fields of theMOT to spatially compress the sample (Petrich et al.,1994).

The dipole trap is filled by simply overlapping it withthe atomic cloud in the MOT, before the latter is turnedoff. In this procedure, it is advantageous to switch offthe magnetic field of the MOT a few ms before the laserfields are extinguished, because the short resulting opti-cal molasses cooling phase establishes the lowest possi-ble temperatures and a quasi-thermal distribution in thetrap. For practical reasons, the latter is important be-cause a MOT does not necessarily load the atoms intothe very center of the dipole trap. When MOT positionand dipole trap center do not coincide exactly, loading re-sults in excess potential energy in the dipole trap. Whenthe MOT light is extinguished, it is very important toshield the dipole trap from any resonant stray light, inparticular if very low scattering rates (<∼ 1 s−1) are to bereached.

The MOT itself can be loaded in a simple vapor cell(Monroe et al., 1990). In such a set-up, however, the life-time of the dipole trap is typically limited to less than 1 sby collsions with atoms in the background gas. If longerlifetimes are requested for a certain application, the load-ing of the MOT under much better vacuum conditionsbecomes an important issue, similar to experiments onBose-Einstein condensation. Loading can then be ac-complished from a very dilute vapor (Anderson et al.,1994), but more powerful concepts can be realized witha Zeeman-slowed atomic beam (Phillips and Metcalf,1982), with a double-MOT set-up (Myatt et al., 1997),or with slow-atom sources based on modified MOTs (Luet al., 1996, Dieckmann et al., 1998).

Regarding trap loading, a dipole trap with red detun-ing can offer an important advantage over a blue-detunedtrap: When MOT and dipole trap are simultaneously

8The lightest alkali atom Li behaves in a completely dif-ferent way: Here optimum loading is accomplished at largerdetunings and optimum cooling is obtained relatively close toresonance (Schunemann et al., 1998)

turned on, the attractive dipole potential leads to a lo-cal density increase in the MOT, which can substantiallyenhance the loading process. In very deep red-detunedtraps, however, the level shifts become too large for thecooling light and efficient loading requires rapid alterna-tion between cooling and trapping light (Miller et al.,1993).

2. Diagnostics

The atomic sample in a dipole trap is characterized bythe number of stored atoms, the motional temperatureof the ensemble (under the assumption of thermal dis-tribution), and the distribution of population among thedifferent ground-state sub-levels. Measurements of theseimportant quantities can be made in the following ways.

Number of atoms. A very simple and efficientmethod, which is often used to determine the numberof atoms in a dipole trap is to recapture them into theMOT and to measure the power of the emitted fluores-cence light with a calibrated photo-diode or CCD camera.In this way, it is quite easy to detect down to about onehundred atoms, but even single atoms may be observedin a more elaborate set-up (Haubrich et al., 1996). Thisrecapture method works particularly well if it is ensuredthat the MOT does not capture any other atoms thanthose released from the dipole trap. This is hardly pos-sible in a simple vapor-cell set-up, but quite easy if anatomic beam equipped with a mechanical shutter is usedfor loading the MOT.

In contrast to the completely destructive recapture,several other methods may be applied. The trappedatoms can be illuminated with a short resonant laserpulse of moderate intensity to measure the emitted flu-orescence light. This can be done also with spatial res-olution by using a CCD camera; see Fig. 18(b) for anexample. If the total number of atoms is not too low, thedetection pulse can be kept weak enough to avoid traploss by heating. Furthermore, absorption imaging canbe used (see Fig. 7), or even more sensitive and less de-structive dark-field or phase-contrast imaging methods asapplied to sensitively monitor Bose-Einstein condensates(Andrews et al., 1996; Bradley et al., 1997; Andrews etal., 1997).

Temperature. In a given trapping potential U(r) thethermal density distribution n(r) direct follows from theBoltzmann factor,

n(r) = n0 exp(−U(r)

kBT

). (32)

The temperature can thus be derived from the measuredspatial density distribution in the trap, which itself canbe observed by various imaging methods (fluorescence,absorptive, and dispersive imaging). For a 3D harmonicpotential U(r) = 1

2m (ω2xx2 + ω2

yy2 + ω2zz2) the resulting

distribution is Gaussian in all directions,

12

n(r) = n0 exp(− x2

2σ2x

)exp

(− y2

2σ2y

)exp

(− z2

2σ2z

),

(33)

with σi = ω−1i

√kBT/m. The temperature is thus re-

lated to the spatial extensions of the trapped atom cloudby

T =m

kBσ2

i ω2i . (34)

Obviously, it is very important to know the exact trap fre-quencies to precisely determine the temperature; a prac-tical example for such a measurement is discussed in con-text with Fig. 7(c). This way of measuring the temper-ature is limited by the resolution of the imaging systemand therefore becomes difficult for very tightly confiningpotentials.

A widely used and quite accurate, but completely de-structive way to measure temperatures is the time-of-flight method. The trap is turned off to release the atomsinto a free, ballistic flight. This has to be done in a rapid,completely non-adiabatic way as otherwise an adiabaticcooling effect (see Sec. III A 1) would influence the mea-surement. After a sufficiently long ballistic expansionphase, the resulting spatial distribution, which can againbe observed by the various imaging methods, directlyreflects the velocity distribution at the time of release.Another method is to detect the Doppler broadening ofRaman transitions between ground states, using a pairof counterpropagating laser beams (Kasevich and Chu,1991), which is not limited by the natural linewidth ofthe optical transition.

Internal distribution. The relative population of thetwo hyperfine ground states of an alkali atom (see levelscheme in Fig. 2) can be measured by application of aprobe pulse resonant to the closed sub-transition F =I + 1/2 → F ′ = 3/2 in the hyperfine structure, which iswell resolved for the heavier alkali atoms (see Table I).The fluorescence light is then proportional to the num-ber of atoms in the upper hyperfine state F = I + 1/2.If, in contrast, a repumping field is present in the probelight (as it is always used in a MOT), the fluorescence isproportional to the total number of atoms, as all atomsare immediately pumped into the closed excitation cy-cle. The normalized fluorescence signal thus gives therelative population of the upper hyperfine ground state(F = I + 1/2); such a measurement is discussed inSec. IVA2. A very sensitive alternative, which worksvery well in shallow dipole traps, is to blow the totalupper-state population out of the trap by the radiationpressure of an appropriate resonant light pulse. Subse-quent recapture into the MOT then shows how manyatoms have remained trapped in the shelved lower hy-perfine ground state.

The distribution of population over different magneticsub-states can be analyzed by Stern-Gerlach methods.When the atomic ensemble is released from the dipole

trap and ballistically expands in an inhomogeneous mag-netic field, then atoms in different magnetic sub-levelscan be well separated in space. Such an analysis hasbeen used for optically trapped Bose-Einstein conden-sates (Stamper-Kurn et al., 1998a; Stenger, 1998); anexample is shown in Fig. 8(b). Another possibility, whichcan be easily applied to shallow dipole traps, is to pullatoms out of the trap by the state-dependent magneticforce, as has been used by Zielonkowski et al. (1998b)for measuring the depolarizing effect of the trap photonscattering; see discussion in Sec. IVB 3.

C. Collisions

It is a well-known experimental fact in the field of lasercooling and trapping that collisional processes can leadto substantial trap loss. Detailed measurements of traploss under various conditions provide insight into ultra-cold collision phenomena, which have been subject of ex-tensive research (Walker and Feng, 1994; Weiner, 1995).Here we discuss the particular features of dipole trapswith respect to ultracold collisions.

The decay of the number N of atoms in a trap can bedescribed by the general loss equation

N(t) = −αN(t)− β

∫V

n2(r, t) d3r − γ

∫V

n3(r, t) d3r .

(35)

Here the single-particle loss coefficient α takes into ac-count collisions with the background gas in the vacuumapparatus. As a rule of thumb, the 1/e lifetime τ = 1/αof a dipole trap is ∼ 1 s at a pressure of 3 × 10−9 mbar.This is about three times lower than the correspondinglifetime in a MOT because of the larger cross sections forcollisional loss at lower trap depth.

FIG. 4. Decay of Cs atoms measured in a crossed-beamdipole trap (see Sec. IVC) realized with the 1064-nm light ofa Nd:YAG laser. The initial peak density is 2.5× 1012 cm−3.When all atoms are in the lower hyperfine ground state(F = 3), the purely exponential decay (1/e-lifetime 1.1 s) isdue to collisions with the background gas. When the atomsare in the upper hyperfine level (F = 4), a dramatic loss isobserved as a result of hyperfine-changing collisions (loss co-efficient β ≈ 5× 10−11 cm3/s). Unpublished data, courtesy ofC. Salomon.

13

The two-body loss coefficient β describes trap loss dueto ultracold binary collisions and reveals a wide range ofinteresting physics. In general, such trap loss becomesimportant if the colliding atoms are not in their abso-lute ground state. In an inelastic process the internalenergy can be released into the atomic motion, causingescape from the trap. Due to the shallowness of opti-cal dipole traps, even the collisional release of the rel-atively small amount of energy in the ground-state hy-perfine structure of an alkali atom will always lead totrap loss9. Hyperfine-changing collisions, which occurwith large rate coefficients β of typically 5×10−11 cm3/s(Sesko et al., 1989; Wallace et al., 1992), are thus ofparticular importance for dipole trapping. Alkali atomsin the upper hyperfine state (F = I + 1/2) can showvery rapid, non-exponential collisional decay, in contrastto a sample in the lower ground state (F = I − 1/2).This is impressively demonstrated by the measurementin Fig. 4, which shows the decay of a sample of Cs atomsprepared either in the upper or lower hyperfine groundstate. For the implementaion of laser cooling schemes indipole traps, it is thus a very important issue to keep theatoms predominantly in the lower hyperfine state; sev-eral schemes fulfilling this requirement are discussed inSecs. IV and V.

Trap loss can also occur as a result of light-assisted bi-nary collisions involving atoms in the excited state. Theradiative escape mechanism (Gallagher and Pritchard,1989) and excited-state fine-structure changing collisionsstrongly affect a MOT, but their influence is negligiblysmall in a dipole trap because of the extremely low op-tical excitation. It can become important, however, ifnear-resonant cooling light is present. Another impor-tant mechanism for trap loss is photoassociation (Lett etal., 1995), a process in which colliding atoms are excitedto bound molecular states, which then decay via bound-bound or bound-free transitions. Dipole traps indeed rep-resent a powerful tool for photoassociative spectroscopy(Miller et al., 1993b).

Three-body losses, as described by the coefficient γ inEq. 35, become relevant only at extremely high den-sities (Burt et al., 1997; Stamper-Kurn et al., 1998a),far exceeding the conditions of a MOT. In a collisionof three atoms, a bound dimer can be formed and thethird atom takes up the released energy, so that all threeatoms are lost from the trap. As a far-detuned dipoletrap allows one to completely suppress binary collisionlosses by putting the atoms into the absolute internalground state, it represents an interesting tool for mea-surements on three-body collisions; an example is dis-cussed in Sec. IVA4.

9In contrast, a MOT operated under optimum capture con-ditions is deep enough to hold atoms after hyperfine-changingcollisions.

In contrast to inelastic collisions releasing energy, elas-tic collisions lead to a thermalization of the trappedatomic ensemble. This also produces a few atoms withenergies substantially exceeding kBT . A loss of theseenergetic atoms in a shallow trap leads to evaporativecooling (see Sec. III A 1) and is thus of great interest forthe attainment of Bose-Einstein condensation. Regard-ing the basic physics of elastic collisions, dipole trapsare not different from magnetic traps, but they offer ad-ditional experimental possibilities. By application of ahomogeneous magnetic field atomic scattering propertiescan be tuned without affecting the trapping itself. Usingthis advantage of dipole trapping, Feshbach resonanceshave been observed with Bose-condensed Na atoms (In-ouye et al., 1998) and thermal Rb atoms (Courteille etal., 1998). Moreover, an intriguing possibility is to studycollisions in an arbitrary mixture of atoms in differentmagnetic sub-states (Stenger et al., 1998).

IV. RED-DETUNED DIPOLE TRAPS

The dipole force points towards increasing intensity ifthe light field is tuned below the atomic transition fre-quency (red detuning). Therefore, already the focus of alaser beam constitutes a stable dipole trap for atoms asfirst proposed by Ashkin (1978). The trapping forces gen-erated by intense focused lasers are rather feeble whichwas thhe main obstacle for trapping neutral atoms indipole traps. Attainable trap depths in a tightly fo-cused beam are typically in the millikelvin range, ordersof magnitude smaller than the thermal energy of room-temperature atoms. One therefore had to first developefficient laser cooling methods for the preparation of coldatom sources (see Sec. III A 1) to transfer significant num-bers of atoms into a dipole trap.

In their decisive experiment, S. Chu and coworkers(1986) succeeded in holding about 500 sodium atoms forseveral seconds in the tight focus of a red-detuned laserbeam. Doppler molasses cooling was used to load atomsinto the trap, which was operated at high intensities andconsiderable atomic excitation in order to provide a suf-ficiently deep trapping potential. Under these circum-stances, the radiation pressure force still significantly in-fluences the trapping potential due to a considerable rateof spontaneous emission. With the development of sub-Doppler cooling (see Sec. III A 1) and the invention of themagneto-optical trap as a source for dense, cold atomicsamples (see Sec. III B 1), dipole trapping of atoms re-gained attention with the demonstration of a far-off res-onant trap by Miller et al. (1993). In such a trap, sponta-neous emission of photons is negligible, and the trappingpotential is given by from the equations derived in Sec.II.

Since then, three major trap types with red-detunedlaser beams have been established, all based on combi-nations of focused Gaussian beams: Focused-beam traps

14

consisting of a single beam, the standing-wave trapswhere atoms are axially confined in the antinodes of astanding wave, and crossed-beam traps created by of twoor more beams intersecting at their foci. The differenttrap types are schmetically depicted in Fig. 5. We dis-cuss these trap configurations and review applicationsof the different trap types for the investigation of inter-esting physical questions. Sec. IVA deals with focused-beam traps, Sec. IVB presents standing-wave traps, andSec. IVC discusses crossed-beam traps. Far-detuned op-tical lattices trapping atoms in micropotentials formedby multiple-beam interference represent a trap class oftheir own. Atoms might get trapped in the antinodes ofthe interference pattern at red detuning from resonance,but also in the nodes when the light field is blue-detuned.Sec. IVD at the end of this chapter is devoted to recentdevelopments on far-detuned optical lattices.

FIG. 5. Beam configurations used for red-detuned far-offresonance traps. Shown below are the corresponding calculatedintensity distributions. (a) Horizontal focused-beam trap. (b)Vertical standing-wave trap. (c) Crossed-beam trap. Thewaist w0 and the Rayleigh length zR are indicated.

A. Focused-beam traps

A focused Gaussian laser beam tuned far below theatomic resonance frequency represents the simplest wayto create a dipole trap providing three-dimensional con-finement [see Fig. 5(a)]. The spatial intensity distributionof a focused Gaussian beam (FB) with power P propa-gating along the z-axis is described by

IFB(r, z) =2P

πw2(z)exp

(−2

r2

w2(z)

)(36)

where r denotes the radial coordinate. The 1/e2 radiusw(z) depends on the axial coordinate z via

w(z) = w0

√1 +

(z

zR

)2

(37)

where the minimum radius w0 is called the beam waistand zR = πw2

0/λ denotes the Rayleigh length. From theintensity distribution one can derive the optical potentialU(r, z) ∝ IFB(r, z) using Eq. 10, 12, or 19. The trapdepth U is given by U = |U(r = 0, z = 0)|.

The Rayleigh length zR is larger than the beam waistby a factor of πw0/λ. Therefore the potential in the ra-dial direction is much steeper than in the axial direction.To provide stable trapping one has to ensure that thegravitational force does not exceed the confining dipoleforce. Focused-beam traps are therefore mostly alignedalong the horizontal axis. In this case, the strong radialforce ∼ U/w0 minimizes the perturbing effects of gravity10.

