Two-color photoassociation spectroscopy of ytterbium atoms and the precise determinations of s-wave scattering lengths
Masaaki Kitagawa, Katsunari Enomoto, Kentaro Kasa, Yoshiro Takahashi, Roman Ciurylo, Pascal Naidon, Paul S. Julienne
aa r X i v : . [ phy s i c s . a t o m - ph ] A ug Two-color photoassociation spectroscopy of ytterbium atoms and the precisedeterminations of s -wave scattering lengths Masaaki Kitagawa, Katsunari Enomoto, Kentaro Kasa, YoshiroTakahashi,
1, 2
Roman Ciury lo, Pascal Naidon, and Paul S. Julienne , Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan Instytut Fizyki, Uniwersytet Miko laja Kopernika, ul. Grudzi¸adzka 5/7, 87–100 Toru´n, Poland Atomic Physics Division and Joint Quantum Institute,National Institute of Standards and Technology, 100 Bureau Drive,Stop 8423, Gaithersburg, Maryland 20899-8423, USA. (Dated: October 22, 2018)By performing high-resolution two-color photoassociation spectroscopy, we have successfully de-termined the binding energies of several of the last bound states of the homonuclear dimers of sixdifferent isotopes of ytterbium. These spectroscopic data are in excellent agreement with theoreticalcalculations based on a simple model potential, which very precisely predicts the s -wave scatteringlengths of all 28 pairs of the seven stable isotopes. The s -wave scattering lengths for collision of twoatoms of the same isotopic species are 13 . Yb, 3 . Yb, − . Yb, − . .
4) nm for
Yb, 10 . Yb, 5 . Yb, and − . Yb. The coefficient of the lead term of the long-range van der Waals potential of the Yb molecule is C = 1932(30) atomic units ( E h a ≈ . × − J nm ). PACS numbers: 34.50.Rk, 34.20.Cf, 32.80.Pj, 34.10.+x
I. INTRODUCTION
A collision between two atoms is a fundamental phys-ical process and can be described by a few partial scat-tering waves for ultracold atoms. At a sufficiently lowtemperature the kinetic energy of colliding atoms be-comes less than the centrifugal barrier, and only the s -wave scattering is possible. The s -wave scattering lengthis an essential parameter for describing ultracold colli-sions. It also governs the static and dynamic propertiesof quantum degenerate gases like a Bose-Einstein conden-sate (BEC) or a degenerate Fermi gas (DFG) of fermionicatoms in different spin states. Since the s -wave scatteringlength is very sensitive to the ground state interatomicpotential, especially at short internuclear distance, theprecise ab initio calculation of the scattering length isvery difficult, and therefore we must resort to experi-mental determination. The most powerful approach fordetermining the scattering length is to measure the bind-ing energy ( E b ) of the last few bound states in the molec-ular ground state, since the energy E b is closely relatedto the s -wave scattering length. So far, the binding ener-gies were measured via two-color photoassociation (PA)spectroscopy for Li [1], Na [2], K [3], Rb [4], Cs [5] andHe [6]. A schematic description is shown in Fig. 1. If alaser field L is resonant to a bound-bound transition, itcauses an Autler-Townes doublet. This effect is detectedas a reduction of a rate of a free-bound PA transitiondriven by the other laser field L . This scheme is calledAutler-Townes spectroscopy, and it is also explained interms of the formation of a dark state. If both lasersare off-resonant and the frequency difference matches E b ,these lasers drive a stimulated Raman transition from thecolliding atom pair to a molecular state in the electronic ground state. In this Raman spectroscopy, the resonanceis detected as an atom loss.The s -wave scattering length of two colliding atoms isdetermined by the adiabatic Born-Oppenheimer interac-tion potential V ( r ) between the two atoms, which is verywell approximated at large interatomic separation r bythe van der Waals contribution, V ( r ) ≈ − C /r , where C is the van der Waals coefficient due to the dipole-dipole interaction. The s -wave scattering length is givenby the following formula, based on a quantum correctionto the WKB approximation so as to be accurate for thezero-energy ( E = 0) limit [7, 8, 9], a = ¯ a h − tan (cid:16) Φ − π (cid:17)i . (1)Here ¯ a = 2 − / / / (cid:0) µC / ¯ h (cid:1) is a characteristiclength associated with the van der Waals potential, whereΓ is the gamma-function, µ is the reduced mass, and ¯ h is the Planck constant divided by 2 π . The semiclassicalphase Φ is defined byΦ = √ µ ¯ h Z ∞ r p − V ( r ) dr, (2)where r is the inner classical turning point of V ( r ) atzero energy. The number of bound states N in the po-tential is [8] N = (cid:20) Φ π − (cid:21) + 1 , (3)where [ . . . ] means the integer part. As was mentionedabove, the scattering length is very sensitive to the phaseΦ and can take on any value between ±∞ when Φ varies ∆ E b L L S + P S + S f f FIG. 1: (color online) Schematic description of the two-colorPA spectroscopy. The laser L drives one-color PA transition.The laser L couples the bound state in the excited molecularpotential to the one in the ground molecular potential. Thedetuning ∆ of the PA laser with respect to the one-color PAresonance is set to several MHz for the Raman spectroscopy,while ∆ is set to zero for the Autler-Townes spectroscopy. over a range spanning π . While the accurate calcula-tion of Φ is very difficult since it requires a knowledge ofthe whole potential, Φ is proportional to √ µ , and so theformula in Eq. (1) gives a simple mass-scaling of the iso-topic variation of the scattering length once a and N areknown for one isotopic combination. Mass scaling appliesas long as small mass-dependent corrections to the Born-Oppenheimer potential [12, 13, 14] can be ignored. Whilemass scaling is often used for bound and scattering statesfor cold atomic systems [10, 11], and seems applicableto large-mass systems within experimental uncertainties[15], exceptions are known [16], and its accuracy shouldbe carefully tested for different kinds of systems.Ytterbium (Yb) is a rare-earth element with an elec-tronic structure similar to that of the alkaline-earthatoms. One of the unique features of Yb atoms is arich variety of isotopes with five spinless bosons ( Yb,
Yb,
Yb,
Yb,
Yb) and two fermions (
Ybwith the nuclear spin I = 1 / Yb with I = 5 / S symmetry. Therefore, the ground molec-ular state of Yb has only one potential of Σ g molec-ular symmetry with no electronic orbital and spin an-gular momenta. This is in contrast to the case of analkali dimer which has spin-singlet Σ g and spin-triplet Σ g ground states with a complicated hyperfine struc-ture. These two unique features of the Yb system, thatis, the existence of rich variety of isotopes and one sim-ple isotope-independent molecular potential, offers thepossibility to systematically check the scattering lengththeory with an unprecedented precision. If we can beconfident in the theory, then it would enable us to deter- vertical FORThorizontal FORT From Dye laser
PBS λ /2 plate AOM 1mirror λ /4 plate PBSfiber couplerfiber coupler mirror λ /4 plate AOM 2PBS PBSPBSOptical fiberOptical fiber Yb atoms FIG. 2: (color online) Schematic description of an experi-mental setup for PA lasers. The two lasers were prepared bysplitting one laser, and two double-pass acousto-optic modu-lators (AOMs) were employed to tune them. Then the twolasers were aligned in the same path. The solid lines andbroken lines indicate the laser beams before and after thedouble-pass, respectively. A polarizing beam splitter (PBS)and a λ/ mine the scattering lengths of all possible isotope pairswhich are not measured experimentally. So far, to deter-mine the s -wave scattering length of Yb, one-color PAspectroscopy was performed [19], which gave the result of5 . Yb was estimated as 6(2) nm [18], assuming thatthe spin was completely unpolarized. However, these twovalues are not enough for a rigorous test of the theory.In this paper, we report an accurate determinationof the s -wave scattering lengths for all Yb isotopes in-cluding those for different isotope pairs. By using two-color PA spectroscopy with the intercombination transi-tion S − P , we successfully determined the binding en-ergy E b of twelve bound states near the dissociation limitof four homonuclear dimers comprised of bosonic atoms( Yb , Yb , Yb , and Yb ) and two dimerscomprised of fermionic atoms ( Yb and Yb ). Thespectroscopically measured binding energies are in excel-lent agreement with theoretical calculations based on asimple model potential that was fit to the data. The cal-culated s -wave scattering lengths for the six isotopes obeythe mass-scaling law with very good precision. Moreover,this excellent agreement allows us to accurately deter- N u m be r o f a t o m s Frequency difference [MHz]
Yb v=1 J=0
Yb v=2 J=0
Yb v=2 J=2
Yb v=1 J=0
Yb v=1 J=2
Yb v=1 J=0
Yb v=1 J=2
