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arXiv:0903.3570v2 [cond-mat.other] 23 Jun 2009

Combination of a magnetic Feshbach resonance

and an optical bound-to-bound transition

Dominik M. Bauer, Matthias Lettner, Christoph Vo, Gerhard Rempe, and Stephan D¨ urr

Max-Planck-Institut f¨ ur Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

We use laser light near resonant with an optical bound-to-bound transition to shift the magnetic

field at which a Feshbach resonance occurs. We operate in a regime of large detuning and large

laser intensity. This reduces the light-induced atom-loss rate by one order of magnitude compared

to our previous experiments [D. M. Bauer et al. Nature Phys. 5, 339 (2009)]. The experiments are

performed in an optical lattice and include high-resolution spectroscopy of excited molecular states,

reported here. In addition, we give a detailed account of a theoretical model that describes our

experimental data.

PACS numbers: 34.50.Cx, 33.20.Kf, 33.40.+f, 03.75.Hh

I.INTRODUCTION

Many properties of ultracold gases are determined by

the interparticle interaction which is characterized by

the s-wave scattering length a.

able to tune this parameter. A much-used method for

this purpose is a magnetic Feshbach resonance [1, 2, 3].

An alternative method is a photoassociation resonance,

which is sometimes also called optical Feshbach resonance

[4, 5, 6, 7, 8, 9].A major advantage of photoassoci-

ation resonances is that the light intensity can be var-

ied on short length and time scales, thus offering more

flexible experimental control over the scattering length.

The problem with photoassociation resonances is that

the light induces inelastic collisions between atoms which

lead to rapid loss of atoms. This is why photoassociation

resonances have only rarely been used to tune the scat-

tering length [7, 8]. A solution for this problem exists for

alkali earth atoms where narrow intercombination lines

allow for tuning of a with only moderate loss [10, 11].

But this is not feasible in the large number of experi-

ments with alkali atoms.

In a recent experiment [12] we explored an alternative

scheme for controlling the scattering length with laser

light. This scheme uses the existing coupling between an

atom-pair state |a?⊗|a? and a molecular state |g? near a

Feshbach resonance, as illustrated in Fig. 1. By adding

a light field that is somewhat detuned from a bound-to-

bound transition between state |g? and an electronically

excited molecular state |e?, one can induce an ac-Stark

shift of state |g?. This results in a shift of the magnetic

field Bresat which the Feshbach resonance occurs. If the

magnetic field B is held close to the Feshbach resonance,

then spatial or temporal variations of the light intensity

affect the scattering length. This scheme also suffers from

light-induced inelastic collisions. However, in Ref. [12]

we reported a two-body loss rate coefficient as small as

K2∼ 10−11cm3/s at parameters where the real part of

the scattering length is changed by Re(a)/abg− 1 = ±1

with respect to its background value abg. This represents

a reduction of the loss rate by one order of magnitude

compared to a photoassociation resonance in87Rb where

This makes it desir-

FIG. 1:

are coupled to a dimer state |g? belonging to the electronic

ground state. This coupling has a strength α and causes a

magnetic Feshbach resonance. Laser light is applied to drive

a bound-to-bound transition from state |g? to an electroni-

cally excited dimer state |e? with Rabi frequency ΩR. This

causes an ac-Stark shift (not shown) of state |g? which leads to

a shift of the magnetic field at which the Feshbach resonance

occurs. Typically, the same light can also drive photoassocia-

tion from state |a?⊗|a? to state |e? with coupling strength β.

The photoassociation is a nuisance for the scheme used here.

The light frequency is typically somewhat detuned from both

transitions.

Level scheme. Pairs of atoms, each in state |a?,

K2∼ 10−10cm3/s was reported for the same change of

Re(a) [7, 8].

Here we show experimentally that a value as low as

K2∼ 10−12cm3/s at Re(a)/abg− 1 = 1 can be reached

with our scheme, thus reducing the loss rate by one more

order of magnitude compared to our previous results.

This improvement is achieved by increasing the detun-

ing and the intensity of the laser light, which is a com-

monly used method for reducing incoherent rates in ex-

periments, which rely on the ac-Stark shift. When follow-

ing this approach, two new issues have to be addressed.

First, the detunings are no longer small compared to the

typical splitting between the various hyperfine and mag-

netic substates of the excited state. Achieving a large

detuning with respect to all excited states thus requires

knowledge of the positions of all nearby excited states, so

that excited-state spectroscopy has to be performed, on

which we report in Sec. IIIA. Second, in our previous ex-

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2

periment [12] we used a model for the temporal evolution

of the cloud size to determine Re(a) and K2. If the laser

that drives the bound-to-bound transition is now oper-

ated at much larger power, then it creates a noticeable

dipole trap. Its rapid turn-on induces large-amplitude

oscillations of the cloud size which are difficult to model.

We therefore explore an alternative way of measuring

Re(a) and K2. To this end, we pin the positions of the

atoms with a deep optical lattice which minimizes the

effect of the additional dipole trap. In the lattice, we

use excitation spectroscopy and a loss measurement to

determine Re(a) and K2, respectively. These results are

reported in Sec. IIIB.

In Ref. [12] we also studied the behavior of the system

when the laser light is tuned close to the bound-to-bound

resonance. In this regime, we observed an Autler-Townes

doublet in the loss rate coefficient as a function of mag-

netic field K2(B). In the same work, we presented a sys-

tematic study of the dependence of the positions Bresof

the loss resonances on laser power and detuning. In Sec.

IIIC of the present paper, we complement these measure-

ments with a systematic study of the height and width

of the resonances in this Autler-Townes doublet. These

experimental data agree well with the theoretical model

that we use, thus showing that the relevant physics is

well understood.

Before turning to the experiment, we begin in Sec. II

with a detailed discussion of the theoretical model that

we use to describe our experiments.

II. THEORY

We consider a system with three internal states, as

shown in Fig. 1. State |a? represents a single unbound

atom. States |g? and |e? represent a dimer in the elec-

tronic ground and excited state, respectively. The atom-

molecule coupling between the atom-pair state |a? ⊗ |a?

and the dimer state |g? relevant for the Feshbach reso-

nance is described by a coupling strength α. Laser light

couples states |g? and |e? on a bound-to-bound transition

with a Rabi frequency ΩR. The same laser light drives

photoassociation from state |a?⊗|a? to |e? characterized

by a coupling strength β.

