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Journal of Statistical Mechanics Theory and Experiment

Exact analytical study of ideal Bose atoms in a two-dimensional isotropic harmonic trap

Ze Cheng

Published Jun 1, 2017• 5 citations
Publisher

Abstract

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.

DOI
10.1088/1742-5468/aa727b

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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.
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.
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.
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.
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 .
Our experimental scheme is based on a barrel optical microresonator filled with a dye solution. The barrel mirror provides a confining potential, a chemical potential, and an effective mass for a photon, making the system formally equivalent to a one-dimensional gas of harmonically trapped, number-conserving, and massive bosons. Within the framework of quantum statistical mechanics, we propose an exact analytical solution to the problem of Bose-Einstein condensation in harmonically trapped, one-dimensional, and ideal photons. It is found that the photon number of vapor is characterized by an analytical function, which involves a qqq-digamma function in mathematics. The numerical calculation of the analytical solution gives many interesting results. In the thermodynamic limit, the analytical expressions of the critical temperature and the condensate fraction are derived. We find that the spectral radiance of a one-dimensional barrel cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.