Download 1. Introduction - Univerza v Ljubljani

Univerza v Ljubljani
Fakulteta za Matematiko in Fiziko
Oddelek za Fiziko
Seminar za 4. letnik
L A S E R
C O O L I N G
Jurij Pahor
Mentor: prof. Martin Čopič
Ljubljana, maj 2002
1. Introduction
This paper will present a brief overview of methods of laser cooling and physical models behind it. The methods
of laser cooling exploit forces which arise from interaction of laser light with neutral atoms. Essentially, there are
two basic mechanisms for deccelerating a beam of neutral atoms.
The first one relies on radiative force as a result of light pressure. Atoms in a beam lose their kinetic energy
by scattering light of a counterpropagating laser beam. The decceleration occurs because atoms gain momentum
in the direction opposite to their motion and lose it in random directions by spontaneous emissions. This process
provides for a dissipative net force acting on neutral atoms in the direction of propagating light.
The second mechanism embodies spatial variation of light field, resulting in a gradient of atomic energy.
The force, driven by this energy gradient is called the dipole force. It can be thought as the force acting on an
optically induced electric dipole moment of the atom in an inhomogeneous light field.
Another very large field of atom manipulation with light is trapping of neutral atoms. This is most readily
accomplished by combining optical forces with magnetic field gradients, although experiments based on purely
optical forces have been carried out.
2. The Radiative Force
The radiative or radiation-pressure force originates from an atom absorbing a photon, thus gaining a momentum

k in the direction of light propagation. The atom is left in the excited state e and eventually returns to the
ground state g by spontaneous emission. The photon is emitted with random polarisation in random direction
and after many scattering processes its average contribution to the atomic momentum equals zero. The resulting


average force is then Frad  R k , where R is the scattering rate.
We shall calculate R using a simple two-level atom model in semiclassical approximation with the light
field as a classical electric field. The result is illustrative and to some extent practically applicable although real
atoms have multiple energy levels. Let us begin the discussion with first defining:
l
laser light frequency
 a  ( Ee  E g ) 
  l  a
  1/ 
atomic transition frequency
detuning of laser frequency from atomic transition frequency
natural radiative decay rate of the excited state e , with  as lifetime of the state
I, I0
intensity and saturation intensity of laser light

the Rabi frequency at which the atomic state oscillates between g and e for zero detuning.
2
The time-dependent Schödinger equation is

(r , t )

,
( H 0  H ' (t ))(r , t )  i
t
(1)
with the field free hamiltonian H 0 and the atom-light interaction hamiltonian H ' (t ) . We write the solution


(r , t ) as an expansion in terms of eigenfunctions  n (r ) of H 0

 (r , t ) 
c
k

(t )  n (r ) e i k t .
(2)
k
After inserting expansion (2) into equation (1), multiplying from left with  *j and integrating over spatial
coordinates we get the equation relating expansion coefficients
i
dc j (t )
dt

c
k
(t ) H 'jk (t ) e
i jk t
,
(3)
k
where H 'jk (t )   j H ' (t )  k
and  jk  ( j   k ) . In our approximation there are only two coefficients in
expantion (2), namely cg and ce , for which eq. (3) becomes
i
dcg (t )
'
 c e (t ) H ge
(t )e i a t
dt
.
dce (t )
i a t
'
i
 c g (t ) H eg (t )e
dt
 
(4)

 

With the interaction hamiltonian H ' (t )  eE (r , t ) r and the light field E (r , t )  E0 e
i ( kz l t )
we write the
diagonal elements as
'
'*
H eg
 H ge
 12  e i ( kz l t ) ,
where   
(5)
2eE0

e r g is the Rabi frequency. Here we neglected the spatial variation of E (r , t ) of order

of several hundred nanometers in the region containing atomic wavefunction, which is of order < 1nm. Next, by
differentiating one of the equations (4) and substituing it in the other, we obtain two uncoupled, second order
differential equations for cg and ce. The initial conditions for this set of equations are c g (0)  1 and c e (0)  0 .
3
Let us write the solution for time dependence of the excited state coefficient:

c e (t )  i
 
2
2
sin  12  2   2 t  e it 2 .


