Spontaneous structural rearrangements play a central role in the organization and function of complex biomolecular systems. In principle, physics-based computer simulations like Molecular Dynamics (MD) enable us to investigate these thermally activated processes with an atomic level of resolution. However, rare conformational transitions are intrinsically hard to investigate with MD, because an exponentially large fraction of computational resources must be invested to simulate thermal fluctuations in metastable states. Path sampling methods like Transition Path Sampling hold the great promise of focusing the available computational power on sampling the rare stochastic transition between metastable states. In these approaches, one of the outstanding limitations is to generate paths that visit significantly different regions of the conformational space at a low computational cost. To overcome these problems we introduce a rigorous approach that integrates a machine learning algorithm and MD simulations implemented on a classical computer with adiabatic quantum computing. First, using functional integral methods, we derive a rigorous low-resolution representation of the system's dynamics, based on a small set of molecular configurations generated with machine learning. Then, a quantum annealing machine is employed to explore the transition path ensemble of this low-resolution theory, without introducing un-physical biasing forces to steer the system's dynamics. Using the D-Wave quantum computer, we validate our scheme by simulating a benchmark conformational transition in a state-of-the-art atomistic description. We show that the quantum computing step generates uncorrelated trajectories, thus facilitating the sampling of the transition region in configuration space. Our results provide a new paradigm for MD simulations to integrate machine learning and quantum computing.

We present a theoretical study of the temporal and spatial coherence properties of a topological laser device built by including saturable gain on the edge sites of a Harper--Hofstadter lattice for photons. For small enough lattices the Bogoliubov analysis applies, the emission is nearly a single-mode one and the coherence time is almost determined by the total number of photons in the device in agereement with the standard Schawlow-Townes phase diffusion. In larger lattices, looking at the lasing edge mode in the comoving frame of its chiral motion, the spatio-temporal correlations of long-wavelength fluctuations display a Kardar-Parisi-Zhang (KPZ) scaling. Still, at very long times, when the finite size of the device starts to matter, the functional form of the temporal decay of coherence changes from the KPZ stretched exponential to a Schawlow-Townes-like exponential, while the nonlinear many-mode dynamics of KPZ fluctuations remains visible as an enhanced linewidth as compared to the single-mode Schawlow-Townes prediction. While we have established the above behaviors also for non-topological laser arrays, the crucial role of topology in protecting the coherence from static disorder is finally highlighted: our ground-breaking numerical calculations suggest the dramatically reinforced coherence properties of topological lasers compared to corresponding non-topological devices. These results open exciting possibilities for both fundamental studies of non-equilibrium statistical mechanics and concrete applications to laser devices.

The existence of boundary states and their protection against symmetry-preserving perturbations are a hallmark feature of topological systems. While this concept originally emerged in the context of sin-gle-particle phenomena in condensed-matter physics, particle interactions have recently been identi-fied as alternative means to establish topological phases. As a consequence, nonlinear topological insu-lators gained much interest as a model system for many interacting particles. However, as their mean-field model inevitably breaks down for small numbers of particles, to date, topological states composed of only few interacting particles remain experimentally largely unexplored. In our work, we explore the physics of extended interaction-induced two-particle topological states, so-called Dou-blons. We experimentally implement non-local-interactions via non-adiabatic periodic driving and dimensional mapping in an artificial photonic solid. The resonant formation of extended Doublon qua-si-particles at specific local interaction strengths is observed, allowing us to probe the topologically protected motion of these entities through the bulk of the system. Our approach is compatible to a number of established experimental platforms and paves the way for studying topological few-particle phenomena with finite interaction strength.

