Showing papers in "New Journal of Physics in 2012"

Encoding many channels on the same frequency through radio vorticity: first experimental test

Abstract: We have shown experimentally, in a real-world setting, that it is possible to use two beams of incoherent radio waves, transmitted on the same frequency but encoded in two different orbital angular momentum states, to simultaneously transmit two independent radio channels. This novel radio technique allows the implementation of, in principle, an infinite number of channels in a given, fixed bandwidth, even without using polarization, multiport or dense coding techniques. This paves the way for innovative techniques in radio science and entirely new paradigms in radio communication protocols that might offer a solution to the problem of radio-band congestion.

Mechanical feedback from active galactic nuclei in galaxies, groups and clusters

Abstract: The radiative cooling timescales at the centers of hot atmospheres surrounding elliptical galaxies, groups and clusters are much shorter than their ages. Therefore, hot atmospheres are expected to cool and to form stars. Cold gas and star formation are observed in central cluster galaxies but at levels below those expected from an unimpeded cooling flow. X-ray observations have shown that wholesale cooling is being offset by mechanical heating from radio active galactic nuclei. Feedback is widely considered to be an important and perhaps unavoidable consequence of the evolution of galaxies and supermassive black holes. We show that cooling x-ray atmospheres and the ensuing star formation and nuclear activity are probably coupled to a self-regulated feedback loop. While the energetics are now reasonably well understood, other aspects of feedback are not. We highlight the problems of atmospheric heating and transport processes, accretion, and nuclear activity, and we discuss the potential role of black hole spin. We discuss x-ray imagery showing that the chemical elements produced by central galaxies are being dispersed on large scales by outflows launched from the vicinity of supermassive black holes. Finally, we comment on the growing evidence for mechanical heating of distant cluster atmospheres by radio jets and its potential consequences for the excess entropy in hot halos and a possible decline in the number of distant cooling flows.

A topological insulator and helical zero mode in silicene under an inhomogeneous electric field

Abstract: Silicene is a monolayer of silicon atoms forming a two-dimensional (2D) honeycomb lattice and shares almost all the remarkable properties of graphene. The low-energy structure of silicene is described by Dirac electrons with relatively large spin–orbit interactions owing to its buckled structure. A key observation is that the band structure can be controlled by applying an electric field to a silicene sheet. In particular, the gap closes at a certain critical electric field. Examining the band structure of a silicene nanoribbon, we show that a topological phase transition occurs from a topological insulator to a band insulator with an increase of electric field. We also show that it is possible to generate helical zero modes anywhere in a silicene sheet by adjusting the electric field locally to this critical value. The region may act as a quantum wire or a quantum dot surrounded by topological and/or band insulators. We explicitly construct the wave functions for some simple geometries based on the low-energy effective Dirac theory. These results are also applicable to germanene, which is a 2D honeycomb structure of germanium.

Resolution-of-identity approach to Hartree?Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions

Abstract: The efficient implementation of electronic structure methods is essential for first principles modeling of molecules and solids. We present here a particularly efficient common framework for methods beyond semilocal density-functional theory (DFT), including Hartree-Fock (HF), hybrid density functionals, random-phase approximation (RPA), second-order Moller-Plesset perturbation theory (MP2) and the GW method. This computational framework allows us to use compact and accurate numeric atom-centered orbitals (NAOs), popular in many implementations of semilocal DFT, as basis functions. The essence of our framework is to employ the 'resolution of identity (RI)' technique to facilitate the treatment of both the two-electron Coulomb repulsion integrals (required in all these approaches) and the linear density-response function (required for RPA and GW). This is possible because these quantities can be expressed in terms of the products of single-particle basis functions, which can in turn be expanded in a set of auxiliary basis functions (ABFs). The construction of ABFs lies at the heart of the RI technique, and we propose here a simple prescription for constructing ABFs which can be applied regardless of whether the underlying radial functions have a specific analytical shape

Quantum Einstein gravity

Abstract: We give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in quantum Einstein gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging space-times is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the causal dynamical triangulation approach.

Surface code quantum computing by lattice surgery

Abstract: 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.

Matter and antimatter in the universe

Abstract: We review observational evidence for a matter-antimatter asymme- try in the early universe, which leads to the remnant matter density we observe today. We also discuss bounds on the presence of antimatter in the present-day universe, including the possibility of a large lepton asymmetry in the cosmic neutrino background. We briefly review the theoretical framework within which baryogenesis, the dynamical generation of a matter-antimatter asymmetry, can occur. As an example, we discuss a testable minimal particle physics model that simultaneously explains the baryon asymmetry of the universe, neutrino oscilla- tions and dark matter.

Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators

Abstract: Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low- rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we 5 Author to whom any correspondence should be addressed.

Maximum-likelihood refinement for coherent diffractive imaging

Abstract: We introduce the application of maximum-likelihood (ML) principles to the image reconstruction problem in coherent diffractive imaging. We describe an implementation of the optimization procedure for ptychography, using conjugate gradients and including preconditioning strategies, regularization and typical modifications of the statistical noise model. The optimization principle is compared to a difference map reconstruction algorithm. With simulated data important improvements are observed, as measured by a strong increase in the signal-to-noise ratio. Significant gains in resolution and sensitivity are also demonstrated in the ML refinement of a reconstruction from experimental x-ray data. The immediate consequence of our results is the possible reduction of exposure, or dose, by up to an order of magnitude for a reconstruction quality similar to iterative algorithms currently in use.

Non-adiabatic holonomic quantum computation

Abstract: We develop a non-adiabatic generalization of holonomic quantum computation in which high-speed universal quantum gates can be realized using non-Abelian geometric phases. We show how a set of non-a ...

Magnetic-field-dependent photodynamics of single NV defects in diamond: an application to qualitative all-optical magnetic imaging

Abstract: Magnetometry and magnetic imaging with nitrogen–vacancy (NV) defects in diamond rely on the optical detection of electron spin resonance (ESR). However, this technique is inherently limited to magnetic fields that are weak enough to avoid electron spin mixing. Here, we focus on the high off-axis magnetic field regime where spin mixing alters the NV defect spin dynamics. We first study, in a quantitative manner, the dependence of the NV defect optical properties on the magnetic field vector B. Magnetic-field-dependent time-resolved photoluminescence (PL) measurements are compared to a seven-level model of the NV defect that accounts for field-induced spin mixing. The model reproduces decreases in (i) ESR contrast, (ii) PL intensity and (iii) excited level lifetime with an increasing off-axis magnetic field. We next demonstrate that these effects can be used to perform all-optical imaging of the magnetic field component |B⊥| orthogonal on the NV defect axis. Using a scanning NV defect microscope, we map the stray field of a magnetic hard disc through both PL and fluorescence lifetime imaging. This all-optical method for high magnetic field imaging at the nanoscale might be of interest in the field of nanomagnetism, where samples producing fields in excess of several tens of milliteslas are typically found.

Negative quasi-probability as a resource for quantum computation

Abstract: A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speed-up and, in particular, for fault-tolerant quantum computation. We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analogue of the Wigner function for quantum systems of odd dimension. This connection allows us to resolve an open question on the existence of bound states for magic state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations. We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.

Electromagnetic multipole theory for optical nanomaterials

Abstract: Optical properties of natural or designed materials are determined by the electromagnetic multipole moments that light can excite in the constituent particles. In this paper, we present an approach to calculating the multipole excitations in arbitrary arrays of nanoscatterers in a dielectric host medium. We introduce a simple and illustrative multipole decomposition of the electric currents excited in the scatterers and connect this decomposition to the classical multipole expansion of the scattered field. In particular, we find that completely different multipoles can produce identical scattered fields. The presented multipole theory can be used as a basis for the design and characterization of optical nanomaterials.

Optimally robust shortcuts to population inversion in two-level quantum systems

Abstract: We examine the stability versus different types of perturbations of recently proposed shortcuts to adiabaticity to speed up the population inversion of a two-level quantum system. We find the optimally robust processes by using invariant-based engineering of the Hamiltonian. Amplitude noise and systematic errors require different optimal protocols.

Coulomb-assisted braiding of Majorana fermions in a Josephson junction array

Abstract: We show how to exchange (braid) Majorana fermions in a network of superconducting nanowires by control over Coulomb interactions rather than tunneling. Even though Majorana fermions are charge-neutral quasiparticles (equal to their own antiparticle), they have an effective long-range interaction through the even–odd electron number dependence of the superconducting ground state. The flux through a split Josephson junction controls this interaction via the ratio of Josephson and charging energies, with exponential sensitivity. By switching the interaction on and off in neighboring segments of a Josephson junction array, the non-Abelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes.

