Optically Polarized Atoms Atomic Transitions

8/10/2019 Optically Polarized Atoms Atomic Transitions

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Optically polarized atomsOptically polarized atoms

Marcis Auzinsh, University of LatviaDmitry Budker, UC Berkeley and LBNL

imon M! "ochester, UC Berkeley


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#reliminaries and de$nitions#reliminaries and de$nitions %ransition amplitude %ransition amplitude %ransition pro&a&ility %ransition pro&a&ilityAnalysis of a t'o(level pro&lemAnalysis of a t'o(level pro&lem

Chapter )* Atomictransitions

ee also*

#ro&lem +! http*--socrates!&erkeley!edu-.&udker-%utorial
http://www.oup.co.uk/isbn/0-19-850949-9


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%'o(level %'o(level

systemsystem

a

b

0

#eriodic pertur&atio#eriodic pertur&atio

/nitialCondition*


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%here are many 'ays to solve for the %here are many 'ays to solve for thepro&a&ility of $ndin0 the system in eitherpro&a&ility of $ndin0 the system in eitherof the t'o states, includin0of the t'o states, includin0

olve time(dependent chr1din0er e2uationolve time(dependent chr1din0er e2uationMake aMake a unitary transformationunitary transformation to 0et rid of timeto 0et rid of timedependence of the pertur&ation 3this isdependence of the pertur&ation 3this is

e2uivalent to 0oin0 into 4e2uivalent to 0oin0 into 4 rotatin0 framerotatin0 frame 5656olve theolve the Liouville e2uationLiouville e2uation for the densityfor the density

matri7matri7

8e 'ill discuss all this in due time, &ut let8e 'ill discuss all this in due time, &ut letus skip to the results for no'9us skip to the results for no'9

olvin0 the pro&lem9


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# : pro&a&ility of $ndin0 system in the upperstate

0

0

0

0

1V

= =

=

=

; Ma7imal(amplitude sinusoidal oscillations

; #t 6


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# : pro&a&ility of $ndin0 system in the upperstate

0

0

10

0

1V

= =

=

=

; Non (ma7imal(amplitude sinusoidal oscillations

; Oscillation fre2uency* E

; For the cases 'here al'ays #3t6GG *


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Heneral solution for any E 3I6


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/ncludin0 the e ect of rela7ation

0

0

0

0.3

1V

= =

=

=

; Decay to uno&served levels 3outside the system6

; Damped oscillations


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/ncludin0 the e ect of rela7ation

0

0

0

10

1V

= =

=

=

; Overdamped re0ime : no oscillations

; %his occurs for IJ=@ "

; %he system &ehaves as if there is no rela7ation for small t

; Heneral analytical

formula *


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election ruleselection rulesCertain 2uantities must remain conserved in a transitionCertain 2uantities must remain conserved in a transitionAn easy 'ay to think a&out it is theAn easy 'ay to think a&out it is the photonphoton picturepictureConserved 2uantities* ener0y, momentum,Conserved 2uantities* ener0y, momentum, total an0ulartotal an0ularmomentummomentum , 9, 98e have many an0ular momenta for atoms*8e have many an0ular momenta for atoms*

L S J I FFor0etFor0et II K< K > transitions arefor&idden

; Henerally, 'e have trian0le rule


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ntertainin0 /nterlude*ntertainin0 /nterlude* cuttin0 acuttin0 astickstickor 0ettin0 to kno' theor 0ettin0 to kno' the trian0le ruletrian0le rule

A stick is randomly cut into threeA stick is randomly cut into three

! 8hat is the pro&a&ility that one can! 8hat is the pro&a&ility that one canmake amake a trian0letrian0le out of the resultantout of the resultantsticks sticks

AA! -P! -P


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election ruleselection rules * 8hat chan0es 'hen J chan0es, L, S , or &oth

A* it isit is LL that chan0esthat chan0es or&ital rearran0ementor&ital rearran0ement /n classical electrodynamics, emission and a&sorption have to do/n classical electrodynamics, emission and a&sorption have to do'ith acceleratin0 char0es'ith acceleratin0 char0esAdditional selection rules 30ood to the e7tent L, are 0oodAdditional selection rules 30ood to the e7tent L, are 0ood2uantum Qs6*2uantum Qs6*

; Another form of the

> > transitions are for&idden rule


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#arity of atomic states#arity of atomic states

patial inversion 3#6patial inversion 3#6 ** , , x x y y z z

Or, inOr, in polarpolar coordinatescoordinates **, ,r r +

x y

z

x y

z


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#arity of atomic states#arity of atomic states /t mi0ht seem that/t mi0ht seem that ## is an operation that mayis an operation that may &e reduced to rotations&e reduced to rotations %his is %his is NOTNOT the casethe case

LetRs see 'hat happens if 'e invert a coordinate frame *LetRs see 'hat happens if 'e invert a coordinate frame *

x y

z

' x

' y

No' apply aNo' apply a rotationrotation around zRaround zR

' z

" x" y

'' z

"i0ht(handed frame"i0ht(handed frame left handedleft handed ## doesdoes NOTNOT reduce to rotations Sreduce to rotations S


