我們使用兩種方法(矩陣方法、拉比方法)來分析微中子在介質中參數式共振
的行為，並將這兩種方法做延伸與應用到不同密度週期性變化的物質系統。對
於物質密度週期變化的相位，我們有相當重要的觀察，相位差會大幅影響參數
式共振的條件(對角項等於零)，這與以前的研究矛盾，而我們也找到矛盾的原因
在於過去研究關於初始條件的設定(微中子波函數的純態與混態)，在基底轉換的
過程中(弱作用味基底、質量基底、共旋轉基底)，並未跟著適時地轉換。
再者，矩陣的表示方法中，我們使用微擾技巧，發現可以推導出和拉比方法
一樣的共振條件，並有數值的驗證，如此便能將兩種方法的關係連接在一起。
最後，我們將矩陣方法延伸到任意週期性密度變化的物質環境，並基於包立
矩陣的特性，找出機率振福的通解，這其中也包含著將參數式共振行為在向量
的空間中圖像化的洞見。以上的方法都將為微中子在各式環境中的參次式共振
行為提供更詳盡的解讀。

In this paper, we explore two methods for analyzing neutrino parametric resonance
and apply them to different systems in order to precisely determine the
behavior of neutrinos. We make an important observation regarding the phase
difference in a fluctuating matter density profile, which plays a significant role in
determining the resonance criteria. This finding contradicts previous conclusions
that overlooked the influence of mixed initial conditions after transformation into
a rotating frame.
Furthermore, we utilize a perturbation method in conjunction with matrix
analysis, demonstrating the equivalence between the resonance criteria derived
from our approach and another criterion obtained using the Rabi method. This
establishes a connection and enhances our understanding of the phenomenon.
Lastly, by examining the general form of arbitrary density profiles, we gain valuable
insights and interpretations regarding neutrino parametric resonance. This
provides a broader understanding of the subject matter and contributes to the
body of knowledge in this field.

[1] J. J. Thomson. Cathode rays. Phil. Mag. Ser. 5, 44:293–316, 1897.
[2] E. Rutherford. Collision of α particles with light atoms. IV. An anomalous
effect in nitrogen. Phil. Mag. Ser. 6, 37:581–587, 1919.
[3] J. Chadwick. The Existence of a Neutron. Proc. Roy. Soc. Lond. A,
136(830):692–708, 1932.
[4] Frederick Reines and Clyde L. Cowan. The neutrino. Nature, 178:446–449,
1956.
[5] L. D. Landau. On the conservation laws for weak interactions. Nucl. Phys.,
3:127–131, 1957.
[6] T. D. Lee and Chen-Ning Yang. Parity Nonconservation and a Two Component
Theory of the Neutrino. Phys. Rev., 105:1671–1675, 1957.
[7] B. Pontecorvo. Mesonium and anti-mesonium. Sov. Phys. JETP, 6:429, 1957.
[8] B. Pontecorvo. Neutrino Experiments and the Problem of Conservation of
Leptonic Charge. Zh. Eksp. Teor. Fiz., 53:1717–1725, 1967.
[9] V. N. Gribov and B. Pontecorvo. Neutrino astronomy and lepton charge.
Phys. Lett. B, 28:493, 1969.
[10] G. Danby, J. M. Gaillard, Konstantin A. Goulianos, L. M. Lederman, Nari B.
Mistry, M. Schwartz, and J. Steinberger. Observation of High-Energy Neutrino
Reactions and the Existence of Two Kinds of Neutrinos. Phys. Rev.
Lett., 9:36–44, 1962.
[11] K. Kodama et al. Observation of tau neutrino interactions. Phys. Lett. B,
504:218–224, 2001.
[12] S. M. Bilenky. Neutrino. History of a unique particle. Eur. Phys. J. H,
38:345–404, 2013.
[13] W. C. Haxton, R. G. Hamish Robertson, and Aldo M. Serenelli. Solar Neutrinos:
Status and Prospects. Ann. Rev. Astron. Astrophys., 51:21–61, 2013.
[14] Alessandro Mirizzi, Irene Tamborra, Hans-Thomas Janka, Ninetta Saviano,
Kate Scholberg, Robert Bollig, Lorenz Hudepohl, and Sovan Chakraborty.
Supernova Neutrinos: Production, Oscillations and Detection. Riv. Nuovo
Cim., 39(1-2):1–112, 2016.
[15] E. Akhmedov. Parametric resonance of neutrino oscillations and passage
of solar and atmospheric neutrinos through the earth. Nuclear Physics B,
538(1):25–51, 1999.
[16] L. Wolfenstein. Neutrino Oscillations in Matter. Phys. Rev. D, 17:2369–2374,
1978.
[17] R. F. Sawyer. Neutrino oscillations in inhomogeneous matter. Phys. Rev. D,
42:3908–3917, 1990.
[18] H. Nunokawa, Anna Rossi, V. B. Semikoz, and J. W. F. Valle. The Effect
of random matter density perturbations on the MSW solution to the solar
neutrino problem. Nucl. Phys. B, 472:495–517, 1996.
48
[19] F. N. Loreti, Y. Z. Qian, G. M. Fuller, and A. B. Balantekin. Effects of
random density fluctuations on matter enhanced neutrino flavor transitions
in supernovae and implications for supernova dynamics and nucleosynthesis.
Phys. Rev. D, 52:6664–6670, 1995.
[20] P. I. Krastev and A. Yu. Smirnov. Parametric Effects in Neutrino Oscillations.
Phys. Lett. B, 226:341–346, 1989.
[21] J. P. Kneller, G. C. McLaughlin, and K. M. Patton. Stimulated Neutrino
Transformation in Supernovae. J. Phys. G, 40:055002, 2013.
[22] James P. Kneller, Gail C. Mclaughlin, and Kelly M. Patton. Stimulated
neutrino transformation in supernovae. AIP Conf. Proc., 1560(1):176–178,
2013.
[23] James P. Kneller and Neel V. Kabadi. Sensitivity of neutrinos to the supernova
turbulence power spectrum: Point source statistics. Phys. Rev. D,
92(1):013009, 2015.
[24] Kelly M. Patton, James P. Kneller, and Gail C. McLaughlin. Stimulated
neutrino transformation through turbulence on a changing density profile and
application to supernovae. Phys. Rev. D, 91(2):025001, 2015.
[25] Shashank Shalgar Lei Ma and Huaiyu Duan. Matter parametric neutrino flavor
transformation through rabi resonances. Physical Review D, 98(1):103011,
2018.
[26] J. Napolitano J. Sakurai. Modern Quantum Mechanics, 2nd Edition. Addison-
Wesley, y, 1301 Sansome Street, San Francisco, CA 941 11, 2011.

[27] T.P. Orlando. Two-Level System with Static and Dynamic Coupling. Massachusetts
Institute of Technology, Massachusetts Institute of Technology,
2004.