$$ \newcommand\Tr{\mathrm{Tr}} \newcommand{\braket}[2]{\langle #1 \mid #2 \rangle} \newcommand\I{\mathbb{I}} \newcommand{\avg}[1]{\left< #1 \right>} \newcommand{\RD}{D} \newcommand{\ri}{\mathrm{i}} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Sign}{Sign} \newcommand{\ii}{\mathrm i} \newcommand{\vv}{\mathrm v} \newcommand{\ff}{\mathrm f} \newcommand{\mm}{\mathrm m} \newcommand{\ee}{\mathrm e} \newcommand{\xx}{\mathrm x} \newcommand{\RR}{\mathrm R} \newcommand{\dd}{\mathrm d} \newcommand{\FF}{\mathrm F} \newcommand{\BB}{\mathrm B} \newcommand{\vph}{v_{\mathrm{ph}}} $$
Neutrino Oscillations

Neutrino Oscillations with Oscillatory Matter Profiles

In certain regions inside a star or a supernova where convection is prominent, the matter density varies rapidly as a function of distance 1 2. The neutrino flavor evolution in such environments is qualitatively different than that with a smooth density profile 3 4 5 6 7 8 9. For example, the flavor conversion of the neutrino is greatly enhanced if the matter density fluctuates with certain wavenumbers. This is known as the parametric resonance 3 10.

In this chapter, I will explain the parametric resonance from the perspective of Rabi oscillations. I will first review the Rabi oscillation phenomenon and derive a criterion when the off-resonance Rabi mode may have a significant impact on the resonance behavior. I will then introduce the background matter basis with which the equation of motion simplifies. After demonstrating the simplest example of the parametric resonance in the presence of a sinusoidal matter profile with the Rabi formula, I will explain the interference effect when there exist multiple Fourier modes in the matter profile. Finally, I will show how to use the Jacobi-Anger expansion to decompose an arbitrary matter profile into an infinite sum of Rabi modes. This decomposition is similar to what J. Kneller et al. did in reference 8 9 but is achieved in a way that makes the physics much more transparent. I will also use the criterion that I have derived for the interference between two Rabi modes to show that only a finite number of Rabi modes are relevant in a real physical system.

  1. B. Muller, "Non-radial instabilities and progenitor asphericities in core-collapse supernovae", Monthly Notices of the Royal Astronomical Society 448, 2141-2174 (2015) . ↩︎

  2. Sean M. Couch, "The role of turbulence in neutrino-driven core-collapse supernova explosions", The Astrophysical Journal 799, 5 (2015) . ↩︎

  3. P.I. Krastev, "Parametric effects in neutrino oscillations", Physics Letters B 226, 341-346 (1989) . ↩︎

  4. F. N. Loreti, "Neutrino oscillations in noisy media", Physical Review D 50, 4762-4770 (1994) . ↩︎

  5. E. Kh Akhmedov, "Parametric resonance in neutrino oscillations in matter", Pramana 54, 47-63 (2000) . ↩︎

  6. Alexander Friedland, "Neutrino signatures of supernova turbulence", , 1-5 (2006) . ↩︎

  7. James Kneller, "Turbulence effects on supernova neutrinos", Physical Review D 82, 123004 (2010) . ↩︎

  8. James P. Kneller, "Stimulated neutrino transformation with sinusoidal density profiles", Journal of Physics G: Nuclear and Particle Physics 40, 055002 (2013) . ↩︎

  9. Kelly M. Patton, "Stimulated neutrino transformation through turbulence", Physical Review D 89, 073022 (2014) . ↩︎

  10. E.Kh. Akhmedov, "Parametric resonance of neutrino oscillations and passage of solar and atmospheric neutrinos through the earth", Nuclear Physics B 538, 25-51 (1999) . ↩︎

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