Quantum adiabatic Markovian master equations

Abstract: We develop from first principles Markovian master equations suited for studying the time evolution of a system evolving adiabatically while coupled weakly to a thermal bath. We derive two sets of equations in the adiabatic limit, one using the rotating wave (secular) approximation that results in a master equation in Lindblad form, the other without the rotating wave approximation but not in Lindblad form. The two equations make markedly different predictions depending on whether or not the Lamb shift is included. Our analysis keeps track of the various time and energy scales associated with the various approximations we make, and thus allows for a systematic inclusion of higher order corrections, in particular beyond the adiabatic limit. We use our formalism to study the evolution of an Ising spin chain in a transverse field and coupled to a thermal bosonic bath, for which we identify four distinct evolution phases. While we do not expect this to be a generic feature, in one of these phases dissipation acts to increase the fidelity of the system state relative to the adiabatic ground state.

A zero-voltage conductance peak from weak antilocalization in a Majorana nanowire

Abstract: We show that weak antilocalization by disorder competes with resonant Andreev reflection from a Majorana zero mode to produce a zero-voltage conductance peak of order e2/h in a superconducting nanowire. The phase conjugation needed for quantum interference to survive a disorder average is provided by particle–hole symmetry—in the absence of time-reversal symmetry and without requiring a topologically nontrivial phase. We identify methods of distinguishing the Majorana resonance from the weak antilocalization effect.

Cavity QED with atomic mirrors

Abstract: A promising approach to merge atomic systems with scalable photonics has emerged recently, which consists of trapping cold atoms near tapered nanofibers. Here, we describe a novel technique to achieve strong, coherent coupling between a single atom and photon in such a system. Our approach makes use of collective enhancement effects, which allow a lattice of atoms to form a high-finesse cavity within the fiber. We show that a specially designated 'impurity' atom within the cavity can experience strongly enhanced interactions with single photons in the fiber. Under realistic conditions, a 'strong coupling' regime can be reached, wherein it becomes feasible to observe vacuum Rabi oscillations between the excited impurity atom and a single cavity quantum. This technique can form the basis for a scalable quantum information network using atom–nanofiber systems.

Optical helicity, optical spin and related quantities in electromagnetic theory

Abstract: We examine the optical helicity, the optical spin and the ij-infra-zilches in electromagnetic theory and show that these conserved quantities can be combined to form a new description of the angular momentum associated with optical polarization: one that is analogous to the familiar description of optical energy and linear momentum. The symmetries of Maxwell's equations that underlie the conservation of our quantities are presented and discussed. We explain that a similar but distinct set of quantities, Lipkin's zilches, describe the 'angular momentum' of the curl of the electromagnetic field, rather than the angular momentum of the electromagnetic field itself.

A k-shell decomposition method for weighted networks

Abstract: We present a generalized method for calculating the k-shell structure of weighted networks. The method takes into account both the weight and the degree of a network, in such a way that in the absence of weights we resume the shell structure obtained by the classic k-shell decomposition. In the presence of weights, we show that the method is able to partition the network in a more refined way, without the need of any arbitrary threshold on the weight values. Furthermore, by simulating spreading processes using the susceptible- infectious-recovered model in four different weighted real-world networks, we show that the weighted k-shell decomposition method ranks the nodes more accurately, by placing nodes with higher spreading potential into shells closer to the core. In addition, we demonstrate our new method on a real economic network and show that the core calculated using the weighted k-shell method is more meaningful from an economic perspective when compared with the unweighted one.

Strongly correlated quantum fluids: ultracold quantum gases, quantum chromodynamic plasmas and holographic duality

Abstract: Strongly correlated quantum fluids are phases of matter that are intrinsically quantum mechanical and that do not have a simple description in terms of weakly interacting quasiparticles. Two systems that have recently attracted a great deal of interest are the quark-gluon plasma, a plasma of strongly interacting quarks and gluons produced in relativistic heavy ion collisions, and ultracold atomic Fermi gases, very dilute clouds of atomic gases confined in optical or magnetic traps. These systems differ by 19 orders of magnitude in temperature, but were shown to exhibit very similar hydrodynamic flows. In particular, both fluids exhibit a robustly low shear viscosity to entropy density ratio, which is characteristic of quantum fluids described by holographic duality, a mapping from strongly correlated quantum field theories to weakly curved higher dimensional classical gravity. This review explores the connection

Lensing and x-ray mass estimates of clusters (simulations)

