On Condensations in the Bogoliubov Weakly Imperfect Bose Gas

We show that condensation in the Bogoliubov weakly imperfect Bose gas (WIBG) may appear in two stages. If interaction is such that the pressure of the WIBG does not coincide with the pressure of the perfect Bose gas (PBG), then the WIBG may manifest two kinds of condensations: nonconventional Bose condensation in zero mode, due to the interaction (the first stage), and conventional (generalized) Bose–Einstein condensation in modes next to the zero mode due to the particle density saturation (the second stage). Otherwise the WIBG manifests only the latter kind of condensation.     

Spontaneous U（1） symmetry breaking and Bose-Einstein condensation

Based on Bogoliubov's truncated Hamiltonian H_B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term $\sqrt{V}(\lambda a₀+\lambda^*a₀⁺) to H_B, we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose--Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schr\"{o}dinger eigenvalue equation and using the self-consistent condition.     

量子玻色流体中的压缩玻色子对数态

导出参量相关的Bogoliubov幺正变换的封闭性表达式，在原粒子数表象中对角化量子玻色流体的约化Hamiltonian．将两模辐射场的非经典特性的表述推广到与流体的相互作用机制相应的量子玻色场，从波动和粒子统计的角度确认Hamiltonian本征态是非经典的压缩玻色子对数态． 关键词：     

Reconciling phase diffusion and Hartree–Fock approximation in condensate systems

Despite the weakly interacting regime, the physics of Bose–Einstein condensates is widely affected by particle–particle interactions. They determine quantum phase diffusion, which is known to be the main cause of loss of coherence. Studying a simple model of two interacting Bose systems, we show how to predict the appearance of phase diffusion beyond the Bogoliubov approximation, providing a self-consistent treatment in the framework of a generalized Hartree–Fock–Bogoliubov perturbation theory.     

Exact phase diagram of the Bogoliubov weakly imperfect Bose gas

We present the exact solution of the Bogoliubov truncated Hamiltonian known as the weakly imperfect Bose gas model. To that end we prove the exactness of the Bogoliubov approximation for this Hamiltonian and we find sufficient and necessary conditions for the interaction potential which ensure a nontrivial phase diagram.     

Quantum fluctuations and collective oscillations of a Bose-Einstein condensate in a 2D optical lattice

We use Bogoliubov theory to calculate the beyond mean field correction to the equation of state of a weakly interacting Bose gas in the presence of a tight 2D optical lattice. We show that the lattice induces a characteristic 3D to 1D crossover in the behavior of quantum fluctuations. Using the hydrodynamic theory of superfluids, we calculate the corresponding shift of the collective frequencies of a harmonically trapped gas. We find that this correction can be of the order of a few percent and hence easily measurable in current experiments. The behavior of the quantum depletion of the condensate is also discussed.     

Condensate density and superfluid mass density of a dilute Bose-Einstein condensate near the condensation transition

We derive via diagrammatic perturbation theory the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation temperature, T(c). Sufficiently below T(c) particle excitations are described by mean field (Bogoliubov). Near T(c), however, mean field fails, and the system undergoes a second order phase transition, rather than first order as predicted by Bogoliubov theory. Both condensation and superfluidity occur at the same T(c), and have similar scaling functions below T(c), but different finite size scaling at T(c) to leading order in the system size. A self-consistent two-loop calculation yields the condensate fraction critical exponent, 2beta approximately 0.66.     

Large Deviations in the Superstable Weakly Imperfect Bose-Gas

The Superstable Weakly Imperfect Bose-Gas (Sup-WIBG) was originally proposed to solve some inconsistencies of the Bogoliubov theory based on the WIBG. The grand-canonical thermodynamics of the Sup-WIBG has been recently studied in details but only out of the point of the (first order) phase transition. The present paper closes this gap. The key technical tools are the Large Deviations (LD) formalism and in particular the analysis of the Kac distribution function. It turns out that the condensate fraction discontinuity as a function of the chemical potential (that occurs at the phase transition point) disappears if one considers it as a function of the total particle density. We prove that at this point the equilibrium state of the Sup-WIBG is a mixture of two (low- and high-density) pure phases related to two critical particle densities. Non-zero Bose-Einstein condensate starts at the smaller critical density and continuously grows (for a constant chemical potential) until the second critical density. For higher particle densities, the Bose condensate fraction as well as the chemical potential both increase monotonously.     

