Rydberg atomspart 1

Tobias Thiele

References

• Part 1: Rydberg atoms

• Part 2: 2 typical (beam) experiments

• T. Gallagher: Rydberg atoms

Content

Introduction – What is „Rydberg“?

• Rydberg atoms are (any) atoms in state with high principal quantum number n.

• Rydberg atoms are (any) atoms with exaggerated properties

equivalent!

Introduction – How was it found?

• In 1885: Balmer series:

– Visible absorption wavelengths of H:

– Other series discovered by Lyman, Brackett, Paschen, ...

– Summarized by Johannes Rydberg:

42

2

n

bn

2

~~

n

Ry

Introduction – Generalization

• In 1885: Balmer series:

– Visible absorption wavelengths of H:

– Other series discovered by Lyman, Brackett, Paschen, ...

– Quantum Defect was found for other atoms:

42

2

n

bn

2)(

~~

ln

Ry

• Energy follows Rydberg formula:

22)( n

hRy

n

hRyEE

l

Introduction – Rydberg atom?

0

0

Ener

gy

0

Hydrogen=13.6 eV

• Energy follows Rydberg formula:

2)( ln

hRyEE

Quantum Defect?

Quantum Defect

Ener

gy

0Hydrogen n-Hydrogen

Rydberg Atom Theory

• Rydberg Atom

• Almost like Hydrogen

– Core with one positive charge

– One electron

• What is the difference?

– No difference in angular momentum states

A

e

Radial parts-Interesting regionsW

r

rrV

1)(

0

)(1

)( rVr

rV core

0rr Ion core

lδ

nH

H

Interesting Region For Rydberg Atoms

(Helium) Energy Structure

• usually measured

– Only large for low l (s,p,d,f)

• He level structure

• is big for s,p

2)(2

1

lnW

l

l

Excentric orbits penetrate into core.Large deviation from Coulomb.Large phase shift-> large quantum defect

(Helium) Energy Structure

• usually measured

– Only large for low l (s,p,d,f)

• He level structure

• is big for s,p

•

2)(2

1

lnW

l

l

3)(

1

lndn

dW

Electric Dipole Moment

• Electron most of the time far away from core

– Strong electric dipole:

– Proportional to transition matrix element

• We find electric Dipole Moment

–

• Cross Section:

red

ififif rered )cos(

2)cos(1 nllrd if

42nr

3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n

• For non-Hydrogenic Atom (e.g. Helium)

– „Exact“ solution by numeric diagonalization of

in undisturbed (standard) basis ( ,l,m)

FdHH ififif

0

n~

2)(2

1

lnW

Numerov

3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n

Stark Effect EFdHH

0

Cross perfectRunge-Lenz vector conserved

Levels degenerate

n=12

n=11

n=13

Stark Map Hydrogen

n=12

n=11

n=13

k=-11blue

k=11red

Stark Map Hydrogen

• Rydberg Atoms very sensitive to electric fields

– Solve: in parabolic coordinates

• Energy-Field dependence: Perturbation-Theory

k

nn 21k

nn 21

Hydrogen Atom in an electric Field

EFdHH

0

)(199)(31716

)(2

3

2

1 522

21

24

212nOmnnnn

FnnnF

nW

Levels not degenerate

n=12

n=11

n=13

Stark Map Helium

s-type

Do not cross!No coulomb-potential

n=12

n=11

n=13

k=-11blue

k=11red

Stark Map Helium

Stark Map Helium

n=12

n=11

n=13

Inglis-TellerLimit α n-5

Rydberg Atom in an electric Field

• When do Rydberg atoms ionize?

– No field applied

– Electric Field applied

– Classical ionization:

– Valid only for

• Non-H atoms if F is

Increased slowly

3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT

• When do Rydberg atoms ionize?

– No field applied

– Electric Field applied

– Classical ionization:

– Valid only for

• Non-H atoms if F is

Increased slowly

416

1

nFcl

Rydberg Atom in an electric Field

4

2

16

1

4

1

n

WF

Fzr

V

cl

416

1 nFcl3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT

blue

• When do Rydberg atoms ionize?

– No field applied

– Electric Field applied

– Quasi-Classical ioniz.:

4

4

2

2

2

2

2

9

29

1

4

44

12)(

n

nZ

WF

FmZV

red

(Hydrogen) Atom in an electric Field

416

1 nFcl3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT

416

1

nFcl

49

1

nFr

49

2

nFb

blue

• When do Rydberg atoms ionize?

– No field applied

– Electric Field applied

– Quasi-Classical ioniz.:

4

4

2

2

2

2

2

9

29

1

4

44

12)(

n

nZ

WF

FmZV

red

(Hydrogen) Atom in an electric Field

416

1 nFcl3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT

416

1

nFcl

49

1

nFr

49

2

nFb

(Hydrogen) Atom in an electric Field

classic

Red states

Blue states

Inglis-Teller

Lifetime

• From Fermis golden rule

– Einstein A coefficient for two states

– Lifetime

2

3

3

,,,

2

,,,12

),max(

3

4nlrln

l

ll

c

eA

lnln

lnln

lnln ,,

1

,,

,,,,

lnln

lnlnln A

416

1 nFcl3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT

Lifetime

• From Fermis golden rule

– Einstein A coefficient for two states

– Lifetime

2

3

3

,,,

2

,,,12

),max(

3

4nlrln

l

ll

c

eA

lnln

lnln

lnln ,,

1

,,

,,,,

lnln

lnlnln A

416

1 nFcl3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT

For l≈0:Overlap of WF

23

n

30, nn

For l≈0:Constant (dominated by decay to GS)

Lifetime

• From Fermis golden rule

– Einstein A coefficient for two states

– Lifetime

2

3

3

,,,

2

,,,12

),max(

3

4nlrln

l

ll

c

eA

lnln

lnln

lnln ,,

1

,,

,,,,

lnln

lnlnln A

416

1 nFcl3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT

For l ≈ n:Overlap of WF

2n

53, ,nnln

For l ≈ n:Overlap of WF

3 n

Lifetime2

3

3

,,,

2

,,,12

),max(

3

4nlrln

l

ll

c

eA

lnln

lnln

1

,,

,,,,

lnln

lnlnln A

StateStark State

60 psmall

Circular state60 l=59 m=59

Statistical mixture

Scaling(overlap of )

Lifetime 7.2 μs 70 ms ms

3n 5n 5.4n

)1,1(),( lnln)l,n(

23

n 2nr

416

1 nFcl3)(

1

lndn

dW

2)(2

1

lnW

2

0nad

4n 5 nFIT53

, ,nnln