arXiv:physics/0309076v1 [physics.atom-ph] 16 Sep 2003 h neato ftomxmlysi-tece ratoms the Cr spin-stretched in maximally for two available of data interaction no the is there exists, symmetry) molecular ie.Mr cuaeBr-pehie oeta en- tempera- potential the ultracold Born-Oppenheimer of accurate and structure More cold electronic the the dimer. revisiting in by regimes atoms ture Cr of erties h Cr the trap. the deplete inelastic that is rates collision scattering “bad” large-spin of byproduct so-desirable ntesetocp ftegon lcrncsae( state electronic ground the information of some field spectroscopy While the self-consistent on space 12]. active [11, potential (CASSCF/CASPT2) complete interaction with perturbation its second-order theory multiconfiguration calculate date, a To to is curves attempt bonding. best metal-metal multiple the of cases treme hoimi h rtao nteproi al iha with table periodic Cr(3 the in atom half-filled first the is Chromium ei rp n oldblw100 mag- below Ioffe-Pritchard cooled an into and MOT trap, a netic a from atoms in chromium loaded used ultracold were where also [5], was scheme Chromium cooling us- new gas cooling. degenerate sympathetic fermionic ing obtaining of possibility the cin,myla onvlpeoeai Es[,10]. isotope, [9, fermionic BECs stable in a phenomena of novel existence inter- to The dipole-dipole lead magnetic may chromium’s actions, interac- as long-range such anisotropic magneto-optical tions, for addition, as In well [4]. as in trapping [7], buffer-cooling trap for magnetic atom purely ideal a an it making magneton), osse eylremgei oet 6 moment, magnetic large very a possesses rmispriua rpris nisgon state ground its of in studies properties; stems Cr theoretical particular cooling its for in interest from need The the properties. emphasize inter- scattering Cr 7] of 6, chromium range 5, with extended [4, experiments Recent an separations. over between sen- nuclear potentials systems are interaction colliding regime the of cold the details the the in to occurring condensa- sitive Processes Bose-Einstein high [3]. in and role tion their [2] and spectroscopy [1] atoms precision of trapping and importance cooling their in of because attention considerable ceived nti omncto,w xlr h olsoa prop- collisional the explore we communication, this In ntetertclfot h lcrncsetu of spectrum electronic the front, theoretical the On olsoso tm tutaodtmeaue aere- have temperatures ultracold at atoms of Collisions d 5 2 13 4 s, ie oe osdrbenmrclchallenge. numerical considerable a poses dimer Σ 7 g + d S 3 hl tegon lcrnccngrto is configuration electronic ground (the shell 2 oeua ymty ..ttlspin, total i.e. symmetry, molecular )adteCr the and )) eghi oprdi antd iharcn esrmn at measurement recent a with magnitude in compared is length hydaaial ffc h hraiainrt.Orcalcu Our rate. thermalization resonances shape the of affect effect the dramatically explore they We atoms. Cr colliding o Cr for a n ant-pia oln fcrmu tm.W calcu We atoms. chromium of and cooling cold magneto-optical the and in gas atoms chromium spin-polarized maximally two TM,HradSihoinCne o srpyis 0Ga 60 Astrophysics, for Center Harvard-Smithsonian ITAMP, eateto hoeia hmsr,Ceia etr P.O Center, Chemical Chemistry, Theoretical of Department erpr ncluain fteeatccosscinadth and section cross elastic the of calculations on report We 1 hsc eatet nvriyo onciu,25 Hills 2152 Connecticut, of University Department, Physics 2 n h a e al coefficient Waals der van the and oa