The probability of finding an atom in the excited state ce (t )
2
oscillates with the frequency
(6)
 2   2 and is
equal to the Rabi frequency as defined above for zero detuning of the laser light frequency from the atomic
resonance frequency.
The result (6) doesn't allow any net force but we haven't accounted for spontaneous emission yet. Using the
Schrödinger equation while taking into account all possible states the photon can be emitted to, would be a rather
difficult task. Instead we introduce the density operator     with the matrix elements
 ij   i   j   i    j  ci c *j .
(7)
With this formalism it is possible to describe the evolution of a pure state e into g , which is not a pure state
any more, because the information about the emitted photon is disregarded. The probability for the excited state
of an atom is now  ee (t ) . The time evolution of the density operator is described with
i
d
 H ,   .
dt
(8)
Spontaneous emission is taken into account with an exponential decay of the coefficient  eg (t ) with a constant
rate  2 , thus
d eg
dt


2
 eg . Using
d ij
dc*j
dci *
, along with eqs. (4) we obtain a set of

c j  ci
dt
dt
dt
equations for density matrix coefficients
d ee
i ~
  ee 
(  ge  ~eg )
dt
2
,
d~eg

i
~
 (  i )  eg 
(  gg   ee )
dt
2
2
(9)
~   e it and ~   e it . The terms including  take into account the spontaneous emission
where 
eg
eg
ge
ge
effect. The set of equations (9) can be solved by applying two general relationships between the density matrix
~  ~ * .
coefficients; the conservation of the population  gg   ee  1 and the identity 
eg
ge
4
The solution for  ee is
 ee 
2
(10)
2 2   2  4 2
and with this result we can finally write the scattering rate R as
R    ee 

2
.
2  2   2 2  2 2
(11)
From expression (11) follows that the scattering rate saturates to  2 for high laser intensities with  being
proportional to the optical field amplitude E 0 , thus  2 being proportional to field intensity. The maximum
force based on spontaneous emission is then
Fmax  k

2
(12)
.
The discussion until now included the force exerted on atoms by a traveling wave but this is not a true
cooling force since it is acts in one direction only. A dissipative force that vanishes for an atom at rest is required
for cooling. Such force arises in a standing wave, detuned below resonance, as a consequence of the Doppler

shift. The atoms traveling with velocity v see the frequency of the light to be  '   l  k v (Figure 1), so that

 becomes    '   a   0  k v .
 '    kv

v
 '    kv
Moving atom
Figure 1: A moving atom sees the light traveling in the opposite direction with higher frequency and the light
traveling in the same direction with lower frequency than an atom at rest.
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Now the atoms see the light of the counterpropagating beam Doppler shifted upward, closer to the resonance,
wheras they see the light of the other beam shifted further away from the resonance (Figure 2). Consequently,
they scatter more light from the beam propagating opposite to their motion than from the other beam, which
results in an average force slowing the atoms down.
0,5
R

2
2    2 2  2 2
2
Scattering rate R []
0,4
0,3
0,2
0,1
0,0
-10
-5
0
5
10
Detuning  []
Figure 2: Schematic dependence of the scattering rate R from laser light detuning  in units of natural width
parameter  . The cooling process takes place when te atoms traveling through a standing wave of laser ligth,
detuned below resonance (   0 ), see one beam Doppler shifted towards the resonance and the other beam
shifted away from the resonance.
The light must be originally detuned below resonance, as can be seen from the following. The
magnitude
of the force in one direction is


F  k R
F  
,
k
2
2
2
2
2    2  2( 0  k v)
(13)

where the '+' sign stands for the direction of v and '-' sign stands for the opposite direction. The total force
acting on the atom is the sum F  F  F , so

  2 k 2
F   2
2
2
2
2
2
  2   4  ( 0  kv)  2   4  ( 0  kv)




 0v .


