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 , is sufficient to
describe the interatomic potential, which in this case can be modeled by repulsive
-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 -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.