Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1801.04993

One-Dimensional Quantum Systems - From Few to Many Particles

Amin S. Dehkharghani

Published Jan 15, 2018• 2 citations
arXivHTMLPDF

Abstract

In this thesis, I go through the well-known solutions to the one and two-particle systems trapped in a quantum harmonic oscillator and then continue to the three, four and many-body quantum systems. This is done by developing new analytical models and numerical methods both for the few- and many-body systems. One-dimensional systems are very interesting in a sense that particles aligned on a line can only change seats by going through each other. This property can be exploited in the strongly interacting regime, where particles are forced to sit in a specific configuration, which can be easily manipulated. The knowledge of how and where the particles are can be exploited in future quantum applications. In short, the thesis is about establishing a solid knowledge about everything that one needs to know about the one-dimensional few- and many-component interacting quantum systems trapped in harmonic oscillator potentials.

DOI
10.48550/arxiv.1801.04993

Related papers

Exactly solved quantum many-body systems in one dimension

Martin A. Hallnäs

This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.

•Jan 1, 2005•0 citations

Few-Body Systems in Low-Dimensional Geometries:• Dipoles in Multi-Layer and Multi-Tube Configurations • Strongly-Interacting Particles in One-Dimensional Traps

Artem G. Volosniev

The research in this dissertation is devoted to few-body bound state physics in experimentally relevant systems of trapped atoms and molecules. First, the complexes of tubes containing dipoles are considered. The tubes are assumed to have zero width such that one-dimensional treatment can be applied. For this setup few-body bound structures are found for different polarization an- gles and dipole strengths by using stochastic variational methods. After that a similar analysis is provided for two-dimensional planes filled with dipolar par- ticles. At the end of the thesis, a system of particles in one-dimensional traps is analyzed assuming that two-body interactions are of zero range. It is shown that such a system is solvable in the limit of infinitely strong repulsion between particles.

•Jan 1, 2013•0 citations

Work analysis of one-dimensional driven quantum systems

Maria Arazo Sánchez

In recent years there has been a tremendous advance in the techniques to trap and control systems of a few bosonic and fermionic atoms [1,2]. In these systems the trap properties are usually tunable, thus allowing one to study how the quantum system adapts to the new trap properties. In particular one can consider a simple scenario in which a particle is trapped in a harmonic oscillator potential which trapping frequency is varied in time with a given time dependence. This system represents a simple example where the concepts of work [3] need to be adapted to quantum

•Sep 6, 2018•0 citations

Dynamics in strongly interacting one-dimensional quantum systems

R. E. Barfknecht

In this thesis we study the dynamics of one-dimensional strongly interacting few-body systems. We present the theory which allows us to describe such systems as a spin chain where the exchange coefficients are determined by the trapping geometry, and comment on the experimental feasibility of these mod- els. We then proceed to apply our formalism to different dynamical problems, where the strength of the interactions between the atoms in the trap plays a fundamental role. We find that interesting effects - ranging from changes in magnetic correlations to impurity oscillations and spin-charge separation - arise in this context. Additionally, we perform studies of the static corre- lations and many-body dynamics of bosonic systems away from the strongly interacting limit.

•Jan 1, 2019•1 citation

A study of one-dimensional quantum gases

Andrew G. Sykes

In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.

•Mar 1, 2010•0 citations

Quantum Correlations of a Few Bosons within a Harmonic Trap

Przemysław Kościk

We study two one-dimensional atomic models composed of identical bosons trapped in a harmonic trap: one model with a short-range interaction potential and another model with a long-range one, and compare their entanglement features. Properties of the ground-states of two-, three-, and four-particle systems are explored.

Few-Body Systems•Jun 15, 2011•12 citations
In this thesis, I go through the well-known solutions to the one and two-particle systems trapped in a quantum harmonic oscillator and then continue to the three, four and many-body quantum systems. This is done by developing new analytical models and numerical methods both for the few- and many-body systems. One-dimensional systems are very interesting in a sense that particles aligned on a line can only change seats by going through each other. This property can be exploited in the strongly interacting regime, where particles are forced to sit in a specific configuration, which can be easily manipulated. The knowledge of how and where the particles are can be exploited in future quantum applications. In short, the thesis is about establishing a solid knowledge about everything that one needs to know about the one-dimensional few- and many-component interacting quantum systems trapped in harmonic oscillator potentials.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
The research in this dissertation is devoted to few-body bound state physics in experimentally relevant systems of trapped atoms and molecules. First, the complexes of tubes containing dipoles are considered. The tubes are assumed to have zero width such that one-dimensional treatment can be applied. For this setup few-body bound structures are found for different polarization an- gles and dipole strengths by using stochastic variational methods. After that a similar analysis is provided for two-dimensional planes filled with dipolar par- ticles. At the end of the thesis, a system of particles in one-dimensional traps is analyzed assuming that two-body interactions are of zero range. It is shown that such a system is solvable in the limit of infinitely strong repulsion between particles.
In recent years there has been a tremendous advance in the techniques to trap and control systems of a few bosonic and fermionic atoms [1,2]. In these systems the trap properties are usually tunable, thus allowing one to study how the quantum system adapts to the new trap properties. In particular one can consider a simple scenario in which a particle is trapped in a harmonic oscillator potential which trapping frequency is varied in time with a given time dependence. This system represents a simple example where the concepts of work [3] need to be adapted to quantum
In this thesis we study the dynamics of one-dimensional strongly interacting few-body systems. We present the theory which allows us to describe such systems as a spin chain where the exchange coefficients are determined by the trapping geometry, and comment on the experimental feasibility of these mod- els. We then proceed to apply our formalism to different dynamical problems, where the strength of the interactions between the atoms in the trap plays a fundamental role. We find that interesting effects - ranging from changes in magnetic correlations to impurity oscillations and spin-charge separation - arise in this context. Additionally, we perform studies of the static corre- lations and many-body dynamics of bosonic systems away from the strongly interacting limit.
In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.