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Measuring the Orbital Angular Momentum of a Single Photon (PRL_88_257901_Courtial)

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Measuring the Orbital Angular Momentum of a Single PhotonJonathan Leach,1Miles J.Padgett,1Stephen M.Barnett,2Sonja Franke-Arnold,2and Johannes Courtial1,* 1Department of Physics and Astronomy,University of Glasgow,Glasgow,Scotland 2Department of Physics and Applied Physics,University of Strathclyde,Glasgow,Scotland(Received21January2002;published5June2002)We propose an interferometric method for measuring the orbital angular momentum of single photons.We demonstrate its viability by sorting four different orbital angular momentum states,and are thusable to encode two bits of information on a single photon.This new approach has implications forentanglement experiments,quantum cryptography and high density information transfer.DOI:10.1103/PhysRevLett.88.257901PACS numbers:03.67.–a,42.50.CtIt is well known that photons can carry both spin and orbital angular momentum(OAM)[1,2].The spin is asso-ciated with polarization and the OAM with the azimuthal phase of the complex electricfield.Each photon of a beam with an azimuthal phase dependence of the form exp͑il f͒,for example,carries an OAM of l¯h.The polari-zation of a single photon is described by a state in a two-dimensional space.For this reason,photon polarization provides a useful physical realization of a single qubit and has been widely employed in demonstrations of quantum key distribution[3,4].As recently pointed out[5],the in-finite number of orthogonal states of OAM places no limit on the number of bits that can be carried by a single pho-ton.Moreover,the ability to create states with different OAM and superpositions of these allows the realization of quNits,that is quantum states in an N-dimensional space, with single photons.The problem addressed in this Letter is the realization of a multichannel device for determining the OAM of a single photon.Such a device will allow us to take advantage of the increase in information capacity as-sociated with orbital rather than spin angular momentum. Previous work has shown that the OAM of a laser beam containing many photons in the same mode can be mea-sured.For example,interfering an exp͑il f͒beam with its mirror image produces an interferogram with2l radial spokes[6,7][Fig.1(a)].Although this technique can dis-criminate between an arbitrarily large number of states, the state of one single photon cannot be measured as many photons are required to form the full interference pattern. For individual photons,computer-generated holograms [8,9],when used in combination with a pinhole in a setup similar to the one shown in Fig.1(b),can determine the particular OAM state[10].Holograms are often used in the generation of exp͑il f͒beams,where the fork dislo-cation in the hologram introduces helical phase fronts in the diffracted beam.When operated in reverse,the holo-gram“flattens”the helical phase fronts.The beam,now with planar phase fronts,can be focused through a pinhole and detected.However,this technique allows photons to be tested only for one particular state.A more complex computer-generated hologram was demonstrated that can detect several different l states but with an efficiency that cannot exceed the reciprocal of the number of different l values[11].This low efficiency means that this method is not likely to be useful for quantum information appli-cations.In a different approach,the difference in Gouy phase was used within an interferometer to distinguish be-tween modes of two different orders(the mode order is dependent on l)[12].Although working at the single-photon level such a scheme sorts between only two states. In this Letter we describe an interferometric technique that can distinguish individual photons in arbitrarily many OAM states with a theoretical efficiency of100%.We demonstrate this principle with a device that simultane-ously sorts four different OAM states,correspondingto(a)(b)l2l = 4 fringeshologramFIG.1.Previous work on measuring the OAM of light.(a)A Mach-Zehnder interferometer with a Dove prism inserted into one arm interferes the incoming light beam with its own mirror image.In the case of light with l intertwined helical phase fronts,the interference pattern has2l radial fringes.This setup is capable of distinguishing between an arbitrary number of states, but forming the required fringe pattern needs many photons.