Bogoliubov Type Research Articles

Global Uniform in N Estimates for Solutions of a System of Hartree–Fock–Bogoliubov Type in the Case $$\beta <1$$

Global Uniform in N Estimates for Solutions of a System of Hartree–Fock–Bogoliubov Type in the Case $$\beta <1$$

On coherent states description of exciton systems: NLSE equations and hydrodynamics

The exciton system is described by means of specially constructed coherent states (CS), according to an idea of L.V. Keldysh long ago. The system is analyzed from the geometric point of view and from the relevant symmetry group contained in the displacement operator as generador of the CS system in the Klauder–Perelomov sense. Explicitly important quantities involving the quantum statistics of the system (partition function, mean density n and the Yuen limit) are calculated and briefly discussed in relation to the known case of pure bosons (e.g. photons). The Bogoliubov type transformations for the field operators of the Wannier–Mott type exciton system are presented in an exact way (in sharp contrast with the case of kel where only a rough approximation was given) showing how these transformations induce nonlinearity (cubic at first order) in the Schrodinger equation of the system. The conditions for a hydrodynamic description of the original Keldysh model, as the possibility of a BEC (Bose–Einstein Condensation), are discussed.

Uniform in N estimates for a Bosonic system of Hartree–Fock–Bogoliubov type

We prove local in time, uniform in N, estimates for the solutions ϕ , Λ, and Γ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential v N ( x − y ) = N 3 β v ( N β ( x − y ) ) , with β < 1 and v a Schwartz function (satisfying additional technical requirements). The initial conditions are general functions in a Sobolev-type space, and the expected correlations in Λ develop dynamically in time. As shown in our previous work, as well as the work of Chong (both in the case β < 2 / 3 ), using the conserved quantities of the system of equations, this type of local in time estimates can be extended globally. Also, they can be used to derive Fock space estimates for the approximation of the exact evolution of a Bosonic system by quasi-free states of the form e N A ( ϕ ) e B ( k ) Ω . This will be addressed in detail in future work.

Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide

We theoretically investigate the dispersion relation of small-amplitude optical waves superimposing upon a beam of polarized monochromatic light propagating along a single-mode channel waveguide characterized by an instantaneous and spatially local Kerr nonlinearity. These small luminous fluctuations propagate along the waveguide as Bogoliubov elementary excitations on top of a one-dimensional dilute Bose quantum fluid evolve in time. They consequently display a strongly renormalized dispersion law, of Bogoliubov type. Analytical and numerical results are found in both the absence and the presence of one- and two-photon losses. Silicon and silicon-nitride waveguides are used as examples. We finally propose an experiment to measure this Bogoliubov dispersion relation, based on a stimulated four-wave mixing and interference spectroscopy techniques.

Nonlinear analysis of cylindrical and conical hysteretic whirl motions in rotor-dynamics

The internal friction of a rotor–shaft-support system is mainly due to the shaft structural hysteresis and to some possible shrink-fit release of the assembly. The experimentation points out the destabilizing effect of the internal friction in the over-critical rotor running. Nevertheless, this detrimental influence may be efficiently counterbalanced by other external dissipative sources located in the supports or by a proper anisotropic configuration of the support stiffness. The present analysis considers a rotor–shaft system which is symmetric with respect to the mid-span and is constrained by viscous-flexible supports with different stiffness on two orthogonal planes. The cylindrical and conical whirling modes are easily uncoupled and separately analysed. The internal dissipation is modelled by nonlinear Coulombian forces and moments, which counteract the translational and rotational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are solved by averaging approaches of the Krylov–Bogoliubov type. In both the over-critical whirling motions, cylindrical and conical, stable limit cycles may be attained whose amplitude is as large as the external dissipation applied by the supports is low. The stiffness anisotropy of the supports may be recognised as quite beneficial for the cylindrical whirl.

Approach to rotor-shaft hysteretic whirl using Krylov–Bogoliubov techniques

The internal friction associated with the shaft hysteresis or with the possible release of some shrink-fit coupling exerts a destabilizing effect on the over-critical rotor running, but may be efficiently counteracted by other external dissipative sources or by a proper anisotropic configuration of the support stiffness. The present analysis considers a symmetric rotor-shaft system on viscous-flexible supports with different stiffness on two orthogonal planes containing the bearing axis. The internal friction of the shaft is described either by a linear hysteretic model or by a nonlinear Coulombian force contrasting the rotor motion relative to the shaft ends. The nonlinear equations of motion are solved using an averaging approach of the Krylov–Bogoliubov type, which yields the steady orbit depending on the support dissipation applied to damp the whirl motion, for various working conditions. The beneficial influence of the support stiffness anisotropy is clearly identified.

Superfluidity of Bose-Einstein condensates in toroidal traps with nonlinear lattices

Superfluid properties of Bose-Einstein condensates (BEC) in toroidal quasi-one-dimensional traps are investigated in the presence of periodic scattering length modulations along the ring. The existence of several types of stable periodic waves, ranging from almost uniform to very fragmented chains of weakly interacting and equally spaced solitons, is demonstrated. We show that these waves may support persistent atomic currents and sound waves with spectra of Bogoliubov type. Fragmented condensates can be viewed as arrays of Josephson junctions and the current as a BEC manifestation of the dc-Josephson effect. The influence of linear defects on BEC superfluidity has been also investigated. We found that for subcritical velocities, linear defects that are static with respect to the lattice (while the condensate moves in respect to both the optical lattice and the defect) preserve the BEC superfluidity.