Excitation picture of an interacting Bose gas

Atomic Bose–Einstein condensates (BECs) can be viewed as macroscopic objects where atoms form correlated atom clusters to all orders. Therefore, the presence of a BEC makes the direct use of the cluster-expansion approach–lucrative e.g. in semiconductor quantum optics–inefficient when solving the many-body kinetics of a strongly interacting Bose. An excitation picture is introduced with a nonunitary transformation that describes the system in terms of atom clusters within the normal component alone. The nontrivial properties of this transformation are systematically studied, which yields a cluster-expansion friendly formalism for a strongly interacting Bose gas. Its connections and corrections to the standard Hartree–Fock–Bogoliubov approach are discussed and the role of the order parameter and the Bogoliubov excitations are identified. The resulting interaction effects are shown to visibly modify number fluctuations of the BEC. Even when the BEC has a nearly perfect second-order coherence, the BEC number fluctuations can still resolve interaction-generated non-Poissonian fluctuations.     

Interactions of condensate atoms in the process of velocity-selective coherent population trapping

The properties of a one-dimensional atomic Bose condensate are studied under the assumption that the condensation leads to a state of velocity-selective coherent population trapping. This state is characterized by the quantum correlation (entanglement) between the intrinsic angular momentum of an atom and its translational motion underlying nontrivial features of the condensate. The effects of weak interatomic interaction are taken into account. The steady state of above-condensate atoms corresponding to the slow decay of the state with coherent population trapping is found. The dynamic problem concerning the evolution of the system of above-condensate atoms after switching off the optical field forming the state with coherent population trapping is solved. The solution is found by the diagonalization of the Hamiltonian based on introducing the Bogoliubov quasiparticles with the unusual dispersion law.     

New look at thermodynamics of gas and at clusterization

In the paper, by using the example of a rigorous formulation and rigorous proof of the Maxwell distribution, estimates for the distribution in dependence of the parameter N (the number of particles) are established. Further, the problem concerning the creation of dimers in classical gas is regarded as an analog of Bose condensation, and estimates for the lower level of the analog of Bose condensation are proved. The relationship between this level and the theory of “capture” in the scattering problem corresponding to interaction in the form of Lennard-Jones potential is clarified. The equation of state of a nonideal gas as a result of pairwise interaction of particles in the Lennard-Jones and Kihara models is derived. New quantum equations for the transfer of neutral gas consisting of particles with evenly and oddly many neutrons in capillaries and nanotubes are also presented. To the memory of my teacher in number theory, Professor Anatolii Karatsuba     

Algebraic structure of quantum fluctuations

On the basis of the existence of second and third moments of fluctuations, we prove a theorem about the Lie-algebraic structure of fluctuation operators. This result gives insight into the quantum character of fluctuations. We illustrate the presence of a Lie algebra of fluctuation operators in a model of the anharmonic crystal, and show the dependence of the Lie-algebra structure on the fine structure of the fluctuation operator algebra. The result is also applied to construct the normal Goldstone mode in the ideal Bose gas for Bose-Einstein condensation     

Higher-loop integrability in gauge theory

The dilatation operator measures scaling dimensions of local operator in a conformal field theory. Algebraic methods of constructing the dilatation operator in four-dimensional N=4 gauge theory are reviewed. These led to the discovery of novel integrable spin chain models in the planar limit. Making use of Bethe ansätze a superficial discrepancy in the AdS/CFT correspondence was found, we discuss this issue and give a possible resolution. To cite this article: N. Beisert, C. R. Physique 5 (2004).

Résumé

L'opérateur de dilatation mesure les dimensions d'échelles des opérateurs locaux des théories conformes des champs. Nous passons en revue les méthodes algébriques de construction de l'opérateur de dilatation pour la théorie de jauge N=4 en quatre dimensions. Ceci nous a conduit à découvrir, dans la limite planaire, de nouveaux modèles intégrables de chaînes de spin. En utilisant l'ansätze de Bethe une incompatibilité avec la correspondance AdS/CFT fut découverte, nous discutons ce problème et une résolution possible. Pour citer cet article : N. Beisert, C. R. Physique 5 (2004).     

