Download Laser Cooling of Atom

Laser Cooling and Trapping
of Atom
Ying-Cheng Chen, 陳應誠
Institute of Atomic and Molecular Science, Academic Sinica,
中研院原分所
Outline
• Basic idea & concept
–
–
–
–
–
Overview of laser cooling and cold atom study
The light force
Doppler cooling for a two-level atom
Sub-Doppler Cooling
Others cooling scheme
• Practical issues about a Magneto-Optical Trap
(MOT)
–
–
–
–
–
Atomic species
Lasers
Vacuum
Magnetic field
Imaging
Temperature Landmark
To appreciate something is a good motivation to learn something!
core of sun surface of sun L N
2
106
103
L He
room
temperature
1
3He
sub-Doppler
superfluidity
cooling
10-3 MOT
10-6
2003 MIT
Na BEC
10-9
0
(K)
typical TC
of BEC
Laser cooling and trapping of atom is a breakthrough to the exploration of the
ultracold world. A 12 orders of magnitude of exploration toward absolute zero
temperature from room temperature !!!
What is special in the ultracold world?
• A bizarre zoo where Quantum Mechanics governs
– Wave nature of matter, interference, tunneling, resonance
 h
–
–
–
–
2mkBT
~1μm for Na @ 100nk
Quantum statistics
Uncertainty principle, zero-point energy
System must be in an ordered state
Quantum phase transition
Trends in Ultracold Research
Cold Molecule
From atomic to
condensed-matter
Many-body Physics physics
From Physics
to Chemistry
Cold Atom
From fundamental
to application
Quantum Computation
Atom Chips…
Cold Plasma &
Rydberg Gas
From ground to
highly-excited states
From isotropic to
anisotropic interaction
Dipolar Gas
Useful References
• Books,
–
–
–
–
H. J. Metcalf & P. van der Straten, “Laser cooling and trapping”
C. J. Pethick & H. Smith ,“Bose-Einstein condensation in dilute gases”
P. Meystre, “Atom optics”
C. Cohen-Tannoudji, J. Dupont-Roc & G. Grynberg “Atom-Photon interaction”
• Review articles
– V. I. Balykin, V. G. Minogin, and V. S. Letokhov, “Electromagnetic trapping of
cold atoms” , Rep. Prog. Phys. 63 No 9 (September 2000) 1429-1510.
– V S Letokhov, M A Ol'shanii and Yu B Ovchinnikov
Quantum Semiclass. Opt. 7 No 1 (February 1995) 5-40 “Laser cooling of atoms:
a review”
The Light Force: Concept
E  i


p  ki
E '   s
' 
p  ks
absorption
emission
An exchange of momentum &
energy between photon and atom !
Photon posses energy
and momentum !

 dp
F 
dt
Force on atom
Net moentum exchange
from the photon to atom
Energy and Momentum Exchange
between Atom and Photon
• Photon posses momentum and energy.
• Atom absorbs a photon and re-emit another photon.
 
  
p  p' p  (ki  k s )
  2
 2 2
2


  ( ki  k s )
( p'  p )
K  K ' K 
  ( ki  k s )  v 
2m
2m

p'
 
(ki  ks )

p
always positive, recoil heating
If
Criteria of laser cooling
  
( ki  k s )  v
 0 the momentum decrease, and if

ki
avg
  
( ki  k s )  v
avg
  2
 ( ki  k s )

the kinetic energy decrease,
2m
avg
where avg stands for averaging over photon scattering events.
A laser cooling scheme is thus an arrangement of an atom-photo
interaction scheme that satisfy the above criteria!

ks
The Light force : quantum mechanics
•
Ehrenfest theorem, the quantum-mechanical analogue of Newton’s second law,
•
 
Interaction potential: for an atom interacting with the laser field, Vˆ  d  E



d p


d2
F 
  V (r , t )  m 2  r 
dt
dt
2

p
ˆ 
H
 V (r , t ),
2m
where V(r,t) is the interaction potential.
where d is atomic dipole moment operator.
•
Semi-classical treatment of atomic dynamics:
– Atomic motion is described by the averaged velocity
– EM field is treat as a classical field
– Atomic internal state can be described by a density matrix which is determined
by the optical Bloch equation
,
Validity of semi-classical treatment
•
Momentum width p is large compared with photon
momentum k.
k p  1
•
an upper bound on v
Atom travel over a distance smaller than the optical
wavelength during internal relaxation time. (Internal variables
are fast components and variation of atomic motion is slow
components in density matrix of atom ρ(r,v,t))
v 1   ,
or
kv   1
an lower bound on v
•
Two conditions are compatible only if
•
If the above conditions is not fullified, full quantummechanical treatment is needed. e.g. Sr narrow-line cooling,
=27.5kHz ~ ωr=2k/2m=24.7kHz
 2 k 2 2m
 1

