Kerr-enhanced amplification of three-wave mixing and emergent masing regimes

Abstract

Integrated optical microresonators exploiting either second-order ($χ^{(2)}$) or third-order ($χ^{(3)}$) nonlinearities have become key platforms for frequency conversion, low-noise microwave photonics, and quantum entanglement generation. Here, we present an analytic theory of Kerr-enhanced three-wave mixing amplification in an electro-optic microresonator with both $χ^{(2)}$ and $χ^{(3)}$ nonlinearities. We demonstrate that Kerr dressing hybridizes the optical sidebands, renormalizing the $χ^{(2)}$ couplings and detunings. As a result the system exhibits gain in regions where analogous bare $χ^{(2)}$ or $χ^{(3)}$ amplifiers are subthreshold. Time-domain Langevin simulations confirm this threshold reduction, mapping a practical design window for experiments.

Figures (4)

• Figure 1: Description of a hybrid $\chi^{(2)}$--$\chi^{(3)}$ electro--optic resonator. (a) Optical resonances with frequencies $\omega_\nu$, $\nu = 0, \pm 1, \dots$, form an approximately equidistant ladder with spacing $D_1$ around the pumped mode $\omega_0$. The $\chi^{(2)}$ effect couples $\omega_{\nu}$ to a microwave cavity enabling up and down conversion between the $0$ and $\nu = \pm \mu$ modes by absorbing or emmiting a microwave photon. The Kerr ($\chi^{(3)}$) effect weakly couples the $0$ mode to the $\pm \nu$ sidebands by four-wave mixing. (b) We show that in the dressed-mode picture the Kerr coupling renormalizes the $\pm \mu$ modes leading to an effective 4-mode $\chi^{(2)}_{\rm eff}$ microresonator that enables parametric gain by down-converting $0$ mode photons to microwave in regimes where the individual $\chi^{(2)}$ and $\chi^{(3)}$ interactions are too weak to cause any amplification.
• Figure 2: Kerr-induced renormlization of the three-wave mixing parameters. (a) Real and imaginary parts of the normalized coupling matrix elements $\Omega_{\rm eff} / \Omega_{\rm eff}\!(g_{3}=0)$ as a function of the Kerr-induced frequency shift $g_{3}|a_0|^2/\zeta_{\rm \mu}$. (b) Kerr-only amplification lobe in the $(g_3|a_0|^2/\kappa_{\rm \mu},\;\zeta_{\rm \mu}/\kappa_{\rm \mu})$ plane obtained by the condition ${\rm Re}(\lambda_{g_3})>0$. The filled area is bound by the dashed straight lines $\zeta_{\rm \mu} > g_3|a_0|^2$, $\zeta_{\rm \mu} < 3g_3|a_0|^2$ and $g_3|a_0|^2/\kappa_{\rm \mu} > 1/2$. (c) Real and imaginary parts of the effective detuning $\Delta/\zeta_{\rm \mu}$. The shaded region $1/3 < g_{3}|a_0|^2/\zeta_{\rm \mu} < 1$ in (a) and (b) marks the region where Kerr-induced amplification is possible (see also(b)).
• Figure 3: Critical cooperativity for $\chi^{(2)}$ amplification in the presence of Kerr nonlinearity. (a) The critical cooperativity $C_{\mathrm{crit}}$ and (b) the corresponding critical microwave detuning $\zeta_{{\rm e},\mathrm{crit}}/\kappa_{\rm \mu}$, obtained from Eq. \ref{['eq:Ccrit_full_SM']}, as functions of the optical detuning $\zeta_{\rm \mu}/\kappa_{\rm \mu}$ for several Kerr shifts, $g_{3}|a_0|^2/\kappa_{\rm \mu}$ (colors). In (a) the horizontal dotted line marks $C_{\mathrm{crit}}=1$, and vertical dotted lines in (a) and (b) indicate $\zeta_{\mu}= g_3 |a_0|^2$ for each Kerr strength. (c) Minimum value of critical cooperativity $\min_{\zeta_{\rm \mu}} C_{\mathrm{crit}}$, and associated (d) optical, $\zeta_{\rm \mu}/\kappa_{\rm \mu}$, and (e) microwave, $\zeta_{{\rm e},\mathrm{crit}}/\kappa_{\rm \mu}$, detuning where this minimum is attained versus $g_{3}|a_0|^2/\kappa_{\rm \mu}$.
• Figure 4: Time-domain verification of Kerr-enhanced $\chi^{(2)}$ amplification by the linearized quantum Langevin equations. (a) Signal-sideband population $\langle a_+^\dagger a_+ \rangle$ and (b) microwave population $\langle b_+^\dagger b_+ \rangle$ versus time. Three cases are considered: ($\chi^{(2)}+\chi^{(3)}$) both $C = 0.1$ and $g_3 |a_{0}|^{2}/\kappa_{\mu} = 0.49$, ($\chi^{(3)}$) $C = 0$ and $g_3 |a_{0}|^{2}/\kappa_{\mu} \neq 0.49$ and ($\chi^{(2)}$) $C = 0.1$$g_3 = 0$. The exponential growth rate of ($\chi^{(2)}+\chi^{(3)}$) matches the analytically expected $2 {\rm Re}(\lambda_{\rm max})$ indicated by a guide-to-the-eye line. The remaining parameters are $\zeta_{\mu}/\kappa_{\mu} = 1$, $\zeta_{e} = 0$, $\kappa_{\rm \mu}/2\pi = 11~\mathrm{MHz}$, $\kappa_{\rm e}/2\pi = 1.1~\mathrm{MHz}$ and $\eta_{\rm \mu} = \eta_{\rm e} = 0.2$. Optical baths are initialized in vacuum and the internal microwave bath has one thermal photon.