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Rydberg MoleculesA S Dickinson
School of Natural Sciences(Physics), University of Newcastle, Newcastle upon Tyne,
NE1 7RU, UK
NEWCASTLE
UN IVERS ITY OF
Rydberg Molecules – p.1/55
Overview
• Introduction;
• Rydberg Atoms• Hydrogen• Alkali Atoms
• Rydberg Molecules• Low l Levels• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen• Alkali Atoms
• Rydberg Molecules• Low l Levels• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen
• Alkali Atoms
• Rydberg Molecules• Low l Levels• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen• Alkali Atoms
• Rydberg Molecules• Low l Levels• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen• Alkali Atoms
• Rydberg Molecules
• Low l Levels• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen• Alkali Atoms
• Rydberg Molecules• Low l Levels
• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen• Alkali Atoms
• Rydberg Molecules• Low l Levels• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen• Alkali Atoms
• Rydberg Molecules• Low l Levels• High l Levels
• Shape-Resonance Long-Range States
• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Overview
• Introduction;• Rydberg Atoms
• Hydrogen• Alkali Atoms
• Rydberg Molecules• Low l Levels• High l Levels
• Shape-Resonance Long-Range States• Possibilities in Other Systems.
Rydberg Molecules – p.2/55
Introduction
• Most familiar Rydberg molecules are probablythose encountered in ZEKE spectroscopy -internuclear separations generallycomparable to ground-state systems;
• Primarily concerned here with excitedmolecules with large internuclear separation;
• Looking at molecules bound by electronicstructure - exploiting availability of cold atomsin MOTs or optical lattices or condensates togive sufficient density for formation and lowprobability of destruction.
Rydberg Molecules – p.3/55
Introduction
• Most familiar Rydberg molecules are probablythose encountered in ZEKE spectroscopy -internuclear separations generallycomparable to ground-state systems;
• Primarily concerned here with excitedmolecules with large internuclear separation;
• Looking at molecules bound by electronicstructure - exploiting availability of cold atomsin MOTs or optical lattices or condensates togive sufficient density for formation and lowprobability of destruction.
Rydberg Molecules – p.3/55
Introduction
• Most familiar Rydberg molecules are probablythose encountered in ZEKE spectroscopy -internuclear separations generallycomparable to ground-state systems;
• Primarily concerned here with excitedmolecules with large internuclear separation;
• Looking at molecules bound by electronicstructure - exploiting availability of cold atomsin MOTs or optical lattices or condensates togive sufficient density for formation and lowprobability of destruction.
Rydberg Molecules – p.3/55
Alternative long-range molecules
• Not considering molecules supported bylong-range potentials, as in the work ofComparat et al. (2000) orNormand, Zemke, Côté, Pichler, and Stwalley(2002) on Cs;
• or work of Boisseau, Simbotin, and Côté(2002) on long-range interactions betweentwo Rydberg (np) atoms;
Rydberg Molecules – p.4/55
Alternative long-range molecules
• Not considering molecules supported bylong-range potentials, as in the work ofComparat et al. (2000) orNormand, Zemke, Côté, Pichler, and Stwalley(2002) on Cs;
• or work of Boisseau, Simbotin, and Côté(2002) on long-range interactions betweentwo Rydberg (np) atoms;
Rydberg Molecules – p.4/55
More alternative long-range molecules
• or Giant Helium Dimers produced fromHe(23S↑) + He(23P↑) by Léonard et al. (2003)with R 150 – 1150 a0 , binding energies1-2 GHz;
• or work ofGranger, Král, Sadeghpour, and Shapiro(2002) on electron interactions withnanotubes.
• or work of Holmlid (2002, and referencestherein) on Rydberg matter (condensation ofcircular, long-lived, Rydberg states).
Rydberg Molecules – p.5/55
More alternative long-range molecules
• or Giant Helium Dimers produced fromHe(23S↑) + He(23P↑) by Léonard et al. (2003)with R 150 – 1150 a0 , binding energies1-2 GHz;
• or work ofGranger, Král, Sadeghpour, and Shapiro(2002) on electron interactions withnanotubes.
• or work of Holmlid (2002, and referencestherein) on Rydberg matter (condensation ofcircular, long-lived, Rydberg states).
Rydberg Molecules – p.5/55
More alternative long-range molecules
• or Giant Helium Dimers produced fromHe(23S↑) + He(23P↑) by Léonard et al. (2003)with R 150 – 1150 a0 , binding energies1-2 GHz;
• or work ofGranger, Král, Sadeghpour, and Shapiro(2002) on electron interactions withnanotubes.
• or work of Holmlid (2002, and referencestherein) on Rydberg matter (condensation ofcircular, long-lived, Rydberg states).
Rydberg Molecules – p.5/55
Energies of Hydrogenic Systems•
En = −13.6 eVn2
= −15.1 meV(n/30)2
≡ −175 kBK
(n/30)2,
where kB is Boltzmann’s constant.
• Spacing between levels n and n+ 1 is
En+1−En =0.50 meV
[(n+ 1/2)/30]3≡ 244 GHz
[(n+ 1/2)/30]3,
• Fine-structure splitting ( j = l ± 1/2)
∆Enl =α213.6 eVn3l(l + 1)
; α ≈ 1/137.
Rydberg Molecules – p.6/55
Energies of Hydrogenic Systems•
En = −13.6 eVn2
= −15.1 meV(n/30)2
≡ −175 kBK
(n/30)2,
where kB is Boltzmann’s constant.• Spacing between levels n and n+ 1 is
En+1−En =0.50 meV
[(n+ 1/2)/30]3≡ 244 GHz
[(n+ 1/2)/30]3,
• Fine-structure splitting ( j = l ± 1/2)
∆Enl =α213.6 eVn3l(l + 1)
; α ≈ 1/137.
Rydberg Molecules – p.6/55
Energies of Hydrogenic Systems•
En = −13.6 eVn2
= −15.1 meV(n/30)2
≡ −175 kBK
(n/30)2,
where kB is Boltzmann’s constant.• Spacing between levels n and n+ 1 is
En+1−En =0.50 meV
[(n+ 1/2)/30]3≡ 244 GHz
[(n+ 1/2)/30]3,
• Fine-structure splitting ( j = l ± 1/2)
∆Enl =α213.6 eVn3l(l + 1)
; α ≈ 1/137.
