This work is about the properties of several low dimensional, small systems of interacting particles. We demonstrate that interaction between particles in the low dimensional small systems can lead to many unexpected effects. We considered electrons in a Luttinger liquid, in a superconducting state, and atoms in a magneto-optical trap. Using bosonization techniques we calculated the thermopower of a Luttinger liquid wire with an impurity. We predicted the appearance of a phase dependent force and resonant phase dependent magnetization in the nanoscopic superconductor - normal metal superconductor (or superconductor - two dimensional electron gas - superconductor) junction. We also considered plasma oscillations inside thin superconducting tubes and rings and predicted that the velocities of the plasmons in these systems are periodic functions of the magnetic flux. By considering neutral atoms in a harmonic trap we discovered that strongly repelling atoms do not form Bose-Einstein condensate at zero temperature but tend to occupy different orbitals with small mutual overlap, forming crystallite structures similar to Wigner molecules of electrons inside a quantum dot.
This work is about the properties of several low dimensional, small systems of interacting particles. We demonstrate that interaction between particles in the low dimensional small systems can lead to many unexpected effects. We considered electrons in a Luttinger liquid, in a superconducting state, and atoms in a magneto-optical trap. Using bosonization techniques we calculated the thermopower of a Luttinger liquid wire with an impurity. We predicted the appearance of a phase dependent force and resonant phase dependent magnetization in the nanoscopic superconductor - normal metal superconductor (or superconductor - two dimensional electron gas - superconductor) junction. We also considered plasma oscillations inside thin superconducting tubes and rings and predicted that the velocities of the plasmons in these systems are periodic functions of the magnetic flux. By considering neutral atoms in a harmonic trap we discovered that strongly repelling atoms do not form Bose-Einstein condensate at zero temperature but tend to occupy different orbitals with small mutual overlap, forming crystallite structures similar to Wigner molecules of electrons inside a quantum dot.
Strongly correlated systems show a variety of intriguing properties and have been extensively studied over the years. In this thesis two strongly correlated systems are discussed: the two-dimensional antiferromagnet and the Luttinger liquid. The Luttinger liquid has attracted much interest as a paradigm of non-Fermi liquid behavior. In the first part of the thesis, basic properties of Fermi liquids are briefly described while those of Luttinger liquids are considered in more detail. The differences in spectral properties of Luttinger liquids and Fermi liquids are discussed, as are the consequences for photoemission experiments. In this context Luttinger liquids with boundaries, treated in Papers I, II, and III, are introduced. The exact single-electron Green's function at finite temperature is calculated with bosonization methods, for both periodic and finite open-boundary systems. The corresponding local spectral density is analyzed. For a system with open boundaries, the well-known zero temperature bulk behavior always crosses over to a boundary dominated regime for small energies, while at finite temperature the low energy behavior becomes quadratic. For a periodic finite-size system we also calculate the Coulomb-blockade-like oscillations and study their dependence on interaction and temperature. The two-dimensional antiferromagnet has attracted much attention due to its connection to the undoped phase of the ceramic high-temperature superconductors. In the second part of the thesis including Papers IV and V, the two-dimensional Heisenberg antiferromagnet defined on the triangular and the honeycomb lattices is studied. We use the Schwinger-boson mean-field theory to investigate the stability of the ground state with respect to frustration. For the honeycomb lattice model we find an increased Néel stability as compared to the classical case. The dynamic structure factor is considered for various temperatures and the spin-wave velocity and the susceptibility are presented as functions of frustration for spin S=1/2. In the case of the intrinsically frustrated triangular lattice model we obtain, in the limit of infinite spin, results in complete agreement with the linear spin-wave theory. In contrast to the latter, however, two of the three modes acquire a mass for finite spin. The origin of this effect is discussed.
This thesis consists of two projects which are seemingly disconnected, yet closely related. The first part explores the effects of Bose-Einstein condensation at temperatures close to, but slightly above, criticality. Following a general introduction into bosonic condensation we justify why a phenomenological theory, similar to the Ginzburg-Landau theory for fermions, holds for weakly interacting Bose gases. From this theory we predict the divergence of certain observables, in particular the quasi-magnetic susceptibility, and discuss the effects of a trapping potential. The divergence of the magnetic susceptibility motivates the introduction of an original scheme in order to measure it, published in Phys. Rev. A 93, 041602(R). The scheme uses modulated laser fields to create well- controlled gradients of artificial magnetic fields. In addition we discuss how rotational schemes might be helpful in detecting different quantum phases by exploiting different signatures in their moments of inertia. \n \nThe second part investigates binary mixtures in one dimension. We show that in certain limits such systems behave like two simply coupled Luttinger liquids, which effectively describe polaronic modes. We study and calculate explicitly how an impurity immersed in the one dimensional system creates two depletion clouds and a phase drop in each of the liquids. After arguing that these clouds and phase drops necessitate a coupling of the impurity to the low-lying excitation modes of the Luttinger liquids, we derive the edge-state singularities of the bosonic and fermionic dynamical structure factors which depend on the coupling between the liquids.
