The condensation of fermion pairs lies at the heart of superfluidity. However, for strongly correlated systems with reduced dimensionality the mechanisms of pairing and condensation are still not fully understood. In our experiment we use ultracold atoms as a generic model system to study the phase transition from a normal to a condensed phase in a strongly interacting quasi-two-dimensional Fermi gas. Using a novel method, we obtain the in situ pair momentum distribution of the strongly interacting system and observe the emergence of a low-momentum condensate at low temperatures. By tuning temperature and interaction strength we map out the phase diagram of the quasi-2D BEC-BCS crossover.The characteristics of quantum many-body systems are strongly affected by their dimensionality and the strength of interparticle correlations. In particular, strongly correlated two-dimensional fermionic systems have been of interest because of their connection to high-T c superconductivity. Although they have been the subject of intense theoretical studies [1][2][3][4][5][6][7][8], a complete theoretical framework has not yet been established.Ultracold quantum gases are an ideal realization for exploring strongly interacting 2D Fermi gases, as they offer the possibility of independently tuning the dimensionality and the strength of interparticle interactions. Reducing the dimensionality [9] led to the observation of a Berezinskii-Kosterlitz-Thouless (BKT) type phase transition to a superfluid phase in weakly interacting 2D Bose gases [10,11]. Tuning the strength of interactions in a three-dimensional two-component Fermi gas made it possible to explore the crossover between a molecular Bose-Einstein Condensate (BEC) and a BCS superfluid [12][13][14][15].Recently, efforts have been made to combine reduced dimensionality with the tunability of interactions and to experimentally explore ultracold 2D Fermi gases [16][17][18][19][20][21]. However, the phase transition to a condensed phase has not yet been observed. Here, we report on the condensation of pairs of fermions in the quasi-2D BEC-BCS crossover.The BEC-BCS crossover smoothly links a bosonic superfluid of tightly bound diatomic molecules to a fermionic superfluid of Cooper pairs in 2D as well as 3D systems. However, changing the dimensionality leads to some inherent differences. In two dimensions, there is a two-body bound state for all values of the interparticle interaction. Furthermore, because of the enhanced role of * To whom correspondence should be addressed. E-mail: mries@physi.uni-heidelberg.de † These authors contributed equally to this work. ‡ Present address: MIT-Harvard Center for Ultracold Atoms, MIT, Cambridge, MA 02139, USA.fluctuations in 2D, true long-range order is forbidden for homogeneous systems at finite temperature [22,23]. Still, a low temperature superfluid phase with quasi-long-range order can emerge due to the BKT mechanism [24,25]. In a 2D gas with contact interactions, the interactions can be described by the 2D scattering length a 2D . Using th...

We experimentally investigate the first-order correlation function of a trapped Fermi gas in the two-dimensional BEC-BCS crossover. We observe a transition to a low-temperature superfluid phase with algebraically decaying correlations. We show that the spatial coherence of the entire trapped system can be characterized by a single temperature-dependent exponent. We find the exponent at the transition to be constant over a wide range of interaction strengths across the crossover. This suggests that the phase transitions in both the bosonic regime and the strongly interacting crossover regime are of Berezinskii-Kosterlitz-Thouless type and lie within the same universality class. On the bosonic side of the crossover, our data are well-described by the quantum Monte Carlo calculations for a Bose gas. In contrast, in the strongly interacting regime, we observe a superfluid phase which is significantly influenced by the fermionic nature of the constituent particles.Long-range coherence is the hallmark of superfluidity and Bose-Einstein condensation [1,2]. The character of spatial coherence in a system and the properties of the corresponding phase transitions are fundamentally influenced by dimensionality. The two-dimensional case is particularly intriguing as for a homogeneous system, true long-range order cannot persist at any finite temperature due to the dominant role of phase fluctuations with large wavelengths [3][4][5]. Although this prevents Bose-Einstein condensation in 2D, a transition to a superfluid phase with quasi-long-range order can still occur, as pointed out by Berezinskii, Kosterlitz, and Thouless (BKT) [6][7][8]. A key prediction of this theory is the scale-invariant behavior of the first-order correlation function g 1 (r), which, in the low-temperature phase, decays algebraically according to g 1 (r) ∝ r −η for large separations r. Importantly, the BKT theory for homogeneous systems predicts a universal value of η c = 1/4 at the critical temperature, accompanied by a universal jump of the superfluid density [9].Several key signatures of BKT physics have been experimentally observed in a variety of systems such as exciton-polariton condensates [10], layered magnets [11,12], liquid 4 He films [13], and trapped Bose gases [14][15][16][17][18][19][20]. Particularly in the context of superfluidity, the universal jump in the superfluid density was measured in thin films of liquid 4 He [13]. More recently, in the pioneering interference experiment with a weakly interacting Bose gas [14], the emergence of quasi-long-range order and the proliferation of vortices were shown.There are still important aspects of superfluidity in two-dimensional systems that remain to be understood, which we aim to elucidate in this work with ultracold atoms. One question is whether the BKT phenomenology can also be extended to systems with nonuniform density. Indeed, if the microscopic symmetries are the same, the general physical picture involving phase fluctuations should be valid also for inhomogeneous systems. However, it ...

