Three-dimensional Harmonic Potential Research Articles

Thermodynamic properties of ideal Bose gas trapped in different external power-law potentials under generalized uncertainty principle**Project supported by the Major Innovation Projects for Building First-class Universities in China’s Western Region (Grant No. ZKZD2017006); Natural science foundation project of ningxia education department (Grant No. NGY0017054).

Significant evidence is available to support the quantum effects of gravity that lead to the generalized uncertainty principle (GUP) and the minimum observable length. Usually in quantum mechanics, statistical physics does not take gravity into account. Thermodynamic properties of ideal Bose gases in different external power-law potentials are studied under the GUP with a statistical physical method. Critical temperature, internal energy, heat capacity, entropy, particle number of ground state and excited state are calculated analytically to ideal Bose gases in the external potentials under the GUP. Below the critical temperature, taking the rubidium and sodium atoms, ideal Bose gases whose particle densities are under standard and experimental conditions, respectively, as examples, the relations of internal energy, heat capacity and entropy with temperature are analyzed numerically. Theoretical and numerical calculations show that: (1) the GUP leads to an increase in the critical temperature. (2) When the temperature is lower than the critical temperature and slightly higher than 0 K, the GUP’s amendments to internal energy, heat capacity and entropy etc are positive. As the temperature increases to a certain value, these amendments become negative. (3) The external potentials can increase or decrease the influence of the GUP on thermodynamic properties. When ε = 1 J, ε is the quantity that reflects the external potential intensity, and atomic density n = 2.687 × 1025 m−3, the GUP’s amendments to the internal energy, heat capacity and entropy of the rubidium atoms ideal Bose gas first decrease and then increase with the increase of X (where X ≡ Σi1/ti is sum of the reciprocal of the exponents of the power function). In three-dimensional harmonic potential, the relative correction term of the GUP is 26 orders of magnitude larger than that of a free-particle system in a fixed container. (4) When ε ≈10−31 J and n ≈ 1020 m−3 (which are the experimental data when BEC was first verified by sodium atomic gas), the influence of the GUP can be completely ignored. (5) Under certain conditions, GUP may become the dominant factor governing the thermodynamic properties of the system.

On the derivation of the entropy of ideal quantum gases confined in a three-dimensional harmonic potential

In this work, the entropy functions of ideal quantum gases in a three-dimensional harmonic trap are analytically calculated using temperature as an explicit variable. Afterward, the applicability of the analytical formulas is validated by comparison with the numerical calculation. The results illustrate that the obtained functions could be applied for the whole temperature regime with a maximum relative deviation of less than 7.5% in the vicinity of the critical temperature Tc in the case of Bose gases. Meanwhile, for Fermi gases, although the analytical formula fits well at very low- and high-temperature regimes, it cannot be applied at temperature in the range , where TF is the Fermi temperature. In addition, the consistency between our formulas and classical ones at significantly high temperatures is also discussed.

Complex structures generated by competing interactions in harmonically confined colloidal suspensions

We investigate the structural properties of colloidal particle systems interacting via an isotropic pair potential and confined by a three-dimensional harmonic potential. The interaction potential has a repulsive–attractive–repulsive profile that varies with the interparticle distance (also known as a ‘mermaid’ potential). We performed Langevin dynamics simulations to find the equilibrium configurations of the system. We show that particles can self-assemble in complex structural patterns, such as compact disks, fringed disks, rods, spherical clusters with superficial entrances among others. Also, for particular values of the parameters of the interaction potential, we could identify that some configurations were formed by quasi two-dimensional (2D) structures which are stable for 2D systems.

Choice of atomic basis set for the study of two electrons in a harmonic anisotropic quantum dot using a configuration interaction approach

We have developed a computational code based on the Hartree–Fock and full interaction configuration approaches which allows the study of N-electron confined quantum systems with different confining potentials and external conditions. The code employs Cartesian anisotropic Gaussian-type orbitals as the atomic basis set, which enables the use of different exponents for each direction space in order to better exploit the characteristics of the confining potential. As an illustration, we have employed it to study a system consisting of two electrons confined by a three-dimensional harmonic potential for different values of confinement strength, leading to different confinement conditions: an isotropic three-dimensional and an anisotropic oblate (or quasi-two dimensional) quantum dot. A central aspect of this study is to propose efficient procedures for choosing the exponents of the atomic basis functions. In particular, we note that the use of more than one function for each atomic orbital can improve the convergence of the electronic energy levels. The present results are compared with other theoretical values published previously.