Rydberg Molecules – p.6/55
Size of Hydrogenic Atoms
•
• For the size of the level we have
〈r〉nl = [3n2 − l(l + 1)]/2 a0.
• It is useful to bear in mind the classical resultsthat 〈r〉 = n2a0 for a circular orbit, l = n− 1.
• The outer classical turning point increases asl decreases, reaching 2n2 a0 for an s-state.
• For the classical ellipse semi-major axis:ac = n2a0, eccentricity e, where e2 = 1 − l2/n2.
• For any Coulomb system 〈T 〉 = −E.
Rydberg Molecules – p.7/55
Size of Hydrogenic Atoms•
• For the size of the level we have
〈r〉nl = [3n2 − l(l + 1)]/2 a0.
• It is useful to bear in mind the classical resultsthat 〈r〉 = n2a0 for a circular orbit, l = n− 1.
• The outer classical turning point increases asl decreases, reaching 2n2 a0 for an s-state.
• For the classical ellipse semi-major axis:ac = n2a0, eccentricity e, where e2 = 1 − l2/n2.
• For any Coulomb system 〈T 〉 = −E.
Rydberg Molecules – p.7/55
Size of Hydrogenic Atoms•
• For the size of the level we have
〈r〉nl = [3n2 − l(l + 1)]/2 a0.
• It is useful to bear in mind the classical resultsthat 〈r〉 = n2a0 for a circular orbit, l = n− 1.
• The outer classical turning point increases asl decreases, reaching 2n2 a0 for an s-state.
• For the classical ellipse semi-major axis:ac = n2a0, eccentricity e, where e2 = 1 − l2/n2.
• For any Coulomb system 〈T 〉 = −E.
Rydberg Molecules – p.7/55
Size of Hydrogenic Atoms•
• For the size of the level we have
〈r〉nl = [3n2 − l(l + 1)]/2 a0.
• It is useful to bear in mind the classical resultsthat 〈r〉 = n2a0 for a circular orbit, l = n− 1.
• The outer classical turning point increases asl decreases, reaching 2n2 a0 for an s-state.
• For the classical ellipse semi-major axis:ac = n2a0, eccentricity e, where e2 = 1 − l2/n2.
• For any Coulomb system 〈T 〉 = −E.
Rydberg Molecules – p.7/55
Size of Hydrogenic Atoms•
• For the size of the level we have
〈r〉nl = [3n2 − l(l + 1)]/2 a0.
• It is useful to bear in mind the classical resultsthat 〈r〉 = n2a0 for a circular orbit, l = n− 1.
• The outer classical turning point increases asl decreases, reaching 2n2 a0 for an s-state.
• For the classical ellipse semi-major axis:ac = n2a0, eccentricity e, where e2 = 1 − l2/n2.
• For any Coulomb system 〈T 〉 = −E.
Rydberg Molecules – p.7/55
Size of Hydrogenic Atoms•
• For the size of the level we have
〈r〉nl = [3n2 − l(l + 1)]/2 a0.
• It is useful to bear in mind the classical resultsthat 〈r〉 = n2a0 for a circular orbit, l = n− 1.
• The outer classical turning point increases asl decreases, reaching 2n2 a0 for an s-state.
• For the classical ellipse semi-major axis:ac = n2a0, eccentricity e, where e2 = 1 − l2/n2.
• For any Coulomb system 〈T 〉 = −E.
Rydberg Molecules – p.7/55
Alkali Atoms
•
Enl = − 13.6 eV(n− δl)2
,
• δl falls off rapidly with l and, even for arelatively large alkali such as Rb, is negligible(≈ 0.02) by l = 3.
• Hence for a Rydberg n-manifold the majorityof the (nl) levels are degenerate.
• The centrifugal potential excludes theelectron from the region where short-rangeeffects are important.
Rydberg Molecules – p.8/55
Alkali Atoms
•
Enl = − 13.6 eV(n− δl)2
,
• δl falls off rapidly with l and, even for arelatively large alkali such as Rb, is negligible(≈ 0.02) by l = 3.
• Hence for a Rydberg n-manifold the majorityof the (nl) levels are degenerate.
• The centrifugal potential excludes theelectron from the region where short-rangeeffects are important.
Rydberg Molecules – p.8/55
Alkali Atoms
•
Enl = − 13.6 eV(n− δl)2
,
• δl falls off rapidly with l and, even for arelatively large alkali such as Rb, is negligible(≈ 0.02) by l = 3.
• Hence for a Rydberg n-manifold the majorityof the (nl) levels are degenerate.
• The centrifugal potential excludes theelectron from the region where short-rangeeffects are important.
Rydberg Molecules – p.8/55
Alkali Atoms
•
Enl = − 13.6 eV(n− δl)2
,
• δl falls off rapidly with l and, even for arelatively large alkali such as Rb, is negligible(≈ 0.02) by l = 3.
• Hence for a Rydberg n-manifold the majorityof the (nl) levels are degenerate.
• The centrifugal potential excludes theelectron from the region where short-rangeeffects are important.
Rydberg Molecules – p.8/55
Rydberg Molecules• Considering molecules, internuclear
separation R, formed from A* + A, where A isan alkali.
• Very difficult - if Σ state dissociating to, say30s + 5s, there are 29s + 5s, 29p + 5s, 29d +5s etc etc below, ≈ 450 states!
• Simplifications: active electron ’sees’• Coulomb interaction with A+ core;• Short-range interaction with A, falling off as
1/r4A.
• Model this as low-energy elastic scattering byA.
Rydberg Molecules – p.9/55
Rydberg Molecules• Considering molecules, internuclear
separation R, formed from A* + A, where A isan alkali.
• Very difficult - if Σ state dissociating to, say30s + 5s, there are 29s + 5s, 29p + 5s, 29d +5s etc etc below, ≈ 450 states!
