Scattering Polarization and the Hanle Efect. R. Casini High Altitude Observatory

Transcription

1 Scattering Polarization and the Hanle Efect R. Casini High Altitude Observatory

2 Syllabus Line polarization recap The Zeeman assumption NLTE vs NLTE of the 2nd kind Atomic polarization Population imbalance Quantum interference Atoms and external felds Atom-photon interactions The fat-spectrum approximation (CRD theory) Spherical tensors Reducible and irreducible representations Rate-equation solution for the polarized 1-0 atom Hanle efect of the excited state Application: The case of the forbidden coronal lines Rate-equation solution for the polarized 0-1 atom Hanle efect of the lower state Dichroism Application: Prominences and flaments in He I Multi-level vs multi-term atom schemes Fine-structure and hyperfne-structure depolarization The ``magnetic kernel of the rate equations The Alignment-to-Orientation conversion mechanism Application: Diagnostics of prominence/flament magnetism Laboratory verifcation of the CRD theory Coherent vs non-coherent scattering (PRD theory) 2nd-order atom-photon processes Collisional relaxation and coherent/non-coherent branching Application: The Mg II h-k lines

3 Introduction / Summary

4 Why Spectro-Polarimetry? polarized radiation is produced every time the interaction of photons with atoms/molecules is afected by symmetry-breaking processes electric and magnetic felds anisotropic excitation and de-excitation (radiative, collisional) Spectro-Polarimetric Diagnostics interpretation of the polarization signatures of electromagnetic and thermo-dynamic plasma processes

5 Polarization of Spectral Lines (1) spectral lines originate in transitions between energy levels of the atom for weak magnetic felds (Zeeman regime), these levels represent angular-momentum eigenstates atomic transitions satisfy selection rules, e.g., and for dipole transitions diferent ΔM transitions show diferent polarizations in any given direction, their unweighted spectral average is zero if some ΔM transition dominates over others at a given frequency, the line is polarized at that frequency

6 Zeeman Efect atomic transition J = 1 J = 0 normal Zeeman triplet ω0 σ ± : ΔM = ±1 π : ΔM = 0 components at component at ω 0 ± ωb ω0 ω0

7 Zeeman Efect atomic transition J = 1 J = 0 normal Zeeman triplet ω0 σ ± : ΔM = ±1 π : ΔM = 0 components at component at ω 0 ± ωb ω0 ω0

8 Zeeman Efect isotropically populated levels atomic transition J = 1 J = 0 normal Zeeman triplet ω0 σ ± : ΔM = ±1 π : ΔM = 0 components at component at ω 0 ± ωb ω0 ω0

9 Polarization of Spectral Lines (2) for zero felds, a spectral line can be polarized only if the weights of the various ΔM transitions (i.e., the populations of their initial levels) are diferent level population imbalance can be generated by anisotropic excitation processes (radiative or collisional) on the contrary, isotropic excitation processes tend to destroy level population imbalance e.g., thermalization of atomic level populations in collision dominated plasmas (a typical LTE condition) zero-feld polarization is typical of anisotropic radiation scattering in low-density plasmas nd NLTE of the 2 kind

10 st NLTE (of the 1 kind) J = 1/2 J = 1/2, 3/2 the populations of the levels 2S+1LJ may depart from LTE (i.e., Boltzmann distribution based on local temperature) however, the magnetic sublevels M = J,,+J are assumed to be isotropically populated energy (Grotrian) diagram of Na I

11 Polarization of Spectral Lines (3) Example: solar atmosphere competing efects of anisotropic irradiation and isotropic collisions photosphere: radiation anisotropy is low; collision isotropy is high; collisional rates (density) is high zero-feld polarization is negligible chromosphere: radiation anisotropy may be large (mostly dependent on CLV); collision isotropy is high; collisional rates decrease quickly with height zero-feld polarization is important corona: radiation anisotropy is dominant (from both CLV and height); collision isotropy starts to break down; collisional rates are low zero-feld polarization is dominant

12 Zeeman Efect Diagnostics the Zeeman efect is mostly applicable to mediumstrong felds (bar level-crossing efects; see below) splitting of intensity profle for B 103 G polarization measurements for B 102 G efect is linear for circular polarization (longitudinal component, B ) efect is quadratic for linear polarization (transverse component, B ) line-integrated polarization is (tipically) zero for weaker felds (B 102 G) we need other methods scattering polarization Hanle efect polarization efects from level crossing

