ATOMIC PHYSICS 10
H. Narumi, 1. Shimamura (editors)
© Elsevier Science Publishers S.V., 1987
LASER-ATOM
INTERACTIONS
Jean DALIBARD
365
: RECENT THEORETICAL
DEVELOPMENTS
and Claude COHEN-TANNOUDJI
Laboratoire de Spectroscopi e Hertzienne de l'Ecole Normale Supérieure*
Collège de France, 24 rue Lhomond, F 75231 Paris Cedex 05 France
et
ln intense laser beams, when perturbative treatments are no longer val id,
the dressed atom approach provides a quantitative understanding of the
main features of dipole or intensity gradient forces (mean value, fluctuations, velocity dependence). ln this lecture, we present such an approach
and we apply it to atomic motion in an intense standing wave. New efficient
laser cooling schemes taking advantage of stimulated processes are proposed.
They work for a blue detuning and do not saturate at high intensity.
1. INTRODUCTION
During the last few years,
atom interactions
several experiments
provide the possibility
have demonstrated
to control
that laser-
the velocity1,2
position3 of an atom. A new exciting field of research
and the
is emerging which is
called laser cooling and trapping4.
ln order to introduce the subject of this talk, we consider
simplest possible example of an atom irradiated
first the
by a laser plane wave with
wave vector k. Wh en the atom ab sorbs a laser photon,
the momentum
gain is fik.
If the atom falls back to the ground state by stimulated emission,
momentum returns to its initial value. But, if the emission
spontaneous one, the momentum
because spontaneous
emission
directions. It follows
loss during such a process
cycle (absorption + spontaneous
<F> experienced
process
emission)
is a
is zero by symmetry,
can occur with equal probabilities
that the net atomic momentum
the atomic
in opposite
gain in a fluorescence
is fik. Consequently, the mean force
by the atom is given by
<F>
( 1)
= fik r Gee
where r Gee is the mean number of fluorescence
cycles per unit time, equal to
the product of the population Gee of the excited atomic state e by the spontaneous emission rate r , which is also the natural width of e. Such a force is
nothing but the well known radiation
pressure5-7
down atomic beams1. At high intensities,
to the value fik r/2.
which has been used for slowing
Gee tends to 1/2 and <F>
saturates
Suppose now that the atom is irradiated by two counterpropagating
*
Unité associée
au CNRS (UA 18)
laser
366
J. Dalibard and C. Cohen- Tannoudji
beams, with the same intensity,
respect to the atomic one Wo
and with a frequency wL detuned
(the detuning
8 = wL - Wo
tensity of such a standing wave is weak enough,
two counterpropagating
effect,
is negative).
the radiation
waves can be added independently.
resonance
with the copropagating
cooling8,9.
If the in-
pressures of the
Because of the Doppler
the atom gets closer to resonance with the opposing
of usual radiative
to the red with
wave, farther from
one. It is slowed down. This is the principle
Such a scheme leads to velocity damping times of
the order of
To
=
Ii/R
where
R = Ii 2 k2 /2M
is the recoil energy and M the atomic mass,and
to
minimum
kinetic temperatures
T of the order of
r/2
kB T = Ii
where kB is Boltzmann's
few microseconds
constant10.
For sodium atoms, To
is of the order of a
and T is as low as 240 vK. The possibility
with such "optical molasses"
If the intensity of the laser standing wave is increased
meter s»
processes,
1), the physical
responsible
counterpropagating
terpropagating
± 2 lik11,12.
