Systematic study of Optical Feshbach Resonances in an ideal gas
S. Blatt, T. L. Nicholson, B. J. Bloom, J. R. Williams, J. W. Thomsen, P. S. Julienne, J. Ye
aa r X i v : . [ phy s i c s . a t o m - ph ] J un Systematic study of Optical Feshbach Resonances in an ideal gas
S. Blatt , T. L. Nicholson , B. J. Bloom , J. R. Williams , J. W. Thomsen , ∗ P. S. Julienne , and J. Ye JILA and Department of Physics, NIST and University of Colorado, Boulder, CO 80309-0440, USA Joint Quantum Institute, NIST and the University of Maryland, Gaithersburg, MD 20899-8423, USA. (Dated: June 3, 2011)Using a narrow intercombination line in alkaline earth atoms to mitigate large inelastic losses, weexplore the Optical Feshbach Resonance (OFR) effect in an ultracold gas of bosonic Sr. A system-atic measurement of three resonances allows precise determinations of the OFR strength and scalinglaw, in agreement with coupled-channels theory. Resonant enhancement of the complex scatteringlength leads to thermalization mediated by elastic and inelastic collisions in an otherwise ideal gas.OFR could be used to control atomic interactions with high spatial and temporal resolution.
PACS numbers: 34.50.Rk, 34.50.Cx, 32.80.Qk
The ability to control the strength of atomic interac-tions has led to explosive progress in the field of quan-tum gases for studies of few- and many-body quantumsystems. This capability is brought about by magneticfield-induced Feshbach scattering resonances (MFR) [1],where both the magnitude and sign of low-energy atomicinteractions can be varied by coupling free particles toa molecular state. MFR in ultracold alkali atoms havebeen used to realize novel few-body quantum states andstudy strongly correlated many-body systems and phasetransitions [1, 2]. However, magnetic tuning has limitedcurrent experiments to relatively slow time scales and lowspatial resolution. Higher resolution could be achieved bycontrolling MFR optically [3].Scattering resonances can also arise under the influenceof laser light tuned near a photoassociation (PA) reso-nance [4] where free atom pairs are coupled to an excitedmolecular state [5, 6]. This Optical Feshbach Resonance(OFR) is expected to enable new and powerful controlwith high spatial and temporal resolution. OFR has beenstudied in thermal [7] and degenerate [8, 9] gases of Rb,but it was not found useful due to large photoassociativelosses. Much narrower optical intercombination lines areavailable in alkaline earth atoms and are predicted toovercome this loss problem [10]. Independently, ultra-cold alkaline earth atoms have recently emerged to playleading roles for quantum metrology [11–13] where preci-sion measurement and many-body quantum systems arecombined to study new quantum phenomena [14, 15].Degenerate gases of alkaline earth atoms have recentlybecome available [16]. Due to the lack of magnetic struc-ture in the ground state of these atoms, the OFR effectcould become an important tool for controlling their in-teractions. OFR work on Yb [17, 18] has been limitedto studying the induced change in scattering phase shiftsand PA rates. Dominant PA losses are evident in all ofthe OFR experiments listed above. Light-induced elasticcollisions for thermalization were not observed.In this Letter, we study the OFR effect across mul-tiple resonances in a metastable molecular potential of Sr. The aim of this work is to test the practical appli- cability of OFR for engineering atomic interactions in thepresence of loss, similar to the successful application of adecaying MFR [19]. For Sr, OFR is predicted [10] to al-low changes in the scattering length by more than a factorof 100 with low losses by using large detunings ( O (10 )linewidths) from the least-bound vibrational level [20].We tested this proposal and find experimentally that theexisting isolated resonance model [6] only describes theexperiment in the small detuning regime. Large detun-ings from a molecular resonance require a full coupled-channels description of the molecular response. Sup-ported by this new theory framework, we present a sys-tematic experimental study of the OFR-enhanced com-plex scattering lengths and demonstrate OFR-inducedthermalization in an ultracold gas.Bosonic Sr has an s -wave background scatteringlength a bg = − . a [21], where a is the Bohr ra-dius. The small | a bg | makes the sample effectively non-interacting and provides an ideal testing environment forOFR. Figure 1a shows the ground ( S - S g ) and low-est excited state ( S - P u ) molecular potentials of Sr ,which are coupled by a PA laser near the atomic tran-sition at λ a = 689 nm. The vibrational levels investi-gated are labelled by their quantum number n , countedas negative integers from the free particle threshold. Fora given PA laser detuning from threshold, the Franck-Condon principle localizes the atom-light interaction inthe vicinity of the Condon point [6].When detuning the PA laser across a vibrational res-onance, the s -wave scattering length shows a dispersivebehavior, just as for a MFR. However, the finite lifetimeof the excited molecular state leads to loss intrinsic toOFR. This process can be described [22] akin to decay-ing MFR [1, 19] with a complex s -wave scattering length α ( k ) ≡ a ( k ) − ib ( k ) that depends on the relative momen-tum ~ k and a PA line strength factor ℓ opt = λ a πc |h n | E i| k I ,called the optical length [10, 23]. Here, c is the speed oflight, and ℓ opt scales linearly with PA intensity I andfree-bound Franck-Condon factor |h n | E i| per unit colli-sion energy E = ~ k / (2 µ ) at reduced mass µ = m Sr / b − − PA detuning (MHz)0 u n = −
24 mW / cm ℓ opt = 28(2) a Γ sc = 0.77 s − F r a c t i ono f a t o m s r e m a i n i ng − − PA detuning (MHz)0 u n = − / cm ℓ opt = 14(1) a Γ sc = 0.05 s − − − PA detuning (MHz)0 u n = − / cm ℓ opt = 14(1) a Γ sc = 0.69 s − a ≈≈
10 20 30 40 50 60 70 80 90 100Atomic distance ( a )01015 E ne r g y ( c m − ) PA l a s e r @ m S + P u S + S g − D i ff. po t en t i a l ( G H z )
50 100 150 200
