Efficient production of long-lived ultracold Sr2 molecules

Alessio Ciamei,∗ Alex Bayerle, Chun-Chia Chen, Benjamin Pasquiou, and Florian SchreckVan der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam,

Science Park 904, 1098 XH Amsterdam, The Netherlands(Dated: September 3, 2018)

We associate Sr atom pairs on sites of a Mott insulator optically and coherently into weakly-bound ground-state molecules, achieving an efficiency above 80%. This efficiency is 2.5 times higherthan in our previous work [S. Stellmer, B. Pasquiou, R. Grimm, and F. Schreck, Phys. Rev.Lett. 109, 115302 (2012)] and obtained through two improvements. First, the lifetime of themolecules is increased beyond one minute by using an optical lattice wavelength that is furtherdetuned from molecular transitions. Second, we compensate undesired dynamic light shifts thatoccur during the stimulated Raman adiabatic passage (STIRAP) used for molecule association. Wealso characterize and model STIRAP, providing insights into its limitations. Our work shows thatsignificant molecule association efficiencies can be achieved even for atomic species or mixtures thatlack Feshbach resonances suitable for magnetoassociation.

PACS numbers: 67.85.-d, 42.50.Hz, 33.80.-b, 37.10.Jk

I. INTRODUCTION

Over the last fifteen years considerable experimentaleffort has been invested into the realization of ultracoldmolecular samples. Ultracold molecules hold promise forunveiling novel phases of matter near quantum degener-acy, implementing quantum information protocols, andenabling precision measurements beyond atomic physics[1–3]. Ultracold dimers in their rovibrational groundstate can be created in a two-step process from ultra-cold atoms. In the first step, atom pairs are associatedinto weakly-bound molecules and in the second step, themolecules are transferred from a weakly-bound state tothe rovibrational ground state [4, 5]. So far the first stepin these experiments has relied on the existence of mag-netically tuneable Feshbach resonances [6, 7]. By ramp-ing an external magnetic field adiabatically across sucha resonance, a coherent transfer between a pair of freeatoms and a molecular bound state can be accomplished.Such magneto-association is hard or even impossible for avast class of atomic systems of interest, for instance com-binations of an alkali metal and an alkaline-earth metalor pairs of alkaline-earth metal atoms. The former sys-tems possess only extremely narrow magnetic Feshbachresonances [8, 9], the latter none at all.

Production of ultracold weakly-bound ground-stateSr2 molecules was achieved in our previous work [10]and in [11], relying respectively on coherent and non-coherent optical transfer schemes, thus overcoming theabsence of magnetic Feshbach resonances in the non-magnetic ground state of these atoms. More recently,two-photon coherent transfer of cold Rb atom pairs intoground-state Rb2 molecules was demonstrated using afrequency-chirped laser pulse [12]. The coherent popula-tion transfer of [10] was stimulated Raman adiabatic pas-sage (STIRAP), which evolves a dark state from a pair of

∗ Sr2molecules@strontiumBEC.com

atoms into a molecule [13, 14]. Unfortunately, because oflosses by non-adiabatic coupling and short lifetime of themolecules, the molecule association efficiency was only30%, far below the efficiency potentially achievable bySTIRAP. Moreover, the short lifetime hindered furtherusage of the molecular sample.

In this article we show how to overcome these limita-tions. As in [10] we investigate the production of 84Sr2 ul-tracold ground-state molecules by STIRAP starting froma Mott insulator (MI). We increase the lifetime of themolecules to over one minute by using an optical lat-tice wavelength that, unlike before, is far detuned fromany molecular transition. We identify that the result-ing STIRAP efficiency of slightly above 50 % is limitedby the finite lifetime of the dark state arising from un-wanted light shifts. We show how to overcome these lightshifts with the help of an additional compensation beam[15], leading to a STIRAP efficiency above 80 %. Ourwork validates a general way of producing large samplesof weakly-bound molecules without relying on Feshbachresonances. This will open the path for new classes of ul-tracold dimers useful for metrology experiments [16–18],for ultracold chemistry [19, 20] and for quantum simula-tion experiments relying on a strong permanent electricdipole moment [21–23].

This article is organized as follows. In Sec. II wepresent an overview of our experimental strategy. InSec. III we introduce the model used to describe theSTIRAP and we discuss the constraints imposed on therelevant experimental parameters when high transfer effi-ciency is required. In Sec. IV we describe the experimen-tal sequence leading to the initial atomic sample and theoptical scheme for the creation of photoassociation (PA)laser light. In Sec. V A, we measure relevant parametersof the system and we show the effect of the lattice onthe free-bound Rabi frequency. In Sec. V B we use STI-RAP to associate atom pairs into molecules, achievinga transfer efficiency of ∼ 50 %, and we show that thisprocess is limited by the finite lifetime of the dark state,

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arising from unwanted light shifts on the binding energyof the ground-state molecule. This challenge is overcomein Sec. V C by modifying the standard scheme with theaddition of a light-shift compensation beam, reaching anefficiency higher than 80 %. Finally, in Sec. V D we char-acterize the effects of the lattice light shifts on STIRAPand we measure the molecule lifetime.

II. EXPERIMENTAL STRATEGY

We will now discuss how one can optically andcoherently associate pairs of atoms into ground-statemolecules. We will explain how STIRAP works and canreach near-unit molecule association efficiency. We moti-vate the use of a MI as initial atomic sample and specifythe STIRAP implementation used here.

To optically transfer a pair of atoms in state |a〉 intothe molecular state |m〉, we use a Λ scheme, including anoptically excited molecular state |e〉, see Fig. 1. States|a〉 and |e〉 are coupled with Rabi frequency ΩFB by thefree-bound laser LFB and |e〉 is coupled to |m〉 with Rabifrequency ΩBB by the bound-bound laser LBB.

The conceptually simplest method for coherentmolecule association are two consecutive π pulses, thefirst between |a〉 and |e〉 and the second between |e〉and |m〉. To provide efficient transfer, this scheme needsto be executed much faster than the lifetime of the ex-cited state |e〉, τe, which requires high Rabi frequencies(ΩFB,BB γe = 1/τe). Since the Franck-Condon factor(FCF) of the free-bound transition is small, satisfying thecondition ΩFB γe is experimentally very challenging.

STIRAP overcomes this limitation by minimizinglosses from |e〉 and provides coherent population transfereven when the condition ΩFB,BB γe is not satisfied. Tosimplify the discussion we first introduce STIRAP with-out loss from the excited state (γe = 0) and with lasersLFB and LBB on resonance with the transitions |a〉 − |e〉and |m〉 − |e〉, respectively. In this system a dark stateexists, namely an eigenstate orthogonal to |e〉. If onlyone of the two lasers (LFB or LBB) is on, then the darkstate coincides with one of the eigenstates in the absenceof light (|m〉 or |a〉, respectively). When both lasers areon, the dark state is |dark〉 = cos(θ) |a〉 − sin(θ) |m〉 withθ = arctan (ΩFB/ΩBB). If the tuning knob θ is varied intime from 0 to π/2, then the dark state is moved from|a〉 to |m〉. Experimentally this is realized by changingthe intensities IFB,BB of LFB,BB, first switching on LBB,then increasing IFB while ramping off IBB, and finallyswitching off LFB. Since an energy gap exists betweenthe dark state and the other two instantaneous eigen-states of the system, the adiabatic theorem of quantummechanics applies [24, 25], provided that the change inthe Hamiltonian is slow enough compared to the energygap. This means that the population is kept in the darkstate, providing unit transfer efficiency. State |e〉 is onlyused to induce couplings but never significantly popu-lated. As a consequence this scheme works even in the

0 20 40 60 80 100

-0.08

-0.04

0.00

14704.84

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14704.92

ψ (1

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Distance r (Å)

Ene

rgy

(cm

-1)

Distance r (Å)

(a)

(b)

gΧ1Σ+u1 (0+)

ћΔ

ћδ

LFB

ψeψm

ψa x102

LBB

LCOMP

|a⟩|m⟩

|e⟩ |a*⟩ћΔωe

ћΔωCOMP

10 100 1000

-4

-2

0

2

4

Figure 1. (color online) (a) 84Sr2 molecular potential forthe electronic ground state X1Σ+

g (J = 0) and the opticallyexited state 1(0+

u ) (J = 1). The energy is referenced to theground-state asymptote. The Λ scheme (|a〉, |e〉, |m〉) usedfor STIRAP is indicated along with the coupling laser fieldsLFB and LBB and their one- and two-color detunings ∆ andδ. The laser field LCOMP with detuning ∆ωCOMP from the|a〉− |a∗〉 atomic transition is added for light-shift compensa-tion in Sec. V C. In the situation shown ∆, δ, ∆ωCOMP > 0and ∆ωe < 0. (b) Probability amplitude for the nuclear wave-function of state |a〉, |e〉 and |m〉. The effect of an externalharmonic confinement is taken into account to determine ψa,see Sec. V B.

presence of dissipation (γe 6= 0) as will be explained inSec. III.

