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arXiv:1007.0281 · Physical Review A

Phase fluctuations in anisotropic Bose-Einstein condensates: From cigars to rings

Ludwig Mathey, Anand K. Ramanathan, Kevin Wright, Sérgio Ricardo Muniz, William D. Phillips, Charles W. Clark

Published Sep 14, 2010• 22 citations
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Abstract

We study the phase-fluctuating condensate regime of ultracold atoms trapped in a ring-shaped trap geometry, which has been realized in recent experiments. We first consider a simplified box geometry, in which we identify the conditions to create a state that is dominated by thermal phase fluctuations, and then explore the experimental ring geometry. In both cases we demonstrate that the requirement for strong phase fluctuations can be expressed in terms of the total number of atoms and the geometric length scales of the trap only. For the ring-shaped trap we discuss the zero temperature limit in which a condensate is realized where the phase is fluctuating due to interactions and quantum fluctuations. We also address possible ways of detecting the phase-fluctuating regime in ring condensates.

DOI
10.1103/physreva.82.033607

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We study the phase-fluctuating condensate regime of ultracold atoms trapped in a ring-shaped trap geometry, which has been realized in recent experiments. We first consider a simplified box geometry, in which we identify the conditions to create a state that is dominated by thermal phase fluctuations, and then explore the experimental ring geometry. In both cases we demonstrate that the requirement for strong phase fluctuations can be expressed in terms of the total number of atoms and the geometric length scales of the trap only. For the ring-shaped trap we discuss the zero temperature limit in which a condensate is realized where the phase is fluctuating due to interactions and quantum fluctuations. We also address possible ways of detecting the phase-fluctuating regime in ring condensates.
We derive a symmetry property for the Fourier-transform of the density fluctuation modes of Bose condensates in anisotropic traps in the hydrodynamic limit connected with a somewhat hidden conservation law.. We discuss its possible observation by dispersive light scattering, its range of validity, and also experiments using inelastic scattering of light. PACS: 03.75.Fi,05.30.Jp,42.50.Gy Since the achievement of Bose-Einstein condensates in trapped alkali gases [1–3] at ultralow temperatures numerous aspects of these systems have been investigated both from the experimental and the theoretical side. Adding a time dependent perturbation to the trap potential the lowest frequencies of the density fluctuation spectrum were measured by observing condensate shape oscillations either by time of flight measurements [4–6] or by dispersive light scattering [7]. In this paper we want to report on a property of the mode functions of the low lying density fluctuations of condensates in anisotropic traps. Mode functions can be observed in dispersive light scattering experiments [7,8], but also in inelastic light scattering. We discuss how the derived property could be observed in both experiments and discuss the limits of both approaches. For a Bose gas below the Bose-Einstein transition the usual separation of the particle field 1 operator into a classical part φ0(r), the condensate wave function, and a residual operator ˆφ(r), describing fluctuations, yields for the number-density operator ˆρ(r) = |φ0(r) | 2 + ( φ ∗ 0(r) ˆ φ(r) + φ0(r) ˆ φ † (r)) + ˆ φ † (r) ˆ φ(r). (0.1) Thermal averaging of the last term gives the condensate depletion at T = 0 plus the density of thermally excited atoms. At low temperatures for a weakly interacting system the condensate density |φ0(r) | 2 contains most of the particles, and the second term in (0.1) dominates by far the last term. Now we restrict ourselves to the low-lying density fluctuations with ¯hω ≪ µ, which can be described by the hydrodynamic approach to trapped Bose systems [9,10]. In this limit the dominant density fluctuations in (0.1) take the form [10,11] φ ∗ 0 (r) ˆ φ(r) + φ0(r) ˆ φ † (r) = ∑ j
Inspired by recent experiments on Bose–Einstein condensates in ring traps, we investigate the topological properties of the phase of a one-dimensional Bose field in the presence of both thermal and quantum fluctuations—the latter ones being tuned by the depth of an optical lattice applied along the ring. In the regime of large filling of the lattice, quantum Monte Carlo simulations give direct access to the full statistics of fluctuations of the Bose-field phase, and of its winding number W along the ring. At zero temperature the winding-number (or topological-sector) fluctuations are driven by quantum phase slips localized around a Josephson link between two lattice wells, and their susceptibility is found to jump at the superfluid-Mott insulator transition. At finite (but low) temperature, on the other hand, the winding number fluctuations are driven by thermal activation of nearly uniform phase twists, whose activation rate is governed by the superfluid fraction. A quantum-to-thermal crossover in winding-number fluctuations is therefore exhibited by the system, and it is characterized by a conformational change in the topologically non-trivial configurations, from localized to uniform phase twists, which can be experimentally observed in ultracold Bose gases via matter–wave interference.
