1. 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

2. 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

3. 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

4. 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

5. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology
ATOMIC WAVES z y x
TEM00

6. 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

7. 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

8. 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

9. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology
E(p)
n(p)
Recoil energy p

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

11. 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

12. 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

13. 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

14. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology
E(p)
Recoil energy p

15. 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

16. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology
E(p) p

17. 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

18. 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

19. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology

20. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology
ABCDx PROPAGATOR

21. 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

22. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology
FOUNTAIN CLOCK a b

23. 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

24. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology
Atom Interferometer
Laser beams
Atom
beam

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

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

27. 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

28. 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

29. 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

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

31. 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:

32. 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

33. 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

34. 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

35. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology

36. 6th Symposium on Frequency Standards & Metrology
10 September 2001
6th Symposium on Frequency Standards & Metrology

37. 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