Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.     

Goldstone Boson Normal Coordinates in Interacting Bose Gases

Spontaneous symmetry breaking (SSB) is one of the basic aspects of collective phenomena such as phase transitions in statistical mechanics or ground-state excitations in field theory. In general, spectral analysis of SSB is related to the presence of a Goldstone boson particle. The explicit construction of the canonical variables (boson field operator and its adjoint) of this boson has so far been an open problem. In this paper, we consider the SSB of Bose–Einstein condensation in two models: the so-called imperfect or mean field Bose gas (which is nothing but a perfect ideal Bose gas including the property of equivalence of ensembles), and the Bogoliubov weakly interacting Bose gas. For both we construct explicitly the Goldstone boson field variables. The remarkable result is that in both cases the field and its adjoint field are formed as the fluctuation operators respectively of the order parameter operator and of the generator of the broken symmetry. The notion of fluctuation operator is essential for our mathematical construction. We find that although the order parameter has a critical fluctuation, the generator of the broken symmetry has a squeezed fluctuation of the same inverse rate. Furthermore, we prove that this canonical pair of variables decouples from the other variables of the system, and that these fluctuations behave dynamically as long-wavelength sound waves or as oscillator variables.     

Coexisting Condensates of Weakly Interacting Bose Gas in a Harmonic Trap

We study particles in a vortex state driven to a core state with lower energy and zero angular momentum by the trap potential asymmetries. We find that at T=0 when the role of the thermal gas can be ignored, there will be coexisting condensates. We also calculate the fluctuation of the number difference and argue that in certa/n range of the parameters the state of the whole system is the macroscopic quantum serf-trapping in the Josephson tunnelling regime.     

Coexisting Condensates of Weakly Interacting Bose Gas in a Harmonic Trap

We study particles in a vortex state driven to a core state with lower energy and zero angular momentumby the trap potential asymmetries. We find that at T = 0 when the role of the thermal gas can be ignored, there will becoexisting condensates. We also calculate the fluctuation of the number difference and argue that in certain range of theparameters the state of the whole system is the macroscopic quantum self-trapping in the Josephson tunnelling regime.     

Collective excitations of a periodic Bose condensate in the Wannier representation

We study the dispersion relation of the excitations of a dilute Bose-Einstein condensate confined in a periodic optical potential and its Bloch oscillations in an accelerated frame. The problem is reduced to one-dimensionality through a renormalization of the s-wave scattering length and the solution of the Bogolubov-de Gennes equations is formulated in terms of the appropriate Wannier functions. Some exact properties of a periodic one-dimensional condensate are easily demonstrated: (i) the lowest band at positive energy refers to phase modulations of the condensate and has a linear dispersion relation near the Brillouin zone centre; (ii) the higher bands arise from the superposition of localized excitations with definite phase relationships; and (iii) the wavenumber-dependent current under a constant force in the semiclassical transport regime vanishes at the zone boundaries. Early results by Slater [Phys. Rev. 87, 807 (1952)] on a soluble problem in electron energy bands are used to specify the conditions under which the Wannier functions may be approximated by on-site tight-binding orbitals of harmonic-oscillator form. In this approximation the connections between the low-lying excitations in a lattice and those in a harmonic well are easily visualized. Analytic results are obtained in the tight-binding scheme and are illustrated with simple numerical calculations for the dispersion relation and semiclassical transport in the lowest energy band, at values of the system parameters which are relevant to experiment. Received 3 December 1999 and Received in final form 22 March 2000     

Structure factor for a Bose fluid

The structure factor is calculated for a Bose fluid using an expression for S(k), which is approximately valid for all values of k. Marked fluctuations appear for $\text{k = 2.15}\text{A}\text{?}{}^{-1}$, near the roton dip in E_k, and at $\text{k = 3.65}\text{A}\text{?}{}^{-1}$ near the dip in the second branch of E_k. Our calculations are indirectly a theoretical justification for the existence of a second branch in E_k. The results presented here agree fairly well with the experimental values for $\text{k >; 3.0}\text{A}\text{?}{}^{-1}$ along with a dip at $\text{k = 3.67}\text{A}\text{?}{}^{-1}$ which has not been reported earlier. It also suggests the existence of two waves, phonons and rotons, for $\text{k >; 3.0}\text{A}\text{?}{}^{-1}$.     

Matter-wave amplification and collective internal excitations in a trapped Bose gas

Received: 23 May 1997/Revised version: 29 July 1997     

Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

We study a system consisting of a heavy quantum particle, called the tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density ${\to \infty}$ ). Our estimates allow us to interchange the thermodynamic with the mean-field limit.     

Structure factor of Bose fluid

The structure factor which is a measure of the correlation between the positions of the atoms in a fluid is calculated for a Bose fluid. Its dependence on the propagation vectork is obtained in an effort to obtain a formula that may be valid for all values ofk.One of the authors (K. M. K.) is thankful to Professor Abdus Salam and the International Centre for Theoretical Physics, Trieste, Italy, for hospitality when the above work was initiated.     

Bose–Einstein Condensation for Homogeneous Interacting Systems with a One-Particle Spectral Gap

We prove rigorously the occurrence of zero-mode Bose–Einstein condensation for a class of continuous homogeneous systems of boson particles with superstable interactions. This is the first example of a translation invariant continuous Bose-system, where the existence of the Bose–Einstein condensation is proved rigorously for the case of non-trivial two-body particle interactions, provided there is a large enough one-particle excitations spectral gap. The idea of proof consists of comparing the system with specially tuned soluble models.     

Collective excitations of a 1D Bose–Einstein condensate in an anharmonic trap

We investigate the collective excitations of a one-dimension Bose–Einstein condensate trapped in an anharmonic potential by solving the time-dependent Tonks–Girardeau equation. The governing equations of motions for the low-energy excitations are obtained using variational approaches. The motion of a 1D BEC in a harmonic trap is just like the motion of one particle in a harmonic trap. And quartic distortion of the potential causes the blue-shift and red-shift on the excitation frequency while cube distortion only causes the red-shift.     

Visualization of dimensional effects in collective excitations of optically trapped quasi-two-dimensional Bose gases

In quasi-two dimensions (quasi-2D), where excitations are frozen in one direction, the scattering amplitudes exhibit 2D features of the particle motion and a 3D to 2D dimensional crossover emerges in the behavior of scattering. We explore its physical consequences, capitalizing on a hidden connection between the Pitaevskii-Rosch dynamical symmetry and breathing modes. We find broken Pitaevskii-Rosch symmetry by arbitrarily small 2D effects, inducing a frequency shift in breathing modes. The predicted shift rises significantly from the order of 0.5% to more than 5% in transiting from the 3D-scattering to the 2D-scattering regime. Comparisons with other relevant effects suggest our results are observable within current experimental capabilities.     

Amplitude Squeezing of a Weakly Interacting Bose System with Spontaneous U(1) Symmetry Breaking

We study the quantum fluctuations and the amplitude squeezing of a weakly interacting Bose system with spontaneous U(1) symmetry breaking. It is found that this system can exhibit amplitude squeezing.     

Quantum backreaction of quantum fluid in Bose Einstein condensates

In this paper, with the full field operator \hat ψ expressed in terms of a particle-number-conserving mean-field ansatz, we investigate the dynamical behaviour of Bose--Einstein condensates from microscopic physics. Including the first-order term correction from single-particle excitation and the remaining higher-order term correction from collective excitations simultaneously, we obtain the formulation for a closed local expression of quantum backreaction Q, and discuss the influence on static Bose--Einstein condensates. Even though the quantum backreaction is small, it still has some influence on its dynamics.