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Using a quantum gas setup consisting of a Bose-Einstein condensate strongly coupled to a high-finesse optical cavity by a transverse pump laser, we experimentally observe an instability of a dissipative continuous time crystal (CTC) towards a time crystalline state exhibiting two prominent oscillation frequencies. Applying a mean-field approximation model and a Floquet analysis, we theoretically confirm that this transition is a manifestation in a many-body system of a torus bifurcation between a limit cycle (LC) and a limit torus (LT). We theoretically illustrate the LC and LT attractors using the minimal model and experimentally reconstruct them using Takens' embedding theorem applied to the non-destructively measured intracavity photon dynamics.

Discrete (DTCs) and continuous time crystals (CTCs) are novel dynamical many-body states, that are characterized by robust self-sustained oscillations, emerging via spontaneous breaking of discrete or continuous time translation symmetry. DTCs are periodically driven systems that oscillate with a subharmonic of the external drive, while CTCs are continuously driven and oscillate with a frequency intrinsic to the system. Here, we explore a phase transition from a continuous time crystal to a discrete time crystal. A CTC with a characteristic oscillation frequency $\omega_\mathrm{CTC}$ is prepared in a continuously pumped atom-cavity system. Modulating the pump intensity of the CTC with a frequency $\omega_{\mathrm{dr}}$ close to $2\,\omega_\mathrm{CTC}$ leads to robust locking of $\omega_\mathrm{CTC}$ to $\omega_{\mathrm{dr}}/2$, and hence a DTC arises. This phase transition in a quantum many-body system is related to subharmonic injection locking of non-linear mechanical and electronic oscillators or lasers.

We propose a general mechanism for generating limit cycle (LC) oscillations by coupling a linear bosonic mode to a dissipative nonlinear bosonic mode. By analyzing the stability matrix, we show that LCs arise due to a supercritical Hopf bifurcation. We find that the existence of LCs is independent of the sign of the effective nonlinear interaction. The LC phase can be classified as a continuous time crystal (CTC), if it emerges in a many-body system. The bosonic model can be realised in three-level systems interacting with a quantised light mode as realised in atom-cavity systems. Using such a platform, we experimentally observe LCs for the first time in an atom-cavity system with attractive optical pump lattice, thereby confirming our theoretical predictions.

We present an efficient cooling scheme for fermionic $^{40}{\text{K}}$ atoms, using laser light red and blue detuned with respect to the $\mathcal{D}2$ and $\mathcal{D}1$ principle flourescence lines, respectively. The cooling scheme is found to significantly increase the saturation level for loading of a $^{40}{\text{K}}$ magneto-optical trap (MOT), resulting in increased atom numbers or decreased cycle times. While the attainable $^{40}{\text{K}}$ atom number is approximately doubled if exclusively $^{40}{\text{K}}$ atoms are cooled, the scheme is particularly powerful for dual-species MOTs, for example, if $^{40}{\text{K}}$ and $^{87}{\text{Rb}}$ atoms are cooled simultaneously in the same MOT configuration. The typical atom losses due to light-assisted hetero-nuclear collisions between $^{40}{\text{K}}$ and $^{87}{\text{Rb}}$ seem to be reduced giving rise to a threefold improvement of the $^{40}{\text{K}}$ atom number as compared to that in a conventional dual-species MOT, operating merely with $\mathcal{D}2$ light. Our scheme can be a useful extension to most dual-species experiments, aiming to reach simultaneous degeneracy of both species., Comment: 8 pages, 5 figures