In order to detect that atoms have been associatedto molecules, we exploit the fact that STIRAP is re-versible. After the association STIRAP (aSTIRAP) de-scribed above, we push remaining free atoms away witha pulse of resonant light. Molecules are barely affectedby this light pulse because they do not have any strongoptical transition close in frequency to the atomic tran-sition used. To detect the molecules, we dissociate themby a time-mirrored aSTIRAP, which we call dissociationSTIRAP (dSTIRAP), and detect the resulting atoms byabsorption imaging. The sequence “aSTIRAP — pushpulse — dSTIRAP” constitutes a STIRAP cycle and will

3

be used to prepare samples with homogeneous conditionsfor molecule association and to measure the STIRAP ef-ficiency, see Sec. V B.

We exploit a deep optical lattice to create a MI of dou-bly occupied sites with high on-site peak density. Thereason for this is twofold. Firstly, the optical lattice in-creases the free-bound Rabi frequency, see Sec. V A 1,leading to an increase in the STIRAP efficiency. Sec-ondly, the lattice increases the molecule lifetime by sup-pressing collisional losses, see Sec. V D 2, leading bothto an increase in the STIRAP efficiency and to a longerlifetime of the resulting molecular sample.

The Λ scheme used here is the same as the one em-ployed in our previous work [10]. The relevant potentialenergy curves of 84Sr2 are shown in Fig. 1. The elec-tronic ground state is a X1Σ+

g (J = 0) state, asymp-

totically correlating to two ground-state Sr atoms (1S0),and the excited state is a 1(0+

u ) (J = 1) state, corre-lating to one Sr atom in the ground state and one inthe optically excited state 3P1. The initial state |a〉consists of two atoms in the lowest vibrational level ofan optical lattice well. At short range, their relative-motion wavefunction is proportional to a scattering stateabove the dissociation threshold of X1Σ+

g (J = 0). Ourtarget state |m〉 is the second to last vibrational level(ν = −2) supported by this potential, with binding en-ergy Em = h × 644.7372(2) MHz, where h is the Planckconstant. As intermediate state |e〉 we chose the ν = −3vibrational level of the 1(0+

u ) potential, with binding en-ergy Ee = h × 228.38(1) MHz. The STIRAP lasers areappropriately detuned from the 1S0−3P1 transition of Srat 689 nm, which has a linewidth of Γ3P1

= 2π×7.4 kHz.For the push beam we use the 1S0 − 1P1 transition at461 nm, which has a linewidth of Γ1P1

= 2π × 30.5 MHz.

III. THEORY

In this section we will introduce a model for moleculeassociation via STIRAP. We will discuss the conditionson experimental parameters under which STIRAP is ef-ficient. An important insight will be that finite two-color detuning δ can lead to a significant reduction inthe dark state lifetime and therefore efficiency. We givean overview of the sources for this detuning in our exper-iment and ways to overcome it.

A. Model

A model for STIRAP starting from two atoms in theground state of an optical lattice is given by the time-dependent Schrodinger equation applied to the three rel-evant states, which in the rotating wave approximationtakes the form [26, 27]

id

dt

aem

= −1

2

iγa ΩFB 0ΩFB iγe − 2∆ ΩBB

0 ΩBB iγm − 2δ

aem

.

(1)The amplitudes a(t), e(t), and m(t) correspond to thestates |a〉, |e〉, and |m〉, respectively. Losses from thesestates are described by γa, γe and γm, which are respon-sible for the coupling of the three-level system to theenvironment. When loss rates are set to zero, the nor-malization condition is |a|2 + |e|2 + |m|2 = 1. Finally, ∆and δ are the one- and two-color detunings, see Fig. 1. Inthe remainder of this section we will study the STIRAPefficiency under the following assumptions. STIRAP isexecuted during the time interval [0, T ], outside of whichthe two Rabi frequencies ΩFB,BB are zero. For timest ∈ [0, T ] the Rabi frequencies are described by (co)sinepulses of identical amplitude Ωm, ΩBB = Ωm cos(θ) andΩFB = Ωm sin(θ), with θ = πt/2T . The scattering rate γeis the dominant loss term in the system, i.e. γe γa,m.The two-color detuning depends on time only throughθ, i.e. δ = δ(θ) = δmf(θ), where δm is the (signed)extremum of δ(θ) with highest amplitude, and f(θ) de-scribes the time variation caused by the time-dependentlight shifts discussed in Sec. III C. Finally, unless statedotherwise, we will assume ∆ = 0.

B. Parameter constraints

We now analyse the constraints on the (experimentallycontrollable) detunings and Rabi frequencies under whichSTIRAP associates molecules with near-unit efficiency.We will first consider the resonant case (zero detunings)and derive the parameter constraints analytically, follow-ing Ref. [26]. We will then study the effects of deviationsfrom two-color resonance, where an approximate solutionand new parameter constraints will be presented. Finally,the effects of one-color and two-color detunings will becompared.

We analyse the problem in the experimentally relevantcase of strong dissipation Ωm <∼ γe, in which STIRAPhas the potential to outperform two appropriate consec-utive π-pulses. Here dissipation leads to a strong non-hermitian contribution in the hamiltonian [28]. By adi-abatically eliminating the variable e(t) as in Ref. [26],

we derive the effective Hamiltonian Heff for the subspace|a〉, |m〉,

Heff = −i γ2

(sin(θ)2 sin(θ) cos(θ)

sin(θ) cos(θ) cos(θ)2 + 2iAf(θ)

), (2)

where γ = Ω2m/γe and A = δm/γ.

We first examine the case δ = ∆ = 0 and we willshow that adiabatic evolution ensures near-unit transferefficiency [26]. Despite the strong dissipation, this sys-tem supports the same dark state as before, the eigen-state |dark〉 = cos(θ) |a〉 − sin(θ) |m〉. The orthogonal

4

lossy state is the bright state |bright〉. In order to gainsome insight, we work in the adiabatic representation,i.e. in the instantaneous eigenbasis |dark〉, |bright〉of the Hamiltonian (2). A wavevector |Ψ, t〉 in the

basis |a〉, |m〉 is written |Ψ′ , t〉 = U(t)|Ψ, t〉 in thisnew basis, with the appropriate unitary transforma-

tion U . The time-dependent Schrodinger equation reads

i ddt |Ψ′〉 = UHeff U

−1|Ψ′〉 + i(ddt U

)U−1|Ψ′〉 = H

′ |Ψ′〉,

where UHeff U−1 is diagonal and

(ddt U

)U−1 gives rise

to non-adiabatic couplings. If we take s = t/T ∈ [0, 1] asour independent variable, the Schrodinger equation be-

comes i dds |Ψ′〉 = TH

′ |Ψ′〉. In the case of zero detunings,

TH′

is independent of s and the equation is exactly solv-able. The solution for the dark-state amplitude d(s) isd(s) = B−e

−iλ−s + B+e−iλ+s, where λ−,+ are respec-

tively the most and least dissipative eigenvalues of TH′,

and B−,+ are constants. In the adiabatic regime, char-acterized by α = γT 1, we have Im (λ−) ' −α/2and Im (λ+) ' −π2/2α. The transfer efficiency η is then

given by η = |d(s = 1)|2 ' |B+|2e2λ+ ' e−π2/α. Thus,

STIRAP with near-unit efficiency requires α π2, whichcan be interpreted as the adiabaticity condition.

Next we examine the case of non-zero two-color de-tuning, i.e. δ 6= 0, and we show that adiabatic evolu-tion does not guarantee high transfer efficiency. Goingfrom zero to finite δ, the initially dark state is mixedwith |e〉, resulting in a finite lifetime. For simplicitywe all the same continue to use the label “dark state”for this state and, similarly, keep using “bright state”for the other eigenstate. We apply the same method

as used previously, with U being this time the appro-priate non-unitary transformation. The time evolution

of the system is determined by TH′, which now de-

pends on s and the solution is not trivial. However,since we are only interested in small δ and the popu-

lations at the end of the pulse, we replace TH′

with itstime average. The imaginary parts of the most and leastdissipative eigenvalues are, to leading order in An/αm,Im (λ−) ' −α/2 and Im (λ+) = −1/2(π2/α + CA2α),

where C = 8π

∫ π/20

f(θ)2 sin(θ)2 cos(θ)2dθ. Thus, thetransfer efficiency is approximately given for α 1,|A| 1 by

η ' e−(π2/α+CA2α), (3)

which is only a function of the parameters α and A. Weidentify two regimes depending on the value of |A|α =

|δm|T . For |δm|T π/√C, the transfer efficiency is

limited by non-adiabatic transitions of the dark state tothe bright state. For |δm|T π/

√C, the main limita-

tion arises from the now finite lifetime of the dark state.For a non-zero detuning δ, the transfer efficiency fea-tures a maximum when varying pulse time T , which isdetermined only by the parameter |A| = |δm|/γ, and the

maximum efficiency is approximately ηmax = e−2π√C|A|.

