Your browser doesn't support javascript.
loading
Mostrar: 20 | 50 | 100
Resultados 1 - 20 de 51
Filtrar
Mais filtros

Base de dados
Tipo de documento
Intervalo de ano de publicação
1.
Phys Rev Lett ; 133(15): 150201, 2024 Oct 11.
Artigo em Inglês | MEDLINE | ID: mdl-39454170

RESUMO

Standard procedures for entanglement detection assume that experimenters can exactly implement specific quantum measurements. Here, we depart from such idealizations and investigate, in both theory and experiment, the detection of genuine multipartite entanglement when measurements are subject to small imperfections. For arbitrary qubits number n, we construct multipartite entanglement witnesses where the detrimental influence of the imperfection is independent of n. In a tabletop four-partite photonic experiment, we demonstrate first how a small amount of alignment error can undermine the conclusions drawn from standard entanglement witnesses and then perform the correction analysis. Furthermore, since we consider quantum devices that are trusted but not perfectly controlled, we showcase advantages in terms of noise resilience as compared to device-independent models.

2.
Phys Rev Lett ; 133(3): 033604, 2024 Jul 19.
Artigo em Inglês | MEDLINE | ID: mdl-39094150

RESUMO

Indistinguishability between photons is a key requirement for scalable photonic quantum technologies. We experimentally demonstrate that partly distinguishable single photons can be purified to reach near-unity indistinguishability by the process of quantum interference with ancillary photons followed by heralded detection of a subset of them. We report on the indistinguishability of the purified photons by interfering two purified photons and show improvements in the photon indistinguishability of 2.774(3)% in the low-noise regime, and as high as 10.2(5)% in the high-noise regime.

3.
Appl Opt ; 63(7): 1822-1827, 2024 Mar 01.
Artigo em Inglês | MEDLINE | ID: mdl-38437286

RESUMO

Single-mode optical fibers exhibit a small but non-negligible birefringence that induces random polarization rotations during light propagation. In classical interferometry these rotations give rise to polarization-induced fading of the interferometric visibility, and in fiber-based polarimetric sensors as well as quantum optics experiments, they scramble the information encoded in the polarization state. Correcting these undesired rotations is consequently an important part of many experiments and applications employing optical fibers. In this Lab Note we review an efficient method for fully compensating fiber polarization rotations for general input states. This method was not originally devised by us, but does, to the best of our knowledge, not appear in the literature, and our interactions with the community have indicated that it is not well known.

4.
Phys Rev Lett ; 131(6): 060803, 2023 Aug 11.
Artigo em Inglês | MEDLINE | ID: mdl-37625060

RESUMO

The quantum switch is an example of a process with an indefinite causal structure, and has attracted attention for its ability to outperform causally ordered computations within the quantum circuit model. To date, realizations of the quantum switch have made a trade-off between relying on optical interferometers susceptible to minute path length fluctuations and limitations on the range and fidelity of the implementable channels, thereby complicating their design, limiting their performance, and posing an obstacle to extending the quantum switch to multiple parties. In this Letter, we overcome these limitations by demonstrating an intrinsically stable quantum switch utilizing a common-path geometry facilitated by a novel reciprocal and universal SU(2) polarization gadget. We certify our design by successfully performing a channel discrimination task with near unity success probability.

5.
Opt Express ; 29(3): 3425-3437, 2021 Feb 01.
Artigo em Inglês | MEDLINE | ID: mdl-33770941

RESUMO

Both photonic quantum computation and the establishment of a quantum internet require fiber-based measurement and feed-forward in order to be compatible with existing infrastructure. Here we present a fiber-compatible scheme for measurement and feed-forward, whose performance is benchmarked by carrying out remote preparation of single-photon polarization states at telecom-wavelengths. The result of a projective measurement on one photon deterministically controls the path a second photon takes with ultrafast optical switches. By placing well-calibrated bulk passive polarization optics in the paths, we achieve a measurement and feed-forward fidelity of (99.0 ± 1)%, after correcting for other experimental errors. Our methods are useful for photonic quantum experiments including computing, communication, and teleportation.