(saturation para-
picture given above breaks down. Stimulated emission
for a coherent redistribution
waves, become predominant:
plane wave can be followed
of cooling atoms
has been recently demonstrated2,4.
by the stimulated
of photons
the absorption
emission of a photon in the coun-
wave, giving rise to a change of atomic momentum
Such stimulated
processes,
between the two
of a photon in a
equal to
which occur at a rate of the order of
the Rabi frequency w1' are at the origin of dipole forces or intensity gradient
forces5,6
(which have been recently used for trapping
standing wave, it has been shown theoretically
the atoms for a red detuning
contrarily
The purpose of this lecture
pressure mol asses" described above.
is to present recent theoretical
a simple physical
intense laser wave, and in particular
interpretation
is based on the dressed
interpretation
of atomic motion in an
atom approach which is briefly recalled
in terms of two valued dressed
cal Stern and Gerlach effect). The particular
is considered
developments
in an intense standing wave15. Such an
in section 2. We then show in section 3 how dipole or intensity
can be interpreted
ln a strong
(8<0) and a cooling for a blue one (8)0)5,13,14,
to what happens in the "radiation
which have provided
atoms3,4).
that they produce a heating of
gradient forces
state dependent
forces (opti-
case of an intense standing wave
in section 4, and a new efficient
laser cooling mechanism, which
Laser-Atom Interactions
does not saturate at high intensity,
Finally, possible
367
is introduced
("Stimulated
molasses").
appl ications of stimulated molasses are investigated
in sec-
tion 5.
2. THE DRESSED ATOM APPROACH.
CONNECTION
WITH RESONANCE
FLUORESCENCE16-18
ln the high intensity limit, the Rabi frequency w1' characterizing
strength of the laser-atom
rate
r.
coupl"ing, is large compared ta the spontaneous
We therefore considerin a firststep
system atom + laser photons
second step, we
interacting
the
emission
the energy levels of the combined
together
(dressed states). Then, in a
take inta account the coupl ing with the vacuum field which gives
rise to spontaneous
transitions
thus appears as spontaneous
between dressed
states. Resonance fluorescence
emission of radiation
11
Co
Ig n + 1>
..
"
n>
,
.
iQ
•
1
,"
.!!r-16
le,n>
1
atom.
1
"
:l
from the dressed
•
' 12, n>
1
1
1
wd>
1
1
i
11n-1>
"
l
Ig.n>
.
-1--,6
1
1
1
•
le, n -1>
Con_1
/
iQ
1
.•"
1
• 12,n-1>
FIGURE 1
Left part: unperturbed states of the combined atom-laser photons system, in
absence of coupling,bunched in well-separated two dimensional manifolds
Right part: dressed states resulting from the atom-laser coupling
Let us first introduce the dressed
given point~.
ln absence of coupling,
atom + laser photons are labelled
states for a 2-level atom at rest in a
the energy levels of the combined
system
by two quantum numbers, e or 9 for the atom
(excited or ground state), n for the number of laser photons. Such "unperturbed"
states are represented
on the left part on Fig. 1. When the laser frequency wL
is close to the atomic one wo, these states are bunched
manifolds ... &n = {Ig,n + 1 >,
le,n >}, &n-1 = {Ig,n>
into two dimensional
, e,n - 1>} ... ,
368
J. Dalibard and C. Cohen- Tannoudji
the distance
between the two levels of a given manifol~being
and the distance
coupling
between two adjacent manifolds
ho = (WL - wo)
being hwL. The laser-atom
V connects the two states of a given manifold.
+
For example,
1 laser photons
the
can absorb
atom in the ground state 9 and in presence
of n
one laser photon and jump into the excited
state e. This means that V has
a non zero matrix element
Actually,
between the two states
Ig,n
+
1> and le,n> of & n .
one can show that
h
<e,njVlg,n
. (,
(2)
+ 1> ="Z w1(r) el\? rJ
where \?(r) is the phase of the laser field and w1(r) the Rabi frequency
related to the amplitude &o(r) of the laser field and to the transition dipole
moment
d
by
(3)
This coupling
represented
gives rise to two perturbed
on the right part of Fig.
combinations
of the unperturbed
states,
1.
