This tutorial introduces the theoretical and experimental basics of electromagnetically induced transparency (EIT) in thermal alkali vapors. We first give a brief phenomenological description of EIT in simple three-level systems of stationary atoms and derive analytical expressions for optical absorption and dispersion under EIT conditions. Then we focus on how the thermal motion of atoms affects various parameters of the EIT system. Specifically, we analyze the Doppler broadening of optical transitions, ballistic versus diffusive atomic motion in a limited-volume interaction region, and collisional depopulation and decoherence. Finally, we discuss the common trade-offs important for optimizing an EIT experiment and give a brief ‘walk-through’ of a typical EIT experimental setup. We conclude with a brief overview of current and potential EIT applications.

This article is designed as a step-by-step guide to optically pumped magnetometers based on alkali atomic vapor cells. We begin with a general introduction to atomic magneto-optical response, as well as expected magnetometer performance merits and how they are affected by main sources of noise. This is followed by a brief comparison of different magnetometer realizations and an overview of current research, with the aim of helping readers to identify the most suitable magnetometer type for specific applications. Next, we discuss some practical considerations for experimental implementations, using the case of an M_z magnetometer as an example of the design process. Finally, an interactive workbook with real magnetometer data is provided to illustrate magnetometer-performance analysis.

Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as ‘the quantum variational eigensolver’ was developed (Peruzzo et al 2014 Nat. Commun. 5 4213) with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through a relaxation of exponential operator splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.

In recent years, surface codes have become a leading method for quantum error correction in theoretical large-scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural two-dimensional nearest-neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar- and defect-based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance-3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.

Quantum and biological systems are seldom discussed together as they seemingly demand opposing conditions. Life is complex, ‘hot and wet’ whereas quantum objects are small, cold and well controlled. Here, we overcome this barrier with a tardigrade—a microscopic multicellular organism known to tolerate extreme physicochemical conditions via a latent state of life known as cryptobiosis. We observe coupling between the animal in cryptobiosis and a superconducting quantum bit and prepare a highly entangled state between this combined system and another qubit. The tomographic data shows entanglement in the qubit-qubit-tardigrade system, with the tardigrade modelled as an ensemble of harmonic oscillators or collection of electric dipoles. The animal is then observed to return to its active form after 420 h at sub 10 mK temperatures and pressures below $6\times 10^{-6}$ mbar, setting a new record for the conditions that a complex form of life can survive.

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a or a topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via ‘dimensional reduction’ by compactifying one or more spatial dimensions (in ‘Kaluza–Klein’-like fashion). For -topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent -topological insulators in the same class, from which they inherit their topological properties. The eightfold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle–hole symmetries) is a reflection of the eightfold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern–Simons invariant. For lower-dimensional cases, this formula relates the winding number to the electric polarization (d=1 spatial dimensions) or to the magnetoelectric polarizability (d=3 spatial dimensions). Finally, we also discuss topological field theories describing the spacetime theory of linear responses in topological insulators (superconductors) and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).

Double-slit diffraction is a corner stone of quantum mechanics. It illustrates key features of quantum mechanics: interference and the particle-wave duality of matter. In 1965, Richard Feynman presented a thought experiment to show these features. Here we demonstrate the full realization of his famous thought experiment. By placing a movable mask in front of a double-slit to control the transmission through the individual slits, probability distributions for single- and double-slit arrangements were observed. Also, by recording single electron detection events diffracting through a double-slit, a diffraction pattern was built up from individual events.

While measuring the orbital angular momentum state of bright light beams can be performed using imaging techniques, a full characterization at the single-photon level is challenging. For applications to quantum optics and quantum information science, such characterization is an essential capability. Here, we present a setup to perform the quantum state tomography of photonic qubits encoded in this degree of freedom. The method is based on a projective technique using spatial mode projection via fork holograms and single-mode fibers inserted into an interferometer. The alignment and calibration of the device is detailed as well as the measurement sequence to reconstruct the associated density matrix. Possible extensions to higher-dimensional spaces are discussed.

The conical shape of a shuttlecock allows it to flip on impact. As a light and extended particle, it flies with a pure drag trajectory. We first study the flip phenomenon and the dynamics of the flight and then discuss the implications on the game. Lastly, a possible classification of different shots is proposed.