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An amazin0 fact * atomic TamiltonianAn amazin0 fact * atomic Tamiltonian isis rotationally invariant &utrotationally invariant &ut is NOTis NOT P-P-invariantinvariant

8e 'ill discuss8e 'ill discuss parity nonconservationparity nonconservation e ects in detail later on in the course9e ects in detail later on in the course9


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8avefunctions in this form8avefunctions in this formare automatically of certainare automatically of certain parity *parity *

( )1 l P

nlm nlm

/n hydro0en, the electron is in/n hydro0en, the electron is in centro(symmetriccentro(symmetric nuclearnuclear potentialpotential /n more comple7 atoms, an electron sees a more complicated potential/n more comple7 atoms, an electron sees a more complicated potential

/f 'e/f 'e appro7imateappro7imate the potential from nucleus and other electrons asthe potential from nucleus and other electrons ascentro(symmetric 3and not parity violatin06 , then *centro(symmetric 3and not parity violatin06 , then *

inceince multi(electronmulti(electron 'avefunction is a 3properly antisymmetrized6 product of'avefunction is a 3properly antisymmetrized6 product of'avefunctions for each electron, parity of a multi(electron state is a product of parities for'avefunctions for each electron, parity of a multi(electron state is a product of parities foreach electron*each electron*

( )1 ii l

%his is &ecause* %his is &ecause*

C l i( l(


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Comments on multi(electronomments on mu t (e ectronatomsatoms

; #otential for individual electrons is NO% centrosymmetric

; An0ular momenta and parity of individual electrons are not e7act

notions 3 con$0uration mi7in0 , etc!6

; But for the system of all electrons, total an0ular momentum and parityare 0ood S

; #arity of a multi(electron state*

( ) ( ) ( )1 21 1 ... 1 nl l l

W A R N I N G

( )1 L


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#arity of atomic states#arity of atomic statesA &it of formal treatment9A &it of formal treatment9

Tamiltonian is #(invariant 3i0norin0 #NC6 * #Tamiltonian is #(invariant 3i0norin0 #NC6 * # (( HH #


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Back toBack to dipoledipole transitionstransitions %ransition amplitude * %ransition amplitude * GG == dd J , 'hereJ , 'here dd


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#arity selection rule#arity selection rule %ransition amplitude %ransition amplitude **

( )2 1

Odd under #Odd under #

( )32 1 2 1d r = r r

%his means that for the amplitude %his means that for the amplitude not to vanishnot to vanish , the product, the product

must also &emust also &e #(odd#(odd

/nitial and $nal states must &e of/nitial and $nal states must &e of opposite parityopposite parity

0 (


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0 er(mu t po e ra at vetransitions

/f electric(dipole(transition 3/f electric(dipole(transition 3 6 selection rules not satis$ed6 selection rules not satis$ed

for&iddenfor&idden transitionstransitions are due to the are due to the electric(dipole Tamiltonian*electric(dipole Tamiltonian* TT dd


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Ma0netic(dipole transitions Let us estimate the ratio of the transition matri7 elements forLet us estimate the ratio of the transition matri7 elements for MM andand

A typical atomicA typical atomic electric(dipoleelectric(dipole moment ismoment is ee aa

A typical atomicA typical atomic ma0netic(dipolema0netic(dipole moment ismoment is ZZ>>

%ransition pro&a&ility 3assumin0 same 'avelen0th6* %ransition pro&a&ility 3assumin0 same 'avelen0th6*

2

22 2250

2

2

( 1) 2~ ~ 10( 1) 2 2

eW M emc

W E ea ceme

= = =

h

h h


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Ma0netic(dipole transitions 8hat are the8hat are the M selection rulesM selection rules

/ma0ine a transition &et'een levels for 'hich an0ular(momentum selection rules are satis$ed, &ut parity rule/ma0ine a transition &et'een levels for 'hich an0ular(momentum selection rules are satis$ed, &ut parity rule is notis not

Notice*Notice* is ais a pseudo(vectorpseudo(vector 3


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Ma0netic(dipole transitions/mportant M transitions occur */mportant M transitions occur *

&et'een&et'een [eeman su&levels of the same state[eeman su&levels of the same state * NM", optical(pumpin0 ma0netometers, etc!* NM", optical(pumpin0 ma0netometers, etc!

