Nonuniform Bose-Einstein condensate. II. Doubly coherent states

We find stationary excited states of a one-dimensional system of $N$ spinless point bosons with repulsive interaction and zero boundary conditions by numerically solving the time-independent Gross-Pitaevskii equation. The solutions are compared with the exact ones found in the Bethe-ansatz approach. We show that the $j$th stationary excited state of a nonuniform condensate of atoms corresponds to a Bethe-ansatz solution with the quantum numbers $n_{1}=n_{2}=\ldots =n_{N}=j+1$. On the other hand, such $n_{1},\ldots,n_{N}$ correspond to a condensate of $N$ elementary excitations (in the present case the latter are the Bogoliubov quasiparticles with the quasimomentum $\hbar \pi j/L$, where $L$ is the system size). Thus, each stationary excited state of the condensate is ``doubly coherent'', since it corresponds simultaneously to a condensate of $N$ atoms and a condensate of $N$ elementary excitations. We find the energy $E$ and the particle density profile $\rho (x)$ for such states. The possibility of experimental production of these states is also discussed.