Atom interferometry History, fundamentals and recent theories

1. Atom interferometry History, fundamentals and recent theories Christian J. Bordé LPL & SYRTE Académie des sciences

2. OUTLINE OF THE THREE LECTURES • A traditionalapproach to atominterferometry • 5D Relativisticatominterferometry • Application to atomicclocks and to gravito-inertialsensors http://christian.j.borde.free.fr/

3. A traditionalapproach to atominterferometry • Background: neutron interferometrymeets laser spectroscopy • Recoilphysics and beamsplitters • Klein-Gordon and WKB • Schroedingerequation and ABCD framework • Lagrange invariant • General formula for the phase shift • 5D Relativisticatominterferometry • Application to atomicclocks and to gravito-inertialsensors http://christian.j.borde.free.fr/

5. This LLL silicon crystal neutron interferometer was cut with a diamond wheel for Sam Werner in the Missouri Physics Machine Shop by Cliff Holmes in 1977. It was used to observe gravitationally-induced quantum interference, the Aharonov-Casher phase shift, the Neutron Sagnac Effect, and several other fundamental quantum mechanical effects. It is still in use at NIST. Dec.10, 2010

10. b* time b space a a a* b MOLECULES

13. Laser beams space b time cτ a Atoms π/2 Pulses Total phase=Action integral+End splitting+Beam splitters

14. SUB-DOPPLER SPECTROSCOPY Absorption cell or molecular beam LASER SOURCE n, 1/l MIRROR External motion v Amplifier Detector Resonator Transmitted Intensity Internal degrees of freedom Derivative spectrum Line-center lock OPTICAL CLOCK Frequencyn

15. b b* b b b* a a* temps b* b espace a a a* a a* ATOMES ab

16. Magnetic hyperfine components of the methane line at 3.39 m exhibiting recoil doubling: The Royal Society Hall, Bordé and Uehara, PRL 1976.

17. E(p) ENERGY Mass 1905 p • MOMENTUM

18. b Recoil energy E(p) a mbc2 mac2 p//

19. The Royal Society SATURATION SPECTROSCOPY E(p) E(p) p p Recoil doublet

20. Bordé-Ramsey interferometers Laser beams Atom beam

22. MOLECULAR INTERFEROMETRY SF6 1981

24. Florence Physics Colloquium Time-domain Ramsey-Bordé interferences with cold Ca atoms Performances: incertitude relative 2.10-14< 10-15 stabilité 5. 10-17 en 1sec

25. First Atom Inertial sensors (1991) First atom-wave gravimeter: Kasevich and Chu 1991 First atom-wave gyro: Riehle et al. 1991 25

26. Florence Physics Colloquium MAGNESIUM INTERFEROMETER hn2/Mc2 l=457 nm 80 kHz

27. Molecular Interferometry experiments

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

29. E(p) ENERGY Mass 1923 1905 p • MOMENTUM

30. CHEMIN OPTIQUE & PRINCIPE DE FERMAT

31. KLEIN-GORDON EQUATION (Curved space-time)

32. Elementary interval Metric tensor Analogy with: Post-Newtonian parameters (PPN):

33. ATOM WAVES - Non-relativistic approximation: - Slowly-varying amplitude and phase approximation:

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

35. BASICS OF ATOM /PHOTON OPTICS Schroedinger-like equation for the atom (photon) field: phase shift

36. ATOM WAVES

37. Minimum uncertainty wave packet: velocity of the wave packet center of the wave packet width of the wave packet in momentum space complex width of the wave packet in physical space conservation of phase space volume z =

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

39. ABCDx PROPAGATOR

40. ABCDx LAW OF ATOM/PHOTON OPTICS

41. Ehrenfest theorem + Hamilton equations for the external motion

42. GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM INTERFEROMETER k k k β 1 β 2 β N M M M β 1 β 2 β N β 1 β β β 2 N D k α N k k α 2 α 1 M M α N M α 2 α α α 1 α N D α 1 2 t t t t 2 N D 1

43. THE LAGRANGE INVARIANT IN ATOM OPTICS “Optical System” Space or Time The quantity: is conserved by the ABCD transformations Then the action difference cancels the mid-point phase shift

44. The four end-points theorem Ch. Antoine and Ch.J. Bordé, Exact phase shifts for atom interferometry, Phys. Lett. A306, 277-284 (2003) β 2 M β 1 β M α 2 α α 1 T= t - t t t 2 1 1 2

45. GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM INTERFEROMETER k k k β 1 β 2 β N M M M β 1 β 2 β N β 1 β β β 2 N D k α N k k α 2 α 1 M M α N M α 2 α α α 1 α N D α 1 2

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

47. References: Ch.J. Bordé, Atomic clocks and inertial sensors, Metrologia 39 (5), 435-463 (2002) 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 (April 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 (March 2004) 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