If the thermal energy kBT of an atomic ensemble ismuch smaller than the potential depth U , the extensionof the atomic sample is radially small compared to thebeam waist and axially small compared to the Rayleighrange. In this case, the optical potential can be well ap-proximated by a simple cylindrically symmetric harmonicoscillator

UFB(r, z) ' −U

[1− 2

(r

w0

)2

−(

z

zR

)2]

. (38)

The oscillation frequencies of a trapped atom are givenby ωr = (4U/mw2

0)1/2 in the radial direction, and ωz =

(2U/mz2R)1/2 in the axial direction. According to Eq. 25,

the harmonic potential represents an important specialcase of a power law potential for which thermal equilib-rium properties are discussed in Sec. III.

1. Collisional studies

In their pioneering work on far-off resonance traps(FORT), Miller et al. (1993a) from the group at Univer-sity of Texas in Austin have observed trapping of 85Rbatoms in the focus of a single, linearly polarized Gaus-sian beam with detunings from the D1 resonance of up to65 nm. The laser beam was focused to a waist of 10 µmcreating trap depths in the mK-range for detunings. Be-tween 103 and 104 atoms were accumulated in the trapfrom 106 atoms provided by a vapor-cell MOT. Smalltransfer efficiencies are a general property of traps withtightly focused beams resulting from the small spatialoverlap between the cloud of atoms in the MOT. Typi-cal temperatures in the trap are below 1 mK resulting in

10A new type of trap for the compensation of gravity waspresented by Lemonde et al. (1995). It combines a focused-beam dipole trap providing radial confinement with an inho-mogeneous static electric field along the vertical z-axis induc-ing tight axial confinement through the dc Stark effekt.

15

densities close to 1012 atoms/cm3. Peak photon scatter-ing rates were a few hundreds per second leading to neg-ligible loss rates by photon heating as compared to lossesby background gas collisions. High densities achieved ina tightly-focused beam in combination with long storagetimes offer ideal conditions for the investigation of colli-sions between trapped atoms.

The trap lifetime without cooling illustrated in Fig.6(a) showed an increase of the lifetime by about an orderof magnitude for increasing detunings at rather small de-tunings. At larger detunings, the lifetime was found to bedetermined by the Rb background pressure of the vapor-cell to a value of about 200ms. The shorter lifetimesat smaller detunings were explained in a later publica-tion (Miller et al., 1993b) by losses through photoasso-ciation of excited Rb2 dimers which was induced by thetrapping light. This important discovery has inspired awhole series of experiments on ultracold collisions inves-tigated by the Austin group with photoassociation spec-troscopy in a dipole trap. Instead of using the trappinglight, photoassociation was induced by additional lasersin later experiments. The investigations comprise colli-sional properties and long-range interaction potentials ofground state atoms, shape resonances in cold collisions,and the observation of Feshbach resonances (Cline et al.,1994b; Gardner et al., 1995; Boesten et al., 1996; Tsai etal., 1997, Courteille et al., 1998).

(b)

(a)

FIG. 6. Measurement of trap lifetimes and hyperfine relax-ation times for Rb atoms in a far-detuned focused-beam trap.(a) Trap decay time as a function of the trapping beam wave-length. From Miller et al. (1993a). (b) Time constant τrel

for hyperfine population relaxation versus trapping beam wave-length, in comparison to the mean time τs between two spon-taneous scattering events. From Cline et al. (1994 c©OpticalSociety of America).

2. Spin relaxation

If the detuning of the trapping light field is larger thanfinestructure splitting of the excited state, photon scat-tering occurs almost exclusively into the elastic Rayleighcomponent. Inelastic Raman scattering changing the hy-perfine ground state is reduced by a factor ∼ 1/∆2 ascompared to Rayleigh scattering. This effect was demon-strated in the Austin group by preparing all trapped 85Rbatoms in the lower hyperfine ground state and study-ing the temporal evolution of the higher hyperfine state(Cline et al., 1994a). The relaxation time constant as afunction of detuning is plotted in Fig. 6(b) in compari-son to the calculated average time between two photonscattering events τs = Γsc . For large detunings, the re-laxation time constant τrel is found to exceed τs by twoorders of magnitude. This shows the great potential offar-detuned optical dipole traps for manipulation of in-ternal atomic degrees of freedom over long time intervals.Using the spin state dependence of the dipole potential ina circularly polarized light beam (see Sec. II B 2), Corwinet al. (1997) from University of Colorado at Boulder haveinvestigated far-off resonance dipole traps that selectivelyhold only one spin state. The small spin relaxation ratesin dipole traps may find useful applications for the searchfor beta-decay asymmtries and atomic parity violation.

3. Polarization-gradient cooling

Polarization-gradient cooling in a focused-beam traphas been investigated by a group at the ENS in Paris(Boiron et al., 1998). The trapped atoms were sub-jected to blue-detuned molasses cooling in a near-resonant standing wave (see Sec. III A 1). Previously, thesame group had performed experiments on polarization-gradient cooling of free atomic samples which wereisotropically distributed. A density-dependent heat-ing mechanism was found limiting the achievable finaltemperatures for a given density (Boiron et al., 1996).This heating was attributed to reabsorption of the scat-tered cooling light within the dense cloud of cold atoms.Of particular interest was the question, whether theanisotropic geometry of a focused beam influences heat-ing processes through multiple photon scattering.

Cesium atoms were trapped in the focus (w0 = 45 µm)of a 700mW horizontally propagating Nd:YAG laserbeam at 1064nm. The trap had a depth of 50 µK, and theradial trapping force exceeded gravity by roughly one or-der of magnitude. From a MOT containing 3×107 atoms,2×105 atoms were loaded into the trap 11. Polarization-gradient cooling was applied for some tens of milliseconds

11The Nd:YAG beam was continuously on also during theMOT loading.

16

yielding temperatures between 1 and 3µK, depending onthe cooling parameters. To avoid trap losses by inelas-tic binary collisions involving the upper hyperfine groundstate (see Sec. III C), the cooling scheme was chosen insuch a way that the population of the upper hyperfineground state was kept at a low level.

a) b)

600 mm

400 mm

(b)

arriv

al ti

me

(ms)(a)

FIG. 7. Properties of a horizontal focused-beam trap for Csatoms. (a) Absorption image of the atom distribution. Thetransverse rms radius in the focal plane is 6µm. (b) Mea-surement of the vertical oscillation period (see text). Adaptedfrom Boiron et al. (1998).

An absorption image of the trapped cesium atoms atT = 2 µK is depicted in Fig. 7(a) showing the rod-shapedcloud of atoms. The distribution of atoms had a radialextension σr = 6 µm and a axial size of σz = 300 µm cor-responding to a peak density of ∼ 1 × 1012 atoms/cm3.The picture was taken 30ms after the cooling had beenturned off. The time interval is large compared to the ra-dial oscillation period (ωr/2π = 330Hz), but short com-pared to the axial oscillation period (ωz/2π = 1.8Hz). Inthis transient regime, the axial distribution had not yetreached its thermal equilibrium extension of σz = 950 µmwhich leads to a decrease of the density. For the radialextension, the measured value coincided with the expec-tation for thermal equilibrium as determined by the mea-sured temperature and oscillation frequency (see Eq. 34).

To measure the transverse oscillation frequency, theNd:YAG beam was interrupted for a ∼ 1 ms time interval,during which the cold atoms were accelerated by gravityto a mean velocity about 1 cm/s. After the trapping laserhad been turned on again, the atoms vertically oscillate inthe trap. The oscillation in vertial velocity was detectedby measuring the mean arrival time of the atoms at aprobe laser beam 12 cm below the trap [ordinate in Fig.7(b)] as a function of the trapping time intervals [abscissain Fig. 7(b)]. The measured vertical oscillation frequencyωr = 330 Hz is consistent with the value ωr = 390 Hzderived from the trap depth and the beam waist.

The temperatures measured in the dipole trap wereabout 30 times lower than one would expect on the basisof the heating rates found for a isotropic free-space sam-ple at the corresponding densities (Boiron et al., 1995).No evidence was found for heating of the trapped sample.The reduction of density-dependent heating is a bene-fit from the strongly anisotropic trapping geometry of afocused-beam trap. Due to the much smaller volume to

surface ratio of an atomic cloud in the focused-beam trapas compared to a spherical distribution, photons emittedduring cooling have a higher chance to escape without re-absorption from the trap sample which causes less heat-ing through reabsorption.

4. Bose-Einstein condensates

Optical confinement of Bose-Einstein condensates wasdemonstrated for the first time by a group at MIT inCambridge, USA (Stamper-Kurn et al., 1998a). Bosecondensates represent the ultimately cold state of anatomic sample and are therefore captured by extremelyshallow optical dipole traps. High transfer efficienciescan be reached in very far-detuned traps, and the pho-ton scattering rate acquires negligibly small values (seeEq. 14). Various specific features of dipole traps canfruitfully be applied to the investigation of many aspectsof Bose-Einstein condensation which were not accessibleformerly in magnetic traps.

Sodium atoms were first evaporatively cooled in amagnetic trap to create Bose condensates containing5 − 10 × 106 atoms in the 3S1/2(F = 1, mF = 1) state(Mewes et al., 1996). Subsequently, the atoms wereadiabatically transferred into the dipole trap by slowlyramping up the trapping laser power, and then suddenlyswitching off the magnetic trap. The optical trap wasformed by a laser beam at 985 nm (396 nm detuningfrom resonance) focused to a waist of about 6 µm. Alaser power of 4mW created a trap depth of about 4 µKwhich was sufficient to transfer 85% of the Bose con-densed atoms into the dipole trap and to provide tightconfinement. Peak densities up to 3 × 1015 atoms/cm3

were reported representing unprecendented high valuesfor optically trapped atomic samples.

Condensates were observed in the dipole trap evenwithout initially having a condensate in the magnetictrap. This strange effect could be explained by an adi-abatic increase of the local phase-space density throughchanges in the potential shape (Pinkse et al., 1997). Theslow increase of the trapping laser intensity during load-ing leads to a deformation of the trapping potential cre-ated by the combination of magnetic and laser fields. Thetrapping volume of the magnetic trap was much largerthan the volume of the dipole trap. Therefore, phase-space density was increased during deformation, whileentropy remained constant through collisional equilibra-tion. Using the adiabatic deformation of the trappingpotential, a 50-fold increase of the phase-space densitywas observed in a later experiment (Stamper-Kurn et al,1998b).

The lifetime of atoms in the dipole trap is shown in Fig.8(a) in comparison to the results obtained in a magnetictrap. In the case of tight confinement and high densities(triangles in Fig. 8), atoms quickly escape through inelas-tic intra-trap collisions. Long lifetimes (circles in Fig. 8)

17

were achieved in the dipole trap and the magnetic trap,respectively, when the trap depth was low enough for col-lisionally heated atoms to escape the trap. Trap loss wasfound to be dominated by three-body decay with no iden-tifiable contributions from two-body dipolar relaxation.The lifetime measurements delivered the three-body lossrate constant γ = 1.1(3) × 10−31 cm6/s (see Eq. 35) forcollisions among condensed sodium atoms.

Fig. 8(b) demonstrates simultaneous confinement of aBose condensate in different Zeeman substates mF =0,±1 of the F = 1 ground state. To populate thesubstates, the atoms were exposed to an rf field sweep(Mewes et al., 1997). The distribution over the Zeemanstates was analyzed through Stern-Gerlach separation bypulsing on a magnetic field gradient of a few G/cm afterturning off the dipole trap. It was verified that all F = 1substates were stored stably for several seconds.

mF = 1

mF = -1

mF = 0

105

106

107

Num

ber

of A

tom

s

403020100Time (s)

(b)(a)

FIG. 8. Bose-Einstein condensates of Na atoms in fo-cused-beam traps. (a) Trapping lifetime in optical traps andmagnetic traps. The number of condensed atoms versus trap-ping time is shown. Closed symbols represent the data foroptical traps with best transfer efficiency (triangles) and slow-est decay (circles). Open circles represent data for a magnetictrap with optimized lifetime. The lines are fits based on sin-gle-particle losses and three-body decay. (b) Optical trappingof condensates in all hyperfine spin states of the F = 1 groundstate. An absorption image after 340ms of optical confine-ment and subsequent release from the trap is shown. Hyper-fine states were separated by a pulsed magnetic field gradientduring time of flight. The field view of the image is 1.6 by1.8mm. Adapted from Stamper-Kurn et al. (1998a).

After the first demonstration of optical trapping, aspectacular series of experiments with Bose-Einstein con-densates in a dipole trap was performed by the MITgroup. By adiabatically changing the phase-space den-sity in the combined magnetic and optical dipole trap(see above), Stamper-Kurn et al. (1998b) were able toreversibly cross the transition to BEC. Using this tech-nique, the temporal formation of Bose-Einstein conden-sates could extensively be studied (Miesner et al, 1998a).The possibility of freely manipulating the spin of trappedatoms without affecting the trapping potential has led tothe observation of Feshbach resonances in ultracold elas-tic collisions (Inouye et al., 1998) and the investigation ofspin domains and metastable states in spinor condensates

(Stenger et al., 1998, Miesner et al., 1998b).

5. Quasi-electrostatic traps

The very interesting case of quasi-electrostatic dipoletrapping has so far not been considered in this review.When the frequency of the trapping light is much smallerthan the resonance frequency of the first excited stateω � ω0, the light field can be regarded as a quasi-staticelectric field polarizing the atom. Quasi-electrostatictraps (QUEST) were first proposed (Takekoshi et al.,1995) and realized (Takekoshi and Knize, 1996) by agroup at University of Southern California in Los An-geles. In the quasi-electrostatic approximation ω � ω0,one can write the dipole potential as

Udip(r) = −αstatI(r)2ε0c

(39)

with αstat denoting the static polarizability (ω = 0). Thelight-shift potential of the excited states is also attractivecontrary to far-off resonant interaction discussed before.Atoms can therefore be trapped in all internal states bythe same light field. Since the trap depth in Eq. 39 doesnot depend on the detuning from a specific resonance lineas in the case of a FORT, different atomic species or evenmolecules may be trapped in the same trapping volume.

For the ground state of alkali atoms, Eq. 39 is well ap-proximated by applying the quasi-static approximationto Eq. 10 which gives

Udip(r) = −3πc2

ω30

Γω0

I(r) . (40)

Compared to a FORT at a detuning ∆, the potentialdepth for ground state atoms in a QUEST is smaller bya factor 2∆/ω0. Therefore, high power lasers in the far-infrared spectral range have to be employed to create suf-ficiently deep traps. The CO2 laser at 10.6µm which iscommercially available with cw powers up to some kilo-watts is particulary well suited for the realization of aQUEST (Takekoshi et al., 1995).

An important feature of the QUEST is the practicalabsence of photon scattering. The relation between thephoton scattering rate and the trap potential can be de-rived from Eqs. 10 and 11 in the quasi-electrostatic ap-proximation:

hΓsc(r) = 2(

ω

ω0

)3 Γω0

Udip(r). (41)

When compared to the corresponding relation for aFORT given by Eq. 14, the dramatic decrease of thephoton scattering rate in a QUEST becomes obvious.Typical scattering rates are below 10−3s−1 showing thatthe QUEST represents an ideal realization of a purelyconservative trap.

18

Takekoshi and Knize (1996) have realized trapping ofcesium atoms in a QUEST by focussing a 20W CO2 laserto a waist of 100 µm resulting in a trap depth of 115 µK.Around 106 atoms prepared in the F = 3 state wereloaded into the trap from a standard MOT. The atomloss rates of ∼ 1 s−1 were consistent with pure lossesthrough background gas collisions. Hyperfine relaxationtimes were found to exceed 10 s.