This model is closely related to previous studies of the

combination of a Feshbach resonance with a photoasso-

ciation resonance, see e.g. Refs. [13, 14, 15]. Unlike those

references, we are mostly interested in the Feshbach reso-

nance and the bound-to-bound transition. The photoas-

sociation is a nuisance in our scheme, because any useful

change of Re(a) that it induces is inevitably accompanied

by the loss rates that limited the photoassociation exper-

iments in Refs. [7, 8]. Luckily, we find excited states in

our experiment for which photoassociation is negligible.

Before turning to a quantitative model, we present a

qualitative argument that motivates why creating a given

change in Re(a) with our scheme causes a smaller loss

rate than a photoassociation resonance would do. To ex-

plain this, we must first understand the limitation of the

photoassociation resonance. If an infinite amount of laser

intensity were available, then the value of K2caused by

a photoassociation resonance could be reduced without

modifying the change in Re(a). To this end, one would

simply have to increase ∆ealong with the laser intensity,

see Eq. (22). The same improvement could be achieved if

a photoassociation resonance with a larger transition ma-

trix element was available because that would be equiv-

alent to having more laser intensity.

Our scheme relies on the fact that the typical inter-

atomic distance is orders of magnitude larger in the

atomic gas than within a single molecule. As a result,

the transition matrix element is typically orders of mag-

nitude smaller for a photoassociation resonance than for

a bound-to-bound transition, see Eq. (13). Hence at a

given laser intensity, one can detune the laser pretty far

in our scheme and still achieve a significant light-induced

change in Re(a), whereas a photoassociation resonance

driven with the same laser intensity requires a smaller

laser detuning, which results in a larger loss rate.

A. Hamiltonian

According to Refs. [16, 17] the system with the light

off is described by the Hamiltonian

ˆH =

?

?

?

j∈{a,g,e}

1

2

?

d3xˆΨ†

j(x)Hj(x)ˆΨj(x)

+

d3x1d3x2ˆΨ†

p(x1,x2)Ubg(x12)ˆΨp(x1,x2)

+ ¯ hd3x1d3x2

?ˆΨ†

g(X12)α(x12)ˆΨp(x1,x2) + H.c.

?

(1)

with the bosonic field operatorsˆΨj(x) for j ∈ {a,g,e},

the single-particle Hamiltonians Hj(x), the relative co-

ordinate x12 = x1− x2, the center-of-mass coordinate

X12 = (x1+ x2)/2, and the abbreviationˆΨp(x1,x2) =

ˆΨa(x1)ˆΨa(x2) for the annihilation of an atom pair. Elas-

tic two-atom collisions far away from the Feshbach reso-

nance are described by Ubg(x12) and the atom-molecule

coupling relevant for the Feshbach resonance is described

by α(x12).

Collisions between more than two atoms were ne-

glected here assuming that the interatomic potentials

are short ranged compared to the average interatomic

distance. Furthermore, collisions involving at least one

molecule were neglected, assuming that the density of

molecules is low.

α(x12) is the Fourier transform of [18]

α(k) =1

¯ h

?

V

2?g|H|k?,(2)

where V is the quantization volume, |k? is a two-atom

scattering state that has an incoming plane wave with

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3

wave vector k in the relative coordinate, and H is the

Hamiltonian describing the atom-molecule coupling in

the collision of two atoms. Its position representation

has the asymptotic form

?x12|k? =eikx12+ f(k)eikx12/x12

√V

,x12→ ∞,(3)

where f(k) is the scattering amplitude. Combination of

the last two equations shows that α(k) is independent of

V .

If all incoming particles in the gas are slow enough,

then we can replace α(k) by a constant α = limk→0α(k).

In the position representation, this corresponds to a con-

tact potential in the relative coordinate. We use an anal-

ogous approximation for Ubgand obtainˆH =?d3xH(x)

with the Hamiltonian density [18]

H(x) =

?

j∈{a,g,e}

?ˆΨ†

ˆΨ†

j(x)Hj(x)ˆΨj(x) +Ubg

2

ˆΨ†

p(x)ˆΨp(x)

+ ¯ h

g(x)αˆΨp(x) + H.c.

?

.(4)

Note that Ubg= 4π¯ h2abg/m is related to the background

scattering length abgand the atomic mass m.

We now extend this model to include effects caused

by the light, which has an electric field of the form

E = −E0cos(ωLt) with amplitude E0 and angular fre-

quency ωL. In analogy to Eq. (4) we obtain the following

additional terms for the Hamiltonian density

HL(x) = ¯ hcos(ωLt)

+ 2¯ hcos(ωLt)

?ˆΨ†

?ˆΨ†

e(x)ΩRΨg(x) + H.c.

?

e(x)βˆΨp(x) + H.c.

?

.(5)

The Rabi frequency ΩRand the coupling strength β de-

scribe the bound-to-bound transition and the photoasso-

ciation, respectively. ΩRis related to a matrix element

of the matter-light interaction term in the Hamiltonian,

which in the electric dipole approximation reads H(t) =

−dE(t). This yields ¯ hΩRcos(ωLt) = −?e|d|g?E(t). We

abbreviate deg= ?e|d|g? and obtain the well-known rela-

tion

ΩR=E0deg

¯ h

.(6)

For β we obtain in analogy to Eq. (2)

β =E0

2¯ h

?

V

2lim

k→0?e|d|k?. (7)

Finally, we specify the single-particle Hamiltonians

which consist of a kinetic part and an internal part Eint

j

Hj(x) = −¯ h2∇2

2mj

+ Eint

j

(8)

for j ∈ {a,g,e}. Here mj= m(2 − δaj) is the mass of a

particle in state j and δij is the Kronecker symbol. The

internal energy of each state depends nonlinearly on the

magnetic field B. Near the pole of the unshifted Feshbach

resonance Bpolewe approximate this dependence as lin-

ear and obtain

Eint

j

= −µj(B − Bpole) + ¯ hωegδej,(9)

where µj is a magnetic dipole moment. At B = Bpole,

the internal states |a? and |g? are degenerate whereas the

internal state |e? has an energy offset ¯ hωeg.

