Using a scaling transformation we exactly determine the dynamics of an harmonically confined Tonks-Girardeau gas under arbitrary time variations of the trap frequency. We show how during a one-dimensional expansion a "dynamical fermionization" occurs as the momentum distribution rapidly approaches an ideal Fermi gas distribution, and that under a sudden change of the trap frequency the gas undergoes undamped breathing oscillations displaying alternating bosonic and fermionic character in momentum space. The absence of damping in the oscillations is a peculiarity of the truly Tonks regime.One of the most challenging and interesting problems in quantum dynamics involves understanding a temporal behavior of strongly correlated many-body systems beyond the linear-response regime or the adiabatic approximation. Aside from fundamental interest, this issue is of primary importance for current experiments with ultracold atomic gases and their potential applications to quantum information, where control and manipulation of entangled states of many-particle quantum systems is required.The one-dimensional (1D) gas of impenetrable bosons (Tonks-Girardeau gas) corresponds to the limit of infinitely strong repulsive interactions in the Lieb-Liniger (LL) model of 1D bosons interacting through a contact pair potential [1]. It was shown in Ref.[2] that the Tonks-Girardeau limit is applicable for describing the low-density regime of bosonic atomic gases in a quasi-1D geometry. The physical properties of 1D bosons in this limit can be investigated in detail since their wavefunction is known explicitly in terms of the one of a noninteracting fermions in the same external potential [3]. In fact, the density profile, the thermodynamic properties, the collective excitation spectrum and the density correlation functions coincide with those of an ideal Fermi gas, leading to interesting manifestations of fermionization of a Bose gas, such as broadening of the density profiles [4], an increase of the frequency of collective excitations [5] and a dramatic reduction of the three-body recombination rate [6]. However the one-body density matrix, and consequently the momentum distribution, differs considerably from that of a Fermi gas, due to the phase correlations stemming from the bosonic statistics of the Tonks gas. For the homogeneous case the momentum distribution n(p) at the origin has a 1/ √ p peak [7] and for a harmonically trapped gas the population of the lowest single-particle state scales as √ N with N being the particle number [8,9]; this shows that due to the strong interactions the bosons do not form a Bose-Einstein condensate. In both cases at large momenta p >hn, with n being the density at the center, the momentum distribution shows characteristic slowly decaying tails n(p) ∼ p −4 [10].Experiments on cold atomic gases under optical con-finement in a quasi-one dimensional geometry are now starting to explore the strongly interacting regime, demonstrated by the examination of its correlation properties [11] and of the frequency ...
A Fermi-Bose mapping method is used to determine the exact ground states of several models of mixtures of strongly interacting ultracold gases in tight waveguides, which are generalizations of the Tonks-Girardeau (TG) gas (1D Bose gas with point hard cores) and fermionic Tonks-Girardeau (FTG) gas (1D spin-aligned Fermi gas with infinitely strong zero-range attractions). We detail the case of a Bose-Fermi mixture with TG boson-boson (BB) and boson-fermion (BF) interactions. Exact results are given for density profiles in a harmonic trap, single-particle density matrices, momentum distributions, and density-density correlations. Since the ground state is highly degenerate, we analyze the splitting of the ground manifold for large but finite BB and BF repulsions.
We evaluate the small-amplitude excitations of a spin-polarized vapour of Fermi atoms confined inside a harmonic trap. The dispersion law ω = ω f [l + 4n(n + l + 2)/3] 1/2 is obtained for the vapour in the collisional regime inside a spherical trap of frequency ω f , with n the number of radial nodes and l the orbital angular momentum. The low-energy excitations are also treated in the case of an axially symmetric harmonic confinement. The collisionless regime is discussed with main reference to a Landau-Boltzmann equation for the Wigner distribution function: this equation is solved within a variational approach allowing an account for non-linearities. A comparative discussion of the eigenmodes of oscillation for confined Fermi and Bose vapours is presented in an Appendix.
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