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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)