![]() The experiment is based on the generation and analysis of event-ready entanglement between two independently trapped single rubidium atoms located in buildings 400 metre apart14. Here we present an experimental system that enables for DIQKD between two distant users. The realization of DIQKD, however, is extremely challenging-mainly because it is difficult to establish high-quality entangled states between two remote locations with high detection efficiency. This test originates from the foundations of quantum physics and also ensures robustness against implementation loopholes13, thereby leaving only the integrity of the users’ locations to be guaranteed by other means. The proper and secure functioning of the devices can be certified by a statistical test using a Bell inequality10,11,12. Such cold and dense samples of polar molecules open the path to the exploration of many-body phenomena with strong dipolar interactions.ĭevice-independent quantum key distribution (DIQKD) enables the generation of secret keys over an untrusted channel using uncharacterized and potentially untrusted devices1,2,3,4,5,6,7,8,9. This large elastic-to-inelastic collision ratio allows us to cool the molecular gas to 21 nanokelvin, corresponding to 0.36 times the Fermi temperature. ![]() The microwave dressing induces strong tunable dipolar interactions between the molecules, leading to high elastic collision rates that can exceed the inelastic ones by at least a factor of 460. The molecules are protected from reaching short range with a repulsive barrier engineered by coupling rotational states with a blue-detuned circularly polarized microwave. Here we demonstrate evaporative cooling of a three-dimensional gas of fermionic sodium–potassium molecules to well below the Fermi temperature using microwave shielding. However, the intrinsically unstable collisions between molecules at short range have so far prevented direct cooling through elastic collisions to quantum degeneracy in three dimensions. Realizing their full potential requires cooling interacting molecular gases deeply into the quantum-degenerate regime. Ultracold polar molecules offer strong electric dipole moments and rich internal structure, which makes them ideal building blocks to explore exotic quantum matter1,2,3,4,5,6,7,8,9, implement quantum information schemes10,11,12 and test the fundamental symmetries of nature13. In particular, we consider halting and totality oracles, which belong to the most frequently investigated oracle machines in the theory of computation. Furthermore, we relate the problem of bandwidth computation to the theory of oracle machines. Among other things, our analysis includes a characterization of the arithmetic complexity of the bandwidth of such signals and yields a negative answer to the question of whether it is at least possible to compute non-trivial upper or lower bounds for the bandwidth of a bandlimited signal. In this work, we consider the most general class of band-limited signals, together with different computable descriptions thereof. Recently, it has been shown that there exist computable bandlimited signals with finite energy, the actual bandwidth of which is not a computable number, and hence cannot be computed on a digital computer. To this end we employ the concept of Turing computability, which exactly describes what is theoretically feasible and can be computed on a digital computer. ![]() In this paper we study questions related to the computability of the bandwidth of computable bandlimited signals. The bandwidth of a signal is an important physical property that is of relevance in many signal- and information-theoretic applications. Our results constitute an important step toward frequency-multiplexed quantum-network nodes operating directly at a telecommunication wavelength. We further implement spectrally multiplexed coherent control and find an optical coherence time of 0.11(1) milliseconds, approaching the lifetime limit of 0.3 milliseconds for the strongest-coupled emitters. In long-term measurements, they exhibit an exceptional spectral stability of <0.2 megahertz that is limited by the coupling to surrounding nuclear spins. Here, we avoid this limitation when enhancing the photon emission up to 70(12)-fold using a Fabry-Perot resonator with an embedded 19-micrometer-thin crystalline membrane, in which we observe around 100 individual erbium emitters. However, fluctuating charges and magnetic moments at the nearby interface then lead to spectral instability of the emitters. In solid-state devices, the required efficient light-emitter interface can be implemented by confining the light in nanophotonic structures. In a quantum network, coherent emitters can be entangled over large distances using photonic channels.
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