1. Quantum Entanglement of Rb Atoms Using Cold Collisions ( 韓殿君 ) Dian-Jiun Han Physics Department Chung Cheng University

2. Outline Introduction Cold Atoms in the Potential Wells Quantum Entanglement (QE) of Cold Atoms in Optical Lattices (OL) Experimental Realization of the Quantum Entanglement Conclusions

3. Introduction Requirements for a quantum computer : A Stean, Rep. Prog. Phys. 61, 117 (1998) 1. Qubits are sufficiently well-isolated from their environment 2. Possible to prepare the qubits in specified states 3. Apply universal quantum gates to them 4. Measure their states 5. The number of qubits must be scalable 6. There must be way to implement quantum error correction Possible candidates: trapped ions, NMR, high-Q optical cavities, electron spin-based condensed matter system, and cold atoms in optical lattices (a newcomer in this field!!)

4. Cold Atoms in Two Adjacent Potential Wells |a> |b> Atom 1 and atom 2 are in the internal states |a> 1 and |b> 2 and are trapped in the ground states of the potential wells. VaVa VbVb u ab Jaksch et al., Phy. Rev. Lett. 82, 1975 (1999)

5. Implementation of the Entanglement |b> |a> X a 1 (t)X b 2 (t) |b>|a> X b 1 (t) X a 2 (t) adiabatically move the wells  atoms are still in the ground state of the trap potential

6. A Two-qubit Gate Transformation before and after the entanglement: |a> 1 |a> 2 → e -i2φ a |a> 1 |a> 2, |a> 1 |b> 2 → e -i(φ a +φ b +φ ab ) |a> 1 |b> 2, |b> 1 |a> 2 → e -i(φ a +φ b ) |b> 1 |a> 2, |b> 1 |b> 2 → e -i2φ b |b> 1 |b> 2 ← kinetic phase ← collisional phase e.g., 87 Rb atom |a>≡|F=1, m f =1>, |b> ≡|F=2, m f =2>

7. Experimental Realization of the Entanglement 1. Using laser cooling, magnetic trapping and evaporative cooling to reach Bose-Einstein condensation of 87 Rb atoms 2. Loading Bose condensed atoms ( ～ 10 6 atoms) into an optical lattice (optical crystal) 3. Increasing lattice potential to isolate each site (i.e., Mott insulator phase) 4. Bring the adjacent lattice sites together to engage the entanglement via cold controlled collisions

8. The Optical Dipole Force If laser beam is far from resonance, the spontaneous scattering rate is low and dipole light traps are nearly conservative. N e-e- E= E 0 cos(  t) induced dipole moment : μ = αE AC Stark shift : U AC = = -(α/2) E 0 2  laser intensity α > 0  atoms attracted to light α < 0  atoms repelled from light

9. Optical Lattices 1D: 2D: 3D: Calculable, versatile atom traps Light potential is state dependent in phase out of phase phase insensitive

10. 3D Optical Lattice Light Configuration + ⊿

11. Phase Transition from a Superfluid to a Mott Insulator in an ultracold Rb gas Greiner et al., Nature 415, 39 (2002) raising up lattice potential

12. Conclusions Long decoherence time in an optical lattice Allow state selected measurements by shining laser beams High spatial resolution should be possible and allow to address single site Many sites are loaded. It might allow for error correction High fidelity for the quantum entanglement