Classical Fano oscillator

Starting from the quantum-mechanical Fano-Anderson Hamiltonian, we derive classical equations of motion for coordinates associated with the discrete and continuum states. The frequency-dependent absorption spectrum associated with this classical system exhibits the same Fano line shape as the quantum-mechanical system when appropriate correspondences between classical and quantum variables are made. In the time domain, the response of this classical Fano oscillator depends upon the asymmetry parameter $q$ that appears in the expression for the Fano line shape. In particular, under the influence of impulsive driving of the system, the discrete oscillator's phase changes by $\pm \pi /2$ as $q$ increases from zero (maximum asymmetry in the frequency domain) to $\mp \infty$ (minimum asymmetry). Previously published ultrafast-laser-pulse-driven coherent-phonon oscillations in degenerate $p$-type Si [K. Kato et al., Jpn. J. Appl. Phys. 48, 100205 (2009)] are discussed in light of these theoretical results.

Figure 1

Fano line shape for several values of the asymmetry parameter $q$.

Figure 2

Phase $\phi =-\mathrm{atan}\left(q\right)$ vs $q$ for the classical Fano oscillator. The solid circle (with vertical error bars included) is the phase that has been measured for a degenerate $p$-type Si sample ($p$ between $3\times {10}^{19}$ and $1.5\times {10}^{20}{\mathrm{cm}}^{-3}$);[18] the horizontal line spans the range of estimated $q$ values for the sample as deduced from Fig. 3 (see text for details).

Figure 3

Experimental Fano asymmetry parameter $q$ deduced from Raman-scattering data vs photon energy $\hslash \omega$ for degenerate $p$-type Si. Indicated doping density for each curve is in units of ${10}^{19}{\mathrm{cm}}^{-3}$. Triangles are from Ref.8, and circles are from Ref. 7. Dotted lines are extrapolations of the $5\times {10}^{19}$ and $1.6\times {10}^{20}{\mathrm{cm}}^{-3}$ data to $\hslash \omega =1.55\mathrm{eV}$.

Figure 4

Components of Eq. (B6) vs ${\Omega }_{0}t$. Parts (a) and (b) correspond to narrow and wide continua, respectively (see text for details). The individual curves correspond to (i) the sum of terms inside the outermost parenthesis on the right-hand-side of Eq. (B6), (ii) the term containing $\cos \left({\Omega }_{0}t\right)$, (iii) the term containing $\sin \left({\Omega }_{0}t\right)$, and (iv) $\mathrm{Si}\left({\omega }_{\min }t\right)-\mathrm{Si}\left({\omega }_{\max }t\right)$. Curves (ii)–(iv) are offset for clarity. The asymptotic limit $\pi \cos \left({\Omega }_{0}t\right)$ is plotted (dotted curve) with (i).