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

4. MOMENTUM E(p) p atom slope=v photon slope=c rest mass ENERGY

5. CHEMIN OPTIQUE & PRINCIPE DE FERMAT

6. E(p) p // Recoil energy a b

7. 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

8. Optical clocks Laser beams Atom beam

9. Stimulated Raman transitions Raman pulses act as mirrors and beam splitters for matter waves k 2,  2 k 1,  1 |i > |b  |a   ~ 1 GHz Alkali atoms (Rb, Cs) |a  and |b  Hyperfine states Transition Probability  Rabi   Effective two level system Quantum superposition => Rabi oscillations  pulse Atomic mirror  /2 pulse Atomic beam Splitter k2k2 k1k1 |a, p  ħ k eff =ħ(k 1 -k 2 ) |b, p+ ħk eff 

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

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

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

13. 25 July 2003BIPM metrology summer school 2003 ATOM WAVES - Non-relativistic approximation: - Slowly-varying amplitude and phase approximation:

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

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

16. 25 July 2003BIPM metrology summer school 2003 ATOM WAVES

17. 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 =

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

19. ABCD  LAW OF ATOM/PHOTON OPTICS

20. Ehrenfest theorem + Hamilton equations

21. Hamilton’s equations for the external motion

22. 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

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

24. The four end-points theorem Ch. Antoine and Ch.J. Bordé, Exact phase shifts for atom interferometry, Phys. Lett. A306, 277-284 (2003) T= t 2 -t 1 β1 β2 α1 α2 M β M α t 1 t 2

25. 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 INTERFEROMETER

26. Atom Interferometers as Gravito-Inertial Sensors:Analogy between gravitation and electromagnetism 1 00 hg   g  T T Metric tensor Newtonian potential Gravitoelectricfield

27. Atom Interferometers as Gravito-Inertial Sensors: I - Gravitoelectric field case Gravitational phase shift: T T with light: Einstein red shift with neutrons: COW experiment (1975) with atoms: Kasevich and Chu (1991) Phase shift Circulation of potential Mass independent  (time) 2 Ratio of gravitoelectric flux to quantum of flux

28. ABCD matrices for matter-wave optics We add a quadratic potential term (gravity gradient):

30. 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

31. 31 Atom Interferometric Gravimeter Performances : –Resolution: 3x10 -9 g after 1 minute –Absolute accuracy:  g/g<3x10 -9 From A. Peters, K.Y. Chung and S. Chu

32. 32 Gradiometer with cold atomic clouds Yale university  Sensitivity: 3.10 -8 s -2 /  Hz 30 E/  Hz  Potential on earth: 1E/  Hz

33. Stanford/Yale Gravity Gradiometer: Measurement of G Pb mass translated vertically along gradient measurement axis. Typical data: ~1x10 -8 g change in acceleration due to gravitational forces for different Pb positions Present sensitivity/accuracy: G = 3 x 10 -3 G Measurement consistent with accepted value

34. Experimental Set-Up  /2  2D-MOT atom interferometer Raman 2 Detection of |a  et |b  3D-MOT 10 7 Rb-atoms in 50 ms T atoms ~2 µK Raman 1 Mirror 2D MOT Passive isolation Plateform Sismometer Magnetic shields

35. Schéma de principe du gravimètre PMO : Piège Magnéto-Optique Faisceau pousseur  /2  Faisceaux laser Raman Détection Impulsions Raman stimulées temporelles 2T=100 ms = interféromètre Jet d’atomes de 87 Rb refroidis dans un PMO-2D 10 8 atomes piégés en 100 ms refroidis à quelques  K dans un PMO-3D 10 9 à 10 10 atomes.s -1 Sélection de l’état |5S 1/2, F=1, m F =0 > Faisceaux laser Raman

36. Enceinte à vide Chambre à vide du PMO 2D Chambre à vide Vanne d’isolation de la réserve Réserve de rubidium PMO 3D Detection

37. 2 paires de faisceaux Raman dans l'enceinte Plans équiphases solidaires de la position du miroir  Mesure des déplacements des atomes par rapport au miroir Montage expérimental Deux faisceaux superposés et retroréfléchis : Raman 1 Raman 2 miroir σ+σ+ σ-σ- σ-σ- σ+σ+ PMO 3D Tps capture : 50 ms T atomes ~2 µK blindage magnétique Détection N at détectés 2.10 5

38. Interferometer fringes Parameters 2T=100 ms  = 6 µs  v ~ v r N det = 10 6 T c = 250 ms Contrast ~ 45 % Sources of noise - laser phase noise (Phase lock : 3.5 mrad /shot) - mirror vibrations SNR = 25 σ Φ = 1/SNR = 40 mrad/shot  g /g = 10 -7 /shot

39. Free fall → Doppler shift of the resonance condition of the Raman transition   = k eff.g.T 2 - aT 2 Dark fringe : independent of T Ramping of the frequency difference to stay on resonance : Principle of g measurement π/2 π