Pavlovi´cZoran olsoa rpriso rpe odcrmu atoms chromium cold trapped of properties Collisional 2 ie soeo h otex- most the of one is dimer µK 1 jonO Roos Bj¨orn O. , oee,anot a However, . µ B 53 Dtd eray1,2018) 12, February (Dated: ( r opens Cr, µ B S Bohr : 7 S 6. = 1 3 Σ it , C 2 g + 6 oi Cˆot´e Robin , n osrc neato oetasbtentwo between potentials interaction construct and , ahcrewsjie mohyt h xoeta form exponential the to smoothly V joined was curve each band h a e al neato coefficient, interaction Waals were der atoms van Cr C The ground two to obtained. dissociating curves ergy tmctasto arxeeet n photoionization be to and sections elements cross matrix transition atomic n nriswsaot10 about bind- was computed energies in ing accuracy numerical and achieved, was in structure resonance We and 7]. bound the [5, the coefficients investigated re- rate two also elastic against of compared measurements con- and cent newly curves the potential using calculated structed were coefficient rate sion efis computed together. first smoothly We joined regions three from constructed nnepstosadwdh safnto frotational of function a as momenta. widths angular and positions onance 9 aino the of ration oga-rl aitna a sdwt oktp cor- 0 Fock-type rection with used was Hamiltonian Douglas-Kroll a sdt orc nrisfrtebssstsuperpo- set 10 basis to the Convergence for (BSSE). energies error correct sition to used was ucin o orltn h 3 the correlating for functions CST) h ai e sdi h acltosi of is contracted calculations type 9 the (ANO-RCC) to orbital in theory natural used perturbation atomic set the basis order energy The occupied. second correlation orbitals (CASPT2). through derived electron obtained in- 3p dynamical is the and keeping 3s, the remaining while 2p, in The orbitals 2s, electrons active 1s, 12 derived active distributing 4s and by 3d formed wave CASSCF is The function 12]. [11, method CASSCF/CASPT2 eemndb acigbt h oeta uv n its and at curve continuously derivative potential first the both matching by determined the where . λ 6 57 o separations For h oeta uvsfrCr for curves potential The ( a bandsm-miial rmaalbebound available from semi-empirically obtained was , R 13 s binitio ab × 8 = ) Σ p C 10 7 g + 1 6 d raiainrt o olso between collision for rate ermalization . ae au o the for value lated n .R Sadeghpour R. H. and , neatccosscin n n that find and section, cross elastic on 5 − 5 oeta nrycreaddtrie h res- the determined and curve energy potential c B 2 -2 0Ln,See and Sweden Lund, 00 S-221 124 .B. late lrcl eie,rlvn obuffer- to relevant regimes, ultracold stevndrWascecet[5,adthe and [15], coefficient Waals der van the is λ f dnSre,Cmrde A02138. MA Cambridge, Street, rden 80 d d,Sor,C 06269-3046 CT Storrs, Rd., ide ∗ lrcl temperatures. ultracold 3 exp( binitio ab g aawr ace oteaypoi form asymptotic the to matched were data g V Jm binitio ab 1 hsbsssti eaiitcadincludes and relativistic is set basis This . λ e 1] h ulcutros method counterpoise full The [14]. see , ( − R 6 R h lsi rs eto n colli- and section cross elastic The . b = ) λ binitio ab ≤ C R aafrteptnilenergy potential the for data 6 oeta nrycurves energy potential ,wt h coefficients the with ), − R 745 = 1 λ R C where , − 6 6 8 s . + wv scattering -wave s oeta uvsuigthe using curves potential ± A n 3 and 2 R hw nFg ,were 1, Fig. in shown , λ 5au,wee1au = a.u. 1 where a.u., 55 1 λ R R 3 tlrevle of values large At . 1 λ ν p e stesals sepa- smallest the is − lcrn 1] The [13]. electrons βR − 10 , nhartrees, in , c λ and V λ ( R R b ), λ , 2
