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

pdF

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