## 3 Bogoliubov Prescription and Quantum Fluctuations  
In a bosonic system, one can separate Bose-condensed particles from non-condensed ones by using of Bogoliubov prescription [21; 22]:  
\[\hat{\psi}(\mathbf{r},t)=\psi_{0}(\mathbf{r})+\hat{\eta}(\mathbf{r},t)\, \tag{5}\]  
where:  
\[\psi_{0}(\mathbf{r})=\langle\hat{\psi}(\mathbf{r},t)\rangle \tag{6}\]  
is the time-independent, but space-dependent complex order parameter (macroscopic wavefunction) of the Bose-Einstein condensate with \(\langle...\rangle\) the thermal average over an equilibrium ensemble.  
Notice that we work at thermal equilibrium, and consequently, the thermal averages are time independent. The field \(\hat{\eta}(\mathbf{r},t)\) is the operator of quantum and thermal fluctuations, which describes out-of-condensate bosons.  
The Bogoliubov prescription for the field operator \(\hat{\psi}(\mathbf{r},t)\) enables us to write the three-body thermal average in the following way:  
\[\langle\hat{\psi}^{+}(\mathbf{r},t)\hat{\psi}(\mathbf{r},t)\hat{\psi}(\mathbf{r },t)\rangle=|\psi_{0}(\mathbf{r})|^{2}\psi_{0}(\mathbf{r})+2\tilde{n}(\mathbf{ r})\ \psi_{0}(\mathbf{r})+\tilde{m}(\mathbf{r})\ \psi_{0}^{*}(\mathbf{r})+\tilde{s}(\mathbf{r})\, \tag{7}\]  
where \(\tilde{n}(\mathbf{r})=\langle\hat{\eta}^{+}(\mathbf{r},t)\hat{\eta}(\mathbf{ r},t)\rangle\) is the density of non-condensed particles, while \(\tilde{m}(\mathbf{r})=\langle\hat{\eta}(\mathbf{r},t)\hat{\eta}(\mathbf{r},t)\rangle\) is the anomalous density and \(\tilde{s}(\mathbf{r})=\langle\hat{\eta}^{+}(\mathbf{r},t)\hat{\eta}(\mathbf{ r},t)\hat{\eta}(\mathbf{r},t)\rangle\) is the anomalous correlation [22].  
Now, we obtain an equation for \(\psi_{0}(\mathbf{r})\) by taking the thermal average on Equation (1). In this way, we find:  
\[\mu\ \psi_{0}(\mathbf{r})=\Big{[}-\frac{\hbar^{2}}{2m}\
abla^{2}+U(\mathbf{r})+ g|\psi_{0}(\mathbf{r})|^{2}+2g\tilde{n}(\mathbf{r})\Big{]}\psi_{0}(\mathbf{r})+ g\tilde{n}(\mathbf{r})\psi_{0}^{*}(\mathbf{r})+g\tilde{s}(\mathbf{r})\, \tag{8}\]  
which is the exact equation of motion of the Bose-Einstein order parameter \(\psi_{0}(\mathbf{r})\)[22; 23]. This is not a closed equation due to the presence of the non-condensed density \(\tilde{n}(\mathbf{r})\) and of the anomalous densities \(\tilde{m}(\mathbf{r})\) and \(\tilde{s}(\mathbf{r})\). Neglecting the non-condensed density and the anomalous densities, the previous equation becomes:  
\[\mu\ \psi_{0}(\mathbf{r})=\Big{[}-\frac{\hbar^{2}}{2m}\
abla^{2}+U(\mathbf{r})+ g|\psi_{0}(\mathbf{r})|^{2}\Big{]}\psi_{0}(\mathbf{r})\, \tag{9}\]  
which is the familiar Gross-Pitaevskii equation [7; 8]. A less drastic approximation, which is called the Bogoliubov-Popov-Beliaev approximation [22; 23; 24], neglects only the term \(\tilde{s}(\mathbf{r})\). Then, the equation of motion of the Bose-Einstein order parameter \(\psi_{0}(\mathbf{r})\) becomes:  
\[\mu\ \psi_{0}(\mathbf{r})=\Big{[}-\frac{\hbar^{2}}{2m}\
abla^{2}+U(\mathbf{r})+ g|\psi_{0}(\mathbf{r})|^{2}+2g\tilde{n}(\mathbf{r})\Big{]}\psi_{0}(\mathbf{r})+ g\tilde{m}(\mathbf{r})\psi_{0}^{*}(\mathbf{r}). \tag{10}\]  
Furthermore, this equation is not closed. We must add an equation for the non-condensed density \(\tilde{n}(\mathbf{r})\) and the anomalous density \(\tilde{n}(\mathbf{r})\) by studying the fluctuation operator \(\hat{\eta}(\mathbf{r},t)\)[22; 23].  
The equation of motion of the fluctuation operator \(\hat{\eta}(\mathbf{r},t)\) is obtained by subtracting Equation (10) from Equation (1). The standard Bogoliubov-Popov approximation [22; 23] neglects both the non-condensate density and the anomalous terms, and it takes only linear terms of \(\hat{\eta}(\mathbf{r},t)\) and \(\hat{\eta}^{+}(\mathbf{r},t)\). In this way, the linearized equation of motion of the fluctuation operator reads:  
\[i\hbar\frac{\partial}{\partial t}\hat{\eta}(\mathbf{r},t)=\Big{[}-\frac{ \hbar^{2}}{2m}\
abla^{2}+U(\mathbf{r})-\mu+2g|\psi_{0}(\mathbf{r})|^{2}\Big{]} \hat{\eta}(\mathbf{r},t)+g\psi_{0}(\mathbf{r})^{2}\hat{\eta}^{+}(\mathbf{r},t ). \tag{11}\]