where α is the fine-structure constant, a0 is the Bohr 13Σ+ 0 g 7Σ+ radius, and σph is the photonization cross section. The u 11Σ+ dimensionless df/dǫ is obtained by multiplying Eq.(3) by u 5Σ+ |E | (also in a.u.). g 0 9Σ+ -0.02 g We assess the quality of the semi-empirical computa- tion of the van der Waals coefficient by satisfying two 3Σ+ u sum rules, namely, the zero-th and the inverse second -0.04 0 V (a.u.) moments, 1 + Σ -0.001 13Σ+ g g -0.002 S(0) = f0i = N -0.06 i -0.003 X 5 10 15 20 2 2 e ¯h f0i S(−2) = α0 = 2 2 + G(1) , (4) m|E0| v -0.08 ( i i ) 2 3 4 5 6 7 8 9 10 X R (a.u.) where N is the number of electrons (24 for Cr), and α0 is the static polarizability. We obtained i f0i = 22.3 and FIG. 1: Potential curves for the ground state manifold of Cr2 α0 = 85.00 a.u., in agreement with a recommended value computed with the CASSCF/CASPT2 method. The max- P 13 + 82±20% a.u. [21]). The resulting dispersion coefficient is imally spin-streched electronic state, Σg , is shown in red expected to have an accuracy of about 7%, C = 745±55 and in detail in the inset. This state supports 30 vibrational 6 levels. a.u. Using these potential curves, we computed the elastic λ σel. cross section for the maximally spin-aligned molecu- 13 + parameters of the exchange energy are determined ac- lar state, i. e. the Σg state. [Results for other molec- 7 cording to Smirnov and Chibisov [16]: ν = 2I − 1 and ular states and also inelastic and spin relaxation cross β = 2I, where I is the ionization energy of the atom sections will be provided in a future publication.] The 52 (I =0.248664314 a.u. for Cr). The parameters Aλ were elastic cross section for the collision of two Cr atoms, found by fitting the ab initio curves at separations where composite bosons, expanded over the rotational quantum the exchange energy was still considerable (e.g. R be- number, l, is [22] 13 + tween 10 and 14 a.u. for Σ ). The C6 coefficient was g 8π calculated using σλ (E)= (2l + 1) sin2 δλ , (5) el. k2 l l even 2 2 2 X 3 e ¯h 1 f0if0j C6 = 3 2 2 2 m |E | vivj (vi + vj ) where E =¯h k /2µ is the kinetic energy of relative mo- 0 i,j λ X tion, µ is the reduced mass, and δl (k) is the l-th scatter- ∞ f G(1 + v ) (df/dǫ)G(2 + ǫ)dǫ ing phase shift in electronic state λ. In the low-energy +2 0i i + , (1) v (1 + ǫ) limit, the elastic cross section behaves as i i Z0 ) X 2 λ E→0 8πaλ where contributions of bound-bound, bound-free, and σel.(E) −→ , (6) 2 2 1 2 2 free-free transitions are given by the first, second, and k aλ + 1 − 2 rλaλk third term, respectively. This expression is derived from where scattering length aλ is determined at the zero- the London’s [17] formula, assuming vi =1 − Ei/E0 and 1 λ th energy s-wave limit, aλ = − limk→0 tan δ (k), while ǫ = −E/E0, where E0, Ei, and E are the ground, the i k 0 the effective range rλ can be found by fitting the cross excited state and continuum energies, respectively, f0i are the oscillator strengths pertaining to transitions to section, from an integral expression [22], or using quan- tum defect theory[23]. the ground states, and df0E accounts for the continuous The cross section as a function of the collision energy spectrum. The auxiliary function G(z) is given by 13 + ∞ is shown in Fig. 2 for three different Σg interaction po- (df/dǫ)dǫ tentials; each constructed by matching at large distances G(z)= . (2) 0 (1 + ǫ)(z + ǫ) to the upper, lower, and mean values of C6. Note that Z