1. ENTANGLEMENT AND QUANTUM CONTROL OF COLD ATOMS CONFINED IN AN OPTICAL LATTICE CARLO SIAS Università di Pisa Quantum Computers, algorithms and chaos – Varenna 4-15/07/2005 with Roberto Franzosi, Matteo Cristiani, and Ennio Arimondo

2. Entanglement in an Optical Lattice D. Jaksch et al., Phys Rev Lett. 82, 1975 (1999) O. Mandel et al. Nature (London) 425, 937 (2003)

3. Entanglement in an Optical Lattice D. Jaksch et al., Phys Rev Lett. 82, 1975 (1999) O. Mandel et al. Nature (London) 425, 937 (2003)

4. Entanglement in an Optical Lattice D. Jaksch et al., Phys Rev Lett. 82, 1975 (1999) O. Mandel et al. Nature (London) 425, 937 (2003)

5. – Lattice

7. Experimental Realizations with BEC Morsch et al. PRL 87,140402 (2001) = 780 nm, d = 1.56 m, = 29 deg Hadzibabic et al. PRL 93,180403 (2004) = 532 nm, d = 2.7 m, = 11.5 deg Albiez et al. PRL 95,010402 (2005) = 811 nm, d = 5.2 m, = 9 deg

8. Single site addressing? F=2 F=1 2 E m F =2 m F =1 3( i+1 - i )=3m F g F B b(x i+1 - x i ) xixi x i+1 B x

9. Single site addressing? F=2 F=1 2 E m F =2 m F =1 3( i+1 - i )=3m F g F B b(x i+1 - x i ) xixi x i+1 B x Albiez et al. b=10 Gcm -1 ( i+1 - i ) 30Khz

10. Filling the lattice Hadzibabic et al. PRL 93,180403 (2004) 30 condensates of 10 4 atoms

11. Filling the lattice Hadzibabic et al. PRL 93,180403 (2004) 30 condensates of 10 4 atoms = 15Hz x 2 2R 17m

12. Analytical calculation F=1 (+) (-) (+) E x =0 m F =+1m F =0m F =-1 x Y z

13. Analytical calculation F=1 (+) (-) (+) E x =0 m F =+1m F =0m F =-1 x Y z F=1 m F =+1m F =0m F =-1 E x 0

14. Analytical calculation

16. Conclusions - lattice for single site addressing in an optical lattice Interactions with single atoms by use of a magnetic field gradient N=1 filling factor in a 3D lattice Strong dependence on the polarization of the lattice beams