• Simplifications: active electron ’sees’• Coulomb interaction with A+ core;• Short-range interaction with A, falling off as
1/r4A.
• Model this as low-energy elastic scattering byA.
Rydberg Molecules – p.9/55
Rydberg Molecules• Considering molecules, internuclear
separation R, formed from A* + A, where A isan alkali.
• Very difficult - if Σ state dissociating to, say30s + 5s, there are 29s + 5s, 29p + 5s, 29d +5s etc etc below, ≈ 450 states!
• Simplifications: active electron ’sees’
• Coulomb interaction with A+ core;• Short-range interaction with A, falling off as
1/r4A.
• Model this as low-energy elastic scattering byA.
Rydberg Molecules – p.9/55
Rydberg Molecules• Considering molecules, internuclear
separation R, formed from A* + A, where A isan alkali.
• Very difficult - if Σ state dissociating to, say30s + 5s, there are 29s + 5s, 29p + 5s, 29d +5s etc etc below, ≈ 450 states!
• Simplifications: active electron ’sees’• Coulomb interaction with A+ core;
• Short-range interaction with A, falling off as1/r4
A.
• Model this as low-energy elastic scattering byA.
Rydberg Molecules – p.9/55
Rydberg Molecules• Considering molecules, internuclear
separation R, formed from A* + A, where A isan alkali.
• Very difficult - if Σ state dissociating to, say30s + 5s, there are 29s + 5s, 29p + 5s, 29d +5s etc etc below, ≈ 450 states!
• Simplifications: active electron ’sees’• Coulomb interaction with A+ core;• Short-range interaction with A, falling off as
1/r4A.
• Model this as low-energy elastic scattering byA.
Rydberg Molecules – p.9/55
Rydberg Molecules• Considering molecules, internuclear
separation R, formed from A* + A, where A isan alkali.
• Very difficult - if Σ state dissociating to, say30s + 5s, there are 29s + 5s, 29p + 5s, 29d +5s etc etc below, ≈ 450 states!
• Simplifications: active electron ’sees’• Coulomb interaction with A+ core;• Short-range interaction with A, falling off as
1/r4A.
• Model this as low-energy elastic scattering byA.
Rydberg Molecules – p.9/55
s-Wave Scattering
• Low-energy scattering, wave-vector k: ⇒ justs-wave important, phase shift δs, need onlyscattering length, a: limk→0 k cot δs = −1/a.
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 0.05 0.1 0.15 0.2
δ s (
rad.
)
E (eV)
e+Rb 3S Phase shift
From Chibisov et al.(2002)
Phase shift for e+Rb not really proportional to k!Rydberg Molecules – p.10/55
Modified Effective Range Theory• Because of the large polarizability (319 a3
0) ofRb, usual low-k expansion very limited range.
• Instead, use aeff(k) wherek cot δs(k) = −1/aeff .
• Changing kinetic energy of electron with R:
~2k2(R)
2me= −EH
2n2+
e2
4πε0R,
me electron mass, EH a.u. (27.21 eV).• Note that δs = 0 at small finite k :⇒ aeff
changes sign.
Rydberg Molecules – p.11/55
Modified Effective Range Theory• Because of the large polarizability (319 a3
0) ofRb, usual low-k expansion very limited range.
• Instead, use aeff(k) wherek cot δs(k) = −1/aeff .
• Changing kinetic energy of electron with R:
~2k2(R)
2me= −EH
2n2+
e2
4πε0R,
me electron mass, EH a.u. (27.21 eV).• Note that δs = 0 at small finite k :⇒ aeff
changes sign.
Rydberg Molecules – p.11/55
Modified Effective Range Theory• Because of the large polarizability (319 a3
0) ofRb, usual low-k expansion very limited range.
• Instead, use aeff(k) wherek cot δs(k) = −1/aeff .
• Changing kinetic energy of electron with R:
~2k2(R)
2me= −EH
2n2+
e2
4πε0R,
me electron mass, EH a.u. (27.21 eV).
• Note that δs = 0 at small finite k :⇒ aeff
changes sign.
Rydberg Molecules – p.11/55
Modified Effective Range Theory• Because of the large polarizability (319 a3
0) ofRb, usual low-k expansion very limited range.
• Instead, use aeff(k) wherek cot δs(k) = −1/aeff .
• Changing kinetic energy of electron with R:
~2k2(R)
2me= −EH
2n2+
e2
4πε0R,
me electron mass, EH a.u. (27.21 eV).• Note that δs = 0 at small finite k :⇒ aeff
changes sign.Rydberg Molecules – p.11/55
R-Dep. Effective Scattering Length
-15
-12
-9
-6
-3
0
3
0 300 600 900 1200 1500 1800
a eff
(a0)
R (a0)
e+Rb 3S Effective Scattering Length, n = 30.
Rydberg Molecules – p.12/55
Fermi Model
• Model due to Fermi (1934) - electron-atominteraction
V F (r,R) =2π~
2
mea δ(r − R),
r and R position vectors of electron and ofground-state atom, respectively.
• Originally introduced for describing shapes oflines for transitions between Rydberg levels ofalkalis perturbed by rare gases.
Rydberg Molecules – p.13/55
Fermi Model
• Model due to Fermi (1934) - electron-atominteraction
V F (r,R) =2π~
2
mea δ(r − R),
r and R position vectors of electron and ofground-state atom, respectively.
• Originally introduced for describing shapes oflines for transitions between Rydberg levels ofalkalis perturbed by rare gases.
Rydberg Molecules – p.13/55
Low-l Levels
•
• Assume V F weak ⇒ use pert. theory.
Enlm(R) = 〈nlm | V F (r,R) | nlm〉
=2π~
2a
me
∫
R2nl(r) | Ylm(r) |2 δ(r − R)dr
=2π~
2a
me| Ylm(R) |2 R2
nl(R)
= δm,0~
2a
2me(2l + 1)R2
nl(R),
quantizing along R.
Rydberg Molecules – p.14/55
Low-l Levels•
• Assume V F weak ⇒ use pert. theory.
