Simple theoretical tools for low dimension Bose gases
نویسنده
چکیده
We first consider an exactly solvable classical field model to understand the coherence properties and the density fluctuations of a one-dimensional (1D) weakly interacting degenerate Bose gas with repulsive interactions at temperatures larger than the chemical potential. In a second part, using a lattice model for the quantum field, we explain how to carefully generalize the usual Bogoliubov approach to study a degenerate and weakly interacting Bose gas in 1D, 2D or 3D in the regime of weak density fluctuations. In the last part, using the mapping to an ideal Fermi gas in second quantized formalism, we calculate and discuss physically the density fluctuations and the coherence properties of a gas of impenetrable bosons in 1D. 1 A CLASSICAL FIELD MODEL FOR THE 1D WEAKLY INTERACTING BOSE GAS 1.1 Reminder: quantum theory for the ideal Bose gas 1.1.1 Second quantized formalism We consider the case of spinless, non-relativistic bosons of mass m moving in a space of spatial coordinates of dimension d and stored either in a trapping potential U(r) or in a cubic box of size L with periodic boundary conditions. In this subsection the bosons are not interacting. The grand canonical Hamiltonian has then the following expression in second quantized formalism: Ĥ = ∫ dr ψ̂(r)(h0 − μ)ψ̂(r). (1.1) μ is the chemical potential, and the differential operator h0 includes the kinetic energy operator and the trapping potential U(r): h0 = − h̄ 2m ∆+ U(r) (1.2) where ∆ is the Laplacian operator in dimension d. Note that U ≡ 0 in the spatially homogeneous case of a cubic box. The field operator ψ̂(r) annihilates a boson in point r and obeys the usual bosonic commutation relations: [ψ̂(r), ψ̂(r)] = 0, (1.3) [ψ̂(r), ψ̂(r)] = δ(r − r). (1.4) Pr1-2 JOURNAL DE PHYSIQUE IV It is convenient to expand the field operator on the orthonormal basis of the eigenmodes φα(r) of h0 with eigenenergy ǫα: ψ̂(r) = ∑
منابع مشابه
Bose Condensation and Superfluidity in Finite Rotating Bose Systems
A long standing problem in the theory of superfluidity concerns the question of how close is the connection between Bose condensation and superfluidity[1]. For the case of superfluid He, the Bose condensate fraction is quite small (if not virtually zero), yet the superfluid fraction goes to unity in the limit of low temperatures T → 0. For the superfluid He case, the observation of superfluidit...
متن کاملUltracold gases far from equilibrium
Ultracold atomic quantum gases belong to the most exciting challenges of modern physics. Their theoretical description has drawn much from (semi-) classical field equations. These mean-field approximations are in general reliable for dilute gases in which the atoms collide only rarely with each other, and for situations where the gas is not too far from thermal equilibrium. With present-day tec...
متن کاملDimensional phase transition from an array of 1D Luttinger liquids to a 3D Bose-Einstein condensate.
We study the thermodynamic properties of a 2D array of coupled one-dimensional Bose gases. The system is realized with ultracold bosonic atoms loaded in the potential tubes of a two-dimensional optical lattice. For negligible coupling strength, each tube is an independent weakly interacting 1D Bose gas featuring Tomonaga Luttinger liquid behavior. By decreasing the lattice depth, we increase th...
متن کاملQuasiparticle spectrum for the dilute Bose-gas in an inhomogeneous external field
The paper deals with the properties of Bose-gases in the inhomogeneous external fields at very low temperatures, when a Bose-Einstein condensation occurs. This particularly refers to the trapped alkali gases, which are of great experimental and theoretical interest during last few years. The system is considered here in the frame of the developed approach closed to the Bogoliubov’s one with nec...
متن کامل