Gravitational red shift Atomic clocks versus atomic ... · Claude Cohen-Tannoudji FRISNO 11...
Transcript of Gravitational red shift Atomic clocks versus atomic ... · Claude Cohen-Tannoudji FRISNO 11...
Claude Cohen-Tannoudji
FRISNO 11
Aussois, 27 March 2011
Gravitational red shift
Atomic clocks versus atomic gravimeters
Collège de France
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Gravitational red shift
One of the most important predictions of Einstein’s theory of
general relativity (GR)
Two clocks located at two different gravitational potentials do
not oscillate at the same frequency
Purpose of this lecture
Discuss the potentialities of atomic clocks using ultracold
atoms for measuring this red shift and testing GR
Discuss the arguments presented in a recent publication
according to which atomic interferometers using ultracold
atoms could be also considered as clocks oscillating at the Compton frequency = mc2/h (where m is the rest mass of the
atom) and could measure the red shift with a higher precision
Explain why we don’t agree with this interpretation
Outline
1: Atomic clocks with ultracold atoms
3: Atomic interferometry
4: Atomic gravimeters
2: Tests of the red shift with atomic clocks
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5: A recent controversy
6: Conclusion
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Orders of magnitude
T: Time of flight of the atoms between the 2 cavities
The width of the central Ramsey fringe is on the order of
1 / 2T ~ v / 2L. If L=0.5 m, v=100 m/s, one gets 100Hz
How to go farther?
Instead of increasing L for diminishing v/L, one can try to
diminish v by using ultracold atoms.
Cesium atomic clocks
Principle
The microwave
oscillator is
locked to the frequency of the
central Ramsey
fringe
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Improving atomic clocks with ultracold atoms
Fountains of ultracold atoms
H = 30 cm T = 0.5 s
The width of the Ramsey fringes is 100 times smaller than
the width obtained with thermal atoms
H Throwing this cloud of ultracold atoms
upwards with a laser pulse to have them
crossing the same cavity twice, once in the way up, once in the way down, and obtaining
in this way 2 coherent interactions separated
by a time interval T
Cloud of atoms cooled by laser cooling
to temperatures on the order of 1 μK
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A relative accuracy of 10-16 corresponds to an error smaller
than 1 second in 300 millions years!
Examples of atomic fountains
- Sodium fountains : Stanford S. Chu
- Cesium fountains : BNM/SYRTE C. Salomon, A. Clairon
Stability : 1.6 x 10-16 for an integration time 5 x 104 s
Relative accuracy : 3 x 10-16
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ACES project (ESA, CNES)
From terrestrial clocks to space clocks
• Time reference and global
clock comparison
• Validation of space clocks
• Fundamental tests
(General relativity, Variation
of fundamental constants)
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Optical clocks
Quality factor of the resonance
Q = / = T
Q increases considerably when the frequency is changed from microwave to optical
Two types of optical clocks are being studied
1 – Single ion optical clocks
A single ion is trapped, cooled and a very narrow
optical transition connecting the ground state to a long lived excited state is used as the clock transition
2 – Neutral atoms in an optical lattice
The light producing the lattice has a frequency such that
the light shifts of the 2 states of the clock transition are the same (H. Katori)
Outline
1: Atomic clocks with ultracold atoms
3: Atomic interferometry
4: Atomic gravimeters
2: Tests of the red shift with atomic clocks
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5: A recent controversy
6: Conclusion
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Gravitational shift
of the frequency of a clock
An observer at an altitude z receives the signal of a clock
located at the altitude z+ z and measures a frequency A(z
+ z) different from the frequency, A(z), of his own clock
2 clocks at altitudes differing by 1 meter have apparent
frequencies which differ in relative value by 10-16.
A space clock at an altitude of 400 kms differs from an earth clock by 4 x 10-11 . Possibility to check this effect with a
precision 50 times better than all previous tests with rockets
Another possible application : determination of the “geoid”,
surface where the gravitational potential has a given value
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Previous tests of the gravitational red shift
Pound and Rebka experiment PRL 4, 337 (1960)
The gamma ray emitted by one solid sample of iron (57Fe) is
absorbed by another identical sample located 22.5 m below
Very narrow line width (Mössbauer effect) allowing one to
measure the red shift between the emitter and the receiver
This red shift is measured by moving the emitter in order to
introduce a Doppler shift compensating the red shift
Uncertainty on the order of 10-2
Hydrogen maser launched on rocket at an altitude of 10,000 km
with its frequency compared with another maser one earth
Uncertainty on the order of 10-4
Vessot experiment Gen. Rel. and Grav. 10, 181 (1979)
ACES clock in space
Expected uncertainty on the order of 2x10-6
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Laboratory tests
of a possible variation of fundamental constants
Because of relativistic corrections, the hyperfine structure of
an alkali atom depends on the fine structure constant and on
the atomic number Z.