B. Mean-Field Model

We assume that the population in each state is Bose

condensed and can be described by a mean field ψj(x) =

?ˆΨj(x)?. In addition, we assume that the system is homo-

geneous. We take the expectation value of the Heisenberg

equation of motion i¯ hdˆΨj/dt = [ˆΨj,H] and approximate

the field operators as uncorrelated, so that the expecta-

tion values factorize [18]. Physics beyond this approxi-

mation is discussed in Refs. [17, 19]. We obtain

id

dtψa =

Eint

a

¯ h

+2α∗ψ∗

ψa+Ubg

¯ h

|ψa|2ψa

aψg+ 4β∗ψ∗

a+Eint

¯ h

a+ ΩRψg)cos(ωLt) +Eint

aψecos(ωLt)(10a)

id

dtψg = αψ2

id

dtψe = (2βψ2

g

ψg+ Ω∗

Rψecos(ωLt)(10b)

e

¯ h

ψe.(10c)

We move to an interaction picture by replacing ψe →

ψeeiωLtand perform a rotating-wave approximation by

neglecting coefficients rotating as e±2iωLt.

move to another interaction picture by replacing ψj →

ψjexp((2−δaj)iEint

Ref. [12])

We then

at/¯ h) for j ∈ {a,g,e}. Hence (see also

id

dtψa =

id

dtψg = αψ2

id

dtψe = βψ2

Ubg

¯ h

|ψa|2ψa+ 2α∗ψ∗

a+ ∆gψg+1

aψg+ 2β∗ψ∗

aψe(11a)

2Ω∗

?

Rψe

(11b)

a+1

2ΩRψg+∆e−i

2γe

?

ψe, (11c)

where we abbreviated

∆g =

1

¯ hµag(B − Bpole)

∆e = −∆L+1

∆L = ωL− ωeg

with µag= 2µa−µgand µae= 2µa−µe. In Eq. (11c) we

included an ad hoc decay rate γe that represents spon-

taneous radiative decay from state |e? into states that

are not included in the model, similar to Ref. [20]. A

model similar to Eqs. (11) was used in Ref. [15] to ex-

plain enhanced photoassociation loss rates near a mag-

netic Feshbach resonance [14].

(12a)

¯ hµae(B − Bpole)(12b)

(12c)

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4

Note that the typical interatomic distance is orders of

magnitude larger in the atomic gas than within a single

molecule. For a typical excited state, this results in

|βψa| ≪ |ΩR|.(13)

C.Adiabatic Elimination and Scattering Length

We assume that all the population is initially prepared

in state |a? and that the populations in states |g? and |e?

will remain small at all times so that they can be elim-

inated adiabatically, similar to Refs. [15, 20]. This is a

good approximation, e.g., if the angular frequencies αψa

and βψaare both small compared to ΩRand γeor com-

pared to ∆eand ∆g. This condition is always satisfied

in the low-density limit, but for a very broad Feshbach

resonance it might be difficult to reach this regime ex-

perimentally.

The adiabatic elimination is achieved by formally set-

ting (d/dt)ψg= (d/dt)ψe= 0. This is used to eliminate

ψgand ψefrom the equations. We obtain

id

dtψa =

4π¯ ha

m

|ψa|2ψa

(14)

with the complex-valued scattering length

m

2π¯ h

|α|2(∆e− iγe/2) − Re(α∗Ω∗

(∆e− iγe/2)∆g− |ΩR/2|2

The term Re(α∗Ω∗

Rβ) represents interference between the

two possible ways to go from state |a? to state |e?, either

directly or indirectly through state |g?.

The real part of the scattering length is responsible for

the mean-field energy [21]. We assume that abg is real

and obtain

m

2π¯ h(∆g∆e− |ΩR/2|2)2+ (∆gγe/2)2

×

+|α|2∆g(γe/2)2?

The imaginary part of the scattering length gives rise

to two-body loss with a rate equation [21]

a = abg−

×

Rβ) + |β|2∆g

. (15)

Re(a) = abg−

??

1

|α|2∆e−Re(α∗Ω∗

Rβ) + |β|2∆g

??

∆e∆g− |ΩR/2|2?

.(16)

dn

dt

K2 = −8π¯ h

= −K2n2g(2)

(17a)

mIm(a),(17b)

where n = |ψa|2is the atomic density, K2is the two-body

loss coefficient for a Bose-Einstein condensate (BEC),

and g(2)is the pair correlation function at zero relative

distance. For a BEC with N atoms g(2)= 1 − 1/N.

Insertion of Eq. (15) yields

K2= 2γe

|αΩR/2|2− ∆gRe(α∗Ω∗

(∆g∆e− |ΩR/2|2)2+ (∆gγe/2)2

Rβ) + |β|2∆2

g

.(18)

?

?

?

?

?

?

?

?

?

FIG. 2: (Color online) Predictions for Re(a)/abg and K2 as

a function of B from Eqs. (16) and (18). For all curves, we

choose ¯ hγe/µae = 2 G, β = 0, and ∆B = 0.2 G; with ∆B

defined in Eq. (20). The dotted lines (red) are for resonant

laser light ∆L = 0 and ¯ h|ΩR|2/γeµag = 100 G. They show

two resonances that are symmetrically split around Bpole.

These resonances represent an Autler-Townes doublet. The

solid lines (blue) are for large laser detuning ∆L/γe = 5 and

¯ h|ΩR|2/γeµag = 100 G. They also show two resonances, but

their heights, widths, and distances from Bpole are quite dif-

ferent. The dashed line (black) is a reference without any

light ΩR = 0.

All terms in the numerator are ∝ E2

importance of the terms is independent of laser inten-

sity. If B is held near the unshifted Feshbach resonance,

then ∆g is small and all terms containing β in Eq. (18)

become negligible. Hence if one considers measurements

of K2performed fairly close to the Feshbach resonance,

then photoassociation is negligible and it is impossible

to extract the value of β only from such measurements

(unless ΩRvanishes).

The interference term in Eq. (15) leads to a correspond-

ing interference term in Eq. (18). Note that the minimum

of K2observed in Ref. [14] is a result of destructive in-

terference due to this term [15].

Figure 2 shows predictions for Re(a)/abgand K2as a

function of B. For large |ΩR|, one can clearly see two res-

onances in K2each of which is approximately Lorentzian.

Each of these resonances is accompanied by a dispersive

feature in Re(a)/abg.

We show now that our model reproduces known results

from the literature in the special cases of a pure Feshbach

resonance or a pure photoassociation resonance. For a

pure Feshbach resonance (β = ΩR= 0) Eq. (15) yields

the familiar result [1]

0, so that the relative

a = abg

?

1 −

∆B

B − Bpole

?

(19)

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5

with the width of the Feshbach resonance

∆B =2¯ h2|α|2

Ubgµag.(20)

For a pure photoassociation resonance (α = ΩR= 0) Eq.

