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arXiv:1106.6329 · Frontiers of Physics

Theory of superfluidity and drag force in the one-dimensional Bose gas

Alexander Yu. Cherny, Jean-Sébastien Caux, Joachim Brand

Published Oct 22, 2011• 44 citations
arXivHTMLPDF

Abstract

The one-dimensional Bose gas is an unusual superfluid. In contrast to higher spatial dimensions, the existence of non-classical rotational inertia is not directly linked to the dissipationless motion of infinitesimal impurities. Recently, experimental tests with ultracold atoms have begun and quantitative predictions for the drag force experienced by moving obstacles have become available. This topical review discusses the drag force obtained from linear response theory in relation to Landau’s criterion of superfluidity. Based upon improved analytical and numerical understanding of the dynamical structure factor, results for different obstacle potentials are obtained, including single impurities, optical lattices and random potentials generated from speckle patterns. The dynamical breakdown of superfluidity in random potentials is discussed in relation to Anderson localization and the predicted superfluid-insulator transition in these systems.

DOI
10.1007/s11467-011-0211-2

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The one-dimensional Bose gas is an unusual superfluid. In contrast to higher spatial dimensions, the existence of non-classical rotational inertia is not directly linked to the dissipationless motion of infinitesimal impurities. Recently, experimental tests with ultracold atoms have begun and quantitative predictions for the drag force experienced by moving obstacles have become available. This topical review discusses the drag force obtained from linear response theory in relation to Landau’s criterion of superfluidity. Based upon improved analytical and numerical understanding of the dynamical structure factor, results for different obstacle potentials are obtained, including single impurities, optical lattices and random potentials generated from speckle patterns. The dynamical breakdown of superfluidity in random potentials is discussed in relation to Anderson localization and the predicted superfluid-insulator transition in these systems.
It has been proposed in recent years that an impurity moving in a superfluid will experience drag forces even below the Landau critical velocity [1–3]. Quantum fluctuations that are present in superfluid systems at zero temperature can scatter from impurities present in the fluid. If the impurity is moving, there will be an asymmetric Doppler shift associated with the scattered waves, which will result in a net drag force. This challenges the long-accepted notion of dissipationless flow in superfluids. Analytical estimates of this drag force in superfluid Bose-Einstein condensates using Bogoliubov methods have found a non-zero drag force, resulting from the most sensible choice of boundary conditions in an infinite system. Through stochastic simulations of the 1D Bose gas under the truncated Wigner formalism, we verify the existence of this sub-critical drag force and quantify the drag forces present. We will also determine whether dissipationless flow can be recovered in finite sized systems.
Different aspects of superfluidity in one dimension (1D) are discussed in this paper. It is shown that the Hess-Fairbank effect takes place in 1D at zero temperature. In particular, the rotational inertia of an interacting 1D Bose gas in a finite ring is zero. Nevertheless, our results indicate that the frictionless motion of impurities depends sensitively on the strength of interactions in the gas. In general, this is possible only in a limit of weak interactions. We obtain the phase diagram of frictionless motion for the gas immersed into a moving shallow optical lattice. In this case the drag force, a quantitative measure of superfluidity, can be zero at specific values of lattice velocity and the gas density even for strong
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.
This thesis examines the manifestation of the critical velocity for superfluidity in dilute gas Bose-Einstein condensates (BECs). Superfluidity is a macroscopic quantum phenomenon characterised by unique properties such as frictionless flow, high thermal conductivity and the presence of quantised vortices. Our study focuses on what is arguably the most well-known attribute: the ability for an object to move through a superfluid without experiencing a drag force when travelling below a threshold critical velocity.The Landau criterion for superfluidity predicts the critical velocity in a BEC is equal to the speed of sound, and this conclusion is supported by a large body of theoretical work. However, a number of experiments have observed lower critical velocities than theory predicts. In the first part of this thesis, we numerically model experiments on the superfluid critical velocity of an elongated, harmonically trapped Bose-Einstein condensate as reported by [P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007)]. These experiments swept a repulsive obstacle, formed by a optical dipole potential, through the long axis of a condensate at constant velocity. Their results found an increase in the resulting density fluctuations of the condensate above an obstacle velocity of v _ 0:3 mm s
The existence of frictionless flow below a critical velocity for obstacles moving in a superfluid 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 nonzero 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 analytical solutions to the Bogoliubov-de Gennes equations for a gray soliton. Our results raise questions regarding the quantum dynamics in the formation of persistent currents in superfluids.
Abstract The phase diagram of a spin–orbit-coupled two-component Bose gas includes a supersolid stripe phase, which is featuring density modulations along the direction of the spin–orbit coupling. This phase has been recently found experimentally [31]. In the present work, we characterize the superfluid behavior of the stripe phase by calculating the drag force acting on a moving impurity. Because of the gapless band structure of the excitation spectrum, the Landau critical velocity vanishes if the motion is not strictly parallel to the stripes, and energy dissipation takes place at any speed. Moreover, due to the spin–orbit coupling, the drag force can develop a component perpendicular to the velocity of the impurity. Finally, by estimating the time over which the energy dissipation occurs, we find that for slow impurities, the effects of friction are negligible on a time scale up to several seconds, which is comparable with the duration of a typical experiment.