Atomic clocks with cold atoms

Atomic
Atomic clocks
clocks with
with cold
cold atoms
atoms
Pierre Lemonde
Bureau National de Métrologie – SYRTE (UMR CNRS 8630)
Observatoire de Paris, France
LASER COOLING and APPLICATIONS
September 2004
Les Houches, France
Accuracy
Accuracy of
of atomic
atomic clocks
clocks
SYRTE
Why
Why atomic
atomic clocks
clocks ??
•
Metrology
–
–
•
Fundamental physics
–
–
–
•
Measurements of fundamental
constants
Test QED
Test equivalence principle…
High Tech
–
–
•
Time scales
Most of the units in turn rely on the SI
second…
Navigation
Telecom networks
Atomic physics
–
–
–
Spectroscopy
Cold collisions
Funding…
Definition linked to SI seconds
Practical realisation with atomic clocks
Attempts to redefine
Overview
Overview
•
Introduction
– Principle of operation of atomic
clocks
•
Basics of signal analysis and
metrology
–
–
–
–
•
Phase/frequency
Different types of noise
Stability
Accuracy
Atomic fountains
– Principle of operation
– Lineshape computation
– Sources of noise in a
fountain/stability
– Accuracy overview, collisions
•
Next generation of atomic clocks
– Space projects
– Optical clocks
•
Example of applications
–
Atomic
Atomic clocks
clocks basics
basics
Frequency of an atomic transition
Atomic frequencies are universal and stable (equivalence principle) as opposed
to macroscopic oscillators (quartz cristal, pendulum, earth rotation...)
Idea of a "perfect" frequency
in practice
Pure AM
Time
Pure PM
s(t)
s(t)
s(t)
AM+PM
Time
Time
Signal
Signal analysis
analysis
Instantaneous frequency
Frequency stability: "magnitude" of relative frequency fluctuations
Frequency Stability is characterized with the Allan variance
typical frequency fluctuations averaged over
with
,
converges for most of the types of noise met experimentally
Experimentally
White
White frequency
frequency noise
noise
3
0
-
3
3
0
-
3
3
0
-
3
3
0
-
3
Flicker
Flicker frequency
frequency noise
noise
2
0
-
2
2
0
-
2
2
0
-
2
2
0
-
2
Frequency
Frequency accuracy
accuracy
The signal average frequency differs from the atomic frequency
is the sum of all possible frequency shifts (Zeeman, Stark, Collisions, Doppler...)
which are evaluated from frequency measurements
Accuracy: uncertainty on
Accuracy and stability are strongly linked
Noise of the measurements (stability) ultimately determines the accuracy
e.g.
10-15 is reached after 104 seconds (3 hours)
10-16 after 106 seconds (10 days)
An accuracy evaluation below 10-16 is impossible
Conversely, a good control of
the frequency shifts is necessary to
avoid long term degradation of the
frequency stability (flicker floor,
frequency drift)
Tjoelker et al. 1996
Atomic
Atomic clock
clock principle
principle
atomic resonance
macroscopic oscillator
1
correction
atoms
interrogation
0
atomic quality factor
-as high as possible (Cs: 10 GHz, Hg+: 1 PHz)
+ transition should be insensitive to
external perturbations
-low natural width (Cs: 10-13 Hz, Hg+: 2 Hz)
-Fourier limit, long interaction time (Cs: 1 Hz, Hg+: 6 Hz)
-low oscillator spectral width (Cs: 10-5 Hz, Hg+: 0.1 Hz)
-Large atom number (Cs: 107, Hg+: 1)
-low noise detection scheme
-low noise oscillator
Cs fountain (SYRTE): 10-14 τ -1/2, accuracy 7 10-16
Optical Hg+ (NIST): 4 10-15 τ -1/2, accuracy 10-14
Atomic
Atomic Fountains
Fountains
10 fountains in operation (SYRTE, PTB, NIST, USNO, PENN state, ON, IEN, NPL)
with an accuracy of ~10-15. Several projects (NIJM, NIM, KRIISS, VNIIFTRI...)
See e.g. Proc. 6th Symp. Freq. Standards and Metrology (World Scientific 2002)
BNM-SYRTE (F)
PTB (D)
NIST (USA)
Energy
Energy levels
levels Cs
Cs
133Cs
133Cs
hyperfine splitting is the basis for the
definition of the SI second.
