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arXiv:1712.05424

Correlations in low-dimensional quantum gases

Lang, Guillaume

Published Dec 14, 2017• 9 citations
arXivHTMLPDF

Abstract

Ultracold gases are a versatile platform to simulate condensed matter physics, as virtually any parameter is experimentally tunable. In particular, highly anisotropic traps allow the realization of low-dimensional systems, where the role of quantum fluctuations is enhanced. I investigate the paradigmatic model of a one-dimensional Bose gas with contact interactions, a.k.a the Lieb-Liniger model, using powerful analytical tools such as Bethe Ansatz, Conformal Field Theory and the Tomonaga-Luttinger liquid formalism, as well as the Bose-Fermi mapping. The effect of an additional harmonic trap in the longitudinal direction is studied within the Local Density Approximation. These methods allow to investigate with high accuracy its ground-state thermodynamics of the gas, its excitation spectra and correlation functions, from the k-body local correlations to the momentum distribution and the dynamical structure factor. These quantities characterize the coherence and superfluidity of the gas. In particular, I study the drag force exerted by a weak gaussian barrier stirred into the gas within linear response theory. Then, I investigate the appearance of a multimode structure obtained by releasing a transverse trapping, and the dimensional crossover in the limit of an infinite number of transverse modes.

DOI
10.48550/arxiv.1712.05424

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Ultracold gases are a versatile platform to simulate condensed matter physics, as virtually any parameter is experimentally tunable. In particular, highly anisotropic traps allow the realization of low-dimensional systems, where the role of quantum fluctuations is enhanced. I investigate the paradigmatic model of a one-dimensional Bose gas with contact interactions, a.k.a the Lieb-Liniger model, using powerful analytical tools such as Bethe Ansatz, Conformal Field Theory and the Tomonaga-Luttinger liquid formalism, as well as the Bose-Fermi mapping. The effect of an additional harmonic trap in the longitudinal direction is studied within the Local Density Approximation. These methods allow to investigate with high accuracy its ground-state thermodynamics of the gas, its excitation spectra and correlation functions, from the k-body local correlations to the momentum distribution and the dynamical structure factor. These quantities characterize the coherence and superfluidity of the gas. In particular, I study the drag force exerted by a weak gaussian barrier stirred into the gas within linear response theory. Then, I investigate the appearance of a multimode structure obtained by releasing a transverse trapping, and the dimensional crossover in the limit of an infinite number of transverse modes.
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.
Describing the collective behavior of many body interacting systems is a central challenge in coldatom physics. Systems confined in one dimension are particularly interesting due to the fact thatin many cases they are integrable. In this work, we focus on such systems—specifically, Bose gasessolvable via the Bethe Ansatz. We begin the thesis by preparing the background, the tools and themotivation needed in the following, inlcuding a review of the Lieb-Liniger gas, it's Bethe Ansatz andthe recent advancements in the experimental realization of it. We include a brief introduction of thetheory of Generalized Hydrodynamics as well as effective field theoretical methods, such as theLuttinger liquid theory, used to compute correlation functions in ground states of low dimensionalsystems. We then turn to the study of lattice hard core bosons out- of euqilibrium. By computing analyticalresults for the one particle density matrix of a freely expanding system of hard-core bosons from aninitial double domain wall potential we provide a fully analytical treatment of the phenomenon ofdynamical quasicondensation and discuss the emergence of off-diagonal long range order. Next, we investigate the dynamics of hard-core bosons released from a domain wall initial statein the presence of two conformal defects. The observed density profile differs from the Euler-scalehydrodynamic description due to quantum interference effects. By analyzing boson propagators, weshow that repeated scattering on the defects leads to interference patterns that cannot be capturedby the hydrodynamic approach. We derive an analytic expression for the density profile during freeexpansion and demonstrate that a modified semi-classical scattering framework is required for thedescription of systems with such conformal defects. The final part of thesis deals with the calculation of the one particle density matrix of Lieb-Lingergas in smooth trapping potentials at finite temperature and interaction strengths. We introduce amethod to obtain such correlation functions by combining long-distance asymptotics with knownshort distance expansions. For the limits of weak and strong interactions we derive analytical re-sults, as well as a general protocol for computing one-particle correlations at any arbitrary repulsiveinteraction strengths. We include a detailed explanation of the numerical method used to computeGreen's functions of field of the effective theory needed as an input to our procedure. We benchmarkour results by comparing them to exact numerical calculations and to recent experimental findings.
Ultracold atoms are exceptional tools to explore the physics of quantum matter. In fact, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for investigating macroscopic manifestations of quantum effects, such as superfluidity. In ultracold gas research, a central role is played by collective oscillations. They can be used to study different dynamical regimes, such as superfluid, collisional, or collisionless limits or to test the equation of state of the system. In this thesis, we present a unified description of collective oscillations in low dimensions covering both Bose and Fermi statistics, different trap geometries and zero as well as finite temperature, based on the formalism of hydrodynamics and sum rules. We discuss the different behaviour exhibited by the second excited breathing mode in the collisional regime at low temperature and in the collisionless limit at high temperature in a 1D trapped Bose gas with repulsive contact interaction. We show how this mode exhibits a single-valued excitation spectrum in the collisional regime and two different frequencies in the collisionless limit. Our predictions could be important for future research related to the thermalization and damping phenomena in this low-dimensional system. We show that 1D uniform Bose gases exhibit a non-monotonic temperature dependence of the chemical potential characterized by an increasing-with-temperature behaviour at low temperature. This is due to the thermal excitation of phonons and reveals an interesting analogy with the behaviour of superfluids. Finally, we investigate a gas with a finite number N of atoms in a ring geometry at T = 0. We discuss explicitly the deviations of the thermodynamic behaviour in the ring from the one in the large N limit.
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This article briefly reviews recent theoretical developments in quantum critical phenomena in one-dimensional (1D) integrable quantum gases of cold atoms. We present a discussion on quantum phase transitions, universal thermodynamics, scaling functions and correlations for a few prototypical exactly solved models, such as the Lieb–Liniger Bose gas, the spin-1 Bose gas with antiferromagnetic spin-spin interaction, the two-component interacting Fermi gas as well as spin-3/2 Fermi gases. We demonstrate that their corresponding Bethe ansatz solutions provide a precise way to understand quantum many-body physics, such as quantum criticality, Luttinger liquids (LLs), the Wilson ratio, Tan's Contact, etc. These theoretical developments give rise to a physical perspective using integrability for uncovering experimentally testable phenomena in systems of interacting bosonic and fermonic ultracold atoms confined to 1D.