(15) yields

a = abg−

m

2π¯ h

|β|2

∆e− iγe/2,(21)

which is a Breit-Wigner form [22] as a function of ∆Lor

B. The real and imaginary parts are

Re(a) = abg−

m

2π¯ h∆e

|β|2

∆2

|β|2

∆2

e+ (γe/2)2

(22a)

K2 = 2γe

e+ (γe/2)2,(22b)

which is identical to Eq. (10) in Ref. [5] in the limit

Γstim≪ Γspon.

A quantitative comparison of Eq. (18) with our ex-

perimental data (here and in Ref. [12]) shows that β is

negligible for most of the excited states |e? that we use.

We therefore set β = 0 for the rest of the theory section.

D. Large Detuning

A good part of our experiments is performed in the

limit of large laser detuning where |∆L| ≫ |µae(B −

Bpole)/¯ h| and |∆L| ≫ γe and with β = 0. In Eq. (15)

we can approximate 1/(∆e−iγe/2) ∼ −1/∆L−iγe/2∆2

and obtain a Breit-Wigner form

L

a = abg

?

1 −

∆B

B − Bres− iW/2

?

(23)

with ∆B from Eq. (20). The real and imaginary parts

are a dispersive line shape and a Lorentzian, respectively,

Re(a) = abg

?

1 −

∆B(B − Bres)

(B − Bres)2+ W2/4

Kmax

2

1 + 4(B − Bres)2/W2.

?

(24a)

K2 =

(24b)

The resonance position Bres, the maximum loss rate co-

efficient Kmax

2

, and the full width at half maximum W

of the Lorentzian are given by

Kmax

2

=

¯ h

µag

¯ h

µag

8|α|2

W

|ΩR|2

4∆2

|ΩR|2

4∆L

(25a)

W =

L

γe

(25b)

Bres− Bpole = −

¯ h

µag

.(25c)

The far-detuned bound-to-bound coupling yields the

well-known ac-Stark shift of state |g? and this shifts Bres.

As in most applications of ac-Stark shifts, we wish to

achieve a certain value of |ΩR|2/∆Land at the same time

keep the rates for incoherent processes as low as possible.

Hence, it is advantageous to increase the detuning and

power of the laser in a way that keeps |ΩR|2/∆Lconstant.

This yields W → 0 and K2(B) → (4π¯ h|α|2/µag)δ(B −

Bres), where δ denotes the Dirac delta function. For any

given value of B ?= Bresone can thus decrease K2(B) by

increasing the detuning and the laser power sufficiently

far.

In general, it is possible that several excited states con-

tribute noticeably to a. Our model is easily adapted to

this situation by introducing a separate version of Eq.

(11c) for each excited state and by including sums over

the excited states in Eqs. (11a) and (11b). In the limit

of large laser detuning and with β = 0 for each excited

state, Eqs. (23)–(25a) remain unchanged and a sum over

the excited states appears on the right hand side of Eqs.

(25b) and (25c).

E.Autler-Townes Model for Weak Damping

More insight into the physics of the problem can be

gained from an Autler-Townes model [23, 24]. In ad-

dition, analytic expressions for the position, height and

width of the resonances in K2(B) can be derived.

This approach is based on the assumption that the

dominant frequencies in the problem are ΩR and/or

(∆g− ∆e). In this case, one can first diagonalize the

driven two-level system spanned by |g? and |e? and sub-

sequently treat the coupling to state |a? as a weak probe.

For the first step, we diagonalize the two-level system

spanned by |g? and |e?, setting α = β = γe = 0. We

assume without loss of generality that the relative phase

between states |g? and |e? is chosen such that ΩRis real.

This yields energy eigenvalues and eigenvectors

E± =

¯ h

2(∆e+ ∆g± Ωeff)

|+? = cosϑ

|−? = −sinϑ

(26a)

2|e? + sinϑ

2|e? + cosϑ

2|g?(26b)

2|g?,(26c)

where the effective Rabi angular frequency Ωeff and the

mixing angle ϑ are real-valued and must satisfy the im-

plicit equations

Ωeffcosϑ = ∆e− ∆g

Ωeffsinϑ = ΩR.

(27a)

(27b)

This determines a unique value of ϑ modulo 2π and it

yields

Ωeff=

?

Ω2

R+ (∆e− ∆g)2. (28)

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6

For the second step, we rewrite the mean-field model

(11) in the new basis and obtain with β = 0

id

dtψa =

id

dtψ+ = C+ψ2

id

dtψ− = C−ψ2

Ubg

¯ h

|ψa|2ψa+ 2ψ∗

?E+

¯ h

a(C∗

+ψ++ C∗

−ψ−)(29a)

a+−i

2γ+

?

?E−

¯ h

ψ+−i

−i

2γmixψ−(29b)

?

a−i

2γmixψ++

2γ−

ψ−(29c)

with C+= αsinϑ

γesin2 ϑ

rotated the basis and the model is still exact. We now

approximate the loss as being diagonal in the states |+?

and |−? by setting γmix = 0. Adiabatic elimination of

the populations in states |+? and |−? then yields

2, C−= αcosϑ

2, γ+= γecos2 ϑ

2cosϑ

2, γ−=

2, and γmix = −γesinϑ

2.So far we only

a = abg+ a++ a−, (30)

where the states |+? and |−? each contribute a Breit-

Wigner form as a function of E±

a±= −m

2π¯ h

|C±|2

E±/¯ h − iγ±/2.(31)

Thus, the states |+? and |−? each cause a single res-

onance and their contributions to a are simply added.

This corresponds to the intuitive understanding of an

Autler-Townes doublet [24].

The above approximation γmix= 0 is self-consistent if

the system is close to one resonance and the resonances

are well separated (γe≪ Ωeff), because in this case the

states |±? have very different populations, so that a pos-

sible coherence between these populations has little effect

and γmixis negligible.

F.Properties of the Autler-Townes Resonances

We note that E±, γ±, and C±in Eq. (31) are gener-

ally nonlinear functions of B. But if the resonances are

narrow and well separated, then ϑ will vary only little

within the width of a resonance and we can approximate

γ± and C± as constant across a resonance. According

to Eqs. (17b) and (31), maxima of K2(B) will then oc-

cur at E± = 0. Combination with Eqs. (12a), (12b),

(26a), and (28) yields the magnetic fields Bresat which

the resonances occur

Bres= Bpole+

¯ h

2µae

?

∆L±

?

∆2

L+µae

µagΩ2

R

?

