Rydberg States of TwoValence Electron Atoms

W. E CookeK.A. SafinyaW. SandnerF. GounandP. PilletN. H. TranR. KachruR. R. Jones

Perturbations of bound Rydberg states

Interactions with doubly excited states

Autoionization

Excitation of autoionizing states

Effective quantum numbers of the Ba 6snd states near n=25 a Lu-Fano plot showing the perturbation of energy levels

n=21

n=27

6snd

5d7d

The interaction with the 5d7d state perturbs the 6snd series

Higher lying levels are pushed upand lower lying ones pushed down.

The properties of the 6snd states are also perturbed.

The slope of the Lu-Fano plot gives the character of the states

6 5s snd d dndA Ay y y= +

The squared amplitude ratio is given by the derivative of the Lu Fano plot

2

2d s

s d

A d

A d

nn

-=

21

22

A

Alarge

21

22

A

Asmall

No interaction(----)

It is a level crossing problem.

Effective quantum numbers of the Ba 6snd states near n=25 a Lu-Fano plot

n=21

n=27

5d7d

Both level shifts and perturbed lifetimes are due to the interaction of the Rydberg states with a state converging to a higher limit.

They are related to autoionizationabove the limit

White 1934

The similarity of series perturbations to autoionizationthe phenomenon of Forced Autoionization-Sandner et al

Forced Autoionization Sandner et al

q=0q=∞

Path A Path B

Spectra taken via paths A and B on zero field and 4.8 kV/cm

q=0

q=∞

Ba two electrons outside a closed shell Ba++ core

Ba+ is isoelectronic to Cs, one electron outside the closed shell core,and the energy levels are similar.

7s6d

6p

5d

6s

Ba is simpler than He since the ion levels are nondegenerate.

To each of these Ba+ levels we add the second electron, producing the energy levels shown below.

We can write the Hamiltonian for the Ba atom, ignoring spin, as

12221

21

22

21

0

10

11)(

1)(

22

rrrfH

rrfH

HHH

Where r1 and r2 are the positions of the two electrons, r12 is their separation, and f(r) Is the potential an electron feels from the Ba++ ion. As r→∞ f(r)→2/r.If we use only H0 the Schrodinger equation ),(),( 21210 rrWrrH

is separable.

221

2121

2222

22

1111

21

2

1

)()(),(

)()(1

2

)()()(2

nWWWW

rrrr

rWrr

rWrrf

n

mnmn

hydrogen

Ba+

Without the coupling between the electrons provided by H1 the excitedstates would only decay by radiative decay of each of the two electrons.

12221

11)(

rrrfH

If r1 < r2 we can use f(r2)=2/r2 and write H1 as

)....(cos)(cos

11

12232

21

12122

11

1221

Pr

rP

r

rH

rrH

Interaction between the dipole of the core and the field from the outer electron

Interaction between the quadrupole of the core and the field gradientfrom the outer electron

We introduce the coupling between the electrons

H1 introduces the coupling, between states of the same parity and angular momentum,leading to both series perturbations and autoionization.

6s 6p

The 6pnd state is coupled to the 6sεf and 6sεp continua by the dipole coupling.It is also coupled to continua by the quadrupole coupling, which we ignore for simplicity.

Autoionization broadens a level coupled to a continuumThe full width at half maximum is the autoionization rate

2

12122

1 6)(cos62 fsPr

rpnd

There is also a phase shift of the continuum

The autoionization rate is given by Fermi’s golden rule, for example, the Autoionization rate from the 6pnd state to the 6sεf continuum is

2

12122

1 6)(cos62 fsPr

rpnd

The continuum state is normalized per unit energy.This expression is a product of an angular factor, of order 1,and two radial matrix elements

2/322

01

11

566

nf

rnd

asrp

The matrix element for the outer electron, 2, depends on the smallr part of its wavefunction, which is why it has the 1/n3/2 scaling.

Due to the centrifugal barrier which keeps high ℓ electrons from the core, autoionization rates fall rapidly with ℓ.

From the latter we can see that the autoionization rates scale as 1/n3

A simple classical picture of autoionization

Each time the Rydberg electron comes by the core it has a finite probability of superelastic scattering, deexciting the core from 6p to 6s and leaving with its energy

The frequency with which the elecron comes to the core is 1/n3 The autoionization rate is thus proportional to 1/n3

How likely the outer electron is todeexcite the core on an orbitdepends on the eccentricity of the orbit. Hence the ℓdependence.

Absorption spectrum of Barium ground state atoms, showing the autoionizing 6pns and 6pnd states converging to both the 6p1/2 and 6p3/2 limits.

The spectrum is composed of odd, certainly not Lorentzian, shapes superimposed on a non zero background

6s6s

There are two interfering pathways to the continuum, direct continuum excitation and excitation of the autoionizing state.The result is a Fano profile.

There are two excitation amplitudes, to the broadened discrete state, and to the continuum, which are added, then squared, to obtain the transition probability.

0ampl

itude

Photon energy

discrete

continuum

The ratio of the discrete to the continuum amplitudes is q, which defines the lineshape.

The lineshapes are as shown.

q=∞ and q=0 are Lorentzian Peaks and dips. Any other q resultsIn the asymmetric Fano profiles shown.

They are observed in many contexts.