J. Dalibard & C. Cohen-Tannoudhi, J. Phys. B. 18,1661,1985
T.H. Loftus et.al. PRL 93, 073001,2004
The light force for a two-level atom
 
U  V  d E



F  U  d  E



E  eˆE0 (r ) cos(t   (r ))






it
it
d  Tr ( d )  12d 21   21d12  d12 ( 12e   21e )  2d12 (u cos t  v sin t )
Where d12=d21 are assumed to be real and we have introduced the Bloch vectors u,v, and w.
1
( 12   21 )
2
1
v 
( 12   21 )
2i
1
w
(  22  11 )
2
u 
Remark: dipole moment contain
in phase and in quadrature
components with incident field.
ρij (or σij)can be determined by the optical Bloch equation of atomic density matrix.
Optical Bloch equation
d 1
 d 
 [H ,  ]   
dt i
 dt 
dij
dii

(
) spon  ii , (
) spon   ij
dt
dt
2
Incoherent part due to spontaneous
emission or others relaxation processes
d11
i
  22  ( 21   12 )
dt
2
d 22
i
  22  ( 12   21 )
dt
2
d 12

i
 (  i ) 12  (  22  11 )
dt
2
2
11   22  1, where


(r )   dE0 (r );  12  12 exp( it );     0
S 0  I / I sat ; I sat 
steady state solution
s0 2
i
 22 
; 21 
2
1  s0  (2 )
2( 2  i )(1 
s0
)
2
1  (2 )
hc
33
Isat ~ 1-10 mW/cm2 for alkali
atom
Two types of forces
Without loss of generality, choose
At r =0,

 (r  0)  0

(E ) j  e j (cos tE0  sin tE0 )

d j  2(d12 ) j (u cos t  v sin t )
Take average over one optical cycle


F   ( d j E j ) avg  (eˆ  d12 )(uE0  vE0 )
j


 

dE0 (r )
dE0 (r )
F  Fdip  Frp  (
)( 12   21 )   (
)i ( 12   21 )
2
2
dipole force or
gradient force
a reactive force
Origin of optical trapping
radiation pressure or
spontaneous emission force
a dissipative force
Origin of optical cooling
Light force for a Gaussian beam
Frp
k
Fdip
F
z
Spontaneous emission force
d11
i
From
  22  ( 21   12 )
dt
2
i
( 12   21 )   22
for steady-state
2
Decay rate,

 


 (r )  k  r ;   k ; E0  0!!
For a plane wave



Frp  k  22  k Rsp
Rsp ( )   22 
,where Rsp is the flourescence rate.
S0

2 1  S 0  (2 ) 2
Max deceleration a 
 k
 50000 g , for Na D2 line !
2m
Dipole Force in a standing wave
• A standing wave has an amplitude gradient, but not a phase gradient. So
only the dipole force exists.

E (r , t )  eˆx E0 cos kz cos t


 ( 2 )
Fdip  
4  2   2 4  2 2
Where s0 is the saturation parameter for each of the two beams that form the standing wave.
For δ<0 (red detuning), the force attracts atom toward high intensity regions.
For δ>0 (blue detuning), the force repels atom away from high intensity regions.
Fdip  U

2 2
U 
ln[ 1  2
]
2
2
  4
Velocity dependent force
Atom with velocity v experiences a Doppler shift kv.


s0
 
Frp  k
2 1  s0  (2(  k  v ) ) 2
The velocity range of the force is significant for atoms with velocity such that their Doppler
detunings keeps them within one linewidth considering the power broadening factor.



 k v 
2
1  s0
Doppler Cooling
 

F  F  F


s0
k 
F  
2 1  s0  [2 (  kv) ]2


8k 2s0 v
kv 4
F



v
,
if
(
)  1
2 2
(1  s0  (2 ) )

For δ<0, the force slows down the velocity.
[/k]
δ/
Doppler Cooling limit
• Doppler cooling : cooling mechanism; Recoil heating : heating mechanism
• Temperature limit is determined by the relation that cooling rate is equal to
heating rate.
• Recoil heating can be treat as a random walk with momentum step size k.
p x2
  2k 2

