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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
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ATOMS ARE QUANTA OF A MATTER-WAVE FIELD JUST LIKE PHOTONS ARE QUANTA OF THE MAXWELL FIELD QM FOR SPACE / ONERA 2005
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Laser beams Total phase=Action integral+End splitting+Beam splitters Atoms
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MOMENTUM E(p) p atom slope=v photon slope=c rest mass ENERGY QM FOR SPACE
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25 July 2003BIPM metrology summer school 2003 ATOM WAVES - Non-relativistic approximation: - Slowly-varying amplitude and phase approximation:
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E(p) p BASICS OF ATOM /PHOTON OPTICS Parabolic approximation of slowly varying phase and amplitude Massive particles E(p) p Photons
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25 July 2003BIPM metrology summer school 2003 ATOM WAVES
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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 =
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ABCD PROPAGATION LAW Framework valid for Hamiltonians of degree 2 in position and momentum is the classical action where
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ABCD LAW OF ATOM/PHOTON OPTICS
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25 July 2003BIPM metrology summer school 2003 Hamilton’s equations for the external motion
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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
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25 July 2003BIPM metrology summer school 2003 ABCD matrices for matter-wave optics We add a quadratic potential term (gravity gradient):
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25 July 2003BIPM metrology summer school 2003
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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
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Laser beams Atoms COSPAR 2004
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Laser beams Atoms COSPAR 2004
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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
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First atom-wave gyro: Riehle et al. 1991
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ARBITRARY 3D TIME-DEPENDENT GRAVITO-INERTIAL FIELDS COSPAR 2004 Example: Phase shift induced by a gravitational wave
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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
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CLASSICAL ACTION AND PROPER TIME
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Invariant de Lagrange
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E(p) p // a b Mac2Mac2 Mbc2Mbc2
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p Mc E
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x s t
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OPTICAL PATH & FERMAT’S PRINCIPLE IN (4+1)D Landau and Lifshitz, vol. 2, §88
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E(p) p BASICS OF ATOM /PHOTON OPTICS Parabolic approximation of slowly varying phase and amplitude
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HAMILTONIAN & LAGRANGIAN
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KLEIN-GORDON EQUATION in presence of weak gravito-inertial fields
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Schroedinger-like equation for the atom /photon field: BASICS OF ATOM /PHOTON OPTICS
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ABCD LAW OF ATOM OPTICS
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Ehrenfest theorem + Hamilton equations
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The four end-points theorem T= t 2 -t 1 β1 β2 α1 α2 M β M α t 1 t 2 Lagrange Invariant
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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
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GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM/PHOTON INTERFEROMETER
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a a b b Application to fountain clocks q1q1
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a a b b Metrologia 39, 435-463 (2002)
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ATOMES b a a b b a*a* b*b* a b b*b* a b*b* a*a* a*a* abab temps espace
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23 Novembre 2004Collège de France Optical clocks Laser beams Atom beam
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23 Novembre 2004Collège de France
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Christian J. Bordé, M. Weitz and T.W. Hänsch, Laser Spectroscopy XI (1993) p.76
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RELATIVISTIC PHASE SHIFTS http://christian.j.borde.free.fr gr-qc/0008033 for Dirac particles interacting with weak gravitational fields in matter-wave interferometers
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http://christian.j.borde.free.fr
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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
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