"Chirpons": one-dimensional phase singularities as atypical local oscillations

Abstract

In this work, phase singularities embedded in a wavepacket are shown to act as sources of atypical localized oscillations when the packet interacts with a linear system. We refer to these oscillations as \textit{chirpons}, since they arise as strong variations of the instantaneous frequency (chirp). A mathematical expression is then provided to describe \textit{chirpons}, and their behavior is explored through the interaction of a super-bandwidth wavepacket -- containing two singularities -- with a damped harmonic oscillator, a fundamental model for many physical systems. This interaction is analyzed theoretically, and the predictions are verified experimentally using a resonant electrical circuit as a realization of the oscillator. The results show that \textit{chirpons} evolve in a manner fundamentally different from standard Fourier oscillations, revealing features of linear systems that are otherwise inaccessible. This introduces a new approach to analyze and characterize system responses, with potential applications in high-resolution spectroscopy and signal sensing.

Figures (4)

• Figure 1: (a) SB--WP with parameters $\alpha=1$, $\beta=0.5$, and $\Delta\omega=0.5$. The red line corresponds to the term in parentheses in Eq.\ref{['e2']}, while the thick and thin black lines show the real amplitude $\left|E_{SB}(t)\right|$ and the normalized instantaneous frequency $\omega_{SB}(t)/\omega_0$, respectively. Phase singularities appear at $t=\pm t_z$ (black arrows). (b) $\left|E_{SB}(t)\right|$ and $\omega_{SB}(t)/\omega_0$ after propagation through a dispersive medium with $\Omega = 2$. The dashed line shows the fit of $\omega_C^+(t,0.69T_0)/\omega_0$ (Eq.\ref{['e9']} with $\gamma = +0.69T_0$) to $\omega_{SB}(t)/\omega_0$. (c) and (d): Same analysis for a WP given by $E_{sinc}(t)=\mathrm{sinc}(t/\sqrt{2})e^{i\omega_0 t}$. In all panels, time is normalized to the carrier period $T_0=2\pi/\omega_0=1$.
• Figure 2: Response (continuous red line) of a damped harmonic oscillator with quality factor $Q=10$ and varying resonance frequency $\omega_r$, to different SB input signals (continuous black line). Dashed lines show the corresponding normalized instantaneous frequency $\omega_{SB}(t)/\omega_0$: black for the input signals and red for the oscillator response. (a) $\alpha=0$ (Gaussian input) with $\omega_r=1.5\omega_0$. (b) $\alpha=1$ with $\omega_r=1.5\omega_0$. (c) $\alpha=1$ with $\omega_r=0.6\omega_0$. (d) $\alpha=1.6$ with $\omega_r=1.5\omega_0$.
• Figure 3: Response (continuous red line) of a damped harmonic oscillator to different SB input signals (continuous black line) that have previously propagated through a dispersive medium characterized by $\Omega$. Dashed lines show the corresponding normalized instantaneous frequencies $\omega_{SB}(t)/\omega_0$: black for the input signals and red for the oscillator response. The oscillator parameters are fixed at $Q=10$ and $\omega_r=1.5\omega_0$. Panels (a)–(d) correspond to $\Omega=0.5$, 0.75, 1, and 2, respectively.
• Figure 4: Experimental results. Response of the RLC circuit to: an SB state $C$ with $\nu_0 = 6\;\mathrm{MHz} < \nu_r$ in (a) and $\nu_0 = 12\;\mathrm{MHz} > \nu_r$ in (b); (c) a state $C^{+\gamma^{(a)}}_{-\gamma^{(a)}}$, $\nu_0 = 6\;\mathrm{MHz} < \nu_r$, $\gamma^{(a)} > 0$; (d) a state $C^{+\gamma^{(b)}}_{-\gamma^{(b)}}$, $\nu_0 = 12\;\mathrm{MHz} > \nu_r$, $\gamma^{(b)} < 0$. The blue line at the top of each panel corresponds to the fit of the right chirpon (dashed red line) in the output signal.