Enlm(R) = 〈nlm | V F (r,R) | nlm〉
=2π~
2a
me
∫
R2nl(r) | Ylm(r) |2 δ(r − R)dr
=2π~
2a
me| Ylm(R) |2 R2
nl(R)
= δm,0~
2a
2me(2l + 1)R2
nl(R),
quantizing along R.Rydberg Molecules – p.14/55
What does the low-l Potential look like?• Almost n nodes, attractive if a < 0.
• How deep? Use WKB approx.(Bethe and Salpeter, 1957) to Rnl(R).
Enl(R) ≈ 2l + 1
π
EH
n∗6
aeff
a0
n∗4a20
R2
~
n∗a0 pl(R)cos2 Φl(R)
Φl(R) =1
~
∫ R
R1
pl(R) dR− π/4,
pl(R), radial momentum for ang. mom. l.
• c.f. Boisseau et al. (2002) R ≈ 0.3n3,E ≈ 1/n7.
Rydberg Molecules – p.15/55
What does the low-l Potential look like?• Almost n nodes, attractive if a < 0.
• How deep? Use WKB approx.(Bethe and Salpeter, 1957) to Rnl(R).
Enl(R) ≈ 2l + 1
π
EH
n∗6
aeff
a0
n∗4a20
R2
~
n∗a0 pl(R)cos2 Φl(R)
Φl(R) =1
~
∫ R
R1
pl(R) dR− π/4,
pl(R), radial momentum for ang. mom. l.
• c.f. Boisseau et al. (2002) R ≈ 0.3n3,E ≈ 1/n7.
Rydberg Molecules – p.15/55
What does the low-l Potential look like?• Almost n nodes, attractive if a < 0.
• How deep? Use WKB approx.(Bethe and Salpeter, 1957) to Rnl(R).
Enl(R) ≈ 2l + 1
π
EH
n∗6
aeff
a0
n∗4a20
R2
~
n∗a0 pl(R)cos2 Φl(R)
Φl(R) =1
~
∫ R
R1
pl(R) dR− π/4,
pl(R), radial momentum for ang. mom. l.
• c.f. Boisseau et al. (2002) R ≈ 0.3n3,E ≈ 1/n7.
Rydberg Molecules – p.15/55
Vibrational levels in low-l Potential?
• Consider a harmonic approximation to theminimum, using just the fast cos2 variation:
cos2 Φ(R) ≈ 1 − [(R−Re)pl(R)/~]2.
• Comparing vibrational spacing to well depth:
~ω
|Enl(R)| = n∗2
√
2π
2l + 1
|aeff |a0
me
µ
[
n∗pl(R)a0
~
]3/2R
n∗2a0
,
µ nuclear reduced mass (≈ 2822 me for Rb).• Bound vibrational levels may exist!
Rydberg Molecules – p.16/55
Vibrational levels in low-l Potential?
• Consider a harmonic approximation to theminimum, using just the fast cos2 variation:
cos2 Φ(R) ≈ 1 − [(R−Re)pl(R)/~]2.
• Comparing vibrational spacing to well depth:
~ω
|Enl(R)| = n∗2
√
2π
2l + 1
|aeff |a0
me
µ
[
n∗pl(R)a0
~
]3/2R
n∗2a0
,
µ nuclear reduced mass (≈ 2822 me for Rb).
• Bound vibrational levels may exist!
Rydberg Molecules – p.16/55
Vibrational levels in low-l Potential?
• Consider a harmonic approximation to theminimum, using just the fast cos2 variation:
cos2 Φ(R) ≈ 1 − [(R−Re)pl(R)/~]2.
• Comparing vibrational spacing to well depth:
~ω
|Enl(R)| = n∗2
√
2π
2l + 1
|aeff |a0
me
µ
[
n∗pl(R)a0
~
]3/2R
n∗2a0
,
µ nuclear reduced mass (≈ 2822 me for Rb).• Bound vibrational levels may exist!
Rydberg Molecules – p.16/55
The 30d potential. Greene et al. (2000)
Rydberg Molecules – p.17/55
Rydberg Molecules – p.18/55
The high-l Potential• Use degenerate pert. theory: 30 × 30 matrix?
hnl,nl′(R) = Rnl(R)Rnl′(R)√
(2l + 1)(2l′ + 1)/4π,
•
En(R) = EHaeff
2a0
n−1∑
l=l1
(2l + 1)R2nl(R)a3
0,
where l1 lowest degen. level (Omont, 1977).• Other eigenvalues 0 !• Sum can be performed analytically in terms ofRn0(R), R′
n0(R), (Chibisov et al., 2000).• Individual l oscillations smoothed.
Rydberg Molecules – p.19/55
The high-l Potential• Use degenerate pert. theory: 30 × 30 matrix?
hnl,nl′(R) = Rnl(R)Rnl′(R)√
(2l + 1)(2l′ + 1)/4π,•
En(R) = EHaeff
2a0
n−1∑
l=l1
(2l + 1)R2nl(R)a3
0,
where l1 lowest degen. level (Omont, 1977).
• Other eigenvalues 0 !• Sum can be performed analytically in terms ofRn0(R), R′
n0(R), (Chibisov et al., 2000).• Individual l oscillations smoothed.
Rydberg Molecules – p.19/55
The high-l Potential• Use degenerate pert. theory: 30 × 30 matrix?
hnl,nl′(R) = Rnl(R)Rnl′(R)√
(2l + 1)(2l′ + 1)/4π,•
En(R) = EHaeff
2a0
n−1∑
l=l1
(2l + 1)R2nl(R)a3
0,
where l1 lowest degen. level (Omont, 1977).• Other eigenvalues 0 !
• Sum can be performed analytically in terms ofRn0(R), R′
n0(R), (Chibisov et al., 2000).• Individual l oscillations smoothed.
Rydberg Molecules – p.19/55
The high-l Potential• Use degenerate pert. theory: 30 × 30 matrix?
hnl,nl′(R) = Rnl(R)Rnl′(R)√
(2l + 1)(2l′ + 1)/4π,•
En(R) = EHaeff
2a0
n−1∑
l=l1
(2l + 1)R2nl(R)a3
0,
where l1 lowest degen. level (Omont, 1977).• Other eigenvalues 0 !• Sum can be performed analytically in terms ofRn0(R), R′
n0(R), (Chibisov et al., 2000).