13 Limitations of the Zeeman Efect Weak-Field Approximation: ξ (ωb /ΔωT) 1 I(ω ; ξ ) ~ I(ω ;0) + O(ξ2) Q(ω ; ξ ) ~ O(ξ2) U(ω ; ξ ) ~ O(ξ2) V(ω ; ξ ) ~ ξ I'(ω ; 0) + O(ξ2) when just a fraction fb (flling factor) of the pixel is magnetic I(ω ; ξ ) (1 fb) I(ω ;0) + fb I(ω ;0) = I(ω ;0) V(ω ; ξ ) 0(1 fb) I'(ω ;0) + ξ fb I'(ω ;0) = (ξ fb)i'(ω ;0) (ξ fb) quantifes the magnetic fux (not strength!)

14 Scattering Polarization and Quantum Mechanics

15 Scattering Polarization (semi-classical view) electron 3-D (damped) oscillator (H. A. Lorentz, The Theory of Electrons, 1916) atom forward scattering 90-deg scattering unpolarized incident light e.g., Rayleigh scattering

16 Atomic Polarization level population imbalance ω0 ω0

17 Atomic Polarization level population imbalance anisotropically populated levels ω0 ω0

18 Atomic Polarization level population imbalance anisotropically populated levels ω0 ω0

19 Atomic Polarization level population imbalance anisotropically populated levels ω0 σ+ σ atomic orientation π σ + + σ atomic alignment ω0

20 Atomic Polarization quantum level interference ψ( 1) 2 ψ(+1) 2 ψ(0) 2 A10 ωb ωb ψ(m ) + ψ(m') 2 ψ(m ) 2 + ψ(m') 2 ψ(m ) + ψ(m') 2 ψ(m ) 2 + ψ(m') 2 when ωb A10

21 Quantum Mechanics of Atoms (1) the possible energies of a quantum system correspond to the spectrum (i.e., the set of eigenvalues) of the Hamiltonian operator H (this is one of the quantum postulates) the Hamiltonian of atoms is an unbounded operator (because of the kinetic term), having both discrete and continuous spectra the line (Fraunhofer) spectrum of an atom corresponds to energy jumps within the discrete spectrum of the Hamiltonian (bound-bound transitions) in this lecture, we ignore transitions within the continuous spectrum (bound-free and free-free transitions)

22 Quantum Mechanics of Atoms (2) the eigenstates of a quantum system are parametrized by a maximal set of good quantum numbers (this is one of the corollaries of the quantum postulates!) determine all non-trivial quantum operators Ai with spectrum {ai} that commute with the Hamiltonian: the eigenstates are then a pure state of a quantum system is any linear (complex) combination of its eigenstates NOTE: this is a coherent superposition of states of a single atom; it cannot describe the incoherent superposition of states of two (or more) non-interacting atoms!

23 Quantum Mechanics of Atoms (3) to any given pure state of a quantum system, we can associate a projection operator, which projects a quantum system in any arbitrary state into that pure state χ ν ν χ direction projection a statistical ensemble of identical, non-interacting quantum systems can be thought of as occupying all possible pure states with probabilities we describe it through an incoherent superposition of projection operators, i.e., the statistical operator

24 Quantum Mechanics of Atoms (4) NOTE: we are considering a quantum statistical ensemble of identical, non-interacting systems (e.g., a gas of atoms where atom-atom correlations can be neglected; NOT a Bose-Einstein condensation!) this is properly represented by an incoherent (real) superposition of projection operators (a mixture ), as opposed to a coherent (complex) superposition of pure states (i.e., a new pure state) problem analogous to that of representing partially polarized radiation by the coherency matrix rather than by a Jones vector) NOTE: pure states are elements of the Hilbert space; projection operators (and the statistical operator) are functionals on that space

25 Density Matrix we describe an ensemble of identical, non-interacting atoms via a single atom statistical operator, i.e., a linear combination of projectors of the single-atom energy eigenstates (possibly including external felds)

26 Density Matrix we describe an ensemble of identical, non-interacting atoms via a single atom statistical operator, i.e., a linear combination of projectors of the single-atom energy eigenstates (possibly including external felds) eigenstates of the zero feld energy

27 Density Matrix we describe an ensemble of identical, non-interacting atoms via a single atom statistical operator, i.e., a linear combination of projectors of the single-atom energy eigenstates (possibly including external felds) eigenstates of the zero feld energy the density matrix is the representation of the statistical operator in any eigenbasis of choice

