Time-dependent conformal transformations and the propagator for quadratic systems

Abstract

The method proposed by Inomata and his collaborators allows us to transform a damped Caldiroli-Kanai oscillator with time-dependent frequency to one with constant frequency and no friction by redefining the time variable, obtained by solving a Ermakov-Milne-Pinney equation. Their mapping ``Eisenhart-Duval'' lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

Figures (5)

• Figure 1: The generelized Niederer map \ref{['XTtrans']} maps each interval ${\bf I_k=(r_k,r_{k+1})}$ onto the entire real line $-\infty < T < \infty$. Its inverse mapping is therefore multivalued, labeled by an integer $k$. The classical motions and the propagator are both regular at the separation points ${\bf r_k}$. All classical trajectories are focused at the caustic points ${\bf t_{\ell}}$, where the propagator diverges.
• Figure 2: The phase factor ${\bf P}(t)$ of the propagator in \ref{['Ksplitting']} lies on the unit circle of the complex plane plotted vertically along a classical path $\bar{\gamma}(t)$. The orentation is positive if it is clockwise when seen from $t = +\infty$. In the time interval $J_{\ell}$ labeled by the Maslov index $\ell={\rm Ent}[t/\pi]$, the factor ${\bf P}(t)$ precesses around ${\bf P}_\ell=\exp[-i\frac{\pi}{4}(1+2\ell)]$ with double frequency w.r.t. the classical path, $\bar{\gamma}(t)$. Arriving at a caustic the phase jumps by $(-\pi/2)$ ( red becoming purple) and then continues until the next caustic when it jumps again (and becomes magenta), and so on.
• Figure 3: The analytic solution of the Mathieu equation with $a=2,q=1$ for $x(t)$ ( dotted in red), lies on the black curve got by \ref{['xAnsatz']} from combining the numerically obtained $\rho(t)$ (in green) and $\tau(t)$ (in blue), which are solutions of the pair \ref{['genErmakov']}-\ref{['tauint']}. The black curve is also obtained by pulling back the free solution \ref{['aTb']} by the inverse Niederer map \ref{['Nkdef']}.
• Figure 4: The probability density $|K(x,t)|^2$\ref{['probadens']} does not depend on $x$ and is regular in each interval $\textcolor{cyan}{\bf J}_{\ell}$ between the adjacent points $\textcolor{cyan}{\bf t_{\ell}}$\ref{['caustic']}, where it diverges. The $\textcolor{rgb(0,128,0)}{\bf r_k}$ which determine the domains $\textcolor{rgb(0,128,0)}{{\bf I}_k}$ of the generalized Niederer map \ref{['XTtrans']} lie between the $\textcolor{cyan}{\bf t_{\ell}}$ and conversely.
• Figure 5: For $0 < t < t_1$ the Mathieu phase factor ${\bf P}(t)$ plotted along a classical path $\bar{\gamma}(t)=(\bar{x}(t),t)$ precesses around $e^{-i\pi/4}$. Arriving at the caustic point $\tau(t_1)=\pi$ its phase jumps by $(-\pi/2)$, then oscillates around $e^{-3i\pi/4}$ until $\tau(t_2)=2\pi$, then jumps again, and so on.