Exploring Dual-Phonon Lasing in a Two-Mode Optomechanical System

Dual-phonon lasing arises when two mechanical modes simultaneously lase, driven by interactions with a shared optical mode. Contrary to what is shown in literature [1], this phenomenon can be achieved for specific values of the optomechanical parameters that are experimentally feasible. The governing normalized equations for the system are: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\frac{da_{N}}{d\tau}=(i(\Delta_{N}+2x_{N_{1}}+2x_{N_{2}})-1)a_{N}+S_{N}, \frac{d^{2}x_{N_{1}}}{d\tau^{2}}=-\Omega_{N_{1}}^{2}x_{N_{1}}-\Gamma_{N_{1}}\frac{dx_{N_{1}}}{d\tau}+\vert a_{N}\vert ^{2}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\frac{d^{2}x_{N_{2}}}{d\tau^{2}}=-\Omega_{N_{2}}^{2}x_{N_{2}}- \Gamma_{N_{2}}\frac{dx_{N_{2}}}{d\tau}+\vert a_{N}\vert^2$</tex>, with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$a_{N}$</tex> the normalized optical field amplitude, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Delta_{N}$</tex> the normalized detuning, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x_{N_{\mathrm{1}}}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x_{N_{2}}$</tex> the normalized displacements of the first and second mechanical modes, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S_{N}$</tex> the normalized optical driving force, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega_{N_{1}}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega_{N_{2}}$</tex> the normalized mechanical frequencies and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Gamma_{N_{1}}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Gamma_{N_{2}}$</tex> the normalized damping rates. For sufficiently high input powers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(S_{N} &gt; &gt;)$</tex> the system undergoes self-sustained oscillations (phonon lasing) after Hopf bifurcations [2]. Figure 1(a) presents a parameter sweep over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S_{N}$</tex>, identifying two Hopf bifurcations (HB1 and HB2) that lead to periodic branches <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{1}$</tex> (orange) and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{2}$</tex> (green), corresponding to self-sustained oscillations of the first and second mechanical modes, respectively. The blue branch <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(P_{DL})$</tex> represents a dual-mode lasing regime, emerging from a branching point (BP) of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{1}$</tex> at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S_{N}=0.0314$</tex>. Figures 1(b-d) show the optical intensity for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S_{N}=0.035$</tex> across the three branches, while Figs. 1(e-g) depict the mechanical oscillations. Only one mechanical mode amplifies in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{1}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{2}$</tex>, but in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{DL}$</tex>, both modes oscillate with comparable amplitudes. Finally, Figs. 1(h-j) display frequency combs for each branch: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{1}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{2}$</tex> show single-frequency combs, while <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{DL}$</tex> reveals a dual-frequency comb characterized by peaks at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega_{N_{1}}=2$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega_{N_{2}}=3$</tex>, along with their harmonics and combinations. These were obtained by performing the FFT on the optical intensity signals shown in Figs. 1(b-d).