Rydberg States of Two Valence Electron Atoms W. E Cooke K.A. Safinya W. Sandner F. Gounand P. Pillet N. H. Tran R. Kachru R. R. Jones
Jan 06, 2016
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.