.(32)

The condition E±= 0 combined with Eqs. (17b) and

(31) yields the maximum of the loss rate coefficient

Kmax

2

=8|α|2

γe

?

tanϑ

2

?±2

.(33)

Insertion of cos2 ϑ

Eqs. (26a), (27a), and E±= 0 yields

2=1

2(1 + cosϑ), sin2 ϑ

2=1

2(1 − cosϑ),

Kmax

2

=8|α|2

γe

∆e

∆g.(34)

Insertion of Eqs. (12a), (12b), (32), and B = Bresyields

Kmax

2

=8|α|2

γe

µae

µag

1 −

2∆L

∆2

∆L±

?

L+µae

µagΩ2

R

. (35)

The width of the resonance can also be calculated. For

this, we recall that the resonance is narrow and approx-

imate E± as linear in B around Bres. As a result, Eq.

(31) becomes a Breit-Wigner form as a function of B.

Correspondingly K2(B) becomes a Lorentzian as in Eq.

(24b).

In this linear approximation E± = ǫ(B − Bres) with

ǫ = dE±/dB|Bres. Eqs. (12a), (12b), (26a) and (28) yield

?

ǫ =1

2

µae+ µag± (µae− µag)∆e− ∆g

Ωeff

?

.(36)

According to Eqs. (17b) and (31), the full width at half

maximum of the Lorentzian K2(B) is W = ¯ hγ±/ǫ. Using

Eq. (27a) we obtain

¯ hγe

W

=

2ǫ

1 ± cosϑ= µae+ µag

?±2can be evaluated as in Eq. (33),

?

tanϑ

2

?±2

.(37)

The term

yielding

?tanϑ

2

W =

¯ hγe

2µae

1 ±

∆L

?

∆2

L+µae

µagΩ2

R

.(38)

We will compare these results with experimental data in

Sec. IIIC.

III.EXPERIMENT

Our experiments are performed with

Feshbach resonance that is characterized by the param-

eters [25, 26, 27, 28] Bpole = 1007.4 G, ∆B = 0.21 G,

µa/2π¯ h = 1.02 MHz/G, µag/2π¯ h = 3.8 MHz/G, and

abg= 100.5a0, where a0is the Bohr radius. With these

parameters Eq. (20) yields |α|/2π = 1.8 mHz cm3/2. The

peak density of the BEC is typically n = |ψa|2∼ 2×1014

cm−3. The quantity |√8αψa| = 2π × 0.07 MHz can be

regarded as a Rabi angular frequency. For typical param-

eters of our experiment, this value is small compared to

γeand |ΩR|, so that the adiabatic elimination performed

in Sec. IIC is justified. To set the scale for K2in Fig. 2,

we note that Ubg/¯ h = 4.9 × 10−11cm3/s.

87Rb using a

Page 7

7

?

?

????

?

??

FIG. 3:

Bound-to-bound spectrum taken at B = 1000.0 G. Arrows indicate the resonances characterized in Tab. I.

(Color online) Excited-state spectroscopy for

87Rb. (a) Photoassociation spectrum taken at B = 1000.0 G. (b)

A. Excited-State Spectroscopy

The starting point for all experiments in this paper is

an essentially pure BEC of87Rb atoms in the hyperfine

state |F,mF? = |1,1?. These atoms are held in a crossed-

beam optical dipole trap with both beams operated at

1064 nm and with trap frequencies of (ωx,ωy,ωz)/2π =

(74,33,33) Hz. Gravity acts along the x axis. The mag-

netic field B points along the z axis and is held several

gauss away from the Feshbach resonance.

For the excited-state spectroscopy, we use two tech-

niques which complement each other. The first technique

is ordinary photoassociation spectroscopy. For this, we

simply illuminate the BEC with photoassociation light.

The light intensity is slowly increased within 80 ms to

a final power of ∼ 10 mW and held there for 100 ms.

The slow increase is needed to avoid large-amplitude os-

cillations of the cloud shape. Next, the photoassociation

light, B, and the dipole trap are switched off simultane-

ously. Finally, the remaining atom number is extracted

from a time-of-flight image.

The photoassociation light is implemented as a

traveling-wave laser beam with a waist (1/e2radius of

intensity) of w = 0.17 mm and a wavelength of ∼ 784.7

nm. In order to address as many excited states as possible

in the spectroscopy measurements, we let the photoas-

sociation beam propagate along the x axis and choose

a specific linear polarization which corresponds to 1/3

of the intensity in each of the polarizations π, σ+, and

σ−. As the population is initially in the atomic state |a?,

the loss signal is typically dominated by photoassociation

processes so that the technique is particularly sensitive

to excited states that have a large value of the photoas-

sociation coupling strength β.

Figure 3(a) shows a photoassociation spectrum at B =

1000.0 G. We extended this photoassociation scan down

to 382,037.3 GHz but did not find any further loss reso-

nances. We determined the corresponding zero-field fre-

quencies by performing a similar measurement at B ∼ 0.

We took data between 382,034 GHz and 382,051 GHz

and found photoassociation loss resonances in the range

between 382,041.8 GHz and 382,044.5 GHz. A compari-

son with photoassociation data from the Heinzen group

shows that the excited states involved are the hyperfine

and magnetic substates of the vibrational state v = 120

in the attractive 1g potential that is adiabatically con-

nected to the2P3/2+2S1/2threshold [29].

Since our technique for shifting a Feshbach resonance

with laser light relies on a bound-to-bound transition, not

on photoassociation, we developed a second spectroscopy

technique that is particularly sensitive to excited states

with a large value of ΩR. The basic idea is to first use

the Feshbach resonance to associate molecules into state

|g? and then illuminate them with light that resonantly

drives bound-to-bound transitions.

bound-to-bound light and employ the same laser beam

previously used for the photoassociation spectroscopy.

In order to avoid loss of particles due to inelastic colli-

sions between molecules, the atoms must be loaded into

a deep optical lattice before associating the molecules

[30]. The lattice has a light wavelength of 830.440 nm

We call this the

TABLE I: Parameters of four selected bound-to-bound reso-

nances. e is the elementary charge, a0 the Bohr radius. β is

negligible for all these resonances.

Polarizationωeg/2π

(MHz)

|deg|/ea0

γe/2π

(MHz) (MHz/G)

4.4(5)

4.7(5)

4.7(5)

5.3(5)

µae/2π¯ h

σ−

σ+

π

π

382,045,759.4(3)

382,045,818.2(3)

382,046,942.8(3)

382,047,581.8(3)

0.24(5)

0.29(5)

0.28(5)

0.18(5)

2.2(1)

1.7(1)

2.6(1)

2.7(1)

Page 8

8

and a depth of V0∼ 20Er, where Eris the atomic recoil

energy. As in Ref. [31] we prepare an atomic Mott insu-

lator, which contains exactly two atoms at each lattice

site in the central region of the lattice. This core is sur-

rounded by a shell of sites that contain exactly one atom

each. After loading the lattice, the laser power of one of

the dipole trapping beams is ramped to zero, as in Ref.

