A critical Schr\"odinger cat qubit

Encoding quantum information onto bosonic systems is a promising route to quantum error correction. In a cat code, this encoding relies on the confinement of the system's dynamics onto the two-dimensional manifold spanned by Schr\"odinger cats of opposite parity. In dissipative cat qubits, an engineered dissipation scheme combining two-photon drive and two-photon dissipation autonomously stabilize this manifold, ensuring passive protection against, e.g., phase-flip errors regardless of their origin. Similarly, in Kerr cat qubits, where highly-performing gates can be engineered, two-photon drive and Kerr nonlinearity cooperate to confine the system to the ground state manifold spanned by the cats. Dissipative, Hamiltonian, and hybrid confinement mechanisms have been mainly investigated at resonance. Here, we propose a critical cat code, where both two-photon dissipation and Kerr nonlinearity are present and the two-photon drive is allowed to be out of resonance. The competition between nonlinearity and detuning triggers a first-order dissipative phase transition, making the encoding efficient over a wide range of parameters. The performance of the code is benchmarked within the general framework of the Liouvillian spectral theory. We introduce a channel-fidelity decay rate, allowing a fair comparison between our critical stabilization mechanism and its Hamiltonian, dissipative, and resonant-hybrid counterparts in the presence of both photon loss and dephasing noise. The critical cat outperforms the others, and this enhanced performance lies within reach of current experimental setups. Efficiently operating over a broad range of detuning values, the critical cat code is particularly resistant to random frequency shifts characterizing multiple-qubit operations, opening venues for the realization of reliable protocols for scalable and concatenated bosonic qubit architectures.Comment: 21 pages, 15 figure

Steady-state quantum chaos in open quantum systems

We introduce the notion of steady-state quantum chaos as a general phenomenon in open quantum many-body systems. Classifying an isolated or open quantum system as integrable or chaotic relies in general on the properties of the equations governing its time evolution. This however may fail in predicting the actual nature of the quantum dynamics, that can be either regular or chaotic depending on the initial state. Chaos and integrability in the steady state of an open quantum system are instead uniquely determined by the spectral structure of the time evolution generator. To characterize steady-state quantum chaos we introduce a spectral analysis based on the spectral statistics of quantum trajectories (SSQT). We test the generality and reliability of the SSQT criterion on several dissipative systems, further showing that an open system with chaotic structure can evolve towards either a chaotic or integrable steady state. We study steady-state chaos in the driven-dissipative Bose-Hubbard model, a paradigmatic example of out-of-equilibrium bosonic system without particle number conservation. This system is widely employed as a building block in state-of-the-art noisy intermediate-scale quantum devices, with applications in quantum computation and sensing. Finally, our analysis shows the existence of an emergent dissipative quantum chaos, where the classical and semi-classical limits display an integrable behaviour. This emergent dissipative quantum chaos arises from the quantum and classical fluctuations associated with the dissipation mechanism. Our work establishes a fundamental understanding of the integrable and chaotic dynamics of open quantum systems and paves the way for the investigation of dissipative quantum chaos and its consequences on quantum technologies.Comment: 23 pages, 12 figure

Revealing higher-order light and matter energy exchanges using quantum trajectories in ultrastrong coupling