B. Standing-wave traps

A standing wave (SW) trap provides extremely tightconfinement in axial dimension as can be seen from Fig.5(b). The trap can be realized by simply retroreflect-ing the beam while conserving the curvature of the wavefronts and the polarization. Assuming small extensionsof the atomic cloud, one can write the potential in theform

USW (r) ' −U cos2 (kz)

[1− 2

(r

w0

)2

−(

z

zR

)2]

(42)

with the standing wave oriented along the z-axis. Thepotential depth is four times as large than the corre-sponding trap depth for a single focused beam discussedin Sec. IVA. As for a single focused beam, radial con-finement is provided by a restoring force ∼ U/w0. Theaxial trapping potential is spatially modulated with a pe-riod of λ/2. Atoms are strongly confined in the antinodesof the standing wave (restoring force ∼ Uk) resulting ina regular one-dimensional lattice of pancake-like atomicsubensembles. When aligned vertically, the axial confine-ment greatly exceeds the gravitational force mg. One cantherefore use rather shallow trap, just sufficiently deep totrap a pre-cooled ensemble, which results in small photonscattering rates (see Eq. 14) and large loading efficiencies.

The tight confinement along the axial direction leadsto large oscillation frequencies ωz = hk(2U/m)1/2 at thecentre of the trap. The oscillation frequency decreaseswhen moving along the z-axis due to the decreasing lightintensity. At low temperatures, the energy of the ax-ial zero-point motion 1

2 hωz in the centre of the trap canbecome of the same order of magnitude as the thermalenergy 1

2kBT even for moderate trap depths 12. In thisregime, the axial atomic motion can no longer be de-scribed classically but has to be quantized, and the vi-brational ground state of the axial motion is substan-tially populated. The axial spread of the wavepacket ismuch smaller than the wavelength of an optical tran-sition (Lamb-Dicke regime) giving rise to spectral line-narrowing phenomena. One might even enter a regime

12Using the recoil temperature Trec for the cooling transitionat the wavelength λ0 introduced in Sec. III A 1, one can writethe zero-point energy as (λ0/λ)(kBTrecU/2)1/2.

where the wavepacket extension comes close to the s-wavescattering length leading to dramatic changes in the col-lisional properties of the trapped gas.

1. Optical cooling to high phase-space densities

Well-resolved vibrational levels and Lamb-Dicke nar-rowing, as realized along the axis of a standing-wavedipole trap, are necessary requirements for the appli-cation of optical sideband cooling as explained in Sec.III A 1. By employing degenerate-sideband Raman cool-ing, high phase-space densities of an ensemble contain-ing large particle numbers have been achieved by Vuleticet al. (1998) from a group in Stanford. In a verticalfar-detuned standing wave, peak phase- space densitiesaround 1/180 have been obtained with 107 cesium atoms,corresponding to a mean temperature of 2.8 µK and apeak spatial density of 1.4× 1013 atoms/cm3.

The trap was generated by a Nd:YAG laser with 17Wsingle-mode power at λ = 1064nm. The large beamwaist of 260 µm created a trap depth of 160 µK whichresulted in high loading efficiencies (≈ 30% from a blue-detuned molasses released from a MOT) and small pho-ton scattering rates (≈ 2 s−1). Atoms oscillated atωr/2π = 120Hz in the radial (horizontal) direction, andat ωz/2π = 130kHz in the axial (vertical) direction. Adramatic dependence of the trap lifetime on the hyperfinestate of the cesium atoms was observed similar to Fig. 4in Sec. III C. Around 1×107 atoms were contained in thetrap populating 4700 vertical potential wells. Degener-ate Raman sideband cooling was applied between vibra-tional states of a pair of Zeeman shifted magnetic sub-levels in the lowest hyperfine ground state as explainedin Sec. III A 1. Suppressed collisional losses through in-elastic binary collisions at high densities are greatly sup-pressed since population in the upper hyperfine state iskept extremely low in this cooling scheme. The Ramancoupling was provided by the lattice field itself (Deutschand Jessen, 1998).

FIG. 9. Resolved-sideband cooling of Cs atoms in a verti-cal standing-wave trap. Evolution of vertical (solid squares)and horizontal (open squares) temperatures. Cooling is ap-plied only along the tightly confining vertical axis, the horizon-tal degrees of freedom are indirectly cooled through collisionalthermalization. From Vuletic et al. (1998).

19

Axial and radial temperatures evolved differently dur-ing cooling as indicated in Fig. 9. The axial direction wasdirectly cooled causing the axial temperature to quicklydrop to Tz = 2.5 µK. The radial temperature followed bycollisional thermalization with a time constant of 150ms.After sideband cooling was turned off, the temperatureincreased at a rate of 4 µK/s limiting the achievable fi-nal temperatures. This heating rate was much largerthan the rate estimated on the basis of photon scatter-ing which might indicate additional heating sources suchas laser noise. The achieved high thermalization rates incombination with large particle numbers would provideexcellent starting conditions for subsequent evaporativecooling.

A different approach to optical cooling of large parti-cle numbers to high phase-space densities was followedby a group at Berkeley (Winoto et al., 1998), who haverecently applied polarization-gradient cooling to cesiumatoms in a vertical standing-wave trap. The standingwave was linearly polarized resulting in equal light shiftsfor all magnetic sublevels of the atomic ground state(see Sec. II B 2). Therefore, polarization-gradient cool-ing, which relies on optical pumping between the ground-state sublevels (see Sec. III A 1), could be applied to thetrapped sample in a very efficient way. A phase-spacedensity around 10−4 with the large number of 108 atomswas reached.

2. Quantum interference

The regular arrangement of atoms in a standing wave-dipole trap has amazing consequences when a Bose con-densate is loaded into such a trap. Macroscopic interfer-ence of Bose condensed 87Rb atoms tunneling from anextremely shallow 1D lattice under the influence of grav-ity has recently been observed by Anderson and Kasevich(1998) at Yale University. The dipole potential was cre-ated by a vertical standing wave at 850nm (detuning of65 nm from the D1 line) with a waist of 80 µm, an or-der of magnitude larger than the transverse radius of thecondensate. The condensate of 104 atoms was coherentlydistributed among≈ 30 very shallow potential wells. Thewells supported only one bound energy band below thepotential edge which is equivalent to U ' kBTrec whereTrec is the recoil temperature (see Eq. 22) at the wave-length of the trapping field.

The gravitational field induces an offset between ad-jacent wells. For weak potential gradients, the externalfield can be treated as a perturbation th the band struc-ture associated with the lattice. In this limit, wavepack-ets remain confined in a single band. The external fielddrives coherent oscillations at the Bloch frequency as hasbeen demonstrated with ultracold atoms confined in anaccelerated far-detuned 1D standing wave by Ben Dahanet al. (1996) at ENS in Paris and by Wilkinson et al.(1996) at University of Texas in Austin.

The shallow potential wells allow for tunneling of par-ticles into unbound continuum states. In the Yale ex-periment, the lifetime of the atoms confined in the lat-tice was purely determined by the tunneling losses. Fora lattice of depth U = 1.1kBTrec, the observed lifetimewas ∼ 50ms. Each lattice site can be seen as a pointemitter of a deBroglie wave. Interference between thedifferent emitter ouputs lead to the formation of atompulses falling out of the standing wave, quite similar tothe output of a mode-locked pulsed laser. The pulse rep-etition frequency ωrep = mgλ/2h was determined by thegravitational increment between two adjacent walls. Therepetition frequency can be interpreted as the differencein chemical potential divided by h. This indicates theclose relation of the observed effect of coherent atomicdeBroglie waves to the ac Josephson effect resulting fromquantum interference of two superconducting reservoirs.

3. Spin manipulation

Zielonkowski et al. (1998b) from the MPI fur Kern-physik in Heidelberg have used a vertical standing wavetrap for the manipulation of spin-polarized atoms. Byusing a large-volume, shallow red-detuned standing wavetrap, depolarizing effects of photon scattering and atomicinteractions could be kept at a low level while allowingfor large numbers of stored atoms.

Cesium atoms were trapped in a retroreflected 220-mW beam at a wavelength of λ = 859nm (detuning of6.1 nm above the D2 line). The waist of the beam in theinteraction region was 0.50mm. The maximum poten-tial depth amounted to only 17 µK which was sufficientlydeep to directly load atoms from a MOT at sub-Dopplertemperatures. Despite the shallow potential depth, theconfining force exceeded the gravitational force by aboutthree orders of magnitude. The transfer efficiency fromthe MOT into the shallow trap was about 14% resultingin ∼ 105 trapped atoms at a liftime of τ = 1.9 s as shownby the stars in Fig. 10(a).

In a first experiment, the spin state mF = 0 was se-lected by a Stern-Gerlach (SG) force. The force wascreated by horizontally shifting the zero of the MOTquadrupole field with respect to the position of the dipoletrap (see Fig. 10(b)). In this way, only atoms withmF = 0 are trapped by the dipole trap, all other mag-netic substates are pulled out of the trap by the magneticdipole force. The depolarizing effect of the trap light wasdetermined by measuring the lifetime τSG of the mF = 0atoms (see circles in Fig. 10(a)) and comparing this valuewith the trap lifetime without Stern-Gerlach selection.The depolarization rate Γdepol = 1/τSG − 1/τ = 0.9 s−1

allowed a determination of the photon scattering ratefrom calculated values of the mF -state branching ratiosfor spontaneous scattering. The resulting scattering rateagrees well with the expectation based on Eq. 21.

20

+4 +3 +2

-1

+1

m = 0

(a)

F

(b)

FIG. 10. Spin manipulation of Cs in a vertical stand-ing-wave trap. (a) Storage time with all ground-state sublevelspopulated (stars), and for the Stern-Gerlach selected mF = 0state (circles). Shown is the trapped particle number N rela-tive to the number of atoms trapped in the MOT before trans-fer N0. The line represents an exponential fit to the datayielding a time constant of 1.9 s for the unpolarized sampleand 0.7 s for the mF = 0 polarized atoms. (b) Trap poten-tials for the combined magnetic quadrupole and optical trapused for Stern-Gerlach selection of the mF = 0 state fromthe F = 4 hyperfine ground state. The zero-point of the mag-netic quadrupole has been horizontally shifted with respect tothe centre of the standing-wave trap. Atoms with mF 6= 0 areexpelled from the dipole trap by the magnetic field gradient.Adapted from Zielonkowski et al. (1998b).

In a second experiment, spin precession in a ficitiousmagnetic field (Cohen-Tannoudji and Dupont-Roc, 1972,Zielonkowski et al., 1998a) was demonstrated. The fieldwas induced by an additional off-resonant circularly po-larized laser beam which induced a light shift scaling lin-ear with mF as discussed in Sec. II B 2. The result-ing splitting corresponds to a fictitious magnetic field of50mG. The mF = 0 state was selected by a short Stern-Gerlach pulse resulting in a macroscopic magnetizationof the sample. The population of the same state wasanalyzed after a 150ms delay. Between preparation andanalysis, the atoms interacted with a pulse of the ficti-tious field laser in combination with a holding magneticfield. The mF = 0 population oscillates with the dura-tion of the laser pulses which can be directly interpretedas the Larmor precession of the spin in the superpositionof fictitous and holding magnetic field. A 2π and a 4πrotation of the magnetization were observed.

4. Quasi-electrostatic lattices

A standing wave created by the light at 10.6 µm froma CO2 laser creates a QUEST (see Sec. IVA5) with aspacing between the axial potential wells which is largecompared to the transition wavelength of the trappedatoms. In a group at the MPI fur Quantenoptik inGarching, such a lattice of mesoscopic potential wellswas recently realized with rubidium atoms (Friebel et al.,1998a; 1998b). A 5 W CO2 laser beam was focused to awaist of 50µm creating a trap depth of about 360 µK.Up to 3 × 105 atoms could be loaded into the horizon-tal standing wave trap which had a lifetime of 1.8 s lim-ited by background gas collisions. Temperatures around

10µK were achieved by polarization gradient cooling inthe trap.

The vibrational frequencies of atoms inside the poten-tial wells were measured by modulating the laser intensityin order to drive parametric excitation of the atomic oscil-lations. When the modulation frequency equals twice thevibrational frequency 2ωosc (or subharmonics 2ωosc/n),atoms are heated out of the trap leading to a reduced life-time. This effect was demonstrated by varying the mod-ulation frequency and measuring the number of atoms

0

20

40

fluo

resc

en

ce [

arb

.u.]

ωz

ωz

ωr

2ωr

IIII

dipole trap

10 100 1000 10000

0

20

40

fluo

resc

en

ce [

arb

.u.]

2ωz

ωz

2ωr

I I I

modulation frequency [Hz]

lattice

0

40

modulation frequency [Hz]100

0

modulation frequency [Hz]

fluor

esce

nce

[arb

.u.]

10000

40

20

10 1000 10000

20

10 100 1000

FIG. 11. Excitation spectra of Rb atoms trapped with a CO2

laser at 10.6µm creating quasi-electrostatic traps. The spec-trum of a standing-wave trap (left graph) is compared to thespectrum from a single-beam trap (right graph). The horizon-tal axis is the fluorescence of atoms which are illuminated af-ter 0.6 s trapping time. The vertical axis gives the modulationfrequency of the trap light intensity. Parametric resonances atthe oscillation frequencies and twice their value are observed.The jump at 100 Hz in both graphs is an artefact resultingfrom a change of the modulation depth and modulation time.Adapted from Friebel et al. (1998a).

that were left in the trap after a fixed trapping time of600ms. The remaining atoms were detected by switch-ing on a resonant light field and recording the fluores-cence. The left graph in Fig. 11 shows the excitationspectrum for the standing-wave trap. Parametric res-onances at 2 kHz and at 32 kHz can be identified whichare attributed to excitations of the radial and axial vibra-tions, respectively. A subharmonic resonance at 16 kHzis also observed. In the left graph of Fig. 11, the ex-citation spectrum of a single focused-beam dipole trapis presented. The trap was realized by interrupting theretroreflected laser beam which had formed the standingwave. The radial resonance shifts to 1.6 kHz (subhar-monic at 0.8 kHz) because of the four times lower trapdepth. The reduction of the axial vibrational frequencyis more dramatic since it scales as 1/(2kzR) relative tothe standing-wave trap (including the factor 4 in trapdepth). The axial resonance is now found at 80Hz (sub-harmonic at 40Hz).

21

C. Crossed-beam traps

A single focused beam creates a highly anisotropic trapwith relatively weak confinement along the propagationaxis and tight confinement in the perpendicular direc-tion [see Figs. 5(a) and 7(a)]. In a standing-wave trap,the anisotropic atomic distribution of the single beamtrap is split into anisotropic subensembles with extremelytight confinement along the axial direction. Crossing twobeams with orthogonal polarization and equal waist un-der an angle of about 90◦ as indicated in Fig. 5(c) rep-resents an obvious way to create nearly isotropic atomicensembles with tight confinement in all dimensions. Inthis case, the dipole potential for small extensions of theatomic cloud can be approximated as

UCB(x, y, z) ' −U

(1− x2 + y2 + 2z2

w20

). (43)

It should be noted that the effective potential depth isonly U/2 as atoms with larger energy leave the steep trapalong one of the beams.

1. Evaporative cooling

Crossed-beam dipole traps provide a good compromisebetween decent trapping volumes and tight confinement,and are therefore suited for the application of evaporativecooling as explained in Sec. III A 1. Adams et al. (1995)at Stanford used a crossed-beam configuration orientedin the horizontal plane for evaporative cooling of sodium.A single-mode Nd:YAG laser generated two beams of 4Weach, focused to a waist of 15 µm and crossing under 90◦.The trap depth was close to 1mK. The polarizations ofthe two beams were chosen orthogonal which results in aspin-independent trapping potential for the ground statesbecause of the large detuning of the 1064nm light fromthe two fine-structure lines of sodium (see Table I).