Atomic distance ( a ) n = − n = − n = − Bound state E / k B = 1 µ KCondon point R C FIG. 1: (a) Ground (blue) and excited (red) molecular po-tentials of Sr . The inset shows the difference potential aftersubtracting the optical frequency. Horizontal lines indicatebound molecular states n in the excited potential. The freeparticle (bound state) radial wave function is indicated in blue(red). (b) Loss spectra for 0 u n =-2, -3, and -4 for exposuretime τ PA = 200 ms and comparable mean density. I is scaledto keep Γ sc sufficiently small. The similarity of the spectrademonstrates the universal scaling with ℓ opt ∝ |h n | E i| I . collision rate is [22] K in ( k ) = 4 π ~ µ ℓ opt γ m γ (∆ + E/ ~ ) /γ + [1 + 2 k ℓ opt γ m γ ] / , (1)where ∆ is the laser detuning from molecular reso-nance [22]. We have accounted for extra molecularlosses with γ > γ m = 2 γ a , where γ m is the linewidthof the molecular transition and γ a = 2 π × . a bg for Sr gives K el ( k ) ≃ k ℓ opt γ m γ K in ( k ). The elastic-to-inelastic colli-sion ratio K el /K in becomes less favorable for smaller k .We load ∼ × atoms from a magneto-optical trapoperating on the S - P intercombination transitioninto a crossed optical dipole trap formed by tilted hori-zontal (H) and vertical (V) beams (1064 nm), with trapdepths ∼ µ K and ∼ µ K, respectively. The trappedsample shows a clear kinetic energy inhomogeneity be-tween the H and V axes (2-2 . µ K vs. 3-4 µ K), dueto the negligible a bg , consistent with a thermal distribu-tion energy-filtered by the trap potential. Typical in-trapcloud diameters are 45-55 µ m. The PA beam intersectsthe trap with a waist of 41 µ m [22].A representative survey of PA resonances in the S - P u potential is shown in Fig. 1b. The PA laser withintensity I , adjusted to achieve similar ℓ opt for all spectrashown, interacts with the sample for τ PA . Photon-atom scattering at rate Γ sc and subsequent radiation trappingset the maximum usable I for a given detuning from theatomic line [22]. In addition to the vibrational levels in-dicated in Fig. 1a, the n =-1 vibrational state exists at-0 . ℓ opt /I [20].The isolated resonance theory indicates that operatingwith a large I at O (10 γ a ) detuning from the n =-1 stateshould allow modifications to a ( k ) of O (100 a ) [10]. Thisprediction relied on extrapolating the large line strengthof the n =-1 state across multiple intermediate PA reso-nances. However, with I up to 1 kW / cm and detuningsup to -1 . u , 1 u ) havethe form of Ref. [20], with an added imaginary term − i ~ γ m /
2. The ground state potential uses the dispersioncoefficients of Ref. [26], has a scattering length of − . a ,and reproduces the bound state data of Ref. [21] to betterthan 0.4%. Coupled-channels calculations do not assumeisolated resonances, and all 0 u and 1 u molecular eigen-states emerge from the calculation as interfering, decay-ing scattering resonances [1].Figures 2c and d show that the coupled-channels modelreproduces the isolated resonance expressions [1, 6] for α ( k ) and the rate constants as long as ∆ is small com-pared to the spacing between molecular levels. How-ever, the coupled-channels K el returns to its backgroundvalue K bgel in between resonances regardless of their rel-ative strengths. The dotted line indicates K bgel ( a bg ) at E/k B = 4 µ K in Fig. 2a (Fig. 2d). These calculationsshow that each molecular line behaves as an isolated res-onance near its line center. For detunings comparableto the molecular level spacing, the isolated resonance ex-pressions cannot be used.At intermediate detunings, | ∆ | ≫ γ (1 + 2 kℓ opt γ m γ ), α ( k ) can be written in the standard form for MFR [22],lim k → α ( k ) = a bg (cid:18) − w ∆ + i wγ ∆ (cid:19) , (2)where w ≡ − ℓ opt γ m /a bg . To obtain a meaningfulchange in scattering length, ℓ opt γ m / ∆ needs to be suf-ficiently large, and the imaginary part b = ℓ opt γ m γ/ ∆ needs to be sufficiently small. Since K in ≃ (2 × − cm / s) ( b/a ), for a density of ρ = 10 cm − and b = 0 . a , K in ρ = Γ sc for I = 53 W / cm assumed forFigs. 2c,d. Thus, the calculations predict that changesin the scattering length of order 10 a ≫ | a bg | shouldbe possible with O (100 γ m ) detunings on timescales of200 ms. -1 -0.8 -0.6 -0.4 -0.2 0 PA laser detuning (GHz)10 -14 -12 -10 K ( c m / s ) K in K el -40 0 40Detuning ∆/ γ m R a t i o K e l / K i n -80 0 80Detuning ∆/ γ m -2000 0 2000Detuning ∆/ γ m -505 u (-4)
53 W/cm E/k B = 4 µK, I=10 W/cm u (-4) -a ab u (-2) K el (a )(a ) bg u (-2) ab u (-4) 1 u (-1) u (-3) c d FIG. 2: (a) Coupled-channels calculations of K el and K in at E/k B = 4 µ K and I = 10 W / cm , versus PA laser detuningfrom atomic resonance. Each resonance peak is labeled byits electronic symmetry 0 u or 1 u and n . Between resonances, K in is only approximate. (b) Ratio of thermally-averaged rateconstants at 2 µ K for ∆ /γ m near 0 u ( − I = 44 mW / cm gives the same ℓ opt = 360 a as for the conditions in Fig. 4b.(c) Zero energy limit of a ( k ) and b ( k ) for detuning near the0 u ( −
4) feature. (d) Same calculation as in (c) at large detun-ing. Here, the isolated resonance results (solid lines) agreewith the coupled-channels theory (circles).
To investigate the utility of OFR, we systematicallycharacterized three different resonances and determinedtheir universal scaling. Because K el /K in ∝ kℓ opt , inelas-tic collisions dominate the dynamics of the sample forsmall ℓ opt ≪ (2 h k i ) − = ~ p π/ (8 µk B T ), where the an-gled brackets indicate a k -average at temperature T , and k B is the Boltzmann constant. In this regime, the resultof scanning the PA laser across resonance is a loss featurethat shows no dependence on elastic collision processes.A typical PA loss feature for small ℓ opt is shown inFig. 3a, where the final atom number after applica-tion of PA light is shown with respect to PA detuningfrom S - P . The per-axis kinetic energies [22] for thisscan correspond to a horizontal (vertical) temperature T H ( T V ) = 2 µ K (3 µ K), resulting in the typical thermaltail towards the red side of the resonance [27]. The solidline is a result of solving [28]˙ ρ ( x , t ) = − K in ( x ) ρ ( x , t )2 − ρ ( x , t ) τ bg , (3)with single particle density ρ , thermally-averaged in-elastic collision rate ¯ K in ( x ) ≡ h K in ( k, ∆ , x ) i [22], andvacuum-limited trap lifetime τ bg ≃ . x and we integrate the density to obtainthe number of atoms at the end of the PA pulse. Wethen use Eq. 3 to fit to the experimental data [22] andextract ℓ opt and the position of the line center. Figure 3b F i na l a t o m no . ( ) − − − PA laser detuning (MHz) a F i na l a t o m nu m be r Hold time (s)
No PAPA, -23.9 MHz b ℓ op t ( a ) I av (W/cm ) c − − ∆ ν ( M H z ) I av (W/cm ) d theory experiment n ℓ opt / I ℓ opt / I av γ m γ ℓ opt / I av ν n ∆ ν/ I av ( a W / cm ) ( a W / cm ) ( a W / cm ) (MHz) ( kHzW / cm )-2 8.3 × × × -23.98(1) − (251 ± ± ± FIG. 3: (a) Typical PA loss feature for n =-2 in the low inten-sity regime at I av = 7 mW / cm , with density-profile-averagedPA intensity I av [22]. (b) Time evolution of the trappedsample with (circles) and without PA light (squares). Thetwo-body loss curve with PA is fit with a thermally averagedmodel (solid curve). (c) Linear increase of ℓ opt with I av for ℓ opt ≪ / (2 h k i ). (d) Molecular line center shift ∆ ν for large I av and decreased τ PA . For each n , OFR parameters from thecoupled-channels calculation and the experiment are summa-rized in the table at the bottom. Here, ν n is the zero-intensitymolecular line center with respect to S - P , and ∆ ν/I av characterizes the molecular ac Stark shift [23]. shows the time dependence of the same photoassociativeloss process. Two-body loss is