1. Detecting atoms in a lattice with two photon raman transitions Inés de Vega, Diego Porras, Ignacio Cirac Max Planck Institute of Quantum Optics Garching (Germany)

2. Summary 3) Conclusions 1)Motivation:  what is an atom lattice?  why measuring atoms in a lattice? 2)Measuring atoms in a lattice:  Time of flight experiments  Our method

3. “I am not afraid to consider the final question as to whether, ultimately---in the great future---we can arrange the atoms the way we want; the very atoms, all the way down! “ Richard Feynman, December 29th 1959 at the annual meeting of the American Physical Society at the California Institute of Technology (Caltech) American Physical SocietyCalifornia Institute of Technology (Caltech)

4. Richard Feynman, December 29th 1959 at the annual meeting of the American Physical Society at the California Institute of Technology (Caltech) American Physical SocietyCalifornia Institute of Technology (Caltech) What would the properties of materials be if we could really arrange the atoms the way we want them? […] I can't see exactly what would happen, but I can hardly doubt that when we have some control of the arrangement of things on a small scale we will get an enormously greater range of possible properties that substances can have, and of different things that we can do.

5. What is an optical lattice A standing wave in the space gives rise to a conservative force over the atoms Optical potential V0V0

6. What is an optical lattice A standing wave in the space gives rise to a conservative force over the atoms Optical potential V0V0 Space dependent Stark shift: when Laser blue detuned  >0 atoms go to the Potential minima  >0

7. What is an optical lattice Due to the periodic potential, the discrete levels in each well form Bloch bands We consider the atoms placed in the lowest Bloch band Described with creation fuction of a particle of spin α: Wannier functions localiced in each lattice site. Creation operator with bosonic (fermionic) conmutation (anticonmutation) relations

8. Atom Hamiltonian in second quantization Gives rise to a kinetic term, with magnitude “t” Gives rise to a repulsive term, with magnitude ~ U.

9. Atom Hamiltonian in second quantization Gives rise to a kinetic term, with magnitude “t” Gives rise to a repulsive term, with magnitude ~U. Bosons  even terms of M (example F=1 we have proyection to P 0 and P 2 ) Fermions  odd terms of M

10. Atom Hamiltonian in second quantization Gives rise to a kinetic term, with magnitude “t” Gives rise to a repulsive term, with magnitude ~U. Spin-spin interactions (example, for atoms with J=1)

11. Atom Hamiltonian in second quantization Gives rise to a kinetic term, with magnitude “t” Gives rise to a repulsive term, with magnitude ~U. Variating parameters t and U, this hamiltonian undergoes Quantum Phase Transitions

12. An optical lattice is controllable V0V0 We can change the standing wave parameters: V 0 and λ We can apply an external magnetic field to increase scattering length We can use state dependent potentials λ B Mott state very important: 1)To simulate magnetic Hamiltonians (spin-spin interactions) 2)As a quantum register (where highly entangled states, cluster states, can be created) t>>U :Shallow lattice (large kinetic energy), gives rise to a superfluid state T<<U :Deep lattice, strong interactions, gives rise to a Mott state. Atoms are localized in each site.

13. Why measuring atoms in a lattice A lattice is a nice quantum simulator, and may be a nice implementation of a quantum computer but......how can we read out the information from it? Time of flight experiments Off-resonant Ramman scattering of light and more...

14. Time of flight S. Fölling et al. Nature (2005) T. Rom et al. Nature (2006)

15. Off resonance Raman scattering Laser  ge z x y Emited photon  se

16. Interaction between atoms and light Adiabatically eliminating the  e> level Duan, Cirac, Zoller (2002) Laser  ge z x y Emited photon  se

17. Direction of the emitted photons 1)If most atoms are in the same state state  g>, or k L d 0 <<1, then k=k L z x y k=k L 2) Atoms in differerent states  Photons emitted in different ,Φ z x y k

18. Z-polarized laser with spin J atoms Laser z x y Emited photon J J’ We detect atoms with any spin J

19. Z-polarized laser with spin J atoms

20. Photon counting type of measure z x y k Detected correlations of photons Correlations of atom variables in momentum space

21. For example y-polarized photons

22. And if we consider T<<1/Γ we detect atom correlations in the ground states (1) (2) This is our main assumption. We check the relative error between (1) and (2) with respect to the number of photons that are emitted.

23. Checking the assumption T<<1/Γ Even if there were some lattice sites without an atom, this function for large is approximately a delta. Through the Quantum Regression Theorem this is the evolution that correlations have

24. Checking the assumption T<<1/Γ Number of y-polarized photons in θ for T=0.0025 This is the type of things we measure N yy (  ) number of photons detected N yy (  ) number photons comming from ground state

25. Checking the assumption T<<1/Γ J/B=0.001 J/B=-0.05 J/B=-0.5

26. Measuring conbinations of quadratures… …we can detect any correlation!!

27. Conclusions -Not destructive: one can perform measures of the state in the middle of an experiment and then continue -More freedom to compute different correlations and hence to detect more complex phases -More precission with respect to time of flight: Signal to noise ration in Time of flight ~ in Raman scattering~ -3D information

28. Thank You!!

29. I.e., we can measure Cluster states l Є neigh (j) 1D Cluster states:2D Cluster states: JxJx JzJz JzJz JzJz JzJz JzJz JzJz JxJx

30. I.e., we can measure Cluster states l Є neigh (j) Measuring in the forward direction, we know that (for instance 1D cluster states): ~2N