Optically polarized atoms

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Optically polarized atomsOptically polarized atoms

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

Simon M. Rochester, UC Berkeley

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A brief summary of atomic structure A brief summary of atomic structure Begin with Begin with hydrogen atomhydrogen atom TheThe SchrSchröödinger Eqndinger Eqn::

In this approximation (ignoring spin and In this approximation (ignoring spin and relativity):relativity):

Chapter 2: Atomic states

Image from Wikipedia

Principal quant. Number

n=1,2,3,…

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Could have guessed Could have guessed me me 44//22 from dimensions from dimensions me me 44//2 2 == 11 HartreeHartree me me 44//222 2 == 1 Rydberg1 Rydberg EE does not depend on does not depend on ll or or mm degeneracydegeneracy

i.e.i.e. different wavefunction have samedifferent wavefunction have same E E

We will see that the degeneracy is We will see that the degeneracy is nn22

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Angular momentum of the electron in the hydrogen atom

OrbitalOrbital-angular-momentum -angular-momentum quantum numberquantum number l l = 0,1,2,…= 0,1,2,…

This can be obtained, e.g., from the Schrödinger Eqn., or This can be obtained, e.g., from the Schrödinger Eqn., or straight from QM straight from QM commutation relationscommutation relations

The The Bohr modelBohr model: classical orbits quantized by requiring : classical orbits quantized by requiring angular momentum to be integer multiple of angular momentum to be integer multiple of

There is kinetic energy associated with orbital motion There is kinetic energy associated with orbital motion an upper bound on an upper bound on ll for a given value of for a given value of EEnn

Turns out: Turns out: l l = 0,1,2, …, = 0,1,2, …, nn-1-1

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(cont’d) In classical physics, to fully specify orbital angular In classical physics, to fully specify orbital angular

momentum, one needs two more parameters (e.g., two momentum, one needs two more parameters (e.g., two angles) in addition to the magnitudeangles) in addition to the magnitude

In QM, if we know projection on one axis (In QM, if we know projection on one axis (quantization quantization axisaxis), projections on other two axes are ), projections on other two axes are uncertainuncertain

Choosing Choosing zz as quantization axis: as quantization axis:

Note: this is reasonable as we expect projection Note: this is reasonable as we expect projection magnitude not to exceed magnitude not to exceed

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(cont’d) mm – – magnetic quantum number magnetic quantum number because because BB-field can be -field can be

used to define quantization axisused to define quantization axis Can also define the axis with Can also define the axis with EE (static or oscillating), (static or oscillating),

other fields (e.g., gravitational), or nothingother fields (e.g., gravitational), or nothing Let’s count states:Let’s count states:

m = -l,…,l m = -l,…,l i. e. i. e. 22ll+1+1 states states l l = 0,…,= 0,…,nn-1 -1 1

2

0

1 2( 1) 1(2 1)2

n

l

nl n n

As advertised !As advertised !

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(cont’d) Degeneracy w.r.t. Degeneracy w.r.t. m m expected from expected from isotropy of isotropy of

spacespace Degeneracy w.r.t. Degeneracy w.r.t. ll, in contrast,, in contrast, is a special is a special

feature of feature of 1/1/rr (Coulomb) potential (Coulomb) potential

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(cont’d) How can one understand restrictions that QM puts on How can one understand restrictions that QM puts on

measurements of angular-momentum components ?measurements of angular-momentum components ? The basic QM The basic QM uncertainty relationuncertainty relation (*) (*)

leads to (and permutations) leads to (and permutations)

We can also write a We can also write a generalizedgeneralized uncertainty relation uncertainty relation

between between llzz and and φφ (azimuthal angle of the e): (azimuthal angle of the e): This is a bit more complex than (*) because This is a bit more complex than (*) because φφ is is cycliccyclic With definite With definite llzz , , cos 0

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Wavefunctions of the H atom A specific wavefunction is labeled with A specific wavefunction is labeled with n l m n l m :: In In polar coordinatespolar coordinates : :

i.e. separation of i.e. separation of radialradial and and angular angular partsparts

Further separation: Further separation:

Spherical functions

(Harmonics)

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Wavefunctions of the H atom (cont’d)

Separation into radial and angular part is possible for any Separation into radial and angular part is possible for any central potential central potential !!

Things get nontrivial for Things get nontrivial for multielectron atomsmultielectron atoms

Legendre Polynomials

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Electron spin and fine structure

Experiment: electron has Experiment: electron has intrinsicintrinsic angular momentum angular momentum ----spin spin (quantum number (quantum number ss))

It is tempting to think of the spin classically as a spinning It is tempting to think of the spin classically as a spinning object. This might be useful, but to a point. object. This might be useful, but to a point.

