Relativistic Quantum Theory of Microwave and Optical Atomic Clocks 10 September 2001 6th Symposium on Frequency Standards & Metrology Relativistic Quantum Theory of Microwave and Optical Atomic Clocks by Christian J. Bordé Laboratoire de Physique des Lasers, Villetaneuse and Bureau National de Métrologie, Paris

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology ATOMS ARE WAVES ! ldeBroglie v The recoil energy is not negligible any more in Cesium clocks Atom sources may be coherent sources of matter-wave Different from small clocks carried by classical point particles Atomic frame of reference may not be well defined Atomic clocks are fully quantum devices, in which both the internal and external degrees of freedom of the atoms must be quantized Gravitation and inertia play an important role: Atomic clocks are relativistic devices

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology Atom laser Rubidium atoms are extracted from a cold rubidium gas (left) and from a Bose-Einstein condensate(right). An intense low divergence atomic beam falls under the effect of gravity. courtesy of the university of Munich

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology E(p) ENERGY atom slope=v rest mass photon slope=c p MOMENTUM

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology ATOMIC WAVES z y x TEM00

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology ABCD matrices for light and matter-wave optics Space or Time Optical System for light rays for massive particles In Gaussian optics, the matrix ABCD also gives the transformation law for the waves: transforms as

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology ABCD PROPAGATOR For a wave packet moving with the initial velocity

a b b a b a a b RAMSEY FRINGES WITH TWO SPATIALLY 10 September 2001 6th Symposium on Frequency Standards & Metrology RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES z y x a b b a b a a b

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology E(p) n(p) Recoil energy p

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology E(p) p

b a b a RAMSEY FRINGES : y z x ATOMS EM WAVE 10 September 2001 6th Symposium on Frequency Standards & Metrology RAMSEY FRINGES : FIRST-ORDER TRANSITION AMPLITUDE AFTER A SINGLE FIELD ZONE z y x a b b a ATOMS EM WAVE

a b b a a b RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES 10 September 2001 6th Symposium on Frequency Standards & Metrology RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES EM WAVE 1 EM WAVE 2 a b b a a b

b a a,pz b b a b a,pz RAMSEY FRINGES WITH TWO SPATIALLY 10 September 2001 6th Symposium on Frequency Standards & Metrology RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES z y b x a a,pz b b ATOMS a b a,pz EM WAVE 1 EM WAVE 2

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology E(p) Recoil energy p

b a a,pz b b a,p'z RAMSEY FRINGES WITH TWO SPATIALLY 10 September 2001 6th Symposium on Frequency Standards & Metrology RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES b z y a b x a a,pz b b ATOMS a,p'z EM WAVE 1 EM WAVE 2

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology E(p) p

b a a,pz b b a,pz±2k RAMSEY FRINGES WITH TWO SPATIALLY 10 September 2001 6th Symposium on Frequency Standards & Metrology RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES b z y a b x a a,pz b b ATOMS EM WAVE a,pz±2k

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology Rubidium clock with a monomode continuous coherent beam Auxiliary Magnetic shield Microwave Height 1 m Microwave resonator Detection of F=1,m=0 - Flux 107 atoms/s (gain of 10/ present fountains) - Average density 109 atoms/cm3 for Dx=50 mm - Continuous operation - No losses between rise and fall: Dvx=15 mm/s Courtesy of Jean Dalibard and David Guéry-Odelin

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology ABCDx PROPAGATOR

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology Quite generally, the phase shift along each arm is: i.e. minus the time integral of the kinetic energy

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology FOUNTAIN CLOCK a b

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology Gravitational/Relativistic Doppler shift for fountain clocks A quantum mechanical calculation ~ Langevin twin paradox a b

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology Atom Interferometer Laser beams Atom beam

Interféromètres atomiques 10 September 2001 6th Symposium on Frequency Standards & Metrology Jets atomiques Faisceaux laser

SATURATION SPECTROSCOPY 10 September 2001 6th Symposium on Frequency Standards & Metrology SATURATION SPECTROSCOPY E(p) E(p) p p recoil doublet

Optical clocks with cold atoms 10 September 2001 6th Symposium on Frequency Standards & Metrology Optical clocks with cold atoms use the “working horse” of laser cooling: Magneto-optical trap (MOT) In the future new atom sources such as atom lasers

Time-domain Ramsey-Bordé interferences with cold Ca atoms 10 September 2001 6th Symposium on Frequency Standards & Metrology Time-domain Ramsey-Bordé interferences with cold Ca atoms

THEORY OF OPTICAL CLOCKS: 10 September 2001 6th Symposium on Frequency Standards & Metrology THEORY OF OPTICAL CLOCKS: SUCCESSIVE STEPS, RELEVANT STUDIES AND DIRECTIONS OF PROGRESS 1977: Naive, perturbative and numerical approaches 1982: 2x2 ABCD matrices for field pulses/zones and free propagation between pulses/zones : still used 1991: ABCDx formalism for atom wave propagation in a gravitational field 1994: Strong field S-matrix treatment of the e.m. field zones 1995: Rabi oscillations in a gravitational field (analogous to frequency chirp in curved wave-fronts) 1996: Dispersive properties of the group velocity of atom waves in strong e.m. fields To-day we combine all these elements in a new sophisticated and realistic quantum description of optical clocks. This effort is also underway for atomic inertial sensors. Strategies to eliminate inertial field sensitivity of optical clocks

RELATIVISTIC PHASE SHIFTS 10 September 2001 6th Symposium on Frequency Standards & Metrology RELATIVISTIC PHASE SHIFTS

Quite generally, the spin-independent part of the phase shift is: 10 September 2001 6th Symposium on Frequency Standards & Metrology Quite generally, the spin-independent part of the phase shift is:

Gravitational phase shift: 10 September 2001 6th Symposium on Frequency Standards & Metrology Atom Interferometers as Gravito-Inertial Sensors: I - Gravitoelectric field case Laser beams Atoms with light: Einstein red shift with neutrons: COW experiment (1975) with atoms: Kasevich and Chu (1991) T T ’ T Gravitational phase shift: Ratio of gravitoelectric flux to quantum of flux Mass independent  (time)2 Phase shift Circulation of potential

10 September 2001 6th Symposium on Frequency Standards & Metrology Atom Interferometers as Gravito-Inertial Sensors: II - Gravitomagnetic field case Laser beams Atoms with light: Sagnac (1913) with neutrons: Werner et al.(1979) with atoms: Riehle et al. (1991) Sagnac phase shift: Ratio of gravitomagnetic flux to quantum of flux Phase shift Circulation of potential

E(p) p DOPPLER-FREE TWO-PHOTON SPECTROSCOPY 10 September 2001 6th Symposium on Frequency Standards & Metrology DOPPLER-FREE TWO-PHOTON SPECTROSCOPY E(p) p

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology

6th Symposium on Frequency Standards & Metrology 10 September 2001 6th Symposium on Frequency Standards & Metrology

E(p) p RECOIL SHIFT IN DOPPLER-FREE TWO-PHOTON SPECTROSCOPY 10 September 2001 6th Symposium on Frequency Standards & Metrology RECOIL SHIFT IN DOPPLER-FREE TWO-PHOTON SPECTROSCOPY E(p) p