Quantum Interference 1 Claude Cohen-Tannoudji Scott Lectures Cambridge, March 7

Quantum Interference 1
Claude Cohen-Tannoudji
Scott Lectures
Cambridge, March 7th 2011
Collège de France
1
Description of the state of an atom
by populations of the energy levels
Atom in thermodynamic equilibrium
The state is completely described by the populations n of the
energy levels En.
n = E n E n e En / k B T
n : diagonal elements of the density operator in the basis
{En}. The off diagonal elements are equal to zero.
: statistical mixture of states En with weights n.
Master equation for the populations
The time evolution of the populations n of the states En can
often be described in terms of transition rates Wnn‘ from one
state to another one:
n = Wn n n + Wnn n
n n
n n
2
Linear superpositions of states
An atom can also be prepared in a linear superposition of the
states En.
= cn En
n
The physical properties of this atom then depend not only on
the IcnI2, but also on the crossed terms cncn’* (with nn‘ ).
If the state of the atom is not a pure state like , but a
statistical mixture of states, the density matrix can have non
diagonal elements nn‘ (with nn‘).
En En = n n = cn cn* 0
cn cn* : average over the states of the statistical mixture.
The relative phase of the coefficients cn and cn‘ does not
average to zero.
nn‘ is often called “coherence” between the states En and En‘
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Purpose of this series of lectures
Review a few physical effects associated with the existence of
atomic coherences ij
•How to prepare and detect these coherences?
•How do they evolve?
•What are the new effects to which they give rise?
•What are the possible applications?
The states which are linearly superposed can be
- two Zeeman sublevels of the same atomic level e or g in
which case ij is called a “Zeeman coherence”
- two different states corresponding to two different spatial
localizations of the atom, in which case ij is called a
“spatial coherence”
- two different states of an ensemble of 2 systems 1 and 2.
If this state 12 cannot be written as a product of a state 1
of 1 by a state 2 of 2, we have an “entangled state” with
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remarkable properties
Lecture 1 : Zeeman coherences
•Level crossing resonances in excited and ground states
•Quantum beats. Time resolved spectroscopy
•Ramsey fringes and microwave atomic clocks
•Dark resonances. Coherent population trapping
Lecture 2 : Spatial coherences
•Coherence length
•Young’s fringes with atomic de Broglie waves
•Feynman path approach to atomic interferometry
•Atomic interferometers
Lecture 3 : Entangled states
•Examples of entangled states. How to prepare them
•Entangled states and non separability. Bell’s inequalities
•Entangled states and which path information
•Entangled states and measurement process. Decoherence
•Entangles states and two-photon interference
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Outline of lecture 1
1 – Level crossing resonances in excited states
2 – Quantum beats. Time resolved spectroscopy
3 – Extension to ground states
4 – Ramsey fringes and microwave atomic clocks
5 – Dark resonances. Coherent population trapping
6
Level crossing resonance in the excited state e
E
e1
B0
Excitation with
polarization e2
B0C
Detection with
polarization '
B0
A static magnetic field B0 is scanned around a value B0C
where e1 and e2 cross (Ee1=Ee2), for example in B0=0.
Optical excitation with polarization .
Detection of the fluorescence with polarization '
et ' are linear superpositions of the polarizations 1 and 2
associated with transitions g-e1 and g-e2 .
How does the intensity of the fluorescence vary with B0?