states
Il,n> and 12,n>
These dressed
je,n> and
(for &n)'
states are both linear
Ig,n + 1> and are separated
by a splitting h~ given by
+ w](r)]
~(r) = [02
Consider
correspond
now the effect of spontaneous
to transitions
between states connected
operator
allowed
by \e,n>, and both dressed
and 12,n - 1> of &n-1 are contaminated
+ ~,
transition
transitions
li,n> ~
: transition
12,n> ~
resonant
thus provides
laser beam. Similarly, various
ved on the fluorescence
sequence of fluorescence
the dressed
states
[2,n - 1> corresponding
Il,n>
\l,n - 1>
to a frequency
to a frequency wL - ~, and
li,n - 1> (with i = 1,2) corresponding
triplet,,19 emitted
states
by Ig,n>, we find 4 allowed transitions
i1,n> ~
!l,n - 1> corresponding
wL' The dressed atom approach
of the "fluorescence
frequencies
i.e. to transitions
ie,n> and Ig,n>. Since both dressed
and !2,n> of &n are contaminated
wL
The emission
levels,
basis, D, which does not change the number of
connects only
between &n and &n-1
emission.
between dressed
by a non zero matrix element of the atomic dipole
D. ln the uncoupled
laser photons,
(4)
If2
both to a frequency
a straightforward
interpretation
by a 2 level atom irradiated
features
light can be easily understood
photons as being emitted
by a
of photon correlations
by considering
in a "radiative
obser-
the
cascade" by
atom18.
3. DRESSED ATOM INTERPRETATION
We apply now the dressed
forces which are associated
OF DIPOLE FORCES1S
atom approach
to the interpretation
with the intensity gradients
of dipole
of the laser beam.
Laser-Atom Interactions
ln an inhomogeneous
lt follows
369
laser beam, the laser intensity
that w21 (r), and consequently
ln Fig. 2, we have represented
~(r) according
the variations
is position
dependent.
to (4), vary in space.
of the energies
of the dressed
states across a gaussian laser beam. Out of the laser beam, the dressed
coincide with the bare ones, and their splitting
beam, each dressed level
is just ho.
Il,n> or 12,n> is a linear superposition
and le,n> and the splitting
between the two dressed
levels
lnside the laser
of Ig, n + 1>
states of a given manifold
becomes h~(r), which is larger than ho.
-_
Out
'JO-
Position
FIGURE 2
Variations across a gaussian laser beam of the dressed-atom energy levels. Out
of the laser beam, the energy levels connect with the uncoupled states ofng. 2,
segarated by o. ln the laser beam, the splitting
between the dressed
states is
~(r» 0
Within each manifold, the energy diagram of Fig. 2 is similar to the one of
a spin 1/2 magnetic moment
in an inhomogeneous
follows that, in absence of spontaneous
dressed state dependent
static magnetic
emission,
field. lt
we can define a two-valued
force, equal to minus the gradient of the dressed
state energy
7;
-+
-+
-;>.-
-+
-+
Tl = - VEln(r)
T2 = - VE2n(r)
As in the ordinary
= -
-+-+
(h/2) V ~(r)
= +.(h/2)
(5)
-+-+-;>.-
V ~(r) = - Tl
Stern and Gerlach effect, we have a force that depends on
the internal state of the dressed
atom, but the basic interaction
occurs now
370
J. Da/ibard and C. Cohen- Tannoudji
between an optical dipole moment and an inhomogeneous
laser electric field
(optical Stern and Gerlach effect).
The effect of spontaneous
tions between dressed
versa. This changes
dressed
emission
is to produce,
states of type 1 and dressed
states of type Z, or vice
in a random way the sign of the instantaneous
state dependent
two valued
force. Such a picture of an instantaneous
ching back and forth between two opposite
first the mean force.
force swit-
values provides a simple understan-
ding of the mean value and of the fluctuations
Consider
at random times, transi-
of dipole forces.
