Energy Extraction from $Q$-balls and Other Fundamental Solitons

Energy exchange mechanisms have important applications in particle physics, gravity, fluid mechanics, and practically every field in physics. In this Letter we show, both in the frequency and time domain, that energy enhancement is possible for waves scattering off fundamental solitons (time-periodic localized structures of bosonic fields), without the need for rotation nor translational motion. We use two-dimensional $Q$-balls as a test bed, providing the correct criteria for energy amplification, as well as the respective amplification factors, and we discuss possible instability mechanisms. Our results lend support to the qualitative picture drawn in Saffin et al. [preceding Letter, $Q$-ball superradiance, Phys. Rev. Lett. 131, 111601 (2023).]; however, we show that this enhancement mechanism is not of superradiant type, but instead is a “blueshiftlike” energy exchange between scattering states induced by the background $Q$-ball, which should occur generically for any time-periodic fundamental soliton. This mechanism does not seem to lead to instabilities.

Figure 1

Radial profile $f(r)$ of the $Q$-balls studied in this work, with ${\omega }_Q=(0.58,0.70,0.76)$, $m_Q=0$, and $g=1/3$. The values at the origin are, respectively, $f(0)=(1.78,1.70,1.54)$, their charge is $Q={\int }_{\mathrm{\Sigma }}{J}_Q^t=(70.77,13.70,9.76)$ and energy is $E={\int }_{\mathrm{\Sigma }}{J}_E^t=(47.02,12.00,9.14)$. We consider only ground-state (nodeless) $Q$-ball solutions.

Figure 2

Upper panel: relative energy amplification factor, $Z_E$, of an incoming mode ${\varphi }_{+}$ with $m=0$ scattering off a nonspinning $Q$-ball; the coupling is $g=1/3$. The dashed lines are the results from time-domain simulations. Lower panel: ratio $|A_{+}^{\mathrm{out}}{|}^2/|A_{+}^{\mathrm{in}}{|}^2$, which is seen to be $\le 1$ for all $\omega$.

Figure 3

Wave packet scattering off a nonspinning $Q$-ball with ${\omega }_Q=0.76$; the coupling is $g=1/3$ and we use initial conditions with ${\sigma }_r=5$, $r_0=100$, and ${\omega }_0=\{-2.24,3.76\}$. Upper panel: energy flux $J_E^r(r=60)$ as a function of time, with the outgoing flux enlarged. Lower panel: integrated energy flux $E^{\mathrm{flux}}={\int }_0^{t}dt{J}_E^r(r=60)$, with an inset showing attenuation ($Z_E=-0.022$) for $\omega =3$ and amplification ($Z_E=0.037$) for $\omega =-3$. These results are in excellent agreement with the frequency-domain analysis (cf. Fig. 2).

Figure 4

Integrated energy flux $E^{\mathrm{flux}}={\int }_0^{t}dt{J}_E^r$ at $r=20$ for a wave packet scattering off a nonspinning $Q$-ball with ${\omega }_Q=0.76$ inside a cavity; the coupling is $g=1/3$ and we use initial conditions with ${\sigma }_r=5$, $r_0=80$, and ${\omega }_0=\{-2.24,3.76\}$. We impose Dirichlet conditions ${\mathrm{\Phi }}_1=0$ at $r=100$ to model the cavity.