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Interferences between scattering amplitudes
g g e1
e2
g
g
There are two different pathes for going from the same initial state g, to the
same final state g, , the first one passing through e1 , the second one through e 2
They are both open because and are linear superpositions of the
polarizations of the transitions g e1 and g e2
The transition amplitude is a sum of 2 amplitudes, one for each path
A
(
e g
)
e g = Ee Eg / 1
1
1
A1
A2
+
+ i ( e / 2) e g + i ( e / 2)
(
)
2
e g = Ee Eg / 2
2
e : Natural width of e
Both amplitudes can be simultaneously large if e g = e g
Resonant variations of the fluorescence near the crossing point
1
2
8
Equation of evolution of 12
The rate of variation of 12 is given by a sum of 3 terms:
- A term describing the excitation process. For a broad line excitation,
it is given by
d 12 / d t exc = W gg
(
)
W: rate coefficient proportional to the light intensity. For low intensity,
one can neglect the depletion of g and replace gg by 1
- A term describing the free evolution (Larmor precession)
E1 E2
d 12 / d t free evol = i
12 = i 0 12 with E2 E1 = 0
(
)
- A term describing the departure rate from e with a rate (d
12
/ dt
)
spont em
= 12
Adding these 3 rates, one gets
d
12 = W + i 0 12 12
dt
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Steady-state solution
If the light intensity is constant, W is time independent and the
equation of evolution of -1+1 has a steady-state solution:
st
1+1
W
=
i 0
The detection signal contains a term proportional to Re -1+1 equal to
W
2 + 02
which is a Lorentz curve centered around B0=0, with a width much smaller than the spectral width of the exciting light
This zero field level crossing resonance has been observed for the
first time by W.Hanle and it is called the “Hanle effect”
W.Hanle, Z.Phys. 30, 93 (1924) and 35,346 (1926)
Similar level crossing resonances have been also observed around
the non zero magnetic field value where two Zeeman sublevels
originating from 2 different hyperfine levels cross. This is called the
“Franken effect”.
F.Colegrove, P.Franken,R.Lewis,R.Sands, Phys.Rev.Lett. 3, 420 (1959)
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Outline of lecture 1
1 – Level crossing resonances in excited states
2 – Quantum beats. Time resolved spectroscopy
3 – Extension to ground states
4 – Ramsey fringes and microwave atomic clocks
5 – Dark resonances. Coherent population trapping
11
Pulsed excitation. Quantum beats
The rate coefficient W is then no longer constant and must be
replaced by a delta function of t in the equation of evolution of 12
d 12 / d t = W0 (t) i 12 12 12
The solution of this equation is given by:
12 (t) = W0 (t) e i 12 t e t
(t): Heaviside function
The excitation prepares the coherence 12 in a very short time.
This coherence then evolves freely at the frequency 12 and is
damped with a rate .
This evolution is detected as a damped oscillation of the
intensity of the fluorescence light which contains a term
proportional to 12.
Analogy with the sound beats which are heard after a short
percussion of 2 different piano strings
12
First observation of quantum beats
Excited states of cadmium
J. Dodd, R. Kaul,
D. Warrington,
Proc. Phys. Soc.
84, 176 (1964)
See also E.
Aleksandrov Optics
and Spectroscopy
17, 522 (1964)
Time-resolved spectroscopy
An ultra short laser pulse (femtosecond laser) excites a linear
superposition of vibration states of a molecule.
One gets in this way a wave packet which oscillates at the
vibration frequency of the molecule.
A second ultra short laser pulse, shifted in time with respect to
the first one, detects the evolution of the wave packet.
Recent extension to the attosecond range (1as = 10-18 s)
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Excitation with modulated light
The equation of evolution of 12 for an excitation modulated
at frequency d 12 / dt = W0 exp(it) i 12 12 12
has a solution oscillating at frequency :
W0
12 =
exp(it)
+ i 12 The intensity of the fluorescence light contains a component
modulated at frequency , which exhibits resonant variations
when the frequency of the excitation is scanned around 12
in an interval of width .
(
)
First observation of this effect
A. Corney, G. Series, Proc. Phys. Soc., 83, 207 (1964).
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Outline of lecture 1
1 – Level crossing resonances in excited states
2 – Quantum beats. Time resolved spectroscopy
3 – Extension to ground states
4 – Ramsey fringes and microwave atomic clocks
5 – Dark resonances. Coherent population trapping
15
Extension to atomic ground states
Optical pumping allows one to polarize atoms by accumulating
them in a given Zeeman sublevel of the ground state g.
One can also prepare atoms in linear superpositions of
Zeeman sublevels of g if the angular momentum transferred
from the pumping beam to the atoms is perpendicular to the
static magnetic field.
Transverse optical pumping Transverse physical quantities have a non zero average value
only if the density matrix g has non zero off diagonal elements
Equations of evolution can be established for the ground
state density matrix g that are quite similar to those for e
The spontaneous damping rate e in e has to be replaced by
the relaxation rate g in g which is much weaker
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Level crossing resonances in g
Level crossing resonances can be very narrow because g is
very small. They can be used to detect very small magnetic
fields
L is on the order of 1 MHz / Gauss.
