No abstract

The collective excitations of Bose condensates in anisotropic axially symmetric harmonic traps are investigated in the hydrodynamic and Thomas-Fermi limit. We identify an additional conserved quantity, besides the axial angular momentum and the total energy, and separate the wave equation in elliptic coordinates. The solution is thereby reduced to the algebraic problem of diagonalizing finite dimensional matrices. The classical quasi-particle dynamics in the local density approximation for energies of the order of the chemical potential is shown to be chaotic. 03.75.Fi,05.30.Jp,32.80.Pj,67.90.+z The new Bose condensates of alkali atoms in magnetic traps [1][2][3] offer a unique way to investigate the low-lying collective excitations in Bose condensates [4][5][6][7][8][9][10][11][12][13][14][15][16]. Experimentally collective modes with a given symmetry have been excited by time-dependent modulations of the trapping potential, and their evolution has been followed in real time by measurements of the resulting shape oscillations of the condensates. The measurements performed so far have involved turning off the trap after a given time [4][5][6], but in the future they could even be performed non-destructively by elastic off-resonant light scattering [7]. Theoretically the collective modes have been analyzed by using the Bogoliubov-equations or by linearizing the time-dependent Gross-Pitaevskii equation around the time-independent condensate and solving these equations numerically [8,9] or analytically in various approximations [10][11][12][13][14][15][16]. Very good agreement between the numerical and the experimental results has been found.In a seminal paper Stringari [16] has shown how the coupled wave equations for the collective excitations are simplified in the hydrodynamic limit to become a single second-order wave equation for density waves, and he obtained analytical solutions for all its modes in spherically symmetric harmonic traps, and, remarkably, also for some of its modes in axially symmetric harmonic traps. The latter are particularly important, because all experiments have been performed with traps of this symmetry [4][5][6].In the present paper it is our goal to study in more detail by analytical means the hydrodynamic wave equation in the axially symmetric case. We wish to find an explanation why at least some analytical solutions have been possible in this case and intend to use this insight to construct more solutions in a systematic way.In principle the collective mode problem looks very different for isotropic and for axially symmetric traps: In the isotropic case the rotational symmetry ensures that angular momentum conservation gives two good quantum numbers, and therefore the wave equation is separable in spherical coordinates. For axial symmetry, however, only the axial component of angular momentum remains a good quantum number, besides the energy, and one may expect that the system, having three degrees of freedom, is not integrable. In fact, this expectation is born out for collect...

average energy of the electrons, calculated from the Doppler broadening of the / 2 >/ 1 peak (4.35 keV FWHM) compared to the 569.7-keV peak from Bi 207 (353 keV FWHM), is 6.3 eV. That of the I^>1 2 peak (3.98 keV FWHM) is 3.3 eV.If the lifetime spectra are resolved into three components, we obtain the following results:(1) The long-lifetime component (7^ = 2.01 ±0.04 nsec, -20%) has a broad angular distribution.(2) The intermediate-lifetime component (TJ = 0.64 ±0.10 nsec, -13%) has an angular distribution narrower than that of the long-lifetime component. It is believed that this component is due to free-positron annihilations with outer atomic electrons. 6 The extra momentum contribution in the long-lifetime component may be due to the orbital momentum of the positron bound in positronium.(3) The short-lifetime component (TQ = 0.33 ±0.02 nsec, -67%) has a complex origin; besides a narrow component due to the annihila-The purpose of this note is to point out some dynamical consequences of the absence of any characteristic length beyond the atomic dimensions in an extended homogeneous system at its phase transition. This similarity property holds neither below nor above the transition temperature, where a temperature-dependent correlation length, which becomes much greater than atomic dimensions, is manifested. It is precisely at the transition temperature that this length becomes infinite and is no longer relevant as a characteristic unit. The similarity property then provides a useful means of connecting the critical behavior of the system above the transition with that below. In this way we predict an anomalous dispersion of sec-tion of singlet positronium, there is some other decay mechanism with large momentum but short lifetime. . ond sound at the liquid-helium lambda temperature, T^9 of the form wcck 3/2 (where co and k are the frequency and wave number, respectively), and a relation between second-sound damping and the heat conductivity, for temperatures T below and above T^, respectively. Both of these quantities are predicted to vary as IT-T^I ~1 /3 , and recent experimental data are consistent with this singular behavior. 1 ? 2 A complete description of the hydrodynamics of the super fluid helium at long wavelengths entails the kinetically conjugate variables of quantum mechanical phase and mass density for the superfluid, and the corresponding variables of normal-fluid velocity and entropy density for the normal fluid. But because of the DISPERSION INThe absence of a characteristic length at the lambda temperature T^ of liquid helium is used to determine the wave-number dependence of the phase fluctuations and the second-sound dispersion relation oo =ak z/2 where co and £ are the frequency and wave number and «« 0.1 cm 3/2 sec -1 . Further predictions are | T-T\| ~1 /3 singular temperature variation for second-sound damping {T T^). 891

The properties of Green's functions and various correlation functions of density and spin operators are considered in a homogeneous spin-1 Bose gas in different phases. The dielectric formalism is worked out and the partial coincidence of the one-particle and collective spectra is pointed out below the temperature of Bose-Einstein condensation. As an application the formalism is used to give two approximations for the propagators and the correlation functions and the spectra of excitations including shifts and widths due to the thermal cloud.