Low-lying energy levels of a one-dimensional weakly interacting Bose gas under zero boundary conditions

Abstract

We diagonalize the second-quantized Hamiltonian of a one-dimensional Bose gas with a nonpoint repulsive interatomic potential and zero boundary conditions. At weak coupling the solutions for the ground-state energy $E_0$ and the dispersion law $E(k)$ coincide with the Bogoliubov solutions for a periodic system. In this case, the single-particle density matrix $F_1(x,x^{\prime })$ at $T=0$ is close to the solution for a periodic system and, at $T>0$, is significantly different from it. We also obtain that the wave function $⟨\hat{\psi }(x,t)⟩$ of the effective condensate is close to a constant $\sqrt{N_0/L}$ inside the system and vanishes on the boundaries (here, $N_0$ is the number of atoms in the effective condensate, and $L$ is the size of the system). We find the criterion of applicability of the method, according to which the method works for a finite system at very low temperature and with a weak coupling (a weak interaction or a large concentration).

18 pages, 1 figure; v3: minor changes; final version

References in corpus (2)

• Bosonizing one-dimensional cold atomic gases
• Extension of Bogoliubov theory to quasi-condensates

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