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:
bn 2
 2
n 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:
bn 2
 2
n 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:
bn 2
 2
n 4
– Other series discovered by Lyman, Brackett, Paschen, ...
– Summarized by Johannes Rydberg: ~  ~ 
Ry
n2
Introduction – Generalization
• In 1885: Balmer series:
– Visible absorption wavelengths of H:
bn 2
 2
n 4
– Other series discovered by Lyman, Brackett, Paschen, ...
– Quantum Defect was found for other atoms: ~  ~ 
Ry
(n   l ) 2
Introduction – Rydberg formula?
• Energy follows Rydberg formula:
E  E 
hRy
(n   l ) 2
Introduction – Rydberg formula?
• Energy follows Rydberg formula:
E  E 
hRy
13.6 eV
(n   l ) 2
Introduction – Hydrogen?
• Energy follows Rydberg formula:
0
E  E 
13.6 eV
hRy
(n   l ) 2
0
Energy
0

hRy
n2
Quantum Defect?
• Energy follows Rydberg formula:
E  E 
hRy
(n   l ) 2
Quantum Defect
Energy
0
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
V (r )    Vcore (r )
r
0
1
V (r )  
r
r
Radial parts-Interesting regions
W
r  r0
Ion core
1
V (r )    Vcore (r )
r
0
1
V (r )  
r
r
Radial parts-Interesting regions
W
r  r0
Ion core
1
V (r )    Vcore (r )
r
0
1
V (r )  
r
r
Interesting Region
For Rydberg Atoms
Radial parts-Interesting regions
W
r  r0
Ion core
1
V (r )    Vcore (r )
r
1
V (r )  
r
0
r
H
Radial parts-Interesting regions
W
r  r0
Ion core
1
V (r )    Vcore (r )
r
1
V (r )  
r
0
r
H
nH
Radial parts-Interesting regions
W
r  r0
Ion core
1
V (r )  
r
1
V (r )    Vcore (r )
r
0
r
δl
H
nH
(Helium) Energy Structure
•  l usually measured
– Only large for low l (s,p,d,f)
• He level structure
•
is big for s,p
1
W 
2(n   l ) 2
(Helium) Energy Structure
•  l usually measured
– Only large for low l (s,p,d,f)
• He level structure
•
is big for s,p
1
W 
2(n   l ) 2
(Helium) Energy Structure
•  l usually measured
– Only large for low l (s,p,d,f)
• He level structure
•  l is big for s,p
1
W 
2(n   l ) 2
(Helium) Energy Structure
•  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
W 
2(n   l ) 2
(Helium) Energy Structure
•  l usually measured
– Only large for low l (s,p,d,f)
• He level structure
•  l is big for s,p
dW
1

•
dn (n   l )3
1
W 
2(n   l ) 2
Electric Dipole Moment
• Electron most of the time far away from core
– Strong electric dipole:
– Proportional to transition matrix element
W 
dW
1
1

3
2(n   l ) 2 dn (n   l )
Electric Dipole Moment
• Electron most of the time far away from core


– Strong electric dipole: d  er
– Proportional to transition matrix element
W 
dW
1
1

3
2(n   l ) 2 dn (n   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
W 
dW
1
1

3
2(n   l ) 2 dn (n   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

–  f d i  r l  1 cos( ) l  n 2
• Cross Section:
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n 2
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

–  f d i  r l  1 cos( ) l  n 2
4


r

n
• Cross Section:
2
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n 2   n 4
Stark Effect



H  H 0  dF   E
• For non-Hydrogenic Atom (e.g. Helium)
– „Exact“ solution by numeric diagonalization of


f H i  f H 0 i  f d i F
in undisturbed (standard) basis ( n~ ,l,m)
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n 2   n 4
Stark Effect



H  H 0  dF   E
• For non-Hydrogenic Atom (e.g. Helium)
– „Exact“ solution by numeric diagonalization of


f H i  f H 0 i  f d i F
in undisturbed (standard) basis ( n~ ,l,m)
1
W 
2(n   l ) 2
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n 2   n 4
Stark Effect



H  H 0  dF   E
• For non-Hydrogenic Atom (e.g. Helium)
– „Exact“ solution by numeric diagonalization of


f H i  f H 0 i  f d i F
in undisturbed (standard) basis ( n~ ,l,m)
1
W 
2(n   l ) 2
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n 2   n 4
Numerov
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   2  F (nn1

n
)
n

n
17
n

3
(
n

n
)

9
m

19

O
(
n
)
1 n2
2
1
22
n
1
n

2n 2 
16
k
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
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n 2   n 4 FIT  n 5
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
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n 2   n 4 FIT  n 5
Hydrogen Atom in an electric Field
• When do Rydberg atoms ionize?
– No field applied
– Electric Field applied
– Classical ionization:
1
V    Fz
r
W2
1
 Fcl 

