The observation of the superfluid to Mott insulator phase transition of ultracold atoms in optical lattices was an enabling discovery in experimental many-body physics, providing the first tangible example of a quantum phase transition (one that occurs even at zero temperature) in an ultracold atomic gas. For a trapped gas, the spatially varying local chemical potential gives rise to multiple quantum phases within a single sample, complicating the interpretation of bulk measurements. Here we report spatially resolved, in-situ imaging of a two-dimensional ultracold atomic gas as it crosses the superfluid to Mott insulator transition, providing direct access to individual characteristics of the insulating, superfluid and normal phases. We present results for the local compressibility in all phases, observing a strong suppression in the insulator domain and suppressed density fluctuations for the Mott insulator, in accordance with the fluctuation-dissipation theorem. Furthermore, we obtain a direct measure of the finite temperature of the system. Taken together, these methods enable a complete characterization of multiple phases in a strongly correlated Bose gas, and of the interplay between quantum and thermal fluctuations in the quantum critical regime.

The collective behavior of a many-body system near a continuous phase transition is insensitive to the details of its microscopic physics [1]. Characteristic features near the phase transition are that the thermodynamic observables follow generalized scaling laws [1]. The Berezinskii-KosterlitzThouless (BKT) phase transition [2,3] in two-dimensional (2D) Bose gases presents a particularly interesting case because the marginal dimensionality and intrinsic scaling symmetry[4] result in a broad fluctuation regime which manifests itself in an extended range of universal scaling behavior. Studies on BKT transition in cold atoms have stimulated great interest in recent years [5][6][7][8][9][10], clear demonstration of a critical behavior near the phase transition, however, has remained an elusive goal. Here we report the observation of a scale-invariant, universal behavior of 2D gases through in-situ density and density fluctuation measurements at different temperatures and interaction strengths. The extracted thermodynamic functions confirm a wide universal region near the BKT phase transition, provide a sensitive test to the universality prediction by classical-field theory [11,12] and quantum Monte Carlo (MC) calculations [13], and point toward growing density-density correlations in the fluctuation region. Our assay raises new perspectives to explore further universal phenomena in the realm of classical and quantum critical physics.PACS numbers: 64.60.F-,05.70. Jk,67.10.Ba, In 2D Bose gases, critical behavior develops in the BKT transition regime, where an ordered phase with finite-ranged coherence competes with thermal fluctuations and induces a continuous phase transition from normal gas to superfluid with quasi-long range order [3]. In this fluctuation region, a universal and scale-invariant description for the system is expected through the powerlaw scaling of thermodynamic quantities with respect to the coupling strength and a characteristic length scale [12,14], e.g., thermal de Broglie wavelength (Fig. 1a). For weakly interacting gases at finite temperatures, in particular, the scale invariance prevails over the normal, fluctuation, and superfluid regions because of the densityindependent coupling constant[15] and the symmetry of underlying Hamiltonian [4].In this letter, we verify the scale invariance and universality of interacting 2D Bose gases, and identify BKT critical points. We test scale invariance of in situ density and density fluctuations of 133 Cs 2D gases at various temperatures. We study the universality near the BKT transition by tuning the atomic scattering length using a magnetic Feshbach resonance [16] and observing a universal scaling behavior of the equation of state and the quasi-condensate density. Finally, by comparing the local density fluctuations and the compressibility derived from the density profiles, we provide strong evidence of a growing density-density correlation in the fluctuation regime.We begin the experiment by loading a nearly pure 133 Cs Bose condensate of N = 2 × 10 ...

We investigate collective emission from coherently driven ultracold 88Sr atoms. We perform two sets of experiments using a strong and weak transition that are insensitive and sensitive, respectively, to atomic motion at 1 μK. We observe highly directional forward emission with a peak intensity that is enhanced, for the strong transition, by >103 compared with that in the transverse direction. This is accompanied by substantial broadening of spectral lines. For the weak transition, the forward enhancement is substantially reduced due to motion. Meanwhile, a density-dependent frequency shift of the weak transition (∼10% of the natural linewidth) is observed. In contrast, this shift is suppressed to <1% of the natural linewidth for the strong transition. Along the transverse direction, we observe strong polarization dependences of the fluorescence intensity and line broadening for both transitions. The measurements are reproduced with a theoretical model treating the atoms as coherent, interacting radiating dipoles.