6.
Opt Express ; 26(3): 3286-3302, 2018 Feb 05.
Artigo em Inglês | MEDLINE | ID: mdl-29401859

RESUMO

Multi-photon state generation is of great interest for near-future quantum simulation and quantum computation experiments. To-date spontaneous parametric down-conversion is still the most promising process, even though two major impediments still exist: accidental photon noise (caused by the probabilistic non-linear process) and imperfect single-photon purity (arising from spectral entanglement between the photon pairs). In this work, we overcome both of these difficulties by (1) exploiting a passive temporal multiplexing scheme and (2) carefully optimizing the spectral properties of the down-converted photons using periodically-poled KTP crystals. We construct two down-conversion sources in the telecom wavelength regime, finding spectral purities of > 91%, while maintaining high four-photon count rates. We use single-photon grating spectrometers together with superconducting nanowire single-photon detectors to perform a detailed characterization of our multi-photon source. Our methods provide practical solutions to produce high-quality multi-photon states, which are in demand for many quantum photonics applications.

7.
Appl Opt ; 57(3): 377-381, 2018 Jan 20.
Artigo em Inglês | MEDLINE | ID: mdl-29400784

RESUMO

The vast development of integrated quantum photonic technology enables the implementation of compact and stable interferometric networks. In particular, laser-written waveguide structures allow for complex 3D circuits and polarization-encoded qubit manipulation. However, the main limitation in the scaling up of integrated quantum devices is the single-photon loss due to mode-profile mismatch when coupling to standard fibers or other optical platforms. Here we demonstrate tapered waveguide structures realized by an adapted femtosecond laser writing technique. We show that coupling to standard single-mode fibers can be enhanced up to 77% while keeping the fabrication effort negligible. This improvement provides an important step for processing multiphoton states on chip.

8.
Opt Express ; 24(3): 2712-27, 2016 Feb 08.
Artigo em Inglês | MEDLINE | ID: mdl-26906842

RESUMO

Pair creation by spontaneous parametric down-conversion (SPDC) has become a reliable source for single-photon states, used in many kinds of quantum information experiments and applications. In order to be spectrally pure, the two photons within a generated pair should be as frequency-uncorrelated as possible. For this purpose most experiments use narrow bandpass filters, having to put up with a drastic decrease in count rates. This article elaborates (theoretically and by numerical evaluation) the alternative method to engineer a setup such that the SPDC-generated quantum states are intrinsically pure. Using pulsed pump lasers and periodically poled crystals this approach makes bandpass filtering obsolete and allows for significantly higher output intensities and therefore count rates in the detectors. After numerically scanning all common wavelength regimes, polarisation configurations and three different non-linear crystals, we present a broad variety of setups which allow for an implementation of this method.

9.
Int J Urol ; 23(8): 674-8, 2016 08.
Artigo em Inglês | MEDLINE | ID: mdl-27225958

RESUMO

OBJECTIVES: To study the effect of end-expiratory pressure used during anesthesia on blood loss during radical prostatectomy. METHODS: We evaluated 247 patients who underwent either radical retropubic prostatectomy or robot-assisted laparoscopic prostatectomy at a single institution from 2008 to 2013 by one of four surgeons. Patient characteristics were compared using t-tests, rank sum or χ(2) -tests as appropriate. The association between positive end-expiratory pressure and estimated blood loss was tested using linear regression. RESULTS: Patients were classified into high (≥4 cmH2 O) and low (≤1 cmH2 O) positive-end expiratory pressure groups. Estimated blood loss in radical retropubic prostatectomy was higher in the high positive end-expiratory pressure group (1000 mL vs 800 mL, P = 0.042). Estimated blood loss in robot-assisted laparoscopic prostatectomy was lower in the high positive end-expiratory pressure group (150 mL vs 250 mL, P = 0.015). After adjusting for other factors known to influence blood loss, a 5-cmH2 O increase in positive end-expiratory pressure was associated with a 34.9% increase in estimated blood loss (P = 0.030) for radical retropubic prostatectomy, and a 33.0% decrease for robot-assisted laparoscopic prostatectomy (P = 0.038). CONCLUSIONS: In radical retropubic prostatectomy, high positive end-expiratory pressure was associated with higher estimated blood loss, and the benefits of positive end-expiratory pressure should be weighed against the risk of increased estimated blood loss. In robot-assisted laparoscopic prostatectomy, high positive end-expiratory pressure was associated with lower estimated blood loss, and might have more than just pulmonary benefits.