4 16n 4
Fcl 
1
16n 4
– Valid only for
• Non-H atoms if F is
Increased slowly
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Hydrogen Atom in an electric Field
• When do Rydberg atoms ionize?
– No field applied
– Electric Field applied
– Classical ionization:
1
V    Fz
r
W2
1
 Fcl 

4 16n 4
Fcl 
1
16n 4
– Valid only for
• Non-H atoms if F is
Increased slowly
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Hydrogen Atom in an electric Field
• When do Rydberg atoms ionize?
– No field applied
– Electric Field applied
– Quasi-Classical ioniz.:
Fcl 
1
16n 4
 Z 2 m 2  1 F 
V ( )  2  
 
2
4
4 
 
m 0
W2
F 
4Z 2
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Hydrogen Atom in an electric Field
• When do Rydberg atoms ionize?
– No field applied
– Electric Field applied
– Quasi-Classical ioniz.:
 Z 2 m 2  1 F 

V ( )  2  

2
4
4 
 
1
m 0
W 2  4 red
F 
9n
4Z 2
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )
Fcl 
1
16n 4
Fr 
1
9n 4

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Hydrogen Atom in an electric Field
• When do Rydberg atoms ionize?
– No field applied
– Electric Field applied
– Quasi-Classical ioniz.:
 Z 2 m 2  1 F 

V ( )  2 


2
4
4 
 
1
2 
red
4
W
9n
F
4Z 2  2
blue
9n 4
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )
Fb 
2
Fr 
1
9n 4
1
Fcl 
16n 4
9n 4

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
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 
An,l ,n,l
4e 2n3,l ,n,l max( l , l )

nl  r nl
3
3c
2l  1
2
– Lifetime
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Lifetime
• From Fermis golden rule
– Einstein A coefficient for two states n, l  n, l 
An,l ,n,l
4e 2n3,l ,n,l max( l , l )

nl  r nl
3
3c
2l  1
– Lifetime  n,l
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )


   An,l ,n,l 
 n , l   n ,l

2
1

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Lifetime
• From Fermis golden rule
– Einstein A coefficient for two states n, l  n, l 
An,l ,n,l
4e 2n3,l ,n,l max( l , l )

nl  r nl
3
3c
2l  1
– Lifetime  n,l
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )


   An,l ,n,l 
 n , l   n ,l

2
1

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Lifetime
• From Fermis golden rule
– Einstein A coefficient for two states n, l  n, l 
An,l ,n,l
4e 2n3,l ,n,l max( l , l )

nl  r nl
3
3c
2l  1
– Lifetime  n,l
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )


   An,l ,n,l 
 n , l   n ,l

2
1
3
For l0:  n 2
Overlap of WF

4
 n,0  n3
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
Lifetime
• From Fermis golden rule
– Einstein A coefficient for two states n, l  n, l 
An,l ,n,l
4e 2n3,l ,n,l max( l , l )

nl  r nl
3
3c
2l  1
– Lifetime  n,l


   An,l ,n,l 
 n , l   n ,l

2
1
For ln:  n 2
Overlap of WF
For ln:  n 3
Overlap of WF
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

4
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
 n,l  n3 , n5
Lifetime
4e 
2
An,l ,n,l 
3
n , l  , n , l
3
3c
Stark State
60 p
(n,l ) small
State
Scaling
max( l , l )
nl  r nl
2l  1
n
Lifetime
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )
7.2 ms
 n ,l
Circular state
60 l=59 m=59


   An,l ,n,l 
 n , l   n ,l

(n, l )  (n  1, l  1)
3
(overlap of   n
2
3
n
2)
5
r  n2
70 ms
1
Statistical
mixture
n
4.5
ms

4
 n,l  n 3 , n5
d  a0n 2   n 4 FIT  n 5 Fcl  116 n
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 Ionizationrate ~ 70% (fitted)
 1250V

cm 16 
nd  


F0


1
4
Results TOF 100ns
Adiabatic Ionizationrate ~ 70% (fitted)
Pure diabatic Ionizationrate: exp( t )
 1250V

cm 16 
nd  


F0


1
4
 1250V
cm
nd  
F0

9 
2

1
4
Results TOF 100ns
Adiabatic Ionizationrate ~ 70% (fitted)
Pure diabatic Ionizationsaturation
rate: exp( t )
 1250V

cm 16 
nd  


F0


1
4
 1250V
cm
nd  
F0

9 
2

1
4
  r  n3
From ground state
2
Results TOF 15ms
lifetime
  n3
 1250V

cm 16 
nd  


F0


1
4
 1250V
cm
nd  
F0

9 
2

1
4
  r  n3
From ground state
2