2. School of something FACULTY OF OTHER School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES Putting entanglement to work: Super-dense coding, teleportation and metrology Jacob Dunningham Paraty, August 2007

3. Overview What can we do with entanglement? Superdense coding Teleportation Quantum cryptography Quantum computing Quantum-limited measurements Other lectures

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

5. Superdense coding Suppose Alice and Bob want to communicate Classically, if Alice sends Bob one bit - only one “piece” of information is shared. Is it possible to do better using quantum physics?

6. Superdense coding Suppose Alice and Bob want to communicate Classically, if Alice sends Bob one bit - only one “piece” of information is shared. Is it possible to do better using quantum physics? Problem:Can Alice send two classical bits of information to Bob if she is only allowed to physically send one qubit to him? Answer:Yes - by making use of entangled states

7. Superdense coding Alice Bob 1 qubit Third party Alice makes one of four measurements and then sends her qubit to Bob

8. Superdense coding Procedure: - If Alice wants to send… 00 She does nothing 01 She applies the phase flip Z to her qubit 10 She applies the quantum NOT gate X 11 She applies the iY gate

9. Superdense coding Bell States

10. Superdense coding Bell States Finally, Alice sends her (single) qubit to Bob Bob now has one of the Bell states. These are orthonormal and can be distinguished by measurements.

11. Teleportation It is also possible to transfer an unknown quantum state to a distant party using entanglement - teleportation Suppose Alice has a state of the form: Unknown coefficients

12. Teleportation It is also possible to transfer an unknown quantum state to a distant party using entanglement - teleportation Suppose Alice has a state of the form: And that Alice and Bob share a Bell state Unknown coefficients Overall:

13. Teleportation Alice has the first two qubits (on the left) This can be rewritten in terms of the Bell states as:

14. Teleportation Alice has the first two qubits (on the left) This can be rewritten in terms of the Bell states as: If Alice detects in the Bell-state basis, Bob’s state is projected onto one of these

15. Teleportation If Alice measuresBob gets Successful teleportation The other outcomes require state rotations by Bob

16. Teleportation If Alice measuresBob gets Successful teleportation The other outcomes require state rotations by Bob Alice measuresBob getsOperation

17. Teleportation If Alice measuresBob gets Successful teleportation The other outcomes require state rotations by Bob Alice measuresBob getsOperation Alice must communicate her measurement outcome to Bob by classical means. This prevents information being transmitted faster than light.

18. Precision measurements Atomic fountains are today’s best clocks, realizing the SI second to better than 10 -15.  -wave Cavity ~1 cm hole Cold is good Colder is better:

19. For earthbound clocks, laser cooling is sufficient ~ meter high fountains give ~ 1 second return times, so ~1 cm/s velocity spread, i.e. T~1  K Cs is sufficient. Space-borne clocks, with much longer interaction times than possible on earth, will benefit from colder atoms. A BEC can provide the really cold temperatures needed. NIST – U. Colorado – JPL Primary Atomic Reference Clock in Space (PARCS)

20. BEC in space A 100  m diameter trapped 87 Rb condensate… …adiabatically expanded to 1 cm… …would, upon release, expand with less than 1  m/sec. In principle, observation times longer than 1000 seconds would be possible We’re missing a trick…

21. Interferometer

23. The phase shift can be found from: Then

24. What is a beam splitter?

25. A state rotation….

26. What is a beam splitter? “Any physical process that transforms states in the same way as a beam splitter”

27. What is a beam splitter? For atoms, this is equivalent to tunnelling between two potential wells “Any physical process that transforms states in the same way as a beam splitter”

28. What is a beam splitter?

29. Using the identity:

30. What is a beam splitter? Using the identity:

31. Shot noise For a coherent state at 1 and a vacuum at 2:

32. Shot noise Due to discreteness of outcomes - same noise as coin toss For a coherent state at 1 and a vacuum at 2: The uncertainty in the phase can be found from ‘Shot noise’

33. Squeezing Not squeezedSqueezed

34. Squeezing If we use a squeezed state as the input to the beam splitter: i.e. there is no advantage to squeezing the input state Since:

35. Squeezed vacuum Bizarrely, by squeezing the vacuum we can do better For modest squeezing: This reduces to:

36. Squeezed vacuum Bizarrely, by squeezing the vacuum we can do better For modest squeezing: This reduces to: There is an optimum degree of squeezing. This gives: But we can still do better…..

37. Heisenberg limit The ultimate quantum limit is the Heisenberg limit:

38. Heisenberg limit The ultimate quantum limit is the Heisenberg limit: For a system with a fixed total number of particles, N, The maximum number uncertainty in any part of the system is N This mean that the uncertainty in the phase is:

39. Heisenberg limit The ultimate quantum limit is the Heisenberg limit: For a system with a fixed total number of particles, N, The maximum number uncertainty in any part of the system is N This mean that the uncertainty in the phase is: Lower bound (equality) is the Heisenberg limit c.f. Shot noise

40. Heisenberg limit The ultimate quantum limit is the Heisenberg limit: For a system with a fixed total number of particles, N, The maximum number uncertainty in any part of the system is N This mean that the uncertainty in the phase is: Lower bound (equality) is the Heisenberg limit c.f. Shot noise How do we reach this limit?

41. The cat gets the cream Stream of single particles:

42. The cat gets the cream Stream of single particles: N-fold enhancement of phase shift Cat state

43. Bucky-ball

44. C 60 molecules (1999)

45. Bucky-ball PROBLEM: Cats are fragile – any advantage is lost Let’s try something else….. C 60 molecules (1999)

46. Number correlated state Bose-Hubbard model U For simplicity, we can consider a two-site system

47. Adiabatically turn down coupling, J Initially, system in binomial state Then as U/J: the state evolves to:

49. Entangling lots of atoms

50. Measurement scheme What input state should we use to measure,  with Heisenberg limited accuracy? ?

51. Hong-Ou-Mandel effect Input state:

52. Hong-Ou-Mandel effect Input state: The effect of the beam splitter is found by transforming the operators

53. Hong-Ou-Mandel effect Input state: The effect of the beam splitter is found by transforming the operators The beam splitter ‘creates’ entanglement The same approach can be taken for:

55. Beam splitter

56. Independent 2/3 of atoms lost N=20

57. Cat State Independent 2/3 of atoms lost N=20

58. Cat State Independent 2/3 of atoms lost N=20

59. Mott Not Entangled  Detectors Efficiency

60. Classical Limit Reference: T. Kim et al. PRA 60, 708 (1999).

61. Classical Limit Reference: T. Kim et al. PRA 60, 708 (1999). To reach the Heisenberg limit we need:  N

62. How do we resolve this? Mott Not Entangled Entangled ? Pass back through Mott transition Reversible process – adiabatically lower potential barrier Disentangle particles

63. N=20 and  =  /30 Independent particles Entangling and disentangling Both cases are robust to loss and insensitive to detector inefficiencies

64. Mott Not Entangled Entangled Allows sub-SQL resolution but still robust to loss Allows sub-SQL resolution but not destroyed by imperfect detectors

65.  Collapse time 

66. Position Atoms A phase shift of N  =1, i.e.  =1/N, is amplified into a dramatic observable: Interference fringes appear and disappear.

67. Summary What can we do with entanglement? Superdense coding Teleportation Quantum cryptography Quantum computing Quantum-limited measurements