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The Gottesman-Kitaev-Preskill (GKP) code may be used to overcome noise in continuous variable quantum systems. However, preparing GKP states remains experimentally challenging. We propose a method for preparing GKP states by engineering a time-periodic Hamiltonian whose Floquet states are GKP states. This Hamiltonian may be realized in a superconducting circuit comprising a SQUID shunted by a superinductor and a capacitor, with a characteristic impedance twice the resistance quantum. The GKP Floquet states can be prepared by adiabatically tuning the frequency of the external magnetic flux drive. We predict that highly squeezed >11.9  dB (10.8 dB) GKP magic states can be prepared on a microsecond timescale, given a quality factor of 10^{6} (10^{5}) and flux noise at typical rates.

Background: Neurofibromatosis type 1 (NF1) is a genetic neurocutaneous disorder commonly associated with motor and cognitive symptoms that greatly impact quality of life. Transcranial magnetic stimulation (TMS) can quantify motor cortex physiology, reflecting the basis for impaired motor function as well as, possibly, clues for mechanisms of effective treatment. We hypothesized that children with NF1 have impaired motor function and altered motor cortex physiology compared to typically developing (TD) control children and children with attention-deficit/hyperactivity disorder (ADHD)., Methods: Children aged 8-17 years with NF1 (n = 21) were compared to children aged 8-12 years with ADHD (n = 59) and TD controls (n = 88). Motor development was assessed using the Physical and Neurological Examination for Subtle Signs (PANESS) scale. The balance of inhibition and excitation in motor cortex was assessed using the TMS measures short-interval cortical inhibition (SICI) and intracortical facilitation (ICF). Measures were compared by diagnosis and tested using bivariate correlations and regression for association with clinical characteristics., Results: In NF1, ADHD severity scores were intermediate between the ADHD and TD cohorts, but total PANESS scores were markedly elevated (worse) compared to both (P < 0.001). Motor cortex ICF (excitatory) was significantly lower in NF1 than in TD and ADHD (P < 0.001), but SICI (inhibitory) did not differ. However, in NF1, better PANESS scores correlated with lower SICI ratios (more inhibition; ρ = 0.62, P = 0.003) and lower ICF ratios (less excitation; ρ = 0.38, P = 0.06)., Conclusions: TMS-evoked SICI and ICF may reflect processes underlying abnormal motor function in children with NF1., (Copyright © 2023 Elsevier Inc. All rights reserved.)

Quaternary ammonium compounds (QACs), a large class of chemicals that includes high production volume substances, have been used for decades as antimicrobials, preservatives, and antistatic agents and for other functions in cleaning, disinfecting, personal care products, and durable consumer goods. QAC use has accelerated in response to the COVID-19 pandemic and the banning of 19 antimicrobials from several personal care products by the US Food and Drug Administration in 2016. Studies conducted before and after the onset of the pandemic indicate increased human exposure to QACs. Environmental releases of these chemicals have also increased. Emerging information on adverse environmental and human health impacts of QACs is motivating a reconsideration of the risks and benefits across the life cycle of their production, use, and disposal. This work presents a critical review of the literature and scientific perspective developed by a multidisciplinary, multi-institutional team of authors from academia, governmental, and nonprofit organizations. The review evaluates currently available information on the ecological and human health profile of QACs and identifies multiple areas of potential concern. Adverse ecological effects include acute and chronic toxicity to susceptible aquatic organisms, with concentrations of some QACs approaching levels of concern. Suspected or known adverse health outcomes include dermal and respiratory effects, developmental and reproductive toxicity, disruption of metabolic function such as lipid homeostasis, and impairment of mitochondrial function. QACs' role in antimicrobial resistance has also been demonstrated. In the US regulatory system, how a QAC is managed depends on how it is used, for example in pesticides or personal care products. This can result in the same QACs receiving different degrees of scrutiny depending on the use and the agency regulating it. Further, the US Environmental Protection Agency's current method of grouping QACs based on structure, first proposed in 1988, is insufficient to address the wide range of QAC chemistries, potential toxicities, and exposure scenarios. Consequently, exposures to common mixtures of QACs and from multiple sources remain largely unassessed. Some restrictions on the use of QACs have been implemented in the US and elsewhere, primarily focused on personal care products. Assessing the risks posed by QACs is hampered by their vast structural diversity and a lack of quantitative data on exposure and toxicity for the majority of these compounds. This review identifies important data gaps and provides research and policy recommendations for preserving the utility of QAC chemistries while also seeking to limit adverse environmental and human health effects.