at collision energies larger than E ∼ 10−6 a.u., aside Values of oscillator strengths and energy levels for dis- from the shape resonance structure, the cross sections crete transitions were taken from the NIST Atomic Spec- differ little in magnitude, but are dramatically different tra Database [18] and Ref. [19]. The continuous oscillator in the s-wave limit. We find the scattering length for strength, df/dǫ, was found using Verner’s [20] analytic 13 + the Σg state, a13, to be large and negative (see Ta- fits for partial photoionization cross sections. If we mea- ble I), indicating that evaporative cooling should be effi- sure dE in atomic units then cient, but that a Bose-Einstein condensate of 52Cr would df 1 not be stable for large number of atoms. This contrasts = 2 2 σph , (3) dE 2π αa0 Ref. [5] in which the extracted value for the scattering 3
6 -9 10 10
C6=800 -10 10 l=4 5 C =800 10 ) 6
-1 -11 C6=690 s 10 3
-12 C =690 10 6 1e-09 4 Rate (cm 10 Cross Section (a.u.) -13 C =745 C6=745 10 6
-14 1e-10 10 0.01 0.1 -12 -11 -10 -9 -8 -7 -6 -5 -4 -9 -8 -7 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 1 Energy (a.u.) Temperature (K)
FIG. 2: Elastic cross sections for collision between two FIG. 3: The calculated collision rate coefficient as a function bosonic Cr atoms as a function of energy in the cold and of temperature. At temperatures above 100 mK, the different ultracold regimes. The three curves are labeled by the value calculated rates are practically the same. The plateau region of the C6 coefficient used to construct the interaction poten- is shown in the inset. tial in each case. See text for details. The influence of shape resonances is dramatic. tum, l. Due to symmetry, only even values of l appear in the collision. There are 30 bound vibrational levels length, obtained from a fit to the experimental collision for the l = 0 partial wave. The grouping of the levels rate coefficient, Fig. 3, agrees in magnitude with our up- in solid and dashed lines indicates the appearance of an per bound calculated value, but differs in the prediction additional shape resonance. Table II gives our calculated of the sign (see also the discussion of the rates below). position and width for a number of rotational shape reso- Our calculated effective range expansion coefficients also nances. The narrowest resonance is for the l = 19 partial agree very closely with the results obtained using the wave with a lifetime of about two seconds. This reso- quantum-defect method of Gao [23]. nance, however, is not populated in a bosonic collision. As the collision energy increases, the appearance of A plateau in the observed elastic collision rate coeffi- shape resonances can lead to enormously large cross sec- cient in the range 10 mK
6 Science Foundation through a grant for the Institute for Theoretical Atomic Molecular and Optical Physics at 4 Harvard University and Smithsonian Astrophysical Ob- 2 servatory. HRS is grateful to K. Andersson for access to numerical data for an earlier calculation of the potential
a.u.) -0 v=29 curves and to P. O. Schmidt and R. deCarvalho for valu- -6 v=28 able correspondence. BOR thanks the Swedish Science -2 Research Council (VR) for financial support. v=27 -4 Energy (10 -6 C6 a13 r13 -8 v=26 690 49 98 -10 0 2 4 6 8 10 12 14 16 18 20 22 745 12 2600 l (angular momentum number) 800 -176 213
