Prof. dr hab. Wojciech Gawlik

Opracowanie utworu pod tytułem:

"Pułapki atomowe" w ramach kursu zaawansowanego, organizowanego dniach 31.08 – 25.09.09 będącego kontynuacją szkoleń z zakresu eksploatacji i zarządzania dużą infrastrukturą badawczą organizowanego przez Narodowe Laboratorium Technologii Kwantowych

2

1. Optical forces– light pressure, orders of magnitude, comparison with Coulomb

forces in ion traps, classical model – spontaneous and dipole forces

2. Deceleration of atomic beams, gas cooling, optical molasis

3. Magneto-optical traps,– detailed mechanisms (loading, repumping, etc.)

4. 4. TemperatureTemperature & & densitydensity limitslimits– subDoppler cooling

5. Dipole 5. Dipole trapstraps– magnetic and optical, optical lattices (main characteristics, loading,

comparison with other traps)

1

Wojciech GawlikWojciech Gawlik

Atom Atom trapstraps—— principlesprinciples & & realizationsrealizations

www.if.uj.edu.pl/ZF

part I

4

IonIon trapstraps vsvs. atom . atom trapstraps

el-stat. & el-magnet. forces

~iontrapsioniontrapstraps

light pressure & dipole forces(optical & magnetic)

I

I

atomtrapsatomatomtrapstraps

El-stat. forces ≈ 107 x optical forcesEl-stat. forces ≈ 107 x optical forces

3

• 1600-1900: Keppler Newton, Maxwell, Lebedev• Lebedev – light pressure on macroscopic objects

• Einstein (1917) – atomic/molecular gases thermalize in light-field• Compton (1923) – role of recoil in electron-photon scattering• Frisch (1933) – first observation of atomic-beam deflection by light

HistoryHistory

Laser Era:Laser Era:

• Ashkin (1970) – proposed accelerationand trapping of neutral particles

• exp. (1972, 73): Cologne, Orsay– deflection of atomic beams

6

0+=+=∆ ∑∑ Lemabs kNkkp

rh

rh

rh

r

Γ=∆∆

= eL nktpF

rh

rr

sodium atoms (M=23, λ = 590 nm) @ v=600m/s

after absorption of 1 photon: ∆v = hk/M = 3 cm/s⇒20 000 photons will stop atom

possible @ I = 6 mW/cm2

stopping time: 1 msstopping distance: 0,5 mdecelleration: 10 6 m/s2

Γ= enN

5

AtomicAtomic--beambeam decelerationdeceleration by by photonsphotons::

laser beam atom beam

HowHow do we do we coolcool atomsatoms??

S.Chu, C.Cohen-Tannoudji, W.Phillips

Principles of cooling and trapping of atoms by laser light –

Nobel 1997 →

8

( ) )(12/)v()(

22 rSkrSkF sp ++Γ⋅−

= rrhδ

γ

( ) )(12/)v()()v(

2 22 rSkrSkFd ++Γ⋅−

∇⋅−−= rrrrh

δδ

( )

( )

h

rr

h

h

Ed

d

d

⋅≡Ω

Ω+Γ+

Γ≡

Ω+Γ+−≡

22

22

2

222

2

222

2

δκ

δ

δαQuantum model: Quantum model: ω0 δ =ω – ω0

κ

kkkk iκαα +≡

SpectralSpectral/kinemat/kinematic ic ddeependencespendences::

SIrIr(r)S )()(2

2

2

=ΓΩ

=

Force depends on light frequency in the moving atom frameDoppler effect ⇒ velocity dependence

1) problem with atom-beam cooling2) possibility of atom-movement manipulation

Err

α0e=d

7

OpticalOptical forceforce –– clasicalclasical modelmodelAtom in EM field – Lorentz model