The dynamics of open quantum systems is often modeled using master equations, which describe the expected outcome of an experiment (i.e., the average over many realizations of the same dynamics). Quantum trajectories, instead, model the outcome of ideal single experiments - the "clicks"of a perfect detector due to, e.g., spontaneous emission. The correct description of quantum jumps, which are related to random events characterizing a sudden change in the wave function of an open quantum system, is pivotal to the definition of quantum trajectories. In this article, we extend the formalism of quantum trajectories to open quantum systems with ultrastrong coupling (USC) between light and matter by properly defining jump operators in this regime. In such systems, exotic higher-order quantum-state and energy transfer can take place without conserving the total number of excitations in the system. The emitted field of such USC systems bears signatures of these higher-order processes, and significantly differs from similar processes at lower coupling strengths. Notably, the emission statistics must be taken at a single quantum trajectory level, since the signatures of these processes are washed out by the "averaging"of a master equation. We analyze the impact of the chosen unraveling (i.e., how one collects the output field of the system) for the quantum trajectories and show that these effects of the higher-order USC processes can be revealed in experiments by constructing histograms of detected quantum jumps. We illustrate these ideas by analyzing the excitation of two atoms by a single photon [Garziano, Phys. Rev. Lett. 117, 043601 (2016)0031-900710.1103/PhysRevLett.117.043601]. For example, quantum trajectories reveal that keeping track of the quantum jumps from the atoms allows one to reconstruct both the oscillations between one photon and two atoms as well as emerging Rabi oscillations between the two atoms

Influence of an internal standard in axial ICP OES analysis of trace elements in plant materials

Internal standardization is a commonly used approach to deal with the problems posed by non-spectral interferences in inductively coupled plasma optical emission spectrometry (ICP OES). However, in many real cases, the application to routine analysis, e.g. of environmental samples, is still difficult and cumbersome. In this work the use of an internal standard (IS) was verified by determining 17 trace elements on 3 certified reference materials (CRMs) using 2 different ISs, Bi and Pt. The results were compared with those obtained without an IS. The elements certified (or reported as indicative values) are not the same in the 3 CRMs and their concentrations are different; however the matrices used are similar, all of them being plant samples, frequently used in studies of environmental biomonitoring. Nevertheless, significant differences in the accuracy and precision of the results obtained were observed, depending on the CRM and the element considered. In all cases, an improvement was obtained when the IS was added, with a better performance for Pt in comparison to Bi for most of the elements. Moreover, a significant effect of the energy of spectral lines on the quality of the measurements was observed: the greater the similarity between the analyte wavelength and that of the IS, the higher the accuracy of the measurements

Trace Element Concentrations Measured in a Biomonitor (Tree Bark) for Assessing Mortality and Morbidity of Urban Population: A New Promising Approach for Exploiting the Potential of Public Health Data

The usefulness of bioindicators to study the state of the environment in different compartments (air, water, and soil) has been demonstrated for a long time. All persistent pollutants can be measured in some form of bioindicator, and numerous organisms are suitable for the biomonitoring purpose. In most of the works on this topic, bioindicators are used to highlight the impact of human activities. Generally, samples collected from polluted areas are compared with samples from an area considered as clean, or samples from areas characterized by different pollution sources are compared with each other. An approach that has not been attempted consists in correlating directly data on environmental quality obtained by bioindicators with parameters measuring the population health. In the present study, the concentrations of As, Cd, Co, Cu, Fe, Mn, Ni, Pb, V, and Zn measured by atomic emission spectrometry (ICP OES) in 56 samples of holm oak bark from trees located in urban parks and along streets in a Northern Italy city were related to the data describing the health status of the citizens. The concentrations of some of the 10 trace elements in the bioindicator were found significantly correlated with mortality and morbidity data regarding cardiac and respiratory diseases. The results, although preliminary, show the potential of this approach for implementing strategies aimed for disease prevention and health promotion in urban areas at risk, with the objective of reducing environmental and health inequalities