FIG. 4: The bound and shape resonance structure in the Cr2 13 + Σg potential energy curve as a function of the rotational TABLE I: The calculated scattering length and effective quantum number l. range for different interaction potentials, all in atomic units. ful discussions. This work was supported by the National
[1] S. Chu, Rev. Mod. Phys. 70, 685 (1998); C.N. Cohen- 215 (1995); B. O. Roos, Collect. Czech. Chem. Commun. Tannoudji, ibid, 707; W.C. Phillips, ibid, 721. 68, 265(2003); K. Andersson, Chem. Phys. Letters 237, [2] J. Weiner, V.S. Bagnato, S.C. Zilio, and P.S. Julienne, 212 (1995). Rev. Mod. Phys. 71, 1 (1999). [13] The primitive basis set was: 21s15p10d6f4g. These basis [3] See A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001), and sets are under construction for the entire periodic system references therein. (B. O. Roos, to be published). [4] C. C. Bradley, et al., Phys. Rev. A 61, 053407-1(2000). [14] K. Andersson, Theor. Chim. Acta 91, 31 (1995). [5] P. O. Schmidt et al., arXiv:quant-ph/0303069 v2, [15] Higher order dispersion coefficients, such as C8, C10, etc., 2003; P. O. Schmidt et al., arXiv:quant-ph/0211032, were not included in our analysis. Nov. 6 2002; P. O. Schmidt P.O. et al., [16] B. M. Smirnov, and M. I. Chibisov, Soviet Physics Jetp, arXiv:quant-ph/0208129v1, Aug. 20 2002. 21 624 (1965); E. L. Duman, and B. M. Smirnov, Optics [6] Stuhler J. at al., Phys. Rev. A 64, 031405 (2001); S. and Spectroscopy, XXIX 229 (1970). Giovanazzi, A. G¨orlitz, and T. Pfau Phys. Rev. Lett. 89, [17] H. Margenau, Phys. Rev., 56, 1000 (1939). 130401(2002). [18] NIST Atomic Spectra Database : [7] J.M. Doyle et al., Phys. Rev. A 65, 021604 (2002); J. D. http://physics.nist.gov/cgi-bin/AtData/main_asd Weinstein et al., Phys. Rev. A 57, R3173(1998). [19] D. A. Verner, P. D. Barthel, and D. Tytler, A&AS, 108, [8] Robert deCarvalho (private communication). 287 (1994). [9] L. Santos, G.V. Shlyapnikov, P. Zoller, and M. Lewen- [20] D. A. Verner, and D. G. Yakovlev, A&AS, 109, 125 stein, Bose-Einstein Condensation in Trapped Dipolar (1995). Gases, Phys. Rev. Lett. 85, 1791 (2000). [21] CRC Handbook of Chemistry and Physics, 83rd Edition, [10] M. Baranov at al., arXiv:condmat/0201100 (2002) Section 10-163, Atomic and Molecular Polarizabilities, [11] B. O. Roos in Advances in Chemical physics, ab initio CRC Press, LLC, www.crcpress.com. methods in quantum chemistry, Ed. K. P. Lawley, John- [22] N. F. Mott, and H. S. W. Massey, The Theory of Atomic Wiley & Sons, Ltd., Chichester, 1987; B. O. Roos et al. Collisions. Third Edition. (1965) in Advances in chemical physics; new methods in compu- [23] B. Gao, Phys. Rev. A P58, 4222(1998). tational quantum mechanics, Ed. I. Prigogine and S. A. [24] Th. Dohrmann et al., J. Phys. B 29, 4461 (1996). Rice, John-Wiley & Sons, Ltd., New-York, 1996. [25] V. K. Dolmatov, J. Phys. B 26, L393 (1993). [12] B. O. Roos, and K. Andersson, Chem. Phys. Letters 245, 5
l E (a.u) Γ (a.u)
3 8.04(-09) 5.82(-10) 4 4.82(-08) 2.87(-08) 7 5.38(-08) 2.03(-12) 8 2.52(-07) 1.60(-08) 9 4.44(-07) 1.04(-07) 10 6.41(-07) 2.81(-07) 11 1.72(-07) 1.10(-14) 12 6.66(-07) 2.27(-09) 13 1.13(-06) 6.11(-08) 14 1.58(-06) 2.42(-07) 15 4.27(-07) 6.32(-17) 16 1.34(-06) 1.11(-10) 17 2.22(-06) 1.82(-08) 18 3.04(-06) 1.59(-07) 19 9.07(-07) ≪ 1(-17) 20 2.36(-06) 4.08(-12) 21 3.78(-06) 2.70(-09) 22 5.12(-06) 7.10(-08)
TABLE II: The rotational shape resonance parameters, po- 13 + sition and width, in the Σg potential. Note that only even partial waves contribute to the scattering of spin polarized bosons (odd partial waves are included for completeness).