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

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

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

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

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

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

25 July 2003BIPM metrology summer school 2003 ATOM WAVES

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 =

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

ABCD  LAW OF ATOM/PHOTON OPTICS

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

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

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

25 July 2003BIPM metrology summer school 2003

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 , 2001

Laser beams Atoms COSPAR 2004

Laser beams Atoms COSPAR 2004

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

First atom-wave gyro: Riehle et al. 1991

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

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) L

CLASSICAL ACTION AND PROPER TIME

Invariant de Lagrange

E(p) p // a b Mac2Mac2 Mbc2Mbc2

p Mc E

x s t

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

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

HAMILTONIAN & LAGRANGIAN

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

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

ABCD  LAW OF ATOM OPTICS

Ehrenfest theorem + Hamilton equations

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

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

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

a a b b Application to fountain clocks q1q1

a a b b Metrologia 39, (2002)

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

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

23 Novembre 2004Collège de France

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

RELATIVISTIC PHASE SHIFTS gr-qc/ for Dirac particles interacting with weak gravitational fields in matter-wave interferometers

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