evident under resonant PAlight.From the experimental data we extract two indepen-dent quantities: ℓ opt γ m and an increased molecular lossrate γ ≃ . γ m . We have ruled out magnetic field orPA laser noise as a source of broadening. Instead, weconclude that this extra broadening is related to a fastermolecular decay rate, which is consistent with our earliermeasurements [20] and Rb results [8].The measurements were performed for a range of ℓ opt by adjusting I av . Multiple molecular resonances weremeasured and results for n =-2 are shown in Fig. 3c. Theoptical length data is fit with a linear function and theresults are summarized in the table at the bottom. Thefit coefficient ℓ opt /I av is given by the free-bound Franck-Condon factor and decreases drastically with decreasing n . Figure 3d exemplifies similar measurements done todetermine the line shift ∆ ν with I av . Linear shift coef-ficients ∆ ν/I av and zero intensity line positions ν n withrespect to the atomic transition are also shown in thetable. The sign and magnitude of ∆ ν/I av are consistentwith the predictions in Ref. [23].At larger optical lengths ( ℓ opt γ m /γ ∼ a ), elasticcollisions start to influence the dynamics of the system.We show the atom loss with respect to PA laser detuningfor n =-2 in Fig. 4a. Both in-trap size and kinetic energyare measured by absorption imaging [22]. Far detunedfrom the resonance, the gas is almost ideal, as shownby the persistent kinetic energy inhomogeneity along Hand V in Fig. 4b. On resonance (vertical dashed line),inelastic collisions dominate and cause heating. For reddetuning from the molecular resonance, the temperaturesapproach each other by cross-thermalization [29].The measured cloud widths w H and w V confirm thatthe potential energy follows the kinetic energy (Fig. 4c)since particles oscillate in the trap many times betweencollisions. Similar measurements were performed for n =-3 and n =-4, and we find that the same dispersivebehavior in temperatures and widths appears around2 h k i ℓ opt γ m /γ ∼
30% at τ PA = 200 ms. The data canbe understood by a simple picture of competition be-tween K el and K in , which average differently with k ina thermal sample [see Eq. (1)], and thus peak at differ-ent values of ∆. Elastic collisions cause cross-dimensionalthermalization and tend to equalize T H and T V . Since in-elastic collisions predominantly remove cold atoms fromthe densest part of the cloud, the resulting loss increasesthe average system energy via anti-evaporation.This behavior is confirmed by a Monte-Carlo simula-tion, where 55 × particles are simulated and eachparticle undergoes elastic and inelastic collisions with aninitial phase-space distribution matched to the experi-mental conditions [22]. The solid lines overlaid on theexperimental data in Fig. 4 are the simulation results. Anaverage ratio of elastic to inelastic collisions per particlefrom the simulation is shown in Fig. 4d. The disper-sive shapes are also predicted by the coupled-channelsmodel (see Fig. 2b) and their shape is sensitive to γ .Combined with the low ℓ opt data in Fig. 3, the entiresimulation reproduces the experimental data only when γ = 2 π × I and have observed a clear modification of both a ( k ) and b ( k ). For the values of ℓ opt γ m /γ achieved here,inelastic losses still contribute significantly and h K el /K in i becomes even less favorable with decreasing T . However,the OFR effect can modify interactions in a degenerategas of alkaline earth atoms and the desired change of a ( k ) is achieved at the smallest ∆ /γ constrained by bothmolecular and atomic loss processes over a given experi-mental timescale [22]. F i na l a t o m nu m be r( ) − − − blue det.red det. a K i ne t i c ene r g y ( µ K ) − − − T H T V b I n - t r ap F W H M ( µ m ) − − − w H w V c h K e l / K i n i − − − d FIG. 4: Elastic contribution to the scattering cross sectionfor n =-2 at I av = 22 mW / cm (open circles) and results of aMonte-Carlo simulation (solid lines) using Eq. 1 in a crosseddipole trap for ℓ opt γ m /γ = 140 a . (a) Atom loss as a functionof PA laser detuning from the atomic S - P resonance. Inpanels b and c, blue (red) data points and solid lines indi-cate the corresponding quantities for the vertical (horizontal)trap axis. (b) Change in kinetic energy derived from time-of-flight images, and (c) potential energy change correspondingto varying in-trap density profile. (d) The resultant ratio ofelastic and inelastic collisions per particle, averaged over τ PA . We thank C. Greene, P. Zoller, and G. Campbell fordiscussions and contributions. J. R. W. is an NRC Fel-low. Our work is funded by DARPA OLE, NIST, & NSF. ∗ Permanent address: The Niels Bohr Institute, Univer-sitetsparken 5, 2100 Copenhagen, Denmark.[1] C. Chin et al. , Rev. Mod. Phys. , 1225 (2010).[2] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod.Phys. , 885 (2008); S. Giorgini, L. P. Pitaevskii, andS. Stringari, ibid. , 1215; W. Ketterle and M. W. Zwier-lein, in Ultracold Fermi Gases, ”Enrico Fermi” CourseCLXIV (2008), p. 95.[3] D. M. Bauer et al. , Nature Physics , 339 (2009).[4] K. Jones et al. , Rev. Mod. Phys. , 483 (2006).[5] P. Fedichev et al. , Phys. Rev. Lett. , 2913 (1996).[6] J. Bohn and P. Julienne, Phys. Rev. A , 414 (1999).[7] F. Fatemi, K. Jones, and P. Lett, Phys. Rev. Lett. ,4462 (2000).[8] M. Theis et al. , Phys. Rev. Lett. , 123001 (2004).[9] G. Thalhammer et al. , Phys. Rev. A , 033403 (2005).[10] R. Ciury lo, E. Tiesinga, and P. S. Julienne, Phys. Rev.A , 030701 (2005).[11] T. Akatsuka, M. Takamoto, and H. Katori, NaturePhysics , 954 (2008).[12] G. K. Campbell et al. , Science , 360 (2009).[13] N. Poli et al. , Phys. Rev. Lett. , 038501 (2011).[14] M. D. Swallows et al. , Science , 1043 (2011).[15] A. V. Gorshkov et al. , Nature Physics , 289 (2010).[16] Y. Takasu et al. , Phys. Rev. Lett. , 040404 (2003); S. Kraft et al. , Phys. Rev. Lett. , 130401 (2009); S.Stellmer et al. , ibid. , 200401; Y. de Escobar et al. , ibid. ,200402.[17] K. Enomoto et al. , Phys. Rev. Lett. , 203201 (2008).[18] R. Yamazaki et al. , Phys. Rev. Lett. , 050405 (2010).[19] D. Naik et al. , Euro. Phys. J. D online,doi:10.1140/epjd/e2010-10591-2 (2011).[20] T. Zelevinsky et al. , Phys. Rev. Lett. , 203201 (2006).[21] Y. de Escobar et al. , Phys. Rev. A , 062708 (2008).[22] See accompanying supplementary material for details.[23] R. Ciury lo, E. Tiesinga, and P. S. Julienne, Phys. Rev.A , 022710 (2006).[24] P. Julienne and F. Mies, Phys. Rev. A , 3792 (1986).[25] R. Napolitano, J. Weiner, and P. S. Julienne, Phys. Rev.A , 1191 (1997).[26] S. G. Porsev and A. Derevianko, J. Exp. Theor. Phys. , 195 (2006).[27] R. Ciury lo et al. , Phys. Rev. A , 062710 (2004).[28] J. P. Burke, Jr., Ph.D. thesis, Univ. of Colorado (1999).[29] J. Goldwin et al. , Phys. Rev. A , 043408 (2005). SUPPLEMENTARY MATERIALScattering matrix, cross sections, and collision rates
