Quantum droplets are small clusters of atoms self-bound by the balance of attractive and repulsive forces. Here we report on the observation of a novel type of droplets, solely stabilized by contact interactions in a mixture of two Bose-Einstein condensates. We demonstrate that they are several orders of magnitude more dilute than liquid helium by directly measuring their size and density via in situ imaging. Moreover, by comparison to a single-component condensate, we show that quantum many-body effects stabilize them against collapse. We observe that droplets require a minimum atom number to be stable. Below, quantum pressure drives a liquid-to-gas transition that we map out as a function of interaction strength. These ultra-dilute isotropic liquids remain weakly interacting and constitute an ideal platform to benchmark quantum many-body theories.Quantum fluids can be liquids -of fixed volume -or gases, depending on the attractive or repulsive character of the inter-particle interactions and their interplay with quantum pressure. Liquid helium is the prime example of quantum fluid. For small particle numbers it forms self-bound liquid droplets: nanometer-sized, dense and strongly interacting clusters of helium atoms. Understanding their properties, which directly reflect their quantum nature, is challenging and requires a good knowledge of the short-range details of the interatomic potential [1,2]. Very different quantum droplets, more than 2 orders of magnitude larger and 8 orders of magnitude more dilute, have recently been proposed in ultracold atomic gases [3]. Interestingly, these ultra-dilute systems enable a much simpler microscopic description, while remaining in the weakly interacting regime. They are thus amenable to well controlled theoretical studies.The formation of quantum droplets requires a balance between attractive forces, which hold them together, and repulsive ones that stabilize them against collapse. In helium droplets, the repulsion is dominated by the electronic Pauli exclusion principle, which arises from quantum statistics. In contrast, in ultracold atomic droplets the repulsion stems from quantum fluctuations, which are a genuine quantum many-body effect. These can be revealed in systems with competing interactions, where mean-field forces of different origins almost completely cancel out and result in a small residual attraction. There, beyond mean-field effects remain sizeable even in the weakly interacting regime. To first order they lead to the Lee-Huang-Yang repulsive energy [4], comparable in strength to the residual mean-field attraction. Recently, ultracold atomic droplets have been realized in magnetic quantum gases with competing attractive dipolar and repulsive contact interactions [5][6][7][8][9][10]. In this case, the anisotropic character of the magnetic dipole-dipole force leads to the formation of filament-like self-bound droplets with highly anisotropic properties [9,11,12]. Given the generality of the stabilization mechanism, droplets should in fact also exist in si...

Attractive Bose-Einstein condensates can host two types of macroscopic self-bound states: bright solitons and quantum droplets. Here, we investigate the connection between them with a Bose-Bose mixture confined in an optical waveguide. We show theoretically that, depending on atom number and interaction strength, solitons and droplets can be smoothly connected or remain distinct states coexisting only in a bistable region. We measure their spin composition, extract their density for a broad range of parameters, and map out the boundary of the region separating solitons from droplets.

Long-term inertial navigation is currently limited by the bias drifts of gyroscopes and accelerometers. Ultra-stable cold-atom interferometers offer a promising alternative for the next generation of high-end navigation systems. Here, we present an experimental setup and an algorithm hybridizing a stable matter-wave interferometer with a classical accelerometer. We use correlations between the quantum and classical devices to track the bias drift of the latter and form a hybrid sensor. We apply the Kalman filter formalism to obtain an optimal estimate of the bias and simulate experimentally a harsh environment representative of that encountered in mobile sensing applications. We show that our method is more precise and robust than traditional sine-fitting methods. The resulting sensor exhibits a 400 Hz bandwidth and reaches a stability of 10 ng after 11 h of integration.Inertial navigation systems determine the position of a moving vehicle by continuously measuring its acceleration and rotation rate, and subsequently integrating the equations of motion [1]. These systems are limited by slow drifts of the biases inherent to their inertial sensors, which ultimately lead to large speed and position errors after integration. Currently, the long-term bias stability of navigation-grade accelerometers is on the order of 10 µg-which, in the absence of aiding sensors such as satellite navigation systems, leads to horizontal position oscillations of 60 m at the characteristic Schuler period of 84.4 minutes [1,2].Since their first demonstration in the early 1990s, atom interferometers (AIs) have proven to be excellent absolute inertial sensors-having been exploited as ultra-high sensitivity instruments for fundamental tests of physics [3][4][5][6][7][8], and as state-of-the-art gravimeters with accuracies in the range of 1 − 10 ng achieved both in laboratories [9][10][11][12][13][14] and with compact transportable systems [15][16][17][18][19]. As a result, they have been proposed for the next generation of inertial navigation systems [20][21][22][23]. However, cold-atom-based sensors generally possess a small bandwidth, and suffer from low repetition rates (with the exceptions of Refs. [24,25]) and dead times during which no inertial measurements can be made. In comparison, mechanical accelerometers exhibit broad bandwidths compatible with navigation applications [26], but are afflicted by long-term bias and scale factor drifts. These two types of sensors can thus be hybridized [27] in order to benefit from the best of both worlds-in strong analogy with the strategy employed in atomic clocks [28].Here, we use correlations between an AI and a classical accelerometer to track the bias of the latter, and we present an approach based on a non-linear Kalman *