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Journal of Statistical Mechanics Theory and Experiment

Exact spatial density of ideal Bose atoms in a two-dimensional isotropic harmonic trap

Ze Cheng

Published Jan 1, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1742-5468/aa9f4b

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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.
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 .
Abstract Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped, two-dimensional, and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the two-dimensional critical temperature is smaller than the one-dimensional critical temperature by an order of magnitude or more.
Abstract We have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. We employ the analytical solution to calculate the internal energy E , the entropy S , the Helmholtz free energy F , and the heat capacity at constant number C N of ideal Bose atoms in a two-dimensional isotropic harmonic trap. The first main finding in this paper is that the variation with temperature of the internal energy of a finite number of ideal Bose atoms has an inflection point, which occurs at the transition temperature T c . The second main finding is that the internal energy of a finite number of ideal Bose atoms has a classic limit E = 2 N k B T as T ≫ T c . The third main finding is that the variation with temperature of the heat capacity of a finite number of ideal Bose atoms has a maximal value, which occurs at T c . The fourth main finding is that the heat capacity of a finite number of ideal Bose atoms has a classic limit C N / k B = 2 N as T ≫ T c . The fifth main finding is that in the thermodynamic limit, at critical temperature T c the heat capacity at constant number shows a cusp singularity, which is analogous to the λ -transition of liquid helium. Consequently, the transition between normal and condensed states is a second-order phase transition.
Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) in harmonically trapped, one-dimensional and ideal atoms. It is found that the number of atoms of vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. The results of the numerical calculation of the analytical solution agree completely with the previous experimental results in the BEC of harmonically trapped, one-dimensional and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit.
Motivated by quantum statistical mechanics, we propose an accurate analytical solution to the problem of Bose–Einstein condensation (BEC) of ideal bosons in a two-dimensional anisotropic harmonic trap. The study reveals that the number of noncondensed bosons is characterized by an analytical function, which relates to a series expansion of q-digamma functions in mathematics. The q-digamma function is a function of temperature, boson number, and anisotropic parameter. The analytical solution describes fully the experimental results of the BEC of ideal bosons in a two-dimensional anisotropic harmonic trap. We derive the analytical expressions of the critical temperature and the condensate fraction in the thermodynamic limit. The first main conclusion is that for a fixed temperature and boson number, there is a critical anisotropic parameter, which is the precise onset of BEC in this harmonically trapped two-dimensional system. The second main conclusion is that the critical temperature in a two-dimensional anisotropic harmonic trap is larger than that in a two-dimensional isotropic harmonic trap.
Bose–Einstein condensates of repulsive Bose atoms in a one-dimensional harmonic trap are investigated within the framework of a mean field theory. We solve the one-dimensional nonlinear Gross–Pitaevskii (GP) equation that describes atomic Bose–Einstein condensates. As a result, we acquire a family of exact breather solutions of the GP equation. We numerically calculate the number density [Formula: see text] of atoms that is associated with these solutions. The first discovery of the calculation is that at the instant of the saddle point, the density profile exhibits a sharp peak with extremely narrow width. The second discovery of the calculation is that in the center of the trap ([Formula: see text] m), the number density is a U-shaped function of the time [Formula: see text]. The third discovery of the calculation is that the surface plot of the density [Formula: see text] likes a saddle surface. The fourth discovery of the calculation is that as the number [Formula: see text] of atoms increases, the Bose–Einstein condensate in a one-dimensional harmonic trap becomes stabler and stabler.