The collision of two identical bosons in the low energynear-threshold limit can be described in a scattering ma-trix treatment by a single s -wave scattering matrix ele-ment, S ( k ) = e iη ( k ) , where ~ k is the relative momen-tum of the collision pair and η ( k ) is the scattering phaseshift due to interactions. When there is loss of scatteringflux from the entrance channel, as is the case of a decay-ing optical Feshbach resonance, it is convenient to define a complex energy-dependent scattering length α ( k ), re-lated to the complex phase η ( k ) and S -matrix elementby [1, 2] α ( k ) ≡ a ( k ) − ib ( k ) ≡ − tan η ( k ) k = 1 ik − S ( k )1 + S ( k ) . (4)This reduces to the usual definition of complex scatteringlength as k →
0. The respective elastic and inelastic s -wave collision cross sections are [2, 3] σ el ( k ) = 2 πk | − S ( k ) | = 8 π | α ( k ) | f ( k ) ,σ in ( k ) = 2 πk (1 − | S ( k ) | ) = 8 πk b ( k ) f ( k ) , (5)where f ( k ) = [1 + k | α ( k ) | + 2 kb ( k )] − → k → µ is the reduced mass of the pair. The secondrelations in Eq. 5 follow from the definition in Eq. 4.The corresponding collision rate coefficients are K el ( k ) = ~ kµ σ el ( k ) ,K in ( k ) = ~ kµ σ in ( k ) . (6) When the entrance channel is coupled by a single fre-quency laser to a single excited molecular bound stateof the pair, the isolated resonance approximation for thethe field-dressed s -wave scattering matrix element can bewritten as [4, 5], S ( k ) ≃ e iη bg ( k ) (cid:18) − i Γ s ( k ) E/ ~ + ∆ + i ( γ + Γ s ( k )) / (cid:19) . (7)Here E = ~ k / µ is the collision energy with µ = m Sr / s ( k ) ≡ kℓ opt ( k ) γ m is the in-duced coupling between the free particle state | E i andthe excited molecular state | n i . Here γ m = 2 γ a is thelinewidth of the molecular transition and γ a = 2 π × . γ > γ m . The opticallength ℓ opt = λ a πc |h n | E i| k I, (8)where c is the speed of light, is a PA resonance-dependentline strength parameter [5, 7]. Under the Wigner thresh-old law, the free-bound Franck-Condon factor per unitcollision energy |h n | E i| ∝ k , and we checked numeri-cally that it is a very good approximation to take ℓ opt to be independent of collision energy for temperatures < µ K. The optical length is also independent of theatomic oscillator strength, and it scales linearly with PAintensity I .Combining Eq. 7 with Eqs. 4-6 gives the near-thresholdisolated resonance approximation to the OFR inelasticloss rate coefficient: K in ( k ) = 4 π ~ µ ℓ opt γ m γ (∆ + E/ ~ ) /γ + [1 + 2 k ℓ opt γ m γ ] / . (9)The background s -wave scattering length a bg is definedin the absence of a resonance as a bg = − lim k → tan η bg ( k ) k . (10)If we neglect a bg for Sr, we find a simple expression forthe OFR-induced elastic collision rate coefficient for anideal ultracold gas K el ( k ) ≃ k ℓ opt γ m γ K in ( k ) . (11)Power broadening is included via Γ s ( k ) in the denom-inator of Eq. 7, which corresponds to 2 k ℓ opt γ m γ in thedenominator of Eq. 9. From the expressions for elas-tic and inelastic collision rates, we see that the relevantstrength parameter in the presence of extra molecularloss is the rescaled optical length ℓ opt γ m γ . Elastic colli-sions and power broadening only become important forthe dynamics when 2 k ℓ opt γ m γ ∼ . (12) Coupled Channels Calculations
The simplest way to do a calculation of the elastic andinelastic collision properties of an optical Feshbach res-onance is to set up a coupled channels model for thecollision of field-dressed states in a laser field of fixed fre-quency [4, 6]. We use the simplest three-channel modelthat is sufficiently complete to represent nonperturba-tively the optically induced S ( k ) for an interfering spec-trum of excited state resonances. One channel representsthe ground state with potential V g ( R ) and two channelsrepresent the two excited states 0 u and 1 u that corre-late with the separated atom limit S + P with re-spective complex potentials V u and V u . These poten-tials are shifted to include the asymptotic detuning ofthe laser frequency from atomic resonance. It is neces-sary to include both u and 1 u excited states to correctlycalculate the complex scattering length near an isolatedresonance of either state. This is because the molecular J = 1 excited states (J= total angular momentum quan-tum number) must become a mixture of s and d -wavesat long range to properly go to separated atoms with P quantized in a space frame instead of a molecular bodyframe [6]; otherwise, spurious 1 /R resonant dipole termswould get mixed into the ground state potential and giveinvalid threshold scattering lengths.The coupled channel potential matrix V ( R ) is foundusing the asymptotic representation of the two excited J = 1 molecular channels in terms of a pair of excitedstate s - and d -waves for the e parity block in Table 1 ofReference [6]: V = V g V opt V opt 13 ( V u + 2 V u ) √ ( V u − V u )0 √ ( V u − V u ) (2 V u + V u + 6 B ) , (13)where 6 B = 6 ~ / (2 µR ) is the d -wave centrifugal en-ergy. Here the row labels, in order from top to bottom,represent the | jℓJM i = | i ground state and | i and | i excited state channels of Ref. [6], where j isthe atomic angular momentum for the collision partnerto the S atom, ℓ is partial wave, and JM representthe total angular momentum and its projection for thepair of interacting atoms. Since the excited bound statesare relatively short range, we use the nonretarded opti-cal coupling V opt = (2 πI/c ) / d m , where d m = √ d A and the atomic transition dipole d A = 0 . µ s. A complete theory would use the retarded optical coupling.The zero of energy for ground state collisions is taken tobe the lowest eigenvalue of this matrix to account for theenergy shift of the field dressed molecule. The numerical S -matrix is calculated using standard coupled channelsmethods to calculate the wave function for the couplingmatrix in Eq. 13.While Reference [4] simulates the spontaneous radia-tive decay of the excited state by introducing artificialchannels, here we include a complex term − i ~ γ/ γ = γ m .This leads to a non-unitary ground state S -matrix ele-ment such that 0 ≤ − | S ( k ) | ≤ . We have verified that a ( k ) is independent of this choice over the full range ofdetunings, even between resonances, and b ( k ) is indepen-dent of this choice out to many linewidths away from anisolated resonance. However, in the far wings of a reso-nance between isolated resonances, b ( k ) will depend onthis choice, although molecular loss tends to be small inthese regions. This dependence on cutoff is because ournon-unitary theory does not distinguish between atomicand molecular light scattering when the atoms are farapart; the cutoff selects which distance inside of whichan excitation is considered to be a molecular loss processinstead of an atomic one. Thus