Assuntos
Perda Sanguínea Cirúrgica/prevenção & controle , Respiração com Pressão Positiva , Prostatectomia , Neoplasias da Próstata/cirurgia , Procedimentos Cirúrgicos Robóticos , Humanos , Laparoscopia , Masculino
10.
Sci Adv ; 10(24): eado0215, 2024 Jun 14.
Artigo em Inglês | MEDLINE | ID: mdl-38875336

RESUMO

Precision interferometry with quantum states has emerged as an essential tool for experimentally answering fundamental questions in physics. Optical quantum interferometers are of particular interest because of mature methods for generating and manipulating quantum states of light. Their increased sensitivity promises to enable tests of quantum phenomena, such as entanglement, in regimes where tiny gravitational effects come into play. However, this requires long and decoherence-free processing of quantum entanglement, which, for large interferometric areas, remains unexplored territory. Here, we present a table-top experiment using maximally path-entangled quantum states of light in a large-scale interferometer sensitive enough to measure the rotation rate of Earth. The achieved sensitivity of 5 µrad s-1 constitutes the highest rotation resolution ever reached with optical quantum interferometers. Further improvements to our methodology will enable measurements of general-relativistic effects on entangled photons, allowing the exploration of the interplay between quantum mechanics and general relativity, along with tests for fundamental physics.

11.
Sci Adv ; 10(16): eadj0993, 2024 Apr 19.
Artigo em Inglês | MEDLINE | ID: mdl-38640248

RESUMO

The interference of nonclassical states of light enables quantum-enhanced applications reaching from metrology to computation. Most commonly, the polarization or spatial location of single photons are used as addressable degrees of freedom for turning these applications into praxis. However, the scale-up for the processing of a large number of photons of these architectures is very resource-demanding due to the rapidly increasing number of components, such as optical elements, photon sources, and detectors. Here, we demonstrate a resource-efficient architecture for multiphoton processing based on time-bin encoding in a single spatial mode. We use an efficient quantum dot single-photon source and a fast programmable time-bin interferometer to observe the interference of up to eight photons in 16 modes, all recorded only with one detector, thus considerably reducing the physical overhead previously needed for achieving equivalent tasks. Our results can form the basis for a future universal photonics quantum processor operating in a single spatial mode.

12.
Nature ; 445(7123): 65-9, 2007 Jan 04.
Artigo em Inglês | MEDLINE | ID: mdl-17203057

RESUMO

As information carriers in quantum computing, photonic qubits have the advantage of undergoing negligible decoherence. However, the absence of any significant photon-photon interaction is problematic for the realization of non-trivial two-qubit gates. One solution is to introduce an effective nonlinearity by measurements resulting in probabilistic gate operations. In one-way quantum computation, the random quantum measurement error can be overcome by applying a feed-forward technique, such that the future measurement basis depends on earlier measurement results. This technique is crucial for achieving deterministic quantum computation once a cluster state (the highly entangled multiparticle state on which one-way quantum computation is based) is prepared. Here we realize a concatenated scheme of measurement and active feed-forward in a one-way quantum computing experiment. We demonstrate that, for a perfect cluster state and no photon loss, our quantum computation scheme would operate with good fidelity and that our feed-forward components function with very high speed and low error for detected photons. With present technology, the individual computational step (in our case the individual feed-forward cycle) can be operated in less than 150 ns using electro-optical modulators. This is an important result for the future development of one-way quantum computers, whose large-scale implementation will depend on advances in the production and detection of the required highly entangled cluster states.

13.
Nat Commun ; 14(1): 3849, 2023 Jun 29.
Artigo em Inglês | MEDLINE | ID: mdl-37386044

RESUMO

Digital payments have replaced physical banknotes in many aspects of our daily lives. Similarly to banknotes, they should be easy to use, unique, tamper-resistant and untraceable, but additionally withstand digital attackers and data breaches. Current technology substitutes customers' sensitive data by randomized tokens, and secures the payment's uniqueness with a cryptographic function, called a cryptogram. However, computationally powerful attacks violate the security of these functions. Quantum technology comes with the potential to protect even against infinite computational power. Here, we show how quantum light can secure daily digital payments by generating inherently unforgeable quantum cryptograms. We implement the scheme over an urban optical fiber link, and show its robustness to noise and loss-dependent attacks. Unlike previously proposed protocols, our solution does not depend on long-term quantum storage or trusted agents and authenticated channels. It is practical with near-term technology and may herald an era of quantum-enabled security.


