Download Claude Cohen-Tannoudji Scott Lectures Cambridge, March 9 2011

Quantum Interference 2
Claude Cohen-Tannoudji
Scott Lectures
Cambridge, March 9th 2011
Collège de France
1
Outline of lecture 2
1: Introduction. Matter waves
2: Coherence length
3: A few experiments demonstrating the wave nature of atoms
4: Atomic interferometers
4: Two examples of atomic interferometers
2
Waves associated with a matter particle
1924
Louis de Broglie waves
dB = h / m v
1927
Experiments of Davisson and Germer
Electron microscopy
Diffraction of a beam of helium atoms by the surface of a
NaCl crystal
Estermann, Stern 1930
Diffraction and interference effects observed with neutrons
- Structure and dynamics of various types of media
- Interferometers using Bragg
diffraction of neutrons by 3
slices cut in the same crystal.
Development of new methods for making « components »
allowing one to manipulate atomic de Broglie waves and to
observe diffraction and interference effects with atoms.
3
Mirrors for atoms
z
Total reflection of a laser
beam giving rise to an
evanescent wave
If the laser is detuned to
the blue, the light shift is
positive and creates a
potential barrier U(z)
near the surface
If the total energy E of an
atom falling on the surface
is smaller then the height
U0 of the barrier, the atom
bounces on the surface
R.J. Cook, R.K. Hill , 1982
4
Description of external variables
Distribution of positions and momenta
R( r ) = r ˆ r
P( p) = p ˆ p
r : width of R( r ) Dispersion of positions
p : width of P( p) Dispersion of momenta
Case where the state of the center of mass can be described
by a wave function
Pure case : ˆ = 2
2
2
2
R( r ) = r = r
P( p) = p = p
The functions (r ) = r and ( p) = p are Fourier
transfoms of each other.
Warning!
2
2
is not the F.T. of P( p) = p R(r ) = r ()
( )
5
Outline of lecture 2
1: Introduction. Matter waves
2: Coherence length
3: A few experiments demonstrating the wave nature of atoms
4: Atomic interferometers
4: Two examples of atomic interferometers
6
ˆ Spatial coherences: r r Non diagonal elements of between 2 different points in space.
If corresponds to a pure case,
* r ˆ r = r r = ( r ) ( r )
ˆ
The argument of the complex number r r is equal to
the difference of the phases of in r and r Global spatial coherence at a distance
3
G( a) = d r r ˆ r + a
Sum of all spatial coherences between couples of
points separated by a distance
If ˆ = (pure case),
* 3
3
G( a) = d r r r + a = d r ( r ) ( r + a)
Overlap integral between the wave packet (r )
* and the wave packet (r ) tanslated by a
7
Connection between G( a)and P( p)
3
G( a) = d r r ˆ r + a
= d r d p d p r p p ˆ p p r + a
3
r p p r + a = 1 / 2 exp i p p .r / exp i p.a / 3
1 / 2 d 3r exp i p p .r / = p p
3
G( a) = d p P( p) exp i p. a / 3
(
)
(
3
)
3
(
(
)
)
(
(
(
)
)
)
The global spatial coherence is the Fourier transform of the
momentum distribution.
Coherence length is the width of G( a)
Do not confuse coherence length and spatial dispersion x!
For a free particle, x depends on time (spreading of the wave
packet) whereas p, and thus , are constants of the motion.
8
Optical analogy
Time correlation function of an optical field
G( ) = d t E(t)E * (t + )
Overlap integral of a wave packet with the wave packet
translated in time by –.
According to Wiener-Khinchine theorem, G() is the Fourier
transform of the spectral density ().
If the width of () is too small for allowing its accurate
measurement with a spectrometer, it can be more convenient
to measure G() by letting the wave trains interfere in a
Michelson interferometer with the same wave trains retarded
by an adjustable delay. One measures in this way G() and, by
Fourier transform, ().
Principle of Fourier spectroscopy.