Description of the 56Ni ground state with a few symmetry projected general complex Hartree–Fock–Bogoliubov determinants

The ground state of the mid pf-shell nucleus 56Ni is described in the framework of a symmetry-projected complex mean-field theory in which the desired symmetry properties are exactly projected from the most general Hartree–Fock–Bogoliubov type determinants. This few determinant method is applied for the first time in a full 1 p0 f-shell model space. In the calculations the FPD6 effective interaction was used. It turns out that the use of only a few such generalized determinants is sufficient to give an excellent result. We compare the result with that of the quantum Monte Carlo approach, as well as with recently performed exact shell-model diagonalizations.

Dynamical evolution of a fermionic anharmonic oscillator

We use reduced density matrix methods of general applicability in many-body physics to study one-particle properties of a simple, soluble quantum system, the fermionic anharmonic oscillator. The time evolution of a general initial state of this system is worked out and analyzed in detail from this point of view. We use a canonical transformation of the Bogoliubov type to enhance the power of the one-particle reduced description and show that the reduced density matrix corresponds in general to a statistical mixture even if the state of the entire many-body system is described in terms of a state vector (pure state).

Depletion of superfluid fraction in the quasi-two-dimensional Bose gas.

A theory is presented for the depletion of superfluid density in a low-temperature Bose gas adsorbed on a surface. Longitudinal excitations are treated in a Bogoliubov type of approximation. Transverse modes are found to contribute significantly if the excitation energy of adatoms normal to the surface is small.

On the Continuous Dependence on Parameter of the Solution Set of Differential Inclusions

We prove a theorem on continuous dependence of solutions of differential inclusions in Banach spaces on parameters and derive from it the first fundamental Bogoliubov type theorem on averaging.

TheN 7/5 law for charged bosons

Non-relativistic bosons interacting with Coulomb forces are unstable, as Dyson showed 20 years ago, in the sense that the ground state energy satisfies E 0 ⪳ − AN 7/5. We prove that 7/5 is the correct power by proving that E 0 ≧ − BN 7/5. For the non-relativistic bosonic, one-component jellium problem, Foldy and Girardcau showed that E 0 ≦ − CN ρ 1/4. This 1/4 law is also validated here by showing that E 0 ⪴ − DN ρ 1/4. These bounds prove that the Bogoliubov type paired wave function correctly predicts the order of magnitude of the energy.

A possible generalisation of the coherent states for free fields

It is shown explicitly how a Bogoliubov type transformation acting on coherent states gives rise to states where the mean values of the field strengths keep the original values while the mean values of the squared strengths or the correlation functions can, within some bounds be prescribed independently.

Critical analysis of the hard-sphere model for superfluidHe4. I.

The basic goal of this paper is to present the theory which stands at the basis of the calculations of the /sup 4/He excitation spectrum reported by Wong and Huang. The exact N-particle hard-sphere pseudopotential is constructed through the method of non-local-field operators developed earlier. Keeping only the two-body interaction terms, this pseudopotential replaces the exact two-particle boundary conditions which is demonstrated here and thus has been proposed as a model potential for actual helium-helium interaction in the liquid state. A pair state of the Bogoliubov type is used to calculate the ground-state energy of such a system. An expansion-parameter scheme is presented which drastically simplifies the mathematics involved and justifies previous approximations employed by one of the authors. The expansion describes the dilute case by assuming a large portion of the particles in the zero-momentum state and the dense case by assuming only a negligible portion of the particles in the Bose-Einstein state.

Generalized condensation of an interacting bose gas with pair Hamiltonian

An interacting Bose gas atT=0, with a pair Hamiltonian where there exists a generalized condensation in state is investigated up to an upper limitk0 in momentum space. Corresponding to the idea that an ideal Bose gas can be represented by a coherent state, each operator is replaced by a coherent field operator. It is then found that for a particular choice of the potential, the transformations and procedures of Bogoliubov type lead to a stable energy spectrum with a gap at finite volume, but to a photon spectrum at infinite volume. Though a major portion of the interaction in momentum space is negative, a minimum point in the energy-density diagram is indicated.

A Soluble Fermi-Gas Model. Validity of Transformations of the Bogoliubov Type

A linear soluble Fermi-gas model is used to test the validity of perturbation theory and the criteria of instability obtained by applying transformations of the Bogoliubov type. The model Hamiltonian describing the interaction of the electrons in a simple band is H(ρ)=−Σ(cosk+ρ)ck+ck+ρΣcos q ck+q+cu−q+cuck. For a half-filled band, the energies of the low-lying states are calculated exactly, by using previous results concerning a linear chain of spins with interaction. It is shown that perturbation theory must be valid for −1 &amp;lt; ρ &amp;lt; 1, and, for values of ρ belonging to this range, the slope of the specific heat for T = 0 is calculated by direct application of Landau's theory. On the other hand, anomalous Hartree-Fock states can be built for all values of ρ, by using transformations of the Bogoliubov type, and these states are more stable than the normal Hartree-Fock state. Thus, it appears that the Bogoliubov method may lead to completely erroneous interpretations, when the coupling constants are small.

The ground state of the Bose gas

The mathematical formalism describing the Bose gas at zero temperature is analysed with the aid of methods that have recently been successful in relativistic quantum field theory. First the spectrum conditions for an infinitely extended system are given and the algebra of observables and the algebra of field operators are defined. General properties of states over these algebras are discussed and theorems are given which connect the linked cluster property, translation invariance and the purity of the states. It is proved that pure states over the algebra of observables have the property of ‘factorisable off-diagonal long range order’. The class of ‘quasi free states’ is defined and of these states those which are translation invariant and possess the linked cluster property are analysed. It is shown that this class of states contains a subclass of pure states of the Bogoliubov type and a subclass of states which are mixtures of non-translationally invariant pure states. The applications of these ‘quasi free states’ to the interacting Bose gas are summarized.