Assuntos
Fibras Ópticas , Tecnologia , Confiança
14.
Cancer ; 117(17): 3925-32, 2011 Sep 01.
Artigo em Inglês | MEDLINE | ID: mdl-21437885

RESUMO

BACKGROUND: A survival benefit has been observed with salvage radiation therapy (RT) for prostate-specific antigen (PSA) failure after radical prostatectomy (RP) in men with rapid rises in PSA doubling time (DT, < 6 months). Whether such a benefit exits in men with a protracted PSA rise in DT (≥ 6 months) is unclear and was examined in the current study. METHODS: Of 4036 men who underwent RP at Duke University between 1988 and 2008, 519 experienced a PSA failure, had complete data, and were the subjects of this study. Univariate and multivariate Cox regression analyses were performed to evaluate whether salvage RT in men with either a rapid (< 6 months) or a protracted (≥ 6 months) PSA DT was associated with the risk of all-cause mortality adjusting for age at the time of PSA failure, known prostate cancer prognostic factors, and cardiac comorbidity. RESULTS: After a median follow-up of 11.3 years after PSA failure, 195 men died. Salvage RT was associated with a significant reduction in all-cause mortality for men with either a PSA DT of < 6 months (adjusted hazard ratio [AHR], 0.53; P = .02) or a PSA DT of ≥ 6 months (AHR, 0.52; P = .003). In a subset of patients with comorbidity data at the time of PSA failure, salvage RT remained associated with a significant reduction in all-cause mortality for both men with a PSA DT of < 6 months (AHR, 0.35; P = .042) or a PSA DT of ≥ 6 months (AHR, 0.60; P = .04). CONCLUSIONS: Salvage RT for PSA DTs less than or in excess of 6 months is associated with a decreased risk in all-cause mortality.


Assuntos
Antígeno Prostático Específico/metabolismo , Neoplasias da Próstata/radioterapia , Idoso , Antagonistas de Androgênios/uso terapêutico , Causas de Morte , Seguimentos , Cardiopatias/complicações , Humanos , Masculino , Pessoa de Meia-Idade , Prostatectomia , Neoplasias da Próstata/complicações , Neoplasias da Próstata/metabolismo , Neoplasias da Próstata/mortalidade , Recidiva , Estudos Retrospectivos , Medição de Risco , Terapia de Salvação , Falha de Tratamento
15.
J Sex Med ; 8(10): 2935-43, 2011 Oct.
Artigo em Inglês | MEDLINE | ID: mdl-21771284

RESUMO

INTRODUCTION: The association between early and late phase sexual health-related quality of life (HRQoL) following radical prostatectomy (RP) is unclear. Moreover, factors that predict either early or late sexual HRQoL decline have not been fully investigated. AIM: The aim of this study was to evaluate the correlation between early and late phase sexual HRQoL decline, and identify clinical parameters that predict substantial sexual HRQoL decline after surgery in the early phase (3 months) and late phase (20 months) following RP. METHODS: We analyzed data on 2,345 consecutive patients who underwent radical retropubic prostatectomy, radical perineal prostatectomy, or robotic-assisted laparoscopic prostatectomy between 2001 and 2009 from the Duke Prostate Center database. MAIN OUTCOME MEASURE: Sexual HRQoL was assessed using the Expanded Prostate Cancer Index Composite instrument at baseline, early and late phase after surgery. The Spearman rank test was used to calculate correlation coefficients between early and late phase sexual HRQoL decline. Logistic regression analysis was performed to identify factors associated with substantial sexual HRQoL decline during both phases. RESULTS: Of 406 men who met our criteria, 217 (53.5%) men had normal erectile function, whereas 189 (46.5%) men had erectile dysfunction at baseline. Declines of sexual HRQoL during early phase had a significant association with that of a decline during late phase (r = 0.48, P < 0.001). In logistic regression, older age at surgery (odds ratio [OR], 1.06; P = 0.007 and OR, 1.08; P = 0.001), African-American race (OR, 4.32; P = 0.001 and OR, 3.13; P = 0.017), and overall comorbidity (OR, 1.43; P = 0.072 and OR, 1.72; P = 0.010) were consistently associated with substantial decline of sexual HRQoL in both early and late phases. CONCLUSIONS: Sexual HRQoL at early and late phases after RP were strongly correlated. Additionally, several factors were identified to be a predictor for decline of sexual HRQoL. Our findings may be used to advise patients who possess aforementioned risk factors during both phases.


