1. Atomic entangled states with BEC SFB Coherent Control €U TMR A. Sorensen L. M. Duan P. Zoller J.I.C. (Nature, February 2001) KIAS, November 2001.

2. Entangled states of atoms Motivation: Fundamental. Applications: - Secret communication - Computation - Atomic clocks NIST: 4 ions entangled. ENS: 3 neutral atoms entangled. Experiments: j ª i6 = j ' 1 i­j ' 2 i­ ::: j ' N i ' E ' 4 E ' 3 E 10 3 This talk: Bose-Einstein condensate.

3. Outline 1.Atomic clocks 2.Ramsey method 3.Spin squeezing 4.Spin squeezing with a BEC 5.Squeezing and atomic beams 6.Conclusions

4. 1. Atomic clocks To measure time one needs a stable laser click The laser frequency must be the same for all clocks click Innsbruck Seoul The laser frequency must be constant in time click

5. Solution: use atoms to lock a laser detector feed back frequency fixed universal In practice: Neutral atomsions

6. 1 11 1 1 Independent atoms: Entangled atoms: N is limited by the density (collisions). t is limited by the experiment/decoherence. We would like to decrease the number of repetitions (total time of the experiment). Figure of merit: To achieve the same uncertainity: We want

7. 2. Ramsey method # of atoms in |1> single atom Fast pulse: Wait for a time T: Fast pulse: Measurement:

8. Independent atoms Number of atoms in state |1> according to the binomial distribution: 1 1 where If we obtain n, we can then estimate The error will be If we repeat the procedure we will have:

9. Another way of looking at it Initial state: all atoms in |0> First Ramsey pulse: Free evolution:Measurement:

10. In general where the J‘s are angular momentum operators Remarks: We want Optimal: If then the atoms are entangled. That is, measures the entanglement between the atoms

11. 3. Spin squeezing No gain! Product states:

12. Spin squeezed states: (Wineland et al,1991 ) These states give better precission in atomic clocks

13. How to generate spin squeezed states? (Kitagawa and Ueda, 1993) 1) Hamiltonian: It is like a torsion

14. 2) Hamiltonian:

15. Explanation Hamiltonian 1: Hamiltonian 2: are like position and momentum operators

16. 4. Spin squeezinig with a BEC Weakly interacting two component BEC Atomic configuration optical trap A. Sorensen, L.M Duan, J.I. Cirac and P. Zoller, Nature 409, 63 (2001) laser trap AC Stark shift via laser: no collisions + laser interactions FORT as focused laser beam Lit: JILA, ENS, MIT...

17. A toy model: two modes we freeze the spatial wave function Hamiltonian Angular momentum representation Schwinger representation spatial mode function

18. A more quantitative model... including the motion Beyond mean field: (Castin and Sinatra '00) wave function for a two-component condensate with Variational equations of motion the variances now involve integrals over the spatial wave functions: decoherence Particle loss

19. Time evolution of spin squeezing Idealized vs. realistic model Effects of particle loss

20. Can one reach the Heisenberg limit? We have the Hamiltonian: We would like to have: short pulse short evolution Conditions: Idea: Use short laser pulses.

21. Stopping the evolution Once this point is reached, we would like to supress the interaction The Hamiltonian is: Using short laser pulses, we have an effective Hamiltonian:

22. In practice: wait short pulses short pulse

23. 5. Squeezing and entangled beams Atom laser Squeezed atomic beam Limiting cases squeezing sequential pairs atomic configuration collisional Hamiltonian L.M Duan, A. Sorensen, I. Cirac and PZ, PRL '00 atoms condensate as classical driving field pairs of atoms

24. Equations... Hamiltonian: 1D model Heisenberg equations of motion: linear Remark: analogous to Bogoliubov Initial condition: all atoms in condensate

25. Case 1: squeezed beams Configuration Bogoliubov transformation Squeezing parameter r Exact solution in the steady state limit condensate input: vaccum output

27. Case 2: sequential pairs Situation analogous to parametric downconversion Setup: State vector in perturbation theory with wave function consisting of four pieces After postselection "one atom left" and "one atom right" symmetric potential collisions

28. 6. Conclusions Entangled states may be useful in precission measurements. Spin squeezed states can be generated with current technology. - Collisions between atoms build up the entanglement. - One can achieve strongly spin squeezed states. The generation can be accelerated by using short pulses. The entanglement is very robust. Atoms can be outcoupled: squeezed atomic beams.