Quantum technologies exploit entanglement to enhance various tasks beyond their classical limits including computation, communication and measurements. Quantum metrology aims to increase the precision of a measured quantity that is estimated in the presence of statistical errors using entangled quantum states. We present a novel approach for finding (near) optimal states for metrology in the presence of noise, using variational techniques as a tool for efficiently searching the high-dimensional space of quantum states, which would be classically intractable. We comprehensively explore systems consisting of up to 9 qubits and find new highly entangled states that are not symmetric under permutations and non-trivially outperform previously known states up to a constant factor 2. We consider a range of environmental noise models; while passive quantum states cannot achieve a fundamentally superior scaling (as established by prior asymptotic results) we do observe a significant absolute quantum advantage. We finally outline a possible experimental setup for variational quantum metrology which can be implemented in near-term hardware.

In this article, we review some recent theoretical developments on potential high-temperature superconductors and unconventional metallic states that can arise from doping a spin-one Mott insulator in the d⁸ valence. These studies are particularly relevant—though not limited—to the recently discovered bilayer nickelate superconductor La₃Ni₂O₇. We focus on a ferromagnetic Kondo lattice model with mobile electrons in the $d_{x^2-y^2}$ orbital coupled to the localized spin moments in $d_{z^2}$ orbital through a large Hund’s coupling $J_\textrm H$. In the large $J_\textrm H$ limit, the model reduces to the type II t − J model with a mixture of spin-half singlon states and spin-one doublon states. We summarize density matrix renormalization group results on the Luther-Emery liquid in one dimensional chain and two-leg ladder. Then we mainly focus on bilayer square lattice and show that a large inter-layer coupling $J_\perp$ of $d_{z^2}$ orbital can induce strong inter-layer pairing of $d_{x^2-y^2}$ orbital. In the strong $J_\perp$ limit, a kinetic-energy driven high $T_\textrm c$ superconductivity is demonstrated in an ideal model with only a single hopping term. Furthermore, the model predicts a symmetric pseudogap metal—dubbed ‘second Fermi liquid (FL)’—in the underdoped regime, yielding a phase diagram analogous to that of hole-doped cuprates. The bilayer Kondo model therefore presents a promising platform for both realizing higher-$T_\textrm c$ superconductors and exploring non-FL physics. We also comment on the possible limitations of the current models for the bilayer nickelate material and point out some future directions.

The quantum-confined Stark effect (QCSE) is a fundamental phenomenon in quantum dots (QDs) and quantum wells critical to electro-optic modulation. As a bridge between atoms and QDs, the magic-size clusters (MSCs) represent the ultimate in quantum-confined systems, whose response behavior to an external electric field remains largely unknown. Herein, for the first time, we investigate the QCSE in indium phosphide (InP) MSCs and consider the effects of heteroatom doping and inter-cluster coupling. Chemical modifications result in enhanced and highly tunable QCSE in InP MSCs, enabling controlled access to either linear or quadratic Stark shifts. In the coupled InP MSCs, the absorption spectrum is significantly enhanced and broadened in the visible region. The characteristic peaks are effectively tuned by the electric field and meanwhile can maintain the strong absorption and emission intensity owing to the inter-cluster coupling. These results not only fill the knowledge gap in the QCSE under extreme quantum confinement, but also provide valuable guidance for designing high-efficiency optical switches and related devices at atomic precision.

Topological pumping provides a robust mechanism for wave transport protected by band topology, but conventional implementations require strict adiabatic conditions and produce identical transport in both directions. Here, we demonstrate directional topological transport in an elastic Rice-Mele lattice by introducing engineered nonlocal (next-nearest-neighbor) coupling. By treating the modulation phase as a synthetic dimension, we verify the nontrivial gap topology through Chern numbers and show that nonlocal interactions reshape the bulk spectrum, stretching the lower band and compressing the bandgap at specific phases of the modulation cycle. This band deformation induces a strong asymmetry in adiabatic evolution: right-to-left pumping remains localized and topologically protected, whereas left-to-right pumping becomes non-adiabatic and leaks into bulk states. A reduced four-waveguide model and an effective two-level Hamiltonian elucidate this mechanism and reveal a critical wavenumber gap that predicts the transition between adiabatic and non-adiabatic evolution. Nonlocal coupling further relaxes the adiabaticity constraint, enabling comparable pumping performance with 2.75 times fewer spatial layers. Extending the analysis to include both mass and stiffness disorder confirms that the proposed direction selectivity persists under realistic weak disorder. These findings establish engineered nonlocal coupling as a passive and compact route to directional, defect-tolerant topological wave transport in elastic metamaterials and broader synthetic-dimension platforms.