Atomic
Atomic fountains:
fountains: Principle
Principle of
of operation
operation
Nat ~2×109
σr ~1.5−3mm
T ~1µK
∆V ~2 cm.s-1
4m.s-1
Vlaunch ~
H ~1m
T ~500ms
Tc ~0.8-2s
Selection
3
2
1
Detection
BNM
-SYRTE fountain
BNM-SYRTE
fountain ensemble
ensemble
Dual
Dual atomic
atomic fountain
fountain Rb+Cs
Rb+Cs
•
Transverse collimation of an atomic
beam
+ Chirp cooling
î 109 atoms loaded in 200 ms in
optical molasses
Operating parameters
•
•
•
•
•
•
•
Beam collimation
Nat ~ 109
σ ~ 3 mm
T ~ 1 µK
v ~ 4 m/s
H~1m
100 ≤ Tload ≤ 500 ms
1.1 ≤ Tcycle ≤ 1.5 s
AA key
key technique:
technique: The
The moving
moving molasses
molasses
Due to the first order Doppler
effect, the frequency of the
upper and lower laser beam is
the same in the frame moving
upwards at
For Cs:
Measurement
Measurement of
of the
the transition
transition probability
probability
7 0 0
e s ta te
f s ta te
6 0 0
a r b itr a r y u n its
5 0 0
J b/2
J
4 0 0
J b/2
s
3 0 0
2 0 0
1 0 0
0
0
2 0
4 0
tim e ( m s )
6 0
8 0
1 0 0
The area Ae and Af of the time of
flight signals in calculated. The
transition probability is given by
Response of the detection: area per atom
Technical noise ~ 400 atoms
time spent in
laser beam (s)
gain amp. photodiode
collection
(V.A-1)
response (A.W-1) efficiency
radiated
power (W)
vacuum
vacuum chamber
chamber and
and cavity
cavity
The 2 outermost shields are removed
Interrogation region with magnetic
and thermal shields
Compensation coils
Collimator for molasses beams
Optical molasses
optical
optical bench
bench
Laser diode based system, can be made compact and reliable
Acousto-optic
modulator
Optical fibers
Cs vapor cell
Injection
locked diode
laser
External cavity
diode laser
Frequency
Frequency synthesis
synthesis
Typical requirements:
- low phase noise, phase coherence
with the reference signal
- fine PC-controlled frequency tuning,
PC-controlled amplitude (P~ -60dBm)
- give access to « Zeeman »
transitions
- reduced microwave leakage
The power spectral density of the
fractional frequency fluctuations of the
synthesized signal is written
lineshape
lineshape calculation
calculation
Initial hypothesis: - external motion is described by a classical trajectory r(t)
- the two clock levels are spectrally resolved => 2 level atom
- any process causing relaxation is neglected
Hamiltonian:
(magnetic dipole interaction, gI, ge-2 neglected)
(purely stationary field)
We neglect non-resonant terms (rotating wave approximation)
We then rewrite the Hamiltonian in a new representation
Atomic
Atomic evolution
evolution using
using the
the Bloch
Bloch vector
vector
Hamiltonian in the new representation
detuning
Atomic observables are also transformed to the same representation
The Hamiltonian can be written as
Pauli matrices
We define the Bloch vector (or fictitious spin) as
Equation of evolution for the Bloch vector
We used Heisenberg’s
equations for sigma
Atomic observables
and
Rabi
Rabi oscillations
oscillations
The Bloch vector evolves according to
The motion is an instantaneous rotation around
Example: Rabi oscillation on resonance
(δ=0)
π/2 pulses:
Final state
3π/2 pulses:
Ramsey
Ramsey interrogation
interrogation
Time
1st interaction
free flight
2nd interaction
The final state depends mostly on the accumulated phase during the free flight between
the microwave field (ω) and oscillating atomic coherence (ωef):
As long as δ<<Ω0:
Ramsey fringes
One big advantage: most of the time, atoms evolve without being coupled to the exciting
field
Ramsey
Ramsey fringes
fringes in
in atomic
atomic fountain
fountain
1.0
1.0
0.8
transition probability P
0.6
0.4
0.8
0.94 Hz
0.2
0.0
-1.0 -0.5
0.0
0.5
1.0
0.6
0.4
NO AVERAGING
ONE POINT = ONE
MEASUREMENT OF P
0.2
0.0
-100
-50
0
50
100
detuning (Hz)
We alternate measurements on bothe sides of the central fringe to generate an error signal, which
is used to servo-control the microwave source
Fluctuations of the transition probability:
Sensitivity
Sensitivity function
function
Goal: to calculate the effect of temporal frequency fluctuations of the interrogation
oscillator (to first order)
Definition of the sensitivity function
It can be computed (and measured) by looking at the effect of a phase step during the interaction
The sensitivity function can be used (with care) to calculate the effect of a perturbation of
the atomic frequency
The sensitivity function depends on the interaction scheme. It gives simple way to estimate
frequency shifts or the effect of the noise of the interrogation even for complex cases (time
dependent frequency shifts, non-square pulses...).
