Knots are familiar entities that appear at a captivating nexus of art, technology, mathematics, and science [1]. As topologically stable objects within field theories, they have been speculatively proposed as explanations for diverse persistent phenomena, from atoms and molecules [2] to ball lightning [3] and cosmic textures in the universe [4]. Recent experiments have observed knots in a variety of classical contexts, including nematic liquid crystals [5][6][7], DNA [8], optical beams [9, 10], and water [11]. However, no experimental observations of knots have yet been reported in quantum matter. We demonstrate here the controlled creation [12] and detection of knot solitons [13,14] in the order parameter of a spinor Bose-Einstein condensate. The experimentally obtained images of the superfluid directly reveal the circular shape of the soliton core and its accompanying linked rings. Importantly, the observed texture corresponds to a topologically non-trivial element of the third homotopy group [15] and demonstrates the celebrated Hopf fibration [16], which unites many seemingly unrelated physical contexts [17,18]. Our observations of the knot soliton establish an experimental foundation for future studies of their stability and dynamics within quantum systems [19].

Topological defects play important roles throughout nature, appearing in contexts as diverse as cosmology, particle physics, superfluidity, liquid crystals, and metallurgy. Point defects can arise naturally as magnetic monopoles resulting from symmetry breaking in grand unified theories. We devised an experiment to create and detect quantum mechanical analogs of such monopoles in a spin-1 Bose-Einstein condensate. The defects, which were stable on the time scale of our experiments, were identified from spin-resolved images of the condensate density profile that exhibit a characteristic dependence on the choice of quantization axis. Our observations lay the foundation for experimental studies of the dynamics and stability of topological point defects in quantum systems.

We experimentally simulate a quantum-mechanical particle interacting with knotted electromagnetic fields.

We study numerically the detailed structure and decay dynamics of isolated monopoles in conditions similar to those of their recent experimental discovery. We find that the core of a monopole in the polar phase of a spin-1 Bose-Einstein condensate contains a small half-quantum vortex ring. Well after the creation of the monopole, we observe a dynamical quantum phase transition that destroys the polar phase. Strikingly, the resulting ferromagnetic order parameter exhibits a Dirac monopole in its synthetic magnetic field.Comment: 6 pages, 5 figure

We experimentally observe the decay dynamics of deterministically created isolated monopoles in spin-1 Bose-Einstein condensates. As the condensate undergoes a change between magnetic phases, the isolated monopole gradually evolves into a spin configuration hosting a Dirac monopole in its synthetic magnetic field. We characterize in detail the Dirac monopole by measuring the particle densities of the spin states projected along different quantization axes. Importantly, we observe the spontaneous emergence of nodal lines in the condensate density that accompany the Dirac monopole. We also demonstrate that the monopole decay accelerates in weaker magnetic field gradients.

We introduce topologically stable three-dimensional skyrmions in the cyclic and biaxial nematic phases of a spin-2 Bose-Einstein condensate. These skyrmions exhibit exceptionally high mapping degrees resulting from the versatile symmetries of the corresponding order parameters. We show how these structures can be created in existing experimental setups and study their temporal evolution and lifetime by numerically solving the three-dimensional Gross-Pitaevskii equations for realistic parameter values. Although the biaxial nematic and cyclic phases are observed to be unstable against transition towards the ferromagnetic phase, their lifetimes are long enough for the skyrmions to be imprinted and detected experimentally.

We show that quantum dots in photonic nanostructures provide a highly promising platform for deterministic generation of entangled multiphoton states. Our approach utilizes periodic driving of a quantum-dot based emitter and an efficient light-matter interface enabled by a photonic crystal waveguide. We assess the quality of the photonic states produced from a real system by including all experimentally relevant imperfections. Importantly, the protocol is robust against the nuclear spin bath dynamics due to a naturally built-in refocussing method reminiscent to spin echo. We demonstrate the feasibility of producing the Greenberger-Horne-Zeilinger and one-dimensional cluster states with fidelities and generation rates exceeding those achieved with conventional 'fusion' methods in current state-of-the-art experiments. The proposed hardware constitutes a scalable and resource-efficient approach towards implementation of measurement-based quantum communication and computing.

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