Excitation of Autoionizing states from the Rydberg statesIsolated Core Excitation

With the last laser the ion 6s-6p transition is excitedThe outer electron is a spectator.

The Fano q parameter is infinity. Lorentzian lines

The 6s-6p transition is the strongest transition in the Ba+ ion. It is spread over the width of the 6p15d state, yielding a cross section of 10-13 cm2.

The direct photoionization of the 15d state has a cross section of 10-22 cm2

We ignore the direct continuum excitation. Why?

atoms

lasers

ions

detector

Field pulse

lasersIon signal

Time (µs)0 1

Detect the ions from the rapid decay of the autoionizing 6p15d state as the third laser frequency is swept.

The result: a Lorentzian line centered on the 6p15d stateIt is straightforward to determine the width, 15 cm-1 and the energy.

Two photon resonance dueto third laser.

By changing the bound nd state it is straightforward to confirm the 1/n3 dependence of the autoionization rate.

Autoionization widths of the Ba 6pnd states

6sns 6snd

6s6p

6s6s

6snℓ6snp

It is straightforward to populate the low ℓ 6snℓ bound states to study theirAutoionizing 6pnℓ analogues, but can we study the higher ℓ states as well?

?

The Stark switching technique– excite a bound Stark state. Reduce the field Adiabatically to zero, producing the desired high ℓ 6snℓ state.-Freeman and Kleppner

Pruvost et al, Jones

lasersIon signal

Time (µs)0 1

Field ramps

Recordings of the 6s13ℓ to 6p1/213ℓ and 6s13ℓ to 6p3/213ℓ transitions for different ℓ

Splitting of the 6p3/213ℓ statesis due to the quadrupole interactionof H1

Pruvost et al

Scaled Decay rates, n3 Γ, in atomic units of the Ba 6p1/212ℓ statesShowing the rapid decrease with ℓ

Radiative decayrate of the 6p ion

Simple time domain classical picture of autoionization

If the probability of superelastic scattering per orbit is 60% youwould expect in the time domain to see the population decay in linear segments,one per orbit, and the rate to decrease like a stairstep.

time

popu

latio

n

Jones et al

Excite atoms from the Ca 4sndState to the 4pn state with a fs laser

Monitor the population by pumping 4pnd atoms to 4dnd with another fs laserAnd detecting 7.1 eV electrons

The lines are at the Kepler periods

Linear piecewise decay

Can we use the core transition to manipulate bound Rydberg atoms?

Yes, if we can avoid autoionization.

The radiative decay rate is the decay rate of the Ba+ 6p state,1.6x108 s-1.

The autoionization rates decrease with n and ℓ

ℓ

100

10

1

0.1

Dec

ay r

ate

6p1/212ℓ decay rates

Autoionization

radiative

ℓ=10

ℓ

10

1

0.1

0.01

Dec

ay r

ate

6p1/228ℓ decay rates

autoionization

radiative

ℓ=7

For high ℓmany excitations Possible without autoionization

Cooling, trapping, and imaging of high n, high ℓ states using the core transition

6p1/228ℓ>10

6s28ℓ>10

493 nm

Imaging an Interacting Rydberg Imaging an Interacting Rydberg Gas—Killian et al RiceGas—Killian et al Rice

5s5s

5s5p

5s50s

5p50s

wait5s50ℓ

5p50ℓai

fluorescence

5s

5p

493 nm

Populate Sr 5s50s and drive the core transition

Sr+

Sr

Imaging an Interacting Rydberg Imaging an Interacting Rydberg GasGas

3 mm

Ground State5s2 1S0

5s5p 1P1

5s50s 1S0

5s50d 1D2

• Penning ionization

• Collisional l-mixing

• electron-collision ionization

• auto-ionization

Sr neutral

Evolution time (s)0.6

2.9

5.1

7.3

2 s excitation

5s 2P1/2

5s 2S1/2Sr+ ion or

Sr Rydberg core

422 nm

evolution time

Killian et al

Cooling or trapping of high n, high ℓ states using the core transition

6p1/228ℓ>10

6s28ℓ>10

493 nm5d3/228ℓ>10

The autoioization rates of 6p1/228ℓ>10and 5d3/228ℓ>10 states are similar.Radiative decay of the latter is 106 slower.

Cooling, trapping, and imaging of high n, high ℓ states using the core transition

6p1/228ℓ>10

6s28ℓ>10

493 nm5d3/228ℓ>10

The autoioization rates of 6p1/228ℓ>10and 5d3/228ℓ>10 states are similar.Radiative decay of the latter is 106 slower.

650 nm

Far off resonance trap based on ICE

6s15d

6p15d

Laser red detuned from 455 nm

The low power spectrum: a Lorentzian line centered on the 6p15d state

Two photon resonance dueto third laser.

Third laser power 50x higher

The spectrum is due to the ion transition with a spectator electron which is projected from the bound state onto the autoionizing state.

The squared 6s15d-6p15d matrix element, and thus the optical cross section, is

2

222

22

2 15666156 ddApsdpds

Ion dipole matrix element

Spectral density of the autoionizing state

Overlap integral

The center of the cross section looks Like the spectral density.

At high power the center of the cross section is saturated, and the wings become apparent.

The zeroes come from the overlap Integral.

Calculated spectrum for high laser power

Rydberg states of two electron atoms provide easy access to doubly excited autoionizing states.

There are new possibilities for detecting and trapping Rydberg atoms.