Important remark: In true life, the sensitivity function is not exactly the same for all
atomic trajectories. In some cases, this must be accounted for.
Shape
Shape of
of the
the sensitivity
sensitivity function
function
For a single centered trajectory and in the limit δ<<Ω0, this is possible to derive an
analytical expression. For an optimized Ramsey interrogation, we have:
Sensitivity function in atomic fountains
for two different field amplitudes Ω0
In a fountain :
TE011 mode
The
The sensitivity
sensitivity function
function and
and frequency
frequency shifts
shifts
The equilibrium condition of the loop
is developed to first order using g(t)
The fractional frequency shift is then given by
Slightly modified if we account for
actual individual atomic trajectories
Frequency
Frequency noise
noise of
of the
the microwave
microwave source
source
Equation of the servo loop:
Sensitivity function:
Slope of Ramsey fringes
How the frequency fluctuations of the source are sampled and averaged:
Fractional frequency instability (for N>>1)
G. Dick, in Proc. PTTI, Redondo Beach, 1987, pp. 133–147.
G. Santarelli et al., IEEE Trans. UFFC 45, 887 (1998).
=>Limitation at the level of σy(τ)~10-13×τ -1/2 when microwave synthesis is based on a
very good quartz oscillator. Independent on the atom number.
Frequency
-maser
Frequency stability:
stability: Fountain
Fountain vs
vs H
H-maser
Rb fountain vs. H-maser
fractional frequency instability
Limiting term:
phase noise of
the quartz
oscillator
integration time τ(s)
maser autotuning
Typically, when using a very good quartz oscillator:
Frequency
Frequency stability:
stability: Fountain
Fountain vs
vs Fountain
Fountain
« Loose synchronization » (within a few seconds to a few minutes)
Cycle by cycle synchronization with matched sensitivity functions: cancellation of the
high frequency noise of the interrogation oscillator
Cryogenic
Cryogenic oscillator
oscillator
Sapphire cryogenic oscillator from the University of Western Australia
High quality factor sapphire resonator (Q ~ 4×109) at 12 GHz
Turning point of temperature sensibility: dν/dT = 0 near T=6K
SCO has extremely good short term stability
(measured at UWA against a second SCO)
SCO signal is converted to 9.2 GHz
A. Luiten, M. Tobar et al.
Quantum
Quantum projection
projection noise
noise
For a single atom: - transition probability P:
- Variance of quantum fluctuations of P:
For an ensemble of Ndet uncorrelated atoms:
Itano et al. PRA 47, 3554 (1993)
Spin squeezing: Itano, Polzik, Mölmer…
Contribution of the quantum projection noise to the frequency instability:
Contribution of the photon shot noise (an other quantum noise):
nph : number of detected photons per atom
Practically, nph ~ 100 and this contribution is a small (negligible) fraction of the quantum
projection noise
Quantum
Quantum projection
projection noise
noise
-2
σy(τ) Qatπ(τ/Tc)
Quantum
Limited
Operation of a
Cs Fountain up
to Nat∼ 6 105
1/2
10
0.91 N
-1/2
at
-3
10
4
10
5
10
Nat
For Nat∼ 6 105 atoms, the frequency stability is
σy(τ) ~ 4 10-14 τ-1/2
G.Santarelli et al. PRL,82,4619 (1999)
6
10
Frequency
Frequency stability
stability with
with aa cryogenic
cryogenic Oscillator
Oscillator
With a cryogenic sapphire
oscillator, low noise
microwave synthesis
(~ 3×10-15 @ 1s)
FO2 frequency stability
This stability is close to the quantum limit. A resolution of 10-16 is obtained after 6
hours of integration. With Cs the frequency shift is then close to 10-13!
