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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!
106 103 1 10-3 10-6 10-9
0
(K)
core of sun surface of sun
room temperature
L N2
L He3He superfluidity
2003 MIT Na BEC
typical TC
of BEC
MOT
sub-Doppler cooling
Laser cooling and trapping of atom is a breakthrough to the exploration of theultracold 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
– Quantum statistics
– Uncertainty principle, zero-point energy
– System must be in an ordered state
– Quantum phase transition
Tmkh B 2 ~1μm for Na @ 100nk
Cold Atom
Cold Molecule
Cold Plasma &Rydberg Gas
Dipolar Gas
Many-body Physics
Quantum ComputationAtom Chips…
From Physics to Chemistry
From ground to highly-excited states
From isotropic to anisotropic interaction
From fundamentalto application
From atomic tocondensed-matter physics
Trends in Ultracold Research
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 intera
ction”• 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 OvchinnikovQuantum Semiclass. Opt. 7 No 1 (February 1995) 5-40 “Laser cooling of atoms: a review”
– Journal of Opt. Soc. Am. B, Issue 11,1989, special issue on laser cooling
The Light Force: Concept
iE
ikp
dt
sE '
skp
'
Photon posses energy and momentum !
An exchange of momentum &energy between photon and atom !
Force on atom
Net momentum exchange from the photon to atom
absorption emission
Energy and Momentum Exchange between Atom and Photon
•Atom absorbs a photon and re-emit another photon.
m
kkvkk
m
ppKKK
kkppp
sisisi
si
2
)()(
2
)'()('
)('2222
always positive, recoil heating
p
)( si kk
ik
sk
'p
If the momentum decrease, and if then < ΔK > avg <0 or < ωi > > < ωs > ,
where avg stands for averaging over photon scattering events.
0)( avg
si vkk
avg
si
avgsi m
kkvkk
2
)()(
2
Criteria of laser cooling
A laser cooling scheme is thus an arrangement of an atom-photo interaction scheme in which atoms absorb lower energy photon and emit higher energy photon on average!
The Light force : quantum mechanical
• Ehrenfest theorem, the quantum-mechanical analogue of Newton’s second law,
where V(r,t) is the interaction potential.
• Interaction potential: for an atom interacting with the laser field, , 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
EdV
ˆ
FrVdt
pdr
dt
dm
trVm
pH
)(
),,(2
ˆ
2
2
2
Discussion on semi-classical treatment
• Momentum width p is large compared with photon momentum k.
• Considering slow atoms only simplify the formalism. (Internal variables are fast components and variation of atomic motion is slow components in density matrix of atom ρ(r,v,t))
• Two conditions are compatible only if
• If the above conditions is not fullified, full quantum-mechanical treatment is needed. e.g. Sr narrow-line cooling, =27.5kHz ~ ωr=2k/2m=24.7kHz
1pk
,1 v or 1kv
an lower bound on v
an upper bound on v
1222
mk
J. Dalibard & C. Cohen-Tannoudhi, J. Phys. B. 18,1661,1985T.H. Loftus et.al. PRL 93, 073001,2004
Why Density Matrix Not Wavefunction?
• Pure versus Mixed ensemble.– The system that we are studied are usually not in the same state (descri
bed by the same wavefunction) but in a statistical mixture, e.g. atomic population follows Boltzman distribution both in internal states as
well as in external states. Atomic system under preparation (like optical pumping) can be in the same internal state. Bose-Einstein condensate is a system in the same state both in internally and externally .
– When dealing with atom-photon interaction, we usually interest in partial system (e.g. atomic system). Spontaneous emission caused by the coupling of atom with infinite degree of freedom of radiation results in a transition from an initial to a final state and can convert a pure state to a statistical mixture since phase information are lost !
• Density matrix formalism establishes a more direct connection with observables!
• Density matrix is a more powerful method for doing calculations.
Density Matrix• Probablity density to find particle in state |i> is
• The complete basis for state vector
• Diagonal elements are probabilities |cm|2 and off-diagonal terms are coherences cmcn* since they are depend on phase difference.
• Expectation value of operator
• Considering mixed ensemble instead of just pure ensemble, where Pm is classical statistical weight.
• If we are only interested in part of the system, the density matrix has to be average over the other part of the system.
iiiiiiP |||||)(2
m
m mc
nmnmccnm
mnnm
nm ,,
*ˆ
m
mmmP ̂
nm nm m
mmijjinmn
nm
m ATrAAnAmccncAmcA, ,
* )()()()(
)( ARRA Tr
The light force for a two-level atom
)sincos(2)()(
))(cos()(ˆ
1221121212212112
0
tvtudeeddddTrd
rtrEeE
EdUF
EdVU
titi
ρij (or σij)can be determined by the optical Bloch equation of atomic density matrix.
