Images of a Bose-Einstein condensate in position and momentum space

In the Bogoliubov theory, a condensate initially prepared in its ground state described by a stationary Bogoliubov vacuum and later perturbed by a time-dependent potential or interaction strength evolves into a time-dependent excited state which is a dynamical Bogoliubov vacuum. The dynamical vacuum has a simple diagonal form in a time-dependent orthonormal basis of single-particle modes. This diagonal representation leads to a Gaussian probability distribution for possible density-measurement outcomes in position and momentum space.     

Effects of three-body interaction on dynamic and static structure factors of an optically-trapped Bose gas

We investigate how three-body interactions affect the elementary excitations and dynamic structure factor of a Bose- Einstein condensate trapped in a one-dimensional optical lattice. To this end, we numerically solve the Gross-Pitaevskii equation and then the corresponding Bogoliubov equations. Our results show that three-body interactions can change both the Bogoliubov band structure and the dynamical structure factor dramatically, especially in the case of the two-body interaction being relatively small. Furthermore, when the optical lattice is strong enough, the analytical results, combined with the sum-rule approach, help us to understand that： the effects of three-body interactions on the static structure Ihctor can be significantly amplified by an optical lattice. Our predictions should be observable within the current Bragg spectroscopy experiment.     

How to measure the bogoliubov quasiparticle amplitudes in a trapped condensate

We propose an experiment, based on two consecutive Bragg pulses, to measure the momentum distribution of quasiparticle excitations in a trapped Bose gas at low temperature. With the first pulse one generates a bunch of excitations carrying momentum q, whose Doppler line is measured by the second pulse. We show that this experiment can provide direct access to the amplitudes u(q) and v(q) characterizing the Bogoliubov transformations from particles to quasiparticles. We simulate the behavior of the nonuniform gas by numerically solving the time dependent Gross-Pitaevskii equation.     

Berry phase effects on the dynamics of quasiparticles in a superfluid with a vortex

We study quasiparticle dynamics in a Bose-Einstein condensate with a vortex by following the center of mass motion of a Bogoliubov wave packet, and find important Berry-phase effects due to the background flow. We show that the Berry phase invalidates the usual canonical relation between the mechanical momentum and position variables, leading to important modifications of quasiparticle statistics and thermodynamic properties of the condensates. Applying these results to a vortex in an infinite uniform superfluid, we find that the total transverse force acting on the vortex is proportional to the superfluid density. We propose an experimental setup to directly observe Berry phase effects through measuring local thermal atoms' momentum distribution around a vortex.     

Excitations in a Dipolar Bose–Einstein Condensate

We study excitations in a dipolar Bose–Einstein condensate with Green’s function. In Bogoliubov approximation, we obtain the dispersion relation. The excitation energy is dependent on the angle between the momentum and the magnetic moment. In the long-wave limit, the dispersion relation reduces to an anisotropic phonon-like dispersion relation.     

Experimental observation of the Bogoliubov transformation for a Bose-Eeinstein condensed gas

Phonons with wave vector q/(planck constant) were optically imprinted into a Bose-Einstein condensate. Their momentum distribution was analyzed using Bragg spectroscopy with a high momentum transfer. The wave function of the phonons was shown to be a superposition of +q and -q free particle momentum states, in agreement with the Bogoliubov quasiparticle picture.     

基于代数动力学方法实现一类相干态的构建

最近提出的一个构建相干态的方案中,需要精确求解一个时间相关的常微分方程.基于代数动力学理论,利用该方程具有的SU(1,1)动力学对称性,提出了对此方程在含时系数取任意函数形式时的统一的精确求解方法,并且得到了严格的解析解.运用这个精确解,就可以构造相应物理系统的精确相干态的具体表达式.给出了一个解例,即频率取"快变"函数的情形.利用得到的精确结果,讨论了这个系统的量子涨落(量子噪声)随时间演化的情况.针对动量算符不确定度随时间演化的曲线的性态,指出在制备这个系统压缩态时可以利用的一些性质.最后,讨论了这个量子系统的不确定关系随时间演化的情况.     

Bragg spectroscopy of the multibranch Bogoliubov spectrum of elongated Bose-Einstein condensates

We measure the response of an elongated Bose-Einstein condensate to a two-photon Bragg pulse. If the duration of the pulse is long, the total momentum transferred to the condensate exhibits a nontrivial behavior which reflects the structure of the underlying Bogoliubov spectrum. It is thus possible to perform a spectroscopic analysis in which axial phonons with a different number of radial nodes are resolved. The local density approximation is shown to fail in this regime, while the observed data agree well with the results of simulations based on the numerical solution of the Gross-Pitaevskii equation.     

Damping of zero sound in liquid3He

The dynamic structure factor in liquid³He is calculated within the framework of Dissipative Linear Response (DLR) equations. The effective interaction is represented by the Landau parameters up tol=2 and the Random Phase Approximation (RPA)-scattering amplitudes are evaluated in thes–p–d approximation. The dissipative linear response operator is calculated on the basis of RPA solutions of undamped zero sound using integral equation techniques.The dynamical structure factor has a strong resonance structure whose width agrees with measurements of the attenuation coefficient for small momentum transfers. At larger values of momentum transfer, where the dynamical structure factor has been directly measured, the Landau parameters do not give more than a qualitative agreement with the data.     

Fractional statistical mechanics

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Liouville and Bogoliubov equations with fractional coordinate and momenta derivatives are considered as a basis to derive fractional kinetic equations. The Fokker-Planck-Zaslavsky equation that has fractional phase-space derivatives is obtained from the fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.     