Quantum gas systems provide a unique experimental platform to study a fundamental paradigm of quantum many-body physics: the crossover between Bose-Einstein condensed (BEC) molecular pairs and Bardeen Cooper Schrieffer (BCS) superfluidity. Some studies have considered quantum gas samples confined in optical lattices, however, focusing on the case, when only the lowest Bloch band is populated, such that orbital degrees of freedom are excluded. In this work, for the first time, ultracold Feshbach molecules of fermionic $^{40}K$ atoms are selectively prepared in the second Bloch band of an optical square lattice, covering a wide range of interaction strengths including the regime of unitarity. Binding energies and band relaxation dynamics are measured by means of a method resembling mass spectrometry. The longest lifetimes arise for strongly interacting Feshbach molecules at the onset of unitarity with values around 300 ms for the lowest band and 100 ms for the second band. In the case of strong confinement in a deep lattice potential, we observe bound dimers also for negative values of the s-wave scattering length, extending previous findings for molecules in the lowest band. Our work prepares the stage for orbital BEC-BCS crossover physics., Comment: 7 pages, 5 figures

Understanding strongly correlated quantum materials, such as high $T_\textrm{c}$ superconductors, iron-based superconductors, and twisted bilayer graphene systems, remains to be one of the outstanding challenges in condensed matter physics. Quantum simulation with ultra-cold atoms in particular optical lattices, which provide orbital degrees of freedom, is a powerful tool to contribute new insights to this endeavor. Here, we report the experimental realization of an unconventional Bose-Einstein condensate of $^{87}$Rb atoms populating degenerate $p$-orbitals in a triangular optical lattice, exhibiting remarkably long coherence times. Using time-of-flight spectroscopy, we observe that this state spontaneously breaks the rotational symmetry and its momentum spectrum agrees with the theoretically predicted coexistence of exotic stripe and loop current orders. Like certain strongly correlated electronic systems with intertwined orders, as high-$T_\textrm{c}$ cuprate superconductors, twisted bilayer graphene, and the recently discovered chiral density-wave state in kagome superconductors $\textrm{AV}_3 \textrm{Sb}_5$ (A=K, Rb, Cs), the newly demonstrated quantum state, in spite of its markedly different energy scale and the bosonic quantum statistics, exhibits multiple symmetry breakings at ultralow temperatures. These findings hold the potential to enhance our comprehension of the fundamental physics governing these intricate quantum materials., Comment: 7+10 pages, 3+12 figures

Fostered by novel analytical techniques, digitalization and automation, modern bioprocess development provides high amounts of heterogeneous experimental data, containing valuable process information. In this context, data-driven methods like machine learning (ML) approaches have a high potential to rationally explore large design spaces while exploiting experimental facilities most efficiently. The aim of this review is to demonstrate how ML methods have been applied so far in bioprocess development, especially in strain engineering and selection, bioprocess optimization, scale-up, monitoring and control of bioprocesses. For each topic, we will highlight successful application cases, current challenges and point out domains that can potentially benefit from technology transfer and further progress in the field of ML., Comment: Submitted to "Trends in Biotechnology"

We demonstrate condensate formation in a dark state in an ultracold quantum gas coupled to a high-finesse cavity and pumped by a shaken optical lattice. We show experimentally and theoretically that the atoms in the dark state display a strong suppression of the coupling to the cavity. On the theory side, this is supported by solving the dynamics of a minimal three-level model and of the full atom-cavity system. The symmetry of the condensate wave function is anti-symmetric with respect to the potential minima of the pump lattice, and displays a staggered sign along the cavity direction. This symmetry decouples the dark state from the cavity, and is preserved when the pump intensity is switched off.

Hexagonal optical lattices offer a tunable platform to study exotic orbital physics in solid state materials. Here, we present a versatile high-precision scheme to implement a hexagonal optical lattice potential, which is engineered by overlapping two independent triangular optical sublattices generated by laser beams with slightly different wavelengths around 1064 nm. This enables us to precisely control the detailed structure of the hexagonal lattice by adjusting the relative position and the relative lattice depth of the two triangular optical sublattices. Taking advantage of the sensitive dependence of the second Bloch band on small lattice deformations, we propose a strategy to optimize the optical lattice geometry with an extremely high precision. This method can also be extended to other lattice configurations involving more than two sublattices. Our work provides the experimental requirements in the search for novel orbital physics of ultracold atoms, for example, in the flat $p$-band of the hexagonal optical lattice., Comment: 9 pages, 4 figures