Where d12=d21 are assumed to be real and we have introduced the Bloch vectors u,v, and w.
)(2
1
)(2
1
)(2
1
1122
2112
2112
w
iv
u
Remark: dipole moment contain in phase and in quadrature components with incident field.
Note! A general form, can be plane wave,Gaussian beam…
Optical Bloch equation
dt
dH
idt
d ],[
1
Incoherent part due to spontaneous emission or others relaxation processes.The loss of quantum coherence is a bigIssue in quantum computation.
ijsponij
iisponii
dt
d
dt
d
2
)(,)(
012120
2211
11221212
21122222
12212211
);exp();()(
,1
)(2
)2
(
)(2
)(2
tirdEr
where
ii
dt
d
i
dt
d
i
dt
d
steady state solution
))2(1
1)(2(2;
)2(1
2
20
2120
022
s
i
i
s
s30 3
;/
ch
IIIS satsat
Isat ~ 1-10 mW/cm2 for alkali atom
Rabi frequency Ω characterize the magnitude of atom-photon interaction.
Two types of forces0)0( r
)()2
)(())(
2
)(( 2112
02112
0
irdErEd
FFF rpdip
radiation pressure or spontaneous emission forcea dissipative forceRelated to v vector
dipole force orgradient forcea reactive forcerelated to u vector
Without loss of generality, choose
)sincos()(2
)sin(cos)(
12
00
tvtudd
tEEteE
jj
jj
At r =0,
Take average over one optical cycle
))(ˆ()( 0012 vEEudeEdF avgjj
j
Origin of optical trapping Origin of optical cooling
Light force for a Gaussian beam
zk
Frp
FdipF
Optical Tweezers and Dipole Trap• Laser is far off-resonance, the dipole force dominates and trapping of small particles occurs.• For atom, it is called a optical dipole trap. Usually it has a trap depth around 1~1000 μK.
Spontaneous emission force
)(2 122122
11
i
dt
d
20
022 )2(12
)(
S
SRsp
Decay rate,
sprp RkkF
22 ,where Rsp is the flourescence rate.Its maximum value is .
Max deceleration for Na D2 line ! ,500002
gm
ka
From for steady-state 222112 )(2
i
For a plane wave 0;;)( 0 Ekrkr
2
Dipole Force in a standing wave• A standing wave has an amplitude gradient, but not a phase
gradient. So only the dipole force exists.
tkzEetrE x coscosˆ),( 0
24
)(
4 222
2
dipF
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.
]4
21ln[
2 22
2
U
UFdip
Velocity dependent force
Atom with velocity v experiences a Doppler shift kv.
20
0
))(2(12
vks
skFrp
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.
012
svk
Doppler Cooling
δ/
1)(,))2(1(
8
])(2[12
422
0
02
20
0
kvifv
s
vskF
kvs
skF
FFF
For δ<0, the force slows down the velocity.
[/k]
Doppler Cooling, Energy Point of View
• Red-detune laser photons are absorbed by atoms, spontaneously emitted photons have average energy on the resonance frequency.
• On average, atoms absorb lower energy photons and emit higher energy photon.
• Photons from laser are coherent, photons spontaneously emitted are quite random. Entropy of atoms are carried away by spontaneously emitted photons.
Atom
Laser Radiation Reservior
VAL, excite the atom
VAR, Radiation vacuum de-excite atom, Entropy flow
Finite degree of freedom infinite degree of freedom
Coherent photon
Incoherent photon
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.
2
)2(1
4
22
2
)2(12
20
2
2
2
20
0222
sTk
Tkvm
vvFEm
pE
s
skp
B
B
cool
x
heat
x
For low intensity s0<<1
)2
2(
2
TkB
Minimum temperature
2,,2
whenTk DB
TD ~ 100-1000 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
Position-dependent Force in a MOT
1)(,))2(1(
)(
)(
])(2[12
422
0
20
0
cxifxx
s
xBcF
mgmgc
cxs
skF
FFF
ggeeB
Considering v=0,
Sub-Doppler cooling
• Many laser cooling schemes allow one to cool atoms below the Doppler limit, or even down to the recoil limit.
1. Polarization gradient cooling (Sisyphus cooling)• Already exist in the MOT
2. Raman sideband cooling
3. Velocity-selective-coherent-population-trapping (VSCPT) cooling
…
Sisyphus Cooling• Polarization gradient cause a periodic modulation with
period of λ/2 for the ac Stark shift of the ground states.
