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By Gavin K. Brennen and Ivan H. Deutsch

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Lett. 80, 4149 (1998). [19] D. , Phys. Rev. Lett. 81, 3108 (1998). [20] For a review of experiments that perform quantum coherent manipulations of trapped ions see, D. J. , J. Res. Inst. Stand. Techol. 103, 259 (1998). [21] I. H. Deutsch and P. S. Jessen, Phys. Rev. A 57, 1972 (1998). [22] J. Schmeidmayer Eur. Phys. J. D 4, 57 (1998); E. A. Hinds, M. G. Boshier, and I. G. Hughes, Phys. Rev. Lett. 80, 645 (1998), M. , Appl. Phys. Lett. 72, 2906 (1998). [23] For a review see J. , Rev. Mod. Phys. 71, 1 (1999), and references therein.

Lett. 72, 2906 (1998). [23] For a review see J. , Rev. Mod. Phys. 71, 1 (1999), and references therein. [24] C. J. Williams and P. S. Julienne J. Chem. Phys. 101 2634 (1994); P. D. Lett, P. S. Julienne, and W. D. Phillips, Annu. Rev. Phys. Chem. 46, 423 (1996). [25] R. H. Lemberg, Phys. Rev. A 2, 883 (1970); M. Trippenbach, B. Gao, J. Cooper, and K. Burnett, Phys. Rev. A 45, 6555 (1992). Brennen, Deutsch, and Jessen Entangling Dipole-Dipole Interactions and Quantum Logic Gates in Optical Lattices Page 35 [26] E.

D. Levine, and B. Pullman (Dordrecht, Boston, 1991). [42] E. 09 (1999). [43] Ph. , Phys. Rev. Lett. 81, 69 (1998). S. 05 (1999) APPENDIX A Consider two 2-level atoms with ground and excited states field and the vacuum. g , e interacting with a laser After tracing over the vacuum modes, the effective non-Hermitian Hamiltonian is Heff = HA + H AL + Hdd , (A1) where in the rotating wave approximation, the dressed atomic Hamiltonian is ( ) Γ  HA =  −h∆ − i  e1 e1 ⊗ 1ˆ 2 + 1ˆ 1 ⊗ e2 e2 , 2 (A2) and the atom-laser interaction is HAL = − hΩ 2 [( e 1 ] g1 + g1 e1 ) ⊗ 1ˆ 2 + 1ˆ1 ⊗ ( e2 g2 + g2 e2 ) .

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