Free expansion of a Lieb-Liniger gas: Asymptotic form of the wave functions

Abstract

The asymptotic form of the wave functions describing a freely expanding Lieb-Liniger gas is derived by using a Fermi-Bose transformation for time-dependent states, and the stationary phase approximation. We find that asymptotically the wave functions approach the Tonks-Girardeau (TG) structure as they vanish when any two of the particle coordinates coincide. We point out that the properties of these asymptotic states can significantly differ from the properties of a TG gas in a ground state of an external potential. The dependence of the asymptotic wave function on the initial state is discussed. The analysis encompasses a large class of initial conditions, including the ground states of a Lieb-Liniger gas in physically realistic external potentials. It is also demonstrated that the interaction energy asymptotically decays as a universal power law with time, Eint ∝ t −3 .

Single-particle density matrix for a time-dependent strongly interacting one-dimensional Bose gas

“…This work complements the studies of nonequilibrium dynamics of interacting Bose gases which have been addressed by use of the LL model away from [18,19,20,32] and in the TG limit [35,36,43,44,45,46,47,48,49]. The phenomenological relevance of these studies is underlined by the fact that nonequilibrium dynamics is accessible experimentally [7,8].…”

“…Under these circumstances, the time-dependent LL model (1) can be solved by employing the Fermi-Bose transformation [17,19]. This method can be applied, e.g., to exactly study free expansion from an initially localized state [20]. If ψ F (x 1 , .…”

“…[17] (see also Refs. [19,20]). Here, N (c) is a normalization constant, and the differential operator in the square brackets denotes the Fermi-Bose mapping operator [19].…”

“…Even though exact LL many-body wave functions can * Electronic address: rpezer@phy.hr be constructed in some cases (e.g., stationary [1,10,11,12,13,14,15,16] or time-dependent wave functions [17,18,19,20]), the calculation of observables (correlation functions) from such solutions usually poses a major difficulty in practice [21,22,23,24,25,26,27,28,29,30,31,32]. Various methods have been employed to overcome this difficulty including, for example, the quantum inverse scattering method (e.g., see [21,29]) and quantum Monte Carlo integration [25].…”

“…The method is derived by using the 1/c term of the so-called FermiBose (FB) mapping operator introduced by Gaudin [17]. The FB operator method provides us with exact timedependent solutions of a LL model [17,19] in the absence of external potentials and other boundary conditions; it was recently used to study free expansion of a LL gas [20]. We derive an efficient numerical algorithm for calculating the RSPDM for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential.…”