Cold-Atom Elevator: From Edge-State Injection to the Preparation of Fractional Chern Insulators.

Optical box traps offer new possibilities for quantum-gas experiments. Building on their exquisite spatial and temporal control, we propose to engineer system-reservoir configurations using box traps, in view of preparing and manipulating topological atomic states in optical lattices. First, we consider the injection of particles from the reservoir to the system: this scenario is shown to be particularly well suited to activating energy-selective chiral edge currents, but also to prepare fractional Chern insulating ground states. Then, we devise a practical evaporative-cooling scheme to effectively cool down atomic gases into topological ground states. Our open-system approach to optical-lattice settings provides a new path for the investigation of ultracold quantum matter, including strongly correlated and topological phases.

Observation of fragmentation of a spinor Bose-Einstein condensate.

Weakly interacting Bose gases usually form Bose-Einstein condensates in which most particles occupy the same single-particle state. However, when this state cannot realize a continuous symmetry of the many-body Hamiltonian, a fragmented condensate exhibiting the expected symmetry may emerge. Here, we produced a three-fragment condensate for a mesoscopic spin-1 gas of about 100 atoms, with anti-ferromagnetic interactions and vanishing collective spin. Using a spin-resolved detection approaching single-atom resolution, we show that the reconstructed state is close to the expected many-body ground state, whereas one-body observables are the same as for a completely mixed state. Our results highlight how the interplay between symmetry and interactions generates entanglement in a mesoscopic quantum system.

From Many-Body Oscillations to Thermalization in an Isolated Spinor Gas.

The dynamics of a many-body system can take many forms, from a purely reversible evolution to fast thermalization. Here we show experimentally and numerically that an assembly of spin-1 atoms all in the same spatial mode allows one to explore this wide variety of behaviors. When the system can be described by a Bogoliubov analysis, the relevant energy spectrum is linear and leads to undamped oscillations of many-body observables. Outside this regime, the nonlinearity of the spectrum leads to irreversibility, characterized by a universal behavior. When the integrability of the Hamiltonian is broken, a chaotic dynamics emerges and leads to thermalization, in agreement with the eigenstate thermalization hypothesis paradigm.

Probing Spin Correlations in a Bose-Einstein Condensate Near the Single-Atom Level.

Using parametric conversion induced by a Shapiro-type resonance, we produce and characterize a two-mode squeezed vacuum state in a sodium spin 1 Bose-Einstein condensate. Spin-changing collisions generate correlated pairs of atoms in the m=±1 Zeeman states out of a condensate with initially all atoms in m=0. A novel fluorescence imaging technique with sensitivity ΔN∼1.6 atom enables us to demonstrate the role of quantum fluctuations in the initial dynamics and to characterize the full distribution of the final state. Assuming that all atoms share the same spatial wave function, we infer a squeezing parameter of 15.3 dB.

Enhanced Magnetic Sensitivity with Non-Gaussian Quantum Fluctuations.

The precision of a quantum sensor can overcome its classical counterpart when its constituents are entangled. In Gaussian squeezed states, quantum correlations lead to a reduction of the quantum projection noise below the shot noise limit. However, the most sensitive states involve complex non-Gaussian quantum fluctuations, making the required measurement protocol challenging. Here we measure the sensitivity of nonclassical states of the electronic spin J=8 of dysprosium atoms, created using light-induced nonlinear spin coupling. Magnetic sublevel resolution enables us to reach the optimal sensitivity of non-Gaussian (oversqueezed) states, well above the capability of squeezed states and about half the Heisenberg limit.

Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom.

Coherent superposition states of a mesoscopic quantum object play a major role in our understanding of the quantum to classical boundary, as well as in quantum-enhanced metrology and computing. However, their practical realization and manipulation remains challenging, requiring a high degree of control of the system and its coupling to the environment. Here, we use dysprosium atoms-the most magnetic element in its ground state-to realize coherent superpositions between electronic spin states of opposite orientation, with a mesoscopic spin size J = 8. We drive coherent spin states to quantum superpositions using non-linear light-spin interactions, observing a series of collapses and revivals of quantum coherence. These states feature highly non-classical behavior, with a sensitivity to magnetic fields enhanced by a factor 13.9(1.1) compared to coherent spin states-close to the Heisenberg limit 2J = 16-and an intrinsic fragility to environmental noise.