After transfer from a MOT, evaporative cooling wasstarted with ∼ 5000 atoms at a temperature of 140 µKand a peak density of 4 × 1012 atoms/cm3 as indicatedby the dotted lines in Fig. 12. To force evaporation ofhigh-energetic particles, the Nd:YAG power was expo-nentially ramped down from 8W (trap depth ∼ 900 µK)to 0.4W (trap depth ∼ 45 µK) within 2 s . After evapo-ration, ∼ 500 atoms were left in the trap. Temperaturewas reduced by a factor of 35 to 4 µK. Yet, density de-creased by about an order of magnitude since the restor-ing force in the trap is reduced when ramping down theintensity. To keep the density at a high value for effi-cient evaporation, one would have to further compressthe cloud. However, compared to the initial conditions,phase-space density was increased by a factor of 28 in theexperiment indicating the potential of evaporative cool-ing for the enhancement of phase-space density in dipoletraps.

FIG. 12. Evaporative cooling of Na atoms in acrossed-dipole trap. (a) Atom density distribution before (dot-ted line) and after (solid line) evaporative cooling of Na in acrossed dipole trap. The density decreased by a factor of about7. (b) Time-of-flight measurement of the temperature before(dotted line) and after (solid line) evaporative cooling. Thetemperature decreased from 140 µK to 4µK. From Adams etal. (1995).

2. Interference effects

Let us now consider the more general case of two beamsof equal waist w0 propagating in the xy-plane and inter-secting at the focal points under an arbitrary angle. Theangle between the beams and the x-axis is ±φ. This con-figuration includes as special cases the single focussed-beam trap (φ = 0), discussed in Sec. IVA, and thestanding wave trap (φ = 90◦), discussed Sec. IVB. Inthe harmonic approximation, the trapping potential canbe written as

UCB(x, y, z) ' −U(y)(

1− 2x2

w2x

− 2y2

w2y

− 2z2

w20

)(44)

The potential radiiwx,y are given by w2

x =(cos2 φ/w2

0 + sin2 φ/2z2R

)−1and

w2y =

(sin2 φ/w2

0 + cos2 φ/2z2R

)−1. The y-dependence of

the trap depth U reflects interference effects between thetwo waves which lead to a modulation of the trap depthon the scale of an optical wavelength.

If both beams are linearly polarized along the y-direction (lin‖lin), interference results in a pure inten-sity modulation with period D = λ/(2 sin φ) yieldingU(y) = Umax cos2(πy/D). Trapped atoms are thereforebound in the antinodes of the interference pattern form-ing a one-dimensional lattice along the y-direction withlattice constant D. In the case of orthogonal polariza-tion of the two beams (lin⊥lin), the intensity exhibits nointerference effects, but the polarization is spatially mod-ulated between linear and circular with the same periodD. As discussed in Sec. II B 2, the light shift depends onthe spin state of the atoms giving rise to a spatial modu-lation of the potential. For detunings ∆ large comparedto the fine-structure splitting ∆′

FS, the potential depth is

22

modulated with a relative amplitude 13gF mF ∆′

FS/∆ (seeEq. 20).

Making use of these interference effects, the ENS grouphas investigated Raman cooling (Kuhn et al., 1996) andsideband cooling (Perrin et al., 1998; Bouchoule et al.,1998) in a crossed-beam trap for cesium atoms. The trapconsisted of two Nd:YAG laser beams (λ = 1064 nm)propagating along a vertical xy-plane with a power ofabout 5W in each beam. The beams crossed at theircommon waists (w0 ≈ 100 µm) under an angle φ = ±53◦with the horizontal x-axis. The hyperfine splitting ofground and excited state were small compared to thedetuning of the laser from the D1 and D2 lines of cesiumat 894nm and 852nm, respectively. For the lin⊥lin case,the comparatively large fine-structure splitting of cesium(∆′

FS/∆ ≈ 5) lead to spatially modulated potentials witha small, yet significant modulation amplitude of ≈ U/15for the stretched Zeeman states |mF | = I + 1/2.

The trap was loaded from ∼ 107 cesium atoms in aMOT. The loading efficiency and the shape of the atomiccloud strongly depended on the laser polarizations of thedipole trap (Kuhn et al., 1996). The potential wellsformed by interference of the laser beams were used forresolved-sideband cooling with Raman transitions be-tween the F = 3 and F = 4 hyperfine ground states.Great differences in the performance of sideband cool-ing were found for the lin‖lin and the lin⊥lin case dueto the different character of the potential wells beingweakly or strongly modulated, respectively (Perrin etal. 1998). By optimizing the sideband cooling in thelin‖lin configuration, single vibrational states (motionalFock states) could be prepared in the 1D standing wave.Atoms were first sideband-cooled into the lowest vibra-tional state |nosc = 0〉 from where they could be trans-ferred into other pure |nosc〉 states by Raman transitionsat multiples of the resolved vibrational sidebands (Bou-choule et al., 1998).

D. Lattices

When adding more laser beams, one can design awhole variety of interference patterns to create two- andthree-dimensional lattices confining the atoms in microp-otentials of submicron extension (Deutsch and Jessen,1998). When combined with efficient cooling methods,significant population of the vibrational ground state canbe achieved. Many important aspects of these opticallattices have extensively been studied for near-resonanttrapping fields which simultaneously provide tight con-finement and dissipation (Jessen and Deutsch, 1996;Hemmerich et al., 1996; Grynberg and Triche, 1996).

In this chapter, we have so far concentrated on red-detuned traps because of the conceptual differences inthe practical realization of dipole traps as compared toblue-detuned traps discussed in the next chapter. In thecase of three-dimensional far-detuned lattices, this dis-

tinction becomes faint since both lattice types are real-ized through appropriate interference patterns of multi-ple beams. The atoms are trapped either in the antinodes(red detuning) or nodes (blue detuning) of the interfer-ence pattern. The main difference for red and blue detun-ing lies in the photon scattering rates as discussed in Sec.III A 3. Here, we shortly present novel developments forboth types of far-off resonance lattices recently realizedby several groups (Anderson et al., 1996; Muller-Seydlitzet al., 1997; Hamann et al., 1998; Boiron et al., 1998;DePue et al., 1998). Localized wavepackets oscillating inconservative microtraps are promising systems in whichto study fundamental questions related to quantum-statepreparation, coherent control and decoherence of macro-scopic superposition states. Furthermore, optical latticescan serve as prototype systems for the study of condensedmatter models based on periodic arrangements of weaklyinteracting particles.

Anderson et al. (1996) have confined lithium atoms inthree-dimensional optical lattices formed in the intersec-tion of four laser beams. For lithium, the fine-structuresplitting is only 10GHz leading to ground-state opticalpotentials which are independent of the light polariza-tion and the atomic spin state. A face-centered cubiclattice with a nearest-neighbor spacing of 1.13λ was real-ized by a four-beam configuration. Three-dimensionallattices with periodicity much larger than λ could becreated by reducing the angles subtended by each pos-sible pair of the four lattice beams 13. Up to 105 atomswere trapped in the lattice. By reducing the intensity ofthe trapping light so that only the coldest atoms froma MOT were confined in the lattice, a trapped ensem-ble with an rms velocity spread corresponding to 1.8 µKwas prepared. By adiabatically expanding deep potentialwells (see Sec. III A 1), cooling at the expense of smallerspatial density could be achieved.

The temporal evolution of metastable argon atomsstored in the intensity nodes of a blue-detuned opticallattice was investigated by Muller-Seydlitz et al. (1997)at Konstanz University. The lattice of simple-cubic sym-metry was formed by three mutually orthogonal stand-ing waves with othogonal linear polarizations resultingin isotropic potential wells with a lattice constant ofλ/2 = 397 nm. About 104 atoms are initially capturedin the lattice. Fig. 13 shows time-of-flight spectra forvariable trapping times. After a certain storage time,

13A different approach to create structures with large peri-odicities has been used by Boiron et al. (1998) by using in-terference from multiple beams emerging from a holographicphase grating.

23

inte

nsity

(su

mm

ed o

ver

250

cycl

es)

50

100

150

200

250

400

450τ[ms]

40 ms

light intensity

storage time τ

time10 ms

FIG. 13. Time-of-flight spectra of metastable argon atomstrapped in the nodes of a three-dimensional far-off resonanceoptical lattice. The spectra were taken after various storagetimes τ . Atoms were released from the trap by slowly ramp-ing down the trapping light intensity. Higher-energetic boundstates arrive first, deeper confined states later. After longstorage times, only atoms in the lowest bound state are found.From Muller-Seydlitz et al. (1998).

the light intensity is ramped down releasing one boundstate after the other. For increasing storage times, thepopulation of excited bands in the potential wells de-creases faster than the population of the vibrationalground-state band leaving about 50 atoms populatingthe motional ground state after storage times of about450ms. Two processes could be identified for this stateselective loss of particles: Firstly, atoms in higher ex-cited bands have a higher probability of leaving the fi-nite extension of the trapping field (w0 = 0.55 mm) bytunneling. Secondly, atoms interacting with the far-offresonance trapping light (detuning 2 nm from resonancefor the metastables) are optically pumped into the elec-tronic ground state of argon and are therefore lost fromthe trap. The probability for an optical pumping processdepends on the spatial overlap between the optical latticefield and the atomic wavepacket confined to the potentialwell. The smaller the spatial extension of the wavepack-ets, the smaller is the excitation rate being smallest forthe motional ground state. The reduced photon scatter-ing probability of deeper bound states is an importantspecific property of a blue-detuned optical lattice.

FIG. 14. Resolved-sideband Raman cooling of Cs atoms ina two-dimensional far-off resonance lattice. Shown is the in-verse Boltzmann factor 1/qB as a function of the applied mag-netic field. When the magnetic field tunes the lattice-field in-duced Raman coupling to the first-order (at Bz ≈ 0.12 G)) orthe second-order (at Bz ≈ 0.24 G)) motional sidebands, Ra-man cooling is most efficient resulting in the peaks of 1/qB .Solid circles are data points, the solid line is a fit to thesum of two Lorentzians. Inset: Corresponding kinetic tem-peratures measured to determine the Boltzmann factor. Thedashed lines indicates the kinetic temperature of the vibra-tional ground state. From Hamann et al. (1998).

The first demonstration of resolved-sideband Ramancooling in a dipole trap explained in Sec. III A 1 wasperformed with cesium in a two-dimensional lattice bya group at University of Arizona in Tucson (Hamann etal., 1998). The 2D lattice consisted of three coplanarlaser beams with polarizations in the lattice plane. Amagnetic field perpendicular to the lattice plane was ap-plied to Zeeman-shift the motional states |mF = 4; nosc〉and |mF = 3; nosc − 1〉 of the upper hyperfine groundstate F = 4 into degeneracy (see Sec. III A 1). By addinga small polarization component orthogonal to the latticeplane, Raman coupling between magnetic sublevels with∆mF = ±1 was introduced. In Fig. 14, the inverse Boltz-mann factor 1/qB = exp(hωosc/kBT ) is plotted versusthe magnetic field. The inverse Boltzmann factor reachesa maximum when the two motional states are shifted intodegeneracy by the magnetic field, so that sideband cool-ing by the lattice field becomes effective. The second,smaller peak shows well-resolved cooling on the second-order Raman sideband (∆nosc = −2). As shown by theinset in Fig. 14, the thermal energy of the atoms closelyapproaches the zero-point energy in the potential wellsindicating a population > 95% of the vibrational groundstate.

Unity occupation of sites in a three-dimensional far-offresonance optical lattice was very recently realized by theBerkeley group (DePue et al., 1998). In the regime ofunity occupation, interactions between highly localizedatoms have dramatic effects, and studies of collisionalproperties of tightly bound wavepackets become possible.The necessary high densities were achieved by applyingpolarization gradient cooling to the 3D lattice filled with108 cesium atoms, and subsequent adiabatic toggling be-

24

tween the 3D lattice and a 1D standing wave. The 3Dlattice was formed by three mutual orthogonal standingwaves with controlled time-phase differences. After theatoms were cooled in the lattice, the horizontal beams ofthe lattice are adiabatically turned off leaving the atomsin a vertical standing wave trap. Due to their low temper-ature (700 nK), the atoms were essentially at rest at theirrespecive positions. Under the action of the transversetrapping potential, the atoms radially collapsed towardsthe trap centre. All atoms arrived simultaneously atthe centre after about a quarter radial oscillation period.Thereby, the density in the trap centre was transientlyenhanced by a factor of ten reaching 6×1012 atoms/cm3.At the moment of peak density, the horizontal latticebeams were adiabatically turned on again. A substantialfraction of lattice sites was then multiply occupied andunderwent fast inelastic collisions. After multiply occu-pied sites had decayed, 44% of the lattice site were oc-cupied by a single atom cooled close near its vibrationalground state.

V. BLUE-DETUNED DIPOLE TRAPS

Laser light acts repulsively on the atoms when its fre-quency is higher than the transition frequency (“blue”detuning). The basic idea of a blue-detuned dipole trapis thus to surround a spatial region with repulsive laserlight. Such a trap offers the great advantage of atom stor-age in a “dark” place with low influence of the trappinglight, which minimizes unwanted effects like photon scat-tering, light shifts of the atomic levels, and light-assistedcollisional losses. According to the discussion followingEq. 31, this advantage becomes substantial in the case ofhard repulsive optical walls (κ � 1 in Eq. 31) or largepotential depth for tight confinement (U � kBT ).

Experimentally, it is not quite as simple and straight-forward to realize a blue-detuned trap as it is in thered-detuned case, where already a single tightly focusedlaser beam constitutes an interesting dipole trap. There-fore, the development of appropriate methods to producethe required repulsive “optical walls” has played a cen-tral role in experiments with blue-detuned traps. Threemain methods have been applied for this purpose: Lightsheets, produced by strong elliptical focusing of a laserbeam, can be used as nearly flat optical walls (Davidsonet al., 1995; Lee et al., 1996). Hollow laser beams canprovide spatial confinement in at least two dimensions(Yang et al., 1986). Evanescent waves, formed by totalinternal reflection on the surface of a dielectric medium,represent nearly ideal mirrors to reflect atoms (Cook andHill, 1982; Dowling and Gea-Banacloche, 1996). In mostblue-detuned traps, gravity is used to close the confin-ing potential from above. Such traps are referred to asgravito-optical traps. As a further experimental possibil-ity, which we have already discussed in Sec. IVD, atomscan be trapped in the micropotentials of far blue-detuned

optical lattices (Muller-Seydlitz et al., 1997).In this Chapter, we discuss various blue-detuned traps

that have been realized experimentally and their partic-ular features; an overview is given in Table II. In the fol-lowing, these traps are classified according to the mainmethod applied for producing the optical walls: Light-sheet traps (Sec. VA), hollow-beam traps (Sec. VB), andevanescent-wave traps (Sec. V C).

A. Light-sheet traps

In experiments performed at Stanford University,Davidson et al. (1995) and Lee et al. (1996, 1998) haverealized light-sheet traps of various configurations andapplied them for rf spectroscopy on trapped atoms andfor optical cooling to high phase-space densities. Thelight sheets were derived from the two strongest lines ofan all-line argon-ion laser. A laser power of up to 10W at514nm and up to 6W at 488nm was focused with cylin-drical lenses to cross sections of typically 15 µm× 1 mm.For the Na atoms used in the experiments (resonance lineat 589 nm), this leads to maximum light-sheet potentialsin the order of 100µK.

When two light sheets are combined and overlap inspace, there are two ways to avoid perturbing interfer-ence effects, which could open escape channels in the op-tical potential. First, if the two light sheets have differ-ent frequencies then the relevant potential is determinedby the time average over the rapid beat node, in whichthe interference averages out. Second, if the light sheetshave same frequencies but orthogonal polarizations theninterference leads to a spatial modulation of the polariza-tion. In the case of large detunings greatly exceeding thefine-structure splitting, as it was very well fulfilled in theStanford experiments, the polarization modulation hasnegligible effect on the dipole potential (see Eq. 20).

In the first experiment (Davidson et al., 1995), twohorizontally propagating light sheets were combined toform a vertical “V” cross section. This configuration al-ready provides 3D confinement, as the trapping potentialis closed along the propagation direction due to the di-vergence of the tightly focused light, see Fig. 15(a). Ver-tically, the atoms are kept in the trap by gravity. Thegravito-optical trap thus has the form of a boat with alength of about 2mm (for trap parameters see Table II).