[31].

Next, we ramp the magnetic field slowly downward

across the Feshbach resonance so that molecules are as-

sociated [31, 32] in state |g? at sites that contain exactly

two atoms. Sites containing three or more atoms might

exist due to imperfect state preparation. Such sites are

emptied by inelastic collisions as soon as molecules begin

to form. Sites containing one atom are unaffected by the

ramp.

Next, the bound-to-bound light is turned on for 0.2 ms

at a power of ∼ 0.1 µW. This light has the same linear

polarization as for the photoassociation spectroscopy. If

a molecule in state |g? is excited on a bound-to-bound

transition, then it is likely to undergo spontaneous ra-

diative decay into a different internal state. After turn-

ing off the bound-to-bound light, the magnetic field is

ramped back across the Feshbach resonance to dissociate

the molecules that remained in state |g?. Subsequently,

the optical dipole trap at 1064 nm is turned back on,

the lattice depth is slowly ramped to zero, the cloud is

released, B is switched off, and the atom number is deter-

mined from a time-of-flight image. Molecules that were

excited by the laser are not dissociated and thus not de-

tected.

A bound-to-bound spectrum measured at B = 1000.0

G is shown in Fig. 3(b). Comparison with the photoas-

sociation spectrum in part (a) shows that many of the

excited states are visible with both techniques. But iden-

tifying promising candidates with large ΩRis not easily

possible from part (a). The light frequency calibration

for both spectra has a precision of ∼ 30 MHz and can

fluctuate within a single scan. This causes deviations

in the resonance positions between the photoassociation

spectrum and the bound-to-bound spectrum. In addi-

tion, the states |a? and |g? are degenerate at B = 1007.4

G. At B = 1000.0 G, the internal energies of a molecule

in state |g? and a pair of atoms in state |a? differ by

2π¯ h × 20 MHz [27]. In Fig. 3 this yields a 20-MHz shift

of all bound-to-bound resonances with respect to the pho-

toassociation resonances.

Very different values of the light intensity and the illu-

mination time were used when recording the two spectra.

This tremendous difference in the sensitivity of the two

methods is a result of Eq. (13).

In Ref. [12] we developed a method to determine all

the parameters of a bound-to-bound resonance. We now

apply this method to four reasonably strong bound-to-

bound resonances which are fairly close to the high-

frequency end of the spectrum in Fig. 3. Results are listed

in Tab. I. For these measurements, the laser producing

the bound-to-bound light was beat-locked to a frequency

comb, resulting in a much better precision of the fre-

quency calibration. The polarization of each resonance

was determined from a series of measurements in which

the bound-to-bound light had only one of the polariza-

tions π, σ+, or σ−. The latter two polarizations were

implemented with the bound-to-bound beam propagat-

ing along the z axis. Each resonance in Tab. I responded

to only one of these polarizations.

Our choice of the light wavelength for the experiments

in the following Sec. IIIB is based on the spectra ob-

tained here. In order to achieve a large detuning from

all excited states, the light must be detuned either to the

left or to the right of the complete spectrum in Fig. 3.

The three outermost resonances at the low-frequency end

of the spectrum in Fig. 3 have fairly strong photoassoci-

ation loss features, but show hardly any bound-to-bound

features, which is unfortunate. The high-frequency end

of the spectrum looks more promising. We therefore per-

form all the following experiments blue detuned from the

high-frequency end of the spectrum in Fig. 3. According

to Tab. I, the two strongest bound-to-bound resonances

near this end of the spectrum both respond to π polarized

light. Hence, we choose π polarization for the bound-to-

bound light in all the following experiments.

We note as a side remark, that our model, unlike Ref.

[15], assumes the existence of a direct bound-to-bound

coupling term ΩR. In Ref. [15] an indirect bound-to-

bound coupling is constructed by invoking virtual tran-

sitions into the continuum of excited atom-pair states

above threshold. This implies that the bound-to-bound

coupling should be proportional to the photoassociation

coupling β [15]. Our spectroscopy data in Fig. 3 do not

support this prediction of Ref. [15]. There are strong

photoassociation resonance that have hardly any bound-

to-bound coupling. This indicates that the indirect cou-

pling discussed in Ref. [15] is negligible [33].

B.Shifting the Feshbach Resonance with Light

We now use the spectroscopic information gathered

above to shift the Feshbach resonance with far-detuned

light.As discussed in the introduction of this paper,

we minimize the effect of the dipole trap created by the

bound-to-bound light by working in a deep optical lat-

tice.

In order to measure Re(a), we first load the atoms into

the lattice as described in Sec. IIIA. We then use exci-

tation spectroscopy [34] in the lattice, i.e., we modulate

the power of one retro-reflected lattice beam sinusoidally

as a function of time around an average lattice depth of

V0 ∼ 15Er. The modulation amplitude is ∼ 4Er. The

modulation lasts for 10 or 20 ms. During the modulation,

the atoms are illuminated with the bound-to-bound light

and B is held at a specific value close to the Feshbach

resonance. The bound-to-bound light is on for a long

enough time that sites containing two or more atoms are

essentially emptied by light-induced inelastic collisions.

Page 9

9

The signal in the excitation spectrum that is sensitive

to the modulation frequency thus stems from sites that

were initially populated by one atom. For certain modu-

lation frequencies, tunneling of an atom between two such

sites is resonantly enhanced. This leads to a frequency-

dependent loss of atoms and of atomic phase coherence.

At the end of the modulation, we switch the bound-to-

bound light off and simultaneously jump B back to a

value several gauss away from the Feshbach resonance.

Next, the dipole trap at 1064 nm is turned back on and

the lattice depth is slowly reduced to V0 ∼ 6Er, where

the gas is superfluid, thus restoring phase coherence be-

tween neighboring lattice sites. Finally, the dipole trap,

B, and the lattice are simultaneously switched off. The

time-of-flight image shows satellite peaks due to the re-

stored phase coherence.

The visibility [35] of the satellite peaks displays a min-

imum at a modulation frequency where tunneling pro-

cesses between two initially singly occupied sites are res-

onant.This minimum is located at a frequency f =

Re(U)/2π¯ h with the on-site interaction matrix-element

U = g?d3x|w(x)|4, where g = 4π¯ h2a/m and w is a

tight-binding Wannier function. The measurement of f

thus yields Re(a).

