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Chinese Physics Letters

Signature of Critical Point in Momentum Profile of Trapped Ultracold Bose Gases

Qiang Zhu, Bing Wang, Dezhi Xiong, Baolong Lü

Published Jul 1, 2016• 0 citations
Publisher

Abstract

We present a new method to identify the critical point for the Bose-Einstein condensation (BEC) of a trapped Bose gas. We calculate the momentum distribution of an interacting Bose gas near the critical temperature, and find that it deviates significantly from the Gaussian profile as the temperature approaches the critical point. More importantly, the standard deviation between the calculated momentum spectrum and the Gaussian profile at the same temperature shows a turning point at the critical point, which can be used to determine the critical temperature. These predictions are also confirmed by our BEC experiment for magnetically trapped 87 Rb gases.

DOI
10.1088/0256-307x/33/7/070306

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We present a new method to identify the critical point for the Bose-Einstein condensation (BEC) of a trapped Bose gas. We calculate the momentum distribution of an interacting Bose gas near the critical temperature, and find that it deviates significantly from the Gaussian profile as the temperature approaches the critical point. More importantly, the standard deviation between the calculated momentum spectrum and the Gaussian profile at the same temperature shows a turning point at the critical point, which can be used to determine the critical temperature. These predictions are also confirmed by our BEC experiment for magnetically trapped 87 Rb gases.
Quantum criticality emerges when the collective fluctuations of matter undergo a continuous phase transition at zero temperature and has been a research focus in conventional condensed-matter physics over the past several decades. In the quantum critical regime, the exotic and universal properties are expected. These properties are independent of the microscopic details of the system, but depend only on a few general properties of the system, such as its dimensionality and the symmetry of the order parameter. The research of quantum criticality can not only help us to understand quantum phase transitions, but also provide a novel route to new material design and discovery.Ultracold bosonic gases have provided a clean system for studying the quantum critical phenomena. The critical behavior of a weakly interacting three-dimensional (3D) Bose gas should be identical to that of 4He at the superfluid transition, which belongs to the 3D XY universality class. From the normal fluid to the superfluid, the system undergoes a phase transition from completely disorder to long-range order, while in the vicinity of the phase transition point, the system parameters will show some singularity characteristics. In this paper, we observe the critical behavior of 87Rb Bose gas in a quadrupole-Ioffe configuration (QUIC) trap near the phase transition temperature Tc. A novel singularity behavior of the full width at half maximum of momentum distribution (FWHMMD) of atomic gas is discovered in the experiment. Prior to our experiment, we prepare a sample with 7.8105 87Rb atoms in the 5S1/2 |F=2, mF=2 state. Then the sample is held in a QUIC trap for a presetting period of time to control the temperature of atom sample precisely. During the holding time, the sample is heated up due to background gas collisions or fluctuations of the trap potential. In our experiment, the heating rate is deduced to be 0.3480.078 nK/ms from the absorption image. For a bosonic gas in a harmonic trap, critical gas can only cover a finite-size region due to a spatially varying density. We define the finite-size region as a critical region determined by the Ginzburg criterion. Then the FWHMMDs of atomic gas in the critical region are measured for different temperatures near the critical point. To this aim, we first extract the momentum distribution of atomic gas from the absorption image of the atomic clouds released from the QIUC trap after free expansion. Thus momentum distribution of atomic gas in the critical region can be extracted from the absorption image by subtracting the momentum distribution of thermal gas outside the critical region. According to the statistical results of the FWHMMD at different temperatures, we find that the FWHMMD suddenly reduces, thus revealing a very notable singularity behavior when the temperature is very close to the phase transition temperature Tc.
A peculiarity of trapped atomic gases is that Bose-Einstein condensation (BEC) shows up not only in momentum space, where it is usually discussed in traditional textbooks, but also in coordinate space. This is the consequence of the inhomogeneity of these systems, which makes it possible to separate the condensate from the thermal component also in coordinate space. Actually, most of measurements in trapped gases have been so far limited to the study of density profiles and of the effects of BEC in coordinate space. The possibility of a direct measurement of the momentum distribution, as emerged from recent experiments based on two-photon Bragg scattering, is highly appealing. In fact, despite the inhomogeneity of the gas, BEC shows up more deeply in momentum than in coordinate space. In current experiments on harmonically-confined Bose gases, the sizes of the condensate R and of the thermal cloud R_T are in fact typically comparable.
The study of BEC in weakly interacting system holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles, and may advance present understanding of superconductivity and superfluidity. By employing several new experimental techniques, the dark SPOT trap, rf induced evaporation, and the optically plugged magnetic trap, the authors were able to observe BEC in a gas of sodium atoms. The striking signature of Bose condensation was the sudden appearance of a bimodal velocity distribution below the critical temperature of {approximately} 2 {mu}K. The distribution consisted of an isotropic thermal distribution and an elliptical core attributed to the expansion of a dense condensate. Condensates contained up to 5 x 10{sup 5} atoms at densities exceeding 10{sup 14} cm{sup {minus}3}. Such high densities are unprecedented in cold atomic gases and open up new possibilities for studying a weakly interacting Bose gas over a broad range of densities and therefore strengths of interaction.
We have measured the critical atom number in an array of harmonically trapped two-dimensional (2D) Bose gases of rubidium atoms at different temperatures. We found this number to be about 5 times higher than predicted by the semiclassical theory of Bose-Einstein condensation (BEC) in the ideal gas. This demonstrates that the conventional BEC picture is inapplicable in an interacting 2D atomic gas, in sharp contrast to the three-dimensional case. A simple heuristic model based on the Berezinskii-Kosterlitz-Thouless theory of 2D superfluidity and the local density approximation accounts well for our experimental results.
Abstract This chapter describes the main properties of the ideal Bose gas confined by a harmonic potential. It calculates the temperature dependence of the condensate density and identify the value of the critical temperature in terms of the oscillator frequency and the number of atoms. This identification provides an important temperature scale for experiments with ultracold atomic gases. The chapter also discusses the behaviour of the expanding cloud after the release of the trap and how the anisotropy of the gas evolves in time, reflecting the initial anisotropy of the momentum distribution. It finally discusses the possibility of getting Bose–Einstein condensation through adiabatic transformations, working with non-harmonic trapping potentials or with mixtures of two atomic species trapped by different harmonic potentials.
We analyze the results of a recent experiment with bosonic rubidium atoms harmonically confined in a quasi-two-dimensional (2D) geometry. In this experiment a well-defined critical point was identified, which separates the high-temperature normal state characterized by a single component density distribution, and the low-temperature state characterized by a bimodal density distribution and the emergence of high-contrast interference between independent 2D clouds. We first show that this transition cannot be explained in terms of conventional Bose–Einstein condensation of the trapped ideal Bose gas. Using the local density approximation (LDA), we then combine the mean-field (MF) Hartree–Fock theory with the prediction for the Berezinskii–Kosterlitz–Thouless (BKT) transition in an infinite uniform system. If the gas is treated as a strictly 2D system, the MF predictions for the spatial density profiles significantly deviate from those of a recent quantum Monte Carlo (QMC) analysis. However, when the residual thermal excitation of the strongly confined degree of freedom is taken into account, excellent agreement is reached between the MF and the QMC approaches. For the interaction strength corresponding to the experiment, we predict a strong correction to the critical atom number with respect to the ideal gas theory (factor ∼2). Quantitative agreement between theory and experiment is reached concerning the critical atom number if the predicted density profiles are used for temperature calibration.