Bogoliubov Hamiltonian Research Articles

Classical quasiparticle dynamics in trapped Bose condensates

The dynamics of quasiparticles in repulsive Bose condensates in a harmonic trap is studied in the classical limit. In isotropic traps the classical motion is integrable and separable in spherical coordinates. In anisotropic traps the classical dynamics is found, in general, to be nonintegrable. For quasiparticle energies $E$ much smaller than the chemical potential $\ensuremath{\mu}$, besides the conserved quasiparticle energy, we identify two additional nearly conserved phase-space functions. These render the dynamics inside the condensate (collective dynamics) integrable asymptotically for $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$. However, there coexists at the same energy a dynamics confined to the surface of the condensate, which is governed by a classical Hartree-Fock Hamiltonian. We find that also this dynamics becomes integrable for $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$ because of the appearance of an adiabatic invariant. For $E/\ensuremath{\mu}$ of order 1 a large portion of the phase-space supports chaotic motion, both for the Bogoliubov Hamiltonian and its Hartree-Fock approximant. To exemplify this we exhibit Poincar\'e surface of sections for harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps. For $E/\ensuremath{\mu}\ensuremath{\gg}1$ the dynamics is again governed by the Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes quasiintegrable because the remaining small chaotic components in phase space are tightly confined by tori.

The Density of States for Fermi Superfluids with Layered Structures in the Order Parameter: With Application to a Solution 3He in 4He at Low Temperatures

The density of states of Fermionic excitations in s-wave Fermi superfluids is calculated. The order parameter is assumed to have a layered structure with a variation along only a single direction and the Fermions are allowed to move over the full three-dimensional space. The density of states for such a system is calculated using the Bogoliubov Hamiltonian in which the quadratic momentum dependence of the kinetic energy is approximated by a linear dependence. A representation of the density of states in terms of a Fredholm determinant is derived. For the case of a piecewise constant order parameter, the Fredholm determinant is calculated exactly. Explicit results for the density of states are presented for an order parameter with a periodic variation along a single direction.

The Low-Energy, Local Density of States of an Isolated Vortex in an Extreme Type II Superconductor

The local density of states of low energy electronic excitations in a vortex is investigated. Starting from the Bogoliubov Hamiltonian, we make the approximation of replacing the quadratic momentum dependence of the free particle kinetic energy by a linear dependence. We then work with a linearized order parameter profile of the vortex Δ(r); Δ(r) ≍ Δ′(0)r, where r is the distance from the vortex axis. The resulting theory has a simple operator structure and is exactly diagonalizable. It leads to a sequence of "oscillator-like" excitations in the core of the vortex. The low energy behaviour is contained in the zeroth oscillator level and when the excitation energy E and vortex profile obey Δ(r)2 − E2 ⪢ vF Δ′(0), where vF is the Fermi velocity, we find that the leading behaviour in the local density of states is of the form (Δ(r)2 − E2)−1/2 Θ(Δ(r)2 − E2) with Θ(x) a Heaviside step function. This "edge" behaviour is clearly discernible in numerical calculations of the local density of states.

On the Validity of Collective Variable Description of Bose Systems

The validity of Sunakawa, Yamasaki and Kebukawa's Hamiltonian and that of Bogoliubov and Zubarev's Hamiltonian are examined. Perturbational expansion of the ground state energy by these Hamiltonians disagrees with the exact solution of Lieb and Liniger for one-dimensional Bose system with repulsive delta-function interaction. This fact suggests that these Hamiltonians are not microscopic descriptions of the many-Boson system. Mathematical inconsistency in Bogoliubov and Zubarev's theory is also pointed out. Moreover analytic expression of high density expansion for the ground state energy density e0 is found out to be e0n-3=γ-(4/3π)γ3/2+(1/6-1/π2)γ2+O(γ5/2), γ≡c/n, for one-dimensional Bose system with delta function interaction (density n, strength 2c, ħ=2m=1) by the use of the correlated basis function method.