28 Density Matrix we describe an ensemble of identical, non-interacting atoms via a single atom statistical operator, i.e., a linear combination of projectors of the single-atom energy eigenstates (possibly including external felds) eigenstates of the zero feld energy the density matrix is the representation of the statistical operator in any eigenbasis of choice population of level m quantum coherence between levels m and m'

29 Density Matrix we describe an ensemble of identical, non-interacting atoms via a single atom statistical operator, i.e., a linear combination of projectors of the single-atom energy eigenstates (possibly including external felds) eigenstates of the zero feld energy the density matrix is the representation of the statistical operator in any eigenbasis of choice compound probability population of level m quantum coherence between levels m and m'

30 Atoms and External Fields atoms can carry electric-dipole (d) and magnetic-dipole (m) momenta, depending on the dynamics of its charged components (nucleus and electrons) For a displacement vector r, total angular momentum J and spin S of the electron, d = e0 r, m = μ 0 (J+S) where e0 is the electron charge and μ0 is Bohr's magneton in the presence of external felds E and B, the energy of the atom (i.e., its Hamiltonian) is modifed by new terms: WE = d E, WB = m B

31 Atom in an Electric Field the electric Hamiltonian HE (of eigenvalue WE) is an unbounded operator (purely continuous spectrum) for small electric felds, the bound states of the atom are almost discrete (the energy levels look like narrow energy resonances) NOTE: the atom-photon interaction Hamiltonian produces a similar broadening of the atomic bound states (natural broadening) not surprising, because the E1 approximation of the interaction Hamiltonian has the same form as HE

32 Atom in a Magnetic Field the magnetic Hamiltonian HB (of eigenvalue WB) is a bounded operator (purely discrete spectrum) this property follows from the quantization of the angular momentum, e.g., assuming the direction of B as the quantization axis, only M is strictly a good quantum number (this is the general case of the incomplete Paschen-Back regime) the Zeeman limit assumes that J is also approximately good, hence becomes an eigenstate of HB

33 nd NLTE of the 2 Kind and Quantum Electrodynamics

34 Requirements for a NLTE Theory 1) quantum-mecanical foundation radiative process in complex atoms that do not admit a semi-classical interpretation unifed scheme to describe polarization phenomena (both radiative and atomic) 2) separable formalism between atomic and radiative parts iterative numerical schemes for the solution of the Polarized Radiative Transfer (PRT) problem in optically thick plasmas

35 NLTE Polarized Radiative Transfer (PRT) forward problem (local) PRT problem (non-local) self-consistency loop (Landi Degl Innocenti & Landolf 2004)

36 Atom-Photon Interaction (1) an ensemble of atoms (A) interacting with the radiation feld (R) is governed by a Hamiltonian operator H = HA + HR + HI where HI is the atom-photon interaction Hamiltonian in the electric-dipole (E1) approximation HI = d ER(0) where ER(0) is the radiation feld evaluated in the origin of the atomic frame of rest note the resemblance of electric-dipole interaction Hamiltonian with HE

37 Atom-Photon Interaction (2) the interacting system is described by the statistical operator ρ(t) the evolution equation of ρ(t) is the quantum-mechanical Liouville equation (derives directly from the Schrödinger equation) this has the formal solution

38 Atom-Photon Interaction (3) Alternatively, the operator ρ(t) evolves according to where U(t,t0) is the evolution operator this has the formal expression time-ordering this second approach allows a diagrammatic description of atom-photon interactions (Feynman diagrams) requires a formal procedure of second quantization of the atomic Hamiltonian!

39 Atom-Photon Interaction (4) The statistical operator of the interacting A+R system satisfes an initial condition of factorization ρ(t0) = ρa(t0) ρr(t0) i.e., matter and radiation are initially uncorrelated the truncation order of the perturbation solution sets the physical order of the possible atom-photon processes

40 Evolution Equations the time-dependent solution separates into an atomic part (density matrix) and a radiation part (coherency matrix)

41 Evolution Equations the time-dependent solution separates into an atomic part (density matrix) and a radiation part (coherency matrix) Atoms Photons

42 Evolution Equations the time-dependent solution separates into an atomic part (density matrix) and a radiation part (coherency matrix) Atoms Photons SEE RTE

43 Evolution Equations the time-dependent solution separates into an atomic part (density matrix) and a radiation part (coherency matrix) Atoms Photons SEE RTE substitute truncated solution here

44 One-Photon Processes 1st order of perturbation: absorption and emission time theory is well established (Landi Degl'Innocenti & Landolf 2004) valid in the regime of Complete Re-Distribution (CRD) non-coherent scattering (collision dominated and/or fatspectrum radiation) inadequate approximation for many chromospheric lines (e.g., Mg II h-k, Ca II H-K, Na I D) where efects of partially coherent re-distribution (PRD) are important

