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Conference on Lasers and Electro-Optics

Quasimomentum Distribution and Free Expansion of an Anyonic Gas

Tena Dubček, Bruno Klajn, Robert Pezer, Hrvoje Buljan, Dario Jukić

Published Jan 1, 2018• 0 citations
Publisher

Abstract

We point out that the momentum distribution is not a proper observable for anyons in two dimensions. We define the quasimomentum distribution as the asymptotic single-particle density after free expansion (time-of-flight measurements), as a means to measure the anyonic statistics [1].
DOI
10.1364/cleo_qels.2018.fth1h.7

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We point out that the momentum distribution is not a proper observable for a system of anyons in two dimensions. In view of anyons as Wilczek's composite charged flux tubes, this is a consequence of the fact that the orthogonal components of the kinetic momentum operator do not commute at the position of a flux tube, and thus cannot be diagonalized in the same basis. As a substitute for the momentum distribution of an anyonic (spatially localized) state, we propose to use the asymptotic single-particle density after the expansion of anyons in free space from the state. This definition is identical to the standard one when the statistical parameter approaches that for bosons or fermions. Exact examples which underpin our proposal are shown. They reveal that the quasimomentum distribution can be used to identify anyonic statistics in standard time-of-flight measurements.
We investigate the ground state of the one-dimensional interacting anyonic system based on the exact Bethe ansatz solution for arbitrary coupling constant $(0\ensuremath{\leqslant}c\ensuremath{\leqslant}\ensuremath{\infty})$ and statistics parameter $(0\ensuremath{\leqslant}\ensuremath{\kappa}\ensuremath{\leqslant}\ensuremath{\pi})$. It is shown that the density of state in quasimomentum kkk space and the ground-state energy are determined by the renormalized coupling constant ${c}^{\ensuremath{'}}$. The effect induced by the statistics parameter $\ensuremath{\kappa}$ exhibits in the momentum distribution in two aspects: Besides the effect of renormalized coupling, the anyonic statistics results in the nonsymmetric momentum distribution when the statistics parameter $\ensuremath{\kappa}$ deviates from 0 (Bose statistics) and $\ensuremath{\pi}$ (Fermi statistics) for any coupling constant ccc. The momentum distribution evolves from a Bose distribution to a Fermi one as $\ensuremath{\kappa}$ varies from 0 to $\ensuremath{\pi}$. The asymmetric momentum distribution comes from the contribution of the imaginary part of the nondiagonal element of reduced density matrix, which is an odd function of $\ensuremath{\kappa}$. The peak at positive momentum will shift to negative momentum if $\ensuremath{\kappa}$ is negative.
We present a systematic study of the Green functions of a one-dimensional gas of impenetrable anyons. We show that the one-particle density matrix is the determinant of a Toeplitz matrix whose large N asymptotic is given by the Fisher–Hartwig conjecture. We provide a careful numerical analysis of this determinant for general values of the anyonic parameter, showing in full details the crossover between bosons and fermions and the reorganization of the singularities of the momentum distribution function. We show that the one-particle density matrix satisfies a Painlevé VI differential equation that is then used to derive the small distance and large momentum expansions. We find that the first non-vanishing term in this expansion is always k −4 , that is proved to be true for all couplings in the Lieb–Liniger anyonic gas and that can be traced back to the presence of a delta function interaction in the Hamiltonian.
Free expansion following the removal of axial confinement represents a fundamental nonequilibrium scenario in the study of many-body ultracold gases. Using the stationary phase approximation, we analytically demonstrate that for all one-dimensional spinor gases with repulsive contact interactions, whether bosonic or fermionic, the asymptotic density and momentum distribution can be directly determined from the quasimomentum distribution (Bethe rapidities) of the trapped gas. We efficiently obtain the quasimomentum distribution numerically by solving the integral equations that characterize the ground state of the integrable system within the local density approximation. Additionally, we derive analytical solutions for both weakly and strongly interacting regimes. Unlike in bosonic gases, where rapidity distributions and density profiles vary significantly across interaction regimes, fermionic gases maintain similar profiles in both weakly and strongly interacting limits. Notably, the gas expands self-similarly only when released from a harmonic trap. For other power-law trapping potentials, the asymptotic density profile is strongly influenced by the initial confinement geometry. Our results extend readily to Bose-Fermi mixtures and finite temperatures.
We have obtained an expansion of the reduced density matrices (or, equivalently, correlation functions of the fields) of impenetrable one-dimensional anyons in terms of the reduced density matrices of fermions using the mapping between anyon and fermion wavefunctions. This is the generalization to anyonic statistics of the result obtained by Lenard (1966 J. Math. Phys. 7 1268) for bosons. In the case of impenetrable but otherwise free anyons with statistics parameter κ, the anyonic reduced density matrices in the grand canonical ensemble are expressed as Fredholm minors of the integral operator with the complex statistics-dependent coefficient γ = (1 + e ±iπκ )/π. For κ = 0 we recover the bosonic case of Lenard γ = 2/π. Due to nonconservation of parity, the anyonic field correlators ⟨Ψ† A ( x ')Ψ A ( x )⟩ are different depending on the sign of x ' − x .