We propose and experimentally explore a method for realizing frustrated lattice models using a Bose-Einstein condensate held in an optical square lattice. A small lattice distortion opens up an energy gap such the lowest band splits into two. Along the edge of the first Brillouin zone for both bands a nearly flat energy-momentum dispersion is realized. For the excited band a highly degenerate energy minimum arises. By loading ultracold atoms into the excited band, a classically frustrated $XY$ model is formed, describing rotors on a square lattice with competing nearest and next nearest tunnelling couplings. Our experimental optical lattice provides a regime, where a fully coherent Bose-Einstein condensate is observed, and a regime where frustration is expected. If we adiabatically tune from the condensate regime to the regime of frustration, the momentum spectra shows a complete loss of coherence. Upon slowly tuning back to the condensate regime, coherence is largely restored. Good agreement with model calculations is obtained., Comment: 5 pages, 3 figures

Time crystals are classified as discrete or continuous depending on whether they spontaneously break discrete or continuous time translation symmetry. While discrete time crystals have been extensively studied in periodically driven systems since their recent discovery, the experimental realization of a continuous time crystal is still pending. Here, we report the observation of a limit cycle phase in a continuously pumped dissipative atom-cavity system, which is characterized by emergent oscillations in the intracavity photon number. We observe that the phase of this oscillation is random for different realizations, and hence this dynamical many-body state breaks continuous time translation symmetry spontaneously. The observed robustness of the limit cycles against temporal perturbations confirms the realization of a continuous time crystal.

Star formation induced by a spiral shock wave, which in turn is generated by a spiral density wave, produces an azimuthal age gradient across the spiral arm, which has opposite signs on either side of the corotational resonance. An analysis of the spatial separation between young star clusters and nearby HII regions made it possible to determine the position of the corotation radius in the studied galaxies. Fourier analysis of the gas velocity field in the same galaxies independently confirmed the corotation radius estimates obtained by the morphological method presented here., Comment: 15 pages, 13 figures, 2 tables. Accepted for publication in MNRAS

A periodically driven open three-level Dicke model is realized by resonantly shaking the pump field in an atom-cavity system. As an unambiguous signature, we demonstrate the emergence of a dynamical phase, in which the atoms periodically localize between the antinodes of the pump lattice, associated with an oscillating net momentum along the pump axis. We observe this dynamical phase through the periodic switching of the relative phase between the pump and cavity fields at a small fraction of the driving frequency, suggesting that it exhibits a time crystalline character.

We investigate the three-level Dicke model, which describes a fundamental class of light-matter systems. We determine the phase diagram in the presence of dissipation, which we assume to derive from photon loss. Utilizing both analytical and numerical methods we characterize the incommensurate time crystalline, light-induced, and light-enhanced superradiant states in the phase diagram for the parametrically driven system. As a primary application, we demonstrate that a shaken atom-cavity system is naturally approximated via a parametrically driven-dissipative three-level Dicke model.

Spin-polarized samples and spin mixtures of quantum degenerate fermionic atoms are prepared in selected excited Bloch bands of an optical chequerboard square lattice. For the spin-polarized case, extreme band lifetimes above $10\,$s are observed, reflecting the suppression of collisions by Pauli's exclusion principle. For spin mixtures, lifetimes are reduced by an order of magnitude by two-body collisions between different spin components, but still remarkably large values of about one second are found. By analyzing momentum spectra, we can directly observe the orbital character of the optical lattice. The observations demonstrated here form the basis for exploring the physics of Fermi gases with two paired spin components in orbital optical lattices, including the regime of unitarity., Comment: 10 pages, 7 figures

The competition between short-range and cavity-mediated infinite-range interactions in a cavity-boson system leads to the existence of a superfluid phase and a Mott-insulator phase within the self-organized regime. In this work, we quantitatively compare the steady-state phase boundaries of this transition measured in experiments and simulated using the Multiconfigurational Time-Dependent Hartree Method for Indistinguishable Particles. To make the problem computationally feasible, we represent the full system by the exact many-body wave function of a two-dimensional four-well potential. We argue that the validity of this representation comes from the nature of both the cavity-atomic system and the Bose-Hubbard physics. Additionally we show that the chosen representation only induces small systematic errors, and that the experimentally measured and theoretically predicted phase boundaries agree reasonably. We thus demonstrate a new approach for the quantitative numerical determination of the superfluid--Mott-insulator phase boundary., Comment: 30 pages, 9 figures, 1 table. Submission to SciPost

