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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.

Overview

• Introduction;• Rydberg Atoms

• Hydrogen• Alkali Atoms

Overview

• Hydrogen

• Alkali Atoms

Overview

• Rydberg Molecules

• Low l Levels• High l Levels

Overview

• Rydberg Molecules• Low l Levels

• High l Levels

Overview

• Shape-Resonance Long-Range States

• Possibilities in Other Systems.

Overview

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.

Introduction

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;

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;

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).

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.

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,

∆Enl =α213.6 eVn3l(l + 1)

; α ≈ 1/137.

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,

∆Enl =α213.6 eVn3l(l + 1)

; α ≈ 1/137.

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.

Size of Hydrogenic Atoms•

〈r〉nl = [3n2 − l(l + 1)]/2 a0.

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.

Alkali Atoms

Enl = − 13.6 eV(n− δl)2

Alkali Atoms

Enl = − 13.6 eV(n− δl)2

Alkali Atoms

Enl = − 13.6 eV(n− δl)2

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.

1/r4A.

• Simplifications: active electron ’sees’

• Coulomb interaction with A+ core;• Short-range interaction with A, falling off as

1/r4A.

• Simplifications: active electron ’sees’• Coulomb interaction with A+ core;

• Short-range interaction with A, falling off as1/r4

1/r4A.

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 0.05 0.1 0.15 0.2

δ s (

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

4πε0R,

me electron mass, EH a.u. (27.21 eV).• Note that δs = 0 at small finite k :⇒ aeff

changes sign.

~2k2(R)

2me= −EH

4πε0R,

changes sign.

~2k2(R)

2me= −EH

4πε0R,

me electron mass, EH a.u. (27.21 eV).

• Note that δs = 0 at small finite k :⇒ aeff

changes sign.

~2k2(R)

2me= −EH

4πε0R,

changes sign.Rydberg Molecules – p.11/55

R-Dep. Effective Scattering Length

0 300 600 900 1200 1500 1800

R (a0)

e+Rb 3S Effective Scattering Length, n = 30.

Fermi Model

• Model due to Fermi (1934) - electron-atominteraction

V F (r,R) =2π~

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.

Fermi Model

• Model due to Fermi (1934) - electron-atominteraction

V F (r,R) =2π~

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.

Low-l Levels

• Assume V F weak ⇒ use pert. theory.

Enlm(R) = 〈nlm | V F (r,R) | nlm〉

R2nl(r) | Ylm(r) |2 δ(r − R)dr

me| Ylm(R) |2 R2

= δm,0~

2me(2l + 1)R2

nl(R),

quantizing along R.

Low-l Levels•

• Assume V F weak ⇒ use pert. theory.

Enlm(R) = 〈nlm | V F (r,R) | nlm〉

R2nl(r) | Ylm(r) |2 δ(r − R)dr

me| Ylm(R) |2 R2

= δm,0~

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

n∗4a20

n∗a0 pl(R)cos2 Φl(R)

Φl(R) =1

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.

Enl(R) ≈ 2l + 1

n∗4a20

Φl(R) =1

pl(R) dR− π/4,

Enl(R) ≈ 2l + 1

n∗4a20

Φl(R) =1

pl(R) dR− π/4,

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:

|Enl(R)| = n∗2

2l + 1

|aeff |a0

n∗pl(R)a0

n∗2a0

µ nuclear reduced mass (≈ 2822 me for Rb).• Bound vibrational levels may exist!

cos2 Φ(R) ≈ 1 − [(R−Re)pl(R)/~]2.

|Enl(R)| = n∗2

2l + 1

|aeff |a0

n∗pl(R)a0

n∗2a0

µ nuclear reduced mass (≈ 2822 me for Rb).

• Bound vibrational levels may exist!

cos2 Φ(R) ≈ 1 − [(R−Re)pl(R)/~]2.

|Enl(R)| = n∗2

2l + 1

|aeff |a0

n∗pl(R)a0

n∗2a0

µ nuclear reduced mass (≈ 2822 me for Rb).• Bound vibrational levels may exist!

The 30d potential. Greene et al. (2000)

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

n−1∑

(2l + 1)R2nl(R)a3

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.

(2l + 1)(2l′ + 1)/4π,•

En(R) = EHaeff

n−1∑

(2l + 1)R2nl(R)a3

where l1 lowest degen. level (Omont, 1977).

• Other eigenvalues 0 !• Sum can be performed analytically in terms ofRn0(R), R′

(2l + 1)(2l′ + 1)/4π,•

En(R) = EHaeff

n−1∑

(2l + 1)R2nl(R)a3

where l1 lowest degen. level (Omont, 1977).• Other eigenvalues 0 !

• Sum can be performed analytically in terms ofRn0(R), R′

(2l + 1)(2l′ + 1)/4π,•

En(R) = EHaeff

n−1∑

(2l + 1)R2nl(R)a3

n0(R), (Chibisov et al., 2000).

• Individual l oscillations smoothed.

(2l + 1)(2l′ + 1)/4π,•

En(R) = EHaeff

n−1∑

(2l + 1)R2nl(R)a3

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

aeffpl1(R)

~≈ EH

na0pl1(R)

En(R) =EH

aeffpl1(R)

~≈ EH

na0pl1(R)

En(R) =EH

aeffpl1(R)

~≈ EH

na0pl1(R)

More Properties of High-l Potential

• Still some small residual ’ripples’.

• Supports many more vibrational levels,including a few in the ’ripples’.

More Properties of High-l Potential

• Still some small residual ’ripples’.• Supports many more vibrational levels,

including a few in the ’ripples’.

