“Classical entanglement” and cat states

1. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES “Classical entanglement” and cat states Jacob Dunningham Paraty, August 2007

2. Overview • The consequences of entanglement: • The emergence of classicality from the quantum world • Number and phase of BEC • Position and momentum of micro-mirrors • Energy and time? • Schrodinger cat states • How can we make them • How can we see them • What can we do with them

3. Everyday World Quantum Information Multi-particle Entanglements WILD PEDIGREE Cats Bats Bunnies

4. Annihilation and creation operators (bosons) annihilation creation

5. Annihilation and creation operators (bosons) annihilation creation

6. Annihilation and creation operators (bosons) • Eigenvalue equation annihilation creation is the number operator

7. Annihilation and creation operators (bosons) • In the Fock (number state) basis, these can be written as the matrices: annihilation creation

8. Annihilation and creation operators (bosons) • In the Fock (number state) basis, these can be written as the matrices: annihilation creation • An exercise in matrix multiplication confirms the bosonic commutation relation:

9. Emergence of classicality • One of the most perplexing aspects of quantum theory is that microscopic objects can be in superpositions but macroscopic objects cannot Schrödinger’s cat To ‘see’ a coherent superposition, we need interference

10. How do we see them?

11. Macroscopic variables Detect interference of probe state corresponding to phase 

12. Macroscopic variables Detect interference of probe state corresponding to phase  No interference if the macroscopic states are orthogonal Need coupling between them - “Lazarus operator”

13. The key is to wash out the which-way information NOON state

14. The key is to wash out the which-way information Described in detail by A. Ekert yesterday There is the problem of the environment Tracing over the environment gives:

15. Classical entanglement • Can also understand the emergence of classicality in terms of entanglement

16. Classical entanglement • Can also understand the emergence of classicality in terms of entanglement • First it is helpful to consider BECs • Macroscopic quantum entity • Can probe quantum / classical divide • Cold enough to enable quantum phase transitions

17. What is a BEC? Predicted 1924......Created 1995 S. Bose A. Einstein

18. What is a BEC? Bose-Einstein distribution:

19. What is a BEC? Bose-Einstein distribution: Take For consistency:

20. What is a BEC? Bose-Einstein distribution: Take For consistency: Onset of BEC: Cold and dilute

21. How do we make them? • Trap them with magnetic and/or optical fields • Cool them using two main techniques: • Laser Cooling (link) 2. Evaporative Cooling (link)

22. What is a BEC? For our purposes, a BEC is a ‘macroscopic’ quantum entity - thickness of a human hair All the atoms (~103 - 109) are in the same quantum state

23. Phase of a BEC • Coherent state: “Most classical” quantum state  DNDF~1 

24. BEC Localisation Conservation of atom number: N N ? DNDF~1

25. BEC Localisation Conservation of atom number: N N Experiment ? DNDF~1

26. BEC Localisation First detection: b a N N We don’t know which BEC the atom came from Position-dependent phase x DNDF~1

27. BEC Localisation First detection: b a N N We don’t know which BEC the atom came from Position-dependent phase x DNDF~1

28. BEC Localisation b Probability density of second detection:: a N N x DNDF~1

29. BEC Localisation b Probability density of second detection:: a N N Feedback gives fringes with visibility ~ 0.5 x After ~ N measurements: DNDF~1

30. Robust relative phase state - classical The phase of each condensate is still undefined:

31. Fluffy bunny state

32. Phase standard b c a N N N

33. Phase standard b c a N N N

34. Phase standard b c a N N N

35. Phase standard

36. Properties • Robustness: subsequent measurements do not change the result – classical-like • Transitivity: ingrained in our classical perception of the world a c b • Absolute versus relative variables Entanglement is all around us – not just a “quantum phenomenon”!

37. Position Localisation Can do the same for position and momentum Initial state of the mirrors: Relative position Flat distribution

38. Position Localisation Can do the same for position and momentum Initial state of the mirrors: Relative position Flat distribution Photon with momentum k, state before N:

39. Position Localisation

40. Position Localisation Detection at D1: Detection at D2:

41. Position Localisation q 1. Rau, Dunningham, Burnett, SCIENCE 301, 1081 (2003) 2. Dunningham, Rau, Burnett, SCIENCE 307, 872 (2005)

42. Time Barbour view: Angle of hour hand Position of sun ‘time’ ‘time’ No need to go through ‘middle-man’ of time Angle of hour hand Position of sun

43. Entanglement of three particles H| |cn,m |n, m, E-n-m> x23 ? x12

44. Don’t need measurements For every sequence of scattering events, a well-defined relative position (or phase) builds up If we don’t measure the scattered particles the relative position is uncertain (classically) Tracing over the scattered particles gives:

45. Don’t need measurements • Just by shining light on particles they acquire a classical relative position - yet each particle remains highly quantum! For every sequence of scattering events, a well-defined relative position (or phase) builds up If we don’t measure the scattered particles the relative position is uncertain (classically) Tracing over the scattered particles gives: Classical mixture Well-localised state

46. Everyday World Quantum Information Multi-particle Entanglements WILD PEDIGREE Cats Bats Bunnies

47. Experimental progress C60 molecules (1999) 4 Be+ ions (2000) ~ 109 Cooper pairs (2000)

48. Experimental progress Future C60 molecules (1999) 4 Be+ ions (2000) Micro-mirrors Biological systems? (E. Coli) ~ 109 Cooper pairs (2000)

49. Superfluid cats a Ref: Boyer et al, PRA 73, 031402 (2006) b c Coupling between wells Interactions between atoms

50. a Ref: Boyer et al, PRA 73, 031402 (2006) b c Coupling between wells Interactions between atoms