Dynamic Optical Lattices of Subwavelength Spacing for Ultracold Atoms.

We propose a scheme for realizing lattice potentials of subwavelength spacing for ultracold atoms. It is based on spin-dependent optical lattices with a time-periodic modulation. We show that the atomic motion is well described by the combined action of an effective, time-independent lattice of small spacing, together with a micromotion associated with the time modulation. A numerical simulation shows that an atomic gas can be adiabatically loaded into the effective lattice ground state, for time scales comparable to the ones required for adiabatic loading of standard optical lattices. We generalize our scheme to a two-dimensional geometry, leading to Bloch bands with nonzero Chern numbers. The realization of lattices of subwavelength spacing allows for the enhancement of energy scales, which could facilitate the achievement of strongly correlated (topological) states.

Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas.

Phase transitions are ubiquitous in our three-dimensional world. By contrast, most conventional transitions do not occur in infinite uniform low-dimensional systems because of the increased role of thermal fluctuations. The crossover between these situations constitutes an important issue, dramatically illustrated by Bose-Einstein condensation: a gas strongly confined along one direction of space may condense along this direction without exhibiting true long-range order in the perpendicular plane. Here we explore transverse condensation for an atomic gas confined in a novel trapping geometry, with a flat in-plane bottom, and we relate it to the onset of an extended (yet of finite-range) in-plane coherence. By quench crossing the transition, we observe topological defects with a mean number satisfying the universal scaling law predicted by Kibble-Zurek mechanism. The approach described can be extended to investigate the topological phase transitions that take place in planar quantum fluids.

Determination of scale-invariant equations of state without fitting parameters: application to the two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition.

We present a general "fit-free" method for measuring the equation of state (EoS) of a scale-invariant gas. This method, which is inspired from the procedure introduced by Ku et al. [Science 335, 563 (2012)] for the unitary three-dimensional Fermi gas, provides a general formalism which can be readily applied to any quantum gas in a known trapping potential, in the frame of the local density approximation. We implement this method on a weakly interacting two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition and determine its EoS with unprecedented accuracy in the critical region. Our measurements provide an important experimental benchmark for classical-field approaches which are believed to accurately describe quantum systems in the weakly interacting but nonperturbative regime.

Reaching fractional quantum Hall states with optical flux lattices.

We present a robust scheme by which fractional quantum Hall states of bosons can be achieved for ultracold atomic gases. We describe a new form of optical flux lattice, suitable for commonly used atomic species with ground state angular momentum J(g) = 1, for which the lowest energy band is topological and nearly dispersionless. Through exact diagonalization studies, we show that, even for moderate interactions, the many-body ground states consist of bosonic fractional quantum Hall states, including the Laughlin state and the Moore-Read (Pfaffian) state. These phases are shown to have energy gaps that are larger than temperature scales achievable in ultracold gases.

Direct imaging of topological edge states in cold-atom systems.

Detecting topological order in cold-atom experiments is an ongoing challenge, the resolution of which offers novel perspectives on topological matter. In material systems, unambiguous signatures of topological order exist for topological insulators and quantum Hall devices. In quantum Hall systems, the quantized conductivity and the associated robust propagating edge modes--guaranteed by the existence of nontrivial topological invariants--have been observed through transport and spectroscopy measurements. Here, we show that optical-lattice-based experiments can be tailored to directly visualize the propagation of topological edge modes. Our method is rooted in the unique capability for initially shaping the atomic gas and imaging its time evolution after suddenly removing the shaping potentials. Our scheme, applicable to an assortment of atomic topological phases, provides a method for imaging the dynamics of topological edge modes, directly revealing their angular velocity and spin structure.

Exploring the thermodynamics of a two-dimensional Bose gas.