2

c

(1)Presumably, we want finiteThe surface of the object has linear velocity (2)

If we have , (1,2) = 3.9 1

L I mr

r c

L rmc

110 cm

Experiment: electron is pointlike down to ~ 10-18 cm

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Electron spin and fine structure (cont’d)

Another issue for classical picture: it takes a Another issue for classical picture: it takes a 44ππ rotation rotation to bring a half-integer spin to its original state. to bring a half-integer spin to its original state. Amazingly, this does happen in classical world:Amazingly, this does happen in classical world:

from Feynman's 1986 Dirac Memorial Lecture (Elementary Particles and the Laws of Physics, CUP 1987)

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Another amusing classical pictureAnother amusing classical picture: spin angular : spin angular momentum comes from the electromagnetic field of the momentum comes from the electromagnetic field of the electron:electron:

This leads to electron sizeThis leads to electron size

Experiment: electron is pointlike down to ~ 10-18 cm

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ss=1/2 =1/2

““Spin up” and “down” should be used with understanding Spin up” and “down” should be used with understanding that the length (modulus) of the spin vector is that the length (modulus) of the spin vector is >>/2/2 ! !

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Both orbital angular momentum and spin have Both orbital angular momentum and spin have associated associated magnetic momentsmagnetic moments μμl l and and μμs s

Classical estimate of Classical estimate of μμl l : : current loopcurrent loop

For orbit of radius For orbit of radius rr, speed , speed p/m, p/m, revolution raterevolution rate is is

Gyromagnetic ratio

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In analogy, there is also In analogy, there is also spin magnetic moment spin magnetic moment ::

Bohr magneton

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The factor The factor 2 2 is important !is important ! Dirac equation for spin-1/2 predicts exactly Dirac equation for spin-1/2 predicts exactly 22 QED QED predicts deviations from 2 due to predicts deviations from 2 due to vacuum vacuum

fluctuationsfluctuations of the E/M field of the E/M field One of the most precisely measured physical One of the most precisely measured physical

constants: constants: 2=22=21.00115965218085(76)1.00115965218085(76)

Prof. G. Gabrielse, Harvard

(0.8 parts per trillion)

New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron, B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse, Phys. Rev. Lett. 97, 030801 (2006)

http://collargroup.uchicago.edu/pdf/phys-rev_2006.pdf

http://collargroup.uchicago.edu/pdf/phys-rev_2006.pdf

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When both When both ll and and ss are present, these are not conserved are present, these are not conserved separatelyseparately

This is like planetary spin and orbital motionThis is like planetary spin and orbital motion On a short time scale, conservation of individual angular On a short time scale, conservation of individual angular

momenta can be a good approximationmomenta can be a good approximation ll and and ss are coupled via are coupled via spin-orbit interactionspin-orbit interaction: interaction of : interaction of

the the motional magnetic field motional magnetic field in the electron’s frame with in the electron’s frame with μμss

Energy shift depends on relative orientation of ll and and ss, i.e., on , i.e., on

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QM parlance: states with fixed ml and ms are no longer eigenstates

States with fixed j, mj are eigenstates Total angular momentum is a constant of motion of

an isolated system

|mj| j If we add l and s, j ≥ |l-s| ; j l+s s=1/2 j = l ½ for l > 0 or j = ½ for l = 0

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Spin-orbit interaction is a relativistic effect Including rel. effects :

Correction to the Bohr formula 2

The energy now depends on n and j

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1/137 relativistic corrections are small

~ 10-5 Ry

E 0.366 cm-1 or 10.9 GHz for 2P3/2 , 2P1/2

E 0.108 cm-1 or 3.24 GHz for 3P3/2 , 3P1/2

, 1/ 2 , 1/ 2n j l n j lE E

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The Dirac formula :

predicts that states of same n and j, but different l remain degenerate

In reality, this degeneracy is also lifted by QED effects (Lamb shift)

For 2S1/2 , 2P1/2: E 0.035 cm-1 or 1057 MHz

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Example: n=2 and n=3 states in H (from C. J. Foot)

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Vector model of the atom Some people really need pictures… Recall: for a state with given j, jz

We can draw all of this as (j=3/2)

0; = ( 1)x yj j j j 2j

mj = 3/2 mj = 1/2

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Vector model of the atom (cont’d)

These pictures are nice, but NOT problem-free Consider maximum-projection state mj = j

Q: What is the maximal value of jx or jy that can be measured ?