40. Long term stability Bias fluctuations : ± 15.10 -9 g Fluctuations of the systematic effects Earth tides : ± 1.10 -7 g Model accurate to a few 10 -9 g Continuous measurements

41. Earthquake! 2007, January 13 - 04:23 UTC Kuril Islands Magnitude 8.1 Period 17 s

42. Intégration sans isolation passive - Fonctionnement hors du régime linéaire - Nouvel algorithme d’asservissement : trois mesures consécutives de (P,  vib s ) permettent de déduire l’erreur de phase - Robustesse vis-à-vis des modifications du bruit de vibration Séisme du 20 Mars 2008, Chine, Magnitude 7.7

43. Atom Interferometers as Gravito-Inertial Sensors: Analogy between gravitation and electromagnetism Metric tensor Gravitomagnetic field Pure inertial rotation

44. with light: Sagnac (1913) with neutrons: Werner et al.(1979) with atoms: Riehle et al. (1991) Atom Interferometers as Gravito-Inertial Sensors: II - Gravitomagnetic field case Phase shift Circulation of potential Ratio of gravitomagnetic flux to quantum of flux Sagnac phase shift:

45. Laser beams Atoms COSPAR 2004

46. Laser beams Atoms

47. COSPAR 2004 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 

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

49. 49 Atomic Beam Gyroscope Sensitivity: 6.10 -10 rad.s -1 /  Hz (Yale University) Magnetic shield Cs oven Wave packet manipulation Atomic beams State preparation Laser cooling Detection Rotation rate (x10 -5 ) rad/s -10-50510 15 20 Normalized signal 0 1 Interference fringes

50. 50

51.  =    2  2  3 Saut de phase  3 (°) Probabilité de transition 2T= 20 ms C v = 0,55 C f = 0,46 F.Leduc, D. Holleville, J.Fils, A. Clairon, N. Dimarcq, A. Landragin, P. Bouyer and Ch.J. Bordé, ICOLS 2003 Gyro-accéléromètre à césium froid du SYRTE

52. MOT 1 MOT 2 Z X Y Cold atoms Good control of the mean velocity Small velocity dispersion Unique laser beam modulated on time Good stability and knowledge of the scaling factor probe Experimental setup PARAMETERS Cs atoms T atoms ~1 µK Launch velocity 2.4 m/s Angle 8° T c = 0.58 s

53. Six axes of inertia

54. Vertical measurements yy Z X Y   /2 azaz 2T = 80 ms Sum of the signals: Acceleration Difference of the signals: Rotation Rejection of the acceleration

55. Sensitivity Rotation noise (rad/s) 2.4 10 -7 rad/s @ 1 second 1.4 10 -8 rad/s Integration Time [sec] Acceleration ( m.s -2 ) 2.7 10 -9 g 5.5 10 -8 g @ 1 second Integration Time [sec] Acceleration limited by vibrations Best signal to noise to rotation: 200 With seismometer correction : 3.5 10 -8 g in 1 s Rotation is limited by QPN Competitive with best commercial FOG Sensitivity characterized by the Allan standard deviation

56. Test of the linearity of the scale factor Changing the orientation of the experiment - modulates the projection of the Earth rotation - changes the rotation rate in a controled way Excellent linearity No quadratic term at the 10 -5 level Fit with a free offset : 29 mrad North South EE x y  z East West yy

57. Testing the scale factor vs T 2 Rotation signal vs interaction time For two opposite orientations Difference :  rot Sum : bias

58. Conclusions GRAVIMETER Short term stability 2 10 -8 g/Hz 1/2 (under noisy environment) Systematic shifts many controlled at the 10 -9 g level, Coriolis & aberrations remain a challenge First comparison showed  g ~16 10 -9 g difference GYROSCOPE Short term sensitivity 2.4 10 -7 rad/s/√Hz (limited by atomic shot noise) Long term stability 0.6 to 2 10 -8 rad/s (limited by wavefront aberrations and fluctuation of the sources) Linearity of the scale factor

59. 3-D COMBINATION OF GRAVITO-INERTIAL FIELDS Exact phase shift for Gravitation+Field gradient+Rotation 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

60. HYPER HYPER -precision cold atom interferometry in space

61. 61 HYPER Atomic Sagnac Unit Interferometer length 60 cm Atom velocity 20 cm/s Drift time 3 s 10 9 atoms/shot Sensitivity 2x10 -12 rad/s Area 54 cm 2

62. LENSE-THIRRING FIELD Gravitomagnetic field lines Gravitomagnetic field generated by a massive rotating body: Field lines ~ to magnetic dipole:

63. 63 HYPER HYPER Lense-Thirring measurement Signal vs time Hyper carries two atomic Sagnac interferometers, each of them is sensitive to rotations around one particular axis. The two units will measure the vector components of the gravitomagnetic rotation along the two axes perpendicular to the telescope pointing to a guide star.

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

65. 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

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