45 Some Successes of the CRD Theory... Magnetic diagnostics of prominences/flaments (Hanle efect, level-crossing polarization, dichroism) Spectro-polarimetry of M1 coronal lines; atomic alignment efects on circular polarization Spectro-polarimetry of hydrogen lines in magnetic and electric felds (magnetic/electric Hanle efect, anomalous NCP) Scattering polarization and magnetic Hanle efect in molecular lines Experimental verifcation of the theory applied to the polarization of the Na I D doublet (see Sodium Scattering Experiment)

46 Interlude: Tensor Algebra

47 Spherical Tensors (1) the characteristic symmetries of the atomic and radiation Hamiltonians correspond to invariance properties within specifc transformation groups to fully exploit these symmetries, it is convenient to use a spherical (instead of Cartesian) representation of tensors Example: rank-1 tensor (vector) (Cartesian) (spherical) Where and similarly for eq

48 Semi-Classical 3-D Oscillator equivalent description in spherical basis

49 Spherical Tensors (2) the spherical basis eq is the natural eigenbasis for the radiation (hence, an obligated choice!) NOTE: the spherical components of a rank-1 tensor (vector) behave just like the spherical harmonics Y1q(ϑ,φ) given two vectors a and b, we can build a rank-2 tensor with the 9 components this is a reducible representation of a rank-2 tensor the components can be combined into new quantities with characteristic transformation properties under the rotation group SO(3) this makes algebraic manipulation a LOT easier!

50 Spherical Tensors (3) Example: the scalar (i.e., rotational invariant) analogously, we can introduce a rank-1 tensor (which behaves like Y1Q(ϑ,φ)) and a rank-2 tensor (which behaves like Y2Q(ϑ,φ)) the total number of components is still the same 1+3+5=9 but now,, and have distinct transformation properties this is an irreducible (block-diagonal) representation of bringing both atoms and photons into this representation highlights the symmetries of atom-photon interactions

51 Irreducible Density Matrix Tensor given the density matrix elements in the energy representation, the corresponding irreducible spherical tensor representation is easily obtained (e.g., Fano 1957; Landi Degl Innocenti & Landolf 2004) population orientation alignment } atomic polarization

52 Irreducible Radiation Tensor similarly, we can introduce a (reducible) radiation tensor Jqq'(k) starting from the coherency matrix Iλλ'(k) (see Landi Degl Innocenti & Landolf 2004 for details), and derive its irreducible representation intensity circular pol. linear pol., anisotropy

53 Spherical Tensors (4) because of their transformation properties under the rotation group, the behavior of irreducible spherical tensors becomes physically intuitive Example: 1-0 atom where ω 0 is the frequency of the spectral line

54 Hanle Efect The modifcation of scattering polarization in the presence of magnetic felds

55 Hanle Efect The modifcation of scattering polarization in the presence of magnetic felds Wilhelm Hanle

56 Hanle Efect: 1-0 Atom Solution of the SEE in the formalism of the irreducible spherical tensors where: A10 and B01 are the Einstein coefcients for emission and absorption ω is the frequency of the incident radiation (constant across the line transition; fat-spectrum approximation) ω B is the Larmor frequency of the applied magnetic feld NOTE: both and are expressed in the magnetic reference frame (direction of B as the quantization axis)

57 Hanle Efect: 1-0 Atom Radiation Anisotropy (Fractional) Atomic Polarization

58 Hanle Efect: 1-0 Atom Radiation Anisotropy (Fractional) Atomic Polarization rotation matrix The rotation matrix contains the B-vector information

59 Quantum Coherence Relaxation (from Casini & Landi Degl Innocenti 2008, Astrophysical Plasmas, in Plasma Polarization Spectroscopy, ed. Fujimoto & Iwamae, Springer)

60 Quantum Coherence Relaxation magnetic regime of the Hanle efect ωb ~A10 ΔωT (from Casini & Landi Degl Innocenti 2008, Astrophysical Plasmas, in Plasma Polarization Spectroscopy, ed. Fujimoto & Iwamae, Springer)