A: that might be inferred from the picture is wrong…

mj = 3/2

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Vector model of the atom (cont’d)

So how do we draw angular momenta and coupling ? Maybe as a vector of expectation values, e.g., ?

Simple

Has well defined QM meaning

BUT

Boring

Non-illuminating

Or stick with the cones ? Complicated Still wrong…

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Vector model of the atom (cont’d) A compromise :

j is stationary l , s precess around j

What is the precession frequency? Stationary state – quantum numbers do not change Say we prepare a state withfixed quantum numbers |l,ml,s,ms This is NOT an eigenstatebut a coherent superposition of eigenstates, each evolving as Precession Quantum Beats l , s precess around j with freq. = fine-structure splitting

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Multielectron atoms Multiparticle Schrödinger Eqn. – no analytical soltn. Many approximate methods We will be interested in classification of states and

various angular momenta needed to describe them SE:

This is NOT the simple 1/r Coulomb potential Energies depend on orbital ang. momenta

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Gross structure, LS coupling Individual electron “sees” nucleus and other e’s Approximate total potential as central: φ(r) Can consider a Schrödinger Eqn for each e Central potential separation of angular and radial

parts; li (and si) are well defined ! Radial SE with a given li set of bound states Label these with principal quantum number

ni = li +1, li +2,… (in analogy with Hydrogen) Oscillation Theorem: # of zeros of the radial

wavefunction is nni i - - lli i -1-1

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Gross structure, LS coupling (cont’d) Set of Set of ni , , li for all electrons for all electrons electron configuration electron configuration Different configuration generally have different Different configuration generally have different

energiesenergies In this approximation, energy of a configuration is In this approximation, energy of a configuration is

just sum of just sum of Ei

No reference to projections of li or to spins degeneracy If we go beyond the central-field approximation some of the

degeneracies will be lifted Also spin-orbit (ls) interaction lifts some degeneracies In general, both effects need to be considered, but the

former is more important in light atoms

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Gross structure, LS coupling (cont’d)Beyond central-field approximation (Beyond central-field approximation (cfacfa))

Non-centrosymmetric part of electron repulsion (Non-centrosymmetric part of electron repulsion (11/r/rijij ) ) = = residualresidual Coulomb interaction ( Coulomb interaction (RCIRCI))

The energy now depends on how The energy now depends on how lli i andand ssi i combinecombine Neglecting Neglecting (ls) interaction LS coupling or Russell-

Saunders coupling This terminology is potentially confusing…..This terminology is potentially confusing….. …….. but well motivated !.. but well motivated ! Within Within cfacfa, individual orbital angular momenta are , individual orbital angular momenta are

conserved; conserved; RCI RCI mixes states with different mixes states with different projections of projections of lli i

Classically, Classically, RCI RCI causes precession of the orbital planes, so causes precession of the orbital planes, so the direction of the orbital angular momentum changesthe direction of the orbital angular momentum changes

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Gross structure, LS coupling (cont’d)Beyond central-field approximation (Beyond central-field approximation (cfacfa))

Projections of Projections of lli i are not conserved, but the are not conserved, but the totaltotal orbital orbital momentum momentum LL isis, along with its projection !, along with its projection !

This is because This is because lli i form sort of an form sort of an isolated systemisolated system

So far, we have been ignoring So far, we have been ignoring spinsspins One might think that since we have neglected One might think that since we have neglected (ls)

interaction, energies of states do not depend on spins

WRONG !

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Gross structure, LS coupling (cont’d)The role of the spinsThe role of the spins

Not all configurations are possible. For example, Not all configurations are possible. For example, U U has 92 has 92 electrons. The simplest configuration would be electrons. The simplest configuration would be 11ss9292

Luckily, such boring configuration is Luckily, such boring configuration is impossibleimpossible. Why ?. Why ? e’s are e’s are fermions fermions Pauli exclusion principlePauli exclusion principle: : no no

two e’s can have the same set of quantum numberstwo e’s can have the same set of quantum numbers This determines the richness of the periodic systemThis determines the richness of the periodic system Note: some people are looking for rare violations of Pauli Note: some people are looking for rare violations of Pauli

principle and Bose-Einstein statistics… principle and Bose-Einstein statistics… new physicsnew physics

So how does So how does spinspin affect energies (of allowed configs) ? affect energies (of allowed configs) ? Exchange Interaction

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Gross structure, LS coupling (cont’d)Exchange Interaction

The value of the The value of the total total spinspin S S affects the symmetry of affects the symmetry of the the spin wavefunctionspin wavefunction