We present an efficient tensor-network based algorithm for finding the optimal adaptive quantum channel discrimination strategies inspired by recently developed numerical methods in quantum metrology to find the optimal adaptive channel estimation protocols. We examine the connection between channel discrimination and estimation problems, highlighting in particular an appealing structural similarity between models that admit Heisenberg scaling estimation performance, and models that admit perfect channel discrimination in finite–number of channel uses.

The out-of-equilibrium character of active systems is often twofold, arising from both the activity itself and from non-reciprocal couplings between constituents. A well-established measure to quantify the system’s distance from equilibrium is the informatic entropy production rate. Here, we ask the question whether and how the informatic entropy production rate reflects collective behaviors and transitions in an active mixture with non-reciprocal polar couplings. In such systems, non-reciprocal orientational couplings can induce chiral motion of particles. At the field-theoretical level, transitions to these time-dependent chiral states are marked by so-called critical exceptional points. We show that at a particle level, the entropy production rate within the chiral states increases with the degree of non-reciprocity, provided it is sufficiently strong. Moreover, even at small degrees of non-reciprocity, the transitions via exceptional points leave clear signatures in the entropy production rate, which exhibits pronounced peaks at coupling strengths corresponding to the field-theoretical exceptional points. Overall, the increase and peaks of the entropy production rate mirror the susceptibility of the polarization vector at the particle level. This correspondence is supported by a field-theoretical analysis, which reveals that, in the long-wavelength limit, the entropy production rate scales with the susceptibilities of the polarization fields.

In this article, we review some recent theoretical developments on potential high-temperature superconductors and unconventional metallic states that can arise from doping a spin-one Mott insulator in the d⁸ valence. These studies are particularly relevant—though not limited—to the recently discovered bilayer nickelate superconductor La₃Ni₂O₇. We focus on a ferromagnetic Kondo lattice model with mobile electrons in the $d_{x^2-y^2}$ orbital coupled to the localized spin moments in $d_{z^2}$ orbital through a large Hund’s coupling $J_\textrm H$. In the large $J_\textrm H$ limit, the model reduces to the type II t − J model with a mixture of spin-half singlon states and spin-one doublon states. We summarize density matrix renormalization group results on the Luther-Emery liquid in one dimensional chain and two-leg ladder. Then we mainly focus on bilayer square lattice and show that a large inter-layer coupling $J_\perp$ of $d_{z^2}$ orbital can induce strong inter-layer pairing of $d_{x^2-y^2}$ orbital. In the strong $J_\perp$ limit, a kinetic-energy driven high $T_\textrm c$ superconductivity is demonstrated in an ideal model with only a single hopping term. Furthermore, the model predicts a symmetric pseudogap metal—dubbed ‘second Fermi liquid (FL)’—in the underdoped regime, yielding a phase diagram analogous to that of hole-doped cuprates. The bilayer Kondo model therefore presents a promising platform for both realizing higher-$T_\textrm c$ superconductors and exploring non-FL physics. We also comment on the possible limitations of the current models for the bilayer nickelate material and point out some future directions.

Topological phases have been a central focus of condensed-matter physics for over 50 years. Along with many experimental applications, they have provided much intellectual interest due to their characterization via some form of topological ordering, as opposed to the symmetry-breaking ordering of conventional continuous phase transitions. This distinction is most subtle in the case of the Berezinskii–Kosterlitz–Thouless (BKT) transition as its experimental realizations appear to break U(1) symmetry at low temperature. It also presents two further paradoxes: (i) its prototypical short-range interacting planar XY spin model behaves as an emergent long-range interacting electrolyte; (ii) its topological ordering is not accompanied by a topological nonergodicity within the BKT picture. This review paper addresses these three interconnected questions. We review a series of papers that demonstrate that U(1) symmetry is indeed broken, but within a broader framework than that traditionally used to characterize broken symmetry. We discuss recovery of this symmetry by breaking velocity-symmetry in a deterministic Markov process. We then expand on a modern field theory of the emergent electrolyte that maps directly from the spin field to an emergent lattice electric field governed by an augmented electrostatic Boltzmann distribution. This local model of electrolyte physics resolves both the short-range–long-range paradox and the question of topological nonergodicity—as in contrast with the BKT picture, it describes global topological defects and their nonergodic freezing by the topological ordering. It also connects the broken U(1) symmetry with the topological ordering, providing a comprehensive framework for broken symmetry at the transition. We introduce long-time topological stability as a measure of topological nonergodicity—within a general framework for weakly broken ergodicity.