We now address the case of non-zero one-color detun-ing ∆ 6= 0, but zero two-color detuning δ = 0, and weshow that, as for δ = ∆ = 0, adiabatic evolution ensuresnear-unit transfer efficiency. In this case, the Hamil-tonian (1) supports a dark state, and by following the

method used for δ = ∆ = 0, it can be shown that TH′

is

independent of s. The eigenvalues of TH′

are, to leadingorder in 1/α, given by Im (λ−) ' −α/2

(1 + 4(∆/γe)

2)

and Im (λ+) ' −π2/2α. This implies that, even for∆ 6= 0, we can have |Im (λ+) | 1 and thus η ' 1for large enough values of α. To compare the effects ofone-color and two-color detunings, we analyze the con-ditions to ensure high transfer efficiency at large butfixed α. For a constant one-color detuning we derive∆ γeα/2π, while for a constant two-color detuning

we have |δm| γ√

2/α <∼ γe√

2/α, which is a muchstronger constraint. The STIRAP efficiency is thereforemuch more sensitive to the two-color detuning than tothe one-color detuning, which is also expected from en-ergy conservation.

It is noteworthy to point out that population trans-fer between |a〉 and |m〉 can be obtained by exploitingthe coherent dark state superposition in a diabatic evo-lution, i.e. projection, as opposed to STIRAP. How-ever the efficiency in this case is bound to be at mostηproj = 25 %, which is the value for equal free-bound andbound-bound Rabi frequencies, as obtained from ηproj =|〈dark|a〉|2 × |〈m|dark〉|2 =

(1− sin(θ)2

)sin(θ)2 ≤ 1/4.

As a consequence surpassing the maximum possible ef-ficiency of ηproj can be regarded as a requirement forSTIRAP to be relevant.

C. Improving STIRAP efficiency

Finally, we discuss how we can improve the moleculeproduction efficiency based on this theoretical descrip-tion. We will focus on the limitations arising from lightshifts induced on the states of the Λ scheme that con-tribute to δ and ∆. We distinguish between static anddynamic light shifts, where static and dynamic refer totheir behavior during the STIRAP sequence. The lasersLFB and LBB, whose intensities vary during STIRAP, in-duce time- and space-dependent shifts ∆FB and ∆BB onthe free-bound transition. In a similar way, these lasersalso induce shifts δFB and δBB on the two-photon tran-sition. These shifts correspond to changes −hδFB,BB ofthe binding energy of molecules in state |m〉, which is theenergy difference between |a〉 and |m〉. The optical lat-tice, which provides the external confinement, inducesonly the static, space-dependent light shifts ∆Lattice

and δLattice, respectively, onto the free-bound transitionand the two-photon transition. Thus, we decompose∆ and δ into ∆ = ∆Lattice + ∆FB + ∆BB + ∆0 andδ = δLattice + δFB + δBB + δ0. The offsets ∆0 and δ0 canbe freely adjusted by tuning LFB,BB and depend neitheron space nor on time.

These light shifts can influence the molecule produc-

5

tion efficiency η. In our system the effect of ∆ on η isnegligible, see Sec. V C 3, justifying the approximation∆ = 0. Since we also fulfill γa,m γe (see Sec. V A 2and V D 2), we can use Eqn. 3, which shows that the max-imum transfer efficiency for optimal pulse time dependsonly on |A| = |δm|/γ. The efficiency is highest for small|A|, which means we have to achieve low |δm|. The con-tribution δLattice to δm could be reduced by increasing thelattice beam diameter and adjusting δ0 to compensate forthe average remaining light shift across the sample. How-ever in this work we reduce the inhomogeneity by simplyremoving atoms on sites with large light shift using a STI-RAP cycle, see Sec. V B and Sec. V C 2. The contributionδFB is proportional to IFB ∝ sin(θ)2. Lowering A = δm/γby lowering IFB is not possible since also γ ∝ IFB. Astraightforward way to half the effect of this contribu-tion is to adjust δ0 such that δ = δm(sin(θ)2 − 1/2). Togo further and essentially cancel δFB we demonstrate inSec. V C the use of an additional laser beam (LCOMP)that produces the exact opposite light shift of LFB. Thelast light shift, δBB, can be neglected in our case becauseLBB has low intensity.

IV. EXPERIMENTAL SETUP AND CREATIONOF THE MOTT INSULATOR

We will now describe the experimental setup and theprocedure used to prepare the atomic sample that is thestarting point for molecule creation, and describe thelaser system that generates the PA light.

The experimental apparatus is our Sr quantum gas ma-chine, which is described in depth in [29]. The experi-mental sequence leading to a Bose-Einstein condensate(BEC) of 84Sr starts with a magneto-optical trap operat-ing on the 1S0 − 1P1 transition, capturing atoms comingfrom a Zeeman slower. We further cool the atomic cloudby exploiting the intercombination transition 1S0 − 3P1

at 689 nm, reaching a temperature of 1.0µK. We thenload the atoms into a crossed-beam dipole trap (DT),consisting of one horizontal beam with vertical and hori-zontal waists of 20µm and 330µm, respectively, and onenear-vertical beam with a waist of 78µm. After evapora-tive cooling we obtain a BEC of about 3.5 × 105 atoms,with a peak density of 6.0(5) × 1013 cm−3. The trap-ping frequencies are ωx = 2π × 16 Hz, ωy = 2π × 11 Hz,ωz = 2π × 95 Hz, where the y-axis points along thehorizontal DT beam and the z-axis is vertical. TheThomas-Fermi radii are Rx = 23µm, Ry = 35µm andRz = 4.1µm, and the chemical potential is µ = 30(2) nK.These parameters are chosen to maximize the number ofdoubly-occupied sites in the optical lattice.

The Mott insulator (MI) is realized by adiabati-cally loading the BEC into a 3D optical lattice, whosestanding-wave interference pattern is obtained from threeorthogonal, retro-reflected laser beams, with waists of78µm, 206µm and 210µm, derived from a single-modelaser with λLattice = 1064 nm wavelength (Innolight

Mephisto MOPA). Orthogonal polarizations and fre-quency offsets of about 10 MHz between the three laserbeams ensure that the optical potential experienced bythe atoms can be well described by the sum of three in-dependent 1D lattices. The lattice depth is calibratedthrough Kapitza-Dirac diffraction [30].

The BEC is adiabatically loaded into the optical latticeby increasing the lattice potential in three consecutiveexponential ramps. A first ramp of 500 ms duration isused to reach a depth of about 5 Er in each lattice direc-tion, where Er = h× 2.1 kHz is the recoil energy. Duringthe second ramp, which takes 150 ms and reaches 20 Er,the system undergoes the superfluid to MI phase transi-tion. We then switch off the crossed-beam DT. The finalramp increases the trap depth to 200 Er in 5 ms in orderto reduce the size of the lattice site ground-state wave-function, which in turn increases the free-bound Rabifrequency, see Sec. V A 1.

Since only atoms in doubly occupied sites will con-tribute to molecule formation, we maximize that numberby varying initial atom number and DT confinement.To determine the number of doubly-occupied sites, weexploit three-body recombination and photoassociation.After loading the lattice, three-body recombination emp-ties triply-occupied sites on a 1/e timescale of 104(15) ms.After 200 ms of wait time, we also empty the doubly-occupied sites by PA using LFB. The difference in atomnumber before and after the PA pulse is twice the numberof doubly-occupied sites. Measuring the atom numberalso directly after the loading sequence we derive the oc-cupation fractions 47(9) %, 30(5) % and 23(4) % for theoccupation numbers n = 1, n = 2 and n = 3, respec-tively, where we assumed that the number of sites withmore than three atoms is negligible. To prepare samplesfor molecule association by STIRAP, we wait 200 ms af-ter loading the lattice, thereby removing most sites withthree or more atoms.