• Atom climbs up the Stark potential and tends to be optically pumped to excited state and then spontaneously emit to the other ground state. It then repeat the same process
• On average, atoms absorb lower energy photons but emit higher energy photon.
Polarization Gradient Cooling• A new friction force mechanism for the low velocity atom (vτp~λ/4 where
τp is the optical pumping time ).
• Equiliurium temperature
Cs
Optical Pumping
Angular Momentum of Photon
Raman Sideband Cooling• Atoms are confined in a tight optical dipole trap and prepared in
polarized states.• Cooling cycle : |3,3;v> →Stimulated Raman transition → |3,1;v-1>
→optical pumping →|3,3;v=0> or |3,3;v> • |3,3;v=0> is dark both to Stimulated Raman transition and to optical
pumping light so population will accumulate here.• Since atoms are tightly trapped, recoil heating is negligible.
PRL81,5768(1998)
πσ+
VSCPT Cooling• Atoms are in the CPT dark states when their velocities are almost
zero.• Atomic velocity distribution are non-thermal (Levy flight). • Longer atom-photon interaction time cause narrower momentum
width.
PRL 61,826(1988)
Beyond Laser Cooling
• Evaporative cooling
• Sympathetic cooling
• Demagnetization cooling
• Stochastic cooling
• Feedback cooling
• ….???
)()( rBrU
Microwave transition
Part II: Practical Issues about a magneto-optical trap
Laser cooling : demonstrated species
Atomic species• Different atomic species has its unique feature !
852.35nm
6 2P3/2
5.2MHz
6 2S1/2
F=5
4
32
4
3
cooling
repumping
133Cs, alkali metal, I=7/2
(5s2)1S0
(5s5p)3P1
4.7kHz
(5s5p)1P1
32MHz
460.73nmBroad-linecooling
689.26nmNarrow-linecooling
88Sr, alkali earth, I=0
1 0S1
2 3S1
metastable
~20eVby discharge
4He, nobel gas, I=0
2 3P2
1.6MHz
1083nm
Lasers• Diode lasers are extensive use in l
aser 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
ECDL in Littman-Metcalf configuration
master
Tampered amplifiier
MOPA
Diode laser
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.
Background subtracted saturation spectrometerlaser
spectrometer
Error signal
Feedback circuit
Frequency Modulation Spectroscopy
• Frequency modulation and lock-in detection obtain dispersive error signal for frequency stabilization.
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.
0 5 10 15 20 25 300
2
4
6
8
10
12
14
16
18
coil distance(cm)
Axi
al m
agne
tic g
radi
ent
(G/c
m)
Coil radius=6 cm
Current=5 A
Turn number=120
MOT Alignment• Counterpropagating lasers are with the same polarizations (handness or helicity) but t
he configuration is referred as σ+σ- configuration in laser cooling. • Be careful the specifications from vendors on the quarter might be wrong or inconsist
ent.• A thumb rule !
B
laser
E
Fast axisslow axis
Imaging and Number of Atoms
I0(x,y) Itransmitted(x,y)
)ln(),(
),(
0
),(0
darkt
dark
yxODt
II
IIyxOD
eyxII
From experiment
Considering the dark count of CCD2
2*
)2(1
1
2
3
),(),,(),(
),(),,(),(
sabs
absabs
abs
II
NdxdyyxlzyxndxdyyxOD
yxlzyxnyxOD
From theory
3* = 0~3, depends on laser polarization and population distribution around Zeeman sublevels
How to determine the temperature?MOT laser
Magnetic field
Image beam
tm
Tkv
tvt
Brms
rms
2220
2 )( t=3 ms
t=7ms
t=15 ms
TOF(ms)
Size(mm)
Our Exploration, Cold Molecules
Buffer-gas cooling toprepare 4K large sampleof molecules.
Stark-guiding and opticalpumping to load molecules into a microwave trap.
Sympathetic cooling of molecules to mK in a microwave trap by ultracold atoms.
1 K 1 mK 1 μK
Evaporative cooling of molecules to μK in a microwave trap.
Why Cold Molecules ?• High-resolution spectroscopy
– Better understanding of molecular structure– Molecular clock
• Cold molecular collision and reaction – Precise determination of molecular potential energy– Controlled reaction by electromagnetic field
• Test of fundamental physics, – e.g. searching for electron dipole moment
• Study of quantum degenerate dipolar gases– Dipolar effect on Bose condensate– Cooper pairing by dipolar interaction
• Quantum computation
Welcome to join us to explore the ultracold world !
Ying-Cheng Chen, 陳應誠Institute of Atomic and Molecular Science, Academic Sinica,
Ultracold Atom and Molecule Labortory中研院原分所 超低溫原子與分子實驗室
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