1,4-Dioxane is a persistent and mobile organic chemical that has been found by the United States Environmental Protection Agency (USEPA) to be an unreasonable risk to human health in some occupational contexts. 1,4-Dioxane is released into the environment as industrial waste and occurs in some personal-care products as an unintended byproduct. However, limited exposure assessments have been conducted outside of an occupational context. In this study, the USEPA simulation modeling tool, Stochastic Human Exposure and Dose Simulator-High Throughput (SHEDS-HT), was adapted to estimate the exposure and chemical mass released down the drain (DTD) from drinking water consumption and product use. 1,4-Dioxane concentrations measured in drinking water and consumer products were used by SHEDS-HT to evaluate and compare the contributions of these sources to exposure and mass released DTD. Modeling results showed that compared to people whose daily per capita exposure came from only products (2.29 × 10 -7 to 2.92 × 10 -7 mg/kg/day), people exposed to both contaminated water and product use had higher per capita median exposures (1.90 × 10 -6 to 4.27 × 10 -6 mg/kg/day), with exposure mass primarily attributable to water consumption (75-91%). Last, we demonstrate through simulation that while a potential regulatory action could broadly reduce DTD release, the proportional reduction in exposure would be most significant for people with no or low water contamination.

Scalable quantum processors require tunable two-qubit gates that are fast, coherent and long-range. The Heisenberg exchange interaction offers fast and coherent couplings for spin qubits, but is intrinsically short-ranged. Here, we demonstrate that its range can be increased by employing a multielectron quantum dot as a mediator, while preserving speed and coherence of the resulting spin-spin coupling. We do this by placing a large quantum dot with 50-100 electrons between a pair of two-electron double quantum dots that can be operated and measured simultaneously. Two-spin correlations identify coherent spin-exchange processes across the multielectron quantum dot. We further show that different physical regimes of the mediated exchange interaction allow a reduced susceptibility to charge noise at sweet spots, as well as positive and negative coupling strengths up to several gigahertz. These properties make multielectron dots attractive as scalable, voltage-controlled coherent coupling elements.

Surfactants are high production volume chemicals used in numerous domestic and industrial applications and, after use, the most abundant organic contaminants in wastewater. Their discharge might jeopardize the receiving aquatic ecosystems, including sediments, where they tend to accumulate. This is the first comprehensive study on their distribution and fate in this environmental compartment as we performed simultaneous analysis of the three main classes of surfactants (anionic: LAS; nonionic: NPEO and AEO; cationic: DTDMAC, DADMAC, BAC, and ATMAC) and some of their transformation products (SPC, NP, NPEC, and PEG). To account for spatial and time trends, surface sediments and dated cores were collected from Jamaica Bay, a heavily sewage-impacted estuary in New York City. The concentrations of surfactants in surface sediments were between 18 and > 200 μg g -1 and showed slight variation (<10%) over different sampling years (1998, 2003 and 2008). Cationic surfactants were found at the highest concentrations, with DTDMAC accounting for between 52 and 90% of the total sum of target compounds. Vertical concentration profiles in dated cores from the most contaminated station, in the vicinity of the biggest local sewage treatment plant (STP), indicated two sub-surface surfactant peaks in the mid-1960s (469 μg g -1 ) and late 1980s (572 μg g -1 ) coinciding with known STP upgrades. This trend was observed for most target compounds, except for DADMAC, C22ATMAC, and PEG, which showed a continuous increase towards the top of the cores. In-situ degradation was studied by comparing sediment core samples taken 12 years apart (1996 and 2008) and revealed a net decrease in PEG and specific surfactants (BAC, ATMAC, NPEO, and AEO) accompanied by growing concentrations of metabolites (SPC, NP, and NPEC). DTDMAC, DADMAC, and LAS, however, remained stable over this period, suggesting recalcitrant behavior under the anaerobic conditions in Jamaica Bay sediments., Main Finding: Chronology of major synthetic surfactants are illustrated in the dated sediment cores, as well as their different diagenetic fates., (Copyright © 2018 Elsevier Ltd. All rights reserved.)