Current research in statistical mechanics mostly concerns the investigation of out-of-equilibrium, irreversible processes, which are ubiquitous in nature and still far from being theoretically understood. Even the precise characterization of irreversibility is the object of an open debate: while in the context of Hamiltonian systems the one-century-old proposal by M. Smoluchowski looks still valid (a process appears irreversible when the initial state has a recurrence time that is long compared to the time of observation (Smoluchowski 1916 Z. Phys.17 557–85)), in dissipative systems, particularly in the case of stochastic processes, the problem is more involved, and quantifying the ‘degree of irreversibility’ is a pragmatic need. The most employed strategies rely on the estimation of entropy production: this quantity, although mathematically well-defined, is often difficult to compute, especially when analyzing experimental data. Moreover, being a global observable, entropy production fails to capture specific aspects of irreversibility in extended systems, such as the role of different currents and their spatial development. This review aims to address various conceptual and technical challenges encountered in the analysis of irreversibility, including the role of the coarse-graining procedure and the treatment of data in the absence of complete information. The discussion will be mostly based on simple models, analytically treatable, and supplemented by examples of complex, more realistic non-equilibrium systems.

There are several mathematical formulations of quantum mechanics. The Schrödinger picture expresses quantum states in terms of wavefunctions over, e.g. position or momentum. Alternatively, phase-space formulations represent states with quasi-probability distributions over, e.g. position and momentum. A quasi-probability distribution resembles a probability distribution but may have negative and non-real entries. The most famous quasi-probability distribution, the Wigner function, has played a pivotal role in the development of a continuous-variable quantum theory that has clear analogues of position and momentum. However, the Wigner function is ill-suited for much modern quantum-information research, which is focused on finite-dimensional systems and general observables. Instead, recent years have seen the Kirkwood–Dirac (KD) distribution come to the forefront as a powerful quasi-probability distribution for analysing quantum mechanics. The KD distribution allows tools from statistics and probability theory to be applied to problems in quantum-information processing. A notable difference to the Wigner function is that the KD distribution can represent a quantum state in terms of arbitrary observables. This paper reviews the KD distribution, in three parts. First, we present definitions and basic properties of the KD distribution and its generalisations. Second, we summarise the KD distribution’s extensive usage in the study or development of measurement disturbance; quantum metrology; weak values; direct measurements of quantum states; quantum thermodynamics; quantum scrambling and out-of-time-ordered correlators; and the foundations of quantum mechanics, including Leggett–Garg inequalities, the consistent-histories interpretation and contextuality. We emphasise connections between operational quantum advantages and negative or non-real KD quasi-probabilities. Third, we delve into the KD distribution’s mathematical structure. We summarise the current knowledge regarding the geometry of KD-positive states (the states for which the KD distribution is a classical probability distribution), describe how to witness and quantify KD non-positivity, and outline relationships between KD non-positivity, coherence and observables’ incompatibility.

We investigate known mechanisms for enhancing nuclear fusion rates at ambient temperatures and pressures in solid-state environments. In deuterium fusion, on which the paper is focused, an enhancement of >40 orders of magnitude would be needed to achieve observable fusion. We find that different mechanisms for fusion rate enhancement are known across the domains of atomic physics, nuclear physics, and quantum dynamics. Cascading multiple such mechanisms could lead to an overall enhancement of 40 orders of magnitude or more. We present a roadmap with examples of how hypothesis-driven research could be conducted in—and across—each domain to probe the plausibility of technologically-relevant fusion in the solid state.

