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Introduction

Recent progress achieved in techniques of confining Bose condensates has lead to experimental realizations of quasi-one dimensional systems[GVL+01,SKC+01,DHR+01,RGT+03,MSKE03,TOH+04]. The 1D regime is reached in highly anisotropic traps, where the axial motion of the atoms is weakly confined while the radial motion is frozen to zero point oscillations by the tight transverse confinement. Another possible realization of a quasi one-dimensional system could be a cold gas on a chip. The possibility of an experimental observation revived interest in analytical description of the properties of a one-dimensional bose gas. To first approximation one parameter, the one-dimensional scattering length $a_{1D}$, is sufficient to describe the interatomic potential, which in this case can be modeled by repulsive $\delta $-function pseudopotential. Many properties of this integrable model like the ground state energy[LL63], excitation spectrum[Lie63], thermodynamic functions at finite temperature[YY69] were obtained exactly already in 60-ies using the Bethe ansatz method. Gaudin in his book [Gau83] devoted to this powerful method writes that so far almost nothing is known about the correlation functions, apart from the case of impenetrable bosons [Len64,VT79,JMMS80]. Lately active work was carried out in this direction, there are recent calculations of short-range expansion of the one-body density matrix[OD03], the value at zero of the two-body correlation function[GS03b]. Still there are no exact calculations of the correlation functions present in the literature.

We use Diffusion Monte Carlo method (Sec. 2.3), which is exact apart from the statistical uncertainty, in order to address the problem of $\delta $-interacting bosons in the ground state of a homogeneous system. We argue that the trial wave function we propose provides a very good description of the ground state wave function. As a benchmark test for our DMC calculation we recover the equation of state which is known exactly[LL63]. For the first time we find complete description of the one-body density matrix and pair distribution function. We show that our results are in agreement with known analytical predictions. We calculate momentum distribution and static structure factor which are accessible in an experiment. We calculate exactly the value of three-body correlation function which is very important quantity as it governs rates of inelastic processes. Also we address effects of the external confinement.

We present study of the correlation functions of a homogeneous system. We find the one-body density matrix at all densities. We calculate the pair-distribution function. Fourier transformation relate those quantities to the momentum distribution and the static structure factor, which are accessible expirementaly. We calculate a momentum distribution in a trapped system. We expand our prediction on homogeneous three-body correlation function, relevant for the estimation of three-body collision rate, pair-distribution function and static structure factor in traps. We discuss in details known analytical limits and propose a precise expression for the decay coefficient of the one-body density matrix. We provide relevant details of our Monte Carlo study. In particular it is argued that the trial wave function we construct provides good description and variational energy is only slightly higher than the exact one.

The structure of this chapter is as follows. In section 5.2 we discuss a model used to describe a cold one-dimensional gas and make a summary of known analytical expressions for correlation functions. Section 5.3 is devoted to a brief description of the Monte Carlo scheme used for numerical solution of the Schrödinger equation and investigation of a finite size errors that is relevant for infinite system simulation. The trial wave function used for the importance sampling is discussed in detail. We present the result for a homogeneous system in section 5.4. One-body density matrix, pair distribution function and three-body correlation function are calculated and compared to known exact results and analytical expansions. The information about the momentum distribution and static structure factor is extracted by means of the Fourier transformation. In section 5.5 we discuss effects of the external trapping. Modifications to the construction of the trial wave function are discussed. The results for the pair distribution function and momentum distribution are presented. Finally, in Section 5.6 we draw our conclusions.


next up previous contents
Next: Lieb-Liniger Hamiltonian Up: Ground state properties of Previous: Ground state properties of   Contents
G.E. Astrakharchik 15-th of December 2004