• Individual l oscillations smoothed.
Rydberg Molecules – p.19/55
The high-l Potential• Use degenerate pert. theory: 30 × 30 matrix?
hnl,nl′(R) = Rnl(R)Rnl′(R)√
(2l + 1)(2l′ + 1)/4π,•
En(R) = EHaeff
2a0
n−1∑
l=l1
(2l + 1)R2nl(R)a3
0,
where l1 lowest degen. level (Omont, 1977).• Other eigenvalues 0 !• Sum can be performed analytically in terms ofRn0(R), R′
n0(R), (Chibisov et al., 2000).• Individual l oscillations smoothed.
Rydberg Molecules – p.19/55
Properties of High-l Potential
• Qualitatively, pseudopotential selects linearcombination maximizing wavefunction atperturber.
• ≈ n terms in sum, each with (2l + 1)weighting, so ≈ n2 stronger:
• Using WKB, sum→ integral, cos2 → 1/2,Presynakov (1970); Omont (1977)
En(R) =EH
πn3
aeffpl1(R)
~≈ EH
πn4
aeff
a0
na0pl1(R)
~.
Rydberg Molecules – p.20/55
Properties of High-l Potential
• Qualitatively, pseudopotential selects linearcombination maximizing wavefunction atperturber.
• ≈ n terms in sum, each with (2l + 1)weighting, so ≈ n2 stronger:
• Using WKB, sum→ integral, cos2 → 1/2,Presynakov (1970); Omont (1977)
En(R) =EH
πn3
aeffpl1(R)
~≈ EH
πn4
aeff
a0
na0pl1(R)
~.
Rydberg Molecules – p.20/55
Properties of High-l Potential
• Qualitatively, pseudopotential selects linearcombination maximizing wavefunction atperturber.
• ≈ n terms in sum, each with (2l + 1)weighting, so ≈ n2 stronger:
• Using WKB, sum→ integral, cos2 → 1/2,Presynakov (1970); Omont (1977)
En(R) =EH
πn3
aeffpl1(R)
~≈ EH
πn4
aeff
a0
na0pl1(R)
~.
Rydberg Molecules – p.20/55
More Properties of High-l Potential
• Still some small residual ’ripples’.
• Supports many more vibrational levels,including a few in the ’ripples’.
Rydberg Molecules – p.21/55
More Properties of High-l Potential
• Still some small residual ’ripples’.• Supports many more vibrational levels,
including a few in the ’ripples’.
Rydberg Molecules – p.21/55
Rb(n=30) Figure: Greene et al. (2000)
Rydberg Molecules – p.22/55
Rb(n=30) Figure: WKB Approx.
-12-10
-8-6-4-2 0 2
500 1000 1500 2000 2500
Ene
rgy
(G
Hz)
R (a0)
WKB Energy for s Σ Hydrogenic State
Rydberg Molecules – p.23/55
Rb (n=30) Vibrational Levels
Rydberg Molecules – p.24/55
Electron Wavefunction
ψn(r, R, r ·R) = Cn−1∑
l=l1
(2l+1)Rnl(r)Rnl(R)Pl(r ·R),
where C is the normalization factor and Pl(x) is
the l−th Legendre polynomial.
Rydberg Molecules – p.25/55
’Trilobite’: Greene et al. (2000)Rb n = 30, R = 1232 a0.
Electron Density for Ultra-Long-Range Rydberg Molecule.
Trilobite Fossil
Rydberg Molecules – p.26/55
’Trilobite’ II
Rydberg Molecules – p.27/55
movie
Click here to view the movie, produced by Dr J P
Hagon and Mr T Harrison.
Rydberg Molecules – p.28/55
Semiclassical Approximation to ψ.
• Semiclassical approximation using WKB forboth Rnl and for Pl, sum→ integral,
•
ψn(r, R, r · R) ≈∫ n−1
l1
dl{exp[iΦl(r)] + exp[−iΦl(r)]}
√
pl(R)pl(r)(2l + 1) sin θ×
{exp[iΦl(R)] + exp[−iΦl(R)]}×{exp[i(l + 1/2)θ] + exp[−i(l + 1/2)θ]} .
Rydberg Molecules – p.29/55
Semiclassical Approximation to ψ.
• Semiclassical approximation using WKB forboth Rnl and for Pl, sum→ integral,
•
ψn(r, R, r · R) ≈∫ n−1
l1
dl{exp[iΦl(r)] + exp[−iΦl(r)]}
√
pl(R)pl(r)(2l + 1) sin θ×
{exp[iΦl(R)] + exp[−iΦl(R)]}×{exp[i(l + 1/2)θ] + exp[−i(l + 1/2)θ]} .
Rydberg Molecules – p.29/55
Stationary Phase
•
• Stationary phase using ∂∂lΦl(R) = θl(R):
θl(r) ± θl(R) ± θ = 0,where θl(r)(R) is the angle of the electron(Rb) at r(R), measured from perihelion,selecting the ellipse on which the electronpasses through the ground-state atom.
• Two allowed values of l generally - fourinterfering classical contributions. Untidyclosed-form expression for (l/n)(R, r, θ).
• Phase Φ(R) also available in closed forminvolving eccentric anomaly.
Rydberg Molecules – p.30/55
Stationary Phase•
• Stationary phase using ∂∂lΦl(R) = θl(R):
θl(r) ± θl(R) ± θ = 0,where θl(r)(R) is the angle of the electron(Rb) at r(R), measured from perihelion,selecting the ellipse on which the electronpasses through the ground-state atom.
• Two allowed values of l generally - fourinterfering classical contributions. Untidyclosed-form expression for (l/n)(R, r, θ).
• Phase Φ(R) also available in closed forminvolving eccentric anomaly.
