Fermi-Bose mapping for one-dimensional Bose gases

Abstract

One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schr\"odinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.Comment: Brief review, Latex file, 15 page

Cold bosons in optical lattices

“…For dilute gases, when the interaction radius is much shorter than the mean interatomic distance [7][8][9][10][11][12][13][14], one uses the local interaction potential …”

Exact Kohn–Sham potential of strongly correlated finite systems

“…This leads us to the last simplifying assumption, namely the limit of infinitely strong δ-repulsion, λ → ∞. Now the problem becomes immediately solvable by the so called Girardeau mapping [16] (see also a more recent review [17]), which allows to map the ground state of strongly interacting "hardcore" bosons (a symmetric wave function) to the ground state of noninteracting fermions (antisymmetric wave function). In our two-particle case the exact ground-state (singlet, i.e.…”

Bose systems in spatially random or time-varying potentials

“…Passing to the momentum representation, we accomplish in Hamiltonian (2) the Bogolubov shift (6) and substitute there the Fourier transforms (17) and (35). As a result, the grand Hamiltonian (10) acquires the form…”

“…At the beginning, the interest was concentrated on the behavior of liquid helium in nanoporous media [1]. In recent years, the physics of dilute Bose gases has gained much interest [2][3][4][5][6][7][8]. Quasidisordered Bose gases have been realized experimentally by creating quasiperiodic optical lattices [9][10][11].…”