J. Prestage, R. Tjoelker, L. Maleki, Phys. Rev. Lett. 74, 3511 (1995)
Other (non laboratory) tests
• Natural nuclear reactor of Oklo (Gabon)
• Absorption spectroscopy of the light emitted by
distant quasars
By comparing the hyperfine frequencies of cesium and
rubidium measured with 2 fountains, and by following the
evolution of their ratio over several years, on can put an upper bound on
Outline
1: Atomic clocks with ultracold atoms
3: Atomic interferometry
4: Atomic gravimeters
2: Tests of the red shift with atomic clocks
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5: A recent controversy
6: Conclusion
Beam splitter for atomic de Broglie waves
A plane de Broglie wave corresponding to atoms in the ground state
g with momentum p crosses at right angle a resonant laser beam
The atom-laser interaction time, determined by the width of the laser
beam, is chosen for producing a /2 pulse
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If the interaction time corresponds to a pulse, the incident state
g,p is transformed into e,p+ k
After the laser beam the incident de Broglie wave is transformed
into a coherent linear superposition with equal weights of 2 de
Broglie waves g, p and e, p+ k
g,p g,p
e,p+ k
tan =
k
p
C. Bordé, Phys. Lett. A 140, 10 (1989)
Extension of Ramsey fringes to the optical domain
Simplest idea: use 2 laser beams
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z
g,p g,p
e,p+ k
e,p+ k
e,p+ k
Difficulty
The 2 final wave packets have the same momentum, the same
internal state e, but are spatially displaced by an amount kT/m
(T : flight time from one wave to the other)The velocity dispersion v along the z-axis gives rise to a coherence
length = /m v and the 2 wave packets cannot interfere if
kT /m = /m v v T 1 / k = / 2
A possible solution
Add extra beams to recombine the 2 wave packets
Two examples of interferometers
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Mach-Zehnder interferometer
Ramsey Bordé interferometer
C. Bordé, C. Salomon, S. Avrillier, A. Van Lerberghe, C. Bréant, D. Bassi, S. Scolès, Phys.Rev A30,1836 (1984)
Atomic interferometry
Input
Output
At the output of an atomic interferometer the wave function of the
outgoing beam is a linear superposition of 2 wave functions
corresponding to 2 possible paths which can be followed by atoms
Can we calculate the phase shift between the 2 wave functions due to
various causes (free propagation, laser, external or inertial fields)?
The 2 possible paths are represented in the figure above by lines
which suggest trajectories of the particles. These trajectories have
no meaning in quantum mechanics. Can we express the phase
shifts of the wave functions as integrals over classical trajectories?
Using a Feynman path approach, one can show that this is possible
in situations which are realized for most atomic interferometry
experiments. We just give here the main results of this approach.
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Quantum propagator K in space time
K zbtb, z
at
a( ) = N exp i S /( ) N : Normalization coefficient
: Sum over all possible paths connecting zat
ato z
btb
S : Action along the path : S = L z(t), z(t)ta
tb
d t
L : Lagrangian
K zbtb, z
at
a( ) =Probability amplitude for the particle to
arrive in zb
at time tb
if it starts from za
at time ta
Feynman has shown that K can be also written:
If L is a quadratic function of z and z,one can show that the sum
over reduces to a single term corresponding to the classical
path zata
zbtb for which the action, S
clis extremal
K zbtb, z
at
a( ) = F(t
b,t
a) exp i S
clz
btb, z
at
a( ) /{ }
F(tb,t
a) : independent of z
aand z
b
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For a review of Feynman’s approach applied to interferometry, see:
P. Storey and C.C-T, J.Phys.II France, 4, 1999 (1994)
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Expression of the phase shift
For an incident particle with momentum p0, N( ) is given by
N( ) M0( ) exp i S
clN , M
0( ) /{ } where Scl
N , M0( ) is the action
along the classical trajectory M0
N having a momentum p0
in M0
ta tb
z0
zb
z
t
M0
N
p0
cl
zbtb
( ) = dza
zat
a( ) exp
iS
clz
btb, z
a,t
a( )
Analogy with Fresnel-Huygens
principle (here in space-time)
M0: Point of the plane t = t
a such that the classical path M
0N
has an initial momentum p0
in M0
N( ) M
0( ) exp i Scl
N , M0( ) /{ }
If Scl >> , and if the initial state of the
particle at t = ta is a plane wave with
momentum p0, the integral over za
reduces to a single term
The curves representing the paths I and II are classical trajectories
used for calculating the phase shifts
5.22
Two important remarks
1. The same Lagrangian must be used for calculating the
classical trajectories and the classical action along these
trajectories. If two different Lagrangians are used, the principle of
least action is violated. The phase shift is not correct, so
that the wave function no longer obeys Schrödinger
equation. Quantum mechanics is violated.