Frequency
Frequency comparisons
comparisons between
between fountains
fountains
1E-14
-14
stabilité de fréquence
4.13 10
-14
2.83 10
1E-15
FO1
FO2
Difference
-14
5.01 10
1E-16
100
1000
10000
temps (s)
Fountain
Fountain Accuracy
Accuracy
Fountain (BNM-SYRTE)
Effect
FO2(Cs)
FO2(Rb)
FO1
Shift and uncertainty (10-16)
second order Zeeman
1927.3(.3)
3207.0(4.7)
1199.7(4.5)
-168.2(2.5)
-127.0(2.1)
-162.8(2.5)
-357.5(2)
0.0(1.0)
-197.9(2.4)
Residual Doppler effect
0.0(3.0)
0.0(3.0)
0.0(3.0)
Recoil
0.0(1.4)
0.0(1.4)
0.0(1.4)
Neighbouring transitions.
0.0(1.0)
0.0(1.0)
0.0(1.0)
-4.3(3.3)
0.0(2.0)
-3.3(3.3)
0.0(1.0)
0.0(1.0)
0.0(1.0)
6.5
7
7.5
Blackbody radiation
Collisions + cavity pulling
Microwave leaks, spectral purity,
synchronous perturbations.
Collisions with residual gaz.
Total
Frequency difference between FO1 and FO2: 3 10-16
Cold
Cold collisions:
collisions: orders
orders of
of magnitude
magnitude
Inter-atomic potential curves (87Rb)
1000
3
-long range: Van Der Waals interaction
Typical inter-atomic distance:
Typical De Broglie wavelength:
0
potential energy (K)
-size of the « molecular » region:
triplet state Σu
-1000
1
singlet state Σg
-2000
0.2
2+2
0.0
-0.2
-3000
1+2
-0.4
-4000
1+1
-0.6
-0.8
-5000
-1.0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
-6000
0.4
0.6
0.8
1.0
inter-nuclear distance (nm)
1.2
In general, collisions tend to be dominated by s-waves:
This is the case for 87Rb at 1µK. Collisions are then characterized by a single parameter
for each atomic state, the scattering length in the s-wave channel:
133Cs
is more complicated. At 1µK, d-waves must be taken into account due to low
energy molecular resonances (Feshbach resonances). Collisional properties become
independent on temperature only below 0.1nK
Typical phase space density: far from quantum degeneracy
Mean
-field energy
Mean-field
energy and
and collisional
collisional shift
shift
Total mean-field interaction energy as a function of scattering length aff, atomic density
n, atom number Nat and atomic mass m:
Energy conservation before/after the interaction (with only the clock states populated
and assuming 100% transfer efficiency)
Resulting clock shift:
In general (other states populated), the instantaneous frequency shift of the atomic
resonance is given by:
In the fountain, the time variation of the atomic density must be taken into account with
the sensitivity function
Collisional
Collisional shift
shift :: 87Rb
87Rb vs
vs 133Cs
133Cs
Differential
measurements
(interrogation
oscillator H-maser)
SYRTE
In a MOT
Cavity pulling is
taken into account
and subtracted
YALE
Th.
Atomic density changed by changing the trapping light intensity (40% uncertainty)
The shift in 87Rb is much (~50 to 100 times) smaller than in 133Cs
SYRTE Y. Sortais et al. , PRL, 85, 3117 (2000)
Th.1 = -11.7×10-23/ (at.cm-3) (S. Kokkelmans et. al., PRA 56, R4389 (1997))
Th.2 = -7.2×10-23/ (at.cm-3) (C.Williams et al.)