We introduce a novel frequency-dependent incoherent pump scheme with a square-shaped spectrum as a way to study strongly correlated photons in arrays of coupled nonlinear resonators. This scheme can be implemented via a reservoir of population-inverted two-level emitters with a broad distribution of transition frequencies. Our proposal is predicted to stabilize a non-equilibrium steady state sharing important features with a zero-temperature equilibrium state with a tunable chemical potential. We confirm the efficiency of our proposal for the Bose-Hubbard model by computing numerically the steady state for finite system sizes: first, we predict the occurrence of a sequence of incompressible Mott-Insulator-like states with arbitrary integer densities presenting strong robustness against tunneling and losses. Secondly, for stronger tunneling amplitudes or non-integer densities, the system enters a coherent regime analogous to the superfluid state. In addition to an overall agreement with the zero-temperature equilibrium state, exotic non-equilibrium processes leading to a finite entropy generation are pointed out in specific regions of parameter space. The equilibrium ground state is shown to be recovered by adding frequency-dependent losses. The promise of this improved scheme in view of quantum simulation of the zero temperature many-body physics is highlighted.

We propose a realistic scheme to detect the 4D quantum Hall effect using ultracold atoms. Based on contemporary technology, motion along a synthetic fourth dimension can be accomplished through controlled transitions between internal states of atoms arranged in a 3D optical lattice. From a semi-classical analysis, we identify the linear and non-linear quantized current responses of our 4D model, relating these to the topology of the Bloch bands. We then propose experimental protocols, based on current or center-of-mass-drift measurements, to extract the topological 2nd Chern number. Our proposal sets the stage for the exploration of novel topological phases in higher dimensions.

Recent technological advances in integrated photonics have spurred on the study of topological phenomena in engineered bosonic systems. Indeed, the controllability of silicon ring-resonator arrays has opened up new perspectives for building lattices for photons with topologically nontrivial bands and integrating them into photonic devices for practical applications. Here, we push these developments even further by exploiting the different modes of a silicon ring resonator as an extra dimension for photons. Tunneling along this synthetic dimension is implemented via an external time-dependent modulation that allows for the generation of engineered gauge fields. We show how this approach can be used to generate a variety of exciting topological phenomena in integrated photonics, ranging from a topologically-robust optical isolator in a spatially one-dimensional (1D) ring-resonator chain to a driven-dissipative analog of the 4D quantum Hall effect in a spatially 3D resonator lattice. Our proposal paves the way towards the use of topological effects in the design of novel photonic lattices supporting many frequency channels and displaying higher connectivities.

We introduce a simple scheme to implement synthetic dimensions in ultracold atomic gases, which only requires two basic and ubiquitous ingredients: the harmonic trap, which confines the atoms, combined with a periodic shaking. In our approach, standard harmonic oscillator eigenstates are reinterpreted as lattice sites along a synthetic dimension, while the coupling between these lattice sites is controlled by the applied time-modulation. The phase of this modulation enters as a complex hopping phase, leading straightforwardly to an artificial magnetic field upon adding a second dimension. We show that this artificial gauge field has important consequences, such as the counterintuitive reduction of average energy under resonant driving, or the realisation of quantum Hall physics. Our approach offers significant advantages over previous implementations of synthetic dimensions, providing an intriguing route towards higher-dimensional topological physics and strongly-correlated states.

We experimentally reveal the emergence of edge states in a photonic lattice with orbital bands. We use a two-dimensional honeycomb lattice of coupled micropillars whose bulk spectrum shows four gapless bands arising from the coupling of $p$-like photonic orbitals. We observe zero-energy edge states whose topological origin is similar to that of conventional edge states in graphene. Additionally, we report novel dispersive edge states that emerge not only in zigzag and bearded terminations, but also in armchair edges. The observations are reproduced by tight-binding and analytical calculations. Our work shows the potentiality of coupled micropillars in elucidating some of the electronic properties of emergent 2D materials with orbital bands.

We reveal a prethermal temporal regime upon suddenly quenching to the vicinity of a quantum phase transition in the time evolution of 1D spin chains. The prethermal regime is analytically found to be self-similar, and its duration is governed by the ground-state energy gap. Based on analytical insights and numerical evidence, we show that this critically prethermal regime universally exists independently of the location of the probe site, the presence of weak interactions, or the initial state. Moreover, the resulting prethermal dynamics leads to an out-of-equilibrium scaling function of the order parameter in the vicinity of the transition.

