Rydberg atoms

Tobias Thiele References

• T. Gallagher: Rydberg atoms Content

• Part 1: Rydberg atoms

• Part 2: A typical beam experiment

Introduction – What is „Rydberg“?

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

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

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:

bn2   n2  4

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

– Summarized by Johannes Rydberg: Introduction – How was it found?

• In 1885: Balmer series:

– Visible absorption wavelengths of H:

bn2   n2  4

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

– Summarized by Johannes Rydberg: Introduction – How was it found?

• In 1885: Balmer series:

– Visible absorption wavelengths of H:

bn2   n2  4

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

Ry – Summarized by Johannes Rydberg: ~ ~   n2 Introduction – Generalization

• In 1885: Balmer series:

– Visible absorption wavelengths of H:

bn2   n2  4

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

~ ~ Ry – Quantum Defect was found for other atoms:     2 (n l ) Introduction – Rydberg formula?

• Energy follows Rydberg formula: hRy E  E   2 (n l )

Introduction – Rydberg formula?

• Energy follows Rydberg formula:

hRy 13.6 eV E  E   2 (n l )

Introduction – Hydrogen?

• Energy follows Rydberg formula: 13.6 eV 0 hRy hRy E  E     2 2 (n l ) n 0

0

Energy Quantum Defect?

• Energy follows Rydberg formula: hRy E  E   2 (n l ) Quantum Defect

0

Energy Rydberg Atom Theory e • Rydberg Atom A

• Almost like Hydrogen – Core with one positive charge – One electron

• What is the difference?

Rydberg Atom Theory e • Rydberg Atom A

• Almost like Hydrogen – Core with one positive charge – One electron

• What is the difference?

Rydberg Atom Theory e • Rydberg Atom A

• Almost like Hydrogen – Core with one positive charge – One electron

• What is the difference?

Rydberg Atom Theory e • Rydberg Atom A

• Almost like Hydrogen – Core with one positive charge – One electron

• What is the difference? – No difference in angular momentum states

Radial parts-Interesting regions W

1 V (r)   r 0 r Radial parts-Interesting regions W

1 1 V (r)   V (r)   V (r) r r core 0 r Radial parts-Interesting regions W r  r0 Ion core

1 1 V (r)   V (r)   V (r) r r core 0 r Radial parts-Interesting regions W r  r0 Ion core

1 1 V (r)   V (r)   V (r) r r core 0 r

Interesting Region For Rydberg Atoms Radial parts-Interesting regions W r  r0 Ion core

1 1 V (r)   V (r)   V (r) r r core 0 r

H Radial parts-Interesting regions W r  r0 Ion core

1 1 V (r)   V (r)   V (r) r r core 0 r

H

nH Radial parts-Interesting regions W r  r0 Ion core

1 1 V (r)   V (r)   V (r) r r core 0 r

δl

H

nH 1 (Helium) Energy Structure W   2 2(n l )

•  l usually measured – Only large for low l (s,p,d,f) • He level structure

• is big for s,p

1 (Helium) Energy Structure W   2 2(n l )

•  l usually measured – Only large for low l (s,p,d,f) • He level structure

• is big for s,p

1 (Helium) Energy Structure W   2 2(n l )

•  l usually measured – Only large for low l (s,p,d,f) • He level structure

•  l is big for s,p

1 (Helium) Energy Structure W   2 2(n l )

•  l usually measured – Only large for low l (s,p,d,f) • He level structure

•  l is big for s,p

Excentric orbits penetrate into core. Large deviation from Coulomb. Large phase shift-> large quantum defect 1 (Helium) Energy Structure W   2 2(n l )

•  l usually measured – Only large for low l (s,p,d,f) • He level structure

•  l is big for s,p

dW 1 •  3 dn (n l )

Electric Dipole Moment

• Electron most of the time far away from core – Strong electric dipole: – Proportional to transition matrix element