Assuntos
Prostatectomia/efeitos adversos , Qualidade de Vida , Comportamento Sexual , Adulto , Fatores Etários , Idoso , Disfunção Erétil/etiologia , Humanos , Modelos Logísticos , Masculino , Pessoa de Meia-Idade , Neoplasias da Próstata/patologia , Neoplasias da Próstata/cirurgia , Grupos Raciais , Estudos Retrospectivos , Inquéritos e Questionários , Fatores de Tempo
16.
JAMA ; 306(14): 1549-56, 2011 Oct 12.
Artigo em Inglês | MEDLINE | ID: mdl-21990298

RESUMO

CONTEXT: The initial report of the Selenium and Vitamin E Cancer Prevention Trial (SELECT) found no reduction in risk of prostate cancer with either selenium or vitamin E supplements but a statistically nonsignificant increase in prostate cancer risk with vitamin E. Longer follow-up and more prostate cancer events provide further insight into the relationship of vitamin E and prostate cancer. OBJECTIVE: To determine the long-term effect of vitamin E and selenium on risk of prostate cancer in relatively healthy men. DESIGN, SETTING, AND PARTICIPANTS: A total of 35,533 men from 427 study sites in the United States, Canada, and Puerto Rico were randomized between August 22, 2001, and June 24, 2004. Eligibility criteria included a prostate-specific antigen (PSA) of 4.0 ng/mL or less, a digital rectal examination not suspicious for prostate cancer, and age 50 years or older for black men and 55 years or older for all others. The primary analysis included 34,887 men who were randomly assigned to 1 of 4 treatment groups: 8752 to receive selenium; 8737, vitamin E; 8702, both agents, and 8696, placebo. Analysis reflect the final data collected by the study sites on their participants through July 5, 2011. INTERVENTIONS: Oral selenium (200 µg/d from L-selenomethionine) with matched vitamin E placebo, vitamin E (400 IU/d of all rac-α-tocopheryl acetate) with matched selenium placebo, both agents, or both matched placebos for a planned follow-up of a minimum of 7 and maximum of 12 years. MAIN OUTCOME MEASURES: Prostate cancer incidence. RESULTS: This report includes 54,464 additional person-years of follow-up and 521 additional cases of prostate cancer since the primary report. Compared with the placebo (referent group) in which 529 men developed prostate cancer, 620 men in the vitamin E group developed prostate cancer (hazard ratio [HR], 1.17; 99% CI, 1.004-1.36, P = .008); as did 575 in the selenium group (HR, 1.09; 99% CI, 0.93-1.27; P = .18), and 555 in the selenium plus vitamin E group (HR, 1.05; 99% CI, 0.89-1.22, P = .46). Compared with placebo, the absolute increase in risk of prostate cancer per 1000 person-years was 1.6 for vitamin E, 0.8 for selenium, and 0.4 for the combination. CONCLUSION: Dietary supplementation with vitamin E significantly increased the risk of prostate cancer among healthy men. TRIAL REGISTRATION: Clinicaltrials.gov Identifier: NCT00006392.


Assuntos
Antioxidantes/efeitos adversos , Suplementos Nutricionais/efeitos adversos , Neoplasias da Próstata/epidemiologia , Selênio/administração & dosagem , Vitamina E/efeitos adversos , Idoso , Antioxidantes/administração & dosagem , Seguimentos , Humanos , Incidência , Masculino , Pessoa de Meia-Idade , Neoplasias da Próstata/prevenção & controle , Risco , Vitamina E/administração & dosagem
17.
Sci Adv ; 7(16)2021 Apr.
Artigo em Inglês | MEDLINE | ID: mdl-33853777

RESUMO

Semiconductor quantum dots are capable of emitting polarization entangled photon pairs with ultralow multipair emission probability even at maximum brightness. Using a quantum dot source with a fidelity as high as 0.987(8), we implement here quantum key distribution with an average quantum bit error rate as low as 1.9% over a time span of 13 hours. For a proof of principle, the key generation is performed with the BBM92 protocol between two buildings, connected by a 350-m-long fiber, resulting in an average raw (secure) key rate of 135 bits/s (86 bits/s) for a pumping rate of 80 MHz, without resorting to time- or frequency-filtering techniques. Our work demonstrates the viability of quantum dots as light sources for entanglement-based quantum key distribution and quantum networks. By increasing the excitation rate and embedding the dots in state-of-the-art photonic structures, key generation rates in the gigabits per second range are in principle at reach.

