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

arXiv:0907.4608v1 [cond-mat.quant-gas] 27 Jul 2009 Single-particle density matrix for a time-dependent strongly interacting one-dimensional Bose gas R. Pezer∗ Faculty of Metallurgy, University of Zagreb, 44103 Sisak, Croatia T. Gasenzer Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany H. Buljan Department of Physics, University of Zagreb, PP 332, Zagreb, Croatia (Dated: June 8, 2018) We derive a 1/c-expansion for the single-particle density matrix of a strongly interacting time- dependent one-dimensional Bose gas, described by the Lieb-Liniger model (c denotes the strength of the interaction). The formalism is derived by expanding Gaudin’s Fermi-Bose mapping operator up to 1/c-terms. We derive an efficient numerical algorithm for calculating the density matrix for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential. We have applied the formalism to study contraction dynamics of a localized wave packet upon which a parabolic phase is imprinted initially. PACS numbers: 05.30.-d,03.75.Kk I. INTRODUCTION One of the most attractive many-body quantum sys- tems nowadays has been introduced by Lieb and Lin- iger in their landmark paper more than forty years ago [1]. The system is composed of N identical δ-interacting bosons in one spatial dimension and is referred to as a Lieb-Liniger (LL) gas. They have presented an explicit form of the many-body wave function for a homogeneous gas with periodic boundary conditions [1], including equations describing the ground state and the excitation spectrum. In the strongly interacting ”impenetrable- core” regime [2], such a one dimensional (1D) system is referred to as the Tonks-Girardeau (TG) gas; exact so- lutions in this limit are obtained by Girardeau’s Fermi- Bose mapping [2]. Following Ref. [1], Yang and Yang [3] have eliminated a possible existence of phase tran- sitions in the LL system by proving analyticity of the partition function. After many recent experimental suc- cesses [4, 5, 6, 7, 8, 9] in realization of effectively one dimensional (1D) interacting gases, from the weak up to the strongly interacting TG regime [5], the LL model has attracted considerable attention of the physics commu- nity. There is a clear reason for this; nontrivial quantum many-body systems are notoriously oblique to a quan- titative analysis, and therefore possibility of an exact treatment in particular cases, together with experimental realization, is of great value. Moreover, exact solutions can be useful as a benchmark for approximate treatments aiming to describe a broader range of physical systems. 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 (correla- tion 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 over- come this difficulty including, for example, the quantum inverse scattering method (e.g., see [21, 29]) and quan- tum Monte Carlo integration [25]. A recent discussion of several exact methods for the calculation of correlation functions of a nonequilibrium 1D Bose gas can be found in Ref. [32]. In the TG limit, the momentum distribution can be analytically studied for a ring geometry, and also for harmonic confinement (e.g., see [33, 34]). Numerical methods for the calculation of the reduced single-particle density matrix (RSPDM) can be performed efficiently for various TG states (ground state, excited and time- dependent states, see Ref. [35] for hard-core bosons on the lattice, and Ref. [36] for the continuous TG model [2]). Ultracold atoms in 1D atomic wave guides enter the strongly interacting regime at low temperatures, in tight transverse confinement, and with strong effective interac- tions [37, 38, 39]. The correlations functions of a LL gas can in this limit be calculated by using 1/c expansions (e.g., see [22, 24, 27]) from the TG (c →∞) regime. These calculations in the strongly interacting limit ex- ploit the fact that a bosonic LL gas is dual to a fermionic system [40], such that weakly interacting fermions cor- respond to strongly interacting bosons and vice versa [40, 41]. A strongly interacting 1D Bose gas was stud- ied in Ref. [42] by using perturbation theory for the dual fermionic system. In Ref. [27], the dynamic structure factor was calculated for zero and finite temperatures. Here we calculate the 1/c correction for the RSPDM of a Lieb-Liniger gas, which adds upon a recently obtained 2 formula for the RSPDM of a TG gas [36]. The method is derived by using the 1/c term of the so-called Fermi- Bose (FB) mapping operator introduced by Gaudin [17]. The FB operator method provides us with exact time- dependent solut

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