Rydberg Molecules – p.30/55
Stationary Phase•
• Stationary phase using ∂∂lΦl(R) = θl(R):
θl(r) ± θl(R) ± θ = 0,where θl(r)(R) is the angle of the electron(Rb) at r(R), measured from perihelion,selecting the ellipse on which the electronpasses through the ground-state atom.
• Two allowed values of l generally - fourinterfering classical contributions. Untidyclosed-form expression for (l/n)(R, r, θ).
• Phase Φ(R) also available in closed forminvolving eccentric anomaly.
Rydberg Molecules – p.30/55
Stationary Phase•
• Stationary phase using ∂∂lΦl(R) = θl(R):
θl(r) ± θl(R) ± θ = 0,where θl(r)(R) is the angle of the electron(Rb) at r(R), measured from perihelion,selecting the ellipse on which the electronpasses through the ground-state atom.
• Two allowed values of l generally - fourinterfering classical contributions. Untidyclosed-form expression for (l/n)(R, r, θ).
• Phase Φ(R) also available in closed forminvolving eccentric anomaly.
Rydberg Molecules – p.30/55
Elliptical Orbit
Critical Orbits
e-
Rbθ
Rb+
Rydberg Molecules – p.31/55
Rydberg Molecules – p.32/55
p-wave Scattering
• Considered so far only s-wave scattering.Might p-wave ever be important?Electron-Alkali interaction generally strong,partly from polarization of alkali.
-25
0
25
50
12 14 16 18 20 22 24 26 28 30
Vef
f (m
eV)
r (a0)
Electron-Rubidium Effective Potential
p-wave
Hence all alkalissupport low-energyp-wave (ns np 3Po)quasi-bound states,trapped behind long-range centrifugalbarrier.
Rydberg Molecules – p.33/55
p-Wave Pseudo-potential• Extension of Fano model provided by Omont
(1977) (used by Masnou-Seeuws (1982)):
〈i|Vp(r,R)|j〉 = −6πEHa20 tan δp
k3(R)∇Ψi(R)·∇Ψj(R),
where δp is the p-wave phase shift.
0
0.5
1
1.5
2
2.5
0 0.05 0.1 0.15 0.2 0.25
δ p (
rad.
)
E (eV)
e+Rb 3P Phase shift
From Bahrim et al.(2001).Clearly at a resonancethis interaction can bevery strong.
Rydberg Molecules – p.34/55
p-wave Interaction• Gradient operator gives 3 separate
contributions for m = 0,±1. Σ term involvesR′
nl(R), Π involves Rnl(R).
• Within degenerate n manifold getfactorization so one strong interaction, rest 0.
• Again quantal sums can be performedanalytically (Chibisov et al., 2000).
• Estimating Σ semiclassically as for s states:
EpΣ
n (R) = −EH
πn3tan δp[k(R)].
Rydberg Molecules – p.35/55
p-wave Interaction• Gradient operator gives 3 separate
contributions for m = 0,±1. Σ term involvesR′
nl(R), Π involves Rnl(R).
• Within degenerate n manifold getfactorization so one strong interaction, rest 0.
• Again quantal sums can be performedanalytically (Chibisov et al., 2000).
• Estimating Σ semiclassically as for s states:
EpΣ
n (R) = −EH
πn3tan δp[k(R)].
Rydberg Molecules – p.35/55
p-wave Interaction• Gradient operator gives 3 separate
contributions for m = 0,±1. Σ term involvesR′
nl(R), Π involves Rnl(R).
• Within degenerate n manifold getfactorization so one strong interaction, rest 0.
• Again quantal sums can be performedanalytically (Chibisov et al., 2000).
• Estimating Σ semiclassically as for s states:
EpΣ
n (R) = −EH
πn3tan δp[k(R)].
Rydberg Molecules – p.35/55
p-wave Interaction• Gradient operator gives 3 separate
contributions for m = 0,±1. Σ term involvesR′
nl(R), Π involves Rnl(R).
• Within degenerate n manifold getfactorization so one strong interaction, rest 0.
• Again quantal sums can be performedanalytically (Chibisov et al., 2000).
• Estimating Σ semiclassically as for s states:
EpΣ
n (R) = −EH
πn3tan δp[k(R)].
Rydberg Molecules – p.35/55
Two 3Σ Potentials!
•
• Must allow for the interaction of thesepotentials.
• Two states of the same symmetry can’t cross.• Mixing quite weak so avoided crossing barely
detectable.
Rydberg Molecules – p.36/55
Two 3Σ Potentials!
•
• Must allow for the interaction of thesepotentials.
• Two states of the same symmetry can’t cross.• Mixing quite weak so avoided crossing barely
detectable.
Rydberg Molecules – p.36/55
Two 3Σ Potentials!
•
• Must allow for the interaction of thesepotentials.
• Two states of the same symmetry can’t cross.
• Mixing quite weak so avoided crossing barelydetectable.
Rydberg Molecules – p.36/55
Two 3Σ Potentials!
•
• Must allow for the interaction of thesepotentials.
• Two states of the same symmetry can’t cross.• Mixing quite weak so avoided crossing barely
detectable.
Rydberg Molecules – p.36/55
Properties of p-wave Potential
• Near resonance, can’t simply use pert. theory- must diagonalize the interaction in a basis ofseveral n manifolds.
• From non-crossing rule, potential can neverbe stronger than n− (n± 1) splitting.
• Omont (1977) showed that for stronginteraction and equally-spaced Rydberglevels:
Ep ≈ −EH
πn3δp.
Rydberg Molecules – p.37/55
Properties of p-wave Potential
• Near resonance, can’t simply use pert. theory- must diagonalize the interaction in a basis ofseveral n manifolds.
• From non-crossing rule, potential can neverbe stronger than n− (n± 1) splitting.
• Omont (1977) showed that for stronginteraction and equally-spaced Rydberglevels:
Ep ≈ −EH
πn3δp.
Rydberg Molecules – p.37/55
Properties of p-wave Potential
• Near resonance, can’t simply use pert. theory- must diagonalize the interaction in a basis ofseveral n manifolds.
• From non-crossing rule, potential can neverbe stronger than n− (n± 1) splitting.