In April 2020, the QUality Assessment and REProducibility for Instruments and Images in Light Microscopy (QUAREP-LiMi) initiative was formed. This initiative comprises imaging scientists from academia and industry who share a common interest in achieving a better understanding of the performance and limitations of microscopes and improved quality control (QC) in light microscopy. The ultimate goal of the QUAREP-LiMi initiative is to establish a set of common QC standards, guidelines, metadata models, and tools, including detailed protocols, with the ultimate aim of improving reproducible advances in scientific research. This White Paper 1) summarizes the major obstacles identified in the field that motivated the launch of the QUAREP-LiMi initiative; 2) identifies the urgent need to address these obstacles in a grassroots manner, through a community of stakeholders including, researchers, imaging scientists, bioimage analysts, bioimage informatics developers, corporate partners, funding agencies, standards organizations, scientific publishers, and observers of such; 3) outlines the current actions of the QUAREP-LiMi initiative, and 4) proposes future steps that can be taken to improve the dissemination and acceptance of the proposed guidelines to manage QC. To summarize, the principal goal of the QUAREP-LiMi initiative is to improve the overall quality and reproducibility of light microscope image data by introducing broadly accepted standard practices and accurately captured image data metrics., Comment: 17 pages, 3 figures, shortened abstract, Co-Lead Authors: Glyn Nelson and Ulrike Boehm, Corresponding author: Roland Nitschke

The formation of a phase of matter can be associated with the spontaneous breaking of a symmetry. For crystallization, this broken symmetry is the spatial translation symmetry, as the atoms spontaneously localize in a periodic fashion. In analogy to spatial crystals, the spontaneous breaking of temporal translation symmetry results in the formation of time crystals. While recent and on-going experiments on driven isolated systems aim to minimize dissipative processes, as it is an undesired source of decay, well-designed dissipation has been put forth as a constitutive ingredient in the formation of dissipative time crystals (DTCs). Here, we present the first experimental realisation of a DTC, implemented in an atom-cavity system. Its defining feature is a period doubled switching between distinct chequerboard density wave patterns, induced by controlled cavity-dissipation and cavity-mediated interactions. We demonstrate the robustness of this phase against system parameter changes and temporal perturbations of the driving. Our work provides a framework for realising phases of matter with spatiotemporal order in presence of dissipation. We note that this is the natural environment of matter, and therefore shapes its physical phenomena profoundly, making its study imperative.

A Bose-Einstein condensate (BEC) of rubidium atoms is prepared in one of two degenerate energy minima in the second Bloch band of an optical square lattice. A subsequent oscillation of the BEC between the two energy minima is observed, which is driven by two distinct collision processes: the conventional Hubbard-type on-site collision and a collision process that changes the orbital flavor. The oscillation frequency scales with the relative strength of these collisional interactions, which can be readily tuned via an experimentally well controlled distortion of the unit cell. The observations are compared to a quantum model of two single-particle modes and to a semi-classical multi-band tight-binding simulation of 12x12 tubular sites of the lattice. Both models reproduce the observed oscillatory quantum many-body dynamics and show the correct dependence of the oscillation frequency on the ratio between the strengths of the on-site and flavor-changing collision processes., Comment: 10 pages, 11 figures, PRL (2021)

We propose the dynamical stabilization of a nonequilibrium order in a driven dissipative system comprised an atomic Bose-Einstein condensate inside a high finesse optical cavity, pumped with an optical standing wave operating in the regime of anomalous dispersion. When the amplitude of the pump field is modulated close to twice the characteristic limit-cycle frequency of the unmodulated system, a stable subharmonic response is found. The dynamical phase diagram shows that this subharmonic response occurs in a region expanded with respect to that where stable limit-cycle dynamics occurs for the unmodulated system. In turning on the modulation we tune the atom-cavity system from a continuous to a discrete time crystal.