YALE -5.6 ( ± 3.3)×10-23/ (at.cm-3); (C.Fertig and K.Gibble, PRL, 85, 1622 (2000))
Pushing
5
6
Detuning (kHz)
Rabi Frequency (kHz)
Collisional
Collisional shift
shift measurement
measurement
4
2
0
-5
0
100%
0
1
2
3
Time (ms)
4
0
1
2
3
Time (ms)
4
50%
Sweeping
Launching
By performing the internal state selection with adiabatic
passage, we vary the atomic density by a factor of 2
without changing velocity and position distribution
|4,n-1>
|+>
|3,n>
|3,n>
|->
|4,n-1>
¾Cold collisions then only depend on the atomic number
To measure the cold collision shift in quasi real time
¾ We alternate measurements with full or interrupted
adiabatic passage (produces n or n/2)
F. Pereira Dos Santos et al., PRL
89,233004 (2002)
δ
The
The “adiabatic
“adiabatic passage
passage method”:
method”: Experiments
Experiments
The value and the stability of the ratio between 50% and 100% configurations is
monitored during each measurement. We find:
ME A
S UR
ED
We have a practical realization of two atomic
samples with a density ratio precisely equal to 1/2
Extrapolation of the collisional
shift:
Adiabatic
Adiabatic passage
passage method
method limitation
limitation
One of the possible limitation of the adiabatic passage method: off-resonant optical
pumping due to the pushing beam
Measured atom number ratios with adiabatic passage
The effect of spurious mF states is studied by
populating them purposely and selectively
With adiabatic passage and differential measurement,
the measured quantity is the ratio between
contributions of mF=0 states and the other mF state
⇒the atomic density cancels out
⇒possibility of precise comparison with theory
Effect
0 states
Effect of
of the
the m
mFF≠≠0
states
Differential method with 3 different configurations (period 60 cycles):
1. Full density on (0-0) 2. half density on (0-0)
3. Full density on (0-0) and on (mF - mF), (additional synthesizer for m F ≠ 0)
Ratio between the shift per atom due to the 0 states (ν00) and to the mF state
(νmFmF)
The contribution of m F ≠ 0 states are at
most 1/3 of the contribution of the clock
states
In the usual clock configuration:
3×10-3 residual populations in mF
states contributes to less than 1×10-3
¾The cold collisional shift can be
controlled at 10-3 of its value
Clock shifts due to |mF| = 3 are not
equal …
Feshbach
Feshbach resonances
resonances at
at low
low magnetic
magnetic field
field (Cs)
(Cs)
Ratio of frequency shift due to |mF| = 1; 2;
3 over clock states frequency shift plotted
as functions of the magnetic field.
Observation of the amplitude (all mF) and
the width (|mF| = 3) of Feshbach
resonances.
Calibration of the relative contribution of
the remaining mF ≠ 0 atoms on the clock
shift with respect to the B field
H. Marion et al. to be published
Black
Black body
body radiation
radiation shift
shift
Spectral density of blackbody radiation
Cs D1 and D2 lines
894 and 852 nm
Peak:1.7×1013 Hz (λ~17µm)
E. Simon et al. PRA 1997
Resulting frequency shift: most of the effect is the AC Stark shift
Sensitivity to temperature fluctuations (Cs): -2.3×10-16 K-1
Magnetic
Magnetic field
field
A static B field is applied on purpose to spectrally resolve de mF=0 to mF=0 transition:
(Earth field ~300mG)
sensitivity to field fluctuations: 9.3 10-16.nT-1
The B field is measured and made homogeneous using the
mF=+1 to mF=+1 transition
Zero crossings must be avoided along the cloud trajectory to
prevent Majorana transitions
Systematic
Systematic effects/evaluation
effects/evaluation methods
methods
Cold collisions + cavity pulling
-real-time measurement and extrapolation with adiabatic passage method
Black body radiation
-static polarizability, calculations, stabilization and measurement of T
Second order Zeeman effect
-proper shielding and homogeneity, spectroscopy of the “Zeeman” transitions
Microwave leaks
-careful design, characterization of synthesizers, π/2-3π/2-…
Microwave spectrum (spurious side bands, synchronous phase modulation,…)
-careful design and characterization of synthesizers, π/2-3π/2… measurements,
change timing and synchronization
Residual first Doppler effect (due to phase gradients in the Ramsey cavity)
-careful design of the cavity, sym.-assym., π/2-3π/2-…, change clock tilt angle
Ramsey pulling, Rabi pulling, Majorana transitions
-careful design and characterization of the B field, calculations
Background gas collisions
-earlier measurements
Recoil effect
-calculations
Second order Doppler effect
-order of magnitude => negligible
PHARAO:
PHARAO: aa Transportable
Transportable Fountain
Fountain
Measurement of 1S-2S transition of Hydrogen at Max Planck Institut für
Quantenoptik in Garching
cryostat
atomic hydrogen
chopper
Faraday cage
2S detector
vacuum chamber
x 2 243 nm
time resolved
photon counting
dye laser
f dye
x4/7
486 nm
ν1S-2S = 2 466 061 413 187 103 (46) Hz
Accuracy : 1.8 10-14
M. Niering et al, PRL 845, 5496 (2000)
x1/2
4/7 x f dye
1/2 x f dye
microwave
interaction
I
λ
70 fs Ti:sapphire
mode locked laser
9.2 GHz
cold atom
source
detection
Measurement of the Ry, test of QED,
fundamental constants stability.