2. The two classical lines representing the 2 paths in the
interferometer cannot be determined by a measurement.
In an interferometer, where a single atom can follow two different paths, trying to measure the path which is
followed by the atom destroys the interference signal
(wave-particle complementarity)
Outline
1: Atomic clocks with ultracold atoms
3: Atomic interferometry
4: Atomic gravimeters
2: Tests of the red shift with atomic clocks
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5: A recent controversy
6: Conclusion
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Atom in a gravitational field
Quadratic Lagrangian L( z,z ) = m z2 / 2 mgz Feynman's approach
z
t 0 T 2T
A0C0B0D0A0
Unperturbed paths (g = 0)
Straight lines
ACBDA
Perturbed paths (g 0)
Parabolas
Free fall of atoms
DD0=CC0=gT2/2
BB0=2gT2
D0C0 =k
mT
M. Kasevich, S. Chu P.R.L. 67, 181 (1991)
The 2 interfering paths propagate at different heights. The phase shift
is thus expected to depend on the gravitational acceleration g
Kasevich-Chu (KC) interferometer
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Calculation of the phase shift
prop =
1Scl(AC)+ Scl(CB) Scl(AD) Scl(DB)[ ]
Propagation along the perturbed trajectories
Using this equation, one finds prop = 0 Exact result
Phase shift due to the interaction with the lasers
Because of the free fall, the laser phases are imprinted on the
atomic wave function, not in C0,D0,B0, but in C,B,D
This phase shift is expected to scale as the free fall in units of
the laser wavelength, i.e. as gT2/ , on the order of kgT2
Result of the calculation laser
= kgT 2
The lasers act as rulers which measure the free fall of atoms
The classical action along a path joining
za,ta to zb,tb can be exactly calculated
Sclzata ,zbtb( ) =
m
2
zb za( )2
tb ta
mg
2zb + za( ) tb ta( )
mg2
24tb ta( )
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Outline
1: Atomic clocks with ultracold atoms
3: Atomic interferometry
4: Atomic gravimeters
2: Tests of the red shift with atomic clocks
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5: A recent controversy
6: Conclusion
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A recent proposed re-interpretation of this experiment
H. Müller, A. Peters and S. Chu, Nature, 463, 926 (2010)
The atom is considered as a clock ‘ticking” at the Compton frequency
C / 2 = mc2 / h 3.2 1025 Hz
The “atom-clock” propagates along the 2 arms of the interferometer at
different heights and experiences different gravitational red shifts along
the 2 paths leading to the phase shift measured by the interferometer
In spite of the small difference of heights between the 2 paths, the
huge value of C provides the best test of Einstein’s red shift
We do not agree with this interpretation
P. Wolf, L. Blanchet, C. Bordé, S. Reynaud, C. Salomon and
C. Cohen-Tannoudji, Nature, 467, E1 (2010)
See also a more detailed paper of the same authors, submitted to
Class. Quant. Gravity and available at arXiv:1012.1194v1 [gr-qc]
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Our arguments
• The exact quantum calculation of the phase shift due to the
propagation of the 2 matter waves along the 2 arms gives zero.
The contributions of the term –mgz of L (red-shift) and mv2/2
(special relativistic shifts) cancel out. The contribution of the term
mv2/2 cannot be determined and subtracted because measuring
the trajectories of the atom in the interferometer is impossible.
• The phase shift comes from the change, due to the free fall, of the
phases imprinted by the lasers. The interferometer is thus a
gravimeter measuring g and not the red shift. The value obtained for
g is compared with the one measured with a falling corner cube
• The interest of this experiment is to test that quantum objects, like
atoms, fall with the same acceleration as classical objects, like
corner cubes. It tests the universality of free fall.