(b)A hologram can be used to“flatten”the phase fronts of light modes with specific values of l,which makes it possible to focus these modes(but no others)through a pinhole,behind which they can be detected.While this latter setup works with individual photons,it can only test for one particular OAM state.two bits of information that can be transferred with a single photon.Our device relies on the exp͑il f͒form of the trans-verse modes.On rotation of the beam through an angle a,this phase dependence becomes exp͓il͑f1a͔͒.This corresponds to a phase shift of Dc෇l a[13],which is a manifestation of a geometrical phase[14].For particular combinations of l and a,the rotated beam may be either in or out of phase with respect to the original.For example, when a෇p,a beam with even l is in phase with the origi-nal but a beam with odd l rotated by the same angle is out of phase by p(Fig.2).If such a rotation is incorporated into the arms of a two-beam interferometer,then the phase shift between the two arms becomes l dependent.It fol-lows that for different angles of rotation,constructive and destructive interference occurs for different values of l. This concept can be realized in the form of a Mach-Zehnder interferometer with a Dove prism inserted into each arm(Fig.3).A Dove prismflips the transverse cross section of any transmitted beam[15].Two Dove prisms, rotated with respect to each other through an angle a͞2,l 1 2 3∆ψ0π0π4exp(ilφ)exp(il(φ+π))FIG.2.Gray-scale representations of phase profiles of nonro-tated and rotated beams with an exp͑il f͒phase structure.After a rotation through p,a beam with even l is unchanged,while one with odd l is out of phase by p with the nonrotated beam. Interfering an l beam with a rotated copy of itself therefore results in constructive interference for even l and destructive in-terference for odd l.rotate a passing beam through an angle a.In the ex-ample shown in Fig.3,a͞2෇p͞2and hence the relative phase difference between the two arms of the interferome-ter is Dc෇l p.By correctly adjusting the path length of the interferometer we can ensure that photons with even l appear in port A1and photons with odd l appear in port B1.If the input state is a mixture of even and odd l com-ponents,then these components are“sorted”into an even channel A1and an odd channel B1.Our principle can be extended further to enable us to test for an arbitrarily large number of OAM states.This is achieved by cascading additional Mach-Zehnder inter-ferometers with different rotation angles(Fig.4).(Note that the scheme outlined in Ref.[12]could be extended in an analogous fashion.)Thefirst interferometer,stage 1,sorts photons with even and odd values of l into ports A1and B1,respectively.Photons with even l are then passed into the second stage where they are sorted further. The angle between the Dove prisms of the second stage is a͞2෇p͞4corresponding to Dc෇l p͞2.Therefore, modes with l෇4n,where n is an integer,go into port A2 and beams with a phase term of l෇4n12go into port B2.Unfortunately,there is no rotation angle that allows us to unambiguously sort odd-l photons in the same way.We solve this problem by placing a hologram in front of one interferometer of the second stage so that we can increase the azimuthal phase of the odd-l photons by1,thereby making their l values even.An additional interferometer with a෇p͞2will now separate the original odd-l pho-tons in the same way as the even-l photons were sorted. Figure4outlines thefirst three sorting stages,which al-low discrimination between eight different values of l.By adding further stages,this procedure can be extended to allow an arbitrarily large number of OAM states to be dis-tinguished.It should be noted that,in the absence of holo-grams,a scheme similar to that illustrated in Fig.4can be constructed to sort beams where l takes on the values of0 or2n,where n is an integer.To demonstrate the viability of our proposed mecha-nism,three Mach-Zehnder interferometers were built toFIG.3.First stage of our OAM sorter.A Mach-Zehnder in-terferometer with a Dove prism placed in each arm.The beams in the two arms are rotated with respect to each other through an angle a,where a͞2is the relative angle between the dove prisms.In the example shown,a͞2෇p͞2,this device sorts photons with even values of l into Port A1and those with odd values of l into Port B1.FIG.4.First three stages of a general sorting scheme.The gray boxes each represent an interferometer of the form shown in Fig.3with different angles between the Dove prisms.Thefirst stage introduces a phase shift of a෇p and so sorts multiples of2:even l s into Port A1and odd l s into Port B1.The odd-l photons then pass though an D l෇1hologram so that they become even-l photons.The second stage introduces a phase shift of a෇p͞2,so it sorts even-l photons into even and odd multiples of2.The D l෇2hologram is required before the photons are sorted further in the third stage.form thefirst two stages of the general OAM sorter out-lined in Fig.4.The light source used in this experiment was a helium-neon laser with a power of,1mW.An intracavity cross wire introduced rectangular symmetry to the laser cavity and forced