Coupled-cluster theory for systems of bosons in external traps

A coupled-cluster approach for systems of $N$ bosons in external traps is developed. In the coupled-cluster approach the exact many-body wave function is obtained by applying an exponential operator $\exp \left\{T\right\}$ to the ground configuration $\mid {\varphi }_0⟩$. The natural ground configuration for bosons is, of course, when all reside in a single orbital. Because of this simple structure of $\mid {\varphi }_0⟩$, the appearance of excitation operators $T={\sum }_{n=1}^N{T}_n$ for bosons is much simpler than for fermions. We can treat very large numbers of bosons with coupled-cluster expansions. In a substantial part of this work, we address the issue of size consistency for bosons and enquire whether truncated coupled-cluster expansions are size consistent. We show that, in contrast to the familiar situation for fermions for which coupled-cluster expansions are size consistent, for bosons the answer to this question depends on the choice of ground configuration. Utilizing the natural ground configuration, working equations for the truncated coupled-cluster with $T=T_1+T_2$, i.e., coupled-cluster singles doubles are explicitly derived. Finally, an illustrative numerical example for a condensate with up to $N=10000$ bosons in an harmonic trap is provided and analyzed. The results are highly promising.

Figure 1

(Color online) Performance of CCSD method: correlation energy. Shown is the percent of correlation energy, denoted by $\%E_{\mathrm{correlation}}$, obtained by the CCSD for $N=100,1000,10000$ bosons and several values of interaction strength ${\lambda }_0$. The correlation energy is defined as the difference between the Gross-Pitaevskii energy $E_{\mathrm{GP}}$ and the exact energy. The exact energy is obtained in our model by diagonalizing the many-body Hamiltonian within the full configuration-interaction space. For comparison, the corresponding values obtained by CISD are also plotted.

Figure 2

(Color online) Performance of CCSD method: many-body wave function. Shown are the coefficients $C_{2m}$ of the normalized many-body wave function ${\sum }_{m=0}^{N∕2}{C}_{2m}\mid N-2m,2m⟩$, for $N=10000$ bosons and $\lambda ={\lambda }_0(N-1)=100$ obtained by the CCSD method, see Eqs. (53, 54). Although the coupling constant is large, it is remarkable that the CCSD $C_{2m}$ coefficients almost perfectly match the exact coefficients, namely the red curve “sits” atop the black curve. For comparison, the two coefficients of the CISD are also shown, which deviate much from the exact solution.

Figure 3

(Color online) Performance of CCSD method: ground-state depletion. Shown is the average number of bosons in orbital ${\phi }_2$, $⟨n_2⟩$, for $N=100,1000,10000$ bosons and several values of interaction strength. Because of the different spatial symmetry of ${\phi }_1$ and ${\phi }_2$, these orbitals are the eigenfunctions of the reduced one-particle density matrix (natural orbitals) and the respective eigenvalues are $⟨n_1⟩$ and $⟨n_2⟩$ with $⟨n_1⟩+⟨n_2⟩=N$. It is seen that the CCSD is extremely successful in obtaining the depletion $⟨n_2⟩$, which is a very sensitive measure of the exactness of the many-body wave function, for all $N$ up to a large coupling constant $\lambda ={10}^2$. The CISD results are also shown for comparison. See text for more details.