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

Rabi Basis

Previously I have ignored the oscillating $\sigma_3$ terms in the Hamiltonian such as in Eqn. \eqref{chap:matter-sec:single-fequency-eq:hamiltonian-bg-matter-basis-single-frequency}. Using the Jacobi-Anger expansion, J. Kneller et al. showed that these terms result in an infinite number of resonance conditions. In the rest of the chapter, I will present a much simpler derivation of their results from the perspective of Rabi oscillations. For this purpose I will define a new basis called Rabi basis $$ \begin{equation} \begin{pmatrix} \ket{\nu_{\mathrm{r1}}}\\ \ket{\nu_{\mathrm{r2}}} \end{pmatrix} = \mathsf{U}^\dagger \begin{pmatrix} \ket{\nu_{\mathrm{L}}} \\ \ket{\nu_{\mathrm{H}}} \end{pmatrix}, \label{eq-rabi-basis} \end{equation} $$ where $$ \begin{equation} \mathsf{U} = \begin{pmatrix} e^{-\ri \eta (r)} & 0 \\ 0 & e^{\mathrm i \eta (r)} \end{pmatrix}. \label{eq-rabi-transformation} \end{equation} $$ The unitary transformation in Eqn.~\eqref{eq-rabi-transformation} is a rotation in flavor space about the third axis. It commutes with $\sigma_3$ terms in the Hamiltonian, but, due to its dependence on $r$, the left side of the Schr"{o}dinger equation obtains an extra term: $$ \begin{align*} &\begin{pmatrix} \frac{\mathrm d\eta}{\mathrm dr} & 0 \\ 0 & - \frac{\mathrm d\eta}{\mathrm d r} \end{pmatrix} \begin{pmatrix} \psi_{\mathrm R1} \\ \psi_{\mathrm R2} \end{pmatrix} + \mathrm i \frac{\mathrm d}{\mathrm dr} \begin{pmatrix} \psi_{\mathrm R1} \\ \psi_{\mathrm R2} \end{pmatrix} \\ =& \left[ -\frac{\omega_{\mathrm m} }{2} \sigma_3 + \frac{\delta \lambda}{2} \cos 2\theta_{\mathrm m} \sigma_3 - \frac{\delta \lambda}{2} \sin 2\theta_{\mathrm m} \begin{pmatrix} 0 & e^{2\mathrm i\eta} \\ e^{-2 \mathrm i\eta } & 0 \end{pmatrix} \right] \begin{pmatrix} \psi_{\mathrm R1} \\ \psi_{\mathrm R2} \end{pmatrix}. \end{align*} $$ The oscillating $\sigma_3$ terms in Hamiltonian can be eliminated by choosing $\eta(r)$ properly, i.e., $$ \begin{equation} \eta(r) - \eta(0) = \frac{\cos 2\theta_{\mathrm{m}}}{2} \int_0^r \delta\lambda (\tau) \dd\tau. \end{equation} $$ With this definition, the Schr"{o}dinger equation simplifies: $$ \begin{equation} \ri \frac{\dd}{\dd r} \begin{pmatrix} \psi_{\mathrm r1} \\ \psi_{\mathrm r2} \end{pmatrix} = \left[ - \frac{\omega_{\mathrm m}}{2} \sigma_3 - \frac{\delta \lambda}{2} \sin 2\theta_{\mathrm m} \begin{pmatrix} 0 & e^{2\ri\eta} \\ e^{-2 \ri \eta } & 0 \end{pmatrix}\right] \begin{pmatrix} \psi_{\mathrm r1} \\ \psi_{\mathrm r2} \end{pmatrix}. \label{chap:matter-sec:jacobi-subsec:rabi-basis-eqn:eom-rabi-basis-matter} \end{equation} $$ Because the states $\ket{\nu_{r1}}$ and $\ket{\nu_{r2}}$ are related to $\ket{\nu_{\mathrm L}}$ and $\ket{ \nu_{\mathrm H} }$ by phase factors only, the transition probability between the former two states is the same as that between the latter.

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