the laser to oscillate in high-order Hermite-Gaussian͑HG m,n͒modes.Such modes are characterized by the indices m and n which correspond to zeros of intensity in the electricfield in the x and y di-rections,respectively.The Hermite-Gaussian modes were then converted to Laguerre-Gaussian modes by passing them through a p͞2mode converter based on cylindri-cal lenses[16].The resulting Laguerre-Gaussian modes have an exp͑il f͒phase structure and corresponding OAM of l¯h per photon.This conversion of Hermite-Gauss-ian͑HG m,n͒beams gives Laguerre-Gaussian͑LG l p͒beams characterized by l෇j m2n j and p෇min͑m,n͒.Ad-justments to the intracavity cross wire allowed us to gen-erate HG m,0modes with m෇0,1,2,...,which in turn gave rise to LG l0beams with l෇0,1,2,....The inter-ferometers had an arm length of approximately30cm and were built from standard optical components.The D l෇1 hologram was manufactured using standard photographic techniques[17].Note that such a hologram increases the l value of any exp͑il f͒mode by1[18].The four ports were directed onto a screen so a camera could take an image of the output.Figure5shows the output from the two-stage sorting process.As can be seen,we succeeded in sorting modes from l෇0to l෇4into different ports.The l෇4mode appears in the same port as the l෇0beam,as one wouldl1234A2B2C2D2PortFIG.5.Experimental results of a2-stage sorting scheme.The four different output ports correspond to exp͑il f͒modes with the values l͑mod4͒෇0,1,2,3,respectively.expect.In thisfirst experiment,the overall efficiency of the OAM sorter was limited by the poor optical efficiency of the particular hologram used to approximately10%. To demonstrate that our device works at the single-photon level,a further experiment was carried out at in-tensities so low that on average less than one photon was present in each interferometer at any one time.This was achieved by inserting neutral-densityfilters to attenuate the power of the laser beam to,0.3nW.This experiment used a1-stage interferometer.The output ports of the inter-ferometer were directed into a camera that averaged over a number of frames.As anticipated,this interferometer still sorts between odd and even l s with an efficiency lim-ited only by the quality of the optical components(Fig.6). Although the device proposed in this Letter can sort in-dividual photons according to their OAM,we did not de-tect photons individually.We plan to do this in the future with the use of a single photon source and single photon detectors.Our OAM sorter is the analog of the polarizing beam splitter in that it selects the optical path on the basis of OAM,one path for each of the distinguishable states.In this way,our sorter can be used to generate entanglement between the optical path and OAM in the same way that a polarizing beam splitter can create entanglement between the optical path and polarization[19].This will make itl1 2 3 40A1B1PortFIG.6.Experimental results of a1-stage(even-odd)sorting scheme at the single-photon level.Neutral densityfilters were used to reduce the number of photons so that the intensity cor-responded to one photon in the interferometer at one time.useful in generating highly entangled states and extending the optical realization of quantum logic elements to OAM quNits[20].We have demonstrated experimentally that a single pho-ton in an OAM eigenstate can be measured in any one of a number of different orthogonal states corresponding to dif-ferent values of l(the OAM in units of¯h).Our approach is in principle100%efficient,limited only by the efficiency of the components.The ability to measure a single photon to be in any one of an arbitrarily large number of orthogonal states has a number of potential implications for quantum information processing.The efficient measurement of the OAM of a single photon allows us access to a larger state space than that associated with optical polarization.This provides the possibility of a greater density of information transfer along with the generation and analysis of entanglement involving large numbers of states[10].The implications of this work for entanglement based applications such as superdense coding[21],teleportation[22],and quantum computation[23]remain to be explored.This work was supported by the Glasgow-Strathclyde University Synergy fund,the Royal Society,the Lever-hulme Trust,the Royal Society of Edinburgh,and the Scottish Executive Education and Lifelong LearningDepartment.*Electronic address:j.courtial@[1]L.Allen,M.W.Beijersbergen,R.J.C.Spreeuw,and J.P.Woerdman,Phys.Rev.A45,8185(1992).[2]L.Allen,M.J.Padgett,and M.Babiker,in Progress inOptics XXXIX,edited by E.Wolf(Elsevier Science B.V., New York,1999),pp.291–372.[3]D.Bouwmeester,A.Ekert,and A.Zeilinger,The Physics ofQuantum Information(Springer,Berlin,Germany,2000).[4]S.Pheonix and P.Townsend,Contemp.