1 dW 1 W    2 dn (n  )3 2(n l ) l Electric Dipole Moment

• Electron most of the time far away from core   – Strong electric dipole: d  er – Proportional to transition matrix element

1 dW 1 W    2 dn (n  )3 2(n l ) l Electric Dipole Moment

• Electron most of the time far away from core   – Strong electric dipole: d  er – Proportional to transition matrix element   f d i  e f r i  e f r cos() i

1 dW 1 W    2 dn (n  )3 2(n l ) l Electric Dipole Moment

• Electron most of the time far away from core   – Strong electric dipole: d  er – Proportional to transition matrix element   f d i  e f r i  e f r cos() i • We find electric Dipole Moment  2 – f d i  r l 1 cos()l  n • Cross Section:

1 dW 1  W    d  a n2 2 dn (n  )3 0 2(n l ) l Electric Dipole Moment

• Electron most of the time far away from core   – Strong electric dipole: d  er – Proportional to transition matrix element   f d i  e f r i  e f r cos() i • We find electric Dipole Moment  2 – f d i  r l 1 cos()l  n • Cross Section:   r 2  n4

 1 dW 1 2 4 W    3 d  a n   n 2 dn (n  ) 0 2(n l ) l  Stark Effect H  H0  dF   E

• For non-Hydrogenic Atom (e.g. Helium) – „Exact“ solution by numeric diagonalization of   f H i  f H0 i  f d i F in undisturbed (standard) basis ( n~ ,l,m)

 1 dW 1 2 4 W    3 d  a n   n 2 dn (n  ) 0 2(n l ) l  Stark Effect H  H0  dF   E

• For non-Hydrogenic Atom (e.g. Helium) – „Exact“ solution by numeric diagonalization of   f H i  f H0 i  f d i F in undisturbed (standard) basis ( n~ ,l,m)

1 W   2 2(n l )

 1 dW 1 2 4 W    3 d  a n   n 2 dn (n  ) 0 2(n l ) l  Stark Effect H  H0  dF   E

• For non-Hydrogenic Atom (e.g. Helium) – „Exact“ solution by numeric diagonalization of   f H i  f H0 i  f d i F in undisturbed (standard) basis ( n~ ,l,m)

1 W   2 Numerov 2(n l )

 1 dW 1 2 4 W    3 d  a n   n 2 dn (n  ) 0 2(n l ) l Stark Map Hydrogen

n=13

n=12

n=11 Stark Map Hydrogen

n=13

Levels degenerate

n=12

n=11 Stark Map Hydrogen

n=13

k=-11 blue

n=12

k=11 red

n=11 Stark Map Hydrogen

n=13

Levels degenerate

n=12

Cross perfect Runge-Lenz vector conserved

n=11 Stark Map Helium

n=13

n=12

n=11 Stark Map Helium

n=13

Levels not degenerate

n=12

n=11 Stark Map Helium

n=13 k=-11 blue

n=12

k=11 red

n=11 Stark Map Helium

n=13

k=-11 blue

n=12 s-type k=11 red

n=11 Stark Map Helium

n=13

Levels degenerate

n=12

Do not cross! No coulomb-potential

n=11 Stark Map Helium Inglis-Teller Limit n-5

n=13

n=12

n=11 Hydrogen Atom in an electric Field • Rydberg Atoms very sensitive to electric fields  – Solve: H   H 0  dF    E  in parabolic coordinates

• Energy-Field dependence: Perturbation-Theory

1 3 F 4 2 2 2 5 W    F(n1  n2 )n  n 17n 3(n1  n2 ) 9m 19 O(n ) 2n2 2 16 n1 n2 k Hydrogen 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 1 dW 1  W    d  a n2 4 5 2 3 0   n FIT  n 2(n  ) dn (n l ) l Hydrogen 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 1 dW 1  W    d  a n2 4 5 2 3 0   n FIT  n 2(n  ) dn (n l ) l Hydrogen Atom in an electric Field

