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In page 8, last paragraph the equation $$L_\mathrm{I}+L_\mathrm{II}\approx2\langle a^{\dagger2}a^2\rangle$$ should read $$L_\mathrm{I}+L_\mathrm{II}\approx2\int_0^{\infty}dt\,\langle a^{\dagger2}a^2\rangle(t)$$ as the dynamics are always time integrated.
An $i$ is missing in Eq. (10c), so these coefficient should read:
$$L_{\pm}+iK_\pm=\frac{\frac{8\Omega_\mathrm{L}^2}{\gamma_\sigma(\gamma_\sigma+\gamma_\phi)}\big[1 \pm i \frac{5\gamma_\sigma-\gamma_\phi}{4 R_\mathrm{L}}\big]-\frac{\gamma_\sigma-\gamma_\phi}{\gamma_\sigma+\gamma_\phi}\big[1\pm i\frac{\gamma_\sigma-\gamma_\phi}{4R_\mathrm{L}}\big]}{4\big(1+\frac{8 \Omega_\mathrm{L}^2}{\gamma_\sigma(\gamma_\sigma+\gamma_\phi)}\big)}$$