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

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

large small 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 autoionization above the limit White 1934

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

Forced Autoionization Sandner et al q=0 q=∞

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. 7s 6d 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 Where r 1 and r 2 are the positions of the two electrons, r 12 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 H 0 the Schrodinger equation is separable. hydrogen Ba +

Without the coupling between the electrons provided by H 1 the excited states would only decay by radiative decay of each of the two electrons.

If r 1 < r 2 we can use f(r 2 )=2/r 2 and write H 1 as Interaction between the dipole of the core and the field from the outer electron Interaction between the quadrupole of the core and the field gradient from the outer electron We introduce the coupling between the electrons

H 1 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 continuum The full width at half maximum is the autoionization rate 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 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 The matrix element for the outer electron, 2, depends on the small r part of its wavefunction, which is why it has the 1/n 3/2 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/n 3

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/n 3 The autoionization rate is thus proportional to 1/n 3 How likely the outer electron is to deexcite the core on an orbit depends on the eccentricity of the orbit. Hence the ℓ dependence.

Absorption spectrum of Barium ground state atoms, showing the autoionizing 6pns and 6pnd states converging to both the 6p 1/2 and 6p 3/2 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. 0 amplitude 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 results In the asymmetric Fano profiles shown. They are observed in many contexts.

Excitation of Autoionizing states from the Rydberg states Isolated Core Excitation With the last laser the ion 6s-6p transition is excited The 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 cm 2. The direct photoionization of the 15d state has a cross section of cm 2 We ignore the direct continuum excitation. Why?

atoms lasers ions detector Field pulse lasers Ion 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 state It is straightforward to determine the width, 15 cm -1 and the energy. Two photon resonance due to third laser.

By changing the bound nd state it is straightforward to confirm the 1/n 3 dependence of the autoionization rate. Autoionization widths of the Ba 6pnd states

6sns 6snd 6s6p 6s6s 6snℓ 6snp It is straightforward to populate the low ℓ 6snℓ bound states to study their Autoionizing 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 lasers Ion signal Time (µs) 0 1 Field ramps

Recordings of the 6s13ℓ to 6p 1/2 13ℓ and 6s13ℓ to 6p 3/2 13ℓ transitions for different ℓ Splitting of the 6p 3/2 13ℓ states is due to the quadrupole interaction of H 1 Pruvost et al

Scaled Decay rates, n 3 Γ, in atomic units of the Ba 6p 1/2 12ℓ states Showing the rapid decrease with ℓ Radiative decay rate of the 6p ion

Simple time domain classical picture of autoionization If the probability of superelastic scattering per orbit is 60% you would 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 population Jones et al

Excite atoms from the Ca 4snd State to the 4pn state with a fs laser Monitor the population by pumping 4pnd atoms to 4dnd with another fs laser And 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, 1.6x10 8 s -1. The autoionization rates decrease with n and ℓ ℓ Decay rate 6p 1/2 12ℓ decay rates Autoionization radiative ℓ=10

ℓ Decay rate 6p 1/2 28ℓ decay rates autoionization radiative ℓ=7 For high ℓ many excitations Possible without autoionization

Cooling, trapping, and imaging of high n, high ℓ states using the core transition 6p 1/2 28ℓ>10 6s28ℓ> nm

Imaging an Interacting Rydberg Gas—Killian et al Rice 5s5s 5s5p 5s50s 5p50s wait 5s50ℓ 5p50ℓ ai fluorescence 5s 5p 493 nm Populate Sr 5s50s and drive the core transition Sr + Sr

Imaging an Interacting Rydberg Gas Evolution time (  s) mm Ground State 5s 2 1 S 0 5s5p 1 P 1 5s50s 1 S 0 5s50d 1 D 2 Penning ionization Collisiona l l-mixing electron- collision ionization auto- ionization Sr neutral 3  s Excitation Evolution time (  s)  s excitation 5s 2 P 1/2 5s 2 S 1/2 Sr + 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 6p 1/2 28ℓ>10 6s28ℓ> nm 5d 3/2 28ℓ>10 The autoioization rates of 6p 1/2 28ℓ>10 and 5d 3/2 28ℓ>10 states are similar. Radiative decay of the latter is 10 6 slower.

Cooling, trapping, and imaging of high n, high ℓ states using the core transition 6p 1/2 28ℓ>10 6s28ℓ> nm 5d 3/2 28ℓ>10 The autoioization rates of 6p 1/2 28ℓ>10 and 5d 3/2 28ℓ>10 states are similar. Radiative decay of the latter is 10 6 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 due to 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 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.