## 3 Field Quantization
It is possible furthermore to use the approximate replacement  
\[\frac{1}{\hat{q}^{n}}\cong\frac{1}{l^{n}}\bigg{(}1-n\frac{\hat{x}}{l}\bigg{)}, \tag{37}\]  
to obtain  
\[\mathbb{H}=\frac{1}{2m}\hat{p}^{2}+V(\hat{q})+\frac{1}{2}\bigg{[}\hat{p}^{2}+ \omega^{2}\bigg{(}1-2\frac{\hat{x}}{l}+\frac{4}{l^{2}}\hat{x}^{2}+\cdots \bigg{)}\hat{Q}^{2}\bigg{]}-\frac{r}{8m^{2}l^{2}}\mathcal{S}\bigg{\{}\hat{p}^ {2}\bigg{(}1-2\frac{\hat{x}}{l}\bigg{)}\bigg{\}}\hat{Q}^{2}, \tag{38}\]where further substitutions should be taken as  
\[\hat{P}\cong\sqrt{\hbar\omega}\bigg{(}1-\frac{1}{2l}\hat{x}+\frac{3}{8l^{2}}\hat{ x}^{2}+\cdots\bigg{)}\mathbb{P},\;\hat{Q}\cong\sqrt{\frac{\hbar}{\omega}}\bigg{(}1+ \frac{1}{2l}\hat{x}-\frac{1}{8l^{2}}\hat{x}^{2}+\cdots\bigg{)}\mathbb{Q}. \tag{39}\]  
This can be decomposed to the terms  
\[\mathbb{H}=\mathbb{H}_{012}+\mathbb{H}_{3}+\mathbb{H}_{4}+\mathbb{H}_{5}+ \cdots,\;\mathbb{H}_{012}=\frac{1}{2}\hbar\Omega\mathfrak{P}^{2}+U(\mathfrak{X })+\frac{1}{2}\hbar\omega\Big{(}\mathbb{P}^{2}+\mathbb{Q}^{2}\Big{)}. \tag{40}\]  
Hence, there are several distinct types of nonlinear optomechanical multi-phonon/multi-photon interactions, which by defining \(R=r/4\approx 0.95\) are respectively given by  
\[ \begin{array}{c}\mathbb{H}_{3}=-\frac{\hbar\omega}{2l}\sqrt{\frac{\hbar}{m \Omega}}\mathfrak{X}\big{(}\mathbb{P}^{2}+\mathbb{Q}^{2}\big{)},\;\mathbb{H}_ {4}=\frac{\hbar^{2}}{2^{2}m}\Big{[}-R\frac{\Omega}{\omega}\mathfrak{P}^{2} \mathbb{Q}^{2}+\frac{\omega}{\Omega}\mathfrak{X}^{2}\big{(}\mathbb{P}^{2}+ \mathbb{Q}^{2}\big{)}\Big{]},\\ \mathbb{H}_{5}=-\frac{\hbar^{\frac{5}{2}}}{2m^{\frac{5}{2}}l^{3}\sqrt{\Omega}} \Big{\{}-2R\frac{\Omega}{\omega}S\big{\{}\mathfrak{P}^{2}\mathfrak{X}\big{\}} \mathbb{Q}^{2}+\frac{\omega}{\Omega}\mathfrak{X}^{3}\big{(}\mathbb{P}^{2}+ \mathbb{Q}^{2}\big{)}\Big{\}},\end{array}  \tag{41}\]  
and so on for higher order interactions. Here, the expansion of symmetrized terms, for instance, gives  
\[\mathcal{S}\Big{\{}\mathfrak{P}^{2}\mathfrak{X}\Big{\}}=\frac{1}{3}\Big{(} \mathfrak{P}^{2}\mathfrak{X}+\mathfrak{X}\mathfrak{P}^{2}+\mathfrak{X} \mathfrak{P}\mathfrak{P}\Big{)}. \tag{42}\]  
Now, it is noted that since usually \(\omega\gg\Omega\), it may observed that the first terms are much weaker than the second terms. Hence, using the identity \(\mathbb{P}^{2}+\mathbb{Q}^{2}=\frac{1}{2}\hbar+\frac{1}{4}\) where \(\hbar\) is photon number operator, the following is obtained  
\[ \begin{array}{c}\mathbb{H}_{3}=-\frac{\hbar\omega}{l}\sqrt{\frac{\hbar}{m \Omega}}\mathfrak{X}\Big{(}\hat{n}+\frac{1}{2}\Big{)}=-\hbar\alpha\mathfrak{X }\Big{(}\hat{n}+\frac{1}{2}\Big{)},\\ \mathbb{H}_{4}=\frac{\hbar^{2}}{l^{2}m}\frac{\Omega}{\Omega}\mathfrak{X}^{2} \big{(}\hat{n}+\frac{1}{2}\big{)}=+\hbar\beta\mathfrak{X}^{2}\Big{(}\hat{n}+ \frac{1}{2}\Big{)},\\ \mathbb{H}_{5}=-\frac{\omega}{\Omega}\frac{\hbar^{2}}{m^{\frac{5}{2}}l^{3} \sqrt{\Omega}}\mathfrak{X}^{3}\Big{(}\hat{n}+\frac{1}{2}\Big{)}=-\hbar\gamma \mathfrak{X}^{3}\Big{(}\hat{n}+\frac{1}{2}\Big{)},\end{array}  \tag{43}\]  
where \(\mathbb{H}_{3}\equiv\mathbb{H}_{\text{OM}}\) is the simple optomechanical interaction, and \(\mathbb{H}_{4}\) is known as the quadratic interaction. It has to be emphasized that while \(\mathbb{H}_{3}\equiv\mathbb{H}_{\text{OM}}\) is actually nonlinear in the exact mathematical sense, it is the quadratic interaction \(\mathbb{H}_{4}\) which is mostly referred to as the nonlinear interaction in the literature [55; 56; 57]. Since it is possible to make \(g_{0}\) and therefore \(\mathbb{H}_{3}\) identically vanish by appropriate optomechanical design [55; 56; 57; 58] in which the overlap integral of optical and mechanical modes sums up to zero, hence the quadratic interactions \(\mathbb{H}_{4}\) can then find physical significance. The quadratic interaction has been a subject of growing importance in the recent years in optomechanical systems [59; 60; 61] and beyond [62]. In [59] the photon statistics and blockade under \(\mathbb{H}_{4}\) interactions has been studied and analytical expressions were derived.