The Harmonic Oscillator Potential Perturbed by a Combination of Linear and Non-linear Dirac Delta Interactions with Application to Bose-Einstein Condensation

Abstract

In this paper, we study the bound state analysis of a one dimensional nonlinear version of the Schrödinger equation for the harmonic oscillator potential perturbed by a $\delta$ potential, where the nonlinear term is taken to be proportional to $\delta (x)|\psi (x){|}^2\psi (x)$. The bound state wave functions are explicitly found and the bound state energy of the system is algebraically determined by the solution of an implicit equation. Then, we apply this model to the Bose-Einstein condensation of a Bose gas in a harmonic trap with a dimple potential. We propose that the many-body interactions of the Bose gas can be effectively described by the nonlinear term in the Schrödinger equation. Then, we investigate the critical temperature, the condensate fraction, and the density profile of this system numerically.

16 pages, 17 figures, published version

References in corpus (18)

• Realization of Bose-Einstein condensates in lower dimensions
• An Atom Laser with a cw Output Coupler
• Reversible Formation of a Bose-Einstein Condensate
• Laser cooling to quantum degeneracy
• Adiabatically changing the phase-space density of a trapped Bose gas
• Continuous Bose-Einstein condensation
• Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates
• A Quantum Tweezer for Atoms
• Optimized production of large Bose Einstein Condensates
• Bose-Einstein condensate in a harmonic trap decorated with Dirac delta functions
• Growth dynamics of a Bose-Einstein condensate in a dimple trap without cooling
• The attractive nonlinear delta-function potential
• The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
• The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit
• Statics and dynamics of quasi one-dimensional Bose-Einstein condensate in harmonic and dimple trap
• Measurement-enhanced determination of BEC phase transitions
• Kinetics of Bose-Einstein condensation in a dimple potential
• Green's Function Formulation of Multiple Nonlinear Dirac $\delta$-Function Potential in One Dimension