The phenomenon of metastability can shape dynamical processes on all temporal and spatial scales. Here, we induce metastable dynamics by pumping ultracold bosonic atoms from the lowest band of an optical lattice to an excitation band, via a sudden quench of the unit cell. The subsequent relaxation process to the lowest band displays a sequence of stages, which include a metastable stage, during which the atom loss from the excitation band is strongly suppressed. Using classical-field simulations and analytical arguments, we provide an explanation for this experimental observation, in which we show that the transient condensed state of the atoms in the excitation band is a dark state with regard to collisional decay and tunneling to a low-energy orbital. Therefore the metastable state is stabilized by destructive interference due to the chiral phase pattern of the condensed state. Our experimental and theoretical study provides a detailed understanding of the different stages of a paradigmatic example of many-body relaxation dynamics.

We theoretically and experimentally explore the emergence of a dynamical density wave order in a driven dissipative atom-cavity system. A Bose-Einstein condensate is placed inside a high finesse optical resonator and pumped sideways by an optical standing wave. The pump strength is chosen to induce a stationary superradiant checkerboard density wave order of the atoms stabilized by a strong intracavity light field. We show theoretically that, when the pump is modulated with sufficient strength at a frequency $\omega_{d}$ close to a systemic resonance frequency $\omega_{>}$, a dynamical density wave order emerges, which oscillates at the two frequencies $\omega_{>}$ and $\omega_{<} = \omega_{d} - \omega_{>}$. This order is associated with a characteristic momentum spectrum, also found in experiments in addition to remnants of the oscillatory dynamics presumably damped by on-site interaction and heating, not included in the calculations. The oscillating density grating, associated with this order, suppresses pump-induced light scattering into the cavity. Similar mechanisms might be conceivable in light-driven electronic matter., Comment: 8 pages, 9 figures

We propose an experimental realization of a time crystal using an atomic Bose-Einstein condensate in a high finesse optical cavity pumped with laser light detuned to the blue side of the relevant atomic resonance. By mapping out the dynamical phase diagram, we identify regions in parameter space showing stable limit cycle dynamics. Since the model describing the system is time-independent, the emergence of a limit cycle phase indicates the breaking of continuous time translation symmetry. Employing a semiclassical analysis to demonstrate the robustness of the limit cycles against perturbations and quantum fluctuations, we establish the emergence of a time crystal., Comment: 6 pages, 6 figures

Cold samples of calcium atoms are prepared in the metastable $^{3}$P$_{1}$ state inside an optical cavity resonant with the narrow band (375 Hz) $^{1}$S$_{0} \rightarrow ^{3}$P$_{1}$ intercombination line at 657 nm. We observe superradiant emission of hyperbolic secant shaped pulses into the cavity with an intensity proportional to the square of the atom number, a duration much shorter than the natural lifetime of the $^{3}$P$_{1}$ state, and a delay time fluctuating from shot to shot in excellent agreement with theoretical predictions. Our incoherent pumping scheme to produce inversion on the $^{1}$S$_{0} \rightarrow ^{3}$P$_{1}$ transition should be extendable to allow for continuous wave laser operation., Comment: 13 pages, 7 Figures

A quantum model is considered for $N$ bosons populating two orthogonal single-particle modes with tunable energy separation in the presence of flavour-changing contact interaction. The quantum ground state is well approximated as a coherent superposition (for zero temperature) or a mixture (at low temperature) of two quasi-classical states. In a mean field description, the systems realizes one of these states via spontaneous symmetry breaking. Both mean field states, in a certain parameter range, possess finite angular momentum and exhibit broken time-reversal symmetry in contrast to the quantum ground state. The phase diagram is explored at the mean-field level and by direct diagonalisation. The nature of the quantum ground state at zero and finite temperature is analyzed by means of the Penrose Onsager criterion. One of three possible phases shows fragmentation on the single-particle level together with a finite pair order parameter. Thermal and quantum fluctuations are characterized with respect to regions of universal scaling behavior. The non-equilibrium dynamics shows a sharp transition between a self-trapping and a pair-tunneling regime. A recently realized experimental implementation is discussed with bosonic atoms condensed in the two inequivalent energy minima $X_{\pm}$ of the second band of a bipartite two-dimensional optical lattice., Comment: 12 pages, 12 figures, typo in Eq.(2) removed

