We study the ground-state properties and the excitation spectrum of bosons which, in addition to a short-range repulsive two body potential, interact through the exchange of some dispersionless bosonic modes. The latter induces a time dependent (retarded) boson-boson interaction which is attractive in the static limit. Moreover the coupling with dispersionless modes introduces a reference frame for the moving boson system and hence breaks the Galilean invariance of this system. The ground state of such a system is depleted linearly in the boson density due to the zero point fluctuations driven by the retarded part of the interaction. Both quasiparticle (microscopic) and compressional (macroscopic) sound velocities of the system are studied. The microscopic sound velocity is calculated up the second order in the effective two body interaction in a perturbative treatment, similar to that of Beliaev for the dilute weakly interacting Bose gas. The hydrodynamic equations are used to obtain the macroscopic sound velocity. We show that these velocities are identical within our perturbative approach. We present analytical results for them in terms of two dimensional parameters---an effective interaction strength and an adiabaticity parameter---which characterize the system. We find that due the presence of several competing effects, which determine the speed of the sound of the system, three qualitatively different regimes can be in principle realized in the parameter space and discuss them on physical grounds.
We study the superfluid properties of a system of interacting bosons on a lattice, which, moreover, are coupled to the vibrational modes of this lattice, treated here in terms of Einstein phonon modes. The ground state corresponds to two correlated condensates: that of the bosons and that of the phonons. Two competing effects determine the common collective sound-wave-like mode with sound velocity
v, arising from gauge symmetry breaking. (i) The sound velocity
v0 (corresponding to a weakly interacting Bose system on a rigid lattice) in the lowest-order approximation is reduced due to reduction of the repulsive boson-boson interaction, arising from the attractive part of the phonon-mediated interaction in the static limit. (ii) The second-order correction to the sound velocity is enhanced as compared to that of bosons on a rigid lattice when the boson-phonon interaction is switched on due to the retarded nature of the phonon-mediated interaction. The overall effect is that the sound velocity is essentially unaffected by the coupling with phonons, indicating the robustness of the superfluid state. The induction of a coherent state in the phonon system driven by the condensation of the bosons could be of experimental significance, permitting spectroscopic detection of superfluid properties of bosons. Our results are based on an extension of the Beliaev-Popov formalism for a weakly interacting Bose gas on a rigid lattice to one on a deformable lattice with which it interacts.
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy
E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where
n is the density and
f0 is the scattering length for the assumed two-body interaction between the bosons. The coefficients
a and
b are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient
b is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
We theoretically investigate various beyond mean-field effects on Bose gases at zero temperature featuring the anisotropic and long-range dipole-dipole interaction in addition to the isotropic and short-range contact interaction. Within the realm of the Bogoliubov--de Gennes theory, we consider static properties and low-lying excitations of both homogeneous and harmonically trapped dipolar bosonic gases. For the homogeneous system, the condensate depletion, the ground-state energy, the equation of state, and the speed of sound are discussed in detail. Making use of the local density approximation, we extend these results in order to study the properties of a dipolar Bose gas in a harmonic trap and in the regime of large particle numbers. After deriving the equations of motion for the general case of a triaxial trap, we analyze the influence of quantum fluctuations on important properties of the gas, such as the equilibrium configuration and the low-lying excitations in the case of a cylinder-symmetric trap. In addition to the monopole and quadrupole oscillation modes, we also discuss the radial quadrupole mode. We find that the latter acquires a quantum correction exclusively due to the dipole-dipole interaction. As a result, we identify the radial quadrupole as a reasonably accessible source for the signature of dipolar many-body effects and stress the enhancing character that dipolar interactions have for quantum fluctuations in the other oscillation modes.
We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finite-temperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due to both quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes.
We study the ground-state phase diagram of a mixture of bosonic and fermionic cold atoms confined on two- and three-dimensional optical lattices. The coupling between bosonic fluctuations and fermionic atoms can be attractive or repulsive and has similarities with electron-phonon coupling in crystals. We investigate behavior of the mixtures in the limit, where the Bogoliubov sound velocity that dictates bosonic dynamics is comparable to the Fermi velocity, hence the retardation effects are an important part of the physics. The dynamic Lindhard response function of the fermionic density to changes in the bosonic number of particles above some critical frequency can alter the sign, and consequently the interspecies interaction between particles becomes repulsive in contrast to the static limit (instantaneous and always attractive). Considering the above, we show that the structure of the phase diagrams crucially depends on the difference in masses of the bosons and fermions. We discuss the situations where integrating out the fermionic field provides an additional interaction that can decrease or increase bosonic coherence.
The spectral properties of a mixture of bosons hybridized with fermion pairs are studied in the superconducting state. Such a model exhibits two limiting behaviors. One is reminiscent of a BCS system, provided that the total concentration n of particles is below a certain critical concentration
nc, in which case bosons exist only in thermally excited states. The other limit for n\ensuremath{\geqslant}
nc describes a situation largely controlled by the condensation of the bosons. The variation of the single-boson as well as collective excitations as a function of n and for the entire temperature regime from T=0 K to
Tc will be discussed, both with and without long-range Coulomb interaction between the charge carriers. The method for calculation applied here is based on the dielectric formalism and the use of Ward identities, currently used in the theory of the interacting Bose gas, which allows us to derive the density and current correlation functions consistent with local charge-current conservation laws. In the absence of a Coulomb interaction the behavior of the system is characterized by the existence of a sound-wave-like branch in the Bose single-particle and collective excitation spectrum. In the BCS regime the sound velocity is proportional to the Fermi velocity while in the Bose regime it is proportional to the strength of the boson-fermion hybridization and decreases with the superfluid density as we approach
Tc. In the presence of a long-range Coulomb interaction and in the Bose regime one observes, besides the usual optical plasmon mode at high frequency and corresponding to in-phase density oscillation of the two subsytems, an out-of-phase collective mode in the two particle continuum just above the gap. Moreover, upon approaching
Tc a relatively well-defined quasiparticle feature starts to emerge within the gap and is seen in both the Bose single-particle excitation and the density oscillation spectrum. In the BCS regime, due to the very small number of occupied bosonic states, the low-energy quasiparticle features are absent. \textcopyright{} 1996 The American Physical Society.