Cilia and flagella are micron-sized filaments that actively beat with remarkable precision in a viscous medium, driving microorganism movement and efficient flow. We study the rower model to uncover how cilia activity and dissipation enable this precise motion. 
In this model, cilia motion is represented by a micro-bead’s Brownian movement between two distant harmonic potentials. At specific locations, energy pumps trigger potential switches, capturing cilia activity and generating oscillations. 
We quantify precision of oscillation using a quality factor, identifying its scaling with activity and oscillation amplitude, finding precision maximization at an optimal amplitude. The data collapse is not accurate for noisy oscillations. An exact analytic expression for the precision quality factor, based on first passage time fluctuations, and derived in the small noise approximation, explains its optimality and scaling. Energy budget analysis shows the quality factor's consistency with the thermodynamic uncertainty relation. 
Finally, we demonstrate that asymmetric beating reduces oscillation precision compared to the symmetric model: although the optimal amplitude remains unchanged, the overall scaling of the quality factor depends explicitly on the asymmetry parameter.

Motivated by experimental initiatives such as the Helmholtz International 
Beamline for Extreme Fields (HIBEF), we study Compton scattering of x-rays 
at electrons in a strong external field (e.g., a strong optical laser) with 
special emphasis on the polarization-changing contribution on the amplitude level. 
Apart from advancing our theoretical understanding,
the goal of this study is two-fold.
On the one hand, Compton scattering is a potential background process
for the planned vacuum birefringence experiments.
On the other hand, this effect could be used for diagnostic purposes.
Since the polarization-changing signal from free electrons (i.e., without the external field)
vanishes in forward direction, the ratio of the
polarization-changing and the normal (polarization conserving) contribution yields information
about the field strength at the interaction point.

We investigate the impact of space-charge effects on the ponderomotive interaction between electron pulses and laser fields in the context of ponderomotive lenses. We present a numerical framework that self-consistently models both the ponderomotive electron-light interaction and the electron-electron Coulomb repulsion within multi-electron, ultrashort pulses. By comparing these simulations with a single-electron, wave-based description, we demonstrate that space-charge effects significantly degrade the performance of ponderomotive lenses for electron beam shaping and focusing. Our results show that this deterioration appear already at very low bunch charges, setting clear limits for the operation of ponderomotive optics in dense electron ensembles.

We propose a paradigm for quantum enhanced axion dark matter search, which does not rely on power measurements. We propose to measure directly the axion amplitude and phase in an interferometric protocol at the quantum limit, using a non-linear cavity. In addition, we introduce gyromagnetic modes as wide mass range transducers for axion signals compatible with standard haloscope designs. We expect this scheme to offer an improvement of at least 4 orders of magnitude in figure of merit and at least 2 orders of magnitude in mass window with respect to standard haloscopes. Owing to its generality, our proposed protocol has the potential to speed up axion search but also the search for dark photons or other cosmological objects, such as galactic masers.

We develop a method to search for the optimal code space, induced decay rates and control Hamiltonian to implement autonomous quantum error correction (AQEC) for a general open quantum system. The system is defined by a free-evolution Lindbladian superoperator, which contains the free Hamiltonian and naturally occurring decoherence terms, as well as the control superoperators. The performance metric for optimization in our algorithm is the fidelity between the projector onto the code space and the same projector after Lindbladian evolution for a specified time. We use a gradient-based search to update the code words, induced decay matrix and control Hamiltonian matrix. We apply our algorithm to optimize AQEC codes for a variety of few-level systems. The four-level system with uniform decay rates offers a simple example for testing and illustrating the operation of our approach. The algorithm reliably succeeds in finding the optimal code in this case, while success becomes probabilistic for more complicated cases. For a five-level system with photon loss decay, the algorithm finds good AQEC codes, but these codes are not as good as the well-known binomial code. We use the binomial code as a starting point to search for the optimal code for a perturbed five-level system. In this case, the algorithm finds a code that is better than both the original binomial code and any other code obtained numerically when starting from a random initial guess. Our results demonstrate the promise of using computational techniques to discover and optimize AQEC codes in future real-world quantum computers.

Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as ‘the quantum variational eigensolver’ was developed (Peruzzo et al 2014 Nat. Commun. 5 4213) with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through a relaxation of exponential operator splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a or a topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via ‘dimensional reduction’ by compactifying one or more spatial dimensions (in ‘Kaluza–Klein’-like fashion). For -topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent -topological insulators in the same class, from which they inherit their topological properties. The eightfold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle–hole symmetries) is a reflection of the eightfold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern–Simons invariant. For lower-dimensional cases, this formula relates the winding number to the electric polarization (d=1 spatial dimensions) or to the magnetoelectric polarizability (d=3 spatial dimensions). Finally, we also discuss topological field theories describing the spacetime theory of linear responses in topological insulators (superconductors) and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).

