Rydberg atoms
part 1
Tobias Thiele
Content
• Part 1: Rydberg atoms
• Part 2: typical (beam) experiments
References
• T. Gallagher: Rydberg atoms
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: ~  ~ 
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 atom?
• Energy follows Rydberg formula:
0
E  E 
=13.6 eV
hRy
(n   l ) 2
0
Energy
0

hRy
n2
Hydrogen
Quantum Defect?
• Energy follows Rydberg formula:
E  E 
Energy
0
hRy
Quantum Defect
(n   l ) 2
n-Hydrogen
Hydrogen
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
r  r0
Ion core
1
V (r )  
r
1
V (r )    Vcore (r )
r
0
r
δl
Interesting Region
For Rydberg Atoms
H
nH
(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: 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  a0n2   n4
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
Stark Effect



H  H 0  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( n   l ) 2
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n2   n4
Numerov
Stark Map Hydrogen
n=13
Levels
degenerate
n=12
Cross perfect
Runge-Lenz vector
conserved
n=11
Stark Map Hydrogen
n=13
k=-11
blue
n=12
k=11
red
n=11
Stark Map Helium
n=13
Levels not
degenerate
n=12
s-type
Do not cross!
No coulomb-potential
n=11
Stark Map Helium
n=13
k=-11
blue
n=12
k=11
red
n=11
Stark
Map
Helium
Inglis-Teller
Limit α n-5
n=13
n=12
n=11
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
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

d  a0n2   n4 FIT  n 5
Rydberg 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  a0n2   n4 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
red
2 
4
W
9n
F
4Z 2  2
blue
9n 4
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )
Fb 
2
9n 4
1
Fcl 
16n 4
Fr 
1
9n 4

4
d  a0n2   n4 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
red
2 
4
W
9n
F
4Z 2  2
blue
9n 4
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )
Fb 
2
9n 4
1
Fcl 
16n 4
Fr 
1
9n 4

4
d  a0n2   n4 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
– 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  a0n2   n4 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
3
For l≈0:  n 2
Overlap of WF
For l≈0:
Constant (dominated
by decay to GS)
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

4
 n,0  n3
d  a0n2   n4 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:  n2
For l ≈ n:  n 3
Overlap of WF
dW
1
1

W 
3
2(n   l ) 2 dn (n   l )

4
d  a0n2   n4 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 μs
 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)
Statistical
mixture
5
n
r  n2
70 ms
1
4.5
ms

4
d  a0n2   n4 FIT  n 5 Fcl  116 n
 n,l  n3 , n5
Rydberg atoms
part 2
Tobias Thiele
Part 2- Rydberg atoms
• Typical Experiments:
– Beam experiments
– (ultra) cold atoms
– Vapor cells
Cavity-QED systems
Goal: large g, small G
• Couple atoms to cavities
– Realize Jaynes-Cummings Hamiltonian
• Single atom(dipole) - coupling to cavity:
𝑑 𝐹0
𝑔=
=
ℏ
𝜔0 𝑑2
2 𝜖0 ℏ 𝑉
Increase resonance frequency
Increase dipole moment
Reduce mode volume
Coupling Rydberg atoms
• Couple atoms to cavities
– Realize Jaynes-Cummings Hamiltonian
• Single atom(dipole) - coupling to cavity:
𝑔=
𝑑 𝐹0
ℏ
=
𝜔0 𝑑 2
2 𝜖0 ℏ 𝑉
= 2 𝜋 43.768 𝑘𝐻𝑧
• Scaling with excitation state n?
𝑑 𝐹0
𝑔=
=
ℏ
∝ 𝑛−3
S. Haroche
(Rydberg in microwave cavity)
∝ 𝑛2
𝜔0 𝑑2
≈ 2 𝜋 6 ∗ 1011 𝑛−4
2 𝜖0 ℏ 𝑉
∝ 𝑛9
Advantages microwave regime
for strong coupling g>> G,k
• Coupling to ground state of cavity
– ~0.01 m (microwave, possible) for n=50
–  ~10-7 m (optical, n. possible),
• Typical mode volume: 1mm *(50 mm)2
• Linewidth:
– G ~ 10 Hz (microwave)
– G ~ MHz (optical)
Summary coupling strength
• Microwave: 𝑔~2𝜋 40 𝑘𝐻𝑧, κ ~2𝜋 10 𝐻𝑧
• Optical:
𝑔~2𝜋 5 𝑀𝐻𝑧, κ ~2𝜋 1 𝑀𝐻𝑧
n-4
• What about G?
– Optical G6p~ 2.5 MHz
– Rydberg G50p~300 kHz, G50,50~100 Hz
n-3
n-5
– =𝑔/Gnp ∝ 1/n, =𝑔/Gn,n-1 ∝ n
frequency limited
Mode volume
limited
Optimal n when
g >> G ~ κ
Cavity-QED from groundstates to Rydberg-states
Hopefully we
in future!
BEC in cavity
Single atoms in
optical cavities
Haroche
Cavity-QED systems
Combine best of
both worlds - Hybrid
Hybrid system for optical to microwave conversion
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 Ionization limit 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 50 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( Gt )
 1250V

cm 16 
nd  


F0


1
4
 1250 V

cm 9 
nd  
F0
2


1
4
Results TOF 100ns
Adiabatic Ionizationrate ~ 70% (fitted)
Pure diabatic Ionizationsaturation
rate: exp( Gt )
 1250V

cm 16 
nd  


F0


1
4
 1250 V

cm 9 
nd  
F0
2


1
4
  r  n 3
From ground state
2
Results TOF 15 ms
lifetime
  n3
 1250V

cm 16 
nd  


F0


1
4
 1250 V

cm 9 
nd  
F0
2


1
4
  r  n 3
From ground state
2
Rabi oscillations over a distance of 1.5 mm
between Rydberg states