Occupation numbers from functional integral

Occupation numbers for non-relativistic interacting particles are discussed within a functional integral formulation. For bosons at zero temperature the Bogoliubov theory breaks down for strong couplings as well as for low-dimensional models. We find that the leading behavior of the occupation numbers for small momentum is governed by a quadratic time derivative in the inverse propagator that is not contained in the Bogoliubov theory. We propose to use a functional renormalization group equation for the occupation numbers in order to implement systematic non-perturbative extensions beyond the Bogoliubov theory. We also discuss interacting fermions, in particular the issue of pseudogaps.     

Particles sliding on a fluctuating surface: phase separation and power laws

We study a system of hard-core particles sliding locally downwards on a fluctuating one-dimensional surface characterized by a dynamical exponent z and no overall tilt. In numerical simulations, an initially random particle density is found to coarsen and obey scaling with a growing length scale approximately t(1/z). The structure factor deviates from the Porod law for the models studied. The steady state is unusual in that the density-segregation order parameter shows strong fluctuations. The two-point correlation function has a scaling form with a cusp at small argument which we relate to a power law distribution of particle cluster sizes. Exact results on a related model of surface depths provide insight into this behavior.     

The Boltzmann equation for a polyatomic gas

A formulation of the kinetic theory of dilute, classical polyatomic gases is given which parallels the Waldmann development for structureless molecules. In the first section the Boltzmann equation is written in terms of the specific rates of inelastic collision processes and then the properties of these rates and those of the corresponding collision cross sections are examined. The dependence of the distribution function on the dynamical variables is discussed and the equations of change for the gas are derived. Finally, a study is made of the properties of the linearized Boltzmann collision operation. In the second section the Boltzmann equation is deduced from a rigorous statistical-mechanical point of view and discussed in terms of the basic ideas of Bogoliubov. The computationally important special case of impulsive interactions is then considered.This research was supported in part by a grant from the National Science Foundation and in part by the Ames Laboratory of the U. S. Atomic Energy Commission. Contribution No. 2554.     

Direct observation of the phonon energy in a Bose-Einstein condensate by tomographic imaging

The momentum and energy of phonons in a Bose-Einstein condensate are measured directly from a time-of-flight image by computerized tomography. We find that the same atoms that carry the momentum of the excitation also carry the excitation energy. The measured energy is in agreement with the Bogoliubov spectrum. Hydrodynamic simulations are performed which confirm our observation.     

Scattering of neutrons on a Bose condensate

The spectrum of noncondensate excitations in neutron scattering on bosons is obtained in the framework of the Bogoliubov models both for liquid ⁴He and a dilute gas. The problem is solved using a path-integral representation of the partition function of the system. We describe the influence of scattering of neutrons on a Bose condensate in a stationary (time-independent) picture in the Gibbs equilibrium ensemble. This influence is a stationary boson response, and it depends on the initial neutron momentum k, transfer momentum p, and the neutron-boson interaction λ, which is related to the scattering length. The contribution of the neutrons to the initial Bogoliubov spectrum is found to be important for “quasi-elastic” scattering on the noncondensate, while the contribution of deep inelastic scattering is small; no contribution from elastic scattering on the Bose condensate is found. In the case of liquid Helium, the response is unlikely to be observable for all values p. On the other hand, for a gas one may expect a visible effect, in particular for a small momentum transfer p and a small density of the Bose condensate ϱ.     

The Excitation Spectrum for Weakly Interacting Bosons

We investigate the low energy excitation spectrum of a Bose gas with weak, long range repulsive interactions. In particular, we prove that the Bogoliubov spectrum of elementary excitations with linear dispersion relation for small momentum becomes exact in the mean-field limit.     

Low energy theorems in a nonlocal chiral quark model at finite temperature

We solve the Dyson equation and the Bethe-Salpeter equation for a nonlocal effective quark interaction kernel which is instantaneous and separable. The momentum-dependent dynamical quark mass, the scalar and pseudoscalar meson masses, the pion decay constant and the quark meson coupling constant are calculated at finite temperature in the Hartree approximation for the quark self energy. We obtain relations between these quantities, which coincide to leading order in the current quark mass (m ₀/Δm) with the basic low energy theorems: the Goldstone theorem, the Gell-Mann-Oakes-Renner relation and the Goldberger-Treimann relation at finite temperature. A formula for the σ?π mass gap is obtained which exhibits an additional contribution from the momentum dependence of the quark mass.     

On the Brownian motion of a massive sphere suspended in a hard-sphere fluid. I. Multiple-time-scale analysis and microscopic expression for the friction coefficient

The Fokker-Planck equation governing the evolution of the distribution function of a massive Brownian hard sphere suspended in a fluid of much lighter spheres is derived from the exact hierarchy of kinetic equations for the total system via a multiple-time-scale analysis akin to a uniform expansion in powers of the square root of the mass ratio. The derivation leads to an exact expression for the friction coefficient which naturally splits into an Enskog contribution and a dynamical correction. The latter, which accounts for correlated collisions events, reduces to the integral of a time-displaced correlation function of dynamical variables linked to the collisional transfer of momentum between the infinitively heavy (i.e., immobile) Brownian sphere and the fluid particles.     

Electron-positron pair production in the low-density approximation

Electron–positron pair creation is studied in the low-density approximation by solving the quantum Vlasov equation exactly and the mapping equation approximately. The simpler mapping equation is an approximate treatment of the quantum Vlasov equation in which the continuous external field is regarded as a series of delta kicks. Our study indicates that this new treatment is appropriate because the results of the two methods are in good agreement with each other. However, as the period number increases, interference and a complicated structure in the momentum distribution are observed. Furthermore, we also obtain the square power law relation of the number density to the applied electric field strength.