the grand canonical potential due to the quantum fluctuations of the phonon field. For $T\rightarrow 0$, there are no phonons, and consequently, the Hamiltonian of the system for the vacuum of the phonons is specified by $ {\mathcal H}=\Omega_0+{\mathcal E}_0(R,\bar{n}) $. Roberts and Pomeau also deduced a similar term for a homogeneous condensate in Ref. [@Roberts] based on the original work of Lee, Huang and Yang [@Lee].

Quantum fluctuation part of the Casimir force
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In this subsection we calculate the contribution to the Casimir force arising from the second term in Eq. , [*i.e. *]{} the quantum (vacuum) fluctuation part (${\mathcal E}_0$) of the grand canonical Hamiltonian ($\hat{{\mathcal H}}$). Before proceeding with further calculations, we anticipate that the Casimir force due to vacuum fluctuations of the phonon field ($\delta\hat{\Psi}$) would be similar to that due to the photon (electromagnetic) field in Eq.  with the replacement of the speed of light by the speed of phonon (sound), $c \to v(\bar{n})=\sqrt{g\bar{n}/M}$. Thus, using Eqs. , , and , we expect the phonon Casimir force to be $$\begin{aligned}
\label{eqn:67}
f_R^{(\text{phn})}=\frac{2L\hbar v(\bar{n})}{R}[N^{(2)}+D^{(2)}]\approx-0.027122686\frac{\hbar v(\bar{n})L}{R^3}.~~\end{aligned}$$

Let us now go back to the Eq.  whose second term ${\mathcal E}_0(R,\bar{n})$, gives the quantum vacuum fluctuations as well 