We now describe the laser setup used to illuminatethe atoms with PA light, see Fig. 2, which also gen-erates the light-shift compensation beam LCOMP, seeSec. V C 1. The two laser fields LFB and LBB are de-rived, by free-space splitting and recombination, from asingle injection-locked slave laser seeded by a master os-cillator with linewidth of less than 2π × 3 kHz. The fre-quencies of the laser fields are tuned by acousto-opticalmodulators (AOMs) and the beams are recombined intothe same single-mode fiber with the same polarization, sothat the main differences on the atomic cloud are theirfrequency and intensity. This setup ensures a good co-herence between the two laser fields, which must matchthe Raman condition, which means that the frequencydifference between the lasers must be equal to the bind-ing energy of state |m〉 divided by h. The quality ofthe beat note of LFB with LBB is essentially set by theelectronics controlling the AOMs. In the spectrum of thebeat note, recorded on a photodiode and analyzed with abandwidth of 2π× 3 Hz, the beat signal rises by ∼ 60 dBabove the background and has a width of 2π × 60 Hz.

6

AO

M1

AO

M2

AO

M C

OM

P

Master laser

Atoms

Slave laser

λ/2 waveplate

Beam splitter

Polarizing beam splitter

Faraday isolator

Acousto-Optic Modulator

LBB

LCOMP

LFB

Figure 2. (color online) Optical setup used to produce pho-toassociation light and to compensate the light shift δFB inboth space and time.

This narrow width can be neglected on the time scale ofthe experiment (hundreds of µs) and doesn’t need to betaken into account in our theoretical model of STIRAP.Frequency fluctuations of the master laser do not changethe two-color detuning δ. They only change the one-colordetuning ∆ to which the STIRAP efficiency has low sen-sitivity, see Sec. III B. Finally, all laser fields used for PAare contained in one beam, which is sent onto the atomiccloud horizontally, under an angle of 30 from the y-axis. The waist of the beam at the location of the atomsis 113(2)µm. The polarization is linear and parallel toa vertically oriented guiding magnetic field of 5.30(5) G,which means that only π transitions can be addressed.The magnetic field splits the Zeeman levels of the state|e〉 by 2π × 1.65(1) MHz γe.

V. MOLECULE CREATION

We will now discuss molecule creation via STIRAP.First, we will characterize the parameters that can bemeasured before attempting STIRAP (PA Rabi frequen-cies, dynamic light shifts from PA light, and static lightshifts from the lattice), see Tab. I. We will then applySTIRAP on our sample and identify the finite lifetimeof the dark state arising from the light shift δFB as themain limitation to the STIRAP efficiency. We will showhow to overcome this limitation by minimizing δFB usinga compensation beam and we will examine the effects ofthis compensation scheme on STIRAP. Finally we willcharacterize the spectral properties of the initial atomicsample and the lifetime of the molecular sample.

√⟨n⟩ (107 cm-3/2)

√IFB

√(W

/cm

2 )

ΩFB/2π

k

Hz

()

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

20

Figure 3. Free-bound Rabi frequency ΩFB as function of√〈n〉, where 〈n〉 is the average on-site density of a single

atom. The line is a linear fit to the data, justified in the text.

A. Parameter characterization

1. Rabi frequencies

We measure the bound-bound Rabi frequency ΩBB

through loss spectroscopy by probing the Autler-Townessplitting induced by ΩBB with the free-bound laser[31]. We derive from our measurements ΩBB = 2π ×234(5) kHz/

√W/cm2.

To measure the free-bound Rabi frequency ΩFB weshine LFB on the MI and detect the resulting decay ofatom number as a function of time. The observed de-cays are well described by exponentials with time con-stants τ . We measure τ for several lattice depths andintensities of LFB. The variation of the atom number

Na ∝ |a|2 is determined by a = −Ω2FB

2Γea, which is ob-

tained by adiabatic elimination of variable e in model (1)and by using the natural linewidth Γe of the free-boundtransition as γe. Thus the measured decay time constantτ can be related to the Rabi frequency by ΩFB =

√Γe/τ .

The measurement of Γe is explained in Sec. V D 1. Fig-ure 3 shows the free-bound Rabi frequency ΩFB as a func-tion of

√〈n〉, where n is the on-site density of a single

atom and 〈·〉 is the spatial average over one site. We ob-

serve that ΩFB is proportional to√〈n〉, and we derive

ΩFB = 2π × 6.0(1) 10−7 kHz cm3/2/√

W/cm2.We now present an argument that explains the ob-

served linear relationship between ΩFB and√〈n〉 [32–34].

The free-bound Rabi frequency depends on the Franck-Condon factor as ΩFB ∝ FCFFB ∝

∫∞0ψ∗e(r)ψa(r)r2dr,

where ψe and ψa are the radial nuclear wavefunctions of

7

the excited bound state and the initial atomic state, seeFig 1b. ψa is determined at short inter-particle distancesby the molecular potential, and at long distances by theexternal harmonic confinement. ψe is non negligible onlyat short distances, and determined by the excited molec-ular potential. Therefore the value of FCFFB is only de-termined by ψe and the short-range part of ψa. We vary√〈n〉 by changing the external confinement provided by

the lattice, which affects strongly the long-range part ofψa. We now need to determine how this change affectsboth the shape and the amplitude of the short-range partof ψa.

The short-range length scales la,e of ψa,e are set by thelocation of the wavefunction node with largest internu-clear distance and here la,e < 10 nm. Since in our casethe two atoms described by ψa are both in the groundstate of a lattice well, the total extent R of ψa is on theorder of the harmonic oscillator size, which here meansR >∼ 100 nm. A modification of the wavefunction at largedistance by a change in the lattice confinement can af-fect the shape of the short-range wavefunction through aphase shift of order δφ ' (la−as)k, where as = 6.6 nm isthe s-wave scattering length and k ' 1/R is the relativewavevector at long distance. Since R la and R as,the phase shift δφ is negligible and does not affect thevalue of the integral FCFFB. The relative wavefunctionof two non-interacting particles in the external potential,φ0(r), is a good approximation of ψa(r) for r la. Sincela R we can choose r such that ψa(r) ' φ0(r) ' φ0(0).As the external trap is changed, the amplitude at shortdistance is then simply scaled by a factor φ0(0) ∝

√〈n〉,

thus leading to ΩFB ∝ FCFFB ∝√〈n〉.

2. Light shifts and loss by LFB,BB light

We now characterize the parameters δFB and δBB,which are the time- and space-dependent light shiftscontributing to the two-color detuning δ that are in-duced by the free-bound laser LFB and the bound-bound laser LBB, respectively. These detunings corre-spond to binding energy changes −hδFB,BB of moleculesin state |m〉. We measure δFB (δBB) by two-colordark-state spectroscopy for different intensities of LFB

(LBB) at fixed intensity of LBB (LFB). We obtainδFB = 2π × 18.3(6) kHz/(W/cm2) and δBB = −2π ×10(10) kHz/(W/cm2). These values agree with a sim-ple theoretical estimation, which takes into account onlytransitions toward the optically excited atomic state|a∗〉 = 3P1(mJ = 0). Indeed the light shift inducedby LFB,BB on |m〉 is weak because LFB,BB do not cou-ple this state to any other bound state out of the Λscheme and because the coupling of |m〉 to |a∗〉 has tobe scaled by a free-bound FCF. By contrast the lightshift on |a〉 is substantial because of the atomic transi-tion |a〉 − |a∗〉, and therefore dominates the light shiftof δ. To calculate the light shifts, we note that for ourparameters the Rabi frequencies ΩFF

FB,BB induced on the

atomic transition by LFB,BB are much smaller than thedetunings of the lasers from the atomic transition. Tak-ing also into account that two atoms contribute, thelight shift of the binding energy can be approximated

by δFB ≈ hΩFF2

FB /2Ee = 2π × 20.0 kHz/(W/cm2) and

δBB ≈ hΩFF2

BB /2(Ee − Em) = −2π × 11.0 kHz/(W/cm2),which is close to the measured values.

The coupling lasers LFB and LBB also induce time-and space-dependent light shifts ∆FB and ∆BB on thefree-bound transition. We neglect ∆BB because LBB haslow intensity. To determine ∆FB, we perform one-colorspectroscopy at several intensities of LFB. We derive∆FB = 2π × 21(1) kHz/(W/cm2). The same reasoningas above shows that the one-color shift is also dominatedby the light shift of |a〉 and that ∆FB ≈ δFB, consistentwith the measurement.

Significant atom loss is caused by LFB through off-resonant scattering of photons on the 1S0 − 3P1 transi-tion, which is the main contribution to γa. We measurethis scattering rate by illuminating an atomic sample ina Mott insulator with LFB detuned from the free-boundresonance by a few MHz. Since the system is in the LambDicke regime, losses occur only through light-assisted in-elastic collisions in sites with at least two atoms. Wederive an effective natural linewidth for the free-atomtransition of Γa = 1.90(8) Γ3P1

, where Γ3P1is the natu-

ral linewidth of the intercombination line. This is con-sistent with superradiant scattering, which is expectedsince the harmonic oscillator length is much smaller thanthe wavelength of LFB.