If g is on the order of 1 s-1,
the width B0 of the resonance,
given by L g is on the
order of 10-6 Gauss.
Magnetometer with a
sensitivity of 510-10 Gauss
J. Dupont-Roc,
S. Haroche,
C. Cohen-Tannoudji,
Phys. Lett. 28A, 638 (1969)
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Outline of lecture 1
1 – Level crossing resonances in excited states
2 – Quantum beats. Time resolved spectroscopy
3 – Extension to ground states
4 – Ramsey fringes and microwave atomic clocks
5 – Dark resonances. Coherent population trapping
18
Ramsey fringes
Atomic beam
velocity v
L
Two-level atoms g1, g2 with Eg2 Eg1 = A cross 2 cavities fed by the same
coherent field with frequency 0 . The transition probability P g1 g2 is
measured versus 0
If a single cavity is used, one gets a resonance with a frequency
width 1/, where = / v is the time of flight through a single cavity.
Time-frequency uncertainty relation
If two cavities are used, interference fringes appear inside this
resonance with a spacing proportional to 1/T, where = L / v is the
time of flight between the 2 cavities. Ramsey fringes.
Analogy with Young’s two-slit interference fringes, with a spacing
determined by the distance between the 2 slits, appearing inside a
diffraction profile whose width is determined by the width of each slit
(
)
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P ( g1 g2 )
2/T
A 0
0
1 / Interference between 2 transition amplitudes
For going from g1 to g2, the atom can follow 2 paths:
- Interaction with the field in the first cavity
- Interaction with the field in the second cavity
g2
g1
t
T
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Cesium atomic clocks
Principle
The microwave
oscillator is
locked to the
frequency of the
central Ramsey
fringe
Orders of magnitude
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.
21
Improving atomic clocks with ultracold atoms
Fountains of ultracold atoms
Cloud of atoms cooled by laser cooling
to temperatures on the order of 1 μK
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
H
H = 30 cm T = 0.5 s
The width of the Ramsey fringes is 100 times smaller
than the width obtained with thermal atoms
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Examples of atomic fountains
- Sodium fountains :
- Cesium fountains :
Stanford S. Chu
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
A relative accuracy of 10-16 corresponds to an error smaller
than 1 second in 300 millions years!
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From terrestrial clocks to space clocks
•Time reference and global
clock comparison
•Validation of space clocks
•Fundamental tests
(General relativity, Variation
of fundamental constants)
ACES project (ESA, CNES)
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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 70 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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Optical clocks
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Outline of lecture 1
1 – Level crossing resonances in excited states
2 – Quantum beats. Time resolved spectroscopy
3 – Extension to ground states
4 – Ramsey fringes and microwave atomic clocks
5 – Dark resonances. Coherent population trapping
27
Discovery of dark resonances
Optically pumped sodium vapor put in a gradient of magnetic field
parallel to the z-axis. The splitting between 2 Zeeman sublevels g1
et g2 depend on z
g1
h
h
z0
h
z
g1
An applied RF field with frequency can induce resonant transitions
between the 2 sublevels only at the point z0 where the splitting
between the 2 sublevels is equal to h
Modification of the populations of g1 and g2 in z0 which results in a
change of the light intensity emitted by the atoms of the beam in z0
Analogy with Magnetic Resonance Imaging (MRI)
Spatially resolved magnetic resonance
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Bright resonances and “dark” resonances
Bright resonance: appears at the point z0 where the splitting between
g1 and g2 is equal to h
1
2
g2
g1
Dark resonance: appears also in the absence of RF, but only if
the laser contains at least 2 modes with frequencies 1 and 2.
The dark resonance appears at the point z'0 where the splitting
between g1 and g2 is equal to (1-2) Raman resonance condition
Eg2 Eg1 = ( 1 2 )
G. Alzetta, A. Gozzini, L. Moi, G. Orriols, Il Nuovo Cimento, 36B, 5 (1976)
29
Basic physical effect
1- We first consider the situation at time t=0.