• Omont (1977) showed that for stronginteraction and equally-spaced Rydberglevels:
Ep ≈ −EH
πn3δp.
Rydberg Molecules – p.37/55
Σ Potentials. Hamilton et al. (2002)
Rydberg Molecules – p.38/55
p-wave Π Potential.• Within degenerate n manifold factorization so
one strong interaction and remainder zero.
• Estimating this semiclassically for Π:
Epn(R) = −EH
πn3tan δp[k(R)],
identical to p contribution to Σ potentials!• Again, near resonance, can’t simply use pert.
theory - must diagonalize the interaction in abasis of several n manifolds.
• From non-crossing rule, potential can neverbe stronger than n− n± 1 splitting.
Rydberg Molecules – p.39/55
p-wave Π Potential.• Within degenerate n manifold factorization so
one strong interaction and remainder zero.• Estimating this semiclassically for Π:
Epn(R) = −EH
πn3tan δp[k(R)],
identical to p contribution to Σ potentials!
• Again, near resonance, can’t simply use pert.theory - must diagonalize the interaction in abasis of several n manifolds.
• From non-crossing rule, potential can neverbe stronger than n− n± 1 splitting.
Rydberg Molecules – p.39/55
p-wave Π Potential.• Within degenerate n manifold factorization so
one strong interaction and remainder zero.• Estimating this semiclassically for Π:
Epn(R) = −EH
πn3tan δp[k(R)],
identical to p contribution to Σ potentials!• Again, near resonance, can’t simply use pert.
theory - must diagonalize the interaction in abasis of several n manifolds.
• From non-crossing rule, potential can neverbe stronger than n− n± 1 splitting.
Rydberg Molecules – p.39/55
p-wave Π Potential.• Within degenerate n manifold factorization so
one strong interaction and remainder zero.• Estimating this semiclassically for Π:
Epn(R) = −EH
πn3tan δp[k(R)],
identical to p contribution to Σ potentials!• Again, near resonance, can’t simply use pert.
theory - must diagonalize the interaction in abasis of several n manifolds.
• From non-crossing rule, potential can neverbe stronger than n− n± 1 splitting.
Rydberg Molecules – p.39/55
p-wave Π Figure. Hamilton et al. (2002)
Rydberg Molecules – p.40/55
Semiclassical Approximation
• Mixing of s-wave and p-wave contributionsagain can be estimated semiclassically.
• Separation at avoided crossing: 2S√
EsEp,where
S(R) =
√3
2n
(
n2a0
R
)2 (
~
na0 p0
)3
.
Rydberg Molecules – p.41/55
Semiclassical Approximation
• Mixing of s-wave and p-wave contributionsagain can be estimated semiclassically.
• Separation at avoided crossing: 2S√
EsEp,where
S(R) =
√3
2n
(
n2a0
R
)2 (
~
na0 p0
)3
.
Rydberg Molecules – p.41/55
Omont Approximation
-250
-200
-150
-100
-50
0
0 300 600 900 1200 1500 1800
Ene
rgy
(GH
z)
R (a0)
3Σ Energies
n=29
n=30
s-wave Σp-wave Σ
UpperLower
Coupled n
Rydberg Molecules – p.42/55
Coulomb Green’s Function Potentials
Chibisov, Khuskivadze, and Fabrikant (2002).
Rydberg Molecules – p.43/55
p Σ-state wavefunction
•
ψn(r, R, r·R) = C
n−1∑
l=l1
(2l+1)Rnl(r)R′
nl(R)Pl(r·R).
• Calculated numerically, including n-manifoldcoupling, by Hamilton et al. (2002) atR = 302 a0.
Rydberg Molecules – p.44/55
p Σ-state wavefunction
•
ψn(r, R, r·R) = C
n−1∑
l=l1
(2l+1)Rnl(r)R′
nl(R)Pl(r·R).
• Calculated numerically, including n-manifoldcoupling, by Hamilton et al. (2002) atR = 302 a0.
Rydberg Molecules – p.44/55
’Butterfly’ State. Hamilton et al. (2002)
Rydberg Molecules – p.45/55
Single n Wavefunction
Within singlen manifoldwavefunc-tion doesn’thave quite’butterfly’structure.
Rydberg Molecules – p.46/55
Wavefunction
• Semiclassical analysis of this single-nwavefunction proceeds similarly to the trilobite- contribution from identical trajectories butwith different signs of the 4 contributions.
• More work needed for the coupled-n case!
Rydberg Molecules – p.47/55
Wavefunction
• Semiclassical analysis of this single-nwavefunction proceeds similarly to the trilobite- contribution from identical trajectories butwith different signs of the 4 contributions.
• More work needed for the coupled-n case!
Rydberg Molecules – p.47/55
Rydberg Molecules – p.48/55
Possible to Observe Rydberg Molecules?
871.1 µ
5S + 5S 5S + 5S
5S + 5P
5S + 30D
5S + 30FGH
5S + 30P
∆
787 nm
479.4 nm
298 nm
Low l State.High l State
Rydberg Molecules – p.49/55
Other Rydberg Dimers Possible?• For Σ states need negative scattering length.
•Mg Ca Sr 3Se Li Na K Cs Fr
a (a0) -2.5 -12 -18 -7.12 -6.19 -15.4 -21.7 -13.4
Mg, Ca, Sr (Bartschat and Sadeghpour,2003); Li, Na (Norcross, 1971); K (Fabrikant,1986); Cs and Fr (Bahrim et al., 2001).
• For Π states need low-energy resonance.• Available in other alkalis: Li (Karule, 1972),
Na (Bartschat, 2000); K (Nesbet, 1975); Cs(Khuskivadze et al., 2002).
Rydberg Molecules – p.50/55
Other Rydberg Dimers Possible?• For Σ states need negative scattering length.
•Mg Ca Sr 3Se Li Na K Cs Fr
a (a0) -2.5 -12 -18 -7.12 -6.19 -15.4 -21.7 -13.4
Mg, Ca, Sr (Bartschat and Sadeghpour,2003); Li, Na (Norcross, 1971); K (Fabrikant,1986); Cs and Fr (Bahrim et al., 2001).