61 Application: 1-0 Atom saturated Hanle efect of M1 coronal lines

62 Application: 1-0 Atom saturated Hanle efect of M1 coronal lines Alignment correction to longitudinal Zeeman efect

63 Hanle vs Zeeman In the Weak-Field Approximation: I(ω ; ξ ) and V(ω ; ξ ) like in the Zeeman efect on the other hand (e.g., 90 scattering along B) Q(ω ; ξ ) ~ fq(w ; ξ )I(ω ;0) + O(ξ2) U(ω ; ξ ) ~ ξ fu(w ; ξ )I(ω ;0) + O(ξ2) even if only a fraction fb of the resolution element is magnetic, ξ and fb are (formally) disentangled because A10 is much smaller than ΔωT, the Hanle efect is typically sensitive to much smaller felds than the Zeeman efect

64 Hanle Efect: 0-1 Atom Lower-Level Polarization In the weak-anisotropy/polarization limit

65 Hanle Efect: 0-1 Atom Lower-Level Polarization In the weak-anisotropy/polarization limit inverse lifetime of lower level

66 Hanle Efect: 0-1 Atom Lower-Level Hanle Efect magnetic regime of the Hanle efect ωb ~B10 J ΔωT B10 J A01 (from Casini & Landi Degl Innocenti 2008, Astrophysical Plasmas, in Plasma Polarization Spectroscopy, ed. Fujimoto & Iwamae, Springer)

67 Hanle Efect: 0-1 Atom Lower-Level Hanle Efect non-linearity (weak anisotropy) magnetic regime of the Hanle efect ωb ~B10 J ΔωT B10 J A01 (from Casini & Landi Degl Innocenti 2008, Astrophysical Plasmas, in Plasma Polarization Spectroscopy, ed. Fujimoto & Iwamae, Springer)

68 Hanle Efect: 0-1 Atom Lower-Level Dichroism 2 3P2,1,0 ϑ = Homogeneous slab He I ϑ = 90 ϑ = S1 ϑ = 0 Trujillo Bueno et al., Nature 415, 403 (2002)

69 Hanle Efect: 1-1 Atom (from Casini & Landi Degl Innocenti 2008, Astrophysical Plasmas, in Plasma Polarization Spectroscopy, ed. Fujimoto & Iwamae, Springer)

70 Beyond the Multi-Level Atom

71 Level-Crossing Spectroscopy Incomplete Paschen-Back Efect (from Casini & Landi Degl Innocenti 2008, Astrophysical Plasmas, in Plasma Polarization Spectroscopy, ed. Fujimoto & Iwamae, Springer)

72 Multi-Term Atom Magnetic Kernel SEE in the energy representation: 1) No feld: 2) With feld:

73 Multi-Term Atom Magnetic Kernel SEE in the energy representation: 1) No feld: 2) With feld: only ρ gets transformed to the irreducible spherical basis

74 Multi-Term Atom Magnetic Kernel SEE in the energy representation: 1) No feld: 2) With feld: only ρ gets transformed to the irreducible spherical basis the full expression must be transformed

75 Multi-Term Atom Magnetic Kernel SEE in the energy representation: 1) No feld: only ρ gets transformed to the irreducible spherical basis the full expression must be transformed 2) With feld: magnetic kernel

76 Magnetic Kernel The case with FS + HFS 1) Diagonal part: coherence relaxation due to Fine- and Hyperfne-Structure splitting (analogous to the Hanle efect due to Zeeman splitting) 2) Non-Diagonal part:

77 Magnetic Kernel The case with FS + HFS 1) Diagonal part: coherence relaxation due to Fine- and Hyperfne-Structure splitting (analogous to the Hanle efect due to Zeeman splitting) 2) Non-Diagonal part: generalized Landé factor K-coupling

78 Magnetic Kernel The case with FS + HFS 1) Diagonal part: coherence relaxation due to Fine- and Hyperfne-Structure splitting (analogous to the Hanle efect due to Zeeman splitting) 2) Non-Diagonal part: generalized Landé factor Hanle efect: level-crossing interference: K-coupling

79 Magnetic Kernel K-Coupling

80 Magnetic Kernel K-Coupling Alignment-to-Orientation conversion mechanism

81 Level-Crossing Spectroscopy Alignment-to-Orientation Transfer Chromospheric lines of He I (S=1) Net Circular Polarization (NCP) induced by K-coupling between alignment (K =2) and orientation (K=1) (Lehmann 1969, Landi Degl Innocenti 1982, Kemp et al. 1984) (from Casini & Landi Degl Innocenti 2008, Astrophysical Plasmas, in Plasma Polarization Spectroscopy, ed. Fujimoto & Iwamae, Springer)