A sequence of such measurements for various values

of B yields Fig. 4(a). For parameters where Re(a) is

reduced drastically, the system becomes superfluid and

the peak in the excitation spectrum is smeared out so

much that its center cannot be determined any more.

Hence, this method is not applicable in this regime. For

comparison, the figure also shows Re(a) measured with

the same method, but in the absence of bound-to-bound

light. Clearly, the position Bres of the Feshbach reso-

nance is shifted by ∼ −0.35 G due to the presence of the

light.

We now turn to the question, how large a loss-rate co-

efficient K2is associated with this shift. In order to de-

termine K2, we load the atoms into the lattice and asso-

ciate molecules as described in Sec. IIIA. Right after as-

sociation, we dissociate the molecules. This association-

dissociation sequence serves the purpose of emptying all

sites that contain three or more atoms, which will be-

come important below. The lattice depth is V0∼ 20Er

so that tunneling is negligible. We then switch on the

bound-to-bound light and simultaneously jump B to a

value close to the Feshbach resonance. These conditions

are maintained for a variable hold time. During this hold

time, the bound-to-bound light causes rapid loss of atoms

in doubly occupied sites. Next, B is switched to a value

several gauss away from the Feshbach resonance and the

bound-to-bound light is switched off. The dipole trap

light at 1064 nm is turned back on and the lattice depth

is slowly lowered to zero. Finally, the cloud is released

and B is switched off. The remaining number of atoms

is extracted from a time-of-flight image.

The two-body loss during the hold time is described by

the master equation of Ref. [36]. We consider a lattice

site initially occupied by exactly two atoms and we ne-

?

?

?

?

?

?

FIG. 4: (Color online) Shifting the Feshbach resonance with

laser light. (a) Elastic and (b) inelastic two-body scattering

properties are shown as a function of magnetic field B. Ex-

perimental data in the presence (•) and absence (◦) of the

light are compared. The light power is 11.2 mW and the fre-

quency is ωL/2π = 382,048,158 MHz, which is 576 MHz blue

detuned from the nearest bound-to-bound transition.

solid lines in (a) and (b) show fits of Eqs. (19) and (24b), re-

spectively, to the data. The Feshbach resonance is shifted by

∼ −0.35 G. At B = 1006.91 G, we measure Re(a)/abg−1 ∼ 1

and K2 ∼ 1 × 10−12cm3/s.

The

glect tunneling between sites. The master equation then

yields the density matrix ρ = p|2??2|+(1−p)|0??0|, where

p = exp(−Γt) is the probability that a decay at this site

occurred and |n? denotes a Fock state with n atoms. The

parameter Γ is given by [36, 37]

Γ = K2

?

d3x|w(x)|4.(39)

The decay of the total atom number N in the experiment

is obtained by taking the sum over a large number of

isolated lattice sites, yielding

N(t) = N1+ N2exp(−Γt),(40)

where N1and N2are the initial atom numbers on singly

and doubly occupied sites, respectively. It is crucial that

there are no sites with three or more atoms, because they

would give rise to an additional term that would decay

more rapidly, thus making it more difficult to extract Γ

from the measured N(t).

We measured N(t) at a fixed hold time t = 2.1 ms for

various values of B and used Eqs. (39) and (40) to extract

K2(B). Results are shown in Fig. 4(b). At B = 1006.91

G, we measure Re(a)/abg− 1 ∼ 1 and K2∼ 1 × 10−12

cm3/s, which is one order of magnitude lower than our

previously published result of Ref. [12] and two orders of

magnitude lower than the corresponding result reported

for photoassociation resonances in87Rb [7, 8].

Page 10

10

We use the results of Sec. IID to calculate the theoret-

ical expectations from the sum of the two π resonances

in Tab. I. We thus expect Bres− Bpole= −0.24 G and

K2= 3 × 10−13cm3/s at |B − Bres| = ∆B which in the

model corresponds to Re(a)/abg− 1 = ±1. Both exper-

imentally observed values are somewhat larger than the

expectation. This might be due to contributions from

other bound-to-bound and photoassociation resonances

that we did not include in this estimate.

We tried to reduce K2 even further by setting the

power of the bound-to-bound light to 66 mW and its

frequency to 382,050,911 MHz which corresponds to a

detuning of ∆L/2π = 3.33 GHz from the nearest bound-

to-bound resonance. The expected and observed shifts

were −0.35 G and ∼ −0.65 G, respectively. But here

we observed K2∼ 3 × 10−12cm3/s at |B − Bres| = ∆B,

which is much worse than the expectation K2= 1×10−13

cm3/s. As the detuning is much larger than in Fig. 4,

other bound-to-bound and photoassociation resonances

contribute even more strongly to the signal, which might

explain the increased deviation between the observed val-

ues and the estimates based on the two π resonances of

Tab. I.

C.Autler-Townes Doublet

Finally, we compare the theoretical results for the

width and height of the Autler-Townes resonances in

K2(B) from Sec. IIF with experimental results. We mea-

sured Autler-Townes doublets in K2(B), as shown in Fig.

2(b). We fit a Lorentzian (24b) to each of the two peaks

in the experimental data. The best-fit values for Kmax

and W are shown in Fig. 5. The experimental procedure

and parameters are identical to Fig. 4 of Ref. [12]; see

this reference for details.

The experiment is performed in a regime where |ΩR| ≫

γe so that the Autler-Townes model of Sec. IIF is ex-

pected to be a good approximation. The dotted lines

show the corresponding predictions (35) and (38). They

agree well with the experimental data.

For comparison, we numerically determined the peaks

in K2(B) from the full model (18) with the parameters

of Tab. I. The corresponding maximum values Kmax

shown as solid lines in Fig. 5(a). As the full model (18)

does not predict Lorentzian lines, a direct comparison

with the width W is not straightforward. We decide to

use the second derivative of K2(B) at the maximum for

a comparison. The solid lines in Fig. 5(b) therefore show

the values of

2

2

are

W =

?

−

1

8K2

d2K2

dB2

?−1/2

(41)

at the peaks calculated from the full model (18). If the

peaks in the model were Lorentzian, this would yield the

width W. The solid lines also agree well with the exper-

imental data and with the dotted lines.

?

?

?

????

?

?

?

?

?