Long
Long distance
distance comparison
comparison between
between PTB
PTB
and
and NIST
NIST Cesium
Cesium Fountains
Fountains
Frequency Comparison NIST F1 - CSF1
(period of overlap, date of measurement)
PTB: A. Bauch, S. Weyers
NIST: S. Jefferts
BIPM circular T
data base of
clock comparisons
using GPS or TWSTFT
(15 days, August 2000)
2
0
-2
10
15
x
y(F1 - CSF1)
4
-4
(20 days,
November 2001)
(10 days, July 2001)
-6
0
1
2
3
Number of Measurement
4
Atomic
Atomic fountains
fountains:: summary
summary
™
Microwave frequencies
• Cs 9.192 631 770 GHz
• 87Rb 6.834 682 610… GHz
™
Magneto-optical trap or molasses.
™
108 atomes, T ~1µK
™
Resonance width ~1Hz
™
Frequency stability: 1 10-14 τ -1/2
™
Frequency accuracy : 7 10 -16
Ultimate accuracy~10-16
Can we go further ?
Two
Two possible
possible ways
ways
atomic resonance
macroscopic oscillator
1
correction
atoms
interrogation
0
atomic quality factor
-as high as possible
-low natural width
-Fourier limit, long interaction time
-low oscillator spectral width
-Large atom number
-low noise detection scheme
-low noise oscillator
+ transition should be insensitive to
external perturbations
Atomic transition in the optical domain
A clock in space
Optical
Optical frequency
frequency standards
standards ??
Frequency stability :
Increase
(x 105)
Optical fountain at the quantum limit
!!!!!!!!!
Frequency accuracy: most of the shifts (expressed in absolute values) don't depend
on the frequency of the transition (Collisions, Zeeman...).
-Ability to compare frequencies (no fast enough electronics )
Three major difficulties
-Recoil and first order Doppler effect
-Interrogation oscillator noise conversion (Dick effect).
The best optical clocks so far exhibit frequency stabilities in the 10-15 τ -1/2 range together
with an accuracy around 10-14.
Optical
Optical frequency
frequency measurements
measurements
No electronics is fast enough to measure the phase of signals @ frequencies > ~100 GHz
Harmonic frequency chains:
-A set (about 13) of non linear stages, each of them performing a basic operation
(frequency multiplication by 2 or3, addition, difference).
-Uncomfortable regions of the spectrum (THz, MIR): alcohol lasers…
-1 apparatus for one particular transition (to within a few 10's GHz).
-very heavy and complex systems (need for secondary standards).
divide by 3 system
Only a few systems have been fully operated (JILA, PTB, SYRTE, NRC).
Only one achieved phase coherent comparison visible-RF
Schnatz H. et al. PRL 76 18 (1996)
Femto
-second frequency
Femto-second
frequency chain
chain
• temporal domain
E(t)
∆φ
2∆φ
t
τr.t = 1/fr
• frequency domain
I(f)
x2
x
f
0
All optical frequencies of the comb are known in terms of RF frequencies
J. Reichert et al. PRL 84, 3232 (2000), S. Diddams et al. PRL 84,5102 (2000)
Femtosecond
Femtosecond Laser
Laser
™
™
™
Kerr effect mode locked
Ti-Sa laser (25fs)
450 mW output
Spectral broadening
using a photonic crystal
fiber (up to 270 mw
output)
Self referenced
frequency comb
40µm
Locking
Locking scheme
scheme
40 MHz
Cryo or
H Maser
RF Synth
Synth.
™ Repetition rate
phase locked to H
maser or Cryo
9 GHz + 200 MHz
or 9.19 GHz
Rep. lock
11. har.
™ Offset frequency
detected, not locked
™ Output to OFS
fs laser
840 MHz
PCF
measurements
KTP
Off. freq.
Noise of the frequency comparisons (optical-optical or optical-microwave) ~ 10-15t-1/2
Accuracy of frequency ratios ~ 10-19
Ma et al. Science 303, 1843 (2004), Telle et al. PRL 2002
Bauch and Telle, Rep. Prog. Phys. 65, 789, 2002 and references therein
Recoil
Recoil effect
effect
Overlap of the wave packets after 1s of
free flight would imply temperatures ~ 1 fK
Ramsey-Bordé interferometer
Experimental realisation: PTB, NIST (Ca atoms)
C. Bordé et al. PRA 30 1836 (1984)
G. Wilpers et al., PRL 89 230801 (2002)
C. W. Oates et al. Opt. Lett. 25, 1603 (2000).
Ca
Ca optical
optical clock
clock
First order Doppler effect (wavefront curvature,
beam misalignement).
