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arXiv:cond-mat/0307033 · Journal of Physics B Atomic Molecular and Optical Physics

Bosonizing one-dimensional cold atomic gases

Miguel A. Cazalilla

Published Mar 24, 2004• 439 citations
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Abstract

We present results for the long-distance asymptotics of correlation functions of mesoscopic one-dimensional systems with periodic and open (Dirichlet) boundary conditions, as well as at finite temperature in the thermodynamic limit. The results are obtained using Haldane's harmonic-fluid approach (also known as 'bosonization'), and are valid for both bosons and fermions, in weakly and strongly interacting regimes. The harmonic-fluid approach and the method of computing the correlation functions using conformal transformations are explained in great detail. As an application relevant to one-dimensional systems of cold atomic gases, we consider the model of bosons interacting with a zero-range potential. The Luttinger-liquid parameters are obtained from the exact solution by solving the Bethe-ansatz equations in finite-size systems. The range of applicability of the approach is discussed, and the prefactor of the one-body density matrix of bosons is fixed by finding an appropriate parametrization of the weak-coupling result. The formula thus obtained is shown to be accurate, when compared with recent diffusion Monte Carlo calculations, within less than 10%. The experimental implications of these results for Bragg scattering experiments at low and high momenta are also discussed.

DOI
10.1088/0953-4075/37/7/051

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We present results for the long-distance asymptotics of correlation functions of mesoscopic one-dimensional systems with periodic and open (Dirichlet) boundary conditions, as well as at finite temperature in the thermodynamic limit. The results are obtained using Haldane's harmonic-fluid approach (also known as 'bosonization'), and are valid for both bosons and fermions, in weakly and strongly interacting regimes. The harmonic-fluid approach and the method of computing the correlation functions using conformal transformations are explained in great detail. As an application relevant to one-dimensional systems of cold atomic gases, we consider the model of bosons interacting with a zero-range potential. The Luttinger-liquid parameters are obtained from the exact solution by solving the Bethe-ansatz equations in finite-size systems. The range of applicability of the approach is discussed, and the prefactor of the one-body density matrix of bosons is fixed by finding an appropriate parametrization of the weak-coupling result. The formula thus obtained is shown to be accurate, when compared with recent diffusion Monte Carlo calculations, within less than 10%. The experimental implications of these results for Bragg scattering experiments at low and high momenta are also discussed.
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.
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We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
Ultracold bosonic and fermionic quantum gases confined to quasi-one-dimensional (1D) geometry are promising candidates for probing fundamental concepts of Luttinger liquid (LL) physics. They can also be exploited for devising applications in quantum information processing and precision measurements. Here, we focus on 1D dipolar Bose gases, where evidence of super-strong coupling behavior has been demonstrated by analyzing the low-energy static and dynamical structures of the fluid at zero temperature by a combined reptation quantum Monte Carlo (RQMC) and bosonization approach. Fingerprints of LL behavior emerge in the whole crossover from the already strongly interacting Tonks–Girardeau at low density to a dipolar density wave regime at high density. We have also shown that a LL framework can be effectively set up and utilized to describe this strongly correlated crossover physics in the case of confined 1D geometries after using the results for the homogeneous system in LL hydrodynamic equations within a local density approximation. This leads to the prediction of observable quantities such as the frequencies of the collective modes of the trapped dipolar gas under the more realistic conditions that could be found in ongoing experiments. The present paper provides a description of the theoretical framework in which the above results have been worked out, making available all the detailed derivations of the hydrodynamic Luttinger equations for the inhomogeneous trapped gas and of the correlation functions for the homogeneous system.