Many-body excitations in trapped Bose gas: A non-Hermitian view

Abstract

We provide the analysis of a physically inspired model for a trapped dilute Bose gas with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). We formally construct a many-body Hamiltonian, ${\mathcal{H}}_{\text{app}}$, that is quadratic in the Boson field operators for noncondensate atoms. This ${\mathcal{H}}_{\text{app}}$ conserves the total number of atoms. Inspired by Wu (J. Math. Phys., 2:105-123, 1961), we apply a non-unitary transformation to ${\mathcal{H}}_{\text{app}}$. Key in this non-Hermitian view is the pair-excitation kernel, which in operator form obeys a Riccati equation. In the stationary case, we develop an existence theory for solutions to this operator equation by a variational approach. We connect this theory to the one-particle excitation wave functions heuristically derived by Fetter (Ann. Phys., 70:67-101, 1972). These functions solve an eigenvalue problem for a $J$-self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter's excitation spectrum. Our approach leads to a description of the excited eigenstates of the reduced Hamiltonian in the $N$-particle sector of Fock space.

arXiv comment38 pages

References in corpus (16)

• Theory of Bose-Einstein condensation in trapped gases
• Feshbach Resonances in Ultracold Gases
• Topological Bands for Ultracold Atoms
• Conserving and Gapless Approximations for an Inhomogeneous Bose Gas at Finite Temperatures
• Cold hybrid ion-atom systems
• A particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas
• Second-order corrections to mean field evolution for weakly interacting Bosons. I
• Ground State and Excited States of a Confined Bose Gas
• Pair excitations and the mean field approximation of interacting Bosons, I
• Observation of Strong Quantum Depletion in a Gaseous Bose-Einstein Condensate
• Phase Dependence of Phonon Tunneling in Bosonic Superfluid-Insulator-Superfluid Junctions
• Beyond mean field: On the role of pair excitations in the evolution of condensates
• Bounds on variation of spectral subspaces under J-self-adjoint perturbations
• Higher order corrections to the mean-field description of the dynamics of interacting bosons
• Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator
• The Excitation Spectrum of the Bose Gas in the Gross-Pitaevskii Regime