Conclusions

This paper presents a thorough study of correlation properties of a one-dimensional gas of bosons at zero temperature. In a homogeneous system the behavior is fixed by the product of linear density $n_{1D}$ and one-dimensional scattering length $a_{1D}$. In the strongly interacting regime $n_{1D}a_{1D}\ll 1$ the bosonic system behaves effectively as a system of non interacting fermions. In this limit the energy, pair distribution function $g_2(z)$, static structure factor $S_k$ are known explicitly and are same as the ones of the corresponding fermionic system. For arbitrary value of the gas parameter no complete description was known so far. Switching on an external harmonic potential leads to modification in properties as new length, the oscillator length $a_z$ is introduced.

Quasi one-dimensional systems have been already realized in a number of experiments with elongated traps. Many new experiments with condensates in a same geometry, in a waveguide or on a chip are expected to appear. The characteristic parameter $n_{1D}a_{1D}$ can be varied by changing number of atoms in the condensate, trapping frequencies or by adjusting the scattering length using the Feshbach resonance. Momentum distribution is accessible from ballistic expansion and static factor can be measured by the Bragg scattering.

We find for the first time full description of the correlation functions in a wide range of the characteristic parameter $n_{1D}a_{1D}$ starting from Tonks-Girardeau regime and up to Gross-Pitaevskii regime. We benchmark our Diffusion Monte Carlo calculations by recovering the ground state energy known from solution of the Lieb-Liniger integral equations. We completely recover all properties of the Tonks-Girardeau gas and known asymptotic behavior of the momentum distribution and correlation functions. We obtain the one-body density matrix $g_1(z)$ and pair distribution function $g_2(z)$ for all densities. In particular we have the description of the most nontrivial regime $n_{1D}a_{1D}\approx 1$ which is relevant for current experiments.

We study the dependence of the value at zero of the three-body correlation function $g_3(0)$ on the density $n_{1D}a_{1D}$. This function is of a great interest as it governs the three-body recombination rate, which leads to loss of the atoms out of the condensate. The data of an experimental measurement of $g_3(0)$ is available [TOH+04] and is compared with the predictions of the Lieb-Liniger theory. An agreement between theory and experiment is found.

By the means of Fourier transformation we extract the momentum distribution $n(k)$ and static structure factor $S(k)$. Low momentum part is described by phonon hydrodynamic theory which is expected to be applicable at distances $\vert z\vert$ larger than the healing length $\xi$. We judge that $n(k)$ shows phononic power-law divergence for values of $\k$ considerably smaller than $1/\xi$.

Finally we discuss how the presence of a harmonic trapping modifies the correlation functions. We plot the pair distribution function in typical experimental configurations. We discuss possibility of finding in $n(k)$ traces of divergent behavior, which is characteristic for a one-dimensional infinite system, in a trapped system.