11

Optically polarized atomsOptically polarized atoms

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

Simon M. Rochester, UC Berkeley

A 12-T superconducting NMR magnet at the EMSL(PNNL) laboratory, Richland, WA

Dr. A. O. Sushkov, May 2007

Censorship

22

1845, Michael Faraday: magneto-optical rotation

Chapter 4: Atoms in external fields

Medium

Linear Polarization

Circular Components

MagneticField

Origin of magneto-optical rotation: the Zeeman effect

33

Faraday looked for effect of magnetic field on spectra, but failed to find it

1896, Pieter Zeeman: sodium lines broaden under B

1897, Zeeman observed splitting of Cd lines into three components (“Normal” Zeeman effect)

1897, Hendrik Lorentz: classical explanation of ZE

1898, discovery of Resonant Faraday Effect by Macaluso and Corbino

Zeeman effect: a brief history

4

Resonant Faraday Rotationrotation of the plane of linear light polarization by a medium in a magnetic field applied in the direction of light propagation in the vicinity of resonance absorption lines

D.Macaluso e O.M.Corbino,

Nuovo Cimento 8, 257 (1898)

Polarizer

Electromagnet

AnalyzerFlames of Na and Li

DiffractionGratingMonochromator

Photographic Plate

55

Energy in external field: Consider an atom with S=0 J=L In this case,

For magnetic field along z:

This is true for other states in the atom If we have an E1 transition, , A transition generally splits into 3 lines This agrees with Lorentz’ classical prediction

(normal modes), not the case for S0

“Normal” Zeeman effect

66

“Normal” Zeeman effectE1 selection rule: DM=0,1

M= -2 -1 0 1 2

Three lines !

77

“Normal” Zeeman effectClassical Model: electron on a spring

Three eigenfrequencies !

B

Eigenmodes:

88

The magnetic moment of a state with given The magnetic moment of a state with given J J is composed of is composed of

Zeeman effect when S0

99

Neglect interaction of nuclear magnetic Neglect interaction of nuclear magnetic moment with external magnetic field (it is moment with external magnetic field (it is ~2000 x smaller)~2000 x smaller)

However, average However, average μμ now points now points along along FF, not , not JJ AA vector-model vector-model calculations calculations a laa la the one we the one we

just did yields:just did yields:

Zeeman effect for hyperfine levels

1010

Definition of Definition of ggF F : : The magnetic moment is dominated by The magnetic moment is dominated by

the electron, for which we have:the electron, for which we have: To find To find μμ, we need to find the average , we need to find the average

projection of projection of J J on on FF, so that, so that

Now, findNow, find

Finally, Finally,

The actual calculation…

/J J Bg μ J

2 /J BgF

J F

μ F

J F 2 2

( 1) ( 1) ( 1)2

F F J J I I

F J Ι F J Ι F J Ι

J F

( 1) ( 1) ( 1)2 ( 1)F J

F F J J I Ig gF F

/F Bg μ F

1111

Consider Consider 2S1/2 atomic states (H, the alkalis, atomic states (H, the alkalis, group 1B--Cu, Ag, and Au ground states)group 1B--Cu, Ag, and Au ground states)

L=0; J=S=1/2 F=I1/2

This can beThis can beunderstood fromunderstood fromthe fact thatthe fact that μcomes fromcomes from J

Zeeman effect for hyperfine levels (cont’d)

1212

Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling

Hyperfine energies are diagonal in the Hyperfine energies are diagonal in the coupled basiscoupled basis::

However, Zeeman shifts are diagonal in the However, Zeeman shifts are diagonal in the ununcoupled basiscoupled basis: because : because

The bases are related, e.g., for The bases are related, e.g., for S=I=S=I=1/2 (1/2 (HH))FF,,MMFF MMSS, , MMII

1313

Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling

F=1,MF =1

F=1,MF =-1

F=1,MF =0

F=0,MF =0

1, 0 0, 0

1 1 1 1 1 1 1 1, , , ,2 2 2 2 2 2 2 2

2z

F F

z

B

BF M F M

B

B

BS

BS

1414

Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling

1515

Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling

Breit-Rabi diagrams• Nonlinear Zeeman Effect (NLZ)

• But No NLZ for

F=I+1/2, |M|=F states

• Looking more closely at the upper two levels for H :

• These levels eventually cross! (@ 16.7 T)

1616

Atoms in electric field: the Stark effector LoSurdo phenomenon

Johannes Stark (1874-1957)

Nazi Fascist

1717

Atoms in electric field: the Stark effector LoSurdo phenomenon

Magnetic: Electric:

