We investigate the momentum distribution of weakly interacting 1D Bose gases at thermal equilibrium both experimentally and theoretically. Momentum distribution of single 1D Bose gases is measured using a focusing technique, whose resolution we improve via a levitation scheme. The momentum distribution compares very well with quantum Monte Carlo calculations for the Lieb-Liniger model at finite temperature, allowing for an accurate thermometry of the gas that agrees with (and improves upon) the thermometry based on {\it in situ} density fluctuation measurements. The quasi-condensation crossover is investigated via two different experimental parameter sets, corresponding to the two different sides of the crossover. Classical field theory is expected to correctly describe the quasi-condensation crossover of weakly interacting gases. We derive the condition of validity of the classical field theory, and find that, in typical experiments, interactions are too strong for this theory to be accurate. This is confirmed by a comparison between the classical field predictions and the numerically exact quantum Monte Carlo calculations.
We investigate the momentum distribution of weakly interacting 1D Bose gases at thermal equilibrium both experimentally and theoretically. Momentum distribution of single 1D Bose gases is measured using a focusing technique, whose resolution we improve via a levitation scheme. The momentum distribution compares very well with quantum Monte Carlo calculations for the Lieb-Liniger model at finite temperature, allowing for an accurate thermometry of the gas that agrees with (and improves upon) the thermometry based on {\it in situ} density fluctuation measurements. The quasi-condensation crossover is investigated via two different experimental parameter sets, corresponding to the two different sides of the crossover. Classical field theory is expected to correctly describe the quasi-condensation crossover of weakly interacting gases. We derive the condition of validity of the classical field theory, and find that, in typical experiments, interactions are too strong for this theory to be accurate. This is confirmed by a comparison between the classical field predictions and the numerically exact quantum Monte Carlo calculations.
We investigate the momentum distribution of weakly interacting one-dimensional (1D) Bose gases at thermal equilibrium both experimentally and theoretically. Momentum distribution of 1D Bose gases is measured using a focusing technique, whose resolution we improve via a guiding scheme. The momentum distribution compares very well with quantum Monte Carlo calculations for the Lieb-Liniger model at finite temperature, allowing for an accurate thermometry of the gas that agrees with (and improves upon) the thermometry based on in situ density fluctuation measurements. The quasicondensation crossover is investigated via two different experimental parameter sets, corresponding to the two different sides of the crossover. Classical field theory is expected to correctly describe the quasicondensation crossover of weakly interacting gases. We derive the condition of validity of the classical field theory, and find that, in typical experiments, interactions are too strong for this theory to be accurate. This is confirmed by a comparison between the classical field predictions and the numerically exact quantum Monte Carlo calculations.
We study, within the framework of the classical-field approximation, the density correlations of a weakly interacting expanding Bose gas for the whole range of temperatures across the Bose-Einstein condensation threshold. We focus on elongated quasi-one-dimensional systems where there is a huge discrepancy between the existing theory and experimental results [A. Perrin et al., Nat. Phys. 8, 195 (2012)]. We find that the density correlation function is not reduced for temperatures below the critical one as it is predicted for the ideal gas or for a weakly interacting system within the Bogoliubov approximation. This behavior of the density correlations agrees with the above-mentioned experiment with the elongated system. Although the system was much larger than that studied here, we believe that the behavior of the density correlation function found there is quite generic. Our theoretical study indicates also large density fluctuations in the trap in the quasicondensate regime where only phase fluctuations were expected. We argue that the enhanced density fluctuations can originate in the presence of interactions in the system, or more precisely in the presence of spontaneous dark solitons in the elongated gas at thermal equilibrium.
We analyze the equilibrium properties of a weakly interacting, trapped quasi-one-dimensional Bose gas at finite temperatures and compare different theoretical approaches. We focus in particular on two stochastic theories: a number-conserving Bogoliubov (NCB) approach and a stochastic Gross-Pitaevskii equation (SGPE) that have been extensively used in numerical simulations. Equilibrium properties like density profiles, correlation functions, and the condensate statistics are compared to predictions based upon a number of alternative theories. We find that due to thermal phase fluctuations, and the corresponding condensate depletion, the NCB approach loses its validity at relatively low temperatures. This can be attributed to the change in the Bogoliubov spectrum, as the condensate gets thermally depleted, and to large fluctuations beyond perturbation theory. Although the two stochastic theories are built on different thermodynamic ensembles (NCB, canonical; SGPE, grand-canonical), they yield the correct condensate statistics in a large Bose-Einstein condensate (BEC) (strong enough particle interactions). For smaller systems, the SGPE results are prone to anomalously large number fluctuations, well known for the grand-canonical, ideal Bose gas. Based on the comparison of the above theories to the modified Popov approach, we propose a simple procedure for approximately extracting the Penrose-Onsager condensate from first- and second-order correlation functions that is both computationally convenient and of potential use to experimentalists. This also clarifies the link between condensate and quasicondensate in the Popov theory of low-dimensional systems.