Using this light-sheet trap, Davidson et al. have storedabout 3000 Na atoms and impressively demonstrated theadvantages of blue-detuned dipole traps for spectroscopicapplications. By using the method of separated oscilla-tory fields, they have measured Ramsey fringes of theF = 1, mF = 0 → F = 2, mF = 0 hyperfine transition ofthe Na ground state. For the excitation of this transitionan rf travelling wave with a frequency of 1.77GHz wasused. The Ramsey fringes were measured by applyingtwo π/2 rf pulses separated by a time delay of up to 4 s.Initially, all trapped atoms were optically pumped into

25

the lower hyperfine state (F = 1). After applying of thetwo rf pulses, the number of atoms transfered into theupper state (F = 2) was measured by applying a shortpulse of light resonant with the cycling F = 2 → F ′ = 3transition and detecting the induced fluorescence. Thetwo central Ramsey fringes observed by varying the rffrequency for a pulse delay of 4 s are shown in Fig. 15(b).By analyzing the dependence of the fringe contrast on thedelay between the two rf pulses, a 1/e coherence decaytime of 4.4 s was obtained.

(b)(a)

FIG. 15. Light-sheet trap used for rf spectroscopy. (a)Laser intensity produced in a horizontal plane 30 µm abovethe intersection of the two focused sheets of light. The x, ydimensions are in microns, and the intensity is normalized tothe peak laser intensity. (b) The central Ramsey fringes ob-served for rf-induced hyperfine transitions for a measurementtime of 4 s. From Davidson et al. (1995).

In the same experiments, the Stanford group has alsomeasured the mean residual light shift (ac Stark shift)of the hyperfine transition frequency as caused by thetrapping light. From the frequency of the central Ramseyfringe, a corresponding shift of 270mHz was observed.The absolute light shifts of the two hyperfine sublevels arelarger by the ratio of the optical detuning (∼ 90THz) tothe hyperfine splitting (1.77GHz) and thus amounted to∼ 14 kHz. This number directly gives the average dipolepotential Udip ≈ h × 14 kHz≈ kB × 0.7 µK experiencedby the atom and also allows one to determine an averagephoton scattering rate of 0.01 s−1 according to equation14.

The authors also mention similar experiments per-formed in a red-detuned dipole trap realized with aNd:YAG laser. In this case, the longest observed co-herence times were ∼300 times lower, which highlightsthe advantage of the blue-detuned geometry for in-trapspectroscopy.

In a later experiment (Lee et al., 1995), the light-sheettrapping was improved by a new trap geometry in theform of an inverted pyramid. As illustrated in Fig. 16(a),this trap was produced by four sheets of light. Due tothe much larger trapping volume provided by the pyra-midal geometry this trap could be loaded with 4.5× 105

atoms, which constitutes an improvement over the previ-ous configuration by more than a factor of 100 (see alsoTable II). Lee et al. have also tested a similar, tetrahe-

dral box trap, the performance of which was inferior tothe inverted pyramid.

(a) (b)

FIG. 16. Inverted-pyramid trap used for Raman cooling.(a) Schematic of the trap geometry, and (b) lifetime measure-ment performed after Raman cooling at a background pressureof ∼ 3× 10−11 mbar. From Lee et al. (1996).

The particular motivation of the experiment by Leeet al. was to obtain high phase-space densities of thetrapped atomic ensemble by optical cooling. The authorstherefore applied 1D Raman cooling (Kasevich and Chu,1992) as a sub-recoil cooling method, see also Sec. III A 1.The geometry of the inverted-pyramid trap is very advan-tageous for getting dense atomic samples, because of astrong spatial compression with decreasing temperature.In an ideal inverted pyramid the density n would scaleas T−3 in contrast to a 3D harmonic oscillator, wheren ∝ T−3/2. Moreover, this trap configuration providesvery fast motional coupling between the different degreesof freedom, which is particularly interesting in case of a1D cooling scheme.

About 4.5×105 atoms were loaded at a temperature of7.7µK and a peak density of 2×1010 cm−3. By applyinga sequence of Raman cooling pulses during the follow-ing 180ms, the temperature of the atoms was reducedto 1µK and, keeping practically all atoms in the trap, apeak density of 4×1011 cm−3 was reached. This densityincrease by a factor of 20 resulting from a temperature re-duction by a factor of about 7 indicates that the trap wasalready in a regime where the confining potential behavesrather harmonically than like in an ideal pyramid. Thismay be explained by the finite transverse decay length ofthe light-sheets potential walls. As a result of the Ramancooling, the atomic phase-space density was increased bya factor of 320 over the initial one and reached a valuewhich was about a factor of 400 from Bose-Einstein con-densation.

After an initial loss of atoms observed in the first sec-ond after the Raman cooling process, an exponential de-cay of the trapped atom number was measured with atime constant of 7 s, see Fig. 16(b). This loss did notsignificantly depend on the background pressure below10−10 mbar, which points to the presence of an additionalsingle-particle loss mechanism. This loss was consistent

26

with an observed heating process in the trap that 30times exceeded the calculated heating by photon scat-tering. This heating of unknown origin was identified asthe main obstacle to implement evaporative cooling inthe inverted pyramid trap. The experiments moreovershowed evidence that ground-state hyperfine-changingcollisions, ejecting atoms out of the trap, were limitingthe maximum density achievable with Raman cooling to∼1012 cm−3.

In a later experiment, Lee and Chu (1998) used theinverted-pyramid trap for preparing a Raman-cooledsample with spin polarization in any of the three mag-netic sub-levels of the lower hyperfine ground state (F =1). A small bias magnetic field was used to lift the degen-eracy of the ground state and appropriate polarizationwas chosen for the Raman cooling light. The attainedtemperature and phase-space density was similar to theunpolarized case. This experiment can be seen as a niceillustration of the general advantage of dipole trappingto leave the full ground-state manifold available for ex-periments.

B. Hollow-beam traps

Hollow blue-detuned laser beams, which provide ra-dial confinement, are particularly interesting and versa-tile tools for the construction of dipole traps. Hollowlaser beams may be divided into two classes: Beamsin pure higher-order Laguerre-Gaussian (LG) modes andother hollow beams which cannot be represented as sin-gle eigenmodes of an optical resonator. The main differ-ence is that LG beams preserve their transverse profilewith propagation (only converging or diverging in width),which is of particular interest for atom guiding (Dholakia,1998; Schiffer et al., 1998). Other, non-LG hollow beamscan change their profile substantially, thus offering addi-tional features of interest for atom trapping.

A Laguerre-Gaussian mode LGp l is characterized by aradial index p and an azimuthal index l. A hollow beamwith a “doughnut” profile is obtained for p = 0 and l 6= 0with an intensity distribution given by

I0 l(r) = P2l+1r2l

πl! w2(l+1)0

exp(−2r2/w20) ; (45)

here P is the power and w0 is the waist of the beam.For l = 0 this equation gives the transverse profile of ausual Gaussian beam (TEM00 mode) as described by Eq.36. The higher the mode index l, the larger is the ratiobetween the beam radius and the width of the ring, i.e.the harder is the repulsive optical wall radially confiningthe atoms and the weaker is heating by photon scattering.

1. Plugged doughnut-beam trap

The dougnut-beam trap shown in Fig. 17(a) was re-cently realized by Kuga et al. (1997) at the Universityof Tokyo; see also Table II. The LG03 doughnut beam(w0=0.6mm) was derived from a laser which was forcedto oscillate in a Hermite-Gaussian HG03 mode by inser-tion of a thin wire into the cavity. An astigmatic modeconvertor (Beijersbergen et al., 1993) then transformedthe beam into the LG03 mode. As a LG beam does notprovide axial confinement, the hollow beam was pluggedby two additional laser beams (diameter 0.7mm), whichwere separated by 2 mm and perpendicularly intersectedthe hollow beam. The plugging beams were derived fromthe recycled doughnut beam.

An exponential decay of the stored atom number wasobserved with a time constant of about 150ms [see Fig.17(b)], which was explained by heating out of the trapby photon scattering. The authors also observed thattrapping in a LG01 mode showed inferior performancewith very short lifetimes of a few milliseconds only, whichhighlights the benefit of “hard” optical walls for reducingheating by photon scattering.

0 50 100 150 200 250 300 350

107

108

3D trap 2D trap

num

ber

of a

tom

s

time (ms)

(a) (b)

500 100 150 200 250 300 350

time (ms)

num

ber

of a

tom

s

108

107

FIG. 17. (a) Plugged doughnut beam trap, and (b) measure-ment of the storage time (filled squares). The filled circles re-fer to a 2D trap formed by the doughnut beam alone withoutplugging beams (no axial confinement). Adapted from Kugaet al. (1997).

In subsequent experiments (Torii et al., 1998), the stor-age time of the trap was improved by application ofa pulsed optical molasses cooling scheme to a value of1.5 s, which then was dominated by collisions with thebackground gas. This non-continuous molasses coolingscheme allows one to cool the atoms down to nearly thesame temperatures as in a continuous cooling scheme,but suppresses trap losses by light-assisted collisions andthe interference of light shifts from the molasses lightwith the trapping potential. Moreover, potentially severelosses by hyperfine-changing collisions (see Sec. III C) canbe strongly reduced by keeping the atoms in the lower hy-perfine ground states in the off-times of the molasses. Insuch a pulsed cooling scheme, the off-time can be as longas heating by photon scattering (see Sec. III A 2) doesnot significantly degrade the trap performance. The on-time can be as short as the typical cooling time in themolasses.

27

2. Single-beam traps

A blue-detuned trap based on a single laser beam wasrecently demonstrated by Ozeri et al. (1998) at the Weiz-man Institute in Israel; see also Table II. The trappingbeam was produced in an experimentally very simple wayby passing a single Gaussian beam through a phase plateof appropriate size, which shifted the center of the beamby a phase angle of π. In the focus of such a beam, de-structive interference leads to a reduced or even vanishinglight intensity. By choosing the proper ratio between thediameter of the phase plate and the laser beam diameter,Ozeri et al. obtained a darkness ratio (central intensitynormalized to maximum intensity) of 1/750. Close to thecenter of this single-beam trap, the intensity increasesradially with the fourth power of the distance (like inthe case of a LG02 beam) and axially the dependence isquadratic.

In order to measure the average light intensity experi-enced by the atoms in the trap, Ozeri et al. have studiedthe relaxation of hyperfine population caused by photonscattering from the trapping light in a similar way as doneby Cline et al. (1994) in a red-detuned far-off-resonancetrap; see also Sec. IVA2. From measurements performedat a detuning of 0.5 nm it was concluded that the averageintensity experienced by a trapped atom was as low as∼1/700 of the maximum intensity. In another series ofmeasurements they observed that, at constant tempera-ture, the photon scattering rate scaled linearly with theinverse detuning. This observed behavior represents anice confirmation of Eq. 29, which for a power-law po-tential directly relates the average scattering rate to thetemperature with an inverse proportionality to the de-tuning.

In a recent proposal, Zemanek and Foot (1998) consid-ered a blue-detuned dipole trap formed by two counter-propagating laser beams of equal central intensities, butdifferent diameters. Along the axis of such a standing-wave configuration, completely destructive interferencewould lead to minima of the dipole potential with zerointensity. These traps would be radially closed becauseof the incomplete destructive interference at off-axis po-sitions. This resulting linear array of three-dimensionaldipole traps could combine the interesting features ofstanding-wave trapping schemes (see Sec. IVB) with theadvantages of blue detuning. Experimentally, it seemsstraightforward to realize such a trap by retroreflectionand appropriate attenuation of a single slightly converg-ing laser beam.

3. Conical atom trap

A single-beam gravito-optical trap was recentlydemonstrated by Ovchinnikov et al. (1998) at the MPIfur Kernphysik in Heidelberg. The “conical atom trap”(CAT), illustrated in Fig. 18(a), is based on a conical hol-

low beam and combines experimental simplicity with sev-eral features of interest for the trapping of a large numberof atoms at high densities: high loading efficiency, tightconfinement, low collisional losses, and efficient cooling.

In the experiment (parameters see Table II) the up-ward directed conical trapping beam was generated byusing an arrangement of two axicons and one sphericallens. In the focal plane, the beam profile was roughlyGaussian with a diameter of about 100µm. Within afew millimeters distance from the focus, the beam evolvedinto a ring-shaped profile with a dark central region re-sembling a higher-order Laguerre-Gaussian mode. Theopening angle of the conical beam was about 150mrad.

The trap was operated relatively close to resonance ascompared to the other blue-detuned traps discussed sofar. With an optical detuning of 3 GHz (12 GHz) withrespect to the lower (upper) hyperfine ground state ofCs a large potential depth was realized, which togetherwith the funnel-like geometry facilitated the transfer of asmuch as 80% of all atoms from the MOT into the CAT.The relatively small detuning requires efficient coolingfor removing the heat resulting from photon scatteringfrom the trapping light. For this purpose, an optical mo-lasses was applied continuously to cool atoms in the up-per hyperfine ground state. This cooling, however, takesplace with an inherently reduced duty cycle and is thussimilar to the pulsed molasses cooling scheme of Torii etal. (1998), which was discussed before. In the molassesscheme applied in the CAT, the short phases of coolingin the upper hyperfine ground state (typically 50µs) areself-terminating by optical pumping into the lower hyper-fine level. There the atoms stay for a much longer time(typically >∼ 1ms) until they are repumped by the lightof the intense conical trapping beam.

(a)

3 m

m

(b)

molassesbeams

lens

axicon

axicon

coldatoms

FIG. 18. (a) Illustration of the conical atom trap (CAT).(b) Fluorescence image of atoms in the CAT (lower, elongatedblob) combined with an image of atoms in the MOT (upperspot). The dashed lines indicate the conical trapping field.Adapted from Ovchinnikov et al. (1998).

This inherent duty cycle for the molasses provides cool-

28

ing phases which are long enough to efficiently removeheat, while the average population of the upper hyper-fine state is kept very low. This efficiently suppressestrap loss due to hyperfine-changing collisions (see Sec.III C). At an atomic number density of ∼ 1011 cm−3 life-time measurements of the trapped atoms (1/e lifetimeof 7.8 s due to background gas collisions) showed no sig-nificant loss due to ultracold collisions, which indirectlyconfirmed the predominant population of the lower hy-perfine ground state.

The CAT was also operated in a pure “reflection cool-ing” mode without any molasses cooling, similar to thesituation that was theoretially considered by Morsch andMeacher (1998). Reflection cooling (Ovchinnikov et al.,1995a and b) is based on the inelastic reflection of anatom from a blue-detuned light field, as discussed in moredetail in the following section in context with evanescentwaves. In a Sisyphus-like process the atom is pumpedfrom the strongly repulsive lower to the weakly repulsiveupper hyperfine state. A closed cooling cycle requiresa weak repumping beam to bring the atom back to thelower state. Such a beam was applied in the CAT experi-ment from above. The detuning of the conical beam wasincreased to a few ten GHz to optimize reflection cooling.Because of the lower potential depth the loading was lessefficient (∼10% transfer from the MOT instead of 80%reached before), but stable background-gas limited trap-ping was achieved. Without the repumping beam, atomswere rapidly heated out of the trap. These observationsclearly demonstrated reflection cooling in a blue-detunedtrap made of free-propagating light fields. However, op-timum reflection cooling requires very steep optical wallsso that only evanescent waves are suited to reach tem-peratures close to the recoil limit by this mechanism, aswill be discussed in Sec. V C3.

