Bose–Einstein Condensation in a One-Dimensional System of Interacting Bosons

2019

Ukr. J. Phys.

PACSWe diagonalize the second-quantized Hamiltonian of a one-dimensional Bose gas with a nonpoint repulsive interatomic potential and zero boundary conditions. At weak coupling the solutions for the ground-state energy E0 and the dispersion law E(k) coincide with the Bogoliubov solutions for a periodic system. In this case, the single-particle density matrix F1(x, x ′ ) at T = 0 is close to the solution for a periodic system and, at T > 0, is significantly different from it. We also obtain that the wave function ψ (x, t) of the effective condensate is close to a constant N0/L inside the system and vanishes on the boundaries (here, N0 is the number of atoms in the effective condensate, and L is the size of the system). We find the criterion of applicability of the method, according to which the method works for a finite system at very low temperature and with a weak coupling (a weak interaction or a large concentration). K e y w o r d s: interacting bosons, Bogoliubov method, zero boundary conditions.

“…It is seen from Fig. 1 that F 1 (x 1 , x (87) is very close to the solution for a periodic system at T = 0, which reads [39,[41][42][43][44][45][46][47][48]]…”

Section: L/mentioning

confidence: 64%

Low-Lying Energy Levels of a One-Dimensional Weakly Interacting Bose Gas under Zero Boundary Conditions

2019

Ukr. J. Phys.

“…The diagonal representation (52) for a periodic 1D Bose system at T = 0 was determined previously by a different method [31]. Instead of χ 2l (43), we obtain close occupation numbers:…”

Section: The Case Of T =mentioning

confidence: 78%

“…At γ ≫ 1 the atoms are apparently distributed over the very large number of states, and there are no macroscopically occupied states. However, we cannot verify these assumptions, since the methods in [2, 31,49] are valid only at small γ.…”

Section: The Case Of T =mentioning

confidence: 95%

“…Next, we should take into account that the equilibrium occupation numbers N k = â + . These formulae imply that the numbers N k and N −k can be macroscopic only for a 1D system (see also [31]).…”

Section: Periodic Bose System: Collective Descriptionmentioning

confidence: 97%

“…The difference between N 0 and √ NN 0 is insignificant, since the methods in [31,49] require N 0 ≈ N. It is worth to note that the density matrix was found in work [31] directly from the ground-state wave function without any assumptions about the condensate. At γ < ∼ 0.01 the solution in [31] is close to the exact one.…”

Section: The Case Of T =mentioning

confidence: 99%

See 2 more Smart Citations

On a Fragmented Condensate in a Uniform Bose System

2019

J Low Temp Phys

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According to the well-known analysis by Noziéres, the fragmentation of the condensate increases the energy of a uniform interacting Bose system. Therefore, at T = 0 the condensate should be nonfragmented. We perform a more detailed analysis and show that the result by Noziéres is not general. We find that, in a dense Bose system, the formation of a crystal-like structure with a fragmented condensate is possible. The effect is related to a nonzero size of real atoms. Moreover, the wave functions studied by Noziéres are not eigenfunctions of the Hamiltonian and, therefore, do not allow one to judge with confidence about the structure of the condensate in the ground state. We have constructed the wave functions in such a way that they are eigenfunctions of the Hamiltonian. The results show that the fragmentation of the condensate (quasicondensate) is possible for a finite one-dimensional uniform system at low temperatures and a weak coupling.Keywords: interacting bosons, fragmented condensate, quasicondensate.for the Bose gas in a double-well potential of a trap, the state with two condensates, which are localized at different minima of a trap, is energy-gained [19,20]. The other examples of a fragmented condensate and the references can be found in [7,21]. The solutions with a fragmented condensate were obtained for one-dimensional (1D) and two-dimensional (2D) Bose gases in a trap [22,23,24,25,26,27,28,29]. The fragmentation of the condensate of quasiparticles is discussed in review [30].In the present work, we will analyze the problem of fragmentation of the condensate in more details than in [17,18]. We will show that the fragmentation of the condensate is possible even for a uniform system (analogous result was obtained previously [31] without general analysis of the problem of fragmentation). In this case, the condensates are not separated in the r-space, in contrast to the solutions in [19,20,22,23,24,25,26,27,29].We will consider the problem step by step, by passing from a more crude description to an accurate one. Periodic Bose system: quasi-single-particle approachIn this section, we will carry on the analysis similar to the analysis by Pollock [16] and by Noziéres [17,18] and will take into account the nonpointness (nonzero intraction radius) of real particles. Consider the periodic system of N bosons with repulsive interaction (ν(0) > 0).

Thermodynamics of a One-Dimensional System of Point Bosons: Comparison of the Traditional Approach with a New One

2017

J Low Temp Phys

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We compare two approaches to the construction of the thermodynamics of a one-dimensional periodic system of spinless point bosons: the Yang-Yang approach and a new approach proposed by the author. In the latter, the elementary excitations are introduced so that there is only one type of excitations (as opposed to Lieb's approach with two types of excitations: particle-like and hole-like). At the weak coupling, these are the excitations of the Bogolyubov type. The equations for the thermodynamic quantities in these approaches are different, but their solutions coincide (this is shown below and is the main result). Moreover, the new approach is simpler. An important point is that the thermodynamic formulae in the new approach for any values of parameters are formulae for an ensemble of quasiparticles with the Bose statistics, whereas a formulae in the traditional Yang-Yang approach have the Fermi-like one-particle form.