Forces ( ) ( ) EdtrEetrEeFC

rrrrrrr∇=−= ,, 01) Coulomb

2) Lorentz ( ) ( ) ( )EdEdEdBt

dBdt

FL

rrrrrrrrrrr∇−∇=×∇×=

∂∂

×−=×∂∂

=

total: ( )EdFFF LC

rrrrr⋅∇=+=

[ ] [ ]titititi erertrderertrE ωωωω −∗−∗ +=+= )()(2

1),(,)()(2

1),( rrrrrrrrrrrrddEE

harmonic

( ) ( )EErrrrr

⋅∇+⋅∇= ∗∗ ddF⇒ kkkk iκαα +≡

kkk

kkk

k IIF ϕκα ∇−∇= ∑∑==

3

1

3

121r

dipole force

rE redrr

−≡

light pressure force(spontaneous force)

),,(0

xyxie ϕ−=EE

Err

α0e=d

10

Slowing down atomic beams

variation of atomicresonance frequencyZeeman slower(W.D.Phillips)

2 methods:

resonance condition: ωL + k⋅v = ω0

Doppler effect – modifies the resonance conditions

When ωL, ω0 const, slowing down is limited →

variation of laser frequencylaser frequency chirping(V.S. Letokhov)

9[V.I. Balykin, et al. Opt. Lett. 13, 958 (1988)]

exp. evidence of atomic-beamdiffraction (channelling) →

Special cases

Light linearly polarized along x, ),,(0Iˆ xyxi

x e(x,y,z)e ϕε −=E

ϕκα ∇−∇= IIF xx21r

a) plane running wave zki0x ee

rr⋅= EE ˆ I = const =I0; ∇I = 0

ex nkIF Γ=∇−=r

hr

ˆ0 ϕκ ← force independent on

b) plane standing wave I = I0 cos2kz; ∇I = kI0 cos(kz- π/4)( )zkizkix eee

rrrr⋅−⋅ += 0ˆ EE

ϕ = const; ∇ϕ = 0

)4cos(ˆ21 πδα −=∇= kznkIF ex

rh

r

kzkrrr

=∇⋅= ϕϕ ;

periodic dependence on z

; intensity I=ε0|E|2

12

Adjustment of the magnetic field → choice of max. decelerated velocity υmax⇒ max. distance

attatttx

−=−=

max

221

max

)()(

υυυ

⇒max

maxmax

max1)(υω

υυk

xxx

B =

−=

Zeeman slower

maxmax 1 xxBB −=ωωmaxBω

[W. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982)J. Prodan et al., Phys. Rev. Lett. 49, 1149 (1982) ]

11

Laser frequency chirp

[ V. Balykin et al. Sov.Phys.JETP 53, 919 (1981);W. Ertmer et al. Phys. Rev.Lett. 54, 996 (1985)]

compare with...

14

kvz0– δ δ

F ∝ -vF ∝ -v „viscosity” → OPTICAL MOLASIS

force

zero force for v=0 cooling

net force – optical molasis

k k

13

frequencyω0

force

ωL

•• ForFor ωωLL<<ωω00,, DopplerDoppler effeff. . tunestunes atomsatoms to to resonanceresonance withwith countercounter--pprropagatingopagatingbeamsbeams –– eacheach beambeam exertsexerts deceleratingdecelerating lightlight--pressurepressure forceforce

•• DecelerationDeceleration = = ccoolingooling•• AbsorbedAbsorbed photonsphotons havehave smallersmaller energyenergy thanthan thethe reemittedreemitted onesones

ωωLL << ωω0 0 -- ccoolingooling

→ twotwo counterpropagatingcounterpropagating laser laser beamsbeams(same freq.; ωL < ω0)

ω0 ωωLL

atom atom „„seessees”” Doppler Doppler –– shiftedshifted lightlight frequencyfrequency

Atom Atom gasgas ??

16

hωL

m=+1

m=–1

m=0

B(z)

z=0 z

F(z) ∝ -z⇒⇒ positionposition--dependentdependent forceforce::atom trapatom trap

HowHow to trap to trap coldcold atomicatomic gasgas??

[J. Dalibard]σ+ σ–

15

• spontaneous forces and spatially inhomogenous beams –spatially inhom. saturation

• static magnetic fields for tailoring κ(r) ⇒ MOT

ϕκα ∇−∇= IIF xx21r

for plane standing wave F= Fdipol periodic dependence Fdipol(z),

for plane running wave F=Fspont – no dependence on z, ⇒ trapping impossible!

trapping requires

dipole forces

→ dipole traps

Trapping

18

Real Real trapstraps

17

RealizationRealization of a MOT of a MOT inin 3D3D

ω0 ω

I

I

-δ δ kvz0

force

F(vz) ∝ - vz

Cooling:

-δ δ βz0

force

F(z) ∝ - z

Trapping:

⇒⇒ velocityvelocity-- and and positionposition--dependentdependent forceforce::

[ S. Chu ]

20

Doppler theory of a MOT

notation:

(1D, beam interference neglected)

ω±= ω0 ± ωB(z)

[ ] zmkzgknzgkn

mF

Tzzz2

11 ˆ)()( ωυβυδυδ +=Γ

−−−++= −+h

damping constantconstant//viscosityviscosity coefficient

( )( )2

2

2222

2

2

2216

mcrec

rec

ωω

δ

δωβ

h=

Ω+Γ+Ω=

trap trap frequencyfrequency

zg

kg B

T ∂∂

==ω

βω 2

-- maximalmaximal dampingdamping 2,@221

max Γ==Ω−≅ δδβ kh

vz0-|δ|/k δ/k

β

z

19

22

Temperature limitation?

Cooling kinkinfric

Emm

pmp

tp

mpE

tβ

β2

==∂∂

=

∂∂

Heating:

1) stochastic direction of spont. em. stochastic recoils accelerate atom → diffusion (Brown movements)

∑∑Γ=

=

Γ=

=

Γ==⇒=tnN

jieji

tnN

ii

ee

tknkktpktp1,

22221

11 )()( h

rrh

rrh

r

2) granular nature of absorption (shot noise) # of abs. acts= N±∆N, ∆N= Ndifferent atoms experience different momentum transfers ⇒ broadening of velocity distrib.

tkntpNkNktp e Γ=⇒=∆= 22222 )()( hhh

total change of kinetic energy after time t: Γ≡=

∂∂ 22knD

mDE

t ebeamskindif

hN

Spontaneous emission → 2 mechanisms:

21

Number of trapped atomsreal atom (e.g. 87Rb) – escapes to other levels (to F=1)

2111

1222

NNrNN

NNrNRN

Γ′+Γ ′′−−=

Γ ′′+Γ′−−=&

&

R = capture rater = rate of collisions with hot-atom backgroundG' = depumping rateG'' = repumping rate

stationary

typical values: ( )( ) 021

2022==′↔=

Γ≈=′↔=

FFFF

repump

trap

δ

δ⇒

2/1600/

Γ≈Γ ′′Γ≈Γ′

r ≈ s-1 r>>Γ′>>Γ ′′ MHzRb 62)( ⋅=Γ π

typical R ≈ 10 8-9 s-1 yields N2 ≈108-9

with Ω ≈ Γ

( ) Γ ′′+Γ′=

Γ ′′+Γ′+Γ ′′+=

rNN

rrrRN 212

„repumper” necessary

F=3

210

F=2

1

trap laser

repumperR

r

rΓ’ Γ”

24

Atomic density limitation ?2 limiting regimes of a MOT:

1. Constant volume regime @ small density ρ2

2

20

rTk

m

B

v

eω

ρρ−

=

⇒ atomic cloud radius @ 1/e 2

2v

Be m

TkRω

= • independent on N• density ρ0 ∝ N

kabs

kem

radiation trappingwhen N grows, ρ,

)(max,0LRLLI

kσσσ

ρ−

≈2. Constant density regime

• atomic density = const, independent on N, • cloud size Re ∝ N

ρmax ≈ 1011-12 at/cm3

Td =T (1 + ?sn2/3N1/3).

23

Doppler temperature

equlibrium cooling – heating:

0=

∂∂

+

∂∂

kinfric

kindif

Et

Et

⇒β2DEkin = ⇒

ΓΓ

Ω+

Γ+

−== hδ

δ22

2

224 dk

DTk BN

(d – # of degrees of freedom)

minimum @ Γ

ΓΩ

+=⇒

ΓΩ

+Γ

= h2

min

2

214

212 d

TkBNδ

for 6 beams, 3 deg. of freedom, with Ω→0 : 2minΓ

→ hTkB

(Na: 240 (Na: 240 µµKK, , Rb:Rb: 140140 µµK)K)

BD k

T2Γ

=h

thethe model model includesincludes neitherneither lightlight interferenceinterference & & polarizationpolarization nor nor realreal atomicatomic structurestructure

( )1

4−+Γ−≈ δδ

dhN

26

Part I resume:

• Optical forcesa) origin of the optical f.b) dipole f.c) light pressure (spontaneous or Doppler) f.

• Application of spont. f. for decelearation of atomic beams• Application of spont. f. for cooling and trapping of atomic gas• MOT realization

a) basicsb) typical conditions performance

• Limitations of a MOT a) temperature (Doppler temp.)b) density (radiation trapping) kabs

kem

25

Wojciech GawlikWojciech Gawlik

Atom Atom trapstraps—— principlesprinciples & & realizationsrealizations

www.if.uj.edu.pl/ZF

part II

28

Examples of a cloud dynamicsafter switching off the quadrupole magnetic field with slightly misaligned beams

27

N ≈ 106 atoms of Rb85, T ≈ 100 µK

@ @ T T ≈≈ 0. 0001 K 0. 0001 K

υυatomatom ≈≈ 30 cm/sec30 cm/sec

TemperatureTemperature measurementmeasurement

[ T. Brzozowski et al. J. of Opt. B (Semiclass. and Quant. Opt.) 4, 62 (2002) ]

Examples of a working MOT

time of flight0

30

Transfer of cold atoms

• why?

• problems?

upper MOT: many atoms – vacum 10-9 mbar

lower MOT: vacum < 10-11 mbar

• how? light pressure force– pushing beam

geometry of differential pumping(distance ~0,5 m, cappilary ? 5 mm, L12 cm)

atoms blown out from the top MOTare recaptured in the lower one

29

MagnetoMagneto--optoptiiccalal trappingtrapping inin MOT1MOT1T ≈ 300 µK, N ≈ 108 – 109 MOT1

MOT2 & MT

ExampleExample applicationapplication –– ourour routeroute to BECto BEC

Transfer Transfer to MT, to MT, ccololllecectiontion inin MOT2MOT2

diff. pumping (10-9 mbar →10-11 mbar)

MagneticMagnetic trappingtrapping ((magneticmagnetic dipole trap)dipole trap)::evaporationevaporation of of thethe hottesthottest atomsatoms,,colisional thermalizationT ≈ 100 nK, N ≈ 105 - 106

32

• optical lattices

Robust 1D optical lattice(captures100 µK atoms)created by blue-detuned(160 MHz) retroreflected, pump (300 µW/mm2)

12 +Γ∆Γ=Ω

SS

V

160 µK 96 µK 75 µK 62 µK

Examples of spectroscopic diagnostics

31

Diagnostics of cold atoms (in detail Michal Zawada)

1) shape, size of atomic cloud ←← imagingimaginga) a) distructivedistructiveb) b) nonnon--distructivedistructive

2) # of atoms ← fluorescence, absorption3) density ← from 1) & 2)4) temperature

- time of flight measurement- trap oscillations- recoil-induced resonances

5) momentum distributions ←← spectroscopyspectroscopy6) local intensity of EM fields ←← spectroscopyspectroscopy7) dynamics / quatization of oscill. movement in optical lattice ←← spectroscopyspectroscopy8) diagnostics of cold-molecule formation (photoassociation) ←← spectroscopyspectroscopy9) diagnostics of quantum matter ←← imagingimaging

34

Dipole forces – classical picture

ω< ω0, (δ <0) – induced atomic dipole moment oscillates in phase with the field⇒ atom attracted to the field maximum

(electrostriction with classical dielectric, ε>1)

ω> ω0, (δ >0) – induced atomic dipole moment lags 180o behind the field⇒ atom expelled from the field maximum

(no analogy to electrostriction – no classical dielectric with ε<1)

33

)(rEdDEVVF rrrrii∇=⋅−==−∇= ∑

k z0

Fd F

FFsspp

)( Ddrr

atTr ρ= E(r, t) = E(r) ê(r) e-i[ωt-Φ(r)] + C.C.

OpticalOptical forcesforces::

∇Φ(r) → FFsspp –spontaneous, dissipative forces (light pressure)

∇ ê(r) → polarization gradient

∇ E(r) → FFdd – dipole, reactive

light pressure forces ––occur with plane waves,

dipole forces –– requireinhomogeneous light field

0

x,y

I Vd

-V0

0 z

I

Vd -V0

36

Cooling below Doppler limit (sub-Doppler cooling)

SysiphusSysiphus coolingcooling, , polarizationpolarization gradient gradient coolingcooling

Interpretation:

model atom with Jg=1/2, Je=3/2, 2 bems linearly polarized lin⊥lin, red detuned (δ <0)

• Selection rules ⇒→state couplings polarization dependent→light-shifts polarization dependent→opt. pumping polarization dependent

• Spatial correlation between light-shifts & optical pumping; opt. pump. populates most shifted Zeeman sublevels

• Interference ? resulting light polarizationposition-dependent

observed

35

Dipole forces – quantum picture

• atomic energy levels are modifiedby EM field

• the modifications depend on localfield intensity

force = UzEnzEn −∇=∇+∇ )()( 2211 ( )( )

( )( )22

2

221

12

)(

Γ+Ω

=

+=

δ

δ

zS

zSlnzU h

37

Recoil limit• when U approaches recoil energy, U → Erec, the model fails, atoms stop and oscillate in potential minima

T → Trec; kBT→ Erec

• atoms are ordered in periodic structures – optical lattices

• for deep lattices, atoms trapped at some lowest oscillatory levels

38

typical parameters:

• power: some W• detuning: ~100nm• focussing: below 100µm• trap depth: ~100µK• scattering constant: 1 s-1

The simples trap– tightly focused, far red detuned (δ<<0) laser beam

Nondissipative optical potential – dipole trappingFspont≈ hkΓ ne

Fdipol ≈ hkδ ne

⇒∝⇒Γ>> 2

1~δ

δ enFspont ≈ 1/ δ2

Fdipol≈ 1/δ

high intensity I and detuning δ causes large light shifts while atomic excitation(and dissipation by spont. emission) is negligible, ne<<1

→ atomic trapping possible in nondissipative dipole trapsinitial sub-Doppler precooling necessary

40

Nd: YAG

MOT

Loading

39

green – ground state of Rb, broken – excited @ 1W with 22 µm waist

Real atom – multilevel structure, complicated potentials

?

various trap realizationsdepending on the trap laser

41

Loading problems a) geometryb) cooling

42

Magnetic trap

• Heart of the MT– set of conical coils(two → quadrupole field, third = Ioffe coil – Majorana losses!).

• Along axis of the Ioffe coil – two offset coils (Helmholtz).

• Typical current 40A → water cooling

44

RF evaporation

N(υ)

υ

Bad scenario:

υ

N(υ)

Good scenario:

rνRF

U(r)

• MT – conservative – no energy dissipation;• The hottest atoms removed by RF resonance to other magnetic state;

43

40A & offset Bo=1,7938 G yields trap frequencies: radial νr = 137 Hzaxial νa = 12,6 Hz

Magnetic trap

46

Dipole traps – resume:

• less effficient than MOTs – need MOT for loading• no temp. limit – perfect for BEC• conservative character – extra cooling mechanism necessary

– evaporation!• optical – insensitive to magnetic states but not trivial evaporation• magnetic – only specific m-state trapped (J?0)

cool down....

45

Dipole trapping of classical (macro) objects4 µm polysterene beads in water [M. Burns et al., Science 249, 749 (1990)]

• beads pulled into high-intensity region (interferencepattern) (green Ar+laser beam is red-detuned fromUV polysterene absorption line)

• room-temp. (water cooling) sufficient for trapping• green Ar+ laser light visible due to Rayleigh

scattering on water molecules

Other important examples – optical tweezers