• For Π states need low-energy resonance.• Available in other alkalis: Li (Karule, 1972),
Na (Bartschat, 2000); K (Nesbet, 1975); Cs(Khuskivadze et al., 2002).
Rydberg Molecules – p.50/55
Other Rydberg Dimers Possible?• For Σ states need negative scattering length.
•Mg Ca Sr 3Se Li Na K Cs Fr
a (a0) -2.5 -12 -18 -7.12 -6.19 -15.4 -21.7 -13.4
Mg, Ca, Sr (Bartschat and Sadeghpour,2003); Li, Na (Norcross, 1971); K (Fabrikant,1986); Cs and Fr (Bahrim et al., 2001).
• For Π states need low-energy resonance.
• Available in other alkalis: Li (Karule, 1972),Na (Bartschat, 2000); K (Nesbet, 1975); Cs(Khuskivadze et al., 2002).
Rydberg Molecules – p.50/55
Other Rydberg Dimers Possible?• For Σ states need negative scattering length.
•Mg Ca Sr 3Se Li Na K Cs Fr
a (a0) -2.5 -12 -18 -7.12 -6.19 -15.4 -21.7 -13.4
Mg, Ca, Sr (Bartschat and Sadeghpour,2003); Li, Na (Norcross, 1971); K (Fabrikant,1986); Cs and Fr (Bahrim et al., 2001).
• For Π states need low-energy resonance.• Available in other alkalis: Li (Karule, 1972),
Na (Bartschat, 2000); K (Nesbet, 1975); Cs(Khuskivadze et al., 2002).
Rydberg Molecules – p.50/55
Heteronuclear Molecules Possible?
• Most of previous discussion applies to X*-Rbmolecules, where X* is any Rydberg atom, aselectron scattering by Rb(5s) provides criticalproperties.
• Assumes high density of both X and Rbavailable!
• Other alkalis or alkaline earths possibleinstead of Rb.
Rydberg Molecules – p.51/55
Heteronuclear Molecules Possible?
• Most of previous discussion applies to X*-Rbmolecules, where X* is any Rydberg atom, aselectron scattering by Rb(5s) provides criticalproperties.
• Assumes high density of both X and Rbavailable!
• Other alkalis or alkaline earths possibleinstead of Rb.
Rydberg Molecules – p.51/55
Heteronuclear Molecules Possible?
• Most of previous discussion applies to X*-Rbmolecules, where X* is any Rydberg atom, aselectron scattering by Rb(5s) provides criticalproperties.
• Assumes high density of both X and Rbavailable!
• Other alkalis or alkaline earths possibleinstead of Rb.
Rydberg Molecules – p.51/55
Cs Potentials (Khuskivadze et al., 2002)
Rydberg Molecules – p.52/55
Semiclassical Na potentials• Phase shifts from Bartschat (2000)
-4
-2
0
2
4
600 900 1200 1500 1800
Energ
y (
GH
z)
R (a0)
Na 3Σ s-wave Energies
s-wave Σp-wave Σ
UpperLower
Coupled n
-250
-200
-150
-100
-50
0
0 300 600 900 1200 1500 1800
Energ
y (
GH
z)
R (a0)
Na 3Σ p-wave Energies
n=29
n=30
s-wave Σp-wave Σ
UpperLower
Coupled n
Rydberg Molecules – p.53/55
Conclusions
• Creation of ultra-long-range molecules withunusual properties with a variety of welldepths and separations should be possible;
• Well-depths can be comparable with splittingbetween adjacent n levels;
• Should exist in many species.• Most favourable conditions probably with
densities 1012 − 1014 cm−3;• Coherence properties of BEC or DFG not
essential to formation of such molecules.
Rydberg Molecules – p.54/55
Conclusions
• Creation of ultra-long-range molecules withunusual properties with a variety of welldepths and separations should be possible;
• Well-depths can be comparable with splittingbetween adjacent n levels;
• Should exist in many species.• Most favourable conditions probably with
densities 1012 − 1014 cm−3;• Coherence properties of BEC or DFG not
essential to formation of such molecules.
Rydberg Molecules – p.54/55
Conclusions
• Creation of ultra-long-range molecules withunusual properties with a variety of welldepths and separations should be possible;
• Well-depths can be comparable with splittingbetween adjacent n levels;
• Should exist in many species.
• Most favourable conditions probably withdensities 1012 − 1014 cm−3;
• Coherence properties of BEC or DFG notessential to formation of such molecules.
Rydberg Molecules – p.54/55
Conclusions
• Creation of ultra-long-range molecules withunusual properties with a variety of welldepths and separations should be possible;
• Well-depths can be comparable with splittingbetween adjacent n levels;
• Should exist in many species.• Most favourable conditions probably with
densities 1012 − 1014 cm−3;
• Coherence properties of BEC or DFG notessential to formation of such molecules.
Rydberg Molecules – p.54/55
Conclusions
• Creation of ultra-long-range molecules withunusual properties with a variety of welldepths and separations should be possible;
• Well-depths can be comparable with splittingbetween adjacent n levels;
• Should exist in many species.• Most favourable conditions probably with
densities 1012 − 1014 cm−3;• Coherence properties of BEC or DFG not
essential to formation of such molecules.Rydberg Molecules – p.54/55
Acknowledgments
• Chris Greene, JILA
• Hossain Sadeghpour; Harvard• Tom Harrison Newcastle• Jerry Hagon Newcastle
Rydberg Molecules – p.55/55
Acknowledgments
• Chris Greene, JILA• Hossain Sadeghpour; Harvard
• Tom Harrison Newcastle• Jerry Hagon Newcastle
Rydberg Molecules – p.55/55
Acknowledgments
• Chris Greene, JILA• Hossain Sadeghpour; Harvard• Tom Harrison Newcastle
• Jerry Hagon Newcastle
Rydberg Molecules – p.55/55
Acknowledgments
• Chris Greene, JILA• Hossain Sadeghpour; Harvard• Tom Harrison Newcastle• Jerry Hagon Newcastle
Rydberg Molecules – p.55/55
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55-1