It can be written
as
(6)
l
where TI. is the proportion
of time spent in a dressed
For 0>0 (case of Fig. Z), the unperturbed
wL > wo, and the dressed
beam. It follows that
state
state of type i (i
Ig,n + 1> is above
state
=
1,2).
le,n>, since
to Ig,n + 1> out of the laser
Il,n> connects
Il,n> is less contaminated
by le,n> than IZ,n>, and is
therefore more stable with respect to spontaneous emission, so that TIl > TI2 '
and the atom spends more time in dressed states of type 1 than in dressed
states of type Z. We conclude
one corresponding
sity regions.
to level
that, for 0>0, the sign of the mean force is the
Il,n> : the atom is expelled out of the high inten-
For 0<0, the conclusions
are reversed. The dressed
connects to Ig,n + 1> out of the laser beam, is more populated
state IZ,n>
than Il,n> and
imposes its sign to the mean dipole force which attracts the atom towards the
high intensity regions.
same admixture
Finally, for 0 = 0, the Z dressed
of le,n>, are equally
force vanishes. We understand
populated
states contain the
(TIl= TI2), so that the mean
in this way why the variations
of the mean dipole
force (for an atom initially at rest) versus the detuning 0 = wL - Wo are of a
dispersive
type. The argument
quantitative.
spontaneous
given above is not only qualitative
If one calculates
transition
but also
TIl and TIZ from the mas ter equation giving the
rates between the various dressed
states, and if one
puts their values in (6), one gets the exact value of the mean dipole force,
(to lowest order
Similarly,
in f/w1)15.
by studying the correlation
function of the instantaneous
force
switching back and forth between *1 and *Z = - *1' it is possible to get a
quantitative interpretation of the diffusion coefficient associated with the
fluctuations
of dipole forces5,15.
4. DRESSED STATES lN A STANDING WAVE. STIMULATED MOLASSES
We come back now to the problem mentioned
concerning
Atomic
the importance of stimulated
motion
at the end of the introduction
processes
in a strong standing wave.
in a strong standing wave has been studied by several
Laser-Atom Interactions
authors20,13,5,21,15.
We have recently
the dressed atom approach,
problem15. We present
proposed
and which provides
371
a physical picture,
some physical
based on
insight in this
in this section a summary of this theoretical
work.
ln a plane standing wave, along Oz, the Rabi frequency w1(z) is a periodic
function of z
(7)
It follows that the 2 dressed
states of a givenmanifold
in space since their splitting
is according
r/(z) = [w~ cos' k z + (wL - wo)']
Fig. 3 represents
these dressed
states
Il,n> and
w1(z) is different
(8)
(dashed lines).
... ), w1(z) vanishes
12,n> respecti~ely
states Ig,n + 1> and \e,n>, separated
periodically
I/i
states for a positive detuning
At a node of the standing wave (z = \/4 ,3\/4
two dressed
oscillate
to (4) and (7).
by 0 (dotted lines). Out of anode,
from zero, the dressed
states are linear combinations
Ig,n + 1> and je,n> and their splitting
is maximum
at the antinodes
(z = 0, \/2, \ ... ) where w21 (z) reaches
its maximum
value. Consider
effect of spontaneous
emission.
and the
coincide with the unperturbed
of
now the
As we have seen in section 2 above, an atom in
level Il,n> or 12,n> can emit a spontaneous
photon and decay to levels
Il,n - 1> or 12,n - 1>. The key point is that, in a standing wave, the various
rates for such spontaneous
of the wave functions.
processes
For example,
vary in space because of the z dependence
if 0 is positive
level j1,n>,its decay rate is zero at anode
where
maximum at an antinode where the contamination
On the contrary,
nodes where
for an atom in level
12,n> coincides with
The previous considerations
and if the atom is in the
Il,n> = Ig,n + 1> and
of \l,n> by le,n> is maximum.
[2,n>, the decay rate is maximum
will allow us now to understand
slowed down in an intense standing wave, when 0 is positive,
happens in usual radiation
pressure molasses
example follow the "trajectory"
at the
le,n>.
why an atom is
contrarily
to what
(weak standing waves). We can for
of a moving atom starting at a node of the
standing wave, in level Il,n + 1> (full lines of Fig. 3). Starting from this
valley, the atom climbs uphill until it approaches
its decay rate is maximum.
the top (antinode) where
It may jump either in level Il,n> (which does not
change anything from a mechanical
point of view) or in level
case the atom is again in a valley.
reaches a new top (node) where
\2,n> is the most unstable,
It is clear that the atomic velocity
12,n>, in which
It has now to climb up again until it
is decreased
and so on ...
in such a process,
since
the atom sees on the average more "uphill" parts than "downhill" ones. Su ch a
scheme can be actually
considered
as a microscopic
realization
of the
372
J. Dalibard and C. Cohen- Tannoudji
"Sisyphus myth"
: every time the atom has cl imbed a hill, it may be put back at
the bot tom of another one by spontaneous
emission and it has to climb up again.