Boltzmann avenged

An experiment with cold atoms confined in an isotropic three-dimensional harmonic potential confirms the long-predicted non-damping oscillations of the breathing mode.

Analytical solitonlike solutions and the dynamics of ultracold Fermi gases in a time-dependent three-dimensional harmonic potential

We present a theoretical study of solitonlike solutions and their dynamics of ultracold superfluid Fermi gases trapped in a time-dependent three-dimensional (3D) harmonic potential with gain or loss. The 3D analytical solitonlike solutions are obtained without introducing any additional integrability constraints used elsewhere. The propagation of both bright- and dark-soliton-like solutions is investigated. We show that the amplitudes of dark-soliton-like solutions exhibit periodic oscillation, whereas those of the bright-soliton-like ones do not show such behavior. Moreover, we highlight that the oscillation periods of dark-soliton-like solutions predicted by our approach are matched very well with those observed in a recent experiment carried out by Yefsah etal. [T. Yefsah, A. T. Sommer, M. J. H. Ku, L. W. Cheuk, W. Ji, W. S. Bakr, and M. W. Zwierlein, Nature (London) 499, 426 (2013)NATUAS0028-083610.1038/nature12338] in both Bose-Einstein condensation and unitarity regimes.

Thermodynamics of a correlated confined plasma. I. Macroscopic system

We consider classical and quantum systems of charge carriers which are con.ned by a three-dimensional harmonic potential. Thermodynamic functions are determined, and the transition to an uncon.ned system is discussed.

Phase transition of trapped interacting Bose gases and the renormalization group

We apply perturbative renormalization-group theory to the symmetric phase of a dilute interacting Bose gas which is trapped in a three-dimensional harmonic potential. Using Wilson's energy-shell renormalization and the $\ensuremath{\epsilon}$ expansion, we derive the flow equations for the system. We relate these equations to the flow for the homogeneous Bose gas. In the thermodynamic limit, we apply our approach to examine the transition temperature of the harmonically trapped Bose gas as a function of the scattering length. The results are compared to previous studies of the problem.

Counting statistics and fluctuations in Bose–Einstein condensation

For a system of N Bose atoms, we propose a natural probability distribution for the occupation numbers. The counting statistics only depends on the probabilities of occupation of the single-particle states. Fluctuations of the occupation numbers are related in a simple way with the corresponding probabilities. The case of N ideal bosons trapped in a symmetric three-dimensional harmonic potential is analyzed. In this case, fluctuations of the condensate display a physically expected behavior. When the thermodynamic limit is considered, we obtain analytic expressions for the counting statistics and fluctuations of the ground estate occupation number. The temperature of maximum fluctuations of the ground state occupation number is proposed in order to characterize in an unambiguous way the apparition of a macroscopic condensate fraction.

Complementarity and Young’s interference fringes from two atoms

The interference pattern of the resonance fluorescence from a $J=1/2$ to $J=1/2$ transition of two identical atoms confined in a three-dimensional harmonic potential is calculated. The thermal motion of the atoms is included. Agreement is obtained with experiments [U. Eichmann et al., Phys. Rev. Lett. 70, 2359 (1993)]. Contrary to some theoretical predictions, but in agreement with the present calculations, a fringe visibility greater than 50% can be observed with polarization-selective detection. The dependence of the fringe visibility on polarization has a simple interpretation, based on whether or not it is possible in principle to determine which atom emitted the photon.

Statistical mechanics of a confinement electron gas in a uniform electromagnetic field

In this paper we study an ideal electron gas in the presence of a uniform electromagnetic field and confinement by a three-dimensional harmonic potential. We find the partition function of this system and in the sequel we examine the Boltzmann statistics and Fermi-Dirac statistics applying the grand canonical ensemble method.