Chapter 5 Experimental Apparatus II

40 Pages·2001·1.37 MB·English

Chapter 5
Experimental Apparatus II
159
160 Chapter5.ExperimentalApparatusII
5.1 Overview
In this chapter we discuss several modifications to the experimental apparatus described in
Chapter 3. These improvements were necessary to prepare localized atomic wave packets in
phase space for the experiments in Chapter 6. The first step towards such localized initial
states is further cooling of the atoms beyond what is possible in a typical MOT. We accom-
plishedthis additionalcoolinginathree-dimensional,far-detunedoptical lattice, aswediscuss
in Section 5.2. Further velocity selection well below the recoil limit was accomplished using
two-photon, stimulated Raman transitions. We will examine the theory of stimulated Raman
transitions as well as their experimental implementation in Section 5.3. It was also necessary
to have control over the spatial phase of the optical lattice, so that the wave packet could be
shiftedtovariousinitiallocationsinphasespace. Thisspatialcontrolwasaccomplishedthrough
an electro-optic phase modulator placed before the standing-wave retroreflector, as described
in Section 5.4. Finally, wetrace through the entire state-preparationsequence, usingall these
atom-optics tools, inSection 5.5, andwe discuss the calibration ofthe optical-lattice potential
inthemodifiedsetupinSection5.6.
5.2 Coolingin a Three-DimensionalOptical Lattice
UsingthestandardtechniquesofcoolingandtrappinginaMOT,asdescribedinChapter3,we
werelimitedtotemperaturesontheorderof10µKfortheinitialconditionsoftheexperiment.
It is desirable, however, to have much lower temperatures for the initial conditions, especially
lookingtowardsexperimentswithminimum-uncertaintywavepacketsinphasespace. Although
it has been shown that temperatures below 3 µK can be achieved in cesium using a standard
six-beamMOT[Salomon90],ourMOTtemperaturesweresubstantially higherduetoresidual
magnetic fields from eddycurrents in the stainless steel vacuum chamber after the field coils
were switched off. One successful approach to achieving additional cooling beyond that of a
standardMOTis coolinginathree-dimensional optical lattice. Severalmethodsfor cooling in
three-dimensionalopticallatticeshavebeendemonstrated[Kastberg95;Hamann98;Vuleti´c98;
Kerman00],butthemethodimplementedherewasbasedonthesetupdevelopedbythegroup
5.2CoolinginaThree-DimensionalOpticalLattice 161
ofDavidWeiss[Winoto99a;Winoto99b;DePue99;Wolf00;Han01].
The 3D optical lattice was formed by five beams, as illustrated in Fig. 5.1. Three of
the beamswereinthe horizontalplane;twoof these beamscounterpropagate, andthethird is
perpendicular to the other two. These beams formed a two-dimensionalinterference pattern,
consistingofalatticeofspotswithmaximumintensity. Thispatternthusformsconfiningpoten-
tialwellsforred-detunedlight,butnotforblue-detunedlight,wheretheintensitymaximaform
scattering barriers for the atoms, resemblingthe Lorentzgas. The use of three beamsfor this
two-dimensionallatticeisimportant,inthatusingtheminimumnumberofbeamstodetermine
alattice ensuresthatthe structure ofthe interferencepatternwillbestabletophaseperturba-
tions[Grynberg93]. Intheoriginalimplementationofthis lattice [Winoto99b;DePue99],four
beams(intwocounterpropagatingpairs)wereusedtoformthehorizontalpartofthelattice. Be-
causetheinterferencepatterncouldchangeitsperiodicitybyafactoroftwoasthephaseofone
of the beams varied, the authors inthat experiment implementedinterferometric stabilization
ofthe beamphases[Han01]. Inthe realizationhere, wesimplyomitted oneofthefour beams
to gain relatively easy stability at the expense of lattice intensity. The omission of one of the
Figure5.1: Configurationofthebeamsformingthethree-dimensionallatticeforadditionalcool-
ingoftheatoms. Fivetotalbeamsformthelattice, andthedirectionsofthelinearpolarizations
of each beam areindicated. Each of the beams isorthogonalto orcounterpropagatingwith re-
specttotheotherbeams. Thetwoverticalbeamsaredecoupledfromthethreehorizontalbeams
byan80MHzfrequencyshift. (GraphicsrenderedbyW.H.Oskay.)
162 Chapter5.ExperimentalApparatusII
beams was important in allowing long-term storage of the atoms in the lattice, as we describe
below,aswellasrepeatableatomictemperatures.