the cutoff is taken to beon the order of but larger than the outer turning pointof the excited molecular bound states.Finally, we ignore coupling of the excited J = 1 molec-ular levels back to d -waves in the ground state. Thismeans that our model only gets the s -wave but not the d -wave contributions to the excited state light shifts [5].It would be straightforward to add an extra ground statechannel to the numerical calculation to account for this.While there are a number of ways our simple modelcan be improved, it gives a practical way to calculatethe needed complex scattering length for characterizingthe intercombination line OFR spectrum of alkaline earthatoms. The model gives two key results. First, it shouldcorrectly describe the variation of the real part of thescattering length as laser frequency is tuned across mul-tiple resonances. Second, the isolated resonance approx-imation should correctly describe the molecular loss rateout to order of hundreds of line widths away from indi-vidual isolated resonances. Experimental Setup
Details of the experimental setup can be found inRef. [8]. Atoms are loaded into an optical dipole trapfrom a magneto-optical trap operating on the S - P intercombination transition. A horizontal (H) and a ver-tical (V) beam intercept in the ˆ x − ˆ z plane to form acrossed dipole trap at the origin shown in Fig. 5 (draw-ing is to scale). Both beams are derived from the same1064 nm laser and are linearly polarized along the ˆ y axiswith 1 /e waists 63 µ m and 53 µ m, respectively. Theoptical beams point in the directionsˆ k H = − cos θ H ˆ x − sin θ H ˆ z , ˆ k V = − cos θ V ˆ z + sin θ V ˆ x , ˆ k PA = − cos θ PA ˆ x + sin θ PA ˆ y , (14)with θ H = 16 . ◦ , θ V = 14 . ◦ , and θ PA = π/
8. The beamsare linearly polarized along ˆ ǫ H = ˆ ǫ V = ˆ y and ˆ ǫ PA = ˆ z .The image plane is spanned by cos π ˆ x − sin π ˆ y and ˆ z .A bias magnetic field of B z ≃
100 mGauss defines theatomic quantization axis. P o t e n t i a l ( µ K ) − −
50 0 50 100
V beam coord. ( µ m) P o t e n t i a l ( µ K ) − −
50 0 50 100
H beam coord. ( µ m) FIG. 5: Geometry of the experiment in the absorption imageplane. The directions ˆ x , ˆ y , and ˆ z define the lab frame, whereboth gravity ˆ g and bias magnetic field ˆ B are parallel to ˆ z .Symbols ˆ k are beam directions, and ˆ ǫ are beam polarizationvectors, where subscripts H , V , and PA indicate horizontal,vertical, and PA beams. H and V Gaussian beam profilesare shown in blue outline, PA Gaussian beam profile in redoutline.
The focal condition is not critical to model the poten-tial sufficiently well because of the large Rayleigh ranges( > U i ( x ) ≡ − U iT exp (cid:26) − w i (cid:16) [ˆ ǫ i · x ] + [(ˆ k i × ˆ ǫ i ) · x ] (cid:17)(cid:27) , (15)with individual Gaussian beam trap depth [9] U iT = P i πcǫ w i Re α S (1064 nm) , (16)and total laser power P i . The real part of the dynamic S polarizability at 1064 nm is Re α S (1064 nm) ≃
239 a . u . [10], with atomic unit of polarizability 4 πǫ a . The full model potential includes the gravitational ac-celeration g pointing along − ˆ z and is given by U ( x ) = U H ( x ) + U V ( x ) + m Sr gz. (17)Gravity sets the trap depths of ∼ µ K and ∼ µ K alongV and H. The two graphs on the right hand side of Fig. 5show cuts through the model potential, which has beenadjusted to match the measured trapping frequencies.An isosurface (dark blue) of the model potential at 7 µ Kis shown in the zoomed-out portion.Typical kinetic energies are 2-4 µ K, and in-trap FullWidths at Half Maximum (FWHM) are 45-55 µ m. ThePA beam (red outline) propagates along the horizontalimage axis in the ˆ x − ˆ y plane with a waist w PA = 41 µ mand is linearly polarized along ˆ z . Although the PA beamdiameter 2 √ × w PA ≃ µ m is larger than the typ-ical cloud FWHM, we use a density-averaged intensity I av = R d xρ ( x ) I ( x ) / R d xρ ( x ) to characterize the PAintensity interacting with the atoms. Typical values are I av ≃ (0 . − . × I pk , where I pk = Pπw is the Gaussianbeam peak intensity for total beam power P .In addition to scattering from the atomic transitionwith rate Γ sc , the PA beam also adds to the optical trapslightly. For large intensities and small w PA and espe-cially in a standing wave configuration, this effect can be-come important. In this work, the additional trap depthintroduced by the PA beam is typically < . µ K. Imaging Procedure and Analysis
We use a two-component long-working-distance micro-scope with 3 µ m resolution at 532 nm. The imaging lensis placed outside the vacuum chamber at a distance of150 mm from the atomic cloud. The second part of thetelescope is an eyepiece mounted to a CCD camera with1024 × µ m width. The eyepieceincludes an interference filter at 461 nm (intensity trans-mission 0.45 within 10 nm bandwidth, ND 5.0 otherwise).Time of flight measurements were used to determine animaging magnification of 6.5, corresponding to 2 . µ mper pixel. We tested the imaging system resolution ina separate test setup that included the effect of an anti-reflection (AR) coated vacuum viewport, same as the oneused on our vacuum chamber. Using incoherent whitelight filtered by the blue interference filter with transmis-sive test targets, resolutions of ∼ µ m were measured.Absorption imaging with coherent light causes etaloningbetween the CCD chip and the glass plate that covers it.To reduce fringing, the glass plate surface is wedged at ∼ ◦ and AR coated for the imaging wavelength.Without the elastic contribution of the OFR effect,the Sr sample does not thermalize on experimentaltimescales due to its small a bg , and it is important toobtain information about both the potential and kineticenergies of the gas. We interleave experimental runs be-tween in-situ imaging of the sample and imaging after theatoms were dropped for 1 . ◦ about thecenter of the atom cloud to transform the images intothe eigenframe of the trap. The pictures are then inte-grated along the horizontal (vertical) axis and are fit toextract w H ( w V ) with one-dimensional Gaussian distri-butions including a background offset and linear slope toaccount for residual fringing and CCD readout noise inthe absorption image.A common problem in imaging atomic clouds withlarge optical depth is light that does not interact withatoms hitting the camera in the same position. Commoncauses are forward scattering of photons into the imag-ing path or mixed probe polarization. Probing the S - P transition is insensitive to probe polarization evenunder a small bias magnetic field. To identify the pos-sibility of a limiting optical depth OD ceiling , a series ofimages were taken with increasing atomic density. Wecompared the peak optical depth in-situ versus the pixel-summed optical depth either of an in-situ or a corre-sponding time-of-flight image, and we estimate a con-servative limit OD ceiling ≥ .
