Gravitational red shift Atomic clocks versus atomic ... · Claude Cohen-Tannoudji FRISNO 11 Aussois, 27 March 2011 Gravitational red shift Atomic clocks versus atomic gravimeters

Claude Cohen-Tannoudji

FRISNO 11

Aussois, 27 March 2011

Gravitational red shift

Atomic clocks versus atomic gravimeters

Collège de France

1

2

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

3

5: A recent controversy

6: Conclusion

4

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

5

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

6

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

7

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)

8

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)

9

Optical clocks

Outline





10


6: Conclusion

11

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

12

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

13

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

14

Recent results obtained by the NIST-Boulder group

Science, 319, 1808 (2008)

Outline





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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

16

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

17

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

18

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.

19

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

20

For a review of Feynman’s approach applied to interferometry, see:

P. Storey and C.C-T, J.Phys.II France, 4, 1999 (1994)

21

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





23


6: Conclusion

24

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

25

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( )

3

Outline





26


6: Conclusion

27

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]

28

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’

29

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.

30

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

31

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

32

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).

33

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)

34

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

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Gravitational red shift Atomic clocks versus atomic ... · Claude Cohen-Tannoudji FRISNO 11 Aussois, 27 March 2011 Gravitational red shift Atomic clocks versus atomic gravimeters