Ca Vs Maser
Ca Vs cavity
Ca Vs Hg+
Dick effect (very low duty cycle)
Trapped
-Dicke regime)
Trapped particles
particles (Lamb
(Lamb-Dicke
regime)
classical picture
To first order, no effect of the atomic motion
quantum picture
Momentum conservation
if the atom is confined to well below λ,
is a broad function of p
The atomic transition is not shifted or broadened (Recoil, Doppler)
Very detailed calculations in Wineland, Itano PRA 20, 1521 (1979)
For atoms, however, conservative traps require large trapping fields inducing large shifts of the levels
ions can be trapped with much lower fields, but with only a few ions in the trap for a clock
Single
Single ion
ion clock
clock
Hg+, NIST (Bergquist et al.)
Other experiments:
Yb+, PTB (Tamm, Peik...)
NPL : Yb+, Sr+, NRC : Sr+, MPQ : In+…
NIST
NIST Hg+
Hg+ Clock
Clock
R. Rafac, B. Young, J. Beall, W. Itano, D. Wineland and J. Bergquist, PRL 85, 2462 (2000)
Width
40 Hz
10 Hz
Quality factor: Q = ν/∆ν=1.6
Highest Q ever achieved
Current accuracy: 10-14
20 Hz
6 Hz
1014
++ Clock accuracy
NIST
Hg
NIST Hg Clock accuracy
Effect
Second order Zeeman effect
@ B=2.7 G
Black Body radiation shift
@ T=300K
Black Body radiation shift
@ T=4K
Second order Doppler
secular motion
Second order Doppler
micromotion
4
Background collisions ( He)
Quadrupole shift
-2
@ ∇E=10 V.mm
Correction Uncertainty
2.6
+1 386
+0.079
<0.08
0
0
<0.003
0.003
<0.1
0.1
0
0
<1
10
Quadratic sum
10 Hz
At PTB sub Hz comparison of Yb+ single ion clocks (Tamm et al. Proc. ICOLS 2003, arXiv 0403120)
Optical
Optical clock
clock with
with trapped
trapped atoms
atoms
•
Trap ions:
– Accuracy. Régime de Lamb-Dicke, Effects of traooing field
– Stability limited by Nat : one or a few ions
– Problem of the ultra-stable laser
•
Free atoms:
– Stability Nat~106, fast response time
– Accuracy limited by atomic motion: Doppler effect
•
Trapped atoms:
– Combine the advantages of both methods
– Trapping the atoms implies strong field ?
– Maintain coherence ?
AA future
future optical
optical standard
standard with
with
trapped
trapped Atoms
Atoms
3S
1P
1
461 nm
λtrap
1S
0
1
688 nm
λtrap
679 nm
689 nm
698 nm
(87Sr: 1 mHz)
3P
1
0
Combine advantages of single trapped ion and
Free fall neutral atoms optical standards
Katori, Proc. 6th Symp. Freq. Standards and Metrology (2002)
Pal’chikov, Domnin and Novoselov J. Opt. B. 5 (2003) S131
Katori et al. PRL 91, 173005 (2003)
Clock transition 1S0-3P0 transition (Γ =1mHz)
11S -33P transition
Sr
Sr S00- P00 transition
MOT fluorescence (%)
Free Falling atoms
100.0
99.8
ν1S0-3P0=429 228 004 235 (20) kHz
99.6
99.4
99.2
Courtillot et al., PRA 68, 030501 (2003).
99.0
-4
-3
-2
-1
0
1
2
3
4
698 nm laser detuning from resonance (MHz)
Trapped atoms
Takamoto et al. PRL 91, 223001 (2003)
Magic wavelength measurement (813 nm)
Very recently linewidth 50 Hz (Katori ICAP 2004)
Other experiements (JILA, PTB, LENS, NIST…)
Other atoms: Yb, Hg…
Einstein
Einstein Equivalence
Equivalence Principle
Principle
and
and the
the stability
stability of
of fundamental
fundamental constants
constants
In any free falling local reference frame, the result of a non
gravitational measurement should not depend upon when it is
performed and where it is performed.
EEP ensures the universality of the definition of the second
It implies the stability of fundamental constants: α=e2/hc,
me, mp,…
In particular: the ratio of the transition frequencies in different atoms and
molecules should not vary with space and time
The EEP can be tested by high resolution frequency
measurements regardless of any theoretical assumption
EEP revisited by modern theories: gµν ⇒ gµν,ϕ,…
Fundamental constants depend upon local value of ϕ : α(ϕ), m(ϕ),…
Violations of EEP are expected at some level !!