Achieving error rates that meet or exceed the fault-tolerance threshold is a central goal for quantum computing experiments, and measuring these error rates using randomized benchmarking is now routine. However, direct comparison between measured error rates and thresholds is complicated by the fact that benchmarking estimates average error rates while thresholds reflect worst-case behavior when a gate is used as part of a large computation. These two measures of error can differ by orders of magnitude in the regime of interest. Here we facilitate comparison between the experimentally accessible average error rates and the worst-case quantities that arise in current threshold theorems by deriving relations between the two for a variety of physical noise sources. Our results indicate that it is coherent errors that lead to an enormous mismatch between average and worst case, and we quantify how well these errors must be controlled to ensure fair comparison between average error probabilities and fault-tolerance thresholds.

We use a simple real-space renormalization-group approach to investigate the critical behavior of the quantum Ashkin-Teller model, a one-dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously. This approach, which is based on exploiting the on-site symmetry of the model, has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the quantum transverse-field Ising model. Our investigation explores this approach in more generality, in a model in which the critical behavior has a richer structure but which reduces to the simpler Ising case at a special point. We demonstrate that the correlation length critical exponent as predicted from this real-space renormalization-group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality. Near the Ising special point, the error in the estimated critical exponent from this simple method is comparable to that of numerically intensive simulations based on much more sophisticated methods, although the accuracy decreases away from the decoupled Ising model point.

Unwanted interaction between a quantum system and its fluctuating environment leads to decoherence and is the primary obstacle to establishing a scalable quantum information processing architecture. Strategies such as environmental and materials engineering, quantum error correction and dynamical decoupling can mitigate decoherence, but generally increase experimental complexity. Here we improve coherence in a qubit using real-time Hamiltonian parameter estimation. Using a rapidly converging Bayesian approach, we precisely measure the splitting in a singlet-triplet spin qubit faster than the surrounding nuclear bath fluctuates. We continuously adjust qubit control parameters based on this information, thereby improving the inhomogenously broadened coherence time (T2*) from tens of nanoseconds to >2 μs. Because the technique demonstrated here is compatible with arbitrary qubit operations, it is a natural complement to quantum error correction and can be used to improve the performance of a wide variety of qubits in both meteorological and quantum information processing applications.

The compound semiconductor gallium-arsenide (GaAs) provides an ultra-clean platform for storing and manipulating quantum information, encoded in the charge or spin states of electrons confined in nanostructures. The absence of inversion symmetry in the zinc-blende crystal structure of GaAs however, results in a strong piezoelectric interaction between lattice acoustic phonons and qubit states with an electric dipole, a potential source of decoherence during charge-sensitive operations. Here we report phonon generation in a GaAs double quantum dot, configured as a single- or two-electron charge qubit, and driven by the application of microwaves via surface gates. In a process that is a microwave analogue of the Raman effect, phonon emission produces population inversion of the two-level system and leads to rapid decoherence of the qubit when the microwave energy exceeds the level splitting. Comparing data with a theoretical model suggests that phonon emission is a sensitive function of the device geometry.

Defects in topologically ordered models have interesting properties that are reminiscent of the anyonic excitations of the models themselves. For example, dislocations in the toric code model are known as twists and possess properties that are analogous to Ising anyons. We strengthen this analogy by using the topological entanglement entropy as a diagnostic tool to identify properties of both defects and excitations in the toric code. Specifically, we show, through explicit calculation, that the toric code model including twists and dyon excitations has the same quantum dimensions, the same total quantum dimension, and the same fusion rules as an Ising anyon model.

We derive dynamical equations for a driven, dissipative quantum system in which the environment-induced relaxation rate is comparable to the Rabi frequency, avoiding assumptions on the frequency dependence of the environmental coupling. When the environmental coupling varies significantly on the scale of the Rabi frequency, secular or rotating wave approximations break down. We avoid these approximations, yielding dynamical steady states which account for the interaction between driven quantum dots and their phonon environment. The theory, which is motivated by recent experimental observations, qualitatively and quantitatively describes the transition from asymmetric unsaturated resonances at weak driving to population inversion at strong driving.

