A new method of description of Bose systems in external fields is proposed. Equations derived on the basis of this method are the Bose analog of the Bogolyubov–De Gennes equations for Fermi systems with pairing. An exact solution of the equations is obtained for the case when a Bose system is in a field varying linearly with the coordinate. The possibility of propagation of previously unknown surface oscillations in a nonuniform field is predicted. The effect of the nonuniform field on the long-range order in two- and three-dimensional Bose systems is investigated.
A new method of description of Bose systems in external fields is proposed. Equations derived on the basis of this method are the Bose analog of the Bogolyubov–De Gennes equations for Fermi systems with pairing. An exact solution of the equations is obtained for the case when a Bose system is in a field varying linearly with the coordinate. The possibility of propagation of previously unknown surface oscillations in a nonuniform field is predicted. The effect of the nonuniform field on the long-range order in two- and three-dimensional Bose systems is investigated.
For two-component Fermi gases at zero temperature, through a general derivation based on the BCS wave function, our research shows that the nonuniform molecular Bose-Einstein condensate has off-diagonal long-range order and can be described well by an order parameter when the size of molecules is much smaller than the size of the whole system. We also give the equation of state and nonlinear evolution equation for the order parameter of nonuniform molecular condensates, which is similar to the Gross-Pitaevskii equation of atomic Bose condensate. The nonlinear evolution equation is applied to consider the Josephson effect for two weakly linked molecular condensates in the presence of nonuniform magnetic field. We find clear particle-number oscillation where spatially dependent binding energy of molecules plays an important role.
Abstract This chapter develops the theory of weakly interacting nonuniform Bose gases at zero temperature. Interaction effects are accounted for by a single coupling constant fixed by the value of the s-wave scattering length. The nonlinear Gross–Pitaevskii equation is derived in both the stationary and time-dependent cases. The irrotational nature of the flow and the crucial role played by the chemical potential are highlighted. The classical nature of the equation is emphasized and the analogy with the Maxwell equations of classical electrodynamics is pointed out. The stationary solutions of the Gross–Pitaevskii equation, corresponding to quantized vortex lines, vortex rings, and solitons, are discussed. The time-dependent solutions corresponding to small-amplitude oscillations are derived and the corresponding dispersion relation is shown to coincide with the predictions of Bogoliubov theory for uniform media. The Thomas–Fermi limit is also discussed.
We develop an approach for a description of collective excitations in a two-dimensional gas of interacting Bose particles in an external potential. We present a method of finding an approximate analytical solution for the spectra of collective excitations of a Bose gas in a linear potential and in a potential of the form u(r)=m-u₀cosh²x/l, where m is the chemical potential. Numerical study shows that the analytical solution corresponds to collective modes localized at the edge or at the low-density region. We investigate the influence of the external potential on a critical velocity of a superfluid flow. It is shown that the effect of strong suppression of the critical velocity takes place in a nonuniform Bose system. We discuss a possibility of Bose-Einstein condensation (BEC) in the systems under investigations at nonzero temperatures and find that in case of a finite number of the particles BEC can emerge.
We derive a description of the spatially inhomogeneous Bose-Einstein condensate, treating the system locally as a homogeneous Bose gas. This approach, similar to the Thomas-Fermi model for the inhomogeneous many-particle fermion system, is well suited to describe the atomic Bose-Einstein condensates that have recently been obtained experimentally through atomic trapping and cooling. In this paper, we confine our attention to the zero-temperature case, although the treatment can be generalized to finite temperatures, as we shall discuss elsewhere. Several features of this approach, which we shall call the Thomas-Fermi-Bogoliubov description, are very attractive. (i) It is simpler than the Hartree-Fock-Bogoliubov technique. We can obtain analytical results in the case of weakly interacting bosons for quantities such as the chemical potential, the local depletion, pairing, pressure, and density of states. (ii) The method provides an estimate for the error due to the inhomogeneity of the Bose-condensed system. This error is a local quantity so that the validity of the description for a given trap and a given number of trapped atoms can be tested as a function of position. We see, for example, that at the edge of the condensate the Thomas-Fermi-Bogoliubov theory always breaks down. (iii) The Thomas-Fermi-Bogoliubov description can be generalized to treat the statistical mechanics of the Bose gas at finite temperatures.
We extend classical-field methods [Blakie et al., Adv. Phys. 57, 363 (2008)] to provide a description of atom-molecule systems. We use a model of Bose-Einstein condensation of atoms close to a Feshbach resonance, in which the tunable scattering length of the atoms is described using a system of coupled atom and molecule fields [Holland et al., Phys. Rev. Lett. 86, 1915 (2001)]. We formulate the basic theoretical methods for a coupled atom-molecule system, including the determination of the phenomenological parameters in our system, the Thomas-Fermi description of Bose-Einstein condensate, the Bogoliubov--de Gennes equations, and the Bogoliubov excitation spectrum for a homogenous condensed system. We apply this formalism to the special case of Bragg scattering from a uniform condensate and find that for moderate and large scattering lengths, there is a dramatic difference in the shift of the peak of the Bragg spectra, compared to that based on a structureless atom model. The result is compatible with the experimental results of Papp et al. [Phys. Rev. Lett 101, 135301 (2008)] for Bragg scattering from a nonuniform condensate.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas–Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. In Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. Finally, in Sec. 6.5 we show how the magnetic dipole–dipole interaction, which is long-ranged and anisotropic, may be included in the Gross–Pitaevskii equation and determine within the Thomas–Fermi approximation its effect on the density distribution. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4π ħ 2 a/m .