5. Typical in-situ opticaldepths in this paper were below 2, so that we can neglectthe effect.For probe intensities larger than 0 . I sat , with satu-ration intensity I sat ≃
40 mW / cm , saturation of theimaging transition becomes important. To account forthe saturated absorption, we correct the full expressionfor the OD to the linear Beer-Lambert regime [8, 11].For each picture the saturation correction is applied ona pixel-by-pixel basis.While the shadow image is forming and the atoms arescattering light, their momentum undergoes a randomwalk due to the spontaneous emission of blue photons.By examining the shadow image versus a reference im-age without atoms, we extract the number of photonsthat the atoms removed from the probe pulse. By vary-ing the probe time, we find the cloud expansion as afunction of the number of photons scattered and apply acorresponding correction to the cloud widths from bothin-situ images and TOF images [8].We extract the integrated OD of the mean image toobtain the atom number. Corrections for probe heatingare then applied to the fitted widths. The in-situ image widths are used to calculate the mean atomic density. Fi-nally, we calculate a time-of-flight temperature correctedfor finite size effects by the in-situ widths. Loss Spectra ( kℓ opt γ m /γ ≪ ) To extract ℓ opt γ m from the loss spectra, data wasfit to an approximate expression for the integral of theatomic density, where the density after the PA pulse wascalculated via the differential equation ˙ ρ = − ¯ K in ρ − ρ/τ bg [12]. The initial density distribution was modelledas Gaussian with w H and w V . The width into the planeof the images was extrapolated from a Monte-Carlo sim-ulation of the trap model potential. The fitting functionis N = 2 √ π Z ∞ du √ u e − u K in (∆ , u, ℓ opt γ m , T ) τ eff ρ e − u , (18)with normalized dimensionless trap length scale u . Thisexpression describes the number signal normalized to themeasured atom number after the PA pulse of duration τ PA and when the detuning from molecular resonanceis large. The quantity τ eff = τ bg e τ hold /τ bg ( e τ PA /τ bg − τ bg and that wehold the atoms for τ hold = 100 ms between the end ofthe PA pulse and the imaging. The number ρ is thepeak density after the PA pulse and at large detuning.The definition of all other symbols follows the main text.To make the fit numerically tractable, the integral wasapproximated as a ten-term sum using Gauss-Laguerrequadrature.The thermally-averaged inelastic collision rate per par-ticle ¯ K in was derived from the s -wave scattering matrixof Bohn and Julienne, using the definition of the opticallength ℓ opt = Γ s / kγ m in terms of the stimulated widthΓ s [4, 7]. Here, we use a thermally-averaged inelastic ratecoefficient ¯ K in ≡ h ~ kσ in /µ i , where σ in = 2 πk (1 −| S | ) [2].The angular brackets denote a thermal average over theinitial collision momenta. This average was performedon the inelastic rate coefficient (rather than the densityafter the PA pulse) under the assumption that prior tothe PA pulse, many single-particle velocities exist in ev-ery differential volume in the trap. Since single collisionstake place in a small volume, the spatial dependence of¯ K in was carried through into the differential equation forthe density rather than averaged out.To fit the inelastic loss spectra, ¯ K in was expressed as¯ K in (∆ , u, ℓ opt γ m , T ) = 8 √ π ~ ℓ opt γ m µ Z ∞ dη γ √ η e − η D + Γ / ,D ≡ ∆ + k B Th η − ν rec − ν s e − u/u , Γ ≡ γ + 2 k th ℓ opt γ m √ η, (19)with dimensionless relative momentum magnitude η ≡ k /k , thermal momentum ~ k th = √ µk B T , center-of-mass PA photon recoil energy hν rec , trapping laser acStark shift at the center of the trap ν s , and Planck’sconstant h = 2 π ~ . The PA laser with optical frequency ν l is detuned from the PA resonance by∆ ≡ ν l − [ ν ( S − P ) + ν n + ∆ ν ( I )] , (20)where ν ( S − P ) is the atomic transition frequency,and hν n is the energy of state n with respect to the freethreshold. The detuning term ∆ ν ( I ) accounts for the acStark shift of the molecular resonance with respect to I .The integral in Eq. 19 was approximated as a 53-termsum using Gauss-Laguerre quadrature. The quantities( ℓ opt γ m ), T , u , and a term added to the detuning torepresent the line center were allowed to vary. The pa-rameter T is used as a check against the experimentallymeasured temperatures and agrees well with the experi-ment. The Stark shift term ν s e − u/u was included to ac-count for the broadening of the atomic loss profile to theblue side of a molecular resonance due to the ac Starkshift induced by the trap. The molecular line width γ was extracted from the Monte-Carlo simulation in thenext Section. Monte-Carlo Simulation ( kℓ opt γ m /γ ∼ ) To model the thermodynamic effects caused by the in-terplay of elastic and inelastic collisions in an anharmonictrap with evaporation, we use a Monte-Carlo simula-tion since analytic expressions such as those presentedin the previous Section are not available. The simula-tion is based on classical particles moving in a conser-vative model potential that includes the Gaussian beamshapes as well as gravity. A commonly used method [13–15] to include collisions in such simulations is due toBird [16] and detailed discussions of the method in thecontext of ultracold atoms in optical traps can be foundin Refs. [17, 18]. The simulation procedure here is de-scribed in more detail in Ref. [8].The simulation uses the elastic and inelastic crosssection formulas derived in Section “Scattering Matrix,Cross Sections, and Collision Rates”: σ in ( k ) = 4 πk ℓ opt γ m γ (∆ + E/ ~ ) /γ + (1 + 2 k ℓ opt γ m γ ) / ,σ el ( k ) ≃ k ℓ opt γ m γ σ in ( k ) , (21)where the second relation is an approximation valid inan ideal ultracold gas like Sr where the backgroundscattering length a bg can be neglected. For a gas at non-zero temperature, the detuning ∆ includes the atomicmotion and the trap ac Stark shift as in the previoussection. The initial particle distribution is synthesized by drop-ping atoms into the model potential and letting themevolve for several hundred ms without collisions (usingan embedded Runge-Kutta method). Each particle is ini-tially generated from independent Gaussian distributionsalong the trap eigenaxes, both