Point bosons in a one-dimensional box: the ground state, excitations and thermodynamics

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.

Section: Introductionmentioning

confidence: 86%

“…. coincide with the corresponding levels [8,9] of a system of point bosons, which is described via the Bethe ansatz.…”

Section: Methods 2: Diagonalization Of the Hamiltonianmentioning

confidence: 99%

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

2019

Ukr. J. Phys.

“…We emphasize that the Bogoliubov method describes well a finite 1D system at weak coupling and T → 0. This follows from the facts that the criterion of applicability of the method is satisfied [49], the solutions for E 0 and E(k) coincide with the solutions in the exactly solvable approach based on the Bethe ansatz [35,50,51,52,53], and the solution for F 1 (x, x ′ )| T =0 is close to the solution for a periodic system, obtained by different methods (see references in [49]). The solution for the density matrix of a 1D Bose gas under zero BCs reads [49]:…”

Section: One-dimensional Bose Gas Under Zero Boundary Conditionsmentioning

confidence: 59%

“…We note that the method works for sufficiently large N: N > ∼ N cr . For a 1D system the Bogoliubov solutions [2, 49] agree with the exact ones [35,50,51,52,53] for N > ∼ 100 under periodic BCs and for N > ∼ 1000 under zero BCs. Therefore, N cr ≃ 100 for periodic BCs, and N cr ≃ 1000 for zero BCs.…”

Section: Periodic Bose System: Collective Descriptionmentioning

confidence: 83%

“…Interestingly, for a finite system the order parameter ψ (x, t) does not generally coincide with the unique condensate defined with the help of criterion (1). Under periodic BCs, the function F 1 (x, x ′ ) is set by formula (52), and the number of atoms in the effective condensate ψ (x, t) ,Ñ 0 , is equal to N 0 . If the unique condensate is not fragmented, it coincides with ψ (x, t) .…”

Section: The Case Of T >mentioning

confidence: 99%

See 1 more Smart Citation

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).

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

2016

J Low Temp Phys

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Using the Vakarchuk formulae for the density matrix, we calculate the number N k of atoms with momentumhk for the ground state of a uniform one-dimensional periodic system of interacting bosons.We obtain for impenetrable point bosons N0 ≈ 2is no condensate or quasicondensate on low levels at large N . For almost point bosons with weak. In this case, the quasicondensate exists on the level with k = 0 and on low levels with k = 0, if N is large and β is small (e.g., for N ∼ 10 10 , β ∼ 0.01). A method of measurement of such fragmented quasicondensate is proposed.