C. Evanescent-wave traps

A hard repulsive optical wall with nearly ideal proper-ties can be realized by a blue-detuned evanescent wave(EW), produced by total internal reflection of a laserbeam from a dielectric-vacuum interface. In the vacuumthe EW intensity falls off exponentially within a typicaldistance of λ/2π from the surface and thus provides avery large gradient. The use of an EW as a mirror forneutral atoms was suggested by Cook and Hill (1982),who also proposed to trap atoms in a box realized withEW mirrors. The first experimental demonstration ofan atom mirror was made by Balykin et al. (1987) bygrazing-incidence reflection of a thermal atomic beam.A few years later, Kasevich et al. (1990) observed re-flection of cold atoms at normal incidence. Since thenmany experiments have been conducted with EW atommirrors. An extensive review on EW atom mirrors andrelated trapping schemes has already been given by Dowl-ing and Gea-Banacloche (1996). We thus concentrate on

the essential issues and some interesting, new develop-ments.

1. Evanescent-wave atom mirror

The exponential shape of the repulsive optical poten-tial of a far-detuned EW leads to simple expressions forthe basic properties of such an atom mirror. The EWintensity as a function of the distance z from the surfaceis given by

I(z) = I0 exp(−2z/Λ) , (46)

where the 1/e2 decay length Λ = λ/(2π√

n2 sin2 θ − 1)depends on the angle of incidence θ and the refractiveindex n of the dielectric. The maximum EW intensityI0 is related to the incident light intensity I1 by I0/I1 =4n cos2 θ/(n2 − 1) .

The repulsive dipole potential of an atom mirror is in-dependent of the magnetic substate if the detuning islarge compared to the excited-state hyperfine splittingand the EW is linearly polarized (see Sec. 2); the latteris obtained for an incoming linear polarization perpen-dicular to the plane of incidence (TE polarization). Inthe interesting case of low saturation, the EW dipole po-tential can then be calculated according to Eq. 19.

A very important quantity to characterize the EW re-flection process is its probability to take place coherently,i.e. without spontaneous photon scattering. The prob-ability for an (in)coherent reflection can be calculatedby integrating the intensity-dependent scattering rate Γsc

(Eq. 21) over the classical trajectory of the atom in therepulsive potential. For large enough laser detuning (stillclose to one of the D lines), the resulting small probabil-ity psp � 1 for an incoherent reflection process is givenby

psp =mΛh

Γ∆

v⊥ , (47)

where v⊥ is the velocity component perpendicular to thesurface.

The light shift of the ground-state sublevel integratedover time in a single reflection process corresponds to aphase shift

Φls =mΛh

v⊥ (48)

experienced by the atom. The simple relation psp =(Γ/∆)Φls is a result of the fundamental connection be-tween the absorptive and dispersive effect of the interac-tion with the light field (see also Eq 14). As an interestingconsequence of the exponential shape of the EW poten-tial, both equations Eq. 47 and Eq. 48 do not dependon the intensity of the light field, as long as the poten-tial barrier is high enough. For higher/lower intensities

29

the reflection just takes place at larger/smaller distancesfrom the surface.

Very close to the surface (z <∼ 100 nm), the van-der-Waals attraction becomes significant. Its main effect isto reduce the maximum potential barrier provided by theEW mirror (Landragin et al., 1996b). For small kineticenergies, the reflected atoms do not penetrate deeply intothe EW and thus do not feel the surface attraction. Wehave thus neglected the effect of the van-der-Waals forcein Eqs. 47 and 48, and we will do so in the following.

2. Gravito-optical cavities

A great deal of interest in EW atom mirrors has beenstimulated by the intriguing prospect to build resonatorsand cavities for atomic de-Broglie waves (Balykin andLetokhov, 1989; Wallis et al., 1992). The simplest wayto realize such a scheme is a single atom mirror on whichthe atoms classically bounce like on a trampoline. Sucha trapping scheme is referred to as gravito-optical cavity.

The time tb between two bounces in a gravito-opticalcavity is related to the maximum height h and the maxi-mum velocity v⊥ of the atoms by tb = 2

√2h/g = 2v⊥/g,

where g is the gravitational acceleration. Using Eq. 47for the probability for photon scattering per bounce, theaverage photon scattering rate can be expressed as

Γsc =psp

tb=

mgΛ2h

Γ∆

. (49)

Analogously using Eq. 48, the mean light shift δωls =Udip/h experienced by the bouncing atom is obtained as

δωls =Φls

tb=

mgΛ2h

. (50)

It is a remarkable consequence of the exponential shapeof the EW potential that Γsc and δωls do not depend onthe energy of the atom. For an atom with less energy ina gravito-optical cavity, the decrease in scattering proba-bility and light shift is exactly compensated by the higherbounce rate.

The average scattering rate Γsc can be interpreted asthe decoherence rate of gravito-optical resonator due tophoton scattering. It also determines the heating poweraccording to Eq. 23. The mean light shift is of interestfor possible spectroscopic applications of gravito-opticalcavities.

The eigenenergies of the vertical modes in a gravito-optical cavity can be approximately calculated by ide-alizing the EW potential as a hard wall (Wallis et al.,1992). In this case the vertical potential has the shapeof a wedge and the energy of the n-th vertical mode isgiven by

En = hωv

(n− 1

4

)2/3

, (51)

where ωv = (9π2mg2/8h)1/3 is a characteristic frequency.For example, for cesium atoms ωv/2π=2080 Hz, whichcorresponds to a temperature of hωv/kB ' 95 nK. Con-sequently, the population of a single vertical mode withmany atoms, a challenging issue for future experiments,requires cooling below this temperature.

As an important step for EW trapping, the ENS groupin Paris observed the bouncing of atoms in a stablegravito-optical cavity (Aminoff et al., 1993). The atommirror used in the experiment (diameter ∼ 1.5mm) wasproduced on a concave spherical substrate (radius of cur-vature 2 cm) to obtain additional transverse confinementof the atomic motion (Wallis et al., 1992); the resultingtransverse trap depth was about 5µK.

In the set-up sketched in Fig.19(a), about 107 coldatoms were dropped onto the mirror from a MOT lo-cated at a height of 3mm above the prism (correspond-ing bounce period tb = 50 ms). The bouncing atomswere detected by measuring the fluorescence in a reso-nant probe beam, which was applied to the atoms after avariable time delay. The experimental results displayedin Fig. 19(b) show up to ten resolved bounces, after whichthe number of atoms dropped below the detection limit.The measured loss per bounce of ∼ 40% was attributedto photon scattering during the EW reflection process(∼ 5% according to Eq. 47), photon scattering from straylight from the mirror (∼ 10%), collisions with the back-ground gas (∼ 10%), and another, not identified sourceof loss (∼ 20%). An explanation for the latter may be thediffusive reflection from an EW mirror, as observed laterby Landragin et al. (1996a). The ENS gravito-opticalcavity represents the first atom trap realized with evanes-cent waves.

104

105

106

107

5004003002001000

Nu

mb

er

of

ato

ms

time in ms

1

2

3

4

5

6

7

8

9

10

Ti: Saphire

Laser Beam

Probe

beam

PhotodiodeTrapping

Beams

(b)(a)

num

ber

of a

tom

s

time (ms)

FIG. 19. Observation of atoms bouncing in a grav-ito-optical cavity. (a) Experimental set-up, and (b) numberof atoms detected in the probe beam for different times aftertheir release (points). The solid curve is the result of a corre-sponding Monte-Carlo simulation. Adapted from Aminoff etal. (1993).

3. Gravito-optical surface trap

A new step for gravito-optical EW traps was the intro-duction of a dissipative mechanism, following suggestions

30

by Ovchinnikov et al. (1995a) and Soding et al. (1995)to cool atoms by inelastic reflections. In a correspond-ing experiment at the MPI fur Kernphysik in Heidel-berg, Ovchinnikov et al. (1997) have realized the gravito-optical surface trap (GOST) schematically shown in Fig.20(a); see also Table II. This trap facilitates storage andefficient cooling of a dense atomic gas closely above anEW atom mirror.

A flat atom mirror was used in the GOST and hori-zontal confinement was achieved by a hollow, cylindricallaser beam, far blue-detuned from the atomic resonance.This beam with a ring-shaped transverse intensity profileproviding very steep optical walls was generated using anaxicon (Manek et al., 1998). A Laguerre-Gaussian beamof similar performance would require extremly high or-der (about LG0, 100). The optical potentials of the GOSTthus come very close to ideal hard walls, which leads to astrong reduction of photon scattering from the trappinglight even relatively close to resonance.

���������������

���������������

(b)

���������������������������������������������������������������������������������������������������������������������������������������

���������������������������������������������������������������������������������������������������������������������������������������

λ z

vacuumdielectric

repumping

beam

atoms in MOT

atoms in GOST

beam

prism

hollow beam

EW laser

(a)

FIG. 20. (a) Schematic of the gravito-optical surface trap(GOST). (b) Illustration of the evanescent-wave cooling cycle.The dot indicates an atom approaching the dielectric surfacein the lower hyperfine state, scatters an EW photon, leavesthe EW in the upper, less repulsive ground state, and is finallypumped back into the lower state.

Cooling by inelastic EW reflections (evanescent-waveSisyphus cooling) is the key to stable trapping in theGOST. The basic mechanism, which was experimentallystudied before in grazing-incidence reflection of an atomicbeam (Ovchinnikov et al. 1995b; Laryushin et al., 1996)and normal-incidence reflection of cold atoms (Desbiolleset al., 1996), is based on the splitting of the 2S1/2 groundstate of an alkali atom into two hyperfine sublevels. Inthe case of linear EW polarization, the atom can be mod-eled as a three-level scheme (Soding et al., 1995; see alsoSec. II B 2) with two ground states separated by ∆HFS

and one excited state, for which the hyperfine splittingcan be neglected.

An inelastic reflection takes place when the atom en-ters the EW in the lower ground state and, by scat-tering an EW photon during the reflection process, ispumped into the less repulsive upper state; see Fig. 20(b).The cooling cycle is then closed by pumping the atomback into the lower hyperfine state with a weak reso-

nant repumping laser (coming from above in Fig. 20(a)).The mean energy loss ∆E⊥ from the motion perpen-dicular to the surface per inelastic reflection is given by∆E⊥/E⊥ = − 2

3∆HFS/(∆ + ∆HFS), where E⊥ = mv2⊥/2

is the kinetic energy of the incoming atom and ∆ isthe relevant detuning with respect to the lower hyper-fine state. With the probability psp for an incoherentreflection according to Eq. 47, the branching ratio q forspontaneous scattering into the upper ground state, andthe bounce rate t−1

b = g/2v⊥ EW Sisyphus cooling canbe characterized by a simple cooling rate

β =q

3∆HFS

∆ + ∆HFS

mgΛh

Γ∆

. (52)

The vertical motion is damped exponentially because theaverage photon scattering rate is independent of the ki-netic energy (see Eq. 49). The final attainable tempera-ture is recoil-limited to a value of ∼ 10 Trec, similar to apolarization-gradient optical molasses (see Sec. III A 1).

The GOST was experimentally realized for Cs atoms,the high mass of which is very favorable for gravito-optical trapping: In the gravitational potential, the re-coil temperature Trec corresponds to a height hrec =kBTrec/(mg), which for the heavy Cs atoms is partic-ularly low (hrec = 1.3 µm, see also Table I). As a conse-quence, Cs atoms cooled close to the recoil limit can beaccumulated very close to the dielectric surface.

In the experiment, trapped atoms were observed forstorage times up to 25 s, corresponding to more than10.000 (unresolved) bounces. The observed exponentialloss with a 1/e-lifetime of 6 s was completely consistentwith collisions with the background gas. By time-of-flight diagnostics, a temperature of 3.0µK was measured,which is quite close to the theoretical cooling limit. Inthermal equilibrium, the number density as function ofthe distance from the surface follows from Eq. 32 (in thiscase, equivalent to the barometric equation), which forthe measured temperature of 3.0µK corresponds to anexponential decay with a 1/e height as low as 19µm.

FIG. 21. Cooling dynamics in the GOST. The vertical(◦) and horizontal (•) temperatures measured for about 105

trapped Cs atoms are plotted as a function of the storagetime. The solid lines are theoretical fits based on Eq. 52 andthe assumption of an EW reflection with a small diffusive,non-specular component. From Ovchinnikov et al. (1997).

31

The cooling dynamics observed in the experiments isshown in Fig. 21. The initial horizontal temperature isdetermined by the temperature of the MOT, whereasthe much higher initial vertical temperature results fromthe release of the atoms at a height of ∼ 800µm. Withincreasing storage time, the vertical temperature fol-lows a nearly exponential decay with a time constant of400 ms, in good agreement with the cooling rate accord-ing to Eq. 52 (with ∆/2π = 1GHz, ∆HFS/2π = 9.2GHz,q = 0.25). The temperature of the horizontal motion,which is not cooled directly, follows the vertical one witha clearly visible time lag and approaches the same finalvalue of about 3µK. This apparent motional coupling canbe fully explained by a small diffusive component of thereflection from the EW atom mirror, as observed beforeby Landragin et al. (1996a).

A very important feature of the GOST and theevanescent-wave cooling mechanism is the predominantpopulation of the absolute internal atomic ground state:The lower hyperfine level is populated by more than99.99% of the atoms. A unique feature of EW coolingis that the trapping and cooling light does not penetratethe atomic sample. Moreover, a possible reabsorptionof scattered trap photons is strongly reduced becauseof the large surface area for photons to escape. Due tothese facts trap loss by ultracold collisions (see Sec. III C)and other density-limiting mechanisms are strongly sup-pressed as compared to a MOT. One can therefore expectEW Sisyphus cooling to work very well up to number den-sities of 1013 cm−3, or even higher (Soding et al., 1995).In the first GOST experiments, this interesting regimewas out of reach because the trap could be loaded withonly 105 atoms, leading to peak densities of 2×1010 cm−3.In present experiments with the GOST, being performedwith a substantially improved loading scheme, the high-density regime of evanescent-wave cooling is being ex-plored.

The GOST offers several interesting options for futureexperiments on dense atomic gases (Engler et al., 1998).By detuning the EW and the hollow beam very far fromresonance, a situation can be reached in which the photonscattering rate is far below 1 s−1. For a sufficiently densegas, one can then expect very good starting conditionsfor evaporative cooling, which may be implemented byramping down the EW potential. This appears to bea promising route to quantum-degeneracy of Cs, whichbecause of anomalously fast dipolar relaxation seems notattainable in a magnetic trap (Soding et al., 1998; Guery-Odelin et al., 1998).

Other interesting applications of the GOST, arisingfrom its particular geometry, are related to the possi-ble formation of a two-dimensional quantum gas. Thevertical motion is much more strongly confined than thehorizontal one, so that the corresponding level spacingsof the quantized atomic motion differ by nearly six or-ders of magnitude. In such a highly anisotropic situ-ation, Bose-Einstein condensation is predicted to occurin two distinct steps (van Druten and Ketterle, 1997).

First, the vertical motion condenses into its ground state(Wallis, 1996). Then, in a second step occuring at muchlower temperatures, the system condenses to its abso-lute ground state. The highly anisotropic nature of thetrapping potential may be further enhanced by additionof a second, attractive evanescent-wave (Ovchinnikov etal., 1991). This can create a wavelength-sized poten-tial well close to the surface, which could be efficientlyloaded by elastic collisions. The situation would then re-semble the situation of atomic hydrogen trapped on liq-uid helium, for which evidence of a Kosterlitz-Thoulessphase transition was reported very recently (Safonov etal., 1998). The prospect to realize such a system withalkali atoms14 is of particular interest to study effects ofquantum-degeneracy in a 2D system.

VI. CONCLUDING REMARKS

In this review, we have discussed the basic physicsof dipole trapping in far-detuned light, the typical ex-perimental techniques and procedures, and the differentpresently available trap types along with their specificfeatures. In the discussed experiments, optical dipoletraps have already shown their great potential for a va-riety of different applications.

The particular advantages of dipole trapping can besummarized in the following three main points:

• The ground-state trapping potential can be de-signed to be either independent of the particularsub-level, or dependent in a well-defined way. Inthe first case, the internal ground-state dynamicsunder the influence of additional fields behaves inthe same way as in the case of a free atom.