29. Quantum repeaters with atomic ensembles SFB Coherent Control €U TMR €U EQUIP (IST) L. M. Duan M. Lukin P. Zoller J.I.C. (Nature, November 2001)

30. Quantum communication: Classical communication: Quantum communication: Quantum Mechanics provides a secure way of secret communication Alice Bob Alice Bob Classical communication: Alice Bob Quantum communication: Alice Bob Eve

31. Problem: decoherence. We cannot know whether this is due to decoherence or to an eavesdropper. Probability a photon arrives: 2. States are distorted: Alice Bob 1. Photons are absorbed: Quantum communication is limited to short distances (< 50 Km). j ª i ½ P=e _ L=L 0 In practice: photons. laser optical fiberphotons vertical polarization horizontal polarization

32. laser repeater Questions: 1. Number of repetitions 2. High fidelity: 3. Secure against eavesdropping. j ª i j ª i ½ Solution: Quantum repeaters. (Briegel et al, 1998).

33. Outline 1.Quantum repeaters: 2.Implementations: 1.With trapped ions. 2.With atomic ensembles. 3.Conclusions

34. 1. Quantum repeaters The goal is to establish entangled pairs: (i) Over long distances. (ii) With high fidelity. (iii) With a small number of trials. Once one has entangled states, one can use the Ekert protocol for secret communication. (Ekert, 1991)

35. Establish pairs over a short distance Small number of trials Connect repeaters Correct imperfections Long distance High fidelity Key ideas: 1. Entanglement creation: 2. Connection: 3. Pufication: 4. Quantum communication:

36. 2. Implementation with trapped ions ion Aion B laser ion A ion B Internal states - Weak (short) laser pulse, so that the excitation probability is small. - If no detection, pump back and start again. - If detection, an entangled state is created. Entanglement creation: (Cabrillo et al, 1998)

37. Initial state: After laser pulse: Evolution: Detection: Description: ion Aion B

38. Repeater: Entanglement creation Entanglement creation Gate operations: Connection Purification

39. 3 Implementation with atomic ensembles Internal states - Weak (short) laser pulse, so that few atoms are excited. - If no detection, pump back and start again. - If detection, an entangled state is created. Atomic cell

40. Initial state: After laser pulse: Evolution: Detection: photons in several directions (but not towards the detectors) 2 photon towards the detectors and others in several directions 1 photon towards the detectors and others in several directions 2 photon towards the detectors and others in several directions Description: negligible do not spoil the entanglement

41. Atomic „collective“ operators: and similarly for b Entanglement creation: Measurement: Sample A Sample B Apply operator Apply operator: Photons emitted in the forward direction are the ones that excite this atomic „mode“. Photons emitted in other directions excite other (independent) atomic „modes“.

42. (A) Ideal scenareo After click: (1) (2) After click: Thus, we have the state: Sample A Sample R Sample B A.1 Entanglement generation:

43. A.2 Connection: If we detect a click, we must apply the operator: Otherwise, we discard it. We obtain the state:

44. A.3 Secret Communication: - Check that we have an entangled state: One can use this method to send information. Enconding a phase: Measurement in A Measurement in B: The probability of different outcomes +/- depends on

45. (B) Imperfections: - Spontaneous emission in other modes: No effect, since they are not measured. - Detector efficiency, photon absorption in the fiber, etc: More repetitions. - Dark counts: More repetitions - Systematic phaseshifts, etc: Are directly purified

46. (C) Efficiency: Fix the final fidelity: F Number of repetitions: Example: Detector efficiency: 50% Length L=100 L 0 Time T=10 T 0 6 (to be compared with T=10 T 0 for direct communication) 43

47. Advantages of atomic ensembles: 1. No need for trapping, cooling, high-Q cavities, etc. 2. More efficient than with single ions: the photons that change the collective mode go in the forward direction (this requires a high optical thickness). Photons connected to the collective mode. Photons connected to other modes. 4. Purification is built in. 3. Connection is built in. No need for gates.

48. 4. Conclusions Quantum repeaters allow to extend quantum communication over long distances. They can be implemented with trapped ions or atomic ensembles. The method proposed here is efficient and not too demanding: 1. No trapping/cooling is required. 2. No (high-Q) cavity is required. 3. Atomic collective effects make it more efficient. 4. No high efficiency detectors are required.

49. Institute for Theoretical Physics € FWF SFB F015: „Control and Measurement of Coherent Quantum Systems“ EU networks: „Coherent Matter Waves“, „Quantum Information“ EU (IST): „EQUIP“ Austrian Industry: Institute for Quantum Information Ges.m.b.H. P. Zoller J. I. Cirac Postdocs: - L.M. Duan (*) - P. Fedichev - D. Jaksch - C. Menotti (*) - B. Paredes - G. Vidal - T. Calarco Ph D: - W. Dur (*) - G. Giedke (*) - B. Kraus - K. Schulze