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}.

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.

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