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8422 Atomic Physics II Spring2013

Prof WolfgangKetterle

Assignment9

DueWednesdayMay12013

ConsiderthefollowingquestionsforNa Trytoplug innumbersafterobtainingananalyticalexpression

NaoventemperatureT =600K

naturallinewidth

γ = 2πtimes10MHz

Zeemansplitting14MHzgauss

wavelength

λ= 2πk

=589nm

1A

Zeeman

Slower

If youwanttoslowanatomicbeamefficientlyyouhavetocompensateforthechangingDopplershift

(983147 kmiddot983147v)duringthedeceleration Thiscanbedonebysweepingthefrequencyof theslowingbeam iechirping (as discussed in class) A method of producing a continuous beam of cold atoms is Zeeman

slowing

A well collimated beam of atoms is originating from an oven with a temperature T The beam

propagates along a distance L with a longitudinal magnetic field B(x) ( 0lt x lt L ) A laser beam

of intensity I iscounterpropagating Its frequency isdetuned by δ (δ equivωminusω0) from thetransition

frequencyatB=0

a) Calculatethemaximumdecelerationamax

youcanachieve Assumeyoucouldchoosearbitrarily

large laser intensities

Assume

you

want

to

slow

down

atoms

with

speeds

lower

than

the

peak

(most

probable)

velocity

v peak

of the thermal distribution to a stand still using the constant deceleration f amax (0 lt f lt 1) The

Zeeman effect shifts the resonance ω0

rarr ω0

+gmicroB

B(x) (where gmicroB

= (2π)14 MHzgauss in this

case)

b) Calculatethespatialdependenceof themagnetic fieldB(x)andthe lengthL of theslowerasa

functionof f

c) Assumethreedifferentmodelsforspontaneousemission (i)Photonsareemittedalongthe+minus xdirection (ii) isotropic emission or (iii) emission in a dipole pattern What is the momentum

diffusionconstantD forthe(longitudinal)x-componentof themomentum inthethreecases at

agivenphotonscatteringrateΓs

1

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2 Slowinganatom

with

off-resonant

light

Assumeyouwanttoslowanatomof velocity

v peak

withacounterpropagating laserbeamthat ison

resonancewiththeatomatrest (Usethesame

v peak

asinProblem1and

I = 5I sat)

a) Howlongwouldittake

b) Howfarwouldtheatomtravel

(Hint Thinkabout integratingtheequationof motion)

3 Density Limit in aMOT In a 3-D magneto-optical trap the density of the trapped atoms is

limited by a net outward radiation pressure which opposes the trapping force We can divide the

density-dependentphoton-pressure force intotwoparts Firstthere isarepulsive lsquoradiationtrapping

forcersquoduetoatomsabsorbingphotonsscatteredfromotheratomsinthetrap Alsothereisanattractivelsquoattenuationforcersquowhichiscausedbyatomsatthesideof thecloudattenuatingthelaserbeamsthuscreatingan intensity imbalancewhich leadstoaninwardforce

(a) Showthattheradiationtrappingforceobeystheequation

6σLσRIn

nabla middotFR

=

c

where

I is the intensityof one of the trapping laserbeams n

is the numberdensity of atoms in

the cloud σL

is the cross-section for absorption of the laser beam and σR

is the cross-section

forabsorptionof thescatteredlight (Hint Findthemagnitudeof theforcebetweentwoatoms

separatedbyadistancedwhereoneatomre-radiatesalaserphotonandthesecondatomabsorbs

it Nowsincethisisaninverse-square forceyoucanuseGaussrsquolawtofindnablamiddotFR Assumethat

photonsareonlyscatteredtwice)

(b) TheattenuationforcemaybeobtainedsimplybyreplacingσR

withminusσLsothat

6σ2InLnablamiddotFA

=minus

c

Explain

why

this

is

so

(c) The total force is the sum of FRFA and the trapping forceFT =minusκr where κ is the spring

constantof thetrap ForstabilitywerequirethatthetotalforceisattractivenablamiddotFtotal lt0 Find

themaximumdensityof thetrappedcloudatagiven

κfromtheconditionnabla middotFtotal

=0

(d) Give a qualitative argument for whether we expect

σR

=

σL σR

gt σL or

σR

lt σL (

Hintsketchtheabsorptionandemissionspectraforanatominastronglaserfieldwithareddetuning)

(e) Suppose that some of the atoms can be put into a lsquodark statersquo so that only a fraction

f of the

atoms absorb the trapping light How do

F R F A

and

F T

vary with

f What happens to the

limitingdensity

nmax This istheconceptof theDarkSPOTtrap(PRL

702253(1993))

2

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MIT OpenCourseWarehttpocwmitedu

8422 Atomic and Optical Physics II

Spring 2013

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

7232019 MIT8_422S13_hw9

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MIT OpenCourseWarehttpocwmitedu

8422 Atomic and Optical Physics II

Spring 2013

For information about citing these materials or our Terms of Use visit httpocwmiteduterms