The Berry curvature is a geometrical property of an energy band which acts as a momentum space magnetic field in the effective Hamiltonian describing single-particle quantum dynamics. We show how this perspective may be exploited to study systems directly relevant to ultracold gases and photonics. Given the exchanged roles of momentum and position, we demonstrate that the global topology of momentum space is crucially important. We propose an experiment to study the Harper-Hofstadter Hamiltonian with a harmonic trap that will illustrate the advantages of this approach and that will also constitute the first realization of magnetism on a torus.

We study the spin of the localised quasiparticle excitations of lowest-Landau-level quantum Hall states defined on a disk. The spin that we propose satisfies the spin-statistics relation and can be used to reconstruct the topological geometric phase associated to the exchange of two arbitrarily chosen quasiparticles. Since it is related to the quadrupole moment of the quasiparticle charge distribution, it can be measured in an experiment and could reveal anyonic properties in a way that is complementary to the interferometric schemes employed so far. We first discuss our definition for the quasiholes of the Laughlin state, for which we present a numerical and analytical study of our spin, and we proceed with a discussion of several kinds of quasiholes of the Halperin 221 state. Finally, we discuss the link between our spin and the adiabatic rotation of the quasiparticles around their axis and demonstrate that our spin obeys the spin-statistics relation.

The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is mediated by the proliferation of topological defects, governs the critical behaviour of a wide range of equilibrium two-dimensional systems with a continuous symmetry, ranging from superconducting thin films to two-dimensional Bose fluids, such as liquid helium and ultracold atoms. We show here that this phenomenon is not restricted to thermal equilibrium, rather it survives more generally in a dissipative highly non-equilibrium system driven into a steady-state. By considering a light-matter superfluid of polaritons, in the so-called optical parametric oscillator regime, we demonstrate that it indeed undergoes a vortex binding-unbinding phase transition. Yet, the exponent of the power-law decay of the first order correlation function in the (algebraically) ordered phase can exceed the equilibrium upper limit -- a surprising occurrence, which has also been observed in a recent experiment. Thus we demonstrate that the ordered phase is somehow more robust against the quantum fluctuations of driven systems than thermal ones in equilibrium.

Quantum many-body systems are characterized by their correlations. While equal-time correlators and unequal-time commutators between operators are standard observables, the direct access to unequal-time anti-commutators poses a formidable experimental challenge. Here, we propose a general technique for measuring unequal-time anti-commutators using the linear response of a system to a non-Hermitian perturbation. We illustrate the protocol at the example of a Bose-Hubbard model, where the approach to thermal equilibrium in a closed quantum system can be tracked by measuring both sides of the fluctuation-dissipation relation. We relate the scheme to the quantum Zeno effect and weak measurements, and illustrate possible implementations at the example of a cold-atom system. Our proposal provides a way of characterizing dynamical correlations in quantum many-body systems with potential applications in understanding strongly correlated matter as well as for novel quantum technologies.

The wave turbulence theory predicts that a conservative system of nonlinear waves can exhibit a process of condensation, which originates in the singularity of the Rayleigh-Jeans equilibrium distribution of classical waves. Considering light propagation in a multimode fiber, we show that light condensation is driven by an energy flow toward the higher-order modes, and a bi-directional redistribution of the wave-action (or power) to the fundamental mode and to higher-order modes. The analysis of the near-field intensity distribution provides experimental evidence of this mechanism. The kinetic equation also shows that the wave-action and energy flows can be inverted through a thermalization toward a negative temperature equilibrium state, in which the high-order modes are more populated than low-order modes. In addition, a Bogoliubov stability analysis reveals that the condensate state is stable.

