SU(3) phase operators: some solutions and properties

1. SU(3) phase operators:some solutions and properties Hubert de Guise Lakehead University

2. Collaborators: Luis Sanchez-Soto Andrei Klimov

3. Summary • Polar decomposition: • can be easily generalized but many “free parameters” • Normally yields non-commuting phase operators • Complementarity: • cannot be easily generalized but no “free parameters” • Normally yields commuting phase operators

4. The origin: the classical harmonic oscillator Classical harmonic oscillator: Use: Quantize:

5. Two approaches

6. Two approaches • write operator in polar form: • think of as the exponential of a hermitian phase operator

7. Two approaches • write operator in polar form: • think of as the exponential of a hermitian phase operator • Use complementarity condition:

8. What they have in common • Look at rather than

9. What they have in common • Look at rather than • is assumed unitary: is hermitian

10. What they have in common • Look at rather than • is assumed unitary: is hermitian • Must fix some “boundary” problems by hand

11. SU(2) phase operator mod(2j+1)

12. SU(2) phase operator mod(2j+1) Only one “boundary” condition

13. An example: j=1 -2=1mod(3)

14. An example: j=1 -2=1mod(3) -2=1mod(3)

15. A short course on su(3) • There are eight elements in su(3)

16. A short course on su(3) • There are eight elements in su(3) • There are now two relative phases

17. A short course on su(3) • There are eight elements in su(3) • There are now two relative phases • States are of the form

18. Commutation relations

19. Commutation relations

20. Commutation relations

21. Commutation relations

22. Commutation relations

23. Geometry of weight space

24. Geometry of weight space

25. Geometry of weight space

26. Geometry of weight space

27. 3-dimensional case

28. 3-dimensional case

29. 3-dimensional case

30. 3-dimensional case

31. 3-dimensional case

32. 3-dimensional case NOT an su(3) system

33. SU(3) phase operators:polar decomposition

34. Solution 1: commuting solution

35. Solution 1: commuting solution

36. Solution 1: commuting solution

37. Complementaritry The matrices form generalized discrete Weyl pairs, in the sense

38. Solution 2: the SU(2) solution

39. Higher-dimensional cases • No commuting solutions • No complementarity

40. Infinite dimensional limit • The edges are infinitely far • One can find commuting solutions: the phase operator commute, and have common eigenstates of zero uncertainty

41. Summary • Polar decomposition: • can be easily generalized but many “free parameters” • Normally yields non-commuting phase operators • Complementarity: • cannot be easily generalized but no “free parameters” • Normally yields commuting phase operators