Abstract: We present a comparison between weak-lensing and x-ray mass estimates of a sample of numerically simulated clusters. The sample consists of the 20 most massive objects at redshift z = 0.25 and M_vir > 5 × 10^(14) M_☉ h^(−1). They were found in a cosmological simulation of volume 1 h^(−3) Gpc^3, evolved in the framework of a WMAP-7 normalized cosmology. Each cluster has been resimulated at higher resolution and with more complex gas physics. We processed it through Skylens and X-MAS to generate optical and x-ray mock observations along three orthogonal projections. The final sample consists of 60 cluster realizations. The optical simulations include lensing effects on background sources. Standard observational tools and methods of analysis are used to recover the mass profiles of each cluster projection from the mock catalogue. The resulting mass profiles from lensing and x-ray are individually compared to the input mass distributions. Given the size of our sample, we could also investigate the dependence of the results on cluster morphology, environment, temperature inhomogeneity and mass. We confirm previous results showing that lensing masses obtained from the fit of the cluster tangential shear profiles with Navarro–Frenk–White functionals are biased low by ~5–10% with a large scatter (~10–25%). We show that scatter could be reduced by optimally selecting clusters either having regular morphology or living in substructure-poor environment. The x-ray masses are biased low by a large amount (~25–35%), evidencing the presence of both non-thermal sources of pressure in the intra-cluster medium (ICM) and temperature inhomogeneity, but they show a significantly lower scatter than weak-lensing-derived masses. The x-ray mass bias grows from the inner to the outer regions of the clusters. We find that both biases are weakly correlated with the third-order power ratio, while a stronger correlation exists with the centroid shift. Finally, the x-ray bias is strongly connected with temperature inhomogeneities. Comparison with a previous analysis of simulations leads to the conclusion that the values of x-ray mass bias from simulations are still uncertain, showing dependences on the ICM physical treatment and, possibly, on the hydrodynamical scheme adopted.

Beyond Bell's theorem: correlation scenarios

Abstract: Bell's theorem witnesses that the predictions of quantum theory cannot be reproduced by theories of local hidden variables in which observers can choose their measurements independently of the source. Working out an idea of Branciard, Rosset, Gisin and Pironio, we consider scenarios which feature several sources, but no choice of measurement for the observers. Every Bell scenario can be mapped into such a correlation scenario, and Bell's theorem then discards those local hidden variable theories in which the sources are independent. However, most correlation scenarios do not arise from Bell scenarios, and we describe examples of (quantum) non-locality in some of these scenarios, while posing many open problems along the way. Some of our scenarios have been considered before by mathematicians in the context of causal inference.

Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering

Abstract: Tests of the predictions of quantum mechanics for entangled systems have provided increasing evidence against local realistic theories. However, there remains the crucial challenge of simultaneously closing all major loopholes—the locality, freedom-of-choice and detection loopholes—in a single experiment. An important sub-class of local realistic theories can be tested with the concept of ‘steering’. The term ‘steering’ was introduced by Schrodinger in 1935 for the fact that entanglement would seem to allow an experimenter to remotely steer the state of a distant system as in the Einstein–Podolsky–Rosen (EPR) argument. Einstein called this ‘spooky action at a distance’. EPR-steering has recently been rigorously formulated as a quantum information task opening it up to new experimental tests. Here, we present the first loophole-free demonstration of EPR-steering by violating three-setting quadratic steering inequality, tested with polarization-entangled photons shared between two distant laboratories. Our experiment demonstrates this effect while simultaneously closing all loopholes: both the locality loophole and a specific form of the freedom-of-choice loophole are closed by having a large separation of the parties and using fast quantum random number generators, and the fair-sampling loophole is closed by having high overall detection efficiency. Thereby, we exclude—for the first time loophole-free—an important class of local realistic theories considered by EPR. Besides its foundational importance, loophole-free steering also allows the distribution of quantum entanglement secure event in the presence of an untrusted party.

Braess's paradox in oscillator networks, desynchronization and power outage

Abstract: Robust synchronization is essential to ensure the stable operation of many complex networked systems such as electric power grids. Increasing energy demands and more strongly distributing power sources raise the question of where to add new connection lines to the already existing grid. Here we study how the addition of individual links impacts the emergence of synchrony in oscillator networks that model power grids on coarse scales. We reveal that adding new links may not only promote but also destroy synchrony and link this counter-intuitive phenomenon to Braess's paradox known for traffic networks. We analytically uncover its underlying mechanism in an elementary grid example, trace its origin to geometric frustration in phase oscillators, and show that it generically occurs across a wide range of systems. As an important consequence, upgrading the grid requires particular care when adding new connections because some may destabilize the synchronization of the grid—and thus induce power outages.

