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

Single-Frequency Matter Profiles Revisited

Let us consider the single-frequency matter profile that we have studied in Sec. Single-Frequency Matter Profiles again. In the Rabi basis, the Hamiltonian becomes $$ \begin{equation} \mathsf H^{(\RR)} = \frac{\omega_\mm}{2} \sigma_3 - \frac{\sin 2\theta_\mm \lambda_1\cos (k_1 r)}{2} \begin{pmatrix} 0 & e^{2\ri \eta(r)} \\ e^{-2\ri \eta(r)} & 0 \end{pmatrix}, \end{equation} $$ where $$ \begin{equation} \eta(r) = \frac{\lambda_1 \cos 2\theta_{\mathrm m}}{2 k} \sin (k_1 r) . \end{equation} $$ With the Jacobi-Anger expansion $$ \begin{equation} e^{\ri z \sin (\phi)} = \sum_{n=-\infty}^\infty J_n(z) e^{\ri n\phi}, \end{equation} $$ the term $e^{2\ri \eta(r)}$ becomes $$ \begin{equation} \exp\left({\ri \frac{\lambda_1 \cos 2\theta_{\mm}}{k_1} \sin (k_1 r) }\right) = \sum_{n=-\infty}^{\infty} J_n \left( \frac{\lambda_1 \cos 2\theta_\mm}{k_1}\right) e^{\ri n k_1 r}. \end{equation} $$ where $J_n(z)$ is the Bessel function of the first kind of order $n$. The Hamiltonian becomes a Rabi oscillation system with an infinite number of Rabi modes: $$ \begin{equation} \mathsf H^{(\mathrm{R})} = -\frac{\omega_{\mathrm{m}}}{2} \sigma_3 - \frac{1}{2} \sum_{n=-\infty}^\infty A_n \begin{pmatrix} 0 & e^{\ri n k_1 r} \\ e^{ - \ri n k_1 r} & 0 \end{pmatrix}, \label{chap:matter-sec:jacobi-eqn:hamil-jacobi-expanded} \end{equation} $$ where $$ \begin{align} A_n &= \tan 2\theta_{\mathrm m} n k_1 J_{n} \left( \frac{\lambda_1}{k_1}\cos 2\theta_{\mathrm m} \right) \end{align} $$ and $n k_1$ are the amplitude and wavenumber of the $n$th Rabi mode.1 In obtaining Eqn. \eqref{chap:matter-sec:jacobi-eqn:hamil-jacobi-expanded}, I have used the following identity of the Bessel function $$ \begin{equation} J_{n-1}(z) + J_{n+1}(z) = \frac{2 n}{z} J_n(z). \label{eqn:bessel-function-sum-property} \end{equation} $$

Eqn. \eqref{chap:matter-sec:jacobi-eqn:hamil-jacobi-expanded} implies an infinite number of resonance conditions $$ \begin{equation} \omega_\mm = n k_1 \qquad (n=1, 2, \ldots) \label{chap:matter-sec:single-revisted-eqn:resonance-condition} \end{equation} $$ However, the amplitude of the Rabi mode $A_n$ drops quickly as a function of $n$ because $$ \begin{equation} J_n(z) \xrightarrow{z\ll \sqrt{n+1}} \frac{ (z/2)^n }{n!} \qquad \text{if } n>0 \label{chap:matter-sec:single-revisit-eqn:bessel-small-arg} \end{equation} $$ when $\lambda_1/k_1 \ll 1$. For $\lambda_1/k_1\gtrsim 1$, $A_n$ also becomes small for sufficiently large $n$ because $$ \begin{equation} J_n(z) \xrightarrow{n\gg 1} \frac{1}{\sqrt{2\pi n}} \left( \frac{ e z }{ 2n } \right)^n. \end{equation} $$ Any real physical system has a finite size $l$ and the Rabi modes with the oscillation wavelengths $2\pi/\Omega_n\sim 2\pi/A_n \gtrsim l$ can be ignored even if they are on resonance.

In Sec. \ref{chap:matter-sec:single}, I have discussed the scenario where the $n=1$ Rabi mode is on or close to the resonance with $\lambda_1/\omega_\mm \ll 1$. Using Eqn. \eqref{chap:matter-sec:single-revisit-eqn:bessel-small-arg} and identity $$ \begin{equation} J_{-n}(z) =(-1)^n J_{n}(z) \end{equation} $$ I obtain $$ \begin{equation} A_{\pm 1} \approx \frac{ \lambda_1 \sin (2\theta_\mm)}{2}, \end{equation} $$ which agrees with Eqn. \eqref{chap:matter-sec:single-eqn:rabi-amplitudes}. According to Eqn. \eqref{app:chap:matter-eq:relative-detuning-changed}, compared with the case where the first Rabi mode is present, the relative detuning of the Rabi resonance has a change of magnitude $$ \begin{equation} \Delta \RD_{(n)} = \frac{1}{2} \frac{A_n}{A_1} \frac{1}{\RD_n} \end{equation} $$ when another Rabi mode $n$ is present, where $\RD_n$ is the relative detuning when only the $n$th Rabi mode is considered. In Table \ref{table:relative-detunings-single-frequency-example}, I listed the values of $A_n/A_1$, $\RD_n$, and $\Delta \RD_{(n)}$ of a few off-resonance Rabi modes for the three numerical examples plotted in Fig. Neutrino Rabi Oscillations for Single Mode. One can see that all the off-resonance Rabi modes have very little impact on the resonance which explains why we could ignore the oscillating $\sigma_3$ terms in Eqn. \eqref{eq-hamiltonian-bg-matter-basis-single-frequency}.

$k_1/\omega_{\mathrm m}=1$$k_1/\omega_\mm=1-2\times 10^{-5}$$k_1/\omega_\mm=1-10^{-4}$
$n$$A_n/A_1$$\RD_n$$\Delta \RD_{(n)}$
-1$1$$10^5$$5\times 10^{-6}$
2$4.8 \times 10^{-5}$$1.1 \times 10^{9}$$2.2\times 10^{-14}$
$-2$$4.8 \times 10^{-5}$$3.2\times 10^{9}$$7.5\times 10^{-15}$

Table relative-detunings-single-frequency-example shows the amplitudes and the relative detunings of a few Rabi modes and their impact on the Rabi resonance for the three numerical examples shown in Fig. Neutrino Rabi Oscillations for Single Mode.

  1. A phase in the matter potential would contribute to the phases of the Rabi modes, which do not play any role in the resonance for the reason discussed in Sec. Rabi Oscillations. ↩︎

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