This dissertation, based on six included papers, theoretically investigates properties of quantum few-particle systems. An overview of related experimental research - ultra-cold trapped dilute gases, and electrons in quantum dots - is given, followed by a description of some of the studied many-particle phenomena - Bose-Einstein condensation, quantized vortices, Wigner localization and the Tonks-Girardeau gas. As the research results are presented in the included papers, a main part of the text in this thesis sets focus on methodology. Most of the papers involve use of the configuration interaction method, a numerical method which can give approximative eigenvalues and eigenstates of a few-particle Hamiltonian. The research has also involved further development of this method, by use of the Lee-Suzuki approximation. Formal descriptions of the methods are presented, together with a discussion about the numerical implementation. Explicit examples are given in an appendix. Papers I and II investigate properties of a rotating two-component Bose-Einstein condensate, in particular emerging vortex structures and associated wavefunctions. Paper III demonstrates that the Lee-Suzuki approximation, initially developed in the field of nuclear structure theory, can be useful to describe short-range particle-particle correlations in a trapped bosonic gas. Paper IV investigates the possibility to observe Wigner localization in a nanowire quantum dot, and compares predicted electron transport properties with experimental measurements. Paper V analyzes structures of ultra-cold atoms or molecules with dipolar interactions, in a quasi-one-dimensional trap. Paper VI also considers cold atoms or molecules with dipolar interactions, but in a quasi-two-dimensional setup, with a focus on the resulting Wigner states' dependence on the anisotropy of the interaction.
With the recent technical breakthroughs, physicists fully play with the beauty of quantum mechanics. In this theoretical work on the mesoscopic Josephson effect, we take advantage of the collaborations with experimentalists as well as the exchanges between the cold atoms and condensed matter communities. We consider various systems based on the Josephson junction, beginning with its quantum description in the underdamped regime. Using the Keldysh formalism, we obtain the current-voltage characteristics from the classical to the zero temperature regime and the quantum Smoluchowski equation in the quasi-classical limit. We then study the quantum dynamics of a phase qubit realized with a SQUID in a novel configuration where tunneling events occur through two quartic barriers. The escape rate in this camel-back potential, calculated with the instantons technique, allows us to describe the experiments. Circuit quantum electrodynamics predicts that a lasing effect appears when a qubit is coupled to a resonant cavity. We consider the case of a charge qubit and also that of a transmon which exploits the Purcell effect. With the Lindbladian we obtain the density matrix from which we derive the spectrum of the output field. Finally, we study a gas of cold atoms in a ring shape trap comprising a barrier, thus creating a Bose Josephson junction. The low energy physics is described through the correlation functions with the Luttinger liquid theory. We show that the quantum fluctuations in the ring induce a renormalization of the Josephson energy.
The quantum physics of one dimensional systems exhibits many remarkable and exotic physical phenomena. These are the result of the strong correlations which exist in such systems due to the reduced dimensionality and in many cases can be described by Luttinger Liquid theory. Likewise quantum impurity systems, where a bath of particles couples to the same impurity, exhibit well known non perturbative phenomena like the Kondo effect. In this thesis we study the physics of quantum impurities in interacting environments - the intersection of these two areas. Such Luttinger-impurity systems can be realized experimentally in a range of different settings from quantum wires and carbon nano tubes to edges of fractional quantum hall systems and cold atom gases. We show that many of the models used to describe these experiments are integrable and can be studied by means of the Bethe Ansatz. This powerful and exact method provides us with the exact eigenstates and energy levels of the model Hamiltonian. The models considered include two which describe a Lutttinger liquid coupled to a back scattering impurity, the Kane-Fisher and weak tunnelling model as well as Luttinger liquids coupled a quantum dot in various geometries: at the boundary, sidecoupled and embedded. To incorporate both the backscattering nature of the impurities and the interacting bulk particles a new formulation of the Bethe Ansatz is developed and implemented. The eigenstates and spectra of the models are found and used to investigate their ground state and thermodynamic properties.
Investigations of emergent symmetry breaking phenomena occurring in small finite-size systems are reviewed, with a focus on the strongly correlated regime of electrons in two-dimensional semiconductor quantum dots and trapped ultracold bosonic atoms in harmonic traps. Throughout the review we emphasize universal aspects and similarities of symmetry breaking found in these systems, as well as in more traditional fields like nuclear physics and quantum chemistry, which are characterized by very different interparticle forces. A unified description of strongly correlated phenomena in finite systems of repelling particles (whether fermions or bosons) is presented through the development of a two-step method of symmetry breaking at the unrestricted Hartree-Fock level and of subsequent symmetry restoration via post Hartree-Fock projection techniques. Quantitative and qualitative aspects of the two-step method are treated and validated by exact diagonalization calculations.Strongly-correlated phenomena emerging from symmetry breaking include the following.Chemical bonding, dissociation and entanglement (at zero and finite magnetic fields) in quantum dot molecules and in pinned electron molecular dimers formed within a single anisotropic quantum dot, with potential technological applications to solid-state quantum-computing devices.Electron crystallization, with particle localization on the vertices of concentric polygonal rings, and formation of rotating electron molecules (REMs) in circular quantum dots. Such electron molecules exhibit ro-vibrational excitation spectra, in analogy with natural molecules.At high magnetic fields, the REMs are described by parameter-free analytic wave functions, which are an alternative to the Laughlin and composite-fermion approaches, offering a new point of view of the fractional quantum Hall regime in quantum dots (with possible implications for the thermodynamic limit).Crystalline phases of strongly repelling bosons. In rotating traps and in analogy with the REMs, such repelling bosons form rotating boson molecules (RBMs). For a small number of bosons, the RBMs are energetically favored compared with the Gross-Pitaevskii solutions describing vortex formation.We discuss the present status concerning experimental signatures of such strongly correlated states, in view of the promising outlook created by the latest experimental improvements that are achieving unprecedented control over the range and strength of interparticle interactions.