In this thesis we use the stochastic Gross-Pitaevskii equation (SGPE), a finite temperature model for weakly interacting ultracold Bose gases which fully incorporates density and phase fluctuations, to demonstrate ab initio quantitative agreement with a number of quasi-one-dimensional experiments. To achieve this, we propose and numerically solve a quasi-one-dimensional form of the SGPE, supported by a self-consistent treatment of radially-excited thermal modes. The quasi-one-dimensional stochastic Gross-Pitaevskii equation provides an accurate finite temperature description of the dynamical equilibrium of the lowenergy axial modes of a Bose gas, assumed to be highly populated and thus treated within the ‘classical field’ approximation. This treatment allows to selfconsistently account for transverse, quasi-one-dimensional effects, which makes it a valid model in the regime where the chemical potential, μ, is approximately equal to a few times the transverse excitation energy, ~ω⊥. In the regime where the thermal energy, kBT , is also comparable to or larger than ~ω⊥, the transverse excited states play an increasingly important role, and are treated here as onedimensional independent Bose gases at static equilibrium. Firstly, we demonstrate that this is an excellentmodel for ab initio investigation of equilibrium properties, such as density profiles and density fluctuations. This is shown by accurately reproducing the in situ density profiles recently obtained in the experiments of Trebbia et al. [Phys. Rev. Lett. 97, 250403 (2006)] and van Amerongen et al. [Phys. Rev. Lett. 100, 090402 (2008)], and the density fluctuation data reported by Armijo et al. [Phys. Rev. Lett. 105, 230402 (2010)]. Moreover, we perform an ab initio analysis of the temperature dependence of the phase coherence of finite temperature, quasi-one-dimensional Bose gases measured in the experiments of Richard et al. [Phys. Rev. Lett. 91, 010405 (2003)] and Hugbart et al. [Eur. Phys. J. D 35, 155 (2005)]. We find very good agreement across the entire observed temperature range in both experiments, and improve upon previous theoretical modelling of the latter.
By measuring the density fluctuations in a highly elongated weakly interacting Bose gas, we observe and quantify the transition from the ideal gas to a quasi-condensate regime throughout the dimensional crossover from a purely 1D to an almost 3D gas. We show that that the entire transition region and the dimensional crossover are described surprisingly well by the modified Yang-Yang model. Furthermore, we find that at low temperatures the linear density at the quasi-condensate transition scales according to an interaction-driven scenario of a longitudinally uniform 1D Bose gas, whereas at high temperatures it scales according to the degeneracy-driven critical scenario of transverse condensation of a 3D ideal gas. We also perform measurements of the third moment of atom number fluctuations. We find a positive skewness of the atom number distribution in the ideal gas regime and a reduced skewness compatible with zero in the quasi-condensate regime. For our parameters, the third moment is a thermodynamic quantity whose measurement constitutes a sensitive test of the equation of state and our results are in agreement with a modified Yang-Yang thermodynamic prediction. Moreover, we show that the measured skewness reveals the presence of true three body correlations in the system.
This Thesis discusses approaches to calculating static and dynamic correlation functions in onedimensional gases of interacting bosons. The first part of the Thesis deals with the momentummomentum correlation function in the weakly interacting 1D Bose gas. In the regime of phase fluctuating quasicondensate this correlator is found to differ qualitatively from the form predicted by Bogoliubov theory of the true condensate in that correlations between any two values of momentum become finite. Linear hydrodynamics used to calculate the quasicondensate correlation function prove to be adequate. A classical field approximation is used to smoothly interpolate between the quasicondensate and strongly degenerate gas regimes. In the second part the focus is shifted to the dynamical structure factor and it is shown that hydrodynamics is generally inapplicable to calculating the dynamical correlators. The hydrodynamic treatment is enhanced using impurity theory and an exact model-dependent boundary is obtained for the region of applicability of hydrodynamics on the momentum-energy plane. Numerical estimates show that non-hydrodynamic behavior should be observable for the currently available values of experimental parameters.