Semiclassical quantization of the Bogoliubov spectrum

We analyze the Bogoliubov spectrum of the three-site Bose-Hubbard model with a finite number of Bose particles by using a semiclassical approach. The Bogoliubov spectrum is shown to be associated with the low-energy regular component of the classical Hubbard model. We identify the full set of the integrals of motion of this regular component and, quantizing them, obtain the energy levels of the quantum system. The critical values of the energy, above which the regular Bogoliubov spectrum evolves into a chaotic spectrum, is indicated as well.     

The solution of a chain of Bogoliubov quantum kinetic equations for Bose systems interacting by delta potential

The Bogoliubov's hierarchy of quantum kinetic equations that describes the system of Bose particles interacting by delta potential is solved with the help of nonlinear Schrödinger's equations. The solution of the hierarchy is defined in terms of the Bethe ansatz.     

Stability properties of vector solitons in two-component Bose-Einstein condensates with tunable interactions

By using a unified theory of the formation of various types of vector-solitons in two-component Bose-Einstein condensates with tunable interactions,we obtain a family of exact vector-soliton solutions for the coupled nonlinear Schro¨dinger equations.Moreover,the Bogoliubov equation shows that there exists stable dark soliton in specific situations.Our results open up new ways in considerable experimental interest for the quantum control of multi-component Bose-Einstein condensates.     

On the stochastic dynamics of Ising models

With the help of recent results in the mathematical theory of master equations, we present a rigorous derivation of the stochastic Glauber dynamics of Ising models from Hamiltonian quantum mechanics. A thermal bath is explicitly constructed and, as an illustration, the dynamics of the Ising-Weiss model is analyzed in the thermodynamic limit. We thus obtain an example of a nonequilibrium statistical mechanical system for which a link without mathematical gap can be established from microscopic quantum mechanics to a macroscopic irreversible thermodynamic process.     

Superfluid phases and excitations in a cold gas of d-wave interacting bosonic atoms and molecules

Motivated by recent advances in orbitally tuned Feshbach resonance experiments, we analyze the ground-state phase diagram and related low-energy excitation spectra of a d-wave interacting Bose gas. A two-channel model with d-wave symmetric interactions and background s-wave interactions is adopted to characterize the gas. The ground state is found to have three interesting superfluid phases: atomic, molecular, and atomic–molecular. In great contrast to what was previously known about the p-wave case, the atomic superfluid is found to be momentum-independent for the d-wave case discussed here. The Bogoliubov spectra above each superfluid phase are obtained both analytically and numerically.     

Superconductivity and Temperature Effects on Davydov Model of α-Helix Protein

We study the thermodynamic properties of α-helix protein on the basis of Davydov theory. The Hamiltonian of Davydov theory can be represented as a form of pairing Hamiltonian in the momentum space. We prove that the quasi-classical Davydov theory is essentially a theory with Bose condensation no matter whether the molecular excitations are treated as fermions or bosons. The transition temperature of superconductivity theory gives the existence condition of the Davydov soliton.     

The Equilibrium States for a Model with Two Kinds of Bose Condensation

We study the equilibrium Gibbs states for a Boson gas model, defined by Bru and Zagrebnov, which has two phase transitions of the Bose condensation type. The two phase transitions correspond to two distinct mechanisms by which these condensations can occur. The first (non-conventional) Bose condensation is mediated by a zero-mode interaction term in the Hamiltonian. The second is a transition due to saturation quite similar to the conventional Bose–Einstein (BE) condensation in the ideal Bose gas. Due to repulsive interaction in non-zero modes the model manifests a generalized type III; i.e., non-extensive BE condensation. Our main result is that, as in the ideal Bose gas, the conventional condensation is accompanied by a loss of strong equivalence of the canonical and grand canonical ensembles whereas the non-conventional one, due to the interaction, does not break the equivalence of ensembles, at least not on the level of the gauge invariant states. It is also interesting to note that the type of (generalized) condensate, I, II, or III (in the terminology of van den Berg, Lewis, and Pulé), has no effect on the equivalence of ensembles. These results are proved by computing the generating functional of the cyclic representation of the Canonical Commutation Relation (CCR) for the corresponding equilibrium Gibbs states.