Critical Schrödinger Cat Qubit

Encoding quantum information onto bosonic systems is a promising route to quantum error correction. In a cat code, this encoding relies on the confinement of the dynamics of the system onto the two-dimensional manifold spanned by Schrödinger cats of opposite parity. In dissipative cat qubits, an engineered dissipation scheme combining two-photon drive and two-photon loss has been used to autonomously stabilize this manifold, ensuring passive protection against, e.g., bit-flip errors regardless of their origin. Similarly, in Kerr-cat qubits, where highly performing gates can be engineered, two-photon drive and Kerr nonlinearity cooperate to confine the system to a twofold-degenerate ground-state manifold spanned by cat states of opposite parity. Dissipative, Hamiltonian, and hybrid confinement mechanisms have been investigated at resonance, i.e., for driving frequencies matching that of the cavity. Here, we propose a critical cat code, where both two-photon loss and Kerr nonlinearity are present and the two-photon drive is allowed to be out of resonance. The performance of this code is assessed via the spectral theory of Liouvillians in all configurations ranging from the purely dissipative to the Kerr limit. We show that large detunings and small, but non-negligible, two-photon loss rates are fundamental to achieve optimal performance. We further demonstrate that the competition between nonlinearity and detuning results in a first-order dissipative phase transition, leading to a squeezed vacuum steady state. We show that to achieve the maximal suppression of the logical bit-flip rate requires initializing the system in the metastable state emerging from the first-order transition and we detail a protocol to do so. Efficiently operating over a broad range of detuning values, the critical cat code is particularly resistant to random frequency shifts characterizing multiple-qubit operations, opening avenues for the realization of reliable protocols for scalable and concatenated bosonic qubit architectures

Quantum error correction using squeezed Schrodinger cat states

Bosonic quantum codes redundantly encode quantum information in the states of a quantum harmonic oscillator, making it possible to detect and correct errors. Schrodinger cat codes-based on the superposition of two coherent states with opposite displacements-can correct phase-flip errors induced by dephasing, but they are vulnerable to bit-flip errors induced by particle loss. Here, we develop a bosonic quantum code relying on squeezed cat states, i.e., cat states made of a linear superposition of displaced-squeezed states. Squeezed cat states allow to partially correct errors caused by particle loss, while at the same time improving the protection against dephasing. We present a comprehensive analysis of the squeezed cat code, including protocols for code generation and elementary quantum gates. We characterize the effect of both particle loss and dephasing and develop an optimal recovery protocol that is suitable to be implemented on currently available quantum hardware. We show that with moderate squeezing, and using typical parameters of state-of-the-art quantum hardware platforms, the squeezed cat code has a resilience to particle loss errors that significantly outperforms that of the conventional cat code.LTP

Trace elements in Plantago lanceolata L., a plant used for herbal and food preparations: new data and literature review

Plantago lanceolata L. is a common grassland and roadside plant, widely used in many countries in food and herbal preparations. In this study, samples of this wild plant were collected from rural, suburban/urban, and industrial environments; the concentrations of As, Cd, Co, Cu, Fe, Mn, Ni, P, Pb, V, and Zn were measured in the edible parts of the plant (leaves), in the roots, and in the soils in order to calculate the bioaccumulation and translocation factors. The data obtained were compared with literature data available. Except for samples collected near mines and smelting plants, where Cd, Pb, and Zn concentrations were up to 15 times higher, in all other cases, no differences were observed with respect to samples from rural areas, except for Pb concentration, which was 3 times higher in urban areas. In the samples collected in our study area, the metal content does not pose particular health risks; however, even within a quite restricted region like the investigated area, high metal concentrations, possibly due to the presence of particular substrates, were observed in some samples collected from areas considered "clean" and suitable for wild food plant gathering

Urban and industrial contribution to trace elements in the atmosphere as measured in holm oak bark.

The concentrations of As, Cd, Co, Cu, Fe, Mn, Ni, Pb, V and Zn were measured by ICP-OES in samples of bark of the holm oak (Quercus ilex L.) collected from trees in different urban environments (residential and mixed residential/industrial). The use of tree bark as a bioindicator makes it easy to create maps that can provide detailed data on the levels and on the spatial distribution of each trace element. For most of the elements considered (As, Co, Fe, Mn, Ni, V and Zn), the concentrations in the industrial sites are about twice (from 1.9 to 2.8 times higher) of those in the residential area. Arsenic, Fe and Zn show the highest concentrations near a steel plant (operational until 2005), but for the other elements it is not possible to identify any localized source, as evident from the maps. In areas where urban pollution is summed up by the impact of industrial activities, the population is exposed to significantly higher amounts of some metals than people living in residential areas