E
heat
m v2

s0

2 1  s0  ( 2 ) 2
p x2
2m

 E
cool
 
  F  v  v 2
k BT
2
2
  1  s0  ( 2 ) 2
k BT 
4
2 

For low intensity s0<<1
k BT  


2
(

)
2
2

Minimum temperature
k BTD 

, when,     2
2
TD ~ 100-200 K for alkali atom
Magneto-optical trap (MOT)
• Cooling, velocity-dependent force: Doppler effect
• Trapping, position-dependent force: Zeeman effect
1-D case
3-D case
SubDoppler cooling
•
1.
2.
3.
Many cooling schemes allow one to cool atoms below the
Doppler limit, or even down to the recoil limit.
Polarization gradient cooling (Sisyphus cooling)
Raman cooling
Velocity-selective-coherent-population-trapping(VSCPT)
cooling
…
But we won’t discuss in this course.
Part II: Practical Issues about a
magneto-optical trap
Laser cooling : demonstrated species
Atomic species
• Different atomic species has its unique feature !
(5s5p)1P1
32MHz
F=5
6 2P3/2
5.2MHz
2 3P2
1.6MHz
4
(5s5p)3P1
4.7kHz
3
2
1083nm
2 3S1
metastable
cooling
852.35nm
repumping
460.73nm
Broad-line
cooling
689.26nm
Narrow-line
cooling
~20eV
by discharge
4
6 2S1/2
133Cs,
3
alkali metal, I=7/2
(5s2)1S0
88Sr,
alkali earth, I=0
1 0S1
4He,
nobel gas, I=0
Lasers
•
•
•
Diode lasers are extensive use in laser
cooling community due to inexpensive
cost and frequency tunability.
Diode lasers in external cavity
configuration are used to reduce the
laser linewidth.
Master oscillator power amplifier
(MOPA) configuration is used to
increase the available laser power.
ECDL in Littrow configuration
master
Diode laser
MOPA
Tampered
amplifiier
ECDL in Littman-Metcalf configuration
Laser frequency stabilization
• Frequency-modulated saturation
spectroscopy is the standard setup
to generate the error signal for
frequency stabilization.
• Feedback circuits are usually built
to lock the laser frequency.
laser
Background subtracted saturation spectrometer
spectrometer
Error signal
Feedback
circuit
Vacuum
• Two different kinds of vacuum setup are mainly used, one is glass vapor
cell, the other is stainless chamber.
• Ion pump and titanium sublimation pump are standard setup to achieve
ultrahigh vacuum.
Vapor-cell MOT
Chamber MOT
Magnetic field
• Anti-Helmholtz coils for the MOT
– Magnetic field reach maximum if the distance between two coils equal to the
radius of the coil
– Arial field gradient is twice the radial field gradient.
• Helmholtz coils for earth-compensation
– Magnetic field is most uniform ~ x4 when the distance between two coils equal
to the radius of the coil
– Earth compensation is critical to get good polarization gradient cooling.
• The magnitude of magnetic field scales ~  for different atomic species.
18
16
Axial magnetic gradient (G/cm)
14
12
10
Coil radius=6 cm
8
Current=5 A
6
Turn number=120
4
2
0
0
5
10
15
coil distance(cm)
20
25
30
Imaging
n ( x, y , z )
CCD camera
Itransmitted(x,y)
I0(x,y)
z
From experiment
From theory
I t  I 0 ( x, y )e OD( x , y )
OD ( x, y )  n( x, y, z ) absl ( x, y )
I  I dark
OD ( x, y )  ln( 0
)
I t  I dark
 OD( x, y)dxdy    n( x, y, z)l ( x, y)dxdy  
Considering the dark count of CCD
abs
abs
N
3* 2
1
 abs 
2 1  I I s  (2 ) 2
3* = 0~3, depends on laser polarization and
population distribution around Zeeman sublevels
How to determine the temperature?
 2 (t )   02  v 2t 2
v 
2 k BT
m
t=200 s t=500 s t=1000 s
MOT laser
1.88
t=2100 s
x 10-3
data
fit
1.86
1.84
Magnetic field
t
Sigma X (m)
1.82
1.8
1.78
1.76
Image beam
1.74
1.72
1.7
1.68
200
400
600
800
1000
delay (us)
1200
1400
1600
1800