Yb v=1 J=0
Yb v=1 J=2
Yb v=1 J=2
Yb v=1 J=0
Yb v=1 J=0
FIG. 3: (color online) Two-color PA spectra at about 1 µ K. The horizontal and vertical axes are the frequency differencebetween the two lasers and the number of remained atoms, respectively. The vertical error bars represent the fluctuationsof the number of atoms. The solid lines are fits of the Lorentz functions. The Autler-Townes spectroscopy is applied to the v = 1 , J = 0 , Yb and the v = 1 , J = 0 state of Yb . while the Raman spectroscopy is applied to the others. mine the scattering lengths of all twenty-eight differentisotopic combinations. In addition, we can reveal scat-tering properties of other partial waves such as p - and d -wave scatterings and energy dependence of the elasticcross sections. These results are an important foundationfor future research, such as the efficiency of evaporativecooling, stability of quantum gases and their mixtures,and the clock shift [20]. II. EXPERIMENT
The experimental setup was almost the same as ourprevious experiment of one-color PA spectroscopy [21].All the experiments were performed at about 1 µ K,where only the s -wave scattering is possible. Atomswere first collected in a magneto-optical trap (MOT)with the intercombination transition S − P at 556nm. The linewidth and saturation intensity of the transi-tion were 182 kHz and 0 .
14 mW/cm , respectively. Thelaser beam for the MOT was generated by a dye laserwhose linewidth was narrowed to less than 100 kHz. Thedye laser was stabilized by an ultralow expansion cavity,whose frequency drift was typically less than 20 Hz/s.The number, density and temperature of atoms in theMOT were about 2 × , 10 cm − and 40 µ K, respec-tively. Then the atoms were transferred into a crossed far-off resonant trap (FORT). The number, density andtemperature of atoms in the FORT were about 2 × ,10 cm − , and 100 µ K, respectively. To reach lower tem-peratures, evaporative cooling was carried out by grad-ually decreasing the potential depth of the horizontalFORT beam to several tens of µ K in several seconds.Evaporative cooling worked rather well for the bosonicisotopes,
Yb,
Yb and
Yb and for the fermionicisotope
Yb. Typically 1 × atoms at the temper-ature of about 1 µ K finally remained in the trap andthe density was between 10 cm − and 10 cm − , al-though optimized evaporation ramps and efficiency foreach isotope were different. In particular, we observedrapid atom decay for Yb due to three-body recom-bination. The steeper evaporation ramp was needed for
Yb to obtain enough number of atoms. It is also notedthat an unpolarized sample of
Yb enabled us to per-form efficient evaporative cooling even at a low temper-ature via elastic collisions between atoms with differentspins. For the bosonic
Yb and fermionic
Yb iso-topes, however, the evaporative cooling did not work well.In order to cool
Yb and
Yb, we performed sympa-thetic cooling with bosonic
Yb [22, 23]. BichromaticMOT beams for simultaneous trapping of two isotopes inthe MOT were generated by an electro-optic modulator(EOM), of which the modulation frequency correspondsto the isotope shift. Typically 6 × atoms for Yband 2 × atoms for Yb at the temperature of about1 µ K finally remained in the trap. The bosonic isotopeof
Yb was hard to collect in the MOT due to its smallnatural abundance of 0.13 percent, although there are nofundamental difficulties for
Yb in principle.After the evaporative cooling, the two lasers, L for thefree-bound transition and L for the bound-bound tran-sition, were simultaneously applied to the atoms in thetrap for about 30 ms. These beams were focused to a 100 µ m diameter. The schematic setup for the PA lasers isrepresented in Fig. 2. The two laser beams were preparedby splitting one laser beam for the MOT and thereforehad the same frequency linewidth and stability as theMOT beam. The relative frequency was controlled byacousto-optic modulators (AOMs). The two laser beamswere coupled in the same optical fiber. A polarizing beamsplitter (PBS) and a half-wave ( λ/
