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Atomic entangled states with BEC

Atomic entangled states with BEC. A. Sorensen L. M. Duan P. Zoller J.I.C. (Nature, February 2001). KIAS, November 2001. SFB Coherent Control €U TMR. ª. =. '. '. :. :. :. '. j. i. 6. j. 1. i. ­. j. 2. i. ­. j. N. i. Entangled states of atoms. Motivation:.

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Atomic entangled states with BEC

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

  2. ª = ' ' : : : ' j i 6 j 1 i ­ j 2 i ­ j N i Entangled states of atoms Motivation: • Fundamental. • Applications:- Secret communication - Computation - Atomic clocks Experiments: • NIST: 4 ions entangled. • ENS: 3 neutral atoms entangled. 4 ' E 3 ' E This talk:Bose-Einstein condensate. 3 1 0 E '

  3. Outline • Atomic clocks • Ramsey method • Spin squeezing • Spin squeezing with a BEC • Squeezing and atomic beams • Conclusions

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

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

  6. 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 1 1 1 1 1

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

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

  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 • Product states: No gain!

  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 are like position and momentum operators Hamiltonian 1: Hamiltonian 2:

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

  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 condensatewith • 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: Idea: Use short laser pulses. short evolution short evolution short pulse short pulse Conditions:

  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 pulse short pulses

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

  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 input: vaccum condensate output

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

  27. 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.

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

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

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

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

  32. Outline • Quantum repeaters: • Implementations: • With trapped ions. • With atomic ensembles. • Conclusions

  33. 1. Quantum repeaters The goalis 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)

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

  35. 2. Implementation with trapped ions Entanglement creation: (Cabrillo et al, 1998) Internal states ion A ion A ion B laser ion B laser - 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.

  36. Description: Initial state: ion A ion B After laser pulse: Evolution: Detection:

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

  38. 3 Implementation with atomic ensembles Atomic cell Internal states Atomic cell - 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.

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

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

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

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

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

  44. (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

  45. (C) Efficiency: Fix the final fidelity: F Number of repetitions: Example: Detector efficiency: 50% Length L=100 L0 6 Time T=10 T0 43 (to be compared with T=10 T0 for direct communication)

  46. 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. 3. Connection is built in. No need for gates. 4. Purification is built in.

  47. 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: • No trapping/cooling is required. • No (high-Q) cavity is required. • Atomic collective effects make it more efficient. • No high efficiency detectors are required.

  48. Institute for Theoretical Physics 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 P. Zoller J. I. Cirac 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. €

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