However, things are as different as they can be…

Permanent dipole:

OK NOT OK

(P and T violation)

First-order effect Second-order effect

1818

Atoms in electric field: the Stark effectPolarizability of a conducting sphere

Outside the sphere, the electric field is a sum of Outside the sphere, the electric field is a sum of the applied uniform field and a dipole fieldthe applied uniform field and a dipole field

Field lines at the surface are normal, for Field lines at the surface are normal, for example, at equator: example, at equator:

1919

Atoms in electric field: the Stark effectClassical insights

Natural scale for atomic polarizability is the Natural scale for atomic polarizability is the cube of cube of Bohr radiusBohr radius

(a(a00))33 is also the is also the atomic unit of polarizabilityatomic unit of polarizability

In practical units: In practical units:

2020

Atoms in electric field: the Stark effectHydrogen ground state

n l m n l m Neglect spin!Neglect spin!

Polarizability can be found from Polarizability can be found from

2121

Atoms in electric field: the Stark effectHydrogen ground state (cont’d)

The calculation simplifies by approximatingThe calculation simplifies by approximating

=1

2222

Atoms in electric field: the Stark effectHydrogen ground state (cont’d)

Alas, this is Alas, this is HydrogenHydrogen, so use explicit wavefunction:, so use explicit wavefunction:

Finally, our estimate isFinally, our estimate is

Exact calculation: Exact calculation:

2323

Atoms in electric field: the Stark effectPolarizabilities of Rydberg states

The sum is dominated by terms with The sum is dominated by terms with nni i n nkk Better overlap of wavefunctionsBetter overlap of wavefunctions Smaller energy denominatorsSmaller energy denominators

ddik ik n n2 2 . . Indeed,Indeed,

((EEkk-E-Eii))-1-1 scale as scale as nn33

32 3

1 1 1; ;i k

dEE nn dn n E E

4 3 7n n n

2424

Atoms in electric field: the linear Stark effect

Stark Stark shifts increase, while energy intervals decrease for shifts increase, while energy intervals decrease for largelarge nn When shifts are comparable to energy intervals – the When shifts are comparable to energy intervals – the nondegenerate perturbation theory nondegenerate perturbation theory no longer works no longer works

even for lab fields <100 kV/cm even for lab fields <100 kV/cm use use degenerate perturbation theorydegenerate perturbation theory Also in molecules, where opposite-parity levels are separated by rotational energy ~10Also in molecules, where opposite-parity levels are separated by rotational energy ~10-3-3 Ry Ry Also in some special cases in non-Rydberg atoms: Also in some special cases in non-Rydberg atoms: HH, , Dy, Ba…Dy, Ba… In some In some BaBa states, polarizability is states, polarizability is >10>1066 a.u. a.u.

C.H. Li, S.M. Rochester, M.G. Kozlov, and D. Budker, Unusually large polarizabilities and "new" atomic states in Ba, Phys. Rev. A 69, 042507 (2004)

2525

The bizarre Stark effect in Ba

Chih-Hao Li Misha Kozlov

2626

The bizarre Stark effect in Ba (cont’d)

2727

The bizarre Stark effect in Ba (cont’d)

C.H. Li, S.M. Rochester, M.G. Kozlov, and D. Budker, Unusually large polarizabilities and "new" atomic states in Ba, Phys. Rev. A 69, 042507 (2004)

2828

Atoms in electric field: the linear Stark effect

Hydrogen 2s-2p states Opposite-parity levels are separated only by the Opposite-parity levels are separated only by the Lamb shiftLamb shift Secular equationSecular equation with a 2x2 Hamiltonian: with a 2x2 Hamiltonian:

Eigenenergies:Eigenenergies:

Quadratic Linear

Not EDM !

2929

Atoms in electric field: the linear Stark effect

Hydrogen 2s-2p states (cont’d) Linear shift occurs forLinear shift occurs for

Lamb ShiftLamb Shift: : ωωspsp/2/21058 GHz1058 GHz

Neglect spin!Neglect spin!

3030

Atoms in electric field: polarizability formalism

Back to quadratic Back to quadratic StarkStark, neglect hfs, neglect hfs Quantization axis along Quantization axis along E E MMJJ is a good is a good

quantum #quantum # Shift is Shift is quadratic inquadratic in E E same for same for MMJJ and and --MMJJ

A slightly involved A slightly involved symmetry argumentsymmetry argument based on based on tensors leads to the most general form of shifttensors leads to the most general form of shift

Scalar polarizability Tensor polarizability