9
Outline of lecture 2
1: Introduction. Matter waves
2: Coherence length
3: A few experiments demonstrating the wave nature of atoms
4: Atomic interferometers
4: Two examples of atomic interferometers
10
Young’s two slit experiment with atoms
v , v, v
Transverse collimation ensures
v / v 1 resulting in an improved
transverse coherence of the source
The fringe period is dictated
by the de Broglie wavelength
dB 1 / v
Number of fringes determined by
v / v i.e. the longitudinal
coherence of the source. This is why
supersonic beams and laser cooled
atoms are of interest : v / v 1
11
Interference fringes obtained
with the de Broglie waves of a supersonic
beam of metastable Helium atoms
O. Carnal, J. Mlynek
Phys. Rev. Lett.
66, 2689 (1991)
12
Interference fringes obtained
with the de Broglie waves associated
with metastable laser cooled Neon atoms
Cloud of cold atoms
F.Shimizu, K.Shimizu, H.Takuma Phys.Rev. A46, R17 (1992)
13
Experimental results
Each atom gives rise to a localized impact on the detector
The spatial repartition of the impacts is spatially modulated
Wave-particle duality for atoms
The wave associated with the atom allows one to
calculate the probability to find the atom at a given point
14
Increasing the size
of the interfering objects
15
Other examples of effects demonstrating
the wave nature of atomic motion
Frequency modulation of de Broglie waves
- Reflection of atomic wave packets by a mirror for atoms
(evanescent laser wave) vibrating at frequency .
- Frequency modulation of the reflected waves giving rise to
sidebands ± n (n=1,2,3…) in the energy spectrum of the
atoms bouncing off the mirror.
- Time of flight measurement of the energy spectrum
A. M. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard,
Phys. Rev. Lett. 74, 4972 (1995)
16
Experimental results
/2
0
950 KHz
880 KHz
800 KHz
Pure quantum effect
17
Outline of lecture 2
1: Introduction. Matter waves
2: Coherence length
3: A few experiments demonstrating the wave nature of atoms
4: Atomic interferometers
4: Two examples of atomic interferometers
18
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
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
e,p+k
g,p g,p
k
tan =
p
C. Bordé, Phys. Lett. A 140, 10 (1989)
If the interaction time corresponds to a pulse, the incident state
g,p is transformed into e,p+k
19
Extension of Ramsey fringes to the optical domain
Simplest idea: use 2 laser beams
e,p+k
z
e,p+k
Difficulty
g,p
e,p+k
g,p
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
20
Two examples of interferometers
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) 21
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.
22
Quantum propagator K in space time
(
)
K zbtb , z a ta = Probability amplitude for the particle to
arrive in zb at time tb if it starts from z a at time ta
Feynman has shown that K can be also written:
K zbtb , z a ta = N exp i S / N : Normalization coefficient
(
)
(
)
: Sum over all possible paths connecting z a ta to zbtb
tb
S : Action along the path : S = L z(t), z(t) d t
ta
L : Lagrangian
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 z a ta zbtb for which the action, Scl is extremal
(
)
{ (
) }
K zbtb , z a ta = F(tb ,ta ) exp i Scl zbtb , z a ta / F(tb ,ta ) : independent of z a and zb
For a review of Feynman’s approach applied to interferometry, see:
P. Storey and C.C-T, J.Phys.II France, 4, 1999 (1994)
23
Expression of the phase shift
i
zbtb = d z a z a ta exp Scl zbtb , z a ,ta Analogy with Fresnel-Huygens
principle (here in space-time)
( )
( )
(
)
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
z
zb
N
p0
z0
cl
M0
ta
tb
t
M0 : Point of the plane t = ta such that the classical path M0 N
has an initial momentum p0 in M0
( )
{ ( ) }
( )
) }
(
)
( )
N M 0 exp i Scl N , M 0 / For an incident particle with momentum p0 , N is given by
N M 0 exp i Scl N , M 0 / where Scl N , M 0 is the action
along the classical trajectory M 0 N having a momentum p0 in M 0
The curves representing the paths I and II are classical trajectories