Since overall Since overall ψ has to be antisymmetric symmetry of spatial wavefunction spatial wavefunction is affected is affected this affects Coulomb repulsion between electrons effect on energies

Thus, energies depend on L and S. Term: 2S+1L 2S+1 is called multiplicity Example: He(g.s.): 1s2 1S

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Gross structure, LS coupling (cont’d)

Within present approximation, energies do not depend on Within present approximation, energies do not depend on ((individually conservedindividually conserved) projections of ) projections of L L andand S S

This degeneracy is lifted by This degeneracy is lifted by spin-orbit interaction spin-orbit interaction (also spin-(also spin-spin and spin-other orbit)spin and spin-other orbit)

This leads to energy splitting This leads to energy splitting within within a term according to the a term according to the value of value of total angular momentum total angular momentum J (fine structure)

If this splitting is larger than the residual Coulomb interaction (heavy atoms) breakdown of LS coupling

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Example: a two-electron atom (Example: a two-electron atom (HeHe)) Quantum numbers:Quantum numbers:

J, mJ “good” no restrictions for isolated atoms

l1, l2 , L, S “good” in LS coupling ml , ms , mL , mS “not good”=superpositions

““Precession” rate hierarchy:Precession” rate hierarchy: l1, l2 around L and s1, s2 around S:

residual Coulomb interaction (term splitting -- fast)

L and S around J (fine-structure splitting -- slow)

Vector Model

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jj and intermediate coupling schemes

Sometimes (for example, in Sometimes (for example, in heavy atomsheavy atoms), ), spin-orbit interaction > residual Coulomb spin-orbit interaction > residual Coulomb LSLS coupling coupling

To find alternative, step back to To find alternative, step back to central-field approximationcentral-field approximation Once again, we have energies that only depend on

electronic configuration; lift approximations one at a time Since spin-orbit is larger, include it first

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jj and intermediate coupling schemes(cont’d)

In practice, atomic states often do not fully conform to In practice, atomic states often do not fully conform to LS LS oror jj jj scheme; sometimes there are different schemes for scheme; sometimes there are different schemes for different states in the same atom different states in the same atom intermediate couplingintermediate coupling

Coupling scheme has important consequences for Coupling scheme has important consequences for selection selection rulesrules for for atomic transitionsatomic transitions, for example, for example LL and and SS rules: rules: approximateapproximate; only hold within ; only hold within LSLS

couplingcoupling JJ, , mmJJ rules: rules: strictstrict; hold for ; hold for anyany coupling scheme coupling scheme

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Notation of states in multi-electron atomsSpectroscopic notation

Configuration (list of ni and li ) ni – integers li – code letters

Numbers of electrons with same n and l – superscript, for example: Na (g.s.): 1s1s222s2s222p2p663s = [Ne]3s3s = [Ne]3s

Term Term 22SS+1+1L L StateState 22SS+1+1LLJJ

22SS+1 = +1 = multiplicity multiplicity (another inaccurate historism)(another inaccurate historism) Complete designation of a state [e.g., Complete designation of a state [e.g., Ba (g.s.)Ba (g.s.)]: ]:

[Xe]6s[Xe]6s22 11SS00

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Fine structure in multi-electron atoms

LS sLS states with different tates with different JJ are split by are split by spin-orbit spin-orbit interactioninteraction

Example: Example: 22PP1/21/2--22PP3/23/2 splittingsplitting in the in the alkalisalkalis Splitting Splitting ZZ22 (approx.)(approx.)

Splitting Splitting with with nn

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Hyperfine structure of atomic states NuclearNuclear spinspin I I magnetic moment magnetic moment Nuclear magnetonNuclear magneton TotalTotal angular momentum: angular momentum:

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Hyperfine structure of atomic states (cont’d)

Hyperfine-structure splitting results from Hyperfine-structure splitting results from interaction of the interaction of the nuclear momentsnuclear moments with fields and with fields and

gradients produced by e’s gradients produced by e’s Lowest terms: Lowest terms:

M1M1 E2 E2

E2 term: E2 term: BB0 only for 0 only for II,,JJ>1/2>1/2

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Hyperfine structure of atomic states A nucleus can only support multipoles of rank A nucleus can only support multipoles of rank

κκ22II E1, M2, …. moments are forbidden by E1, M2, …. moments are forbidden by P P andand T T

BB0 only for 0 only for II,,JJ>1/2>1/2 Example of hfs splitting (not to scale)Example of hfs splitting (not to scale)

85Rb (I=5/2)

87Rb (I=3/2)

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Optically polarized atoms