FIG. 5:

nances. K2(B) was measured for certain values of the laser

frequency at a fixed laser power of 0.47 mW. (a) The max-

imum Kmax

2

and (b) the width W were determined from a

fit to Eq. (24b). The experimental data for the resonances

that occur at the lower (◦) and higher (•) value of B both

agree well with the predictions of the full model Eq. (18)

(solid lines) which is well approximated by Eqs. (35) and (38)

(dotted lines).

(Color online) Systematic study of the loss reso-

IV. CONCLUSION

To summarize, we improved our recently developed

scheme for shifting a magnetic Feshbach resonance with

laser light by exploring the regime of even larger detuning

and laser power. We demonstrated that the light-induced

loss rate can be reduced by one order of magnitude com-

pared to our pervious work [12]. The measurements re-

quired excited-state spectroscopy and an optical lattice.

We also presented a detailed discussion of a model that

describes our experimental data.

Acknowledgments

We thank B. Bernhardt and K. Predehl for provid-

ing light from their frequency comb. We acknowledge

fruitful discussions with T. Bergeman, D. Heinzen, and

C.-C. Tsai. This work was supported by the German

Excellence Initiative through the Nanosystems Initiative

Munich and by the Deutsche Forschungsgemeinschaft

through SFB 631.

Page 11

11

[1] A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, Phys.

Rev. A 51, 4852 (1995).

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

D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151

(1998).

[3] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, e-print

arXiv:0812.1496.

[4] P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and

J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996).

[5] J. L. Bohn and P. S. Julienne, Phys. Rev. A 56, 1486

(1997).

[6] F. K. Fatemi, K. M. Jones, and P. D. Lett, Phys. Rev.

Lett. 85, 4462 (2000).

[7] M. Theis, G. Thalhammer, K. Winkler, M. Hellwig,

G. Ruff, R. Grimm, and J. Hecker Denschlag, Phys. Rev.

Lett. 93, 123001 (2004).

[8] G. Thalhammer, M. Theis, K. Winkler, R. Grimm, and

J. Hecker Denschlag, Phys. Rev. A 71, 033403 (2005).

[9] K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne,

Rev. Mod. Phys. 78, 483 (2006).

[10] R. Ciury? lo, E. Tiesinga, and P. S. Julienne, Phys. Rev.

A 71, 030701 (2005).

[11] K. Enomoto, K. Kasa, M. Kitagawa, and Y. Takahashi,

Phys. Rev. Lett. 101, 203201 (2008).

[12] D. M. Bauer, M. Lettner, C. Vo, G. Rempe, and S. D¨ urr,

Nature Phys. 5, 339 (2009).

[13] F. A. van Abeelen, D. J. Heinzen, and B. J. Verhaar,

Phys. Rev. A 57, R4102 (1998).

[14] M. Junker, D. Dries, C. Welford, J. Hitchcock, Y. P.

Chen, and R. G. Hulet, Phys. Rev. Lett. 101, 060406

(2008).

[15] M. Mackie, M. Fenty, D. Savage, and J. Kesselman, Phys.

Rev. Lett. 101, 040401 (2008).

[16] S. J. Kokkelmans and M. J. Holland, Phys. Rev. Lett.

89, 180401 (2002).

[17] T. K¨ ohler, T. Gasenzer, and K. Burnett, Phys. Rev. A

67, 013601 (2003).

[18] E. Timmermans, P. Tommasini, M. Hussein, and A. Ker-

man, Phys. Rep. 315, 199 (1999).

[19] M. J. Holland, J. Park, and R. Walser, Phys. Rev. Lett.

86, 1915 (2001).

[20] F. A. van Abeelen and B. J. Verhaar, Phys. Rev. Lett.

83, 1550 (1999).

[21] S. D¨ urr, J. J. Garc´ ıa-Ripoll, N. Syassen, D. M. Bauer,

M. Lettner, J. I. Cirac, and G. Rempe, Phys. Rev. A 79,

023614 (2009).

[22] G. Breit and E. Wigner, Phys. Rev. 49, 519 (1936).

[23] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703

(1955).

[24] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,

Atom-Photon Interactions (Wiley, New York, 1992).

[25] A. Marte, T. Volz, J. Schuster, S. D¨ urr, G. Rempe, E. G.

van Kempen, and B. J. Verhaar, Phys. Rev. Lett. 89,

283202 (2002).

[26] S. D¨ urr, T. Volz, and G. Rempe, Phys. Rev. A 70,

031601(R) (2004).

[27] S. D¨ urr, T. Volz, A. Marte, and G. Rempe, Phys. Rev.

Lett. 92, 020406 (2004).

[28] T. Volz, S. D¨ urr, S. Ernst, A. Marte, and G. Rempe,

Phys. Rev. A 68, 010702(R) (2003).

[29] C.-C. Tsai and D. Heinzen. Personal communication.

[30] G. Thalhammer, K. Winkler, F. Lang, S. Schmid,

R. Grimm, and J. Hecker Denschlag, Phys. Rev. Lett.

96, 050402 (2006).

[31] T. Volz, N. Syassen, D. M. Bauer, E. Hansis, S. D¨ urr,

and G. Rempe, Nature Phys. 2, 692 (2006).

[32] T. K¨ ohler, K. G´ oral, and P. S. Julienne, Rev. Mod. Phys.

78, 1311 (2006).

[33] There is a correlation in the observed line strengths, in-

sofar as almost every resonance with a strong bound-to-

bound feature correlates with a fairly strong feature in

the photoassociation spectrum. But this does not imply

that these lines have a large β. Instead, Eq. (18) shows

that ΩR alone (even for β = 0) can create a strong loss

feature for atoms in state |a? illuminated with light reso-

nant with the photoassociation transition, as long as ∆g

is not too large.

[34] T. St¨ oferle,H. Moritz,

T. Esslinger, Phys. Rev. Lett. 92, 130403 (2004).

[35] F. Gerbier, A. Widera, S. F¨ olling, O. Mandel, T. Gericke,

and I. Bloch, Phys. Rev. Lett. 95, 050404 (2005).

[36] N. Syassen, D. M. Bauer, M. Lettner, T. Volz, D. Dietze,

J. J. Garc´ ıa-Ripoll, J. I. Cirac, G. Rempe, and S. D¨ urr,

Science 320, 1329 (2008).

[37] Here we use a and K2 from Eqs. (15) and (18). We de-

rived these equations assuming that the sample is Bose

condensed which is not the case in a deep optical lattice.

But a and K2 express only properties of two-body scat-

tering processes. They are unaffected by the many-body

state of the system.

C. Schori,M. K¨ ohl,and