## Appendix D Relativistic Correction  
The relativistic Lagrangian density for a light field with normal incidence to a fully reflective and non-compressible moving mirror, correct to the first-order in \(\dot{q}\) and \(\ddot{q}\), reads [68]  
\[\frac{\partial\mathcal{L}}{\partial V}=\frac{1}{2}\big{(}\mathbf{E}\!\cdot \!\mathbf{D}-\mathbf{H}\!\cdot\!\mathbf{B}\big{)}+\frac{\Gamma^{2}\epsilon_{0} }{2}\chi|\mathbf{E}\!\cdot\!\hat{z}-c\mathbf{B}\!\mathbf{B}\!\cdot\!\mathbf{y}| ^{2}, \tag{105}\]  
where \(\mathbf{E}=-\hat{z}\frac{\partial}{\partial t}A\), \(\mathbf{B}=\
abla\times(A\hat{z})=-\hat{y}\frac{\partial}{\partial x}A\), \(\mathbf{D}=\epsilon_{0}\mathbf{E}\), and \(\mathbf{B}=\mu_{0}\mathbf{H}\). Furthermore,  
\[\mathbf{B}=\frac{v}{c}=\frac{\dot{q}}{c}=\frac{p}{mc},\ \Gamma=\frac{1}{\sqrt{1- \mathbf{B}^{2}}}, \tag{106}\]  
and \(\chi\) is a dimensionless shape function independent of \(v\), being zero outside mirror and relative susceptibility of the mirror's dielectric \(\chi_{0}\) inside, and \(\epsilon_{0}\) is the permittivity of vacuum. By expanding in the powers of \(\mathbf{B}\), this Lagrangian gives the first- and second-order corrections to the quadratic Hamiltonian density as  
\[\frac{\partial}{\partial V}\Delta\mathcal{H}=\frac{\partial}{\partial V}\Delta \mathcal{H}^{(1)}+\frac{\partial}{\partial V}\Delta\mathcal{H}^{(2)}=-\left. \frac{\partial}{\partial\mathbf{B}}\frac{\partial\mathcal{L}}{\partial V} \right|_{\mathbf{B}=0}\mathbf{B}-\frac{1}{2}\left.\frac{\partial^{2}}{ \partial\mathbf{B}^{2}}\frac{\partial\mathcal{L}}{\partial V}\right|_{\mathbf{B}= 0}\mathbf{B}^{2}. \tag{107}\]Hence, one may obtain  
\[\Delta\mathcal{H}^{(1)}=-\epsilon_{0}S\int\limits_{0}^{q}\frac{\partial}{\partial \mathbb{B}}\left[\frac{1}{2}\bigg{(}\frac{1}{\sqrt{1-\mathbb{B}^{2}}}\bigg{)}^ {2}\chi|A_{t}+c\mathbb{B}A_{x}|^{2}\right]\Bigg{|}_{\mathbb{B}=0}\mathbb{B}dx, \tag{100}\]  
which further simplifies as  
\[\Delta\mathcal{H}^{(1)}=-\epsilon_{0}S\epsilon\int\limits_{0}^{q}\chi A_{t}A_ {x}\mathbb{B}dx. \tag{101}\]  
It is appropriate to assume the approximation of conducting interface [87; 88; 89] for the mirror, such as the thickness is let to approach zero, while it susceptibility increases proportionally. In that limit, one may set  
\[\chi(x,t)\approx\chi_{0}d\delta[x-q(t)], \tag{102}\]  
where \(d\) is the mirror's thickness. This is similar to the assumption of the locality of interaction by [PERSON] [90], too. Hence, one gets  
\[\Delta\mathcal{H}^{(1)}\approx-\epsilon_{0}Vc\chi_{0}A_{t}(q,t)A_{x}(q,t) \mathbb{B}, \tag{103}\]  
with \(V=Sd\) is the cavity volume. Now, one has from (2), \(A_{x}(q,t)=s\pi\sqrt{2/q^{3}}\sum kQ_{k}\), \(A_{t}(q,t)=-\dot{q}A_{x}(q,t)\), and thus  
\[\Delta\mathcal{H}^{(1)}\approx 2\pi^{2}d\chi_{0}\frac{\dot{q}^{2}}{q^{3}}\sum kjQ _{k}Q_{j} \tag{104}\]  
It is quite remarkable that (104) is purely relativistic, and vanish in the limit of infinite \(c\), as shown below. Here, the dependence on \(t\) is hidden for convenience. This term translates after symmetrization into  
\[\Delta\mathcal{H}^{(1)}=-2\hbar\Big{(}\hat{b}^{\dagger}-\hat{b}\Big{)}^{2} \sum_{k}w_{kj}\Big{(}\hat{a}_{k}^{\dagger}+\hat{a}_{k}\Big{)}\Big{(}\hat{a}_{ j}^{\dagger}+\hat{a}_{j}\Big{)}, \tag{105}\]  
where \(w_{kj}=\sqrt{jk}\chi_{0}\pi\hbar d\Omega/4 mcl^{2}\) are the coupling rates.