• When do Rydberg atoms ionize? – No field applied – Electric Field applied

1 – Classical ionization: F  1 cl 16n4 V    Fz r W 2 1  F   cl 4 16n4 – Valid only for • Non-H atoms if F is Increased slowly

1 dW 1  4 W    d  a n2 4 5 F  1 n 2 3 0   n FIT  n cl 16 2(n  ) dn (n l ) l Hydrogen Atom in an electric Field

• When do Rydberg atoms ionize? – No field applied – Electric Field applied

1 – Classical ionization: F  1 cl 16n4 V    Fz r W 2 1  F   cl 4 16n4 – Valid only for • Non-H atoms if F is Increased slowly

1 dW 1  4 W    d  a n2 4 5 F  1 n 2 3 0   n FIT  n cl 16 2(n  ) dn (n l ) l Hydrogen Atom in an electric Field

• When do Rydberg atoms ionize? – No field applied – Electric Field applied

1 – Quasi-Classical ioniz.: F  cl 16n4  Z m2 1 F  V ()  2 2     2    4 4  m0 W 2  F 

4Z2

 1 dW 1 2 4 5 1 4 W    3 d  a n   n Fcl  n 2 dn (n  ) 0 FIT  n 16 2(n l ) l Hydrogen Atom in an electric Field

• When do Rydberg atoms ionize? – No field applied – Electric Field applied

1 – Quasi-Classical ioniz.: F  cl 16n4

2 1  Z2 m 1 F  F  V ()  2    r 9n4  2    4 4 

2 1 m0 W  red  F  9n4 4Z2

 1 dW 1 2 4 5 1 4 W    3 d  a n   n Fcl  n 2 dn (n  ) 0 FIT  n 16 2(n l ) l Hydrogen Atom in an electric Field

• When do Rydberg atoms ionize? – No field applied

– Electric Field applied 2 Fb  9n4 1 – Quasi-Classical ioniz.: F  cl 16n4

2 1  Z2 m 1 F  Fr  V ()  2    9n4  2    4 4  1 2  W 4 red  F  9n 4 Z 2 2  blue 9n4  1 dW 1 2 4 5 1 4 W    3 d  a n   n Fcl  n 2 dn (n  ) 0 FIT  n 16 2(n l ) l Hydrogen Atom in an electric Field

Blue states

Red states classic Inglis-Teller Lifetime • From Fermis golden rule – Einstein A coefficient for two states n,l  n,l 2 3 4e n,l,n,l max(l,l) 2 An,l,n,l  nl r nl 3c3 2l 1

– Lifetime

 1 dW 1 2 4 5 1 4 W    3 d  a n   n Fcl  n 2 dn (n  ) 0 FIT  n 16 2(n l ) l Lifetime • From Fermis golden rule – Einstein A coefficient for two states n,l  n,l 2 3 4e n,l,n,l max(l,l) 2 An,l,n,l  nl r nl 3c3 2l 1

1     – Lifetime  n,l    An,l,n,l   n,ln,l 

 1 dW 1 2 4 5 1 4 W    3 d  a n   n Fcl  n 2 dn (n  ) 0 FIT  n 16 2(n l ) l Lifetime • From Fermis golden rule – Einstein A coefficient for two states n,l  n,l 2 3 4e n,l,n,l max(l,l) 2 An,l,n,l  nl r nl 3c3 2l 1

1     – Lifetime  n,l    An,l,n,l   n,ln,l 

 1 dW 1 2 4 5 1 4 W    3 d  a n   n Fcl  n 2 dn (n  ) 0 FIT  n 16 2(n l ) l Lifetime • From Fermis golden rule – Einstein A coefficient for two states n,l  n,l 2 3 4e n,l,n,l max(l,l) 2 An,l,n,l  nl r nl 3c3 2l 1