• Photon scattering from the trap light can take placeon an extremely long time scale exceeding manyseconds. The trap then comes close to the idealcase of a conservative, non-dissipative trapping po-tential, allowing for long coherence times of the in-ternal and external dynamics of the stored atoms.

• Light fields allow one to realize a great variety ofdifferent trap geometries, e.g., highly anisotropictraps, multi-well potentials, mesoscopic and mi-croscopic traps, and potentials for low-dimensionalsystems.

Regarding these features, the main research lines for fu-ture experiments in optical dipole trapping may be seenin the following fields:

14Other experimental approaches to two-dimensional sys-tems based on dipole trapping make use of the particularproperties of the optical transition structure in metastablenoble gas atoms (Schneble et al., 1998; Gauck et al., 1998).

32

Ultracold collisions and quantum gases. The behav-ior of ultracold, potentially quantum-degenerate atomicgases is governed by their specific collisional properties,which strongly depend on the particular species, the in-ternal states of the colliding atoms, and possible externalfields. In this respect, dipole traps offer unique possibili-ties as they allow to store atoms in any sub-state or com-bination of sub-states for the study of collisional prop-erties and collective behavior. Experiments along theselines have already been performed (Gardner et al., 1995;Tsai et al., 1997; Stenger et al., 1998; Miesner et al.,1998b), but it seems that this has just opened the doorto many other studies involving other atomic species, oreven mixtures of different species (Engler et al., 1998).

Of particular importance is the trapping of atomsin the absolute internal ground state, which cannot betrapped magnetically. In this state, inelastic binary col-lision are completely suppressed for energetical reasons.In this respect, an ultracold cesium gas represents a par-ticularly interesting situation, as Bose-Einstein conden-sation seems only attainable for the absolute ground state(Soding et al., 1998; Guery-Odelin et al., 1998). As a con-sequence, an optical trap may be the only way to realizea quantum-degenerate gas of Cs atoms.

Tuning of scattering properties by external fields is an-other very interesting subject. In this respect, Feshbachresonances at particular values of magnetic fields play avery important role. In optical traps one is completelyfree to choose any magnetic field without changing thetrap itself. Using this advantage, Feshbach resonanceshave been observed for sodium and rubidium (Inouye etal., 1998; Courteille et al., 1998), and future experimen-tal work will certainly explore corresponding collisionalproperties of other atomic species.

Highly anisotropic dipole traps offer a unique envi-ronment for the realization of low-dimensional quantumgases. In this respect, interesting trapping configura-tions are standing-wave traps (Sec. IVB), optical lattices(Sec. IVD), evanescent-wave surface traps (Sec. VC),and combinations of such trapping fields (Gauck et al.,1998). New phenomena could be related to a step-wiseBose-Einstein condensation (van Druten and Ketterle,1997) and modifications of scattering properties in cases,in which the atomic motion is restricted to a spatial scaleon the order of the s-wave scattering length.

Another fascinating issue would be to study collisionalproperties of ultracold fermions (e.g., the alkali atoms6Li and 40K) with the challenging goal to produce aquantum-degenerate Fermi gas. Direct optical coolingin a standing-wave dipole trap by degenerate sidebandcooling, similar to the method of Vuletic et al. (1998),seems to be a particularly promising route.

Spin physics and magnetic resonance. As dipole trapsallow for confinement with negligible effect on the atomicground-state spin, experiments related to the coherentground-state dynamics in external fields can be per-formed in essentially the same way as in the case offree atoms. In such experiments, the trap would provide

much longer observation times as attainable in atomicbeams or vapor cells. As a further advantage, trappedatoms stay at the same place which keeps inhomogenetiesof external fields very low. First demonstrations of alongthis line are the experiments by Davidson et al. (1995)and Zielonkowski et al. (1998b), as discussed in Sec.VA and IVB 3, respectively. In principle, a dipole trapconstitutes an appropriate environment to perform anykind of magnetic resonance experiment with opticallyand magnetically manipulated ground-state spins (Suter,1997). This could be of interest, e.g., for storing andprocessing quantum information in the spin degrees offreedom.

A possible, very fundamental application would be themeasurement of a permanent electric dipole moment of aheavy paramagnetic atom like cesium (Khriplovich andLamoreaux, 1997). Such an experiment, testing time-reversal symmetry, could greatly benefit from extremelylong spin lifetimes and coherence times of spin polariza-tion, which seem to be attainable in dipole traps.

Trapping of other atomic species and molecules. Op-tical dipole traps do neither rely on a resonant interac-tion with laser light nor on spontaneous photon scatter-ing. Therefore any polarizable particle can be trapped inpowerful, sufficiently far-detuned light. For this purpose,the quasi-electrostatic trapping with far-infrared lasersources, like CO2 lasers (see Secs. IVA5 and IVB4),appears to be very attractive. The trapping mecha-nism could be applied to many other atomic species ormolecules, which are not accessible to direct laser cooling.The problem is not the trapping itself, but the coolingto the very low temperatures required for trap loading.There may be several ways to overcome this problem.

A possible way to load a dipole trap could be basedon buffer-gas loading of atoms or molecules into mag-netic traps (Kim et al., 1997; Weinstein et al., 1998)and subsequent evaporative cooling. The effectiveness ofcryogenic loading and subsequent evaporative cooling isdemonstrated by the recent attainment of Bose-Einsteincondensation of atomic hydrogen confined in a magnetictrap (Fried et al., 1998), without any optical cooling in-volved. Similar strategies based on buffer-gas loadingcould open ways for filling optical dipole traps with manymore atomic species than presently available or even withmolecules.

Another possible way could be the production of ultra-cold molecules by photoassociation of laser-cooled atoms(Fioretti et al., 1998). The translational energies of thesemolecules can be low enough for loading into far-infrareddipole traps. A very recent experiment by Takekoshi etal. (1998) indeed reports evidence of dipole trapping of afew Cs2 molecules in a CO2-laser beam, which may justbe the beginning of a new class of experiments in dipoletrapping.

Optical dipole traps can be seen as storage devices atthe low end of the presently explorable energy scale. Weare convinced that future experiments exploiting the par-

33

ticular advantages of these traps will reveal interestingnew phenomena and show many surprises.

ACKNOWLEDGMENTS

We would like to thank Hans Engler, Markus Hammes,Moritz Nill, Ulf Moslener, M. Zielonkowski, and in partic-ular Inka Manek from the Heidelberg cooling and trap-ping group for many useful discussions and assistancein preparing the manuscript. Further useful discussionsare acknowledged with Steven Chu, Lev Khaykovich,Takahiro Kuga, Heun Jin Lee, Roee Ozeri, ChristopheSalomon, and Vladan Vuletic. We are grateful toDirk Schwalm for supporting the work on laser coolingand trapping at the MPI fur Kernphysik. One of us(Yu.B.O.) acknowledges a fellowship by the Alexandervon Humboldt-Stiftung. Our work on dipole trappingis generously supported by the Deutsche Forschungsge-meinschaft in the frame of the Gerhard-Hess-Programm.

REFERENCES

Adams, C.S., Sigel, M., and Mlynek, J. (1994). Phys.Rep. 240, 143.

Adams, C.S., Lee, H.J., Davidson, N., Kasevich, M., andChu, S. (1995). Phys. Rev. Lett. 74, 3577.

Adams, C.S., and Riis, E. (1997). Prog. Quant. Elec-tron. 21, 1.

Allen, L., and Eberly, J.H. (1975). Optical resonance andtwo-level atoms (Wiley, New York).

Aminoff, C.G., Steane, A., Bouyer, P., Desbiolles, P.,Dalibard, J., and Cohen-Tannoudji, C. (1993). Phys.Rev. Lett. 71, 3083.

Anderson, B.P., Gustavson, T.I., and Kasevich, M.A.(1996). Phys. Rev. A 53, R3727.

Anderson, M.H., Ensher, J.R., Matthews, M.R., Wie-man, C.E., and Cornell, E. (1995). Science 269, 198.

Anderson, B.P., and Kasevich, M.A. (1998). Science, inpress.

Andrews, M.R., Mewes, M.-O., van Druten, N.J., Durfee,D.S., Kurn, D.M., and Ketterle, W. (1996). Science 273,84.

Andrews, M.R., Kurn, D.M., Miesner, H.-J., Durfee,D.S., Townsend, C.G., Inouye, S., and Ketterle, W.(1997). Phys. Rev. Lett. 79, 553.

Arimondo, E., Phillips, W.D., and Strumia, F., eds.(1992). Laser Manipulation of Atoms and Ions Pro-ceedings of the International School of Physics “EnricoFermi”, Varenna, 9 - 19 July 1991, (North Holland, Am-sterdam).

Ashkin, A (1970). Phys. Rev. Lett. 24, 156.

Ashkin, A. (1978). Phys. Rev. Lett. 40, 729.

Askar’yan, G.A. (1962). Sov. Phys. JETP 15, 1088.

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

Balykin, V.I., and Letokhov, V.S. (1989). Appl. Phys.B 48, 517.

Balykin, V.I., Letokhov, V.S., Ovchinnikov, Yu.B., andSidorov, A.I. (1987). JETP Lett. 45, 353; Phys. Rev.Lett. 60, 2137 (1988).

Beijersbergen, M.W., Allen, L., van der Veen, H.E.L.O.,and Woerdman, J.P., Opt. Commun. 96, 123 (1993).

Ben Dahan, M., Peik, E., Reichel, J., Castin, Y., andSalomon, C. (1996). Phys. Rev. Lett. 76, 4508.

Bergeman, T., Erez, G., and Metcalf, H.J. (1987). Phys.Rev. A 35, 1535.

Bergstrom, I., Carlberg, C., and Schuch, R., eds. (1995).Trapped charged particles and fundamental applications(World Scientific, Singapore); also published as PhysicaScripta T59.

Bjorkholm, J.E., Freeman, R.R., Ashkin, A., and Pear-son, D.B. (1978). Phys. Rev. Lett. 41, 1361.

Boesten, H.M.J.M., Tsai, C.C., Verhaar, B.J., andHeinzen, D.J. (1997). Phys. Rev. Lett. 77, 5194.

Boiron, D., Triche, C., Meacher, D.R., Verkerk, P., andGrynberg, G. (1995). Phys. Rev. A 52, R3425.

Boiron, D., Michaud, A., Lemonde, P., Castin, Y., Sa-lomon, C., Weyers, S., Szymaniec, K., Cognet, L., andClairon, A. (1996). Phys. Rev. A 53, R3734.

Boiron, D., Michaud, A., Fournier, J.M., Simard, L.,Sprenger, M., Grynberg, G., and Salomon, C. (1998).Phys. Rev. A 57, R4106.

Bouchoule, I., Perrin, H., Kuhn, A., Morinaga, M., andSalomon, C. (1998). Phys. Rev. A, in press.

Bradley, C.C., Sackett, C.A., Tollet, J.J., and Hulet,R.G. (1995). Phys. Rev. Lett. 75, 1687.

Bradley, C.C., Sackett, C.A., and Hulet, R.G. (1997).Phys. Rev. Lett. 78, 985.

Chen, J., Story, J.G., Tollett, J.J., and Hulet, R.G.(1992). Phys. Rev. Lett. 69, 1344.

Cho, D. (1997). J. Kor. Phys. Soc. 30, 373.

Chu, S., Bjorkholm, J.E., Ashkin, A., and Cable, A.(1986). Phys. Rev. Lett. 57, 314.

Chu, S. (1998). Rev. Mod. Phys. 70, 686.

Cline, R.A., Miller, J.D., Matthews, M.R., and Heinzen,D.J. (1994a). Opt. Lett. 19, 207.

Cline, R.A., Miller, J.D., and Heinzen, D.J. (1994b).

34

Phys. Rev. Lett. 73, 632.

Cohen-Tannoudji, C., and Dupont-Roc, J. (1972). Phys.Rev. A 5, 968.

Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G.(1992). Atom-Photon Interactions: Basic Processes andApplications (Wiley, New York).

Cohen-Tannoudji, C. (1998). Rev. Mod. Phys. 70, 707.

Cook, R.J. (1979). Phys. Rev. A 20, 224; ibid. 22, 1078(1980).

Cook, R.J., and Hill, R.K. (1982). Opt. Commun. 43,258.

Corwin, K.L., Lu, Z.-T., Claussen, N., , Wieman, C., andCho, D. (1997). Bull. Am. Phys. Soc. 42, 1057.

Courteille, P., Freeland, R.S., Heinzen, D.J., van Abee-len, F.A., and Verhaar, B.J. (1998). Phys. Rev. Lett.81, 69.

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

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

Davidson, N., Lee, H.J., Kasevich, M., and Chu, S.(1994). Phys. Rev. Lett. 72, 3158.

Davidson, N., Lee, H.J., Adams, C.S., Kasevich, M., andChu, S. (1995). Phys. Rev. Lett. 74, 1311.

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

DePu, M.T., McCormick, C., Winoto, S.L., Oliver, S.,and Weiss, D.S. (1998). “Unity Occupation of Sites in a3D Optical Lattice”, preprint.

Desbiolles, P., Arndt, M., Szriftgiser, P., and Dalibard,J. (1996a). Phys. Rev. A 54, 4292.

Deutsch, I.H., and Jessen, P. (1997). Phys. Rev. A 57,1972.

Dholakia, K (1998). Contemp. Phys. 39, 351.

Dieckmann, K., Spreeuw, R.J.C., Weidemuller, M., andWalraven, J. (1998). Phys. Rev. A 58, 3891.

Dowling, J.P., and Gea-Banacloche, J. (1996). Adv. At.Mol. Opt. Phys. 37, 1.

Drewsen, M., Laurent, P., Nadir, A., Santarelli, G., Cla-iron, A., Castin, Y., Grison, D., and Salomon, C. (1994).Appl. Phys. B 59, 283.

Engler, H., Manek, I., Moslener, U., Nill, M.,Ovchinnikov, Yu.B., Schloder, U., Schunemann, U.,Zielonkowski, M., Weidemuller, M., and Grimm, R.(1998). Appl. Phys. B 67, 709.

Fioretti, A., Comparat, D., Crubellier, A., Dulieu, O.,Masnou-Seewes, F., and Pillet, P. (1998). Phys. Rev.Lett. 81, 4402.

Foot, C.J. (1991). Comtemp. Phys. 32 369.

Friebel, S., D’Andrea, C., Walz, J., Weitz, M., andHansch, T.W. (1998a). Phys. Rev. A 57, R20.

Friebel, S., Scheunemann, R.,, Walz, J., Hansch,T.W.,and Weitz, M. (1998b). Appl. Phys. B 67, 699.

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

Gallagher, A., and Pritchard, D.E. (1989). Phys. Rev.Lett. 63, 957.

Gardner, J.R., Cline, R.A., Miller, J.D., Heinzen, D.J.,Boesten, H.M.J.M., and Verhaar, B.J. (1995). Phys.Rev. Lett. 74, 3764.

Gauck, H., Hartl, M., Schneble, D., Schnitzler, H., Pfau,T., and Mlynek, J. (1998). Phys. Rev. Lett. 81, 5298.

Ghosh, P.K. (1995). Ion Traps (Clarendon Press, Ox-ford).

Gordon, J.P., Ashkin, A. (1980). Phys. Rev. A 21, 1606.

Guery-Odelin, D., Soding, J., Desbiolles, P., and Dal-ibard, J. (1998a). Opt. Express 2, 323.

Grynberg, G., and Triche, C., (1996), in Coherent andCollective Interactions of Particles and Radiation Beams,Proceedings of the International School of Physics “En-rico Fermi”, Varenna, 11 - 21 July 1995, edited by Aspect,A., Barletta, W. and Bonifacio, R. (IOS Press, Amster-dam), p. 95.

Guery-Odelin, D., Soding, J., Desbiolles, P., and Dal-ibard, J. (1998b). Europhys. Lett., submitted.

Hamann, S.E., Haycock, D.L., Klose, G., Pax, P.H.,Deutsch, I.H., and Jessen, P.S. (1998). Phys. Rev. Lett.80, 4149.

