Observing dynamical phases of BCS superconductors in a cavity QED simulator.

In conventional Bardeen-Cooper-Schrieffer superconductors1, electrons with opposite momenta bind into Cooper pairs due to an attractive interaction mediated by phonons in the material. Although superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system parameters are abruptly changed2-8. The resulting out-of-equilibrium phases are predicted to occur in real materials and ultracold fermionic atoms, but not all have yet been directly observed. Here we realize an alternative way to generate the proposed dynamical phases using cavity quantum electrodynamics (QED). Our system encodes the presence or absence of a Cooper pair in a long-lived electronic transition in 88Sr atoms coupled to an optical cavity and represents interactions between electrons as photon-mediated interactions through the cavity9,10. To fully explore the phase diagram, we manipulate the ratio between the single-particle dispersion and the interactions after a quench and perform real-time tracking of the subsequent dynamics of the superconducting order parameter using nondestructive measurements. We observe regimes in which the order parameter decays to zero (phase I)3,4, assumes a non-equilibrium steady-state value (phase II)2,3 or exhibits persistent oscillations (phase III)2,3. This opens up exciting prospects for quantum simulation, including the potential to engineer unconventional superconductors and to probe beyond mean-field effects like the spectral form factor11,12, and for increasing the coherence time for quantum sensing.

Exploring dynamical phase transitions with cold atoms in an optical cavity.

Interactions between atoms and light in optical cavities provide a means of investigating collective (many-body) quantum physics in controlled environments. Such ensembles of atoms in cavities have been proposed for studying collective quantum spin models, where the atomic internal levels mimic a spin degree of freedom and interact through long-range interactions tunable by changing the cavity parameters1-4. Non-classical steady-state phases arising from the interplay between atom-light interactions and dissipation of light from the cavity have previously been investigated5-11. These systems also offer the opportunity to study dynamical phases of matter that are precluded from existence at equilibrium but can be stabilized by driving a system out of equilibrium12-16, as demonstrated by recent experiments17-22. These phases can also display universal behaviours akin to standard equilibrium phase transitions8,23,24. Here, we use an ensemble of about a million strontium-88 atoms in an optical cavity to simulate a collective Lipkin-Meshkov-Glick model25,26, an iconic model in quantum magnetism, and report the observation of distinct dynamical phases of matter in this system. Our system allows us to probe the dependence of dynamical phase transitions on system size, initial state and other parameters. These observations can be linked to similar dynamical phases in related systems, including the Josephson effect in superfluid helium27, or coupled atomic28 and solid-state polariton29 condensates. The system itself offers potential for generation of metrologically useful entangled states in optical transitions, which could permit quantum enhancement in state-of-the-art atomic clocks30,31.

Squeezing Multilevel Atoms in Dark States via Cavity Superradiance.

We describe a method to create and store scalable and long-lived entangled spin-squeezed states within a manifold of many-body cavity dark states using collective emission of light from multilevel atoms inside an optical cavity. We show that the system can be tuned to generate squeezing in a dark state where it will be immune to superradiance. We also show more generically that squeezing can be generated using a combination of superradiance and coherent driving in a bright state, and subsequently be transferred via single-particle rotations to a dark state where squeezing can be stored. Our findings, readily testable in current optical cavity experiments with alkaline-earth-like atoms, can open a path for dissipative generation and storage of metrologically useful states in optical transitions.

Bosonic Pair Production and Squeezing for Optical Phase Measurements in Long-Lived Dipoles Coupled to a Cavity.

We propose to simulate bosonic pair creation using large arrays of long-lived dipoles with multilevel internal structure coupled to an undriven optical cavity. Entanglement between the atoms, generated by the exchange of virtual photons through a common cavity mode, grows exponentially fast and is described by two-mode squeezing of effective bosonic quadratures. The mapping between an effective bosonic model and the natural spin description of the dipoles allows us to realize the analog of optical homodyne measurements via straightforward global rotations and population measurements of the electronic states, and we propose to exploit this for quantum-enhanced sensing of an optical phase (common and differential between two ensembles). We discuss a specific implementation based on Sr atoms and show that our sensing protocol is robust to sources of decoherence intrinsic to cavity platforms. Our proposal can open unique opportunities for next-generation optical atomic clocks.

Cavity-QED Quantum Simulator of Dynamical Phases of a Bardeen-Cooper-Schrieffer Superconductor.

We propose to simulate dynamical phases of a BCS superconductor using an ensemble of cold atoms trapped in an optical cavity. Effective Cooper pairs are encoded via the internal states of the atoms, and attractive interactions are realized via the exchange of virtual photons between atoms coupled to a common cavity mode. Control of the interaction strength combined with a tunable dispersion relation of the effective Cooper pairs allows exploration of the full dynamical phase diagram of the BCS model as a function of system parameters and the prepared initial state. Our proposal paves the way for the study of the nonequilibrium features of quantum magnetism and superconductivity by harnessing atom-light interactions in cold atomic gases.

Protocol for Precise Field Sensing in the Optical Domain with Cold Atoms in a Cavity.

