1. VARENNA 2007 Introduction to 5D-Optics for Space-Time Sensors Introduction to 5D-Optics for Space-Time Sensors Christian J. Bordé A synthesis between optical interferometry and matter-wave interferometry

2. ATOMS ARE QUANTA OF A MATTER-WAVE FIELD JUST LIKE PHOTONS ARE QUANTA OF THE MAXWELL FIELD QM FOR SPACE / ONERA 2005

5. Laser beams Total phase=Action integral+End splitting+Beam splitters Atoms

6. MOMENTUM E(p) p atom slope=v photon slope=c rest mass ENERGY QM FOR SPACE

7. 25 July 2003BIPM metrology summer school 2003 ATOM WAVES - Non-relativistic approximation: - Slowly-varying amplitude and phase approximation:

8. E(p) p BASICS OF ATOM /PHOTON OPTICS Parabolic approximation of slowly varying phase and amplitude Massive particles E(p) p Photons

9. 25 July 2003BIPM metrology summer school 2003 ATOM WAVES

10. 25 July 2003BIPM metrology summer school 2003 Minimum uncertainty wave packet: center of the wave packet complex width of the wave packet in physical space velocity of the wave packet width of the wave packet in momentum space conservation of phase space volume z =

11. ABCD  PROPAGATION LAW Framework valid for Hamiltonians of degree  2 in position and momentum is the classical action where

12. ABCD  LAW OF ATOM/PHOTON OPTICS

13. 25 July 2003BIPM metrology summer school 2003 Hamilton’s equations for the external motion

14. k β1 k β2 k α1 k α2 β 1 α 1 β 2 α 2 M α1 M β1 M α2 M β2 t 1 t 2 β N k βN M βN β D α D α N t N t D M αN k αN GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM INTERFEROMETER

15. 25 July 2003BIPM metrology summer school 2003 ABCD matrices for matter-wave optics We add a quadratic potential term (gravity gradient):

16. 25 July 2003BIPM metrology summer school 2003

17. 25 July 2003BIPM metrology summer school 2003 Exact phase shift for the atom gravimeter which can be written to first-order in  with T=T’  Reference: Ch. J. B., Theoretical tools for atom optics and interferometry, C.R. Acad. Sci. Paris, 2, Série IV, p. 509-530, 2001

18. Laser beams Atoms COSPAR 2004

19. Laser beams Atoms COSPAR 2004

20. Reference: Ch. J. B., Atomic clocks and inertial sensors, Metrologia 39 (5), 435-463 (2002) SAGNAC PHASE IN THE ABCD FORMALISM To first order in 

21. First atom-wave gyro: Riehle et al. 1991

22. ARBITRARY 3D TIME-DEPENDENT GRAVITO-INERTIAL FIELDS COSPAR 2004 Example: Phase shift induced by a gravitational wave

23. Atomic phase shift induced by a gravitational wave Ch.J. Bordé, Gen. Rel. Grav. 36 (March 2004) Ch.J. Bordé, J. Sharma, Ph. Tourrenc and Th. Damour, Theoretical approaches to laser spectroscopy in the presence of gravitational fields J. Physique Lettres 44 (1983) L983-990

24. CLASSICAL ACTION AND PROPER TIME

26. Invariant de Lagrange

28. E(p) p // a b Mac2Mac2 Mbc2Mbc2

29. p Mc E

30. x s t

31. OPTICAL PATH & FERMAT’S PRINCIPLE IN (4+1)D Landau and Lifshitz, vol. 2, §88

32. E(p) p BASICS OF ATOM /PHOTON OPTICS Parabolic approximation of slowly varying phase and amplitude

35. HAMILTONIAN & LAGRANGIAN

36. KLEIN-GORDON EQUATION in presence of weak gravito-inertial fields

37. Schroedinger-like equation for the atom /photon field: BASICS OF ATOM /PHOTON OPTICS

38. ABCD  LAW OF ATOM OPTICS

39. Ehrenfest theorem + Hamilton equations

40. The four end-points theorem T= t 2 -t 1 β1 β2 α1 α2 M β M α t 1 t 2 Lagrange Invariant

41. k β1 k β2 k α1 k α2 β 1 α 1 β 2 α 2 M α1 M β1 M α2 M β2 β N k βN M βN β D α D α N M αN k αN GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM/PHOTON INTERFEROMETER

42. GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM/PHOTON INTERFEROMETER

43. a a b b Application to fountain clocks q1q1

44. a a b b Metrologia 39, 435-463 (2002)

45. ATOMES b a a b b a*a* b*b* a b b*b* a b*b* a*a* a*a* abab temps espace

46. 23 Novembre 2004Collège de France Optical clocks Laser beams Atom beam

47. 23 Novembre 2004Collège de France

48. Christian J. Bordé, M. Weitz and T.W. Hänsch, Laser Spectroscopy XI (1993) p.76

49. RELATIVISTIC PHASE SHIFTS http://christian.j.borde.free.fr gr-qc/0008033 for Dirac particles interacting with weak gravitational fields in matter-wave interferometers

51. http://christian.j.borde.free.fr

52. 23 Novembre 2004Collège de France References: Ch.J. Bordé, Atomic clocks and inertial sensors, Metrologia 39 (2002) 435-463 Ch.J. Bordé, Theoretical tools for atom optics and interferometry, C.R. Acad. Sci. Paris, t.2, Série IV (2001) 509-530 Ch. Antoine and Ch.J. Bordé, Exact phase shifts for atom interferometry Phys. Lett. A 306 (2003) 277-284 and Quantum theory of atomic clocks and gravito-inertial sensors: an update Journ. of Optics B: Quantum and Semiclassical Optics, 5 (2003) 199-207 Ch.J. Bordé, Quantum theory of atom-wave beam splitters and application to multidimensional atomic gravito-inertial sensors, General Relativity and Gravitation, 36 (2004) 475-502 Atom Interferometry, ed P. Berman, Academic Press (1997) Ch.J. Bordé, Atomic Interferometry and Laser Spectroscopy, in Laser Spectoscopy X, World Scientific (1991) 239-245