The atom is put in a linear superposition of g1 and g2
such that the 2 absorption amplitudes g1e and g2e interfere
destructively
ci : Amplitude for the atom to be in state gi (i=1,2)
i (Rabi frequencies): Amplitudes for the atom to absorb a
photon i from the state gi (i=1,2). D : atomic dipole moment
The state c1 g1 + c2 g 2
is called "dark state"
The atom is put in a linear superposition of states which cannot
absorb light because of destructive interference
Phenomenon called “Coherent Population Trapping” (CPT)
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Basic physical effect (continued)
2- If a state is dark at t=0, can it remain dark at a later time t?
The coefficients ci acquire phase factors exp(-iEit/) due to the
energies Ei of the states gi
The laser fields Ei acquire phase factors exp(-iit) due to their
frequencies i
The total absorption amplitude thus becomes:
It still vanishes, as in t=0, if and only if the 2 phase factors multiplying
c11 and c22 are the same, i.e. if:
One recovers the Raman resonance condition and one understands
why the dark resonance only appears when this condition is fulfilled.
If it is not fulfilled, a state which is dark at a certain time t=o will no
longer be dark at a later time, and it will be then able to absorb light
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Variations of the fluorescence rate RF with the detuning from the
unperturbed Raman resonance
One varies the splitting between
g1 and g2, 1 and 2 being fixed.
Or one sweeps 2, 1 being fixed
The dark resonance has a width ’ determined by the relaxation
time in the ground state, much smaller than the width determined by the lifetime of the excited state
It is their very small width which explains the interest of dark
resonances for applications
Example of application:
« Electromagnetically Induced Transparency (EIT) »
An atomic vapor with a great optical depth for a laser 1 exciting
the transition g1e becomes transparent for this laser if one adds
a second laser 2 exciting the transition g2e with a frequency
fulfilling the Raman resonance condition.
The index of refraction for the laser 1 varies also very rapidly
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in the neighborhood of the dark resonance slow light
Subrecoil laser cooling by Velocity Selective CPT
R(v)
0
v
0
v
- Because of the Doppler effect the fluorescence rate R is v-dependent
and can be adjusted to vanish for v=0. Atoms with zero are in dark
states and thus protected from the « bad effects » of light (random
recoils due to the spontaneous emission processes).
- Atoms with non zero velocity are no longer dark because the Doppler
shift changes. They absorb light and the velocity changes due to the
the random recoils following spontaneous emission can make them
falling in the region near v=0 where they remain trapped and pile up
- The longer the interaction time , the narrower the interval v in
which the atoms can remain trapped during this time.
One can have p = m v < k (subrecoil cooling)
A. Aspect, R. Kaiser, N. Vansteenkiste, E. Arimondo, C. Cohen-Tannoudji PRL. 61, 826 (1988)
Cooling mechanism leading to temperatures in the nK range
33
Quantum Monte-Carlo simulation of 1D-VSCPT
Calculation by the dressed atom
approach of the distribution of the
time intervals between 2
successive spontaneous
emissions of photons by the atom
Between 2 emissions, the
momentum quantum number
remains constant and changes
in a random way after each
emission
F. Bardou, J-P. Bouchaud,O. Emile,
A.Aspect,C. Cohen-Tannoudji,
PRL. 72, 203 (1994)
Anomalous random walk along the time axis dominated
by a few rare events where p remains close to zero.
Self-similarity at all scales
Analogy with other situations involving ‘Lévy flights”
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35
Stimulated Raman Adiabatic Passage (STIRAP)
Klaas Bergmann
e
2
1
g2
g1
Dark state
D = c1 g1 + c2 g2
c11 + c22 = 0
i : Rabi frequencies
c2
c1
=
2
1
1 2 The dark state coincides with g2
2 1 The dark state coincides with g1
By starting with 1 2 , by reducing adiabatically 1 while
increasing adiabatically 2 until 2 becomes much larger than
1 , atoms are selectively transferred adiabatically from g2 to g1
Anti-intuitive order
The system always remains in a dark state, so that e is never
populated (no spontaneous transitions from e). Selective transfer
of molecules from one state to another
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Conclusion
Atomic physics experiments provide interesting examples of
physical effects demonstrating the importance of linear
superpositions of states
These effects discovered several decades ago are revisited and
are playing now an important role in modern developments
(nonlinear optics, laser cooling, molecular physics, atomic
fountains,…) Atomic clocks with ultracold atoms are now used in all
metrological institutes. They have reached an impressive relative
accuracy (10-17) allowing them to perform very precise tests of
basic theories (general relativity, 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
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