In the context of quantum metrology, optical cavity-QED platforms have primarily been focused on the generation of entangled atomic spin states useful for next-generation frequency and time standards. Here, we report a complementary application: the use of optical cavities to generate nonclassical states of light for electric field sensing below the standard quantum limit. We show that cooperative atom-light interactions in the strong collective coupling regime can be used to engineer generalized atom-light cat states which enable quantum enhanced sensing of small displacements of the cavity field even in the presence of photon loss. We demonstrate that metrological gains of 10-20 dB below the standard quantum limit are within reach for current cavity-QED systems operating with long-lived alkaline-earth atoms.

Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals.

Fully controllable ultracold atomic systems are creating opportunities for quantum sensing, yet demonstrating a quantum advantage in useful applications by harnessing entanglement remains a challenging task. Here, we realize a many-body quantum-enhanced sensor to detect displacements and electric fields using a crystal of ~150 trapped ions. The center-of-mass vibrational mode of the crystal serves as a high-Q mechanical oscillator, and the collective electronic spin serves as the measurement device. By entangling the oscillator and collective spin and controlling the coherent dynamics via a many-body echo, a displacement is mapped into a spin rotation while avoiding quantum back-action and thermal noise. We achieve a sensitivity to displacements of 8.8 ± 0.4 decibels below the standard quantum limit and a sensitivity for measuring electric fields of 240 ± 10 nanovolts per meter in 1 second. Feasible improvements should enable the use of trapped ions in searches for dark matter.

Dispersive Regimes of the Dicke Model.

We study two dispersive regimes of the Dicke model in the dynamics of N two-level atoms interacting with a bosonic mode for long interaction times. Firstly, we analyze the model for the regime in which the qubit frequencies are equal and smaller than the mode frequency, and for values of the coupling strength similar or larger than the mode frequency, namely, the deep strong coupling regime. Secondly, we address an interaction that is dependent on the photon number, where the coupling strength is comparable to the geometric mean of the qubit and mode frequencies. We show that the associated dynamics is analytically tractable and provide useful frameworks with which to analyze the system behavior. In the deep strong coupling regime, we unveil the structure of unexpected resonances for specific values of the coupling, present for N ≥ 2, and in the photon-number-dependent regime we demonstrate that all the nontrivial dynamical behavior occurs in the atomic degrees of freedom for a given Fock state. We verify these assertions with numerical simulations of the qubit population and photon-statistic dynamics.

Desenvolvimento e produtividade de arroz irrigado em resposta a diferentes doses de fósforo e potássio, em várzea de primeiro ano, no estado de Roraima / Growth and productivity of irrigated rice in response to diferent doses of phosphorus and potassium in lowland soil of first year in the state of Roraima

São poucas as pesquisas em Roraima sobre o uso de fósforo e potássio na cultura do arroz irrigado (Oryza sativa L.). Assim, neste trabalho, objetivou-se avaliar a resposta do arroz irrigado à aplicação de diferentes doses de fósforo e potássio (0, 50, 100, 200 e 400 kg ha-1 de P2O5 ou K2O) sobre a produtividade de grãos, altura de plantas e componentes de produção em área de primeiro ano de cultivo. Utilizou-se o delineamento em blocos aos acaso, no esquema de parcelas subdivididas com quatro repetições. Foi feita análise de regressão, por meio da superfície de resposta. O potássio influenciou significativa e positivamente todas as variáveis, exceto a massa de mil grãos, reduzindo-a linearmente, sendo seu efeito o principal responsável pelas produtividades alcançadas. O fósforo influenciou positivamente a massa de mil grãos e a produtividade de grãos. O número de grãos por panícula, a massa de mil grãos e a altura de plantas contribuíram decisivamente para as produtividades atingidas. A maior produtividade encontrada (8.154 kg ha-1 de grãos) ocorreu com uso de 304 kg ha-1 de P2O5 e 228 kg ha-1 de K2O. A máxima eficiência econômica foi obtida com 134 e 158 kg ha-1 de P2O5 e K2O, respectivamente, e produtividade de 7.501 kg ha-1 de grãos, proporcionando lucro líquido de R$ 696,78 ha-1, relação benefício/custo de 1,18 e custo unitário de R$ 0,5271 kg-1 de arroz.

There are few studies about the use of phosphorus and potassium in irrigated rice (Oryza sativa L.) in the State of Roraima. The objetive of this study was to evaluate the response of rice to the application of different doses of phosphorus and potassium (0, 50, 100, 200 and 400 kg ha-1 P2O5 K2O) on productivity, plant height and yield composition in an area in the first year of cultivation. A randomized blocks design was used in a split plot with four replicates. A regression analysis was PERFORMED using surface response. The potassium significantly and positively affected all variables except weight of thousand grains, reducing it linearly. wits its effect being the responsible for the achieved productivity. The phosphorus positively influenced the weight of thousand grains and productivity. The number of grains per panicle, the weight of thousand grains and plant height contributed decisively to the yields achieved. The higher productivity found (8,154 kg ha-1 grains) occurred with the use of 304 kg ha-1 P2O5 and 228 kg ha-1 K2O. The maximum economic efficiency was obtained with 134 and 158 kg ha-1 P2O5 and K2O, respectively, and yield of 7,501 kg ha-1 grain, providing net earnings of R$ 696.78 ha-1, benefit / cost ratio of 1.18 and unit cost of R$ 0.5271 kg-1 rice.