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Expand a weakly depleted Bose gas around its macroscopically occupied mode. Replacing a condensate operator by an amplitude of order $\sqrt{N}$ gives quadratic, cubic and quartic fluctuation interaction scales proportional to $N{N}_{\mathrm{ex}}$, $\sqrt{N}{N}_{\mathrm{ex}}^{3/2}$ and $N_{\mathrm{ex}}^2$, with the same interaction-volume factor. Thus the depletion fraction supplies formal amplitude counting. This is not a uniform bound on infrared modes or a substitute for checking the interaction regime. At fixed particle number, expansion of the condensate interaction must precede the final quadratic truncation; the resulting cancellation is the number-conserving Bogoliubov quadratic Hamiltonian.

For a weakly depleted homogeneous Bose gas with repulsive contact interaction, set $g=uN/V$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$. A zero-mode coherent state times a two-mode squeezed vacuum for each opposite-momentum pair has mean particle number $N=N_0+{\sum }_{k\ne 0}{\sinh }^2{\theta }_k$. Substitution of $N_0=N-\sum {\sinh }^2{\theta }_k$ into the leading condensate interaction energy, followed by a consistent quadratic truncation, gives the displayed variational energy. Omitting this substitution loses the subtraction embodied in the number-conserving Bogoliubov quadratic Hamiltonian.

For each mode set $A_k={ϵ}_k+g$ and $E_k=\sqrt{A_k^2-g^2}$. If $A_k>|g|$, choose $\tanh (2{\theta }_k^{\ast })=g/A_k$. The mode contribution equals $\frac{1}{2}{E}_k\cosh [2({\theta }_k-{\theta }_k^{\ast })]-A_k/2$, proving a unique minimum $(E_k-A_k)/2$. At fixed ultraviolet cutoff this lowers the energy relative to the uncorrelated condensate. The approximation requires small quantum depletion of a Bose condensate; an unregulated three-dimensional ground-state energy also requires renormalization of a three-dimensional dilute-gas contact coupling.

The coherent state in the zero mode has mean occupation number $N_0$. The other occupation numbers follow from part (c), giving exactlyImpose the fixed mean particle number $⟨\hat{N}⟩=N$ by choosingHere $D$ is the quantum depletion of a Bose condensate. The approximation requires $0\le D\ll N$, so $N_0$ remains positive and macroscopically large. The specified coherent state and multimode squeezed vacuum are not exact eigenstates of the total number operator; $N$ in this variational construction is their prescribed mean particle number.

Write $P={\sum }_{\mathbf{k}\ne 0}{s}_k{c}_k$. Substituting into the retained part of the interaction energy before truncating givesThe last two terms quadratic in $D,P$ have no explicit large-$N$ factor when the Bogoliubov transformation parameters are held fixed, and are discarded along with the nonzero-mode quartic contribution already omitted in part (c). This is the same truncation that underlies the number-conserving Bogoliubov quadratic Hamiltonian. Thus, defining the physical number density correctly as $\rho =N/V=N/L^3$,Replacing $N_0$ by $N$ at the outset would give the wrong coefficient $2{\sinh }^2{\theta }_k$: expanding the leading condensate interaction energy supplies the necessary subtraction.

The printed definition $\rho =N/L$ is an error. In three dimensions the contact interaction coupling $u$ has units of energy times volume, so $uN/L$ does not even have units of energy. There is a direct counterexample with no depletion: when ${\theta }_k=0$, the specified coherent state has exactly $⟨H_{\mathrm{int}}⟩=u{N}^2/(2L^3)$. In fixed length and energy units, $L=2$, $u=1$ and $N=100$ give $625$, whereas the printed $u\rho N/2$ with $\rho =N/L$ gives $2500$. The requested formula and the minimization in part (e) hold with $\rho =N/L^3$.

Write $\mathcal{V}=L^3$ and $N_0=N-N_{\mathrm{ex}}$, where $N_{\mathrm{ex}}={\sum }_{\mathbf{k}\ne 0}{b}_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$. Macroscopic occupation permits $b_0,b_0^{†}\simeq \sqrt{N_0}$, with relative commutator corrections of order $1/N_0$. Keep the fixed-total-number correction before replacing $n_0$ by $n$.

The four-condensate term is $V_0{N}_0(N_0-1)/(2\mathcal{V})$. Its leading expansion gives $V_0{N}^2/(2\mathcal{V})-n{V}_0{N}_{\mathrm{ex}}$. Terms with two condensate operators give normal scattering $n(V_0+V_{\mathbf{k}})b_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$ and conversion of two condensate particles into opposite-momentum excitations, with coefficient $n{V}_{\mathbf{k}}/2$. The $n{V}_0$ normal contribution cancels against the change of condensate energy at fixed $N$. Hence the number-conserving Bogoliubov quadratic Hamiltonian isUse a real even interaction transform, $V_{\mathbf{k}}=V_{-\mathbf{k}}$, as for a real inversion-symmetric pair potential. The constant is the condensate interaction energy; the diagonal part combines particle kinetic energy with condensate scattering; the anomalous pair terms create and annihilate correlated opposite momenta while transferring particles to or from the condensate.

At fixed volume, counting replaced condensate operators gives quadratic vertices proportional to $N_0$, omitted cubic vertices proportional to $\sqrt{N_0}$ and quartic vertices with no condensate factor. There are also the finite-$N_0$ correction in $N_0(N_0-1)$ and corrections from the condensate-number expansion. Their physical control parameter is small depletion, $N_{\mathrm{ex}}/N\ll 1$, together with weak interaction; simply dropping them is not justified by low temperature in a strongly interacting gas. This is the weakly interacting Bose gas approximation.

Use a real, even interaction $V_{-\mathbf{k}}=V_{\mathbf{k}}$. Let $\mathcal{V}=L^3$ and $N_{\mathrm{ex}}={\sum }_{\mathbf{k}\ne 0}{b}_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$. In the Bogoliubov theory expansion, keep terms with zero or two noncondensed operators. The zero-momentum term gives $V_0{N}_0(N_0-1)/(2\mathcal{V})$. The terms with two noncondensed operators giveHere $V_0$ is the direct condensate interaction and $V_{\mathbf{k}}$ is its exchange contribution. At fixed total particle number, $N_0=N-N_{\mathrm{ex}}$, soThe second term cancels the direct normal term. Adding the kinetic energy yields the number-conserving Bogoliubov quadratic HamiltonianThe constant is the condensate mean-field theory interaction energy; the normal term describes moving a particle out of the condensate, while the anomalous term creates or annihilates an opposite-momentum pair by exchanging two particles with the condensate. Total particle number remains fixed in the underlying theory; the condensate amplitudes in this notation are treated as classical amplitudes.

For condensate fluctuation power counting, put $\delta =N_{\mathrm{ex}}/N\ll 1$. Formal interaction-amplitude counting gives $H_2-E_{\mathrm{mf}}\sim E_{\mathrm{mf}}\delta$, cubic terms $\sim E_{\mathrm{mf}}{\delta }^{3/2}$ and quartic terms $\sim E_{\mathrm{mf}}{\delta }^2$; these are not uniform bounds on individual modes. Thus the latter are down by $\sqrt{\delta }$ and $\delta$ relative to the quadratic interaction terms. The $N_0-1$ correction is relative $1/N$. Homogeneity eliminates terms linear in nonzero-momentum operators. Small depletion and weak interactions, not low temperature alone, justify the truncation. ^[ref]