2) plate after the op-tical fiber were inserted to fix the polarization of bothlasers in the same direction. Finally, the PA laser beamswere aligned to pass through the atoms in the FORT byusing a CCD camera for absorption imaging. The detun-ing of the PA laser with respect to the atomic resonance S − P was easily checked by observing the frequencyat which the atoms in the MOT disappeared. The powerof the PA laser was monitored by a photodiode. For theRaman spectroscopy, the frequency f of L was fixedwith a certain detuning ∆ from a particular PA reso-nance, and the frequency f of L was scanned to searchfor the bound states of the ground state. For the Autler-Townes spectroscopy, f was fixed to a particular PA res-onance, and f was scanned. Table I shows the excitedstate rovibrational levels used for the free-bound transi-tion, where the resonance position is given in frequencydetuning from the atomic S − P transition. Our setupfor the PA laser beams could provide enough power fordetuning smaller than 1 GHz. To obtain enough powerfor detuning larger than 1 GHz, the sideband of the EOMwas used for the MOT while the carrier is used as the PAlasers. We observed the two-color PA signals by measur-ing the number of surviving atoms with the absorptionimaging method with the S − P transition. III. EXPERIMENTAL RESULTS
All the two-color PA spectra observed are shown inFig. 3, where v is the vibrational quantum number count-ing down from the dissociation limit and J is the rota-tional quantum number. The observed spectral linewidthis typically several tens of kHz due to the finite energydistribution at the 1 µ K temperature and the powerbroadening. The selection rule for the parity determinesthe observable quantum number J in the ground state Σ + g as 0 , , , · · · from the s -wave collision. We observedall resonance positions with detuning less than 300 MHzfrom the dissociation limit except for the v = 2 state for Yb; see Fig. 6 (b).The observed peak positions included the light shift.
TABLE I: Excited state rovibrational Yb dimer levels usedfor the free-bound PA transition, where v e and J e are the re-spective vibrational and rotational quantum numbers; v e isnumbered from the dissociation limit. The position of theexcited state level is given as detuning in MHz from the fre-quency of the atomic S − P transition. For the fermionicisotopes the hyperfine level with the largest total atomic an-gular momentum for the P state defines the dissociationlimit.isotope v e J e position (MHz) Yb 16 1 618 Yb a a Yb 17 1 789
Yb 791
Yb 18 1 99319 1 1972
Yb 17 1 868 a Reference[24]. -10.80-10.75-10.70-10.65-10.60-10.55 f - f [ M H z ] -0.2 -0.1 0.0 0.1 Light shift δ LS [MHz] FIG. 4: (color online) Resonance positions f − f of the v = 1 , J = 0 state of Yb as a function of δ LS . Thetemperature shift is not compensated. The vertical error barsrepresent the uncertainty estimates for the temperature andthe center frequency of the resonance. The horizontal errorbars include the fluctuations of the laser intensities. The dataare fitted with a linear function represented by a solid line. -10.68-10.66-10.64-10.62-10.60 f - f [ M H z ] µ K] FIG. 5: (color online) Resonance positions f − f of the v = 1 , J = 0 state of Yb as a function of the temperature.The light shift is not compensated. The vertical error barsrepresent the uncertainty estimates for the center frequencyof the resonance. The data are fitted with a linear functionrepresented by a solid line. We measured the peak positions with several laser in-tensities for L and L and different detunings ∆, in or-der to compensate the light shift by interpolation or ex-trapolation. We found that the bound-bound coupling isdominant for the light shift for the Raman spectroscopy.Namely, the light shift is expressed as δ LS = β (cid:18) I ∆ + f − f + I ∆ (cid:19) , (4)where β is a constant related to the Franck-Condon fac-tor of the bound-bound transition, I and I are the laserintensities of L and L , respectively. Typical values ofthe detuning ∆ and laser intensity were about 2 MHz and10 − . The contributions of the other ex-cited bound states were negligible, since their resonancefrequencies were typically more than 300 MHz apart,which was much larger than ∆ and ∆ + f − f . Fig-ure 4 shows the light shift δ LS with several intensities for L and L and detuning ∆. The error bars include theuncertainty due to fluctuations of laser intensity and theuncertainty for the temperature of the atom clouds. Wenote that the data are well fitted by a linear function. Itis also noted that the value of β obtained from the fittingis consistent with our estimation of the Franck-Condonfactor of the bound-bound transition, which also ensuresthe validity of our analysis. The resonance position at δ LS = 0 gives the peak position without the light shift.For the Autler-Townes spectroscopy the light shift wasbasically caused by the laser L . When the detuning ∆was not zero, however, the laser L also contributed tothe light shift. The sense and magnitude of the detuning∆ could be estimated from the careful investigation of thespectral shape [25]. Similarly, the peak positions withdifferent laser intensities for L and L were measuredand the light shift was compensated by extrapolation ofthese data.The observed peak positions also suffered from thetemperature shift. The temperature shift is assumed tobe a T k B T , where a T is a constant, k B is the Boltzmannconstant and T is the temperature. The factor a T wasexpected to be 3/2 [6], which was checked experimentallyfor v = 1 , J = 0 state of Yb for the temperature rangefrom 0.5 µ K to 2 µ K. The resonance positions with dif-ferent temperatures are shown in Fig. 5. The data werefitted with a linear function represented in a solid line.The results are consistent with k B T within 9 percent.We also compensated the temperature shift for the otherdata using the k B T dependence. The final results of theexperimentally determined binding energies are listed inTable II.We also measured E b for the v = 1 , J = 0 state of Yb with the two-color PA using the strongly allowed S − P transition. The wavelength, linewidth, and sat-uration intensity of the atomic transition were 399 nm,29 MHz, and 60 mW/cm , respectively. We used the v e = 157 level of the excited Σ + u molecular state, where v e is numbered from the dissociation limit S + P [19]. The resonance position of the level was − . TABLE II: Measured and calculated binding energies E b forhomonuclear isotopic pairs, where v and J are the vibrationaland rotational quantum numbers of the ground state dimerlevel and v is numbered from the dissociation limit. R and ATrespectively represent the Raman and Autler-Townes spectro-scopic method of determining E b .isotope v J method E b (MHz) E b (MHz) Differenceexperiment theory (MHz) Yb 1 0 R -27.661(23) -27.755 0.0942 R -3.651(26) -3.683 0.032
Yb 1 0 AT -64.418(40) -64.548 0.1302 AT -31.302(50) -31.392 0.090
Yb 1 0 AT -123.269(26) -123.349 0.0802 R -81.786(19) -81.879 0.093
Yb 1 0 R -1.539(74) -1.613 0.074
Yb 1 0 R -10.612(38) -10.642 0.0301 0 AT -10.606(17) -10.642 0.0362 0 R -325.607(18) -325.607 0.0002 2 R -268.575(21) -268.576 0.001
Yb 1 0 R -70.404(11) -70.405 0.0011 2 R -37.142(13) -37.118 -0.024
GHz from the dissociation limit. Raman spectroscopywas employed to measure the peak position. The lightshift and temperature shift were also compensated. Themeasured E b of 10 . IV. CALCULATION AND DISCUSSION
The binding energies of the bound states as wellas scattering lengths of all isotopic combinations aredetermined by the reduced mass and a single Born-Oppenheimer potential V ( r ), as long as small mass-dependent adiabatic and non-adiabatic corrections to thepotential are sufficiently small. The key features of thepotential that determine the positions of the last fewbound states are the form of the long range potentialand a phase associated with the short range potential.Consequently, we assume the following simple potentialform to analyze the data: V ( r ) = − C r (cid:18) − σ r (cid:19) − C r + B ( r ) J ( J + 1) , (5)where σ is a constant, C is the van der Waals constantassociated with the dipole-quadrupole interaction, and B ( r ) = ¯ h / (2 µr ) is due to molecular rotation. The firstterm in Eq. (5) gives the Lennard-Jones form for thepotential, for which the short range form can be changedby varying σ . The C term is needed to improve thequality of the fit to the data.By solving the Schr¨odinger equation numerically forthe eigenvalues and comparing to the measured bindingenergies, it is possible to determine an optimum set of TABLE III: Calculated s -wave scattering lengths in nm for Yb isotopic combinations Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb -0.15(19) -4.46(36) -30.6(3.2) 22.7(7) 7.49(8) Yb -31.7(3.4) Yb Yb Yb -1.28(23) potential parameters. The J = 0, v = 2 level of Yb and the J = 0, v = 1 level for Yb were fit to de-termine C and obtain the right number of bound states N for Yb . Then the J = 2, v = 2 level for Yb was added to the fit to determine C and improve thedetermination of C . The results are C = 1931 . E h a , C = 1 . × E h a , σ = 9 . a , N = 72,where a ≈ .
