ρ$_{θ}$ ( θ, t ) = 1 6 ∑ N$_{c}$ ( c$_{N}$$_{c}$ ) 2 ∑ J,m, Π ′ , Π ,q ′ ,q [ D Jm Π ′ q ′ ( t )] ∗ D Jm Π q ( t ) ( − 1) l + l $^{′}$− J × [(2 L ′ + 1)(2 L + 1)(2 l ′ + 1)(2 l + 1)] 1 / 2 [ ∫ r $^{2}$dr f (1 / 2) n ′ l ′ ( r ) f (1 / 2) nl ( r )] × [ ∫ R $^{2}$dR f (2 / 3) N ′ L ′ ( R ) f (2 / 3) NL ( R )] × ∑ λ W ( ll ′ LL ′ ; λJ ) √ π 2 λ + 1 C λ 0 L ,L ′ $_{0}$C λ 0 l ,l ′ $_{0}$Y$_{λ}$$_{0}$ ( θ, 0) (A3) (iv) ζ ( R , t ) = 1 6 ∑ N$_{c}$ ( c$_{N}$$_{c}$ ) 2 ∑ J ′ ,J,m ′ ,m, Π ′ , Π ,q ′ ,q δ$_{n}$ ′ $_{n}$δ$_{l}$ ′ $_{l}$[ D J $^{′}$m$^{′}$Π′ q ′ ( t )] ∗ D Jm Π q ( t ) × k [ C J $^{′}$m′ l ′ k,L ′ m ′ − $_{k}$C Jm lk,Lm − $_{k}$Y ∗ L ′ ,m ′ − $_{k}$( ̂ R ) $^{Y$_{L,m}$}$− $_{k}$( ̂ R )] × f (2 / 3) N ′ L ′ ( R ) f (2 / 3) NL ( R ) (A4) 6 where the Wigner and Clebsch-Gordan coefficients are introduced. of the $^{52}$Cr − $^{52}$Cr interaction via repeating collisions of a pair of atoms in a trap. preprint, arXiv:0908.2929v1. [5] Tobocman, W.: A generalized Talmi-Moshinsky transfor- mation for few-body and direct interaction matrix ele- ments. Nucl. Phys. A. 357 , 293 (1981). [6] M. Baranger, M., Davies, K. T. R.: Oscillator brackets for Hartree-Fock calculations. Nucl. Phys. 79 , 403 (1966). [7] Brody, T. A., Moshinski, M.: Table of Transformation Brackets. Monografias del Instituto de Fisica, Universidad Nacional Autonoma de Mexico (1960). [4] Li, Z. B., Chen, Z. F., He, Y. Z., Bao, C. G.: Evaluation [3] W¨ urtz, P., Langen, T., Gericke, T., Koglbauer, A., Ott, H.: Experimental Demonstration of Single-Site Address- ability in a Two-Dimensional Optical Lattice. Phys. Rev. Lett. 103 , 080404 (2009). [2] Barrett, M. D., Sauer, J. A., Chapman, M. S.: All-Optical Formation of an Atomic Bose-Einstein Condensate. Phys. Rev. Lett. 87 , 010404 (2001). [1] Stenger, J., Inouye, S., Stamper-Kurn, D. M., Miesner, H.-J., Chikkatur, A. P., Ketterle, W.: Spin domains in ground-state Bose-Einstein condensates. Nature (London) 396 , 345 (1998).