1   3   For l0:  n 2 – Lifetime  n,l    An,l,n,l   n,ln,l  Overlap of WF

 1 dW 1 2 4 5 1 4 3 W    3 d  a n   n Fcl  n   n 2 dn (n  ) 0 FIT  n 16 n,0 2(n l ) l Lifetime • From Fermis golden rule – Einstein A coefficient for two states n,l  n,l 2 3 4e n,l,n,l max(l,l) 2 An,l,n,l  nl r nl 3c3 2l 1

1     For ln:  n2 – Lifetime  n,l    An,l,n,l   n,ln,l  Overlap of WF

For ln:  n3 Overlap of WF

 1 dW 1 2 4 5 1 4 3 5 W    3 d  a n   n Fcl  n   n ,n 2 dn (n  ) 0 FIT  n 16 n,l 2(n l ) l Lifetime 2 3 1 4e  max(l,l) 2   A  n,l,n,l nl r nl    A  n,l,n,l 3 n,l   n,l,n,l  3c 2l 1  n,ln,l 

Stark State Circular state Statistical State 60 p 60 l=59 m=59 mixture ( n  , l ) small (n,l)  ( n 1,l 1)

3 5 4.5 Scaling n 3 n n  2 (overlap of   n 2 ) r  n

Lifetime 7.2 ms 70 ms  ms

 1 dW 1 2 4 5 1 4 3 5 W    3 d  a n   n Fcl  n   n ,n 2 dn (n  ) 0 FIT  n 16 n,l 2(n l ) l Part 2- Generation of Rydberg atoms

• Typical Experiments: – Beam experiments – (ultra) cold atoms – Vapor cells ETH physics Rydberg experiment

Creation of a cold supersonic beam of Helium. Speed: 1700m/s, pulsed: 25Hz, temperature atoms=100mK ETH physics Rydberg experiment

Excite electrons to the 2s-state, (to overcome very strong binding energy in the xuv range) by means of a discharge – like a lightning. ETH physics Rydberg experiment

Actual experiment consists of 5 electrodes. Between the first 2 the atoms get excited to Rydberg states up to n=inf with a dye laser. Dye/Yag laser system Dye/Yag laser system

Experiment Dye/Yag laser system

Experiment

Nd:Yag laser (600 mJ/pp, 10 ns pulse length, Power -> 10ns of MW!!) Provides energy for dye laser Dye/Yag laser system

Dye laser with DCM dye: Dye is excited by Yag and fluoresces. A cavity creates light with 624 nm wavelength. This is then frequency doubled to 312 nm.

Experiment

Nd:Yag laser (600 mJ/pp, 10 ns pulse length, Power -> 10ns of MW!!) Provides energy for dye laser Dye laser

Nd:Yag pump laser cuvette

Frequency doubling

DCM dye ETH physics Rydberg experiment

Detection: 1.2 kV/cm electric field applied in 10 ns. Rydberg atoms ionize and electrons are Detected at the MCP detector (single particle multiplier) . Results TOF 100ns Results TOF 100ns

Inglis-Teller limit Results TOF 100ns Adiabatic Ionization- rate ~ 70% (fitted)

 1 1250V  4 n   cm 16 d  F   0  Results TOF 100ns Adiabatic Ionization- Pure diabatic Ionization- rate ~ 70% (fitted) rate: exp(  t)

 1  1 V 4 V 4 1250  1250 9  n   cm 16 n   cm  d  F  d  F 2   0   0  Results TOF 100ns 2 3 Adiabatic Ionization- Pure diabatic Ionization-   r  n rate ~ 70% (fitted) rate: exp(  t) saturation From ground state

 1  1 V 4 V 4 1250  1250 9  n   cm 16 n   cm  d  F  d  F 2   0   0  2 Results TOF 15ms   r  n3 From ground state

lifetime   n3

 1  1 4 4 1250V  1250V   cm 9   cm  nd  nd  16    F  F0 2  0   