We report the experimental measurement of the equation of state of a two-dimensional Fermi gas with attractive s-wave interactions throughout the crossover from a weakly coupled Fermi gas to a Bose gas of tightly bound dimers as the interaction strength is varied. We demonstrate that interactions lead to a renormalization of the density of the Fermi gas by several orders of magnitude. We compare our data near the ground state and at finite temperature to predictions for both fermions and bosons from Quantum Monte Carlo simulations and Luttinger-Ward theory. Our results serve as input for investigations of close-to-equilibrium dynamics and transport in the two-dimensional system. The rich phenomenology of fermionic many-body systems reveals itself on very different scales of energy, ranging from solid state materials and ultracold quantum gases to heavy-ion collisions and neutron stars. Understanding the underlying mechanisms promises substantial advances both on a fundamental and technological level. Ultracold quantum gases provide a platform for the exploration of the macroscopic phases and thermodynamic properties of fermionic many-body Hamiltonians in a highly controlled manner [1]. In particular, using strongly anisotropic traps, it is possible to enter the 2D regime [2][3][4][5][6][7] which is of large interest to the condensed matter community [8,9].The thermodynamic properties of a many-body system are encapsulated in its equation of state (EOS) n(µ, T, {g i }), which expresses the density n as a function of chemical potential µ, temperature T , and further system parameters {g i } characterizing, for instance, the interactions between particles. For ultracold atoms with short-range attraction, the only additional parameter is the s-wave scattering length a. This universality allows one to describe different atomic species by the same EOS n(µ, T, a). The equilibrium EOS is also the basis for studying dynamics close to thermal equilibrium.In this Letter, we report the experimental determination of the EOS of two-component fermions with attractive short-range interactions in the 2D BEC-BCS crossover regime. We tune the interaction strength using a Feshbach resonance to connect the well-known limits of a weakly attractive Fermi gas and a Bose gas of tightly bound dimers. We report the measurement of the finite temperature EOS in the intermediate, strongly correlated region and compare with theoretical predictions.Our experimental setup consists of a populationbalanced mixture of N ∼ 100, 0006 Li-atoms in the lowest two hyperfine states, which we denote by |1 and |2 . The interactions between both species can be tuned by means of a magnetic Feshbach resonance [10,11]. The atoms are trapped in a highly anisotropic trapping potential, which is radially symmetric to a high degree in the xy-plane and provides a tight confinement along the z-direction with the aspect ratio of frequencies ω x : ω y : ω z ≈ 1 : 1 : 310. A detailed description of the experiment is given in [7]. This strong anisotropy induces a quantu...

The frequency of the breathing mode of a two-dimensional Fermi gas with zero-range interactions in a harmonic confinement is fixed by the scale invariance of the Hamiltonian. Scale invariance is broken in the quantized theory by introducing the two-dimensional scattering length as a regulator. This is an example of a quantum anomaly in the field of ultracold atoms and leads to a shift of the frequency of the collective breathing mode of the cloud. In this work, we study this anomalous frequency shift for a two component Fermi gas in the strongly interacting regime. We measure significant upwards shifts away from the scale invariant result that show a strong interaction dependence. This observation implies that scale invariance is broken anomalously in the strongly interacting two-dimensional Fermi gas.Symmetries are an indispensable ingredient to any attempt of formulating a fundamental theory of nature. Yet, it is not allways true that one can make accurate predictions about the behaviour of some complex system based on the symmetries of its Hamiltonian alone. The fundamental reason behind this is the concept of symmetry breaking [1]. Symmetry violations often have drastic effects on the state of the system, for example when some metal breaks rotational invariance and becomes ferromagnetic. There are three different mechanisms through which a given system may violate some of the symmetries of its Hamiltonian: explicit, spontaneous and anomalous symmetry breaking [2].Quantum anomalies are violations of an exact symmetry of a classical action in the corresponding quantized theory [3]. They may occur when a cut-off has to be introduced to regularize divergent terms. This regulator may explicitly break some symmetry of the theory. If this symmetry is not restored even after the cut-off is removed at the end of the renormalization procedure, the symmetry is broken anomalously.Quantum anomalies are ubiquitous in quantum field theories and provide, important constraints on physical gauge theories like the standard model [4, 5] or on string theories [6, 7]. Whereas the formalisms of explicit and spontaneous symmetry breaking are frequently applied across many fields in physics [8][9][10], anomalous symmetry breaking is typically associated only with high energy physics. One exception was found in molecular physics [11,12] and here we report an observation of a quantum anomaly in the field of cold atom physics.A particular class of anomalies, called conformal anomalies, break the scale invariance of a theory, that is invariance of the Hamiltonian under r → λr. The most prominent examples are found in field theories like QED or QCD where the renormalized coupling constants depend on the energy scale and thus break scale invariance explicitly. In ordinary quantum mechanics the 1/r 2 -and the δ 2 -potential in 2D are well-known examples of conformal anomalies [13,14].Notably, the δ 2 -potential is used to model contact in-teractions in cold atom gases in two-dimensions as V int ∝ g 0 δ 2 (r i − r j ). Including the kinetic...

We demonstrate and characterize an experimental technique to directly image the momentum distribution of a strongly interacting two-dimensional quantum gas with high momentum resolution. We apply the principles of Fourier optics to investigate three main operations on the expanding gas: focusing, collimation and magnification. We focus the gas in the radial plane using a harmonic confining potential and thus gain access to the momentum distribution. We pulse a different harmonic potential to stop the rapid axial expansion which allows us to image the momentum distribution with high resolution. Additionally, we propose a method to magnify the mapped momentum distribution to access interesting momentum scales. All these techniques can be applied to a wide range of experiments and in particular to study many-body phases of quantum gases.