Fluctuations are fundamental in physics and important for understanding and characterizing phase transitions. In this spirit, the phase transition to the Bose-Einstein condensate (BEC) is of specific importance. Whereas fluctuations of the condensate particle number in atomic BECs have been studied in continuous systems, experimental and theoretical studies for lattice systems were so far missing. Here, we explore the condensate particle number fluctuations in an optical lattice BEC across the phase transition in a combined experimental and theoretical study. We present both experimental data using ultracold ^{87}Rb atoms and numerical simulations based on a hybrid approach combining the Bogoliubov quasiparticle framework with a master equation analysis for modeling the system. We find strongly anomalous fluctuations, where the variance of the condensate number δN_{BEC}^{2} scales with the total atom number as N^{1+γ} with an exponent around γ_{theo}=0.74(10) and γ_{exp}=0.62(9), which we attribute to the 2D/3D crossover geometry. Our Letter highlights the importance of the trap geometry on the character of fluctuations and on fundamental quantum mechanical properties.
The occurrence of phase fluctuations due to thermal excitations in Bose-Einstein condensates (BECs) is studied for a variety of temperatures and trap geometries. We observe the statistical nature of the appearance of phase fluctuations and characterize the dependence of their average value on temperature, number of particles, and the trapping potential. We find pronounced phase fluctuations for condensates in very elongated traps in a broad temperature range. The results are of great importance for the realization of BEC in quasi-1D geometries, for matter wave interferometry with BECs, as well as for coherence properties of guided atom laser beams.
We present an overview of our recent work describing the treatment of ultracold atomic Bose gases confined to two dimensions in a harmonic trap. We show how the gas goes through a number of regimes as the axially confining potential is increased and discuss the scattering properties of the gas in each of these regimes. We then use the formalism developed to investigate the phase-coherent Bose-Einstein condensate that is present in the trapped two-dimensional gas at temperatures that are low compared to the ideal gas critical temperature and to analyze the first-order coherence as a function of temperature. We identify the regime in which the gas loses its global phase coherence, although a macroscopic occupation of the ground state persists at a temperature well below the ideal gas critical temperature. We comment on the possible connection between this fluctuating regime and the Berezinskii-Kosterlitz-Thouless phase.
The work of this thesis concerns the properties of a Bose gas of polarised dipoles that have long-range and anisotropic interactions. Our work is divided into two parts. First, the stability of a dipolar Bose gas at finite temperature (both above and below the critical Bose-Einstein condensation (BEC) temperature Tc). Second, the fluctuations of a dipolar BEC at zero and small finite temperature (T ≪ Tc) in a regime where rotonic excitations emerge. \n \nPart I: Stability of a Trapped Finite Temperature Dipolar Bose Gas: Above Tc we implement a semiclassical Hartree-Fock theory and characterise the dependence of the stability boundary on temperature, trap geometry and the strength of the dipole-dipole interaction and contact interaction. We find that stability is greatly enhanced above Tc and that trap geometry continues to play a key role. Furthermore, we find that for oblate traps a novel double instability feature emerges. \nTo extend our stability analysis to the low temperature regime, T < Tc, we develop a beyond semiclassical Hartree theory. We use this to characterise the stability boundary as a function of geometry. Interestingly, we find large beyond semiclassical effects above Tc for prolate trapping geometries. We characterise thermal effects on biconcave condensate states. \n \nPart II: Rotons and Fluctuations in a Trapped Dipolar Condensate: To study density fluctuations we implement a numerical scheme to solve the Gross-Pitaevskii equation and the Bogoliubov de Gennes equations. We find that the phonon and roton gases spatially separate and we characterise the role of the anomalous density on the density fluctuations of the thermally activated rotons. We develop a numerical scheme that calculates number fluctuations within cells of various shapes and sizes, and find a strong peak in the fluctuations when the cell size is around half the roton wavelength, which should be detectable by current experiments. By tailoring the cell shape we predict that experiments should be able to detect the effects of individual roton modes. \nFor the study of zero temperature fluctuations we deploy the Gross-Pitaevskii and Bogoliubov de Gennes equations to calculate the dynamic and static structure factors for a highly oblate BEC. We find a clear signature of the roton gas dispersion relation within the structure factors. This signature should be detectible in current experiments using Bragg spectroscopy.