The problem of estimating an unknown force driving a linear oscillator is revisited. When using linear measurement, feedback is often cited as a mechanism to enhance bandwidth, sensitivity or resolution. We show that as long as the oscillator dynamics are known, there exists a real-time estimation strategy that reproduces the same measurement record as any arbitrary feedback protocol. Consequently some form of nonlinearity is required to gain any advantage beyond estimation alone. This result holds true in both quantum and classical systems, with nonstationary forces and feedback, and in the general case of non-Gaussian and correlated noise. Recently, feedback enhanced incoherent force resolution has been demonstrated [E. Gavartin, P. Verlot, and T. J. Kippenberg, Nat. Nano. 7, 509 (2012)], with the enhancement attributed to a feedback induced modification of the mechanical susceptibility. As a proof-of-principle, we experimentally reproduce this result through straightforward filtering.

Quantum-dot spin qubits characteristically use oscillating magnetic or electric fields, or quasi-static Zeeman field gradients, to realize full qubit control. For the case of three confined electrons, exchange interaction between two pairs allows qubit rotation around two axes, hence full control, using only electrostatic gates. Here, we report initialization, full control, and single-shot readout of a three-electron exchange-driven spin qubit. Control via the exchange interaction is fast, yielding a demonstrated 75 qubit rotations in less than 2 ns. Measurement and state tomography are performed using a maximum-likelihood estimator method, allowing decoherence, leakage out of the qubit state space, and measurement fidelity to be quantified. The methods developed here are generally applicable to systems with state leakage, noisy measurements and non-orthogonal control axes.

A new approach to single-qubit operations using exchange interactions of single electrons in gate-defined quantum dots has recently been demonstrated: the resonant exchange qubit. We show that two-qubit operations, specifically the controlled phase gate, can be performed between resonant exchange qubits very straightforwardly, using a single exchange pulse. This is in marked contrast to the best known protocols for exchange qubits where such a gate requires many pulses so that leakage processes arising from the exchange interaction can be overcome. For resonant exchange qubits a simple two-qubit gate is possible because in this mode of operation energy conservation suppresses leakage.

We experimentally surpass the 3 dB limit to steady-state parametric squeezing of a mechanical oscillator. The localization of an atomic force microscope cantilever, achieved by optimal estimation, is enhanced by up to 6.2 dB in one position quadrature when a detuned parametric drive is used. This squeezing is, in principle, limited only by the oscillator Q factor. Used on low temperature, high frequency oscillators, this technique provides a pathway to achieve robust quantum squeezing below the zero-point motion. Broadly, our results demonstrate that control systems engineering can overcome well established limits in applications of nonlinear processes. Conversely, by localizing the mechanical position to better than the measurement precision of our apparatus, they demonstrate the usefulness of mechanical nonlinearities in control applications.

We measure the quantum fluctuations of a pumped nonlinear resonator using a superconducting artificial atom as an in situ probe. The qubit excitation spectrum gives access to the frequency and amount of excitation of the intracavity field fluctuations, from which we infer its effective temperature. These quantities are found to be in agreement with theoretical predictions; in particular, we experimentally observe the phenomenon of quantum heating.

We report the dispersive charge-state readout of a double quantum dot in the few-electron regime using the in situ gate electrodes as sensitive detectors. We benchmark this gate sensing technique against the well established quantum point contact charge detector and find comparable performance with a bandwidth of ∼ 10 MHz and an equivalent charge sensitivity of ∼ 6.3 × 10(-3) e/sqrt[Hz]. Dispersive gate sensing alleviates the burden of separate charge detectors for quantum dot systems and promises to enable readout of qubits in scaled-up arrays.

We revisit the stochastic master equation approach to feedback cooling of a quantum mechanical oscillator undergoing position measurement. By introducing a rotating wave approximation for the measurement and bath coupling, we can provide a more intuitive analysis of the achievable cooling in various regimes of measurement sensitivity and temperature. We also discuss explicitly the effect of backaction noise on the characteristics of the optimal feedback. The resulting rotating wave master equation has found application in our recent work on squeezing the oscillator motion using parametric driving and may have wider interest.

