1. School of something FACULTY OF OTHER 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. Multi-particle Entanglements WILDPEDIGREE Bunnies CatsBats Quantum Information Everyday World

4. Annihilation and creation operators (bosons) annihilation creation

5. Annihilation and creation operators (bosons) annihilation creation

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

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

8. Annihilation and creation operators (bosons) annihilation creation In the Fock (number state) basis, these can be written as the matrices: 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. Detect interference of probe state corresponding to phase  Macroscopic variables

12. Detect interference of probe state corresponding to phase  No interference if the macroscopic states are orthogonal Macroscopic variables 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 There is the problem of the environment Tracing over the environment gives: Described in detail by A. Ekert yesterday

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: 1.Laser Cooling (link)(link) 2. Evaporative Cooling (link)(link)

22. What is a BEC? For our purposes, a BEC is a ‘macroscopic’ quantum entity - thickness of a human hair All the atoms (~10 3 - 10 9 ) are in the same quantum state

23. Phase of a BEC Coherent state:    “Most classical” quantum state

24. BEC Localisation  NN Conservation of atom number: ?

25. BEC Localisation  NN Conservation of atom number: ? Experiment

26. BEC Localisation First detection:  NN a b We don’t know which BEC the atom came from x Position-dependent phase

27. BEC Localisation First detection:  NN a b We don’t know which BEC the atom came from x Position-dependent phase

28. BEC Localisation  NN a b x Probability density of second detection: :

29. BEC Localisation  NN a b x Probability density of second detection: : Feedback gives fringes with visibility ~ 0.5 After ~ N measurements:

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

31. Fluffy bunny state

32. Phase standard NN a c N b

33. NN a c N b

34. NN a c N b

36. Properties Absolute versus relative variables a b c Robustness: subsequent measurements do not change the result – classical-like Transitivity: ingrained in our classical perception of the world 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. Detection at D 1 : Detection at D 2 :

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

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

43. Entanglement of three particles H|  |  c n,m |n, m, E-n-m> x 23 x 12 ?

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: Well-localised state Classical mixture

46. Multi-particle Entanglements WILDPEDIGREE Bunnies CatsBats Quantum Information Everyday World

47. Experimental progress 4 Be + ions (2000) C 60 molecules (1999) ~ 10 9 Cooper pairs (2000)

48. Experimental progress 4 Be + ions (2000) C 60 molecules (1999) ~ 10 9 Cooper pairs (2000) Future Micro-mirrors Biological systems? (E. Coli)

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

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

51. No rotation Clockwise Anticlockwise

52. No rotation Clockwise Anticlockwise Flow is quantized in units of 2  around the loop -- vortices

53. How do we make them?

56. Cat

57. Entanglement witness Separable states How do we see them?

58. Metastable states Spectroscopically scan the energy gap -- see it directly How do we see them?

59. What can we do with them? +

60. + For superfluid flows: Bell state experiments with macroscopic objects Precision measurements - quantum-limited gyroscopes

61. Precision measurements of angular momentum Gyroscopes

62. Precision measurements of angular momentum Can measure  to within 1/N Gyroscopes

63. Summary Next lecture: An even better way of using entanglement to make measurements The emergence of classicality from the quantum world Number and phase of BEC Position and momentum of micro-mirrors Schrodinger cat states How can we make them How can we see them What can we do with them