For instance: T. Damour, G. Veneziano, PRL 2002
Does
Does the
the fine
fine structure
structure constant
constant αα varies
varies ??
Because of large relativistic corrections,
the hyperfine energy of an alkali atom
depends upon Z and α=e2/ħc
Search method for α drift:
Compare hyperfine energy of rubidium
and cesium as function of time
Present
Present tests
tests of
of cosmological
cosmological
Variations
Variations of
of αα
– Oklo test : geochemical analysis of the natural fossil fission
reactor in Oklo (Gabon, 1.8×109 yr ago) :
α now − α Oklo ≤ 1×10 −7 α& α ≤ 5 ×10−17 yr −1
Damour, Poliakov, Nucl. Phys. B 480, 37 (1996)
– Absorption spectroscopy from quasars:
∆ α α = (− 0 . 72 ± 0 . 18 )×10
−5
( 0 . 5< z < 3 . 5 )
J. Webb et al., PRL 87, 091301 (2001)
Laboratory
Laboratory tests
tests versus
versus cosmological
cosmological tests
tests
•
•
•
•
•
•
•
•
A priori loss of factor 1010 in sensitivity !!
~ 1 year versus 1010 years
But:
ultra-stable and accurate clocks:
10-15 Æ10-16 -10-17
repeatable measurements
independent checks in various labs
choice of hyperfine, fine and optical transitions
See S. Karshenboim, Can. J. Phys 78 639, (2000), J.P. Uzan Rev. Mod. Phys., 75, 403 (2002)
Test
Test of
of Local
Local Position
Position Invariance
Invariance
Stability
Stability of
of fundamental
fundamental constants
constants (1)
(1)
[Following the work of Prestage et al., PRL 74, 3511 (1995)]
Frequency of hyperfine transitions as a function of fundamental constants (at lowest
order):
atomic unit of frequency
nuclear g-factor
purely numerical factor
electron to proton mass ratio
relativistic « correction »
factor, function of the fine
structure constant α
Frequency of electronic transitions as a function of fundamental constants (at lowest
order):
Ratio between atomic frequencies:
Sensitivity to variation of fundamental constants:
Test
Test of
of Local
Local Position
Position Invariance
Invariance
Stability
Stability of
of fundamental
fundamental constants
constants (2)
(2)
g(i)
[Following the work of V.V. Flambaum, ArXiv:physics/0302015 (2003)]
and mp are not “fundamental” constants but they can be linked (at least in principle)
to 2 fundamental parameters:
mass scale of QCD:
quark mass:
Any clock comparison can be seen as testing the stability of 3 fundamental parameters:
The K(i) are sensitivity coefficients: hfs:
elec:
molecular vibration:
With 4 well chosen atomic clocks, we constrain the stability of 3 independent frequency
ratios => independent constraint to the variation of
With more than 4 well chosen atomic clocks, redundancy is achieved
Test
Test of
of Local
Local Position
Position Invariance
Invariance
Stability
Stability of
of fundamental
fundamental constants
constants (3)
(3)
Comparisons between 87Rb and 133Cs hyperfine frequencies in atomic fountains over 5
years: early measurement with a fractional uncertainty
one data point on this graph corresponds to
1.3×10-14. Improvement by 104 at the time.
~2 months of measurements, with many
checks on systematic shifts
latest measurements with a fractional
uncertainty 1.3×10-15:
Stability of fundamental constants:
Marion et al. PRL 90, 150801 (2003)
Other recent comparison 199Hg+(elec) vs. 133Cs(hfs) at NIST:
Bize et al. PRL 90, 150802 (2003)
Test
Test of
of Local
Local Position
Position Invariance
Invariance
Stability
Stability of
of fundamental
fundamental constants
constants (4)
(4)
Other Measurements with comparable accuracy (H, Yb+)
Bize et al. PRL 90, 150802 (2003)
Fischer et al. arXiv:physics 0311128 and 0312086, PRL 2004
Peik et al., arXiv:physics 0402132 (2004)
AA FEW
FEW REFERENCES
REFERENCES
Vannier, Audoin, "the quantum physics of atomic frequency standards" Adam Hilger (1989)
A. Bauch, H. Telle,"frequency standards and frequency measurements" Rep. Prog. Phys. 65, 789, (2002)
S. Cundiff and J. Ye "femtosecond optical frequency combs", Rev. Mod. Phys. 75, 325 (2003)
Proc. 6th Symp. Freq. Standards and Metrology, World scientific, (2002)
www.bipm.fr
www.nist.gov
www.ptb.de
www.obspm.fr
….