\
..:.l....
../"\\..II~\
....
\./.,' , .
11,n
+ 1>
i~·~~;::{\~·T\7\··l\
..ï2
'
'~I '''' '
1
n+1>
\11,n-1>
ï~:~~1>"
\·I'\··~i2,n-1>
, ,ï'\..
\.,·r\7...
"
\
,
.•.
...•
FIGURE 3
Laser cooling in a strong standing wave with a blue detuning (8)0). The dashed
lines represent the spatial variations of the dressed atom energy levels which
coincide with the unperturbed levels (dotted lines) at the nodes. The full
lines represent the "trajectory" of a slowly moving atom. Because of the spatial variation of the dressed wave functions, spontaneous emission occurs
preferentially at an antinode (node) for a dressed state of type 1 (2).
Between two spontaneous emissions (wavy lines), the atom sees on the average
more uphill parts than downhill ones and is therefore slowed down.
We have used such a picture
dependence
(k v «
r),
we have a linear dependence
than for usual radiation
The force reaches
results
for the velocity
words, for situations
between
such
two spontaneous
of this friction
depth of the dressed
kv ~
r,
when
emissions.
force is directly
energy levels,
this force increases
After this maximum,
(by a factor of the order of
value for velocities
w1/r).
or in other
in which, as in Fig. 3, the atom travels over a distance
point is that the magnitude
As a consequence,
with a slope which can be much higher
pressure molasses
its maximum
of the order of a wavelength
modulation
to derive quantitative
of the force acting upon the atom15. At very low velocities
The important
related to the
i.e. to the Rabi frequency wl'
indefinitely
with the laser intensity.
kv becomes large compared to
r,
the force decreases
Laser-Atom Interactions
as v-l and finally,
to non-adiabatic
at very large velocities,
Landau-Zener·
transitions
373
resonances
appear which are due
between the two dressed
states of
each manifold2l.
To conclude
this section,
it may be useful to analyze
balance in the cooling process associated
Between two spontaneous
emission
energy is transformed
which redistribute
wl· Atomic momentum
When the atom climbs uphill,
into potential
photons
between
is therefore
energy by stimulated
processes
waves at a rate
to laser photons. The total
by spontaneous
away part of the atomic potential
its kinetic
emission
the two counterpropagating
transferred
atomic energy is then dissipated
molasses".
the total (kinetic + potential)
processes,
energy of the atom is conserved.
the energy-momentum
with these "stimulated
emission
processes which carry
energy.
5. APPLICATIONS
5.1 Transverse
cooling of an atomic beam
By irradiating
at right angle an atomic beam with an intense standing
wave detuned to the blue, it is possible to cool the transverse
cities (along the direction
in the previous
section.
k~v ~ r, the cooling
of the standing wave) by the mechanism
If the transverse
efficiency
with an interactionlength
atomic velo-
is maximum
velocity
described
spread ~v is such that
and it is possible to cool the beam
much shorter than the one required
for usual radia-
tion pressure molasses.
Actually,
such an experiment
has just been done in our group on a
Cesium atomic beam22. An example of experimental
Fig. 4. Fig. 4a gives the transverse
(measured by a hot wire detector
laser. The velocity
velocity
results
is represented
located 2 meters downstream)
in absence of
spread is ± 2 m/s. Fig. 4b shows the effect of a laser
standing wave detuned + 30 MHz above resonance which corresponds
(the power is 70 mW, and the beam waist in the interaction
w = 1,8 mm leading
to wl = 50 r). The width of the velocity
narrower than the one of the unperturbed
is detuned
- 30 MHz below resonance,
exhibits a double peak structure
Such an experiment
Realization
beam. Finally,
the efficiency
stimulated molasses,
more decreased,
profile
is 5 times
if the laser frequency
and
of stimulated molasses.
