MQDT analysis

1. MQDT analysis James Millen

2. Introduction • 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. MQDT analysis – Group meeting 13/09/10

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

4. Two-channel MQDT Channel 1 Channel 2 • Consider the excitation from the 5snd bound series to the 5p3/2nd autoionizing series. • Note: the autoionizing series is above the 5snd ionization limit -> (auto)ionization. • For a total energy EChannel 1: E = I2 - 1/(2ν2) Channel 2: E = I1 + γ εd I1 1/(2ν2) 5p3/2nd E Continuum γ I2 5snd MQDT analysis – Group meeting 13/09/10 MQDT analysis – Group meeting 13/09/10

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

6. General MQDT • By doing the QDT (essentially considering the correct boundary conditions for the problem) we find the following conditions: [tan(πηi)δij + Rij]aj = 0 ai = Zicos(πηi) • 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! MQDT analysis – Group meeting 13/09/10

7. Spectral density μA=0.5 • 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 R12, so open channel sets the width. • R can, in theory, be derived from the Coulombic wavefunctions. MQDT analysis – Group meeting 13/09/10

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

9. Transition • Now we can write the transition between the initial Rydberg state and the autoionizing state, introducing the dipole operator T • ZB (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 MQDT analysis – Group meeting 13/09/10

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

11. Cross section • The optical cross section is proportional to the dipole matrix element. • Its shape depends on the difference between µA and µB. • Its width depends on the energy of the state (i.e. nB ). 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. MQDT analysis – Group meeting 13/09/10

12. Cross section • Here µA = µB (mod1) MQDT analysis – Group meeting 13/09/10

13. Cross section • Here µA = µB + 0.5 (mod1) MQDT analysis – Group meeting 13/09/10

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

15. Six-channel MQDT Channel 2 Channel 3 εd εd Channel 1 εd 5p3/2nd+ 5p3/2nd- 5p1/2nd • There are three channels, each with a coupled continuum, so six channels in total. • Channel 1 is open, only two bound channels. 5snd MQDT analysis – Group meeting 13/09/10 MQDT analysis – Group meeting 13/09/10

16. Six-channel MQDT • Z’: The switch of variable means that mJ 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. MQDT analysis – Group meeting 13/09/10

17. Six-channel MQDT |Z|2 Effective quantum number ν Detuning of autoionizing laser from 5s1/2+ -> 5p3/2+ ion transition • Get interference between channels • Fitting parameters: Amplitude, mJ state mixing angle. R coefficients and quantum defects previously measured, and allowed to vary within errors. MQDT analysis – Group meeting 13/09/10

18. An application • 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. MQDT analysis – Group meeting 13/09/10

19. An application • 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. MQDT analysis – Group meeting 13/09/10

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