Using in situ measurements on a quasi-two-dimensional, harmonically trapped (87)Rb gas, we infer various equations of state for the equivalent homogeneous fluid. From the dependence of the total atom number and the central density of our clouds with chemical potential and temperature, we obtain the equations of state for the pressure and the phase-space density. Then, using the approximate scale invariance of this 2D system, we determine the entropy per particle and find very low values (below 0.1k(B)) in the strongly degenerate regime. This shows that this gas can constitute an efficient coolant for other quantum fluids. We also explain how to disentangle the various contributions (kinetic, potential, interaction) to the energy of the trapped gas using a time-of-flight method, from which we infer the reduction of density fluctuations in a nonfully coherent cloud.

Can a Bose gas be saturated?

We scrutinize the concept of saturation of the thermal component in a partially condensed trapped Bose gas. Using a 39K gas with tunable interactions, we demonstrate strong deviation from Einstein's textbook concept of a saturated vapor. However, the saturation picture can be recovered by extrapolation to the strictly noninteracting limit. We provide evidence for the universality of our observations through additional measurements with a different atomic species, 87Rb.

Sum-frequency generation of 589 nm light with near-unit efficiency.

We report on a laser source at 589 nm based on sum-frequency generation of two infrared laser at 1064 nm and 1319 nm. Output power as high as 800 mW is achieved starting from 370 mW at 1319 nm and 770 mW at 1064 nm, corresponding to converting roughly 90% of the 1319 nm photons entering the cavity. The power and frequency stability of this source are ideally suited for cooling and trapping of sodium atoms.

Measuring the one-particle excitations of ultracold fermionic atoms by stimulated Raman spectroscopy.

We propose a Raman spectroscopy technique which is able to probe the one-particle Green function, the Fermi surface, and the quasiparticles of a gas of strongly interacting ultracold atoms. We give quantitative examples of experimentally accessible spectra. The efficiency of the method is validated by means of simulated images for the case of a usual Fermi liquid as well as for more exotic states: specific signatures of, e.g., a d-wave pseudogap are clearly visible.

Critical point of an interacting two-dimensional atomic Bose gas.

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.

Berezinskii-Kosterlitz-Thouless crossover in a trapped atomic gas.

Any state of matter is classified according to its order, and the type of order that a physical system can possess is profoundly affected by its dimensionality. Conventional long-range order, as in a ferromagnet or a crystal, is common in three-dimensional systems at low temperature. However, in two-dimensional systems with a continuous symmetry, true long-range order is destroyed by thermal fluctuations at any finite temperature. Consequently, for the case of identical bosons, a uniform two-dimensional fluid cannot undergo Bose-Einstein condensation, in contrast to the three-dimensional case. However, the two-dimensional system can form a 'quasi-condensate' and become superfluid below a finite critical temperature. The Berezinskii-Kosterlitz-Thouless (BKT) theory associates this phase transition with the emergence of a topological order, resulting from the pairing of vortices with opposite circulation. Above the critical temperature, proliferation of unbound vortices is expected. Here we report the observation of a BKT-type crossover in a trapped quantum degenerate gas of rubidium atoms. Using a matter wave heterodyning technique, we observe both the long-wavelength fluctuations of the quasi-condensate phase and the free vortices. At low temperatures, the gas is quasi-coherent on the length scale set by the system size. As the temperature is increased, the loss of long-range coherence coincides with the onset of proliferation of free vortices. Our results provide direct experimental evidence for the microscopic mechanism underlying the BKT theory, and raise new questions regarding coherence and superfluidity in mesoscopic systems.

Quantized vortices in the ideal bose gas: a physical realization of random polynomials.

We propose a physical system allowing one to experimentally observe the distribution of the complex zeros of a random polynomial. We consider a degenerate, rotating, quasi-ideal atomic Bose gas prepared in the lowest Landau level. Thermal fluctuations provide the randomness of the bosonic field and of the locations of the vortex cores. These vortices can be mapped to zeros of random polynomials, and observed in the density profile of the gas.

Observation of phase defects in quasi-two-dimensional Bose-Einstein condensates.

We have observed phase defects in quasi-2D Bose-Einstein condensates close to the condensation temperature. Either a single or several equally spaced condensates are produced by selectively evaporating the sites of a 1D optical lattice. When several clouds are released from the lattice and allowed to overlap, dislocation lines in the interference patterns reveal nontrivial phase defects.