529 nm, E h ≈ . × − J. These param-eters then determine without additional adjustment thebinding energies of the other isotopic combinations shownin Table II and Fig 6(b). Taking into account a lack ofprecise knowledge about the short range part of the po-tential and the magnitude of possible retardation effectsand neglect of higher order dispersion energy terms, weestimate the uncertainties in C and C to be about 2 per-cent and 25 percent respectively, or C = 1932(30) E h a and C = 1 . × E h a . Adding a − C /r van derWaals term only very slightly improved the quality ofthe fit, and the optimum values of C and C remainwithin the above stated uncertainties. The C valueis lower than a previous experimental determination of2300(250) E h a [19] and recent theoretical predictions of2291 . E h a [26], 2567 . E h a [27], and 2062 E h a [28].The short range form of the potential should be viewedas a pseudopotential having the right number of boundstates N but not necessarily giving an accurate shape forthe potential. The model well depth D e /h = 32469 GHzis significantly larger than ab initio values; see Ref. [27]and references therein.The agreement shown in Table II with a precision lessthan 100 kHz between the experimentally determined E b and most of the calculated E b values is quite impressivefor such a simple model potential. The use of a singlemass-independent potential with the appropriate reducedmass is thus seen as an excellent approximation for cal-culating the isotopic variation in binding energies. Thefailure to obtain a fit to the data within experimental er-ror in all cases could be indicative of the failure of massscaling, although it may only be due to the limitationsof the simple form we assumed to represent the potentialover its whole range. Adding a small mass-dependentcorrection on the order of 1 GHz to the well depth of thepotential allows us to fit the binding energies for each iso-tope almost within experimental uncertainties. We seeindications that the potential is deeper for heavier iso- topes. Additional work is needed to see if the Yb systemcan be used to make quantitative tests of the accuracyand limitations of mass scaling.When we added terms to account for relativistic re-tardation effects [29], the χ of the overall fit did notimprove, although the energies of the least bound levelsimproved slightly. These relativistic terms take on theform α W /r in the physically interesting region r ≪ λ ,where α is the fine structure constant and λ ≈ isa characteristic length associated with a mean electronicexcitation energy. The lead term in the retarded van derWaals interactions switches to 1 /r behavior in the verylong range region r ≫ λ . Since the outer turning pointof the least bound level is still only around 150 a , the α W /r form is the appropriate form to use in lookingfor the magnitude of the effect. When we added a C /r term to the potential, we find that we get a slightly bet-ter fit for the most weakly bound levels, but not a betteroverall fit, if we take C = 1 . × − E h a , similar inmagnitude to the value 0 . × − E h a estimated in theway proposed in Ref. [29] using the dipole polarizabilityand C coefficient for Yb reported in Ref. [26]. Our re-ported model parameters and scattering lengths includethe uncertainties associated with the lack of knowledgeof the retardation corrections for this system. Conse-quently, there is a need for a theoretical evaluation ofthese corrections.Given the ground state potential energy curve with theform of Eq. (5) and the parameters given above, we cancalculate the s -wave scattering lengths for all isotopiccombinations by numerically solving the Schr¨odingerequation with the appropriate reduced mass. These areshown in Table III and Fig. 6(a). The uncertainties re-flect the uncertainties in the model parameters and theneed to include retardation corrections to the potential.The value for Yb is in excellent agreement with thevalue 5 . Ybis slightly larger than the value estimated from the ther-malization experiments in Ref. [18]. However, the s -waveelastic cross section for Yb decreases by about 10 per-cent between E = 0 and the 6 µ K temperature of theexperiment. The experimental value could also be largerif the unknown distribution of spin populations differedfrom the assumption of uniformity.Since the three-parameter model potential Eq. (5) is