82 magnetic map of a quiescent (?) prominence observed May 25, 2002 He I nm (D3) at NSO Dunn Solar Telescope with HAO Advanced Stokes Polarimeter (Casini et al. 2003)

83 observed July 5, 2005 He I nm at Vacuum Tower Telescope with IAC Tenerife Infrared Polarimeter II (Kuckein et al. 2009) magnetic map of an A-R flament

84 Sodium Scattering Experiment light-level monitoring calibration channel scattered light analysis input channel

85 Experimental Setup light-level monitoring Reference photodiode Light trap Lenses D1/D2 Selector Polarimeter scattered light analysis Helmholtz coils B1 Calibration light source calibration channel B2 Photomultiplier tube Sodium vapor Polarization selectors Stabilized light source spectrally fat radiation input channel Helmholtz-coil pairs provide B-feld in the scattering plane up to 150 G

86 Multi-Level Atom with HFS (1/2) B atom incident radiation 1) broadband polarized emissivity 2) Hanle phase matrix scattered radiation

87 Multi-Level Atom with HFS (2/2) 3) polarizability factor

88 Multi-Level Atom with HFS (2/2) 3) polarizability factor depolarizing collisions (K=1 orientation; K=2 alignment) inelastic collisions (collisional de-excitation) Hanle efect (i = j), level-crossing interference

89 Multi-Level Atom with HFS (2/2) 3) polarizability factor depolarizing collisions (K=1 orientation; K=2 alignment) inelastic collisions (collisional de-excitation) Hanle efect (i = j), level-crossing interference the coupling between K'= 2 and K = 1 is responsible for the so-called Alignment-to-Orientation (A-O) conversion mechanism

90 Experiment and Model Fit 3 free parameters: optical thickness (τ D2 1.3); alignment relaxation rate (δ(2) 19); collisional de-excitation rate (ε 0.44) Li, Casini, Tomczyk, Landi Degl Innocenti & Marsell, 2018, Astrophys. J. Lett. (submitted)

91 Beyond CRD

92 Two-Photon Processes 2nd order of perturbation: coherent scattering; absorption and emission (cascade) of two photons diagrams of partial frequency redistribution (PRD) two-photon absorption is a non-linear process (negligible in the weak-radiation limit that typically applies to the Sun) two-photon cascade does not apply to the Λ-type multi-term atom to which the theory is currently restricted

93 Two-Photon Processes 2nd order of perturbation: coherent scattering; absorption and emission (cascade) of two photons diagrams of partial frequency redistribution (PRD) Self-Energy diagram two-photon absorption is a non-linear process (negligible in the weak-radiation limit that typically applies to the Sun) two-photon cascade does not apply to the Λ-type multi-term atom to which the theory is currently restricted

94 Self-Energy Insertion Formal procedure of dressing of the atomic propagator (Dyson equation) via the self-energy diagram yields lifetimes of excited atomic states for spontaneous emission, and corresponding level widths

95 Self-Energy Insertion Formal procedure of dressing of the atomic propagator (Dyson equation) via the self-energy diagram yields lifetimes of excited atomic states for spontaneous emission, and corresponding level widths Risk of double counting of processes is avoided, relying on Wick's theorem and diagram topology

96 Radiative/Collisional Branching (heuristic) The formal description of partially coherent scattering was given in the absence of collisions In principle, one should develop a parallel, diagrammatic framework for collisions as well! (from Casini, del Pino Alemán, & Manso Sainz 2017)

97 Partial Redistribution (PRD) PRT equation for the Λ multi-term atom (no stimulated emission) Casini et al, 2014, 2016, 2017 } ωk u,u' ω k' } l,l' f

98 Partial Redistribution (PRD) PRT equation for the Λ multi-term atom (no stimulated emission) Casini et al, 2014, 2016, 2017 } ωk u,u' ω k' } l,l' f density matrix of initial state total inverse lifetime of excited state redistribution function

99 Partial Redistribution (PRD) Redistribution Function (in the atomic frame of rest) Casini, Landi Degl Innocenti, Manso Sainz, Landi Degl Innocenti, & Landolf 2014

100 Mg II h-k Modeling Hanle + M-O + PRD B = 20 G ϑb = 30, φb = 180 B = 100 G (from del Pino Alemán, Casini, & Manso Sainz 2017)

101 Mg II h-k Modeling Weak-Field Approximation in Stokes V (from del Pino Alemán, Casini, & Manso Sainz 2017)

102 Conclusions

103 Conclusions Quantum Mechanics is fun...

104 Conclusions Quantum Mechanics is fun... but also useful!