The discovery of topological states of matter has profoundly augmented our understanding of phase transitions in physical systems. Instead of local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example thereof is the two-dimensional integer quantum Hall effect. It is characterized by the first Chern number which manifests in the quantized Hall response induced by an external electric field. Generalizing the quantum Hall effect to four-dimensional systems leads to the appearance of a novel non-linear Hall response that is quantized as well, but described by a 4D topological invariant - the second Chern number. Here, we report on the first observation of a bulk response with intrinsic 4D topology and the measurement of the associated second Chern number. By implementing a 2D topological charge pump with ultracold bosonic atoms in an angled optical superlattice, we realize a dynamical version of the 4D integer quantum Hall effect. Using a small atom cloud as a local probe, we fully characterize the non-linear response of the system by in-situ imaging and site-resolved band mapping. Our findings pave the way to experimentally probe higher-dimensional quantum Hall systems, where new topological phases with exotic excitations are predicted.

In this Chapter, we give a brief review of the state of the art of theoretical and experimental studies of quantum fluids of light. Such systems consist of ensembles of photons that acquire a finite mass from spatial confinement or diffraction and finite binary interactions from the optical nonlinearity of the optical medium. The peculiar properties of these fluids are highlighted in comparison with standard condensed matter systems, with a special emphasis on the novel possibilities that they offer for the generation, the manipulation and the diagnostics of the fluid, as well as on their intrinsically non-equilibrium and/or dynamical nature. Perspectives towards a new generation of experiments on strongly correlated fluids of light and towards opto-electronic applications are finally sketched.

In an atomic Bose-Einstein condensate quenched to the unitary regime, we predict the sequential formation of a significant fraction of condensed pairs and triples. At short-distances, we demonstrate the two-body and Efimovian character of the condensed pairs and triples, respectively. As the system evolves, the size of the condensed pairs and triples becomes comparable to the interparticle distance, such that many-body effects become significant. The structure of the condensed triples depends on the relative size of Efimov states to density scales. Unexpectedly, we find universal condensed triples in the limit where these scales are well-separated. Our findings provide a new framework for understanding dynamics in the unitary regime as the Bose-Einstein condensation of few-body composites.

We consider the ground state properties of a trapped dipolar condensate under the influence of quantum fluctuations. We show that this system can undergo a phase transition from a low density condensate state to a high density droplet state, which is stabilized by quantum fluctuations. The energetically favored state depends on the geometry of the confining potential, the number of atoms and the two-body interactions. We develop a simple variational ansatz and validate it against full numerical solutions. We produce a phase diagram for the system and present results relevant to current experiments with dysprosium and erbium condensates.

Two-dimensional lattices of coupled micropillars etched in a planar semiconductor microcavity offer a workbench to engineer the band structure of polaritons. We report experimental studies of honeycomb lattices where the polariton low-energy dispersion is analogous to that of electrons in graphene. Using energy-resolved photoluminescence we directly observe Dirac cones, around which the dynamics of polaritons is described by the Dirac equation for massless particles. At higher energies, we observe p orbital bands, one of them with the nondispersive character of a flatband. The realization of this structure which holds massless, massive and infinitely massive particles opens the route towards studies of the interplay of dispersion, interactions, and frustration in a novel and controlled environment.

Multipartite entanglement, such as witnessed through the quantum Fisher information (QFI), is a crucial resource for quantum technologies, but its experimental certification is highly challenging. Here, we propose an experimentally friendly protocol to measure the QFI. It relies on recording the short-time dynamics of simple observables after a quench from a thermal state, works for spins, bosons, and fermions, and can be implemented in standard cold-atom experiments and other platforms with temporal control over the system Hamiltonian. To showcase the protocol, we simulate it for the one-dimensional Fermi--Hubbard model. Further, we establish a family of bounds connecting the QFI to multipartite mode entanglement for fermionic systems, which enable the detection of multipartite entanglement at sizable temperatures. Our work paves a way to experimentally accessing entanglement for quantum enhanced metrology.