166 168 170 172 174 176 178-0.1-1-10-100-1000-30-20-100102030 µ /m u E b / h [ M H z ] a [ n m ] (a)(b) FIG. 6: (color online) Calculated scattering lengths (a) andbinding energies E b (b) versus twice the reduced mass, 2 µ inthe atomic mass unit, using the 3-parameter potential energymodel in Eq. (5) with the parameters given in the text. Thevertical lines show the masses for the 7 like-atom pairs. Thesolid and dashed lines in (b) show the J = 0 and 2 eigenvaluesrespectively. The measured values with vertical error bars areshown for the levels for which they have been measured. Themeasured Yb scattering length of Ref. [19] is also shownin (a). The horizontal dashed line in (a) shows the van derWaals length ¯ a . assumed to be common for all possible isotopic combina-tions, the simple analytical formula Eq. (1) also holds.The predictions of this formula are completely indistin-guishable from those of the numerical calculation on thescale of Fig. 6 (a). The actual difference between scatter-ing lengths calculated exactly from the model potentialand those calculated from the analytical formula in Eq.(1) are below 0.03 nm for all like isotope cases except theone with the largest scattering length magnitude, Yb,for which the difference is 0.35 nm. A similar statementapplies to the mixed-isotope cases, for which most differ-ences are below 0.03 nm.As the reduced mass varies from 168/2 to 176/2, thescattering length varies through a complete cycle from −∞ to + ∞ . In fact, we have a rich variety of the scatter- B ( µ K) -15 -14 -13 -12 -11 -10 solid: s +(p)+ddashed: s-only σ ( E ) [ c m ] FIG. 7: (color online) Calculated cross section σ ( E ) versuscollision energy E/k B for two atoms of the same isotope. Thelabel for each curve shows the isotopic mass number. Evenmass number corresponds to identical bosons and odd massnumber corresponds to two fermions with different spin pro-jections. The dashed lines show the s -wave contribution tothe total cross section. The solid lines show the contributionfrom s - and d - waves for the like boson cases, and the con-tribution from s -, p -, and d -waves for the fermion case withdifferent spin components. The E → πa and 4 πa for the respective boson and fermion cases. ing lengths from large negative values for Yb -
Yb,
Yb -
Yb,
Yb -
Yb, and
Yb -
Yb, and al-most zero for
Yb -
Yb,
Yb -
Yb, and
Yb-
Yb, and to large positive for
Yb -
Yb and
Yb-
Yb. Thus, the scattering length can be widely tunedby varying isotopic composition [30].It should be noted that the observed behavior of evap-orative cooling and sympathetic cooling are consistentwith these scattering lengths. The efficient evaporativecooling for
Yb,
Yb, and
Yb is consistent withthe large scattering lengths: 3 . Yb, 11 nmfor
Yb, and 5 . Yb. Inefficient evaporativecooling for
Yb and
Yb is also consistent with thesmall scattering lengths − . Yb and − . Yb. On the other hand, we have successfully usedsympathetic cooling to cool
Yb and
Yb with
Yb,for which the respective mixed-isotope scattering lengthsare 22 nm and 2.9 nm. The extremely large value of − Yb would explain very rapid atom decay ob-served for this system, which could be due to three-bodyrecombination.Using the three-parameter model potential, we can alsocalculate the collisional properties of other partial wavesat non-zero collision energies. Figure 7 shows the energy-dependent cross section for the collision of like isotopicspecies. The cross section can be resonantly enhanced bya shape resonance, caused by the existence of quasiboundrovibrational levels supported by the centrifugal barrier.A d -wave shape resonance exists for Yb as alreadypointed out in Refs. [19, 21]. Our model predicts a broadpeak in the collision cross section near
E/h = 4 . E/k B = 220 µ K, which is off-scale in Fig. 7. We foundthat a low energy p -wave shape resonance also exists for Yb, giving rise to a peak in the cross section near
E/k B = 48 µ K.The s -wave contribution to the cross section becomeszero when the s -wave collisional phase shift is zero. Sucha zero as a function of energy results in a Ramsauer-Townsend minimum in the cross section. This minimumis especially evident when the collision energy is so smallthat other partial waves make negligible contributions tothe cross section, This effect occurs at very low collisionenergy when the scattering length has a small negativevalue, as for Yb and for
Yb. Figure 7 shows thecalculated cross section minima near 2 µ K and 25 µ K forthese respective species. This effect explains why evapo-rative cooling is found to be inefficient for these isotopes.
V. CONCLUDING REMARKS
In conclusion, we report the accurate determinationof the binding energies of the twelve least bound statesin the ground molecular potentials for six Yb isotopesand the accurate determination of the s -wave scatteringlengths for all possible combination of the isotopes basedon a simple three-parameter model potential. The model parameters are based on fitting a very limited set of theexperimental data, making use of only three binding en-ergies: two bound-states of Yb and one bound-stateof Yb . Fitting this limited set of data allows us toreproduce the other nine binding energies with an accu-racy of about 100 kHz and to determine the 28 scatter-ing lengths for the different isotope combinations withan accuracy of about a few percent for most cases. Inaddition, we can calculate the energy-dependent elasticscattering due to other partial waves such as p - and d -waves. These results provide an important foundationfor future research with Yb atoms on such topics as theefficiency of evaporative cooling, the stability of quantumgases and their mixtures, and clock shifts. Acknowledgments
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