Ground states of spin lattices can serve as a resource for measurement-based quantum computation. Ideally, the ability to perform quantum gates via measurements on such states would be insensitive to small variations in the Hamiltonian. Here, we describe a class of symmetry-protected topological orders in one-dimensional systems, any one of which ensures the perfect operation of the identity gate. As a result, measurement-based quantum gates can be a robust property of an entire phase in a quantum spin lattice, when protected by an appropriate symmetry.

Nonlinear forces allow motion of a mechanical oscillator to be squeezed below the zero-point motion. Of existing methods, mechanical parametric amplification is relatively accessible, but previously thought to be limited to 3 dB of squeezing in the steady state. We consider the effect of applying continuous weak measurement and feedback to this system. If the parametric drive is optimally detuned from resonance, correlations between the quadratures of motion allow unlimited steady-state squeezing. Compared to backaction evasion, we demonstrate that the measurement strength, temperature and efficiency requirements for quantum squeezing are significantly relaxed.

We have performed spectroscopic measurements of a superconducting qubit dispersively coupled to a nonlinear resonator driven by a pump microwave field. Measurements of the qubit frequency shift provide a sensitive probe of the intracavity field, yielding a precise characterization of the resonator nonlinearity. The qubit linewidth has a complex dependence on the pump frequency and amplitude, which is correlated with the gain of the nonlinear resonator operated as a small-signal amplifier. The corresponding dephasing rate is found to be close to the quantum limit in the low-gain limit of the amplifier.

We present theory and experiment for the task of discriminating two nonorthogonal states, given multiple copies. We implement several local measurement schemes, on both pure states and states mixed by depolarizing noise. We find that schemes which are optimal (or have optimal scaling) without noise perform worse with noise than simply repeating the optimal single-copy measurement. Applying optimal control theory, we derive the globally optimal local measurement strategy, which outperforms all other local schemes, and experimentally implement it for various levels of noise.

Quantum computation can proceed solely through single-qubit measurements on an appropriate quantum state, such as the ground state of an interacting many-body system. We investigate a simple spin-lattice system based on the cluster-state model, and by using nonlocal correlation functions that quantify the fidelity of quantum gates performed between distant qubits, we demonstrate that it possesses a quantum (zero-temperature) phase transition between a disordered phase and an ordered "cluster phase" in which it is possible to perform a universal set of quantum gates.

Many proposals for quantum information processing are subject to detectable loss errors. In this Letter, we show that topological error correcting codes, which protect against computational errors, are also extremely robust against losses. We present analytical results showing that the maximum tolerable loss rate is 50%, which is determined by the square-lattice bond percolation threshold. This saturates the bound set by the no-cloning theorem. Our numerical results support this and show a graceful trade-off between tolerable thresholds for computational and loss errors.

We investigate the anisotropic quantum orbital compass model on an infinite square lattice by means of the infinite projected entangled-pair state algorithm. For varying values of the Jx and Jz coupling constants of the model, we approximate the ground state and evaluate quantities such as its expected energy and local order parameters. We also compute adiabatic continuations of the ground state, and show that several ground states with different local properties coexist at Jx=Jz. All our calculations are fully consistent with a first order quantum phase transition at this point, thus corroborating previous numerical evidence. Our results also suggest that tensor network algorithms are particularly fitted to characterize first order quantum phase transitions.

One of the most significant and well-known properties of entangled states is that they may lead to violations of Bell inequalities and are thus inconsistent with any local-realistic theory. However, there are entangled states that cannot violate any Bell inequality, and in general the precise relationship between entanglement and observable nonlocality is not well understood. We demonstrate that a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality can be demonstrated in a certain kind of Bell experiment for all entangled states. Our proof of the result consists of two main steps. We first provide a simple characterization of the set of states that do not violate the CHSH inequality even after general local operations and classical communication. Second, we prove that for each entangled state sigma, there exists another state rho not violating the CHSH inequality, such that rhomultiply sign in circlesigma violates the CHSH inequality.

