## 3 Field Quantization
The quantum dissipative master function has been numerically solved and the corresponding correlation functions were obtained. Interestingly, quadratic optomechanical interactions can arise at the single-photon level, too, where rigorous analytical solutions have been devised [60]. Such type of interactions can be also well described using equivalent nonlinear electrical circuits, where a Josephson junction brings in the desired nonlinearity of quadratic interactions and terminate a pair of lumped transmission lines [61]. Finally, ultracold atoms also can exhibit interactions of a comparable type which is mathematically equivalent to the quadratic interaction [62]. The single-photon multi-particle rates are given by  
\[\alpha=\frac{\omega}{l}\sqrt{\frac{\hbar}{m\Omega}}\equiv\sqrt{2}g_{0},\;\beta= \frac{\hbar}{l^{2}m}\frac{\omega}{\Omega},\;\gamma=\frac{\omega}{\Omega}\frac {\hbar^{3}}{m^{\frac{3}{2}}l^{3}\sqrt{\Omega}}. \tag{44}\]This summarizes the Hamiltonian as  
\[\mathbb{H}=\mathbb{H}_{0}+\mathbb{H}_{int}. \tag{45}\]  
in which \(\mathbb{H}_{0}\) and \(\mathbb{H}_{int}\) are respectively the non-interacting and interacting Hamiltonians  
\[\mathbb{H}_{0}=\mathbb{H}_{012}-\frac{1}{2}\hbar\Big{(}\alpha\mathfrak{X}-\beta \mathfrak{X}^{2}+\gamma\mathfrak{X}^{3}+\cdots\Big{)},\,\mathbb{H}_{int}=\hbar \Big{(}\alpha\mathfrak{X}-\beta\mathfrak{X}^{2}+\gamma\mathfrak{X}^{3}+\cdots \Big{)}\hat{n}, \tag{46}\]  
when \(\omega\gg\Omega\). Now, the dimensionless constant \(\theta\) is defined as  
\[\theta=\frac{1}{l}\sqrt{\frac{\hbar}{m\Omega}}=\frac{x_{zp}}{l}, \tag{47}\]  
with \(x_{zp}\) being the r.m.s. value of zero-point fluctuations, by which the following is deduced  
\[\beta=\theta\alpha\ll\alpha,\,\gamma=\theta\beta=\theta^{2}\alpha\ll\beta. \tag{48}\]  
This implies that every kind of higher-order interaction is typically \(\theta\) times weaker than the interaction of the preceding-order. It should be noted that while such interactions are normally expected to rapidly vanish with the order increasing, is a well-known fact that certain physical phenomena such as magnetism in solid \({}_{3}\)He cannot be understood without inclusion of four-particle interaction terms [74; 75]. It is worth here to mention that a detailed theory of optomechanics in superfluid. It has been developed [76], but no expression for the nonlinear terms has been reported. In general, the interaction of mechanical and optical modes is not strictly one-dimensional, implying that the overlap integral of normalized modes should also be taken into account. For instance, odd mechanical modes with even optical modes have zero interaction. In that sense, tuning the interaction to an odd mode and then shining an even optical mode, or vice versa, makes the optomechanical interaction identically zero by setting \(\alpha\equiv\sqrt{2}g_{0}=0\). Then the lowest order surviving interaction would be the \(\mathbb{H}_{4}\) term. This method has been used in [55; 56; 57] to highlight the quadratic interaction and make its measurement much easier. It has been shown that these quadratic terms may be exploited for direct observation of mechanical eigenmode jumps [55; 56], as well as two-phonon cooling and squeezing [57], while the coupling strength \(\beta\) could be increased by three orders of magnitude [56]. Moreover, the origin of mechanical parametric coupling which has recently been phenomenologically hypothesized [58] for the associated physical interactions cannot be understood without the presented analysis, although based on some earlier experimental evidence [77]. It must be added that the condition \(\omega\gg\Omega\) may be violated in carefully designed superconducting microwave circuits and also the recently demonstrated molecular optomechanics [78], which signifies the importance of the \(\mathfrak{P}^{2}\mathbb{Q}^{2}\) term in \(\mathbb{H}_{4}\).