3. Light shifts from lattice light

The lattice light induces static, space-dependent shiftsof the two-color detuning (δLattice) and of the free-boundtransition (∆Lattice), which we now analyze experimen-tally and theoretically. We measure shifts δLattice up to acommon offset through dark-state spectroscopy for sev-eral lattice depths, keeping the lattice well isotropic withωx,y,z within 10 % of each other. Similarly we measureshifts ∆Lattice through one-color spectroscopy up to acommon offset. Both shifts are shown in Fig. 4 as a func-tion of the average trap frequency of the central well. Asbefore, the detuning δLattice corresponds to a change ofthe molecular binding energy of −hδLattice.

In the following we identify two independent compo-nents of the shifts by separating the two-body probleminto its center-of-mass (CM) motion and its relative mo-tion (rel), neglecting mixing terms if present. The firstcomponent is induced on the CM by the difference inpolarizability of atom pairs and molecules. The secondcomponent is induced on the eigenenergies of the relativemotion Hamiltonian by the external confinement and en-ables us to measure the zero-point energy of the latticewells.

We start by deriving the two components of δLattice.Firstly, the difference in polarizability of state |a〉

8

ΔLat

tice/2π

, δ L

attice/2π

(kH

z)

δLattice/2π

ΔLattice/2π

0 10 20 30 40 50 60 70

-100

-50

0

50

100

150

200

ωLattice/2π (kHz)

Figure 4. (color online) Shift δLattice of the two-color detun-ing (red disks) and shift ∆Lattice of the free-bound transition(black squares), as a function of the average trap frequency ofthe central well of the optical lattice. The measurements de-termine δLattice (∆Lattice) only up to a common offset, whichis obtained by fits (curves) described in the text. The er-ror bars of the measurements (i.e. excluding the error in thecommon offsets) are smaller than the symbol sizes.

and |m〉 leads to a differential shift δCMLattice ∝(

2α(1S0) − αm)ILattice as the external potentials expe-

rienced by the CM differ, where α(1S0) is the polariz-ability of a ground-state atom and αm the polarizabil-ity of a molecule in state |m〉. Since the polarizabil-ity of weakly-bound molecules for far detuned light isclose to the sum of the atomic polarizabilities of the con-stituent atoms we expect this shift to be small [35, 36].Secondly, the external confinement induces a differentialshift on the eigenenergies of the relative motion Hamil-tonian, leading to δrel

Lattice. The relative motion compo-nent of state |a〉 occupies the ground state of the lattice

well potential with energy E = 3 hωLattice/2 ∝ I1/2Lattice,

where ωLattice/2π = (ωx + ωy + ωz)/6π is the averagetrap frequency. By contrast, the energy of the relativemotion component of state |m〉 is almost insensitive tothe external confinement because the corresponding Con-don point sets a volume scale which is less than 0.1 %of the ground-state oscillator volume. The light shiftδrelLattice is therefore dominated by the behavior of |a〉,

hence δrelLattice ∝ I

1/2Lattice. In contrast to δCM

Lattice this shiftis even present for αm = 2α(1S0). We neglect the density

dependent interaction shift δCollLattice ∝ n ∝ I

1/4Lattice, be-

cause in our case it is small compared to δrelLattice. The to-

tal light shift is then given by δLattice = δrelLattice+δCM

Lattice =

a1 I1/2Lattice+b1 ILattice, with a1, b1 being constants. We can

Table I. Relevant parameters for molecule production.

Parameter Units Experiment Theory

Em/h MHz 644.7372(2) -

Ee/h MHz 228.38(1) -

ΩFBkHz cm3/2√

W/cm22π × 6.0(1) 10−7 -

ΩBBkHz√W/cm2

2π × 234(5) -

Γe kHz 2π × 17.0(1.5) > 2π × 14.8

Γa/Γ3P1- 1.90(8) 2

τm ms 6(2)× ttunnel ∝ ttunnelγFBm

HzW/cm2 2π × 11(1) -

γCOMPm

HzW/cm2 2π × 60(6) -

δFBkHz

W/cm2 +2π × 18.3(6) +2π × 20.0

δBBkHz

W/cm2 −2π × 10(10) −2π × 11.0

∆FBkHz

W/cm2 +2π × 21(1) +2π × 20.0

δLattice ωLattice −1.50(6) −1.5

∆CMLattice

HzW/cm2 2π × 2.13(2) 2π × 2.98

∆relLattice ωLattice −1.4(5) −1.5

τDarkState ms 2.1(2) -

thus distinguish the two contributions because of theirdifferent scaling with intensity.

The magnitude of the two components of δLattice canbe determined from our experimental data. We fit the

data as a function of ωLattice ∝ I1/2Lattice, i.e. δLattice =

a2ωLattice + b2 ω2Lattice, using also a common offset to all

δLattice data points as fit parameter. The fit gives a2 =1.5(3), which is consistent with the expected zero-pointenergy shift 3h ωLattice/2. The fit result for b2 yields anupper bound for the relative polarizability variation be-tween atom pair and molecule of |(2α(1S0)−αm)/2αa| <1 %. By assuming δrel

Lattice to dominate the shift and re-fitting while keeping b2 = 0, we obtain a2 = 1.50(6).This measurement directly determines the variation ofthe harmonic oscillator zero point energy with trap fre-quency and was possible for two reasons: first, the negli-gible difference in polarizability between atom pairs andmolecules and, second, the high precision achievable withdark-state spectroscopy.

We finally analyze the shift ∆Lattice of thefree-bound transition. Analogous to δCM

Lattice, thecentre-of-mass component of the shift is givenby ∆CM

Lattice ∝(2α(1S0) − αe

)ILattice, where now

the molecular state is |e〉 with polarizability αe.Again assuming that the molecular polarizabil-ity is the sum of atomic polarizabilities, we have∆CM

Lattice ∝(2α(1S0) −

(α(3P1) + α(1S0)

))ILattice =(

α(1S0) − α(3P1)

)ILattice. In contrast to before this shift

is not small since the difference in polarizability of the1S0 and 3P1 (mJ = 0) states at the lattice wavelengthis significant. A smaller shift is induced by changinglattice potentials on the CM zero point energy difference

9

between |a〉 and |e〉, which we neglect. The relativemotion component of the shift ∆rel

Lattice is given by≈ −3ωLattice/2, since also here the relative motionenergy of molecular state |e〉 is barely influenced bythe external potential. The two components add to theone-color detuning shift ∆Lattice = ∆rel

Lattice + ∆CMLattice =

a3 ωLattice + K(α(1S0) − α(3P1)

)ω2

Lattice, where a3 is

a free parameter, K = 3mλ2Lattice/(4πhα(1S0)), m the

mass of 84Sr, and α(1S0) = −234 a.u. (atomic units, here

1 a.u. = 4πε0a30, where ε0 is the vacuum permittivity and

a0 the Bohr radius). Using a3, α(3P1), and a commonoffset to all data points as parameters, we fit thisfunction to the data and retrieve α(3P1) = −188(2) a.u.and a3 = −1.4(5). Since we experimentally vary theintensity for the three lattice beams together, we expectto obtain the mean value of the polarizability for thestate 3P1 (mJ = 0) calculated for the three differentpolarizations of the electric field for the three beams,which is −170.4 a.u. [37], roughly 10 % different from theexperimental value. This deviation might be explainedby our model neglecting the anisotropy in the trapfrequencies.

B. STIRAP

We now apply STIRAP to our Mott insulator sample.We prove the association of molecules and measure themolecule association efficiency after optimization of therelevant parameters. We compare this efficiency with ourtheoretical model and find that dynamic light shifts of thebinding energy limit the current scheme. This limit willbe overcome in Sec. V C.

In order to demonstrate the production of moleculesby STIRAP and to obtain a quantitative measurementof the single-path STIRAP efficiency, we apply a STI-RAP cycle, see Sec. II. Between aSTIRAP and dSTI-RAP, we selectively remove all remaining atoms with apulse of light resonant with the atomic 1S0 − 1P1 tran-sition. Figure 8c shows the intensity profile used for theSTIRAP lasers LFB,BB and the push pulse. The pushpulse doesn’t affect molecules because their binding en-ergy is much bigger than the linewidth of the transitionΓ1P1

' 2π × 30 MHz. Similarly, since absorption imag-ing is also performed using this transition, only atomsare imaged. Reappearance of atoms on images taken af-ter the full STIRAP cycle is the experimental signaturefor the presence of molecules after the aSTIRAP. Assum-ing an equal efficiency for aSTIRAP and dSTIRAP, thesingle-path STIRAP efficiency is η =

√Nf/Ni, where Ni

and Nf are the atom numbers in doubly-occupied sitesbefore and after the STIRAP cycle, respectively.

