Introduction

This Chapter is intended to introduce tools for the subsequent Chapters. Here we define the quantities (correlation functions, static structure factor, etc.) that later are used to describe the properties of quantum systems. We explain the analytical methods (Gross-Pitaevskii approach) and approximations (local density approximation, pseudopotential approximation) used in our study. We review the 2-body scattering problem in three- and one-dimensional systems as it gives insight into the many-body physics and is relevant for the implementation of the Monte Carlo techniques. Most of the content of the Chapter is standart and is presented for the completeness of the discussion. Only in several sections some new results are obtained (Secs. 1.6,1.7).

The structure of the Chapter is as follows.

In Section 1.2 we introduce quantities which characterize a quantum system and can be accessed in experiments. We start by considering the representations of the first and second quantization (Secs. 1.2.1, 1.2.2). A special attention is paid to the relation between mean averages and correlation functions. The calculation of the correlation functions can be largely simplified in a homogeneous system (Sec. 1.2.3), although the case of trapped systems is also considered (Sec. 1.2.5). The momentum distribution and static structure factor are introduced in Sec. 1.2.4.

The scattering theory is addressed in Section 1.3. The scope of our study is general and we consider the theory in a three-dimensional system (Sec. 1.3.2), as well as in a one-dimensional system (Sec. 1.3.3). The scattering problem is solved for a number of potentials that appear in different models. The scattering solutions are used to construct the trial wave function (see Chapter 2) and make comparison of $N$-body and 2-body physics (see Chapter 5). We discuss scattering on a $\delta$-potential (pseudopotential) in a one dimensional system (Sec. 1.3.3.2), where the problem is well posed, and also in a three dimensional system (Sec. 1.3.4.1), where instead a regularization procedure is needed. We relate the coupling constant to the $s$-wave scattering length (in 1D and 3D) for the scattering on the pseudopotential, which is a highly important theoretical tool widely used throughout this dissertation. In the conclusion of Section 1.3 we consider the case of resonant scattering, when the scattering length can be much larger than the range of the potential.

A dilute quantum system of repulsive bosons shows very peculiar properties in 1D. Fermionization of the bosonic system takes place (Tonks-Girardeau gas[Gir60]), and the particles behave as if they were ideal fermions. We address some of the properties of an ideal Fermi gas in Section. 1.4. The Fermi momentum and Fermi energy of an ideal 1D Fermi gas (Sec. 1.4.1) provide an important physical scale not only in the TG regime but in the whole range of densities. The ground-state energy of a gas of impenetrable particles (hard-rod gas) is calculated in Sec. 1.4.2. The properties of a gas of hard-rods are important in the proposed relation of such a system to a short-range attractive potential (super-Tonks) gas (see Chapter 6). Also the HR gas equation of state is related to the expansion of the energy of a Lieb-Liniger gas in the regime of strong correlations and this expansion is relevant for the estimation of the properties of correlation functions in this regime.

In this dissertation the Monte Carlo results are systematically compared to the predictions of the mean-field Gross-Pitaevskii approach (when GP equation is applicable). In Section 1.5.1 the GP equation is derived from a energy functional, which later is used to study the properties of a condensate disturbed by an impurity (see Chapter 7). In a similar way the GP equations in restricted geometries (cigar- and disk- shaped condensates) are derived in Sec. 1.5.2. In this approach virtual excitations in the tight direction are neglected and the resulting expression of the coupling constants is to be compared with the one of an exact solution of a two-body scattering problem in 1D obtained by Olshanii [Ols98].

If the equation of state of the homogeneous system is known, the local density approximation allows one to estimate the properties of a system in the presence of an external confinement. The general idea of this method is explained in Sec. 1.6.1 and the characteristic parameters in three- and one-dimensional systems are discussed. We propose an exact solutions of the LDA problem for a ``perturbative'' equation of state both in one dimension (Sec. 1.6.2) and in three dimensions (Sec. 1.6.3). The obtained formulas are applied to bosonic systems (see Chapter 6) as well as fermionic systems (see Chapter 8). In particular the LDA method together with the sum rule approach (in 1D) and scaling approach (in 1D) can be used to estimate the frequencies of collective excitations. Expansions for those frequencies are obtained and later are compared to the result of the numerical results obtained using LDA (see Figs. 6.4, 8.4). Finally in Section 1.6.4 we consider the LDA applied to the Tonks-Girardeau gas and calculate the static structure factor in a trapped system.

This introductory Chapter is concluded with a newly proposed derivation of the dynamic form factor, pair distribution function and the one-body density matrix of a weakly interacting bosonic gas in 1D. The Haldane description [Hal81] of this system is corrected in order to replace the phononic excitation spectrum with the more precise Bogoliubov spectrum. In this way we eliminate logarithmic divergences present in the problem and estimate the prefactors of the long-range asymptotics. In particular the coefficient of the decay of the OBDM (Sec. 1.7.5) is compared to the exact DMC result (see Sec. 5) and is found to be extremely accurate (less than $0.3\%$ error).