&et'een&et'een hyper$ne(structure levelshyper$ne(structure levels * atomic clocks, the* atomic clocks, the = (cm line= (cm line

%his horn antenna, no' displayed in frontof the Kansky La& at N"AO in Hreen Bank,8\, 'as used &y Tarold 'en and

d'ard #urcell, then at the LymanLa&oratory of Tarvard University, in the$rst detection of the = cm emission fromneutral hydro0en in the Milky 8ay! %heemission 'as $rst detected on March =),

^) !ee* http*--

'''!nrao!edu-'hatisra-hist_e'enpurcell!shtml
http://www.nrao.edu/whatisra/hist_ewenpurcell.shtmlhttp://www.nrao.edu/whatisra/hist_ewenpurcell.shtmlhttp://www.nrao.edu/whatisra/hist_ewenpurcell.shtmlhttp://www.nrao.edu/whatisra/hist_ewenpurcell.shtml


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ome o er mu po etransitionslectric(2uadrupole 3lectric(2uadrupole 3 == 6 transitions6 transitions

No parity chan0e SNo parity chan0e S

8ith L couplin0 , 'e alsohave

%his can &e continued3 +,M=,96

' t t t t


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o' to ca cu ate trans t onpro&a&ility

/n/n 2uantum mechanics2uantum mechanics * transition pro&a&ility &et'een the* transition pro&a&ility &et'een the initialinitial

andand $nal state$nal state is proportional to*is proportional to*

' t t t t


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Let us recall ho' this comes a&out9Let us recall ho' this comes a&out9

For sin0le(electron atom, ne0lectin0 nuclear spinFor sin0le(electron atom, ne0lectin0 nuclear spin

Can this &e simpli$ed Can this &e simpli$ed LetRs relate the li0ht electric $eld and theLetRs relate the li0ht electric $eld and the vector potentialvector potential

8e can relate electron momentum to8e can relate electron momentum to atomic electric $eldatomic electric $eld this sho's that if the li0ht $eld is much 'eaker than the atomic this sho's that if the li0ht $eld is much 'eaker than the atomic

$eld, the term 2uadratic in$eld, the term 2uadratic in AA can &e ne0lected this is usually the case 3can &e ne0lected this is usually the case 3 e7cept modern ultra(short laser pulsese7cept modern ultra(short laser pulses 66

' t t t t
http://www.oup.co.uk/isbn/0-19-850949-9


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/n this appro7imation and ne0lectin0 electron spin */n this appro7imation and ne0lectin0 electron spin *

0

o' to ca cu ate trans t on


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%o calculate transition pro&a&ility, take matri7 elements of %o calculate transition pro&a&ility, take matri7 elements of

pertur&ation &et'eenpertur&ation &et'een com&ined states of li0ht and atomscom&ined states of li0ht and atoms **

ForFor a&sorptiona&sorption ,, n+1n+1 nn

Tere 'e used essential results fromTere 'e used essential results from

D*D* %hese re`ect the essential %hese re`ect the essential &osonic&osonic

properties of li0ht, and relateproperties of li0ht, and relate stimulatedstimulated

emissionemission andand a&sorption 'itha&sorption 'ith spont! em!spont! em!

Only annihilationoperator contributes

+1


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pro&a&ility

Ne7t, 'e apply theNe7t, 'e apply the DipoleDipole

Appro7imationAppro7imation **

and make use of theand make use of the

Teisen&er0 e2nTeisen&er0 e2n

**


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/nterlude* the/nterlude* the Teisen&er0Teisen&er02n2n !!

; Classical momentum*Classical momentum*

; /n M, time derivative of any/n M, time derivative of anyoperator is 0iven &yoperator is 0iven &y commutatorcommutator 'ith the Tamiltonian'ith the Tamiltonian

[ ] ( )

[ ]

,

,

d i i H H H

dt d im

m H dt

= =

= =

rr r r

rp r

h h

h

d m m

dt = = rp v


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pro&a&ility

8ith this 'e have *8ith this 'e have *

8e see that for8e see that for a&sorptiona&sorption , amplitude is, amplitude is

'hile for'hile for emissionemission , amplitude is, amplitude is

calar productcalar product of vectors can &e 'ritten asof vectors can &e 'ritten as


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pro&a&ility

8ith this 'e have *8ith this 'e have *

8e ne7t concentrate on the8e ne7t concentrate on the MM of the components of the dipoleof the components of the dipole

momentmoment

%he %he dynamicdynamic andand an0ularan0ular parts are separated usin0 the all(parts are separated usin0 the all(

importantimportant 8i0ner( ckart %heorem8i0ner( ckart %heorem

3j symbol


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pro&a&ility 8i0ner( ckart %heorem8i0ner( ckart %heorem

Useful property *Useful property *

3j symbolRe duce d matri eleme!t

"ctually# $it% commo! &%ase co!'e!tio!s# ((d(( are real !

+V sym&ols


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+V sym&ols "epresent the 0eometric part of transition amplitude"epresent the 0eometric part of transition amplitude "educed matri7 element : no reference to proVections*"educed matri7 element : no reference to proVections*

dynamic partdynamic part

+V sym&ols are standard functions in+V sym&ols are standard functions in MathematicaMathematica %M %M

Contain selection rules for an0ular(momenta addition,Contain selection rules for an0ular(momenta addition,

includin0 theincludin0 the trian0ular conditiontrian0ular condition

and theand the proVection ruleproVection rule

+V sym&ols* sum rules


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K< KR


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K< KR

Optically Polarized Atoms Atomic Transitions

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