in position and velocity.The initial widths of these Gaussian distributions are ad-justed until the particle distribution after having settledin the model potential matches the experimental in-situand TOF data when the PA laser is far detuned from theOFR resonance.To calibrate the Monte-Carlo simulation, we modelleda three-dimensional isotropic harmonic trap of mK trapdepth and checked the thermodynamic effects of inelasticand elastic collisions independently. Using a Gaussianinitial phase-space density at several µ K and only al-lowing inelastic collisions at a velocity-independent crosssection, we recovered heating rates per particle thatmatch the corresponding analytic expressions [8]. Simi-larly, we check cross-dimensional thermalization rates ata velocity-independent elastic cross section. We foundthat the average number of elastic collisions requiredfor cross-dimensional thermalization in our simulationagrees [8] with the analytic expressions from Ref. [15].The Monte-Carlo simulation also includes the effect ofscattering from the atomic line. For the largest opticallengths in Fig. 4 of the main paper, the atomic scatteringrate is ∼ − corresponding to 1.2 photons scattered peratom during the 200 ms exposure time. At these scatter-ing rates, the photon absorption along the PA laser di-rection and the random reemission change the resultingcloud widths by less than 10%. Due to the spontaneousreemission as a spherical wave, the mean free path of ascattered photon in a sample of OD ∼ µ s, either in the trapor after TOF expansion for 1 . µ m image resolution. The final image isthen analyzed in the same way as the experimental datato extract in-trap widths and TOF temperatures. Byadding the final measurement step instead of calculatingthe covariance matrices of the position and velocity dis-tributions directly, the quantitative agreement with theexperiment was improved significantly in the regime oflarge inelastic losses.0 Maximal scattering length modification
In the following, we build on the understanding fromour current experiment and try to estimate the maxi-mum scattering length under the most ideal conditions.We assume that atomic scattering is the dominant lossmechanism, that k →
0, and that there is no extra molec-ular loss, such that γ = 2 γ a . There is also no spatialinhomogeneity, and thus it does not reflect the currentexperimental conditions. Theoretical calculations of themolecular line strength factors are used to make theseestimates. Also note that the line strength factors areonly order-of-magnitude estimates beyond n < − The Optically-Modified Scattering LengthConstrained by Atomic Light Scatter
In the k → a due to an optical Feshbach resonance is given by∆ a = ℓ opt ( δ − δ ) γ m ( δ − δ ) + γ m / ℓ opt ( δ − δ ) γ a ( δ − δ ) + γ a , (22)where δ is 2 π times the detuning from atomic resonance, δ is the difference between the molecular and atomicresonance frequencies, γ m is the molecular linewidth, γ a is the atomic linewidth, and γ m = 2 γ a . The quan-tity ℓ opt = ξI , where I is the PA laser intensity and ξ ≡ λ a πc |h n | E i| k is a resonance-specific constant that de-scribes the Sr+Sr molecular structure.When using an OFR, δ must be chosen such that in-elastic collisional loss is small. In this regime, loss due toatomic light scattering dominates. The atomic scatteringrate Γ sc is given byΓ sc = γ a s s + 4( δ/γ a ) . (23)Here s = I/I sat , where I sat is the saturation intensityof the atomic transition. Typically the experiment de-termines the maximum allowable value of Γ sc , therebyconstraining I . Under this constraint, I can be expressedas I = I sat sc /γ a − sc /γ a (cid:18) δ γ a (cid:19) ≈ I sat (cid:18) Γ sc γ a (cid:19) (cid:18) δγ a (cid:19) , (24)where it has been assumed that γ a ≫ Γ sc and δ ≫ γ a , thelatter of which is true due to strontium’s narrow inter-combination line that is used in our work. The maximum ℓ opt for a given detuning is simply ℓ opt = ξI = 8 ξI sat (cid:18) Γ sc γ a (cid:19) (cid:18) δγ a (cid:19) . (25) FIG. 6: Equation 26 plotted near molecular resonance. Herean ξ corresponding to the − .
264 MHz line has been used,Γ sc = 1 s − , γ a = 2 π × . I sat = 3 µ W / cm . Thisbehavior is well approximated by ∆ a dis of Eq. 27.FIG. 7: Equation 26 plotted for large detunings. This plotwas generated with the same values for ξ , δ , Γ sc , γ a , and I sat as those used in Fig. 6. This behavior of ∆ a for largedetunings is well approximated by ∆ a lin of Eq. 28. As willbe discussed in Section “The Large Detuning Case,” thesedetunings are unphysical given the molecular structure andthe coupled-channnels theory discussed in the main text. Thus ∆ a , constrained to a certain Γ sc , is∆ a = 16 ξI sat (cid:18) Γ sc γ a (cid:19) δ γ a ( δ − δ ) γ a ( δ − δ ) + γ a . (26)This expression is plotted in Figs. 6 and 7. Within tensof linewidths γ m of δ = δ , the factor δ in the numeratorof Eq. 26 is slowly varying and can be approximated as δ . Let ∆ a dis refer to ∆ a in this regime. Therefore,∆ a dis = 16 ξI sat (cid:18) Γ sc γ a (cid:19) δ γ a ( δ − δ ) γ a ( δ − δ ) + γ a , (27)which has the dispersion shape of Fig. 6. When δ ≫ δ , | ∆ a | varies linearly with δ . Let ∆ a lin be the expression1for ∆ a when δ ≫ δ .∆ a lin = 16 ξI sat (cid:18) Γ sc γ a (cid:19) (cid:18) δγ a (cid:19) , (28)which describes the linear behavior evident in Fig. 7. The Maximum Useful Scattering Length Near aMolecular Resonance
FIG. 8: Equation 26 plotted in blue and Eq. 29 plotted inred. This plot was generated with the same values for ξ , δ ,Γ sc , γ a , and I sat as those used in Fig. 6. Note that K in issignificant near a molecular resonance but that this quantitydrops off faster than ∆ a . Near a molecular resonance where | ∆ a | has dispersivebehavior, it follows from Eq. 27 that | ∆ a | has a localmaximum at | δ − δ | ≈ ± γ a . However, at these smalldetunings inelastic collisional loss must be considered.For k →
0, the inelastic rate coefficient K in is given by K in = 4 π ~ ℓ opt µ γ m ( δ − δ ) + γ m /