We optimize the STIRAP sequence by maximizing thenumber of Sr atoms retrieved after two STIRAP cycles,by varying independently the relevant parameters (ΩFB,ΩBB, T and 〈n〉). We choose to optimize the atom num-ber after two cycles in order to reduce the influence ofthe inhomogeneous lattice light shift δLattice on the op-

Table II. Optimized parameters used for STIRAP, with-out compensated two-color detuning (NO-COMP) or with(COMP).

Parameter Units NO-COMP COMP

ΩFB kHz 2π × 32(2) 2π × 38(2)

ΩBB kHz 2π × 300(10) 2π × 107(3)

A = γeδFB/Ω2FB - −1.4(3) 0.0(1)

∆ωCOMP MHz - 2π × 197

Tpulse µs 400 400

npeak cm−3 4.6× 1015 4.6× 1015

τm ms > 105 > 105

γe kHz 2π × 44(13) 2π × 44(13)

τDarkState ms 2.1(2) 2.1(2)

η - 53.0(3.5) % 81(2) %

timization result. During the first aSTIRAP only atomson a subset of sites with similar light shift are success-fully associated. The push pulse removes all remainingatoms, such that after dSTIRAP we are left with a sam-ple of atom pairs on sites with similar lightshift. Thispurification of the sample is also evident when compar-ing the width of one-color PA spectra taken before andafter the first STIRAP cycle, see Sec. V D 1. The secondSTIRAP cycle is used to measure the efficiency of STI-RAP on this more homogeneous sample. The optimizedparameters are reported in Tab. II in the column labeledNO-COMP. The best single-path STIRAP efficiency isηexp = 53.0(3.5) %, which represents a considerable im-provement compared to our previous work [10]. This im-provement is made possible by a longer molecule lifetime,see Sec. V D 2.

To compare the performance of the experiment withthe theoretical expectation, we model the aSTIRAP us-ing Eq. (1). This model requires two not yet determinedparameters, γe and γm, which can only be characterizedemploying STIRAP. Their measurement will be discussedin Sec. V C 3 and V D 2, respectively. Taking these andall previously determined parameters together with thepulse shape as input for the model, we predict that aSTI-RAP has an efficiency of ηtheory = 55(5) %, which is con-sistent with our measurements.

In order to discriminate whether the main limitationto the STIRAP efficiency is the adiabaticity of the se-quence or the lifetime of the dark state we examine thecriterion |A|α = |δm|T π/

√C derived from Eq. 3 for

δ ≈ δFB(sin(θ)2 − 1/2) and Ωm ≈ ΩFB ' ΩBB. The lat-ter condition is not fulfilled in the experiment. However,the population transfer happens mainly in the time in-terval during which ΩFB and ΩBB are of the same orderof magnitude, which makes the criterion approximatelyvalid. We obtain T = 400µs π/(

√CδFB) = 88µs,

suggesting that we are in the regime where the main lossmechanism is the dissipation from the finite lifetime of

10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

-100

-80

-60

-40

-20

0

20

40

ξ

(δFB

+ δ

CO

MP)/

2π (

kHz)

Figure 5. Light shift δFB +δCOMP induced by the free-boundlaser LFB and the compensation beam LCOMP as a functionof the compensation level ξ. The solid line is a linear fit tothe data.

the dark state due to δFB 6= 0.

C. STIRAP with light-shift compensation

In order to improve the molecule creation efficiency weneed to increase the dark-state lifetime. This lifetime isproportional to A−2, where A = δm/γ ≈ γeδFB/Ω

2FB,

and efficient operation is ensured in the adiabatic regimeonly if |A| 1. We cannot decrease |A| by increasing theintensity of LFB, because δFB ∝ Ω2

FB. In the following,we show how to compensate the shift δFB both in timeand space by using an additional “compensation” laserbeam. After optimization, the compensation scheme al-lows us to reach an efficiency of ηexp = 81(2) %.

1. Compensation beam

As explained in Sec. V A 2, the two-color detuningshift δFB is dominated by the light shift of |a〉 in pres-ence of LFB because of the atomic 1S0 − 3P1 transi-tion, from which LFB is red detuned by only |∆ωe| =(Ee + ∆)/h ≈ Ee/h = 2π × 228 MHz. The resulting

shift δFB ≈ hΩFF2

FB /2Ee ∝ IFB/∆ωe can be exactly can-celled by superimposing an additional compensation laserfield LCOMP with LFB, creating the shift δCOMP = −δFB

[15]. To cancel the shift, LCOMP has to be detuned by∆ωCOMP to the blue of the atomic transition (see Fig. 1a)and must have an intensity ICOMP

IFB= |∆ωCOMP

∆ωe|. This

light-shift cancellation technique is similar to the SCRAP

A

η(%)

-1.5 -1.0 -0.5 0.0 0.5 1.00

20

40

60

80

1000.260 0.63 1.00 1.37 1.74

ξ

Figure 6. Maximum single-path STIRAP efficiency as afunction of the parameter A = δm/γ, varied by changing theintensity ICOMP of the compensation beam. Values of thecompensation level ξ are given on the top axis for reference.Theoretical values (line) are obtained by simulating the STI-RAP process with Eq. (1), using independently determinedparameters and not performing any fit. The error bars repre-sent one standard deviation.

method [38], as it relies on tailoring the light shifts withan off-resonant beam.

Since IFB varies in time during STIRAP the light-shiftcompensation needs to be dynamic as well [39]. Since wewant to keep ∆ωCOMP and ∆ωe constant for convenience,ICOMP has to be varied proportionally to IFB to alwayskeep the light shift canceled. A simple technical solutionto obtain such a coordinated change in intensity is tosplit the free-bound laser beam into two beams, imposeon one of the beams a frequency offset |∆ωe|+ |∆ωCOMP|and recombine the beams with exactly the same polariza-tion and spatial mode, before passing through an AOMto control the intensity of both frequency componentsin common. This composite beam is finally recombinedwith the bound-bound beam into one single mode fiber,from which PA light is shone on the atoms (see Fig. 2).This setup allows us to compensate the light shift δFB

both in time and space.

To validate our technique we measure the light shiftδFB induced by LFB (referenced to the non-shifted ex-trapolated value for IFB = 0 W/cm2) as a function ofthe compensation level defined as ξ = ICOMP

IFB| ∆ωe

∆ωCOMP|,

where we expect perfect compensation for ξ = 1. Thecompensation level is varied by changing the intensityof the compensation beam before AOM 1 while keep-ing all other parameters fixed. For this measurementwe use ∆ωCOMP = 2π × 66 MHz and an intensity ofIFB = 5 W/cm2, which is of the order of the optimumintensity of LFB for STIRAP. The measured δFB is plot-ted as a function of ξ in Fig. 5 together with a linear

11

Time (ms)

Ato

m n

umbe

r

0.0 0.2 0.4 0.6 0.8 1.0

0

5

10

15

20

25 x 103

Figure 7. Time evolution of Sr atom number during the firstSTIRAP cycle, using light-shift compensation.

fit from which we obtain ξ(δFB = 0) = 1.1(1), whichis consistent with our assumption of the light shift beingdominated by the atomic 1S0−3P1 transition. To reducethe off-resonant scattering rate of photons on the atomictransition, we increase ∆ωCOMP to 2π × 197 MHz in allfurther usages of the compensation beam.

2. STIRAP optimization and characterization

We characterize the effect of our dynamic optical com-pensation scheme on the STIRAP efficiency. For severalcompensation levels ξ we optimize STIRAP and show theefficiency achieved during the second STIRAP cycle inFig. 6. The maximum efficiency rises from 53.0(3.5) %without compensation to 81(2) % and is obtained forξ = 1, corresponding to A = 0.0(1). This proves that astrong limitation to the STIRAP efficiency without com-pensation indeed originates from the space- and time-dependent light shift imposed on the binding energy byLFB and that our scheme is able to compensate this unde-sired shift. The optimized parameters, shown in column“COMP” of Tab. II, lead our theoretical model to de-scribe the experimental data well. The model indicatesthat now, for |A| 1, the limitations to the transferefficiency are to similar amounts off-resonant scatteringof photons from |a〉 and |m〉, and the finite dark-statelifetime resulting, e.g., from residual δFB and δBB. AllSTIRAPs discussed in the remainder of this article usethe compensation beam with ξ = 1.