Kwant is a Python package for numerical quantum transport calculations. It aims to be a user-friendly, universal, and high-performance toolbox for the simulation of physical systems of any dimensionality and geometry that can be described by a tight-binding model. Kwant has been designed such that the natural concepts of the theory of quantum transport (lattices, symmetries, electrodes, orbital/spin/electron-hole degrees of freedom) are exposed in a simple and transparent way. Defining a new simulation setup is very similar to describing the corresponding mathematical model. Kwant offers direct support for calculations of transport properties (conductance, noise, scattering matrix), dispersion relations, modes, wave functions, various Greenʼs functions, and out-of-equilibrium local quantities. Other computations involving tight-binding Hamiltonians can be implemented easily thanks to its extensible and modular nature. Kwant is free software available at http://kwant-project.org/.

We have grown an atom-thin, ordered, two-dimensional multi-phase film in situ through germanium molecular beam epitaxy using a gold (111) surface as a substrate. Its growth is similar to the formation of silicene layers on silver (111) templates. One of the phases, forming large domains, as observed in scanning tunneling microscopy, shows a clear, nearly flat, honeycomb structure. Thanks to thorough synchrotron radiation core-level spectroscopy measurements and advanced density functional theory calculations we can identify it as a √3 × √3 R(30°) germanene layer in conjunction with a √7 × √7 R(19.1°) Au(111) supercell, presenting compelling evidence of the synthesis of the germanium-based cousin of graphene on gold.

In the past 30 years, there has been considerable progress in the development of large-eddy simulation (LES) for turbulent flows, which has been greatly facilitated by the substantial increase in computer power. In this paper, we raise some fundamental questions concerning the conceptual foundations of LES and about the methodologies and protocols used in its application. The 10 questions addressed are stated at the end of the introduction. Several of these questions highlight the importance of recognizing the dependence of LES calculations on the artificial parameter Δ (i.e. the filter width or, more generally, the turbulence resolution length scale). The principle that LES predictions of turbulence statistics should depend minimally on Δ provides an alternative justification for the dynamic procedure.

In recent years, surface codes have become a leading method for quantum error correction in theoretical large-scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural two-dimensional nearest-neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar- and defect-based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance-3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.

Recent results on the non-universality of fault-tolerant gate sets underline the critical role of resource states, such as magic states, to power scalable, universal quantum computation. Here we develop a resource theory, analogous to the theory of entanglement, that is relevant for fault-tolerant stabilizer computation. We introduce two quantitative measures—monotones—for the amount of non-stabilizer resource. As an application we give absolute bounds on the efficiency of magic state distillation. One of these monotones is the sum of the negative entries of the discrete Wigner representation of a quantum state, thereby resolving a long-standing open question of whether the degree of negativity in a quasi-probability representation is an operationally meaningful indicator of quantum behavior.

An all-Si photonic structure emulating the quantum-valley-Hall effect is proposed. We show that it acts as a photonic topological insulator (PTI), and that an interface between two such PTIs can support edge states that are free from scattering. The conservation of the valley degree of freedom enables efficient in- and out-coupling of light between the free space and the photonic structure. The topological protection of the edge waves can be utilized for designing arrays of resonant time-delay photonic cavities that do not suffer from reflections and cross-talk.

Electroweak baryogenesis (EWBG) remains a theoretically attractive and experimentally testable scenario for explaining the cosmic baryon asymmetry. We review recent progress in computations of the baryon asymmetry within this framework and discuss their phenomenological consequences. We pay particular attention to methods for analyzing the electroweak phase transition and calculating CP-violating asymmetries, the development of Standard Model extensions that may provide the necessary ingredients for EWBG, and searches for corresponding signatures at the high energy, intensity and cosmological frontiers.

We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems. Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory. The final results are equivalent to those obtained within a different approach (Rahav et al 2003 Phys. Rev. A 68 013820), (Goldman and Dalibard 2014 Phys. Rev. X 4 031027) and can also be related to the Floquet–Magnus expansion (Casas et al 2001 J. Phys. A 34 3379). We discuss that the dependence on the driving phase, which plagues the latter, can lead to artifactual symmetry breaking. The high-frequency approach is illustrated using the example of a periodically driven Hubbard model. Moreover, we discuss the nature of the approximation and its limitations for systems of many interacting particles.