pressure molasses
length one order of magnitude
forces do not saturate,
which is inversely
proportional
by increasing the laser power.
r
region is
the atomic beam is decollimated
of the same cooling with the usual radiation
since stimulated
to 0 = + 6
(Fig. 4c).
demonstrates
would have required an interaction
Furthermore,
in
profile of the atomic beam
larger.
the damping time for
to wl' could be yet
374
J. Oalibard and C. Cohen- Tannoudji
1-
Z
b
W
0:::
0:::
:::J
U
a1u
W
1W
a
0:::
±40 omis
-1
mis
1
-r-'-10
-'-T-
10
OETECTOR
mm
20
POSITION
FIGURE 4
Transverse velocity profile of the Cs atomic beam
a - ln absence of laser beam
b - ln presence of an intense standing wave (w1 = 50 r) perpendicular to the
atomic beam and detuned to the blue (8 = + 6 r)
c - Same conditions as b except for the detuning which is now negative
(8
= -
6 r)
5.2 Longitudinal
Suppose
slowing down of an atomic beam
now that the atomic beam and the laser standing wave are paral-
lel. If the standing wave could be swept in order to have a weak enough relative velocity with respect
the decelerating
to the atoms, corresponding
force (kv ~
beam with an efficiency
r),
it would be possible
to the maximum
value of
to slow down an atomic
much higher than in experiments
using radiation pres-
sure. For Cesium, a laser intensity of 100 mW/mm2 would allow to reduce the
stopping distance
from 1 meter to 10 centimeters.
This might be of special
interest for the real ization of a compact atomic clock.
The realization
of such an experiment would require two counterpropagating
laser beams with frequencies
wave moving with a velocity
w + 6w and w - 6w (in order to have a standing
c 6w/w). The frequency
offset 6w should also vary
in time (in order to have a standing wave moving with the decelerating
The development
way to realize
atoms).
of high power single mode laser diodes could provide a simple
such an experimental
scheme.
Laser-Atom Interactions
375
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1) For a review of these experiments, see for example W.D. Phillips,
J.V. Prodan and H.J. Metcalf, J.O.S.A. B 2 (1985) 1751 (Special issue on
mechanical effects of light).
2) S. Chu, L. Hollberg,
55 (1985) 48.
J.E. Bjorkholm,
3) S. Chu, J.E. Bjorkholm,
4) S. Chu, Demonstration
A. Cable and A. Ashkin,
Phys. Rev. Let.
A. Ashkin and A. Cable, Phys. Rev. Let. 57 (1986)314.
of laser cooling and trapping of atoms, this volume.
5) J.P. Gordon and A. Ashkin,
Phys. Rev. A 21 (1980) 1606.
6) R.J. Cook, Comments At. Mol. Phys. 10 (1981) 267.
7) V.S. Letokhov
and V.G. Minogin,
Phys. Rep. 73 (1981) 1.
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13 (1975) 68.
Bull. Am. Phys. Soc. 20 (1975) 637.
10) D.J. Wineland and W.M. Itano, Phys. Rev. A 20 (1979) 1521.
Il) E. Kyrola and S. Stenholm,
Opt. Commun.
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12) A.F. Bernhardt and B.W. Shore, Phys. Rev. A 23 (1981) 1290.
13) V.G. Minogin and o.T. Serimaa, opt. Commun. 30 (1979) 373.
14) V.G. Minogin, opt. Commun. 37 (1981) 442.
15) J. Dalibard and C. Cohen-Tannoudji,
J.o.S.A.
B 2 (1985) 1707.
16) C. Cohen-Tannoudji and S. Reynaud, Dressed atom approach to resonance
fluorescence, in : Multiphoton Processes, eds. J.H. Eberly and
P. Lambropoulos (Wiley, New York, 1978) 103-118.
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and S. Reynaud, J. Phys. B 10 (1977) 345.
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Phys. Rev. 188 (1969) 1969.
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A. Heidmann, C. Salomon and C. Cohen-Tannoudji
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