Exact Solution of Density Profile of Ideal Bose Gases in One Dimensional Harmonic Trap

Ji-Xuan Hou, Jing Yang

Ze Cheng

Abstract We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, one-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. We employ the q -digamma function to calculate the spatial density n ( z;T , N ) of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits Friedel oscillations. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a sharp peak with extremely narrow width. The third main finding is that the central peak of the spatial density is a monotonically increasing function of the number of atoms N but is a monotonically decreasing function of temperature T .

Ze Cheng

Abstract We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional (2D), and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series expansion of q -digamma functions in mathematics. We employ the analytical solution to calculate the spatial density n ( r ; T , N ) of ideal Bose atoms in a 2D isotropic harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits a 2D dark spot in the center and a bright ring outside. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a 2D sharp peak with an extremely small radius. The third main finding is that the central density is a monotonically increasing function of the number of atoms N , but is a monotonically decreasing function of temperature T . The fourth main finding is that the number density of atoms of condensate exhibits a 2D Gaussian distribution in the center while the number density of atoms of vapor exhibits a 2D bright annulus with zero density at the origin.

Hao, Ya Ya, Jiang

Y. J. Hao, S. Chen

Shah Saad Alam, Timothy Skaras, Li Yang +1 more

Dynamical fermionization refers to the phenomenon in Tonks-Girardeau gases where, upon release from harmonic confinement, the gases' momentum density profile evolves asymptotically to that of an ideal Fermi gas in the initial trap. This phenomenon has been demonstrated theoretically in hardcore and anyonic Tonks-Girardeau gases and was recently experimentally observed in a strongly interacting Bose gas. We extend this study to a one-dimensional spinor gas of arbitrary spin in the strongly interacting regime and analytically prove that the total momentum distribution after the harmonic trap is turned off approaches that of a spinless ideal Fermi gas, while the asymptotic momentum distribution of each spin component takes the same shape of the initial real space density profile of that spin component. Our work demonstrates the rich physics arising from the interplay between the spin and the charge degrees of freedom in a spinor system.