The concept of steering was introduced by Schrödinger in 1935 as a generalization of the Einstein-Podolsky-Rosen paradox for arbitrary pure bipartite entangled states and arbitrary measurements by one party. Until now, it has never been rigorously defined, so it has not been known (for example) what mixed states are steerable (that is, can be used to exhibit steering). We provide an operational definition, from which we prove (by considering Werner states and isotropic states) that steerable states are a strict subset of the entangled states, and a strict superset of the states that can exhibit Bell nonlocality. For arbitrary bipartite Gaussian states we derive a linear matrix inequality that decides the question of steerability via Gaussian measurements, and we relate this to the original Einstein-Podolsky-Rosen paradox.

Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.

We derive a master equation for a driven double quantum dot damped by an unstructured phonon bath, and calculate the spectral density. We find that bath-mediated photon absorption is important at relatively strong driving, and may even dominate the dynamics, inducing population inversion of the double-dot system. This phenomenon is consistent with recent experimental observations.

For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case: Given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state-based) control problems (the stationary linear-quadratic-Gaussian class), this question is a semidefinite program. Moreover, the answer also applies to Markovian (current-based) feedback.

The shot-noise detection limit in current high-precision magnetometry [Nature (London) 422, 596 (2003)] is a manifestation of quantum fluctuations that scale as 1/sqrt[N] in an ensemble of N atoms. Here, we develop a procedure that combines continuous measurement and quantum Kalman filtering [Rep. Math. Phys. 43, 405 (1999)]] to surpass this conventional limit by exploiting conditional spin squeezing to achieve 1/N field sensitivity. Our analysis demonstrates the importance of optimal estimation for high bandwidth precision magnetometry at the Heisenberg limit and also identifies an approximate estimator based on linear regression.

While all bipartite pure entangled states violate some Bell inequality, the relationship between entanglement and nonlocality for mixed quantum states is not well understood. We introduce a simple and efficient algorithmic approach for the problem of constructing local hidden variable theories for quantum states. The method is based on constructing a so-called symmetric quasiextension of the quantum state that gives rise to a local hidden variable model with a certain number of settings for the observers Alice and Bob.

Modern cavity quantum electrodynamics (cavity QED) illuminates the most fundamental aspects of coherence and decoherence in quantum mechanics. Experiments on atoms in cavities can be described by elementary models but reveal intriguing subtleties of the interplay of coherent dynamics with external couplings. Recent activity in this area has pioneered powerful new approaches to the study of quantum coherence and has fueled the growth of quantum information science. In years to come, the purview of cavity QED will continue to grow as researchers build on a rich infrastructure to attack some of the most pressing open questions in micro- and mesoscopic physics.

We present an experimental demonstration of the power of feedback in quantum metrology, confirming the predicted [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)]] superior performance of an adaptive homodyne technique for single-shot measurement of optical phase. For measurements performed on weak coherent states with no prior knowledge of the signal phase, adaptive homodyne estimation approaches closer to the intrinsic quantum uncertainty than any previous technique. Our results underscore the importance of real-time feedback for reaching quantum limits in measurement and control.

We report high time-resolution measurements of photon statistics from pairs of dye molecules coupled by fluorescence resonance energy transfer (FRET). In addition to quantum-optical photon antibunching, we observe photon bunching on a time scale of several nanoseconds. We show by numerical simulation that configuration fluctuations in the coupled fluorophore system could account for minor deviations of our data from predictions of basic Förster theory. With further characterization we believe that FRET photon statistics could provide a unique tool for studying DNA mechanics on time scales from 10(-9)-10(-3) s.

We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.

The motion of individual cesium atoms trapped inside an optical resonator is revealed with the atom-cavity microscope (ACM). A single atom moving within the resonator generates large variations in the transmission of a weak probe laser, which are recorded in real time. An inversion algorithm then allows individual atom trajectories to be reconstructed from the record of cavity transmission and reveals single atoms bound in orbit by the mechanical forces associated with single photons. In these initial experiments, the ACM yields 2-micrometer spatial resolution in a 10-microsecond time interval. Over the duration of the observation, the sensitivity is near the standard quantum limit for sensing the motion of a cesium atom.