Bose-Einstein condensation in a harmonic potential

Mullin, William J.

doi:10.1007/bf02395928

Cited by 141 publications

(172 citation statements)

References 23 publications

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“…[29]) reignited a huge interest in the topic [30][31][32][33]. Relevant to our research, let us mention a theoretical analysis of the influence of the different forms of the confining potential [30,[34][35][36][37][38][39] and/or BC [6][7][8][9][10][11][12][13][14][15][16][17][18] on the properties of the ideal Bose gas. A quantitative measure of the BE condensation is provided by the ground state occupation n 0 (E ; β), i.e., a ratio of the number of bosons N 0 in the ground state Table 1: Peak values of the specific heat per particle c N max and temperatures β −1 max at which they are achieved for the canonical and several numbers of particles N of the two grand canonical ensembles at some weak electric fields to the total number of corpuscles in the system:…”

Section: Bosonsmentioning

confidence: 99%

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

Olendski

2016

Annalen der Physik

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A theoretical analysis of the thermodynamic properties of the Robin wall characterized by the extrapolation length Λ in the electric field E that pushes the particle to the surface is presented both in the canonical and two grand canonical representations and in the whole range of the Robin distance with the emphasis on its negative values which for the voltage-free configuration support negative-energy bound state. For the canonical ensemble, the heat capacity at Λ < 0 exhibits a nonmonotonic behavior as a function of the temperature T with its pronounced maximum unrestrictedly increasing for the decreasing fields as ln 2 E and its location being proportional to (− ln E ) −1 . For the Fermi-Dirac distribution, the specific heat per particle cN is a nonmonotonic function of the temperature too with the conspicuous extremum being preceded on the T axis by the plateau whose magnitude at the vanishing E is defined as 3(N − 1)/(2N )kB, with N being a number of the particles. The maximum of cN is the largest for N = 1 and, similar to the canonical ensemble, grows to infinity as the field goes to zero. For the Bose-Einstein ensemble, a formation of the sharp asymmetric feature on the cN -T dependence with the increase of N is shown to be more prominent at the lower voltages. This cusp-like dependence of the heat capacity on the temperature, which for the infinite number of bosons transforms into the discontinuity of cN (T ), is an indication of the phase transition to the condensate state. Some other physical characteristics such as the critical temperature Tcr and ground-level population of the Bose-Einstein condensate are calculated and analyzed as a function of the field and extrapolation length. Qualitative and quantitative explanation of these physical phenomena is based on the variation of the energy spectrum by the electric field.

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Section: Bosonsmentioning

confidence: 99%

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

Olendski

2016

Annalen der Physik

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“…If, however, the system has repulsive interactions, no such divergence would be allowed and then the Hohenberg theorem would apply and the transition studied in that section would disappear. A similar situation has been discussed for a rotating Bose gas [13] and with the 2D completely trapped gas [38].…”

Section: B the Hohenberg Theoremmentioning

confidence: 52%

“…Having this length scale is also necessary to see the relative sizes of the occupation numbers of various states [37,38]. The harmonic length a 0 = /(mω) is a measure of the rms deviation of a particle from the center of the well and is not the appropriate density length scale as we will see.…”

Section: The Isotropic 3d Oscillatormentioning

confidence: 99%

“…We see that the way to define a density parameter (or any thermodynamic variable) so that it is scale independent [37,38] is in terms of a volume L 3 with the parameter given by…”

Section: The Isotropic 3d Oscillatormentioning

confidence: 99%

“…In the TL, T 1 would go to zero as 1/ ln N if the 1D density N/L were held constant. Such a behavior is indeed characteristic of a 1D pseudo-phase-transition [38,43], which does not exist except in a finite system. While it is possible to consider a 3D TL such that L ⊥ is held constant while N and L increase, it is ρ = N/(L 2 ⊥ L ) and ∆ that are held constant in the standard 3D TL while the two lengths increase.…”

Section: A Standard Boundary Conditionsmentioning

confidence: 99%

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Generalized Bose–Einstein Condensation

Mullin

Sakhel

2011

J Low Temp Phys

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Generalized Bose-Einstein condensation (GBEC) involves condensates appearing simultaneously in multiple states. We review examples of the three types in an ideal Bose gas with different geometries. In Type I there is a discrete number of quantum states each having macroscopic occupation; Type II has condensation into a continuous band of states, with each state having macroscopic occupation; in Type III each state is microscopically occupied while the entire condensate band is macroscopically occupied. We begin by discussing Type I or "normal" BEC into a single state for an isotropic harmonic oscillator potential. Other geometries and external potentials are then considered: the "channel" potential (harmonic in one dimension and hard-wall in the other), which displays Type II, the "cigar trap" (anisotropic harmonic potential), and the "Casimir prism" (an elongated box), the latter two having Type III condensations. General box geometries are considered in an appendix. We particularly focus on the cigar trap, which Van Druten and Ketterle first showed had a two-step condensation: a GBEC into a band of states at a temperature Tc and another "one-dimensional" transition at a lower temperature T1 into the ground state. In a thermodynamic limit in which the ratio of the dimensions of the anisotropic harmonic trap is kept fixed, T1 merges with the upper transition, which then becomes a normal BEC. However, in the thermodynamic limit of Beau and Zagrebnov, in which the ratio of the boundary lengths increases exponentially, T1 becomes fixed at the temperature of a true Type I phase transition. The effects of interactions on GBEC are discussed and we show that there is evidence that Type III condensation may have been observed in the cigar trap.

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Comparative analysis of electric field influence on the quantum wells with different boundary conditions.

Olendski

2015

Annalen der Physik

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Analytical solutions of the Schrödinger equation for the one-dimensional quantum well with all possible permutations of the Dirichlet and Neumann boundary conditions (BCs) in perpendicular to the interfaces uniform electric field are used for the comparative investigation of their interaction and its influence on the properties of the system. Limiting cases of the weak and strong voltages allow an easy mathematical treatment and its clear physical explanation; in particular, for the small , the perturbation theory derives for all geometries a linear dependence of the polarization on the field with the BC-dependent proportionality coefficient being positive (negative) for the ground (excited) states. Simple two-level approximation elementary explains the negative polarizations as a result of the field-induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role. Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically. Hellmann-Feynman theorem reveals a fundamental relation between the polarization and the speed of the energy change with the field. It is proved that zero-voltage position entropies are BC independent and for all states but the ground Neumann level (which has ) are equal to while the momentum entropies depend on the edge requirements and the level. Varying electric field changes position and momentum entropies in the opposite directions such that the entropic uncertainty relation is satisfied. Other physical quantities such as the BC-dependent zero-energy and zero-polarization fields are also studied both numerically and analytically. Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.

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Bose-Einstein condensation in a harmonic potential

Abstract: We examine several features of Bose-Einstein condensation (BEC)

Cited by 141 publications

References 23 publications

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

Generalized Bose–Einstein Condensation

Comparative analysis of electric field influence on the quantum wells with different boundary conditions.

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