Rb(n=30) Figure: Greene et al. (2000)

Rb(n=30) Figure: WKB Approx.

-12-10

-8-6-4-2 0 2

500 1000 1500 2000 2500

R (a0)

WKB Energy for s Σ Hydrogenic State

Rb (n=30) Vibrational Levels

Electron Wavefunction

ψn(r, R, r ·R) = Cn−1∑

(2l+1)Rnl(r)Rnl(R)Pl(r ·R),

where C is the normalization factor and Pl(x) is

the l−th Legendre polynomial.

’Trilobite’: Greene et al. (2000)Rb n = 30, R = 1232 a0.

Electron Density for Ultra-Long-Range Rydberg Molecule.

Trilobite Fossil

’Trilobite’ II

Click here to view the movie, produced by Dr J P

Hagon and Mr T Harrison.

Semiclassical Approximation to ψ.

• Semiclassical approximation using WKB forboth Rnl and for Pl, sum→ integral,

ψn(r, R, r · R) ≈∫ n−1

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]} .

Semiclassical Approximation to ψ.

• Semiclassical approximation using WKB forboth Rnl and for Pl, sum→ integral,

ψn(r, R, r · R) ≈∫ n−1

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]} .

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.

Stationary Phase•

Elliptical Orbit

Critical Orbits

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.

12 14 16 18 20 22 24 26 28 30

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.

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 0.05 0.1 0.15 0.2 0.25

δ p (

E (eV)

e+Rb 3P Phase shift

From Bahrim et al.(2001).Clearly at a resonancethis interaction can bevery strong.

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:

n (R) = −EH

πn3tan δp[k(R)].

n (R) = −EH

πn3tan δp[k(R)].

n (R) = −EH

πn3tan δp[k(R)].

n (R) = −EH

πn3tan δp[k(R)].

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.

Two 3Σ Potentials!

detectable.

Two 3Σ Potentials!

• Two states of the same symmetry can’t cross.

• Mixing quite weak so avoided crossing barelydetectable.

Two 3Σ Potentials!

detectable.

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.

Ep ≈ −EH

πn3δp.

Ep ≈ −EH

πn3δp.

Σ Potentials. Hamilton et al. (2002)

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.

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.

Epn(R) = −EH

πn3tan δp[k(R)],

Epn(R) = −EH

πn3tan δp[k(R)],

p-wave Π Figure. Hamilton et al. (2002)

Semiclassical Approximation

• Mixing of s-wave and p-wave contributionsagain can be estimated semiclassically.

• Separation at avoided crossing: 2S√

EsEp,where

S(R) =

na0 p0

Semiclassical Approximation

• Mixing of s-wave and p-wave contributionsagain can be estimated semiclassically.

• Separation at avoided crossing: 2S√

EsEp,where

S(R) =

na0 p0

Omont Approximation

0 300 600 900 1200 1500 1800

R (a0)

3Σ Energies

s-wave Σp-wave Σ

UpperLower

Coupled n

Coulomb Green’s Function Potentials

Chibisov, Khuskivadze, and Fabrikant (2002).

p Σ-state wavefunction

ψn(r, R, r·R) = C

n−1∑

(2l+1)Rnl(r)R′

nl(R)Pl(r·R).

• Calculated numerically, including n-manifoldcoupling, by Hamilton et al. (2002) atR = 302 a0.

p Σ-state wavefunction

ψn(r, R, r·R) = C

n−1∑

(2l+1)Rnl(r)R′

nl(R)Pl(r·R).

• Calculated numerically, including n-manifoldcoupling, by Hamilton et al. (2002) atR = 302 a0.

’Butterfly’ State. Hamilton et al. (2002)

Single n Wavefunction

Within singlen manifoldwavefunc-tion doesn’thave quite’butterfly’structure.

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!

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!

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

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).

a (a0) -2.5 -12 -18 -7.12 -6.19 -15.4 -21.7 -13.4

• 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).

a (a0) -2.5 -12 -18 -7.12 -6.19 -15.4 -21.7 -13.4

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.

Cs Potentials (Khuskivadze et al., 2002)

Semiclassical Na potentials• Phase shifts from Bartschat (2000)

600 900 1200 1500 1800

R (a0)

Na 3Σ s-wave Energies

s-wave Σp-wave Σ

UpperLower

Coupled n

0 300 600 900 1200 1500 1800

R (a0)

Na 3Σ p-wave Energies

s-wave Σp-wave Σ

UpperLower

Coupled n

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.

Conclusions

essential to formation of such molecules.

Conclusions

• 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.

Conclusions

densities 1012 − 1014 cm−3;

• Coherence properties of BEC or DFG notessential to formation of such molecules.

Conclusions

essential to formation of such molecules.Rydberg Molecules – p.54/55

Acknowledgments

• Chris Greene, JILA

• Hossain Sadeghpour; Harvard• Tom Harrison Newcastle• Jerry Hagon Newcastle

Acknowledgments

• Chris Greene, JILA• Hossain Sadeghpour; Harvard

• Tom Harrison Newcastle• Jerry Hagon Newcastle

Acknowledgments

• Chris Greene, JILA• Hossain Sadeghpour; Harvard• Tom Harrison Newcastle

• Jerry Hagon Newcastle

Acknowledgments

• Chris Greene, JILA• Hossain Sadeghpour; Harvard• Tom Harrison Newcastle• Jerry Hagon Newcastle

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Rydberg Molecules - lac.u-psud.fr · Introduction; Rydberg Atoms Hydrogen Alkali Atoms ... Looking...

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