Quantum Depletion of a Soliton Condensate

We present rigorous results on the diagonalization of Bogoliubov Hamiltonian for a soliton condensate. Using the complete and orthonomalized set of eigenfunction for the Bogoliubov de Gennes equations, we calculate exactly the quantum depletion of the condensate and show that two degenerate zero-modes, which originate from a U(1) guageand a translational-symmetry breaking of the system, induce the quantum diffusion and transverse instability of the soliton condensate. PACS numbers: 03.75.Kk, 03.75.Lm, 05.45.Yv Typeset using REVTEX 1 The study of matter waves and elementary excitations has received much attention because of the remarkable experimental realization of Bose-Einstein condensation in trapped, weakly interacting atomic gases [1–3]. In the past few years, several paths have been explored in studying elementary excitations in Bose-Einstein condensates (BECs). The most followed method is to use the Gross-Pitaevskii (GP) equation which is suitable to describe a zero-temperature BEC. One of the deficiency of this approach, in addition to the constraint of zero temperature, is the neglect of the quantum fluctuations of condensate, which is an important aspect of elementary excitation [4]. The eigen-modes of the elementary excitations thus obtained so far [1,2] are not complete and their orthonormalities have never been proved. A different approach on elementary excitations is to use Bogoliubov theory [5] that was originally formulated for homogeneous Bose systems but is also valid for inhomogeneous ones. In this approach one makes a canonical transformation for boson operators to diagonalize the quantum Hamiltonian of system and hence the quantum fluctuations of condensate are taken into account [6]. An added advantage of the Bogoliubov theory is that it can be easily generalized to the case of finite temperature (with, e. g., the Hatree-Fock and Popov approximations) [1,2], therefore can be compared directly with experiments carried out at non-zero temperature environment. In this Letter we apply the Bogoliubov theory to the investigation of the elementary excitations generated from a quasi one-dimensional(1D) soliton condensate. It is known that a quasi 1D BEC can form in the presence of a transverse confinement, which introduces a cutoff for long wavelength fluctuations in the transverse directions and hence an off-diagonallong-range order can establish, as have been reported in the case of low-dimensions BEC systems [7] and in the case of matter-wave bright solitons in quasi-1D systems [8,9] and the corresponding flourished theoretical activities [10–13]. The main contribution of our present work is the rigorous diagonalization of quantum Bogoliubov Hamiltonian and the exact calculation of quantum depletion of the soliton condensate and related conclusions on soliton stability. 2 We consider a quasi 1D attractive, weakly interacting Bose gas condensed in a trap with tight transverse confinement and negligible trapping potential in the axial direction. The grand canonical Hamiltonian of the system can be written as the following dimensionless from