1. MQDT analysis James Millen

2. Introduction MQDT analysis – Group meeting 13/09/10 In our experiment we measure the population of Rydberg states using autoionization. The excitation of the inner valence electron gives rise to a spectrum. Can we analyse this spectrum and get information about our Rydberg gas? We use multi-channel quantum defect theory (MQDT) to analyse our data. DISCLAIMER: Won’t really talk about MQDT, just how to use its results to analyse our data.

3. Our experiment 5s 5p 5s nlnl nlnl 5p MQDT analysis – Group meeting 13/09/10

4. Two-channel MQDT MQDT analysis – Group meeting 13/09/10 5p 3/2 nd 5snd εdεd Continuum Channel 1Channel 2 Consider the excitation from the 5snd bound series to the 5p 3/2 nd autoionizing series. Note: the autoionizing series is above the 5snd ionization limit -> (auto)ionization. For a total energy E Channel 1:E = I 2 - 1/(2ν 2 ) Channel 2: E = I 1 + γ I2I2 I1I1 E γ 1/(2ν 2 )

5. General MQDT MQDT analysis – Group meeting 13/09/10 i labels the channel. |Z i | 2 : Density of states for channel i (more later). ψ i : Wavefunction of channel i. If E is greater than I i then the channel is “open” (continuum), otherwise it’s “closed” (bound, or sometimes “quasibound”).

6. General MQDT MQDT analysis – Group meeting 13/09/10 By doing the QDT (essentially considering the correct boundary conditions for the problem) we find the following conditions: [tan(πη i )δ ij + R ij ]a j = 0 a i = Z i cos(πη i ) For an open channel η is the phase shift (divided by π) –τ For a closed channel it’s the sum of the effective quantum number and the quantum defect ν+μ R describes the coupling between the states…more later!

7. Spectral density MQDT analysis – Group meeting 13/09/10 Z is only important for closed channels (otherwise its constant). Position is set by the quantum defect μ of the autoionizing state. R sets the width of the peaks. For 2-channels there’s only R 12, so open channel sets the width. R can, in theory, be derived from the Coulombic wavefunctions. μ A =0.5

8. Expansion MQDT analysis – Group meeting 13/09/10 We can expand the autoionization channel wavefunction into: The wavefunction of the ionic core χ The angular part of the wavefunction of the outer electron Φ The Coulombic interaction term Similarly we can do the same with the bound Rydberg state 5snd …where n B is the effective quantum number of the bound state

9. Transition MQDT analysis – Group meeting 13/09/10 Now we can write the transition between the initial Rydberg state and the autoionizing state, introducing the dipole operator T Z B (the spectral density of the Rydberg state) is pretty uninteresting, just a delta function at the energy of the state. Ultimately the ion core dipole matrix element is just a number, as is the overlap of the angular parts. The overlap of the Coulombic terms has an analytic form

10. Overlap integral MQDT analysis – Group meeting 13/09/10 μ B =0.5 O describes the change in wavefunction of the outer electron. Position is set by the quantum defect μ B of the Rydberg state. Width is set by the quantum number n B of the Rydberg state. Has an analytical form.

11. Cross section MQDT analysis – Group meeting 13/09/10 The optical cross section is proportional to the dipole matrix element. Its shape depends on the difference between µ A and µ B. Its width depends on the energy of the state (i.e. n B ). Extremely wide, at n≈20 in Sr ~1 THz wide, at n≈60,10 GHz wide. This tells you a lot about the nature of autoionizing states.

12. Cross section MQDT analysis – Group meeting 13/09/10 Here µ A = µ B (mod1)

13. Cross section MQDT analysis – Group meeting 13/09/10 Here µ A = µ B + 0.5 (mod1)

14. Two-channel model MQDT analysis – Group meeting 13/09/10 Energy of Autoionizing laser (cm -1 ) Doesn’t reproduce our data very well... Normalised ion signal

15. MQDT analysis – Group meeting 13/09/10 Six-channel MQDT MQDT analysis – Group meeting 13/09/10 5p 1/2 nd 5snd εdεd Channel 1 Channel 2 5p 3/2 nd + εdεd 5p 3/2 nd - εdεd Channel 3 There are three channels, each with a coupled continuum, so six channels in total. Channel 1 is open, only two bound channels.

16. Six-channel MQDT MQDT analysis – Group meeting 13/09/10 Z’: The switch of variable means that m J state mixing is included in the spectral density. D: contains singlet-triplet mixing, 6j and 9j symbols to uncouple the bases, Wigner-Eckart theorem to reduce the matirx element.

17. Six-channel MQDT MQDT analysis – Group meeting 13/09/10 Get interference between channels Fitting parameters: Amplitude, m J state mixing angle. R coefficients and quantum defects previously measured, and allowed to vary within errors. Detuning of autoionizing laser from 5s 1/2 + -> 5p 3/2 + ion transition Effective quantum number ν |Z| 2

18. An application MQDT analysis – Group meeting 13/09/10 In our experiment some of the 5s56d state population is transferred to the 5s54f state We fit our data with a combined 6-channel model for the D state, and two-channel model for the F state.

19. An application MQDT analysis – Group meeting 13/09/10 How much of the population is transferred? To get the ratio of cross sections compare 2-channel fits to each set of data (no normalisation worries). To get ratio of signal amplitudes take relative amplitudes from the full 6+2-channel model. In this case we find that (13±3)% of the Rydberg population ends up in the F state.

20. Aside- Ultra-cold plasma formation Gas of cold Rydberg atoms Fast ionization, some electrons leave Positive charge binds electrons. Electrons oscillate, ionizing atoms [2] See a burst of ions. Ion-Rydberg collisions create high-L Rydbergs through Stark mixing [3] MQDT analysis – Group meeting 13/09/10