4= 16 π ~ ℓ opt µ γ a ( δ − δ ) + γ a . (29)Putting Eq. 25 into Eq. 29, it can easily be shown thatthe maximum value of K in , denoted by K maxin , occurs at δ ≈ δ , consistent with the case when ℓ opt in Eq. 22 isnot constrained by atomic light scatter.As Fig. 8 shows, inelastic loss (described by K in ) issignificant near δ . Nevertheless, this figure makes it ap-parent that one can choose a detuning that is just outsideof the influence of K in but that still yields a significant∆ a . This detuning is implied upon setting K in = κK maxin ,where κ describes the fraction of molecular loss deemedacceptable by the experiment. Solving this equation fordetuning yields that δ − δ ≈ ± γ a / √ κ . Evaluating Eq. 26at this detuning produces ∆ a max ,∆ a max ≈ ± ξI sat √ κ (cid:18) Γ sc γ a (cid:19) (cid:18) δ γ a (cid:19) , (30) n δ (GHz) δ lin (GHz)-2 -0.023 -7.22-3 -0.221 -655-4 -1.084 -1.57 × -5 -3.460 -1.60 × -6 -8.400 -9.40 × -7 -17.0 -3.84 × TABLE I: Theoretical values for δ for the n = − u resonances and the associated values for δ lin . Here κ =0 . where ∆ a max is the maximum scattering length changewhen ∆ a is dispersive and inelastic loss and atomic lightscatter are negligible. The intensity corresponding to∆ a max , given by Eq. 24, is I ∆ a max ≈ I sat (cid:18) Γ sc γ a (cid:19) (cid:18) δ γ a (cid:19) . (31) The Large-Detuning Case
Although | ∆ a | in Eq. 26 seems to increase withoutbound when δ ≫ δ , coupled-channels OFR theory(which so far has not been considered in this discussion)dictates that if | ∆ a | is based on the n th resonance, themodification to the scattering length will vanish at somepoint before δ is equal to the detuning of the ( n − δ − δ can be sufficiently large totake advantage of the linear increase with δ but not sofar that coupled-channels effects become a concern.Let δ lin be the detuning at which the magnitude of∆ a lin is equal to | ∆ a max | . It follows that | δ lin | = √ κδ /γ a . If δ lin is comfortably outside the regime wherecoupled-channels effects are a concern, then the maxi-mum useful scattering length occurs when one is manylinewidths γ m detuned from δ but not far enough de-tuned to be close to other resonances. However, theoret-ical values for δ make it clear that detuning to δ lin wouldalways require crossing another molecular resonance, socoupled-channels effects dictate that the maximum scat-tering length one can achieve is given by ∆ a max (Eq. 30),occurring when δ is as close as possible to a molecularline, while molecular losses are constrained to a givenlevel. Table I considers 6 different molecular resonancesand their associated values of δ lin , which are many ordersof magnitude larger than δ for subsequent resonances.2 n δ (GHz) ξ (a / (W / cm )) | ∆ a max | ( a ) I ∆ a max (W / cm )-2 -0.023 6110 5.97 4.89 × − -3 -0.221 32.8 2.92 0.443-4 -1.084 27.9 59.3 10.6-5 -3.460 3.30 71.5 108-6 -8.400 0.272 34.7 637-7 -17.0 0.012 6.21 2.61 × TABLE II: Theoretical values for δ and ξ for the n = − u resonances are used to calculate the maximumuseful scattering length. The intensities required for thesescattering lengths are also included. The optimal detuningfor each resonance is given by | δ − δ | = γ a / √ κ = 10 γ for thevalue κ = 0 .
01 that was used in the table.
Conclusion
In light of Section “The Large-Detuning Case”, we con-clude that for a given molecular line, the best optically-modified scattering length occurs for a laser detunedjust outside the influence of inelastic loss and with anintensity just low enough to prevent heating due toatomic light scatter. This optimal detuning is given by δ − δ = γ a / √ κ . Table II lists theoretical 0 u values for ξ and δ as well the associated values for ∆ a max and I ∆ a max . Values of Γ sc = 1 s − and κ = 0 .
01 have beenused. For the strontium S - P line, γ a = 2 π × . I sat = 3 µ W / cm . These numbers yield an opti-mal detuning of δ − δ = 2 π ×
75 kHz. The intensitycolumn was included to point out that many resonancescannot be used on technical grounds (for instance in-tensities greater than a few kW/cm are impractical forexternal cavity diode lasers).Under the ideal conditions of zero temperature, no spa-tial inhomogeneity, no extra molecular loss and the as-sumption that atomic light scattering is the dominantloss mechanism, Table II shows that the resonance loca-tion of the n = − ξ rapidly decreases with increasing | δ | . Ac-cording to Eq. 30, only moderate gains in ∆ a max can beachieved by allowing for more inelastic loss (when detun-ing closer to a molecular line) since any increase in K in will only be accompanied by a √ K in increase in | ∆ a max | . More significant gains can be achieved by shortening ex-perimental time scales to allow for larger values of Γ sc since | ∆ a max | is linear in Γ sc . ∗ Permanent address: The Niels Bohr Institute, Univer-sitetsparken 5, 2100 Copenhagen, Denmark.[1] J. M. Hutson, New J. Phys. , 152 (2007).[2] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev.Mod. Phys. , 1225 (2010).[3] J. P. Burke, Jr., Ph.D. thesis, Department ofPhysics, University of Colorado (1999), URL http://jila.colorado.edu/thesis .[4] J. Bohn and P. Julienne, Physical Review A , 414(1999).[5] R. Ciury lo, E. Tiesinga, and P. S. Julienne, Phys. Rev.A , 022710 (2006).[6] P. S. Julienne and F. H.. Mies, Physical Review A ,3792 (1986).[7] R. Ciury lo, E. Tiesinga, and P. S. Julienne, Phys. Rev.A , 030701 (2005).[8] S. Blatt, Ph.D. thesis, Department ofPhysics, University of Colorado (2011), URL http://jila.colorado.edu/yelabs/pubs/theses.html .[9] R. Grimm, M. Weidem¨uller, and Y. B. Ovchinnikov,arXiv:physics/9902072v1 (1999).[10] M. M. Boyd, Ph.D. thesis, Department ofPhysics, University of Colorado (2007), URL http://jila.colorado.edu/thesis/ .[11] L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).[12] T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye,R. Ciury lo, P. Naidon, and P. S. Julienne, Phys. Rev.Lett. , 203201 (2006).[13] H. Wu and C. J. Foot, Journal of Physics B: Atomic,Molecular and Optical Physics , L321 (1996).[14] H. Wu, E. Arimondo, and C. Foot, Physical Review A , 560 (1997).[15] J. Goldwin, S. Inouye, M. Olsen, and D. Jin, PhysicalReview A , 043408 (2005).[16] G. A. Bird, Molecular gas dynamics and the direct simu-lation of gas flows (Clarendon Press, Oxford, 1994).[17] M. E. Gehm, Ph.D. thesis, Departmentof Physics, Duke University (2003), URL .[18] J. M. Goldwin, Ph.D. thesis, Department ofPhysics, University of Colorado (2005), URL http://jila.colorado.edu/thesis/http://jila.colorado.edu/thesis/