Haubrich, D., Schadwinkel, H., Strauch, F., Ueberholz,B., Wynands, R., and Meschede, D. (1996). Europhys.Lett. 34, 663.

Hemmerich, A., Weidemuller, M., Esslinger, T., Zimmer-mann, C., and Hansch, T.W. (1995). Phys. Rev. Lett.75, 37.

Hemmerich, A., Weidemuller, M., and Hansch, T.W.(1996), in Coherent and Collective Interactions of Par-ticles and Radiation Beams, Proceedings of the Interna-tional School of Physics “Enrico Fermi”, Varenna, 11 -21 July 1995, edited by Aspect, A., Barletta, W. andBonifacio, R. (IOS Press, Amsterdam), p. 503.

Henkel, C., Molmer, K., Kaiser, R., Vansteenkiste, N.,Westbrook, C.I., and Aspect, A., Phys. Rev. A 55, 1160(1997).

Hess, H.F., Kochanski, G.P., Doyle, J.M., Masuhara, N.,Kleppner, D., and Greytak, T.J. (1987). Phys. Rev.Lett. 59, 672.

Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.-J.,

35

Stamper-Kurn, D.M., and Ketterle, W. (1998). Nature392, 151.

Jackson, J.D. (1962). Classical electrodynamics (Wiley,New York).

Jessen, P.S., and Deutsch, I.H. (1996). Adv. At. Mol.Opt. Phys. 37, 95.

Kasevich, M.A., Weiss, D.S., and Chu, S. (1990). Opt.Lett. 15, 607.

Kasevich, M., and Chu, S. (1991). Phys. Rev. Lett. 67,181.

Kasevich, M.A., and Chu, S., (1992). Phys. Rev. Lett.69, 1741.

Kastberg, A., Phillips, W.D., Rolston, S.L., Spreeuw,R.J.C., and Jessen, P.S. (1995). Phys. Rev. Lett. 74,1542.

Kazantsev, A.P. (1973). Sov. Phys. JETP 36, 861; ibid.39, 784 (1974).

Ketterle, W., and van Druten, N.J. (1996). Adv. At.Mol. Opt. Phys. 37, 181.

Khripolovich, I.B., and Lamoreaux, S.K. (1997). CP Vi-olation Without Strangeness: Electric Dipole Momentsof Particles, Atoms, and Molecules (Springer, Berlin).

Kim, J., Friedrich, B., Katz, D.P., Patterson, D., We-instein, J.D., DeCarvalho, R., and Doyle, J.M. (1997).Phys. Rev. Lett. 78, 3665.

Kuga, T., Torii, Y., Shiokawa, N., and Hirano, T. (1997).Phys. Rev. Lett. 78, 4713.

Kuhn, A., Perrin, H., Hansel, W., and Salomon, C.(1996). OSA Trends in Optics and Photonics 7, 58.

Landragin, A., Labeyrie, G., Henkel, C., Kaiser, R.,Vansteenkiste, N., Westbrook, C.I., and Aspect, A.(1996a). Opt. Lett. 21, 1591.

Landragin, A.,Courtois, J.-Y., Labeyrie, G., Vansteenkiste, N., West-brook, C.I., and Aspect, A. (1996b). Phys. Rev. Lett.77, 1464.

Laryushin, D.V., Ovchinnikov, Yu.B., Balykin, V.I., andLetokhov, V.S. (1997). Opt. Commun. 135, 138.

Lee, H.J., Adams, C., Davidson, N., Young, B., Weitz,M., Kasevich, M., and Chu, S. (1994). In Atomic Physics14 edt. by Wineland, D.J., Wieman, C.E., and Smith,S.J. (AIP, New York).

Lee, H.J., Adams, C.S., Kasevich, M., and Chu, S.(1996). Phys. Rev. Lett. 76, 2658.

Lee, H.J., and Chu, S. (1998). Phys. Rev. A 57, 2905.

Lemonde, P., Morice, O., Peik, E., Reichel, J., Perrin,H., Hansel, W, and Salomon, C. (1995). Europhys. Lett.32, 555.

Letokhov, V.S. (1968). JETP Lett. 7, 272.

Lett, P.D., Watts, R.N., Westbrook, C.I., Phillips, W.D.,Gould, P.L., and Metcalf, H.J. (1988). Phys. Rev. Lett.61, 169.

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

Lett, P.D., Julienne, P.S., and Phillips, W.D. (1995).Annu. Rev. Phys. Chem. 46, 423.

Lu, Z.T., Corwin, K.L., Renn, M.J., Anderson, M.H.,Cornell, E.A., and Wieman, C.E. (1996). Phys. Rev.Lett.77, 3331.

Manek, I., Ovchinnikov, Yu.B., and Grimm, R. (1998).Opt. Commun. 147, 67.

Metcalf, H., and van der Straten, P. (1994). Phys. Rep.244, 203.

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

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

Miesner, H.-J., Stamper-Kurn, D.M., Andrews, M.R.,Durfee, D.S., Inouye, S., and Ketterle, W. (1998a). Sci-ence 279, 1005.

Miesner, H.-J., Stamper-Kurn, D.M., Stenger, J., Inouye,S., Chikkatur, A.P., and Ketterle, W. (1998b). Preprintcond-mat/9811161.

Migdall, A.L, Prodan, J.V., Phillips, W.D., Bergemann,T.H., and Metcalf, H.J. (1985). Phys. Rev. Lett. 54,2596.

Miller, J.D., Cline, R.A., and Heinzen, D.J. (1993a).Phys. Rev. A 47, R4567.

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

Minogin, V.G., and Letokhov, V.S. (1987). Laser lightpressure on atoms (Gordon and Breach, New York).

Monroe, C., Swann, W., Robinson, H., and Wieman, C.(1990). Phys. Rev. Lett. 65, 1571

Morsch O., and Meacher, D.R., Opt. Commun. 148, 49(1998).

Muller-Seydlitz, T., Hartl, M., Brezger, B., Hansel, H.,Keller, C., Schnetz, A., Spreeuw, R.J.C., Pfau, T., andMlynek, J. (1997). Phys. Rev. Lett.78, 1038.

Myatt, C.J., Burt, E.A., ghrist, R.W., Cornell, E.A., andWieman, C.E. (1997). Phys. Rev. Lett.78, 586.

Ovchinnikov, Yu.B., Shul’ga, S.V., and Balykin, V.I.(1991). J. Phys. B: At. Mol. Opt. Phys. 24, 3173.

Ovchinnikov, Yu.B., Soding, J., and Grimm, R. (1995a).JETP Lett. 61, 21.

36

Ovchinnikov, Yu.B., Laryushin, D.V., Balykin, V.I., andLetokhov, V.S. (1995b). JETP Lett. 62, 113.

Ovchinnikov, Yu.B., Manek, I., and Grimm, R. (1997).Phys. Rev. Lett. 79, 2225.

Ozeri, R., Khaykovich, L., and Davidson, N. (1998).“Long spin relaxation times in a single-beam blue-detuned optical trap”, preprint.

Perrin, H., Kuhn, A., Bouchoule, I., and Salomon, C.(1998). Europhys. Lett. 42, 395.

Petrich, W., Anderson, M.H., Ensher, J.R., and Cornell,E.A. (1994). J. Opt. Soc. Am. B 11, 1332.

Phillips, W., and Metcalf, H. (1982). Phys. Rev. Lett.48, 596.

Phillips, W.D. (1998). Rev. Mod. Phys. 70, 721.

Pinkse, P.W.H., Mosk, A., Weidemuller, M., Reynolds,M.W., Hijmans, T.W., and Walraven, J.T.M., Phys.Rev. Lett. 78, 990 (1997).

Pritchard, D.E., Raab, E.L., Bagnato, V.S., Wieman,C.E., and Watts, R.N. (1986). Phys. Rev. Lett. 57, 310.

Raab, E.L., Prentiss, M., Cable, A., Chu, S., andPritchard, D. (1987). Phys. Rev. Lett. 59, 2631.

Reichel, J., Bardou, F., Ben Dahan, M., Peik, E., Rand,S., Salomon, C., and Cohen-Tannoudji, C. (1995). Phys.Rev. Lett. 75, 4575.

Safonov, A.I., Vasilyev, S.A., Yashnikov, I.S., Lukashe-vich, I.I., and Jaakola, S. (1998). Phys. Rev. Lett. 81,4545.

Salomon, C., Dalibard, J., Phillips, W.D., and Guellatti,S. (1990). Europhys. Lett. 12, 683.

Savard, T.A., O‘Hara, K.M., and Thomas, J.E. (1997).Phys. Rev. A 56, R1095.

Schiffer, M., Rauner, M., Kuppens, S., Zinner, M., Sen-gstock, K., and Ertmer, W. (1998). Appl. Phys. B 67,705 (1998).

Schneble, D., Gauck, H., Hartl, M., Pfau, T., andMlynek, J. (1998). in Bose-Einstein condensation ofatomic gases, Proceedings of the International School ofPhysics “Enrico Fermi”, Varenna, 7 - 17 July 1998, editedby Inguscio, M., Stringari, S., and Wieman, C., in press.

Schunemann, U., Engler, H., Zielonkowski, M., Wei-demuller, M., and Grimm, R. (1998). Opt. Commun., inpress.

Sengstock, K., and Ertmer, W., (1995). Adv. At. Mol.Opt. Phys. 35, 1.

Sesko, D., Walker, T., Monroe, C., Gallagher, A., andWieman, C. (1989). Phys. Rev. Lett. 63, 961.

Sobelman, I.I. (1979). Atomic Spectra and RadiativeTransitions (Springer, Berlin).

Soding, J., Grimm, R., and Ovchinnikov, Yu.B. (1995).

Opt. Commun. 119, 652.

Soding, J., Guery-Odelin, D., Desbiolles, P., Ferrari, G.,and Dalibard, J. (1998). Phys. Rev. Lett. 80, 1869.

Stamper-Kurn, D.M., Andrews, M.R., Chikkatur, A.P.,Inouye, S., Miesner, H.-J., Stenger, J., and Ketterle, W.(1998a). Phys. Rev. Lett. 80, 2027.

Stamper-Kurn, D.M., Miesner, H.-J., Chikkatur, A.P.,Inouye, S. Stenger, J., and Ketterle, W. (1998b). Phys.Rev. Lett. 81, 2194.

Steane, A.M., and Foot, C.J. (1991). Europhys. Lett.14, 231.

Stenger, J., Inouye, S., Stamper-Kurn, D.M., Miesner,H.-J., Chikkatur, A.P., and Ketterle, W. (1998). Nature,in press.

Stenholm, S. (1986). Rev. Mod. Phys. 58, 699.

Suter, D. (1997). The Physics of Laser-Atom Interac-tions (University Press, Cambridge).

Takekoshi, T., Yeh, J.R., and Knize, R.J. (1995). Opt.Comm. 114, 421.

Takekoshi, T., and Knize, R.J. (1996). Opt. Lett. 21,77.

Takekoshi, T., Patterson, B.M., and Knize, R.J. (1998).Phys. Rev. Lett. 81, 5105.

Torii, Y., Shiokawa, N., Hirano, T., Kuga, T., Shimizu,Y., and Sasada, H., Eur. Phys. J. D 1, 239 (1998).

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

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

van Druten, N.J., and Ketterle, W., Phys. Rev. Lett.79, 549 (1997).

Vuletic, V., Chin, C., Kerman, A.J., and Chu, S., (1998).Phys. Rev. Lett., in press.

Walker, T. and Feng, P. (1994). Adv. At. Mol. Opt.Phys. 34, 125.

Wallace, C.D., Dinneen, T.P., Tan, K.-Y.N., Grove, T.T.,and Gould, P.L. (1992). Phys. Rev. Lett. 69, 897.

Wallis, H., Dalibard, J., and Cohen-Tannoudji, C. (1992).Appl. Phys. B 54, 407.

Wallis, H. (1996). Quantum Semiclass. Opt. 8, 727(1996).

Walther, H. (1993). Adv. At. Mol. Opt. Phys. 31, 137.

Weiner, J. (1995). Adv. At. Mol. Opt. Phys. 35, 45.

Weinstein, J.D., deCarvalho, R., Guillet, T., Friedrich,B., and Doyle, J.M. (1998). Nature 395, 148.

Wilkinson, S.R., Bharucha, C.F., Madison, K.W., Niu,

37

Q., and Raizen, M.G. (1996). Phys. Rev. Lett. 76,4512.

Winoto, S.L., DePu, M.T., Bramall, N.E., and Weiss,D.S. (1998). “Laser Cooling Large Numbers of Atoms atHigh Densities”, preprint.

Wineland, D.J., Bergquist, J.C., Bollinger, J.J., andItano, W.M. (1995). in Trapped charged particles andfundamental applications edt. by Bergstrom, I., Carl-berg, C., and Schuch, R. (World Scientific, Singapore);also published as Physica Scripta T59, 286.

Zemanek, P., and Foot, C.J. (1998). Opt. Commun.146, 119.

Zielonkowski, M., Steiger, J., Schunemann, U.,DeKieviet, M., and Grimm, R. (1998a). Phys. Rev. A58, in press.

Zielonkowski, M., Manek, I., Moslener, U., Rosenbusch,P., and Grimm, R. (1998b). Europhys. Lett. 44, 700.

TABLE I. Properties of the alkali atoms of relevance for optical dipole trapping. Transition wavelengths λD2 and λD1 ,fine-structure splitting ∆′

FS, nuclear spin I, ground-state hyperfine splitting ∆HFS, excited-state hyperfine splitting ∆′HFS, natural

linewidth Γ, recoil temperature Trec (for D2 line), and corresponding height hrec = kBTrec/(mg) in the field of gravity.

�D2;D1�0

FS=2� I �HFS=2� �0

HFS=2� �=2� Trec hrec

(nm) (GHz) (MHz) (MHz) (MHz) (�K) (�m)

6Li 670.9, 670.9 10 1 228 4.6 5.9 7.1 10007Li 3/2 804 18 6.1 740

23Na 589.0, 589.6 510 3/2 1772 112 9.9 2.4 88

39K 766.7, 769.9 1500 3/2 462 34 6.2 0.84 1840K 3 1286 100 0.81 1741K 3/2 254 17 0.79 16

85Rb 780.0, 794.8 7200 5/2 3036 213 5.9 0.37 3.787Rb 3/2 6835 496 0.36 3.5

133Cs 852.1, 894.3 16600 7/2 9192 604 5.3 0.20 1.3

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TABLE II. Parameters of various experimentally realized blue-detuned dipole traps.

trap con�ning atomic detuning, # atoms, cooling, speciality�eldsa species depth transfer temperature

light-sheet trap LS Na � 100THz 3000 no very longDavidson et al. (1995) gravity 10�K coherence times

inverted pyramid LS Na � 100THz 4� 105 Raman high phase-Lee et al. (1996) gravity 20�K 1.0�K space densities

plugged doughnut beam HB Rb 60GHz 108 molassesb large numberKuga et al. (1997) 40�K 30 % 13�K of atoms

single-beam trap HB Rb 250GHz c 105 no singleOzeri et al. (1998) laser beam

conical atom trap HB Cs 3GHz 106 molassesd high loadingOvchinnikov et al. (1998) gravity � 1mK 80 % � 10�K e�ciency

gravito-optical cavity EW Cs 1 { 10GHz 107 no ten resolvedAmino� et al. (1993) gravity 1mK e bounces

grav.-opt. surface trap EW/HB Cs 1GHz f 105 re ection atoms veryOvchinnikov et al. (1997) gravity 100 �K 30 % 3�K close to surface

aLS: light sheets, HB: hollow beam, EW: evanescent wave.

bsee Torii et al. (1998).

ctrapping studied in a wide detuning range between 50GHz and 15THz.

dpure re ection cooling demonstrated at a trap detuning of 30GHz.

enumber refers to vertical motion only. The horizontal depth was much lower (5�K).

fnumber refers to evanescent wave. The hollow-beam detuning was much larger (100GHz).

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