Reply to Comment on 'Encoding many channels on the same frequency through radio vorticity: first experimental test'

Abstract: Our recent paper (Tamburini et al 2012 New J. Phys. 14 033001), which presented results from outdoor experiments that demonstrate that it is physically feasible to simultaneously transmit different states of the newly recognized electromagnetic (EM) quantity orbital angular momentum (OAM) at radio frequencies into the far zone and to identify these states there, has led to a comment (Tamagnone et al 2012 New J. Phys. 14 118001). These authors discuss whether our investigations can be regarded as a particular implementation of the multiple-input–multiple-output (MIMO) technique. Clearly, our experimental confirmation of a theoretical prediction, first made almost a century ago (Abraham 1914 Phys. Z. XV 914–8), that the total EM angular momentum (a pseudovector of dimension length × mass × velocity) can propagate over huge distances, is essentially different from—and conceptually incompatible with—the fact that there exist engineering techniques that can enhance the spectral capacity of EM linear momentum (an ordinary vector of dimension mass × velocity). Our OAM experiments (Tamburini et al 2012 New J. Phys. 14 033001; Tamburini et al 2011 Appl. Phys. Lett. 99 204102–3) confirm the availability of a new physical layer for real-world radio communications based on EM rotational degrees of freedom. The next step is to develop new protocols and techniques for high spectral density on this new physical layer. This includes MIMO-like and other, more efficient, techniques.

A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains

Abstract: We study quantum transport properties of an open Heisenberg XXZ spin 1/2 chain driven by a pair of Lindblad jump operators satisfying a global 'micro-canonical' constraint, i.e. conserving the total magnetization. We will show that this system has an additional discrete symmetry that is specific to the Liouvillean description of the problem. Such symmetry reduces the dynamics even more than would be expected in the standard Hilbert space formalism and establishes existence of multiple steady states. Interestingly, numerical simulations of the XXZ model suggest that a pair of distinct non-equilibrium steady states becomes indistinguishable in the thermodynamic limit, and exhibit sub-diffusive spin transport in the easy-axis regime of anisotropy1 > 1.

Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics

Abstract: Silicon photonics has offered a versatile platform for the recent development of integrated optomechanical circuits. However, silicon is limited to wavelengths above 1.1µm and does not allow device operation in the visible spectrum range where low-noise lasers are conveniently available. The narrow bandgap of silicon also makes silicon optomechanical devices susceptible to strong two-photon absorption and free carrier absorption, which often introduce strong thermal effects that limit the devices' stability and cooling performance. Further, silicon also does not provide the desired lowest order optical nonlinearity for interfacing with other active electrical components on a chip. On the other hand, aluminum nitride (AlN) is a wide-band semiconductor widely used in micromechanical resonators due to its low mechanical loss and high electromechanical coupling strength. In this paper, we report the development of AlN-on-silicon platform for low loss, wide-band optical guiding, as well as its use for achieving simultaneously high-optical-quality- factor and high-mechanical-quality-factor optomechanical devices. Exploiting AlN's inherent second-order nonlinearity we further demonstrate electro-optic modulation and efficient second harmonic generation in AlN photonic circuits.

Circadian pattern and burstiness in mobile phone communication

Abstract: The temporal communication patterns of human individuals are known to be inhomogeneous or bursty, which is reflected as heavy tail behavior in the inter-event time distribution. As the cause of such a bursty behavior two main mechanisms have been suggested: (i) inhomogeneities due to the circadian and weekly activity patterns and (ii) inhomogeneities rooted in human task execution behavior. In this paper, we investigate the role of these mechanisms by developing and then applying systematic de-seasoning methods to remove the circadian and weekly patterns from the time series of mobile phone communication events of individuals. We find that the heavy tails in the inter-event time distributions remain robust with respect to this procedure, which clearly indicates that the human task execution-based mechanism is a possible cause of the remaining burstiness in temporal mobile phone communication patterns.

Growth from below: bilayer graphene on copper by chemical vapor deposition

Abstract: We evaluate how a second graphene layer forms and grows on Cu foils during chemical vapor deposition (CVD). Low-energy electron diffraction and microscopy is used to reveal that the second layer nucleates and grows next to the substrate, i.e., under a graphene layer. This underlayer mechanism can facilitate the synthesis of uniform single-layer films but presents challenges for growing uniform bilayer films by CVD. We also show that the buried and overlying layers have the same edge termination.