We demonstrate light-induced formation of coherence in a cold atomic gas system that utilizes the suppression of a competing density wave (DW) order. The condensed atoms are placed in an optical cavity and pumped by an external optical standing wave, which induces a long-range interaction mediated by photon scattering and a resulting DW order above a critical pump strength. We show that light-induced temporal modulation of the pump wave can suppress this DW order and restore coherence. This establishes a foundational principle of dynamical control of competing orders analogous to a hypothesized mechanism for light-induced superconductivity in high-$T_c$ cuprates., Comment: 5 page, 3 figures

We demonstrate dynamical control of the superradiant transition of cavity-BEC system via periodic driving of the pump laser. We show that the dominant density wave order of the superradiant state can be suppressed, and that the subdominant competing order of Bose-Einstein condensation emerges in the steady state. Furthermore, we show that additional, non-equilibrium density wave orders, which do not exist in equilibrium, can be stabilized dynamically. Finally, for strong driving, chaotic dynamics emerges., Comment: 5+9 pages, 7+9 figures

We investigate the interaction-driven instability of a quadratic band crossing arising for ultracold bosonic atoms loaded into a two-dimensional optical lattice. We consider the case when the degenerate point becomes a local minimum of both crossing energy bands such that it can support a stable Bose-Einstein condensate. Repulsive contact interaction among the condensed bosons induces a spontaneously time-reversal symmetry broken superfluid phase and a topological gap is opened in the excitation spectrum. We propose two concrete realizations of the desired quadratic band crossing in lattices with either fourfold or sixfold rotational symmetries via suitable tuning of the unit cell leading to reduced Brillouin zones and correspondingly folded bands. In either case, topologically protected edge excitations are found for a finite system., Comment: 6+6 pages, 4+4 figures

Intrinsic spin angular-momentum coupling of an electron has a relativistic quantum origin with the coupling arising from charged-orbits, which does not carry over to charge-neutral atoms. Here we propose a mechanism of spontaneous generation of spin angular-momentum coupling with spinor atomic bosons loaded into $p$-orbital bands of a two-dimensional optical-lattice. This spin angular-momentum coupling originates from many-body correlations and spontaneous symmetry breaking in a superfluid, with the key ingredients attributed to spin-channel quantum fluctuations and an approximate rotation symmetry. The resultant spin angular-momentum intertwined superfluid has Dirac excitations. In presence of a chemical potential difference for adjacent sites, it provides a bosonic analogue of a symmetry-protected-topological insulator. Through a dynamical mean-field calculation, this novel superfluid is found to be a generic low-temperature phase, and it gives way to Mott localization only at strong interactions and even-integer fillings. We show the temperature to reach this order is accessible with present experiments., Comment: 7 pages, 4 figures, 1 table, with supplementary material

We propose a physical scheme for the realization of two-dimensional topological odd-parity superfluidity in a spin-independent bond-centered square optical lattice based upon interband fermion pairing. The D4 point-group symmetry of the lattice protects a quadratic band crossing, which allows one to prepare a Fermi surface of spin-up fermions with odd parity close to the degeneracy point. In the presence of spin-down fermions with even parity populating a different energetically well separated band, odd-parity pairing is favored. Strikingly, as a necessary prerequisite for pairing both Fermi surfaces can be tuned to match well. As a result, topological superfluid phases emerge in the presence of merely s-wave interaction. Due to the Z2 symmetry of these odd-parity superfluids, we infer their topological features simply from the symmetry and the Fermi-surface topology as confirmed numerically., Comment: 6+7 pages, 3+6 figures