Relativistic Quantum Theory of Microwave and Optical Atomic Clocks

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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
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ATOMS ARE WAVES !
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
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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
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E(p)
ENERGY
atom slopev
rest mass
photon slopec
p
MOMENTUM
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ATOMIC WAVES
y
z
x
TEM00
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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
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ABCD PROPAGATOR
For a wave packet moving with the initial velocity
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RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED
FIELD ZONES
b
a
b
a
a
b
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E(p)
p
10
E(p)
p
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RAMSEY FRINGES FIRST-ORDER TRANSITION
AMPLITUDE AFTER A SINGLE FIELD ZONE
EM WAVE
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RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED
FIELD ZONES
EM WAVE 1
EM WAVE 2
a
b
b
a
a
b
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RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED
FIELD ZONES
y
z
b
x
a
a,pz
b
b
ATOMS
a,pz
EM WAVE 1
EM WAVE 2
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E(p)
p
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RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED
FIELD ZONES
b
y
z
a
b
x
a
b
a,pz
b
ATOMS
a,p'z
EM WAVE 1
EM WAVE 2
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E(p)
p
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RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED
FIELD ZONES
b
y
z
a
b
x
a
b
a,pz
b
ATOMS
EM WAVE
a,pz2?k
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Rubidium clock with a monomode continuous
coherent beam
Auxiliary
Magnetic shield
Microwave
Height 1 m
Microwave resonator
Detection of F1,m0
- Flux 107 atoms/s (gain of 10/ present
fountains) - Average density 109 atoms/cm3 for
Dx50 mm - Continuous operation - No losses
between rise and fall Dvx15 mm/s
Courtesy of Jean Dalibard and David Guéry-Odelin
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ABCDx PROPAGATOR
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Quite generally, the phase shift along each arm
is
i.e. minus the time integral of the kinetic energy
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FOUNTAIN CLOCK
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Gravitational/Relativistic Doppler shift for
fountain clocksA quantum mechanical calculation
Langevin twin paradox
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Atom Interferometer
Laser beams
Atom beam
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Interféromètres atomiques
Jets atomiques
Faisceaux laser
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SATURATION SPECTROSCOPY
E(p)
E(p)
p
p
recoil doublet
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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

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Time-domain Ramsey-Bordé interferences with cold
Ca atoms
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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
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RELATIVISTIC PHASE SHIFTS
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Quite generally, the spin-independent part of
the phase shift is
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Atom Interferometers as Gravito-Inertial
Sensors I - Gravitoelectric field case
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
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Atom Interferometers as Gravito-Inertial
Sensors II - Gravitomagnetic field case
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
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DOPPLER-FREE TWO-PHOTON SPECTROSCOPY
E(p)
p
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RECOIL SHIFT IN DOPPLER-FREE TWO-PHOTON
SPECTROSCOPY
E(p)
p