• If g is changed into g’=g(1+ ) to describe possible anomalies of the
red shift, and if the same Lagrangian, which is still quadratic, is used
in all calculations, the previous conclusions remain valid. The signal
is not sensitive to the red shift. It measures the free fall in g’
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Comparison with real clocks
• The red shift measurement uses 2 clocks A and B located at different
heights and locked on the frequency of an atomic transition. The 2
measured frequencies A and B are exchanged and compared.
• The 2 clocks are in containers (experimental set ups, rockets,..)
that are classical and whose trajectories can be measured by radio or
laser ranging. The atomic transition of A and B used as a frequency
standard is described quantum mechanically but the motion of A and B
in space can be described classically because we are not using an
interference between two possible paths followed by the same atom
• The motion of the 2 clocks can thus be precisely measured and the
contribution of the special relativistic term can be evaluated and
subtracted from the total frequency shift to get the red shift
• In the atomic interferometer, we have a single atom whose wave
function can follow 2 different paths, requiring a quantum description of
atomic motion. The trajectory of the atom cannot be measured.
Nowhere a frequency measurement is performed.
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Tests of alternative theories
Most alternative theories use a modified Lagrangian L with parameters
describing corrections to –mgz due to non universal couplings
between gravity and other fields (for example, electromagnetic and
nuclear energies may have different couplings)
If the same Lagrangian L is used in all calculations, our analysis can
be extended to show that the Kasevich Chu (KC) interferometer
measures the gravitational acceleration (test of UFF) whereas atomic
clocks measure the red shift (test of UCR).
Both tests are related because it is impossible to violate UFF without
violating UCR (Schiff’s conjecture). They are however complementary
because they are not sensitive to the same linear combinations of the
i (atomic clock transitions are electromagnetic, but the Pound-Rebka
experiment uses a nuclear transition.)
If the trajectories are not calculated with the same Lagrangian L as the
one used for calculating the phase shift in the KC interferometer, the
phase shift could measure the red shift, but at the cost of a violation of
quantum mechanics. A new consistent reformulation of the theory is
then needed to explain how to calculate the phase shift
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Ramsey Bordé interferometers
P(e) =
1
4
1
8cos2
L A R( )T R= k
2 / 2m
The phase shift between the 2
paths also depends on the
difference of internal and external
energies of the 2 states between
the 2 lasers of each pair because
this interferometer is not symmetric
as the KC interferometer.
P(e) =1
4
1
8cos2
L A+
R( )T
The calculation of the phase shift is straightforward. One finds that the
probability that the atom exits in e is given by:
Another interferometer leading to an exit of the atom in e
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Ramsey Bordé interferometers (continued)
The probability that the system exits the interferometer in state e is thus
given by 2 systems of Ramsey fringes centered in L= A+ R and
L= A- R. This interferometer can now be considered at a clock since it
delivers a signal from which one can extract an atomic frequency A
which is in the microwave or optical domain (not at Compton frequency!)
2=
2Ry
c
mproton
melectron
matome
mproton
h
matome
To measure the red shift with such interferometers, one would need to
build two of them, to put them at different heights and to compare the
2 values of A that they deliver
Using this interferometer for measuring h/m and then
From the 2 systems of Ramsey fringes, one can also extract R and
then a better value of the ratio h/m. This improves the determination of
the fine structure constant which can be written
D. S. Weiss, B. C. Young, and S. Chu, Phys. Rev. Lett. 70, 2706 (1993).
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Most recent measurement of by a variant of this method
-1 = 137.035 999 037 (91)
P. Bouchendira, P. Cladé, S. Guellati, F. Nez, F. Biraben
PRL, 106, 080801 (2011)
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Conclusion
Atomic clocks with ultracold atoms are now used in all institutes
of metrology. They have reached an impressive relative accuracy
(10-17) allowing them to perform very precise tests of basic theories (red shift, variation of fundamental constants,…)
Their very high sensitivity to the gravitational field clearly shows
that a universal time reference should now be delivered, not by
clocks on earth, but by clocks in space
Atomic interferometers reach a high sensitivity and are useful for:
- Basic studies (test of the universality of free fall, measurement
of the fine structure constant) - Practical applications (gravimeters, gyrometers)
We don’t think that atomic interferometers can be considered as
clocks oscillating at the Compton frequency. Atomic clocks and
atomic interferometers provide different and complementary tests of GR which need to be both pursued with equal vigor