18.
Nat Nanotechnol ; 16(3): 318-324, 2021 Mar.
Artigo em Inglês | MEDLINE | ID: mdl-33318642

RESUMO

Nonlinear nanophotonics leverages engineered nanostructures to funnel light into small volumes and intensify nonlinear optical processes with spectral and spatial control. Owing to its intrinsically large and electrically tunable nonlinear optical response, graphene is an especially promising nanomaterial for nonlinear optoelectronic applications. Here we report on exceptionally strong optical nonlinearities in graphene-insulator-metal heterostructures, which demonstrate an enhancement by three orders of magnitude in the third-harmonic signal compared with that of bare graphene. Furthermore, by increasing the graphene Fermi energy through an external gate voltage, we find that graphene plasmons mediate the optical nonlinearity and modify the third-harmonic signal. Our findings show that graphene-insulator-metal is a promising heterostructure for optically controlled and electrically tunable nano-optoelectronic components.

19.
BJU Int ; 106(11): 1618-22, 2010 Dec.
Artigo em Inglês | MEDLINE | ID: mdl-20553253

RESUMO

OBJECTIVE: To investigate whether salvage radiation therapy (RT) for prostate-specific antigen (PSA) failure can provide the same result as adjuvant RT, which decreases the risk of all-cause mortality (ACM) for men with positive margins (R1), or extra-capsular or seminal vesicle extension (pT3). METHODS: We studied 1638 men at Duke University who underwent radical prostatectomy for unfavourable-risk prostate cancer and whose postoperative PSA was undetectable. Cox regression was used to evaluate whether salvage vs adjuvant RT in men with a rapid (<10 months) or slow (≥10 months) PSA doubling time (DT) was associated with the risk of ACM, adjusting for adverse features (pT3, R1, Gleason score 8-10), age, preoperative PSA level, comorbidity and hormonal therapy use. RESULTS: Despite fewer men with two or more adverse features (61 vs 82%; P=0.016), salvage for a rapid PSA DT vs adjuvant RT increased the risk of ACM [adjusted hazard ratio (AHR)=3.42; 95% confidence interval (CI)=1.27-9.20; P=0.015]. There was no difference (AHR=1.39; 95% CI=0.50-3.90; P=0.53) in the risk of ACM among men who received salvage for a slow PSA DT or adjuvant RT. Nearly all (90%) men with a slow PSA DT had Gleason score ≤7 and the majority (59%) had at most pT3 or R1 disease. CONCLUSION: Radiation therapy after PSA failure as compared with adjuvant RT was not associated with an increased risk of ACM in men with Gleason score ≤7 and pT3R0 or pT2R1 disease.


Assuntos
Antígeno Prostático Específico/metabolismo , Prostatectomia , Neoplasias da Próstata/radioterapia , Terapia de Salvação/métodos , Idoso , Idoso de 80 Anos ou mais , Métodos Epidemiológicos , Humanos , Masculino , Pessoa de Meia-Idade , Prostatectomia/mortalidade , Neoplasias da Próstata/mortalidade , Neoplasias da Próstata/cirurgia , Radioterapia Adjuvante/mortalidade , Terapia de Salvação/mortalidade , Resultado do Tratamento
20.
Nature ; 430(7002): 849, 2004 Aug 19.
Artigo em Inglês | MEDLINE | ID: mdl-15318210

RESUMO

Efficient long-distance quantum teleportation is crucial for quantum communication and quantum networking schemes. Here we describe the high-fidelity teleportation of photons over a distance of 600 metres across the River Danube in Vienna, with the optimal efficiency that can be achieved using linear optics. Our result is a step towards the implementation of a quantum repeater, which will enable pure entanglement to be shared between distant parties in a public environment and eventually on a worldwide scale.

SELEÇÃO DE REFERÊNCIAS
Detalhe da pesquisa