We next characterize the evolution of the atom num-ber over several STIRAP cycles, showing the first cyclein Fig. 7 and subsequent cycles in Fig. 8a. We observethat the fraction of atoms reappearing after a STIRAPcycle compared to the atom number at the beginning ofthat cycle is ∼ 65 % for all but the first cycle, for whichit is only 14 %. Ignoring the first STIRAP cycle andassuming equal efficiencies of aSTIRAP and dSTIRAP

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0

10

20

30

x103

0

10

20

30

x103

LBB LFB

push

LFB LBB

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.0

0.2

0.4

0.6

0.8

Ato

m n

umbe

r

Time (ms)

Time (ms)

Inte

nsity

Ato

m n

umbe

r

(b)

(a)

(c)

Figure 8. (color online) (a) Time evolution of the atom num-ber during STIRAP cycles 2 to 5, using light-shift compen-sation. (b) Number of Sr atoms during the second STIRAPcycle as function of time. The lines are the theory curvesbased on Eq. (1) with no fitting parameter, representing theatom (black solid line) and molecule (red dashed line) num-ber for the parameters given in Tab. II.(c) Intensities of LFB,LBB and push pulse beam in arbitrary units. The intensityof LCOMP is not shown.

within each cycle, we find that the single-path STIRAPefficiency is roughly 80 % and constant from the secondcycle onwards.

The lower fraction of returning atoms of the first cycle

12

0 1 2 3 4 5 6 7 8 9 10

0.0

0.2

0.4

0.6

0.8

1.0N

orm

aliz

ed a

tom

num

ber

T (ms)

Figure 9. Number of atoms retrieved after two STIRAPcycles as a function of the pulse time T used for the secondSTIRAP. The line is a fit using our model, from which weobtain the parameter 〈δ〉.

can be explained by two effects, the existence of singly-occupied lattice sites and the lattice induced inhomogene-ity of the molecular binding energy, which leads to a finitedark-state lifetime and therefore loss. An independentone-color spectroscopy measurement shows that our lat-tice contains 1.1 × 105 atoms on singly occupied sites,which matches the number of atoms remaining after thefirst aSTIRAP, letting us conclude that these atoms sim-ply did not have partners to form molecules with. Fol-lowing the first aSTIRAP these atoms are removed bythe push beam pulse. The atom number reduction dur-ing the first aSTIRAP of 1.4 × 105 matches the numberof atoms on doubly occupied sites. Assuming an 80 %single-path dSTIRAP efficiency as observed in STIRAPcycles beyond the first, the 3.5× 104 atoms reappearingafter dSTIRAP correspond to 4.4×104 atoms associatedinto molecules, which is only 30 % of the initial atomnumber on doubly occupied sites, not 80 % as duringlater STIRAP cycles. This difference can be explainedby the decrease of STIRAP efficiency η with increasingtwo-color detuning δ, characterized in Sec. V C 3, in com-bination with the inhomogeneous spread of δ across thesample originating from the lattice light shift, estimatedin Sec. V D 1. Both, η(δ) and the fraction of popula-tion on sites with detuning δ, p(δ), show peaks of similarwidth. The average STIRAP efficiency can be estimatedby averaging η(δ) with weight p(δ) and is about half ofthe maximum efficiency, consistent with our observation.This indicates that the inhomogeneous shift of the bind-ing energy by the lattice light shift is the main limitationto the fraction of lattice sites usable for molecule asso-ciation. In future work this limit could be overcome byincreasing the width of the lattice beams, while keep-ing the lattice depth constant. Here we simply use thefirst STIRAP cycle to remove atoms on sites with largetwo-color detuning δ and atoms on singly-occupied sites,

Δ/2π (MHz)

Nor

mal

ized

ato

m n

umbe

r

-2.0 -1.5 -1.0 -0.5 0.0 0.50.0

0.2

0.4

0.6

0.8

1.0

Figure 10. STIRAP efficiency in dependence of the one-colordetuning ∆. The data points show the normalized number ofatoms retrieved after two STIRAP cycles, depending on thevalue ∆ used for the second STIRAP. The solid line is derivedfrom Eq. (1) using no fit parameters.

providing us with an ideal sample to study STIRAP.As first such study we trace the atom number during

five consecutive STIRAP cycles in Fig. 8a, each identicalto the first. The single-path STIRAP efficiency of the sec-ond cycle is 81(2) %. All cycles have efficiencies around80 %, from which we can take two conclusions. First, thecleaning of the sample realized by the first STIRAP cycleis enough to decrease the binding energy inhomogeneityto a level that is negligible for the STIRAP efficiency.Second, it proves that our MI sample is not heated signif-icantly, which would have reduced the STIRAP efficiencywith each cycle.

Next we study the dependence of STIRAP efficiencyon pulse time, by recording the atom number after thesecond STIRAP cycle, see Fig. 9. Thanks to the light-shift compensation scheme, the pulse time for the STI-RAP can be as high as a few ms while still resulting insubstantial molecule production. For pulse times longerthan 400µs, we observe a decrease of the retrieved atomnumber on a 1/e time of ∼ 2 ms. This decrease can nei-ther be explained by the residual δFB nor by δBB, butit can be explained by the finite dark state lifetime ofτDarkState = 2.1(2) ms measured in Sec. V D 1. This finitelifetime could originate in laser noise or in a small butnon-zero static Raman detuning 〈δ〉 present during theSTIRAP cycle. Fitting our data with the model Eq. (1)using a static Raman detuning as only free parameter weobtain 〈δ〉 = 2π × 2.4(5) kHz. A similar detuning wouldexplain the observed dark-state lifetime.

3. Effect of ∆, δ on efficiency and determination of γe

We now analyze the efficiency of STIRAP in depen-dence of the one-color detuning ∆ and the two-color de-

13

-25 -20 -15 -10 -5 0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

δ/2π (kHz)

Nor

mal

ized

ato

m n

umbe

r

Figure 11. (color online) Dependence of STIRAP efficiencyon the two-color detuning δ. The data points show the nor-malized number of atoms retrieved after two STIRAP cycles.The red dashed line is the shape determined by our modelfor γe = Γe = 2π × 17.0(1.5) kHz, while the grey area spansthe widths between 2π × 30 kHz and 2π × 57 kHz, which arethe fitted values for the positive-detuning and the negative-detuning side, respectively.

tuning δ, see Fig. 10 and 11 respectively. To this end,we keep the parameters of the first, purification STI-RAP constant, and record the atom number N2 after thesecond STIRAP for varying δ while keeping ∆ = 0, orvice versa. The STIRAP efficiency η ∝

√N2 exhibits

peaks around zero detuning, which have a FWHM of2π × 0.5 MHz for η(∆) and 2π × 20 kHz for η(δ). Thewidth of η(∆) is much bigger than the broadened free-bound linewidth, see Sec. V D 1, so that the shift dueto the inhomogeneous broadening induced by the latticelight on the free-bound transition can be neglected. How-ever, as explained in Sec. V C 2, the width of η(δ) is com-parable to the spread in δ, leading to a lower fraction ofatoms in doubly occupied sites being associated duringthe first aSTIRAP compared to consecutive aSTIRAPs.

From our measurement of N2(δ) in Fig. 11, we are ableto derive the loss term γe of Eq. (1), which describes therate at which population is lost from state |e〉 because ofcoupling to states outside the subspace corresponding toour Λ scheme. It can be viewed as an indicator of howopen our quantum system is. The value of γe relevantfor STIRAP can be higher than the natural linewidthof the free-bound transition, i.e. γe ≥ Γe, because ofone- or two-photon processes induced by LFB, LBB, andLCOMP that introduce dissipation. To measure γe, wemake use of the fact that the STIRAP efficiency is onlyhigh for small A ≈ δγe/Ω2

FB, see Sec. III B, which leads toa γe dependence of the N2(δ) peak width. If we assumeγe = Γe, where Γe is determined in Sec. V D 1, our modelgives the dashed line shown in the figure, which is broaderthan the observed peak. Moreover, we observe a slightasymmetry in the data that is not reproduced by the

-200 -150 -100 -50 0 50 100 150 2000.2

0.4

0.6

0.8

1.0

Δ/2π (kHz)

Nor

mal

ized

ato

m n

umbe

r

Figure 12. (color online) One-color molecular line and cor-responding Lorentzian fits measured on an atomic sample inthe lattice before (red disks) and after (black squares) the firstSTIRAP cycle. The narrower width of the latter signal is theresult of the spectral selection imposed by the STIRAP pulse.

model. We therefore fit the model independently to thedata corresponding to positive and negative detuningsusing γe as the only fit parameter and derive γ+

e = 2π ×30(5) kHz and γ−e = 2π × 57(8) kHz, respectively. Infurther uses of our model we fix γe to the average value2π × 44(13) kHz.

D. Sample characterization

1. Inhomogeneous light shifts by lattice lightand dark state lifetime

In order to help understand the origin of the differ-ence in STIRAP efficiencies between the first STIRAPcycle and the following cycles, we measure the latticelight shifts on the free-bound transition and on the bind-ing energy of |m〉 for both initial and purified samples.We use PA spectroscopy for all measurements, but forthe determination of the binding energy spread of thepurified sample. Since this spread is below the resolutionof our two-color PA spectroscopy, we in this case measurethe dark state lifetime and extract an upper bound of thebinding energy spread from that.