(14)
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The force is deccelerating for negative detuning but is accelerative for positive detuning. Also, for low velocites,
the kv terms in the denominator can be neglected towards  0 , thus making the force proportional to the
velocity. This is then a true damping force, thus it compresses the velocity distribution of the atoms, which is the
essence of the cooling process. Because of the viscous damping the technique is referred to as optical molasses.
To make an example, the term kv amounts to ~10 MHz for   500 nm and v  1 m s , whereas the detuning
can be of the order of  , tipically 100 MHz.
Although it may seem that atoms can be cooled to an arbitrary low temperature, it is not so in reality. The
scattering force is oriented towards the direction opposite to the atoms motion only on average. The individual
momentum kicks of spontaneously emitted photons have random directions resulting in heating the sample.
Obtainable temperature is then limited by the competition between damping and heating, both being equal in a
steady state. The lower temperature limit is called the Doppler limit. A rough estimate for this limit is the kinetic
energy, the atom gains, after emitting a photon. The velocity and the kinetic energy of the atom, after
considering the conservation of the momentum and the energy of the atom-photon system, is
 a
mc 2
.
(  a ) 2
Wk1 
2mc 2
v1  c
(15)
2
The typical value for  a is 1 eV and for the mc is several Gev which amounts to a few 10 cm/s for atomic
velocity and several tens μK. This estimate is rather low, since it accounts only for a single photon emission. The
lower temperature limit is in reality larger for a magnitude, that is of the order of several hundred μK.
3. Cooling Below the Doppler Limit – The Dipole Force
In the previous discussion we derived a force, which relies on the process of the spontaneous emission and can
cool a sample of atoms down to the Doppler limit. A different type of force, which is conservative instead of
dissipative, can be produced. The origin of the dipole force lies in a gradient of the atomic energy in the light
field. In an appropriate setup, the potential can be harmonic. An ansamble of atoms, trapped in the potential, is
being cooled by evaporation of the atoms out of the trap with simultaneously lowering the depth of the trap.
We shall begin the discussion with calculating the energy shifts of a two-level atom states under the
influence of the light field. Because of their origin they are called light shifts. The light shifts are most readily
found by eliminating the explicit time dependence in eqs. (4).
7
This is done by the rotating frame transformation, as in eqs. (9), that is
c~g (t )  c g (t )
.
c~ (t )  c (t ) e it
e
(16)
e
Inserting the new coefficients (16) into eqs. (4) gives
dc~g (t )
 ~
ce (t )
dt
2
.
dc~e (t )  ~
~
i

c g (t )   c e (t )
dt
2
i

(17)
This set of equations is basically the Schrödinger time-dependent equation for the coefficients (16) with the
hamiltonian
H
 
 0
  2  .
2

(18)
The shifted energies are given as the eigenvalues of the hamiltonian (18)
E g ,e 

     2   2  .

2
(19)
For the sake of simplicity we shall now concentrate on the case of    , so that the energy of the
ground state shifts to E g 
 2
(without the light field it is set to zero), while the energy of the excited state
4
remains unchanged E e   . In a standing wave, formed by two light beams of equal polarisation, the
intensity I  E02 sin 2 (kz) changes from zero at the nodes to maximum value at the antinodes. The Rabi
frequency is proportional to the field amplitude, thus enabling a gradient force to arise
Fdip  
E g
z

  2
k

sin( 2kz) .
4 z
4
(20)
The dipole force is, in contrast to the radiative force, position dependent and is thus appropriate for
trapping the atoms. Effective optical traps can be build with first cooling a sample of atoms with the radiative
force and then trapping it with the dipole force. The atoms in the trap gets in thermal equilibrium by collisions.
The atoms with higher velocities evaporate from the trap which results in lower temperature of the remaining
sample. Eventually, lower temeratures can be achieved by lowering the potential well of the trap.
8
4. Conclusion
We have considered the interaction of the neutral atoms with light, accounting for two diffrerent types of forces.
The radiative force is a dissipative, velocity-dependent force, which cools the atoms. The mechanism producing
the radiative force is absorption, folowed by spontaneous emission of the light. The reachable temperature is of
order of several hundred μK. The dipole force is on the other hand conservative, since it originates from a
gradient of atomic energy in the light field. The energy gradient emerges from the light shifts of the atom. The
dipole force potential is appropriate for trapping the atoms, whereas the cooling process includes evaporation of
the atoms out of the trap. With this technique, lower temeratures are achievable, of the order of μK.
References
H. Metcalf, P. van der Straten, Laser Cooling and Trapping, Springer-Verlag, New York (1999)
W. D. Phillips, Optical manipulatoin of atoms at the nanoscale, Proc. of the international school of physics
»Enrico Fermi«, Società Italiana di Fisica (2000)
Y. Shimizu, H. Sasada, Mechanical force in laser cooling and trapping, Am. J. Phys. 66 (11), Nov. 1998
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