used for calculating the phase shifts
( )
( )
{ (
24
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)
5.25
Outline of lecture 2
1: Introduction. Matter waves
2: Coherence length
3: A few experiments demonstrating the wave nature of atoms
4: Atomic interferometers
4: Two examples of atomic interferometers
26
Atom in a gravitational field
Kasevich-Chu (KC) interferometer
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
Quadratic Lagrangian L( z, z ) = m z2 / 2 mgz Feynman's approach
z
A0C0B0D0A0
Unperturbed paths (g = 0)
Straight lines
ACBDA
Perturbed paths (g 0)
Parabolas
Free fall of atoms
DD0=CC0=gT2/2
BB0=2gT2
0
T
2T
t
k
D0C0 =
T
m
27
Calculation of the phase shift
Propagation along the perturbed trajectories
1
prop = [ Scl (AC) + Scl (CB) Scl (AD) Scl (DB)]
The classical action along a path joining
za,ta to zb,tb can be exactly calculated
m ( zb za )
mg
mg 2
3
Scl ( za t a , zb tb ) =
zb + za ) ( tb t a ) tb t a )
(
(
24
2 tb t a
2
2
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
28
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 = mc 2 / h 3.2 10 25 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]
29
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’
30
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 contained in devices (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.
31
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
32
Atom in a rotating frame
Ox yz : Galilean frame R Oxyz : Rotating frame R
= ez
1 2
Free particle in R L = mv Lagrangian
2
v = v + r
Relation between v and v
Lagrangian L in R
L(r, v) = L(r , v )
1
1
2 1 2
L(r, v) = m v + r = mv + m. ( r v ) + m r
2
2
2
(
)
(
)
2
Calculation of the phase shift
Since the Lagrangian is quadratic, one can use Feynman’s approach
leading to m
rot = 2
S
S : surface between the 2 arms of the interferometer
Sagnac effect for matter waves
33
Comparison of the rotational phase shifts
for atoms and for light
atoms
2mS / mc 2
rot
=
=
1
light
2
2S L / c
rot
L
L / 2 : frequency of light
This result is true only if the 2 interferometers have the same surface.
In fact, atomic gyrometers have a much smaller surface than laser
interferometers. Furthermore, laser gyros use optical fibers which can
make several turns and enclose a much larger surface
First experimental observation of the Sagnac effect for atoms
F. Riehle, Th. Kisters, A. Witte, J. Helmcke, C. Bordé P.R.L. 67, 177 (1991)
Atomic gyros reach now a sensitivity of 6 x 10-10 rad/s for an integration
time of 1s (the rotation speed of the earth is 7.3 x 10-5 rad/s)
T. L. Gustavson, A. Landragin, and M. A. Kasevich,
Class. Quantum Grav. 17, 2385 (2000).
34
Ramsey Bordé interferometers
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.
The calculation of the phase shift is straightforward. One finds that the
probability that the atom exits in e is given by: 1 1
P(e) = cos 2 L A R T
R = k 2 / 2m
4 8
Another interferometer leading to an exit of the atom in e (
)
(
)
1 1
P(e) = cos 2 L A + R T
4 8
35
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!)
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
2Ry mproton matome h
2
=
c melectron mproton matome
D. S. Weiss, B. C. Young, and S. Chu, Phys. Rev. Lett. 70, 2706 (1993).
36
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)
37
Conclusion
Young’s interference fringes may be observed with the de Broglie
waves associated with atoms and large molecules
Tests of quantum mechanics with mesoscopic systems
Ramsey fringes can be observed in the optical domain using
beam splitters based on the atomic recoil. Quantum interference
using an interplay of internal and external atomic variables
Atomic interferometers reach a very high sensitivity
They are useful for
- Basic studies (test of the universality of free fall, measurement
of the fine structure constant)
- For practical applications (gravimeters, gyrometers)
In space, they could be used, in combination with atomic clocks,
for testing the gravitational field at very large distances
38