In order to measure the inhomogeneous broadening in-duced by the lattice on the free-bound transition, wetake one-color PA spectra using a MI sample, either di-rectly after lattice loading or after one STIRAP cycle,see Fig. 12. These data sets are modelled by Eqs. (1),where γe is the fit parameter and all other quantities inthe Hamiltonian, except for ΩFB, are set to zero. Weretrieve γe = 63(8) kHz and 17.0(1.5) kHz for the free-bound transition linewidth before and after STIRAP, re-spectively. The latter value is consistent with our mea-surement of the natural linewidth of the molecular tran-

14

sition in a BEC [40], Γe = 2π × 19.2(2.4) kHz, whilethe former is roughly a factor of 4 wider. The reduc-tion of the apparent linewidth after one STIRAP cycleis a result of the sample purification by the STIRAP cy-cle. Since both the free-bound transition shift and thebinding energy shift are strongly dependent on the shiftof state |a〉 by the lattice light, the STIRAP cycle notonly reduces the spread in δ, but also the inhomoge-neous broadening of the one-color PA line. Deconvolvingthe measured linewidths with twice the atomic 1S0− 3P1

transition 2 × Γ3P1= 2π × 14.8 kHz [11, 41], we derive

that the inhomogeneous differential light shifts on thefree-bound transition of the initial and the final sampleare 2π × 61(10) kHz and 2π × 8(3) kHz, respectively.

In order to measure the inhomogeneous broadening in-duced by the lattice on the binding energy of |m〉, we per-form two distinct measurements depending on the sam-ple. We perform two-color spectroscopy on the initialsample using low intensity of LFB,BB, which ensures neg-ligible light shifts from the PA light. The width of thedark resonance is then ∆ωDarkState = 2π×18(4) kHz. Todeduce an upper bound on the binding energy spreadof the purified sample, we measure the dark-state life-time. We prepare a roughly equal superposition of atomicand molecular state by applying only half of the secondaSTIRAP pulse. We wait for a variable time while hold-ing LFB,BB at constant intensity, and finally apply thesecond half of the dSTIRAP. The measured lifetime isτDarkState = 2.1(2) ms, and the experimental data agreewith our model when adding a static two-color detuningof 〈δ〉 = 2π×2.2(2) kHz. This value is an upper bound ofthe binding energy spread induced by the lattice on thepurified sample.

The ratio of the free-bound transition linewidth andthe two-color detuning spread is consistent with the light-shift calibration, both before and after STIRAP (seeTab. I). Using this calibration, we conclude that the ini-tial sample was populating lattice sites located off axisby up to ' 30 % of the lattice beam waist.

2. Molecule lifetime

The molecules produced in our experiment have a fi-nite lifetime as a result of two loss mechanisms, inelasticcollisions with other molecules in the lattice and dissi-pation caused by optical coupling of state |m〉 to stateslying outside our Λ scheme. Thus, the lifetime is givenby τm = 1/γm = 1/(γcoll

m +γoptm ), where γcoll

m is the decayrate for collisional losses and γopt

m is the scattering ratefor optically induced losses.

We measure the molecule lifetime τ collm = 1/γcoll

m thatresults from inelastic collisions by varying the hold timebetween the push pulse and the dSTIRAP at several lat-tice depths. During these measurements the lattice wellpotential is kept isotropic with ωx,y,z within 10 % fromeach other. The molecule lifetime depends strongly onthe lattice depth, see Fig. 13. The observed change of

10 20 30 40 50 60 70 80

1 10 100 1000 100000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Mol

ecul

e lif

etim

e (m

s)

Mean depth per direction (units of Er)

104

103

102

101

105

100

Nor

mal

ized

ato

m n

umbe

r

Time (ms)

Figure 13. (color online) Molecule lifetime as function of thelattice depth averaged over the three directions in the centralregion of the lattice. The depth is that of the potential expe-rienced by the molecules, expressed in units of recoil energiesof the molecule Er/h = 8.4 kHz. Inset: atom number decay asa function of hold time between association and dissociationSTIRAP pulses for different lattice depths.

four orders of magnitude over the lattice depth is consis-tent with loss of molecules by tunneling to neighboringsites followed by effectively instantaneous chemical reac-tion with the molecule already present on that site. Col-lisions between molecules and atoms do not play a rolehere since all atoms have been removed by the push pulseprior to lowering the lattice depth. Under these assump-tions the lifetime is given by τ coll

m = D× ttunnel, where Dis a constant and ttunnel = h/

∑i 2Ji is the total tunnel-

ing time, which depends on the tunneling rates along the3 lattice directions J1,2,3 [42, 43]. We fit the experimen-tal data with this single-parameter model and retrieveD = 6(2). We observe that the lifetime measured for ourdeepest lattice are lower than predicted by our model.This can be explained by loss through scattering of lat-tice photons becoming dominant over the low inelasticcollision loss, since here the lattice light intensity is highand the tunnel time scale long.

For the deepest lattice the molecule lifetime is100(20) s, representing a major improvement comparedto our previous paper, where the lifetime was about 60µs[10]. The main difference between this work and our pre-vious work is the wavelength of the light used for therealization of the optical lattice. In our previous work,the lattice laser source was a Coherent Verdi at 532 nmwavelength, whereas here we use an Innolight MephistoMOPA at 1064 nm. By shining 532-nm light derived from

15

another Verdi laser on our sample, we confirm a strongreduction of the molecule lifetime within a factor of 10of what was measured before. An explanation for the re-duced lifetime is that the 532-nm light is close to molec-ular lines, thus inducing inelastic processes.

The second loss mechanism arises from optical cou-plings of molecules toward states outside our Λ scheme.One contribution is the scattering of lattice photons,as mentioned above. Other significant contributionsarise from LFB,COMP, whereas the low intensity LBB

plays a minor role. We measure the scattering timeτoptm = 1/γopt

m = 1/(γFBm + γCOMP

m ) by shining LFB andLCOMP on the molecules during the hold time betweenthe push pulse and dSTIRAP, at several intensities ofLFB,COMP. We derive γFB

m = 2π × 11(1) Hz/(W/cm2)and γCOMP

m = 2π × 60(6) Hz/(W/cm2).

VI. CONCLUSION AND OUTLOOK

In conclusion, we have used STIRAP to associate pairsof Sr atoms in the ground state of lattice sites intoweakly-bound ground-state molecules. This associationprocess was efficient despite operating in the regime ofstrong dissipation ΩFB

<∼ γe. By making use of a deepoptical lattice using 1064-nm light, but without compen-sating dynamic light shifts created by the STIRAP lasers,we were able to reach a transfer efficiency of roughly 50 %.The improvement from 30 % reached in our previous work[10] is mainly due to the increase in molecule lifetimefrom 60µs up to 100(20) s, which was possible by usinglattice light of 1064 instead of 532 nm. The lifetime isnow proportional to the tunneling time in the lattice.

The efficiency of the STIRAP scheme without com-pensation beam is limited by the finite lifetime of thedark state. This lifetime is mainly the result of time-dependent light shifts induced by the free-bound laser

on the binding energy of the ground-state molecule. Wehave identified the coupling to the atomic 1S0−3P1 tran-sition as the main source of light shift, and we have shownhow to cancel it with an auxiliary compensation beam,leading to an efficiency higher than 80 %. This efficiencyis limited in equal parts by scattering from off-resonantlight and finite dark-state lifetime. We have shown the ef-fect of the time-independent inhomogeneous lattice lightshifts on STIRAP. In further work these detrimental ef-fects could be suppressed by using lattice beams withbigger waists, leading to larger molecular samples.

We thus demonstrated that, by use of STIRAP, wecan optically associate atoms into molecules with anefficiency comparable to that obtained with magneto-association through a Feshbach resonance. This generaltechnique can represent a valuable alternative for asso-ciating molecules containing non-magnetic atoms, andin particular for the creation of alkali — alkaline-earthdimers [44–47]. Such dimers have been attracting greatattention as they can allow fascinating quantum simula-tions, thanks to their permanent magnetic and electricdipole moments [21]. Finally, coherent, efficient and con-trolled creation of long-lived ultracold Sr2 molecules inthe ground state could be useful in metrology experi-ments, for instance as a probe for the time variation ofthe electron-to-proton mass ratio [48, 49].

ACKNOWLEDGMENTS

We gratefully acknowledge funding from the Euro-pean Research Council (ERC) under Project No. 615117QuantStro. B.P. thanks the NWO for funding throughVeni grant No. 680-47-438 and C.-C. C. thanks the Min-istry of Education of the Republic of China (Taiwan) fora MOE Technologies Incubation Scholarship.

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