The other two beams in the 3D lattice counterpropagated in the vertical direction,
andthey were approximatelyperpendicular to the three horizontal beams. These beamswere
offset infrequency by 80MHz with respect tothe horizontalbeams. Inthis arrangement,the
interferences with the horizontal lattice oscillate on a time scale that is very fast comparedto
atomic motiontime scales, andthus itisappropriatetoregardthevertical beamsasdecoupled
from the horizontal beams in terms of analyzingthe interference pattern. Hence, the vertical
beamsproducedanormal1Dstanding-wavelattice,whichconfinedtheatomsvertically,andthe
threehorizontalbeamsconfinedtheatomsintheothertwodimensions.
Coolingin3Dlattices proceedsbyapplyingtheusualMOTbeamstotheatomsinthe
lattice. There are several mechanisms by which lattice cooling achieves much lower tempera-
turesthanastandardMOT.Thefirstmechanismisthatof“adiabaticcooling”[Jessen96],where
the applicationofthe lattice actsasaneffective refrigeratorcycle forcoolingthe atoms. When
theatomsareloadedintothelatticefromtheinitialMOT,theyareheatedbytheincreasingpo-
tentialinordertogainlocalconfinementinthelattice wells. LasercoolingbytheMOTbeams
proceeds asusual, coolingthe atomsfrom the heated temperature back downto normalMOT
temperatures. Whenthe lattice isthen adiabaticallyshut off(together withthe MOTbeams),
the temperature is further lowered at the expense of local confinement, in which we are not
necessarily interested. An important feature of the lattice configuration implemented here is
thatbecauseallthelightislinearlypolarizedandfar-detuned,themagnetic(Zeeman)sublevels
allexperiencethesameenergyshiftduetothelight,andsub-Dopplercoolingmechanismsthat
rely onsuch degeneratelevelstructure (polarization-gradientcooling[Dalibard89])proceedas
in the free-MOT case. This mechanism was especially important for the setup here, as the
atoms could be storedin the lattice until after the magneticfields decayed, allowingfor much
betterpolarization-gradientcoolingthanwecouldachieveinthestandardMOT.Itwasalsoim-
portanttoextinguishtheMOTbeamsadiabatically,astheylikewiseproducedanopticallattice
duetothesix-beaminterference. Thesecondmechanismforbettercoolinginthelatticerelates
5.2CoolinginaThree-DimensionalOpticalLattice 163
to suppressionof the absorptionof rescattered light inthe MOT.The second-handabsorption
of photons that have alreadybeen spontaneously scattered by MOT atoms, or “radiationtrap-
ping,” leadstotemperature anddensitylimitations infree-spaceMOTs[Sesko91; Ellinger94].
These rescattering events are particularly problematic in that they may be much more likely
to be absorbed than regular MOT photons, because their cross section for absorption is inde-
pendentofdetuningduetothe possibilityoftakingpartinatwo-photonstimulatedscattering
event [Castin98; Wolf00]. Inthefestinalenteregime[Castin98], however, where the photon
scatteringrate(duetolatticephotons,aswewillmentionbelow)issmallcomparedtothetrap
oscillation frequency (andthus the vibrational-level splitting), the recoil heating due to these
reabsorption eventsis suppressed[Castin98; Wolf00]. This is because most ofthe rescattered
photons in this regime are scattered elastically in the tight-confinement (Lamb-Dicke) limit,
andtheprobabilityofanatomchangingitsvibrationallevelbyscatteringsucharescatteredpho-
tonissmall. Thissuppressionofrescatter heatingisfurther enhancedbyathirdmechanismin
lattice cooling,wherethecoolingproceedsinanalogytoadarkMOT[Ketterle93]. Thismech-
anism obtains because the normal repumping light used in the regular MOT is extinguished
after the initial cooling phase in the lattice. Most of the atoms are thus in the dark(F = 3)
hyperfinelevel,andsothecoolinglightonlyaffectsasmallfractionoftheatomsatagiventime.
Thefar-detunedlatticelightprovidesslowrepumpingtothetrappingtransition. Thus,thelife-
timeforagivenvibrationallevelissetbythescatteringrateofoptical-lattice light, andnotthe
near-resonantMOTlight. Finally,coolinginthelatticehastheadditionalbenefitthatatomsare
separatedinindividuallattice sites, andthus light-assistedcollision losses andothercollisional
effects aresuppressed,resultinginanearlydensity-independentcoolingrate[Winoto99a].
Fortherealizationhere,thelightwasproducedbythesameTi:sapphirelaserthatpro-
videdthe1Dtime-dependentinteractionlattice. An80MHzAOMpickedofflightforthe3D
lattice justbeforethesimilarpickoffAOMforthe1Dlatticelight. Another80MHzAOMsplit
this beam into two parts, the first order (+80 MHz) having about 1/3 of the light, with the
remainder in the unshifted zeroth order. These two beams were spatially filtered by focusing
through 50µmdiameterpinholes. Theupshiftedlight formedtheverticallattice beams,while
164 Chapter5.ExperimentalApparatusII
the unshifted portion was further split in two with a half-wave plate and a polarizing beam-
splitter cube to form the horizontal beams. These three beams were all focused onto retrore-
flectingmirrorsontheoppositesidesofthechambersothatthe beamwaistw was500µmat
0
theirintersection;oneofthehorizontal,retroreflectedbeamswasblockedtoformthefive-beam
geometrydescribedabove. Each ofthe beamshadapproximately90mWof power. The lattice
had a typical detuning of 50 GHzto the red of the F = 3 −→ F(cid:1) transition multiplet (or 40
GHzto the redof F = 4 −→ F(cid:1)), leadingto anoscillation frequency in the vertical direction
of around170kHz(intheharmonic-oscillator approximation)andascatteringrate ofaround1
kHzatbeamcenter.
Theprocedureforlatticecoolingbeganwithabout5sofloadingtheregularMOTfrom
the background vapor. The optical molasses light intensity was then lowered to 60% of the
loading value, andthe detuning was increased to 37 MHz (from the 13MHz used during the
loading phase). At the same time, the 3D lattice was turned on adiabatically to minimize the
heatingof theatoms. Theintensity followedthetemporalprofileI(t) = I (1−t/τ)−2 (for
max
−800 µs< t < 0)[Kastberg95; DePue99], where the time constantτ was 30µs. Duringthis
lattice-loadingphase,theanti-Helmholtzfieldsandrepumplightwerebothleftontoencourage
rapid binding of the atoms to the 3D lattice. After a total of 22 ms in this loading phase, the
magneticfieldsandrepumplightwereextinguished, andthe molasseslight wasraisedbackup
to 100%intensity. The 3D lattice wasmaintained atfull intensity duringthe subsequent 298
msstoragetime, but the molasseslight wasrampedlinearly downto77%intensity bythe end
ofthisperiod. Thislongstoragetimewassufficienttoallowthemagneticfieldstodecaymostly
away(to70mGorbetter,whencompensatedproperlybytheHelmholtzcoils),althoughaslowly
varyingmagneticfieldwasstill detectable usingthe stimulated Ramanspectroscopydescribed
below. Then the MOT and 3D lattice beams were ramped down adiabatically according to a
similarprofile,I(t) = I (1+t/τ)−2(withthesametimeconstant),over800µs. Themolasses
0
lightbeganitsrampingdownabout20µsbeforethe3Dlatticebeams,givingtheoptimumfinal
temperature.
Thislattice-coolingprocedureledtoanatomicpopulationintheF =3levelwitha1D
5.3StimulatedRamanVelocitySelection 165
temperature(inthehorizontaldirection)of400nK,orσ /2(cid:1)k =0.7.Between50%and90%of
p L
theatomsremainedtrappedinthelatticeduringthecoolingcycle,dependingsensitivelyonhow
wellthelatticewasaligned. Theverticaltemperatureof500nK(σ /2(cid:1)k =0.8)wassomewhat
p L
higher;thetemperaturecouldbemademoreisotropicbychangingtherelativebeampowers,but
attheexpenseofthehorizontaltemperature,whichwastheonlyimportanttemperatureforthe
experimentshere. Thelatticeworkedwelloverdetuningsof25-70GHz(fromF =3 −→F(cid:1));
for closer detunings the final temperature began to rise, and at larger detunings, the fraction
retainedinthelatticedroppedoff.
For some experiments, it wasnecessary to prepare the atoms in the F = 4 hyperfine
level. This could be conveniently achieved by pulsing on the repumping light for 100 µs af-
ter the lattice andmolasses fields wereextinguished, at the expenseof temperature (the final
temperature wastypically 700nKafter repumping). Toimplementstimulated Ramanvelocity
selection, as we discuss in the next section, further optical pumping to the F = 4,m = 0
F
Zeemansublevelwasnecessary,aswediscussinSection5.3.5.
5.3 Stimulated Raman Velocity Selection
Now we consider the implementation of two-photon, stimulated Ramantransitions in cesium
forsubrecoil (i.e., smallerthanthe single-photonmomentum)velocity selection. After givinga
generaloverview of the theory behind stimulated Ramantransitions andvelocity selection, we
willgivethedetailsofourimplementationaswellasadiscussionofopticalpumpingandinternal
stateselectionnecessaryforacleanvelocity-selection method.
5.3.1 StimulatedRamanTransitions: GeneralTheory
Weconsidertheatomicenergylevelstructure showninFig.5.2,wheretwogroundstates|g (cid:4)
1,2
are coupled to a manifold of excited states |e (cid:4) by two optical fields. Our goal is to show that
n
under suitable conditions, the atomic population can be driven between the ground states as
in a two-level system. We restrict our attention to the case where the fieldspropagatealonga
166 Chapter5.ExperimentalApparatusII
commonaxis. Inthecounterpropagatingcase,thecombinedopticalfieldhastheform
E(x,t)=(cid:15)ˆ E cos(k x−ω t)+(cid:15)ˆ E cos(k x+ω t)
1 01 1 L1 2 02 2 L2
(5.1)
=E(+)(x,t)+E(−)(x,t) ,
whereE(±)(x,t)arethepositiveandnegativerotatingcomponentsofthefield,givenby
(cid:1) (cid:2)
E(1±)(x,t)= 12 ˆ(cid:15)1E01e±ik1xe∓iωL1t+(cid:15)ˆ2E02e∓ik2xe∓iωL2t , (5.2)
andˆ(cid:15) arethe unit polarizationvectors of the twofields. The results that wewill derivealso
1,2
applytothecopropagatingcaseaswelluponthesubstitution k →−k .
2 2
ThefreeatomicHamiltoniancanthenbewritten
(cid:3)
p2
H = +(cid:1)ω |g (cid:4)(cid:5)g |+(cid:1)ω |g (cid:4)(cid:5)g |+ (cid:1)ω |e (cid:4)(cid:5)e | , (5.3)
A 2m g1 1 1 g2 2 2 en n n
n
andtheatom-fieldinteractionHamiltonianis
H =−d(+)·E(−)−d(−)·E(+) , (5.4)
AF
wherewehavemadetherotating-waveapproximation,wehaveassumedthatω :=ω −ω (cid:7)
21 g2 g1
ω :=max{ω }−ω ,andwehaveinmindthatthe|e (cid:4)arenearlydegenerate.Additionally,
egj en gj n
}
den |e(cid:150)æ
w
eg
2
w
eg1 w
L2
w
L1
|g“æ w
21
|g`æ
Figure 5.2: Energy level diagram for stimulated Raman transitions. Each ground level |g (cid:4) is
j
coupled to the excited-state manifold |e (cid:4) via two laser fields, which are tuned so that their
n
detuningsfromtheexcited-statemanifoldarenearlythesame.
5.3StimulatedRamanVelocitySelection 167
wehavedecomposedthedipoleoperatordintoitspositive-andnegative-rotatingcomponents,
d=d(+)+d(−)
(cid:3)(cid:1) (cid:2) (cid:3)(cid:1) (cid:2) (5.5)
= a (cid:5)e |d|g (cid:4)+a (cid:5)e |d|g (cid:4) + a† (cid:5)e |d|g (cid:4)+a† (cid:5)e |d|g (cid:4) ,
1n n 1 2n n 2 1n n 1 2n n 2
n n
wherea :=|g (cid:4)(cid:5)e |isanannihilationoperator.Substituting(5.5)into(5.4),wefind
jn j n
(cid:3) (cid:1) (cid:2)
HAF =− 12(cid:5)en|ˆ(cid:15)1·d|g1(cid:4)E01 a1neik1xe−iωL1t+a†1ne−ik1xeiωL1t
(cid:3)n (cid:1) (cid:2) (5.6)
− 12(cid:5)en|ˆ(cid:15)2·d|g2(cid:4)E02 a1ne−ik2xe−iωL2t+a†2neik2xeiωL2t .
n
Inwritingthis expression,wehaveassumedthedetunings∆ := ω −ω arenearlyequal;
Lj Lj egj
hence, to make this problem more tractable, we assume that the field E couples only |g (cid:4) to
j j
the |e (cid:4). After solving this problem we will treat the cross-couplings as a perturbation to our
n
solutions. IfwedefinetheRabifrequency
−(cid:5)e |ˆ(cid:15) ·d|g (cid:4)E
Ω := n k j 0k , (5.7)
jkn (cid:1)
whichdescribesstrengthofthecouplingfromlevel|g (cid:4)throughfieldE tolevel|e (cid:4),wearrive
j k n
at
HAF =(cid:3)(cid:1)Ω211n(cid:1)a1neik1xe−iωL1t+a†1ne−ik1xeiωL1t(cid:2)
+n(cid:3)(cid:1)Ω222n(cid:1)a1ne−ik2xe−iωL2t+a†2neik2xeiωL2t(cid:2) (5.8)
n
asaslightlymorecompactformfortheinteractionHamiltonian.
Now, before examining the equations of motion, we transform the ground states into
therotatingframeofthelaserfield,asinChapter2:
|˜gj(cid:4):=e−iωLjt|gj(cid:4)
(5.9)
E˜(±) :=e±iωLktE(±) .
k k
Also, forconcreteness, wewilltakemax{ω }= 0. Thentherotating-frame,free-atomHamil-
en
tonianis
(cid:3)
p2
H˜ = +(cid:1)∆ |˜g (cid:4)(cid:5)˜g |+(cid:1)∆ |˜g (cid:4)(cid:5)˜g |+ (cid:1)δ |e (cid:4)(cid:5)e | , (5.10)
A 2m L1 1 1 L2 2 2 en n n
n
168 Chapter5.ExperimentalApparatusII
whereδ :=ω −max{ω }(i.e.,δ ≤0).TheinteractionHamiltonianintherotatingframe
en en en en
is
H˜ =−d˜(+)·E˜(−)−d˜(−)·E˜(+)
AF
=(cid:3)(cid:1)Ω211n(cid:1)˜a1neik1x+a˜†1ne−ik1x(cid:2)+(cid:3)(cid:1)Ω222n(cid:1)˜a1ne−ik2x+a˜†2neik2x(cid:2) , (5.11)
n n
wheretheannihilationoperatora˜ isdefinedinthesamewayasa ,butwith|g (cid:4)replacedby
jn jn j
|˜g (cid:4).
j
Turningto the equationsof motion, wewill manifestly neglect spontaneousemission,
since ∆ (cid:11) Γ, where Γis the decayrateof |e (cid:4), byusingaSchro¨dinger-equationdescription
Lj n
oftheatomicevolution. Thenwehave
i(cid:1)∂ |ψ(cid:4)=(H˜ +H˜ )|ψ(cid:4) , (5.12)
t A AF
wherethestatevectorcanbefactoredintoexternalandinternalcomponentsas
(cid:3)
|ψ(cid:4)=|ψ (cid:4)|˜g (cid:4)+|ψ (cid:4)|˜g (cid:4)+ |ψ (cid:4)|e (cid:4) . (5.13)
g1 1 g2 2 en n
n
Thenifψ (x,t):=(cid:5)x|ψ (cid:4),weobtaintheequationsofmotion
α α
p2 (cid:1)Ω (cid:1)Ω
i(cid:1)∂tψen = 2mψen + 211ne−ik1xψg1 + 222neik2xψg2 +(cid:1)(δen −∆L)ψen
p2 (cid:3)(cid:1)Ω
i(cid:1)∂tψg1 = 2mψg1 + 211neik1xψen +(cid:1)(∆L1−∆L)ψg1 (5.14)
n
p2 (cid:3)(cid:1)Ω
i(cid:1)∂tψg2 = 2mψg2 + 222ne−ik2xψen +(cid:1)(∆L2−∆L)ψg2 ,
n
where we have boostedall energies by−(cid:1)∆ , with ∆ := (∆ +∆ )/2(i.e., weappliedan
L L L1 L2
overallphaseofei∆Lttothestatevector). Sinceweassumethat|δen|(cid:7)|∆L|and|∆L2−∆L1|(cid:7)
|∆ |, itis clear that the ψ carry the fasttime dependenceat frequencies oforder|∆ | (cid:11) Γ.
L en L
We are interested in motion on timescales slow comparedto 1/Γ, andthe fast oscillations are
dampedbycouplingto thevacuum ontimescales of 1/Γ,sowecanadiabatically eliminate the
ψ bymakingthe approximationthatthey dampto equilibrium instantaneously(∂ ψ = 0).
en t en
Also,weusep2/2m(cid:7)(cid:1)|∆ |,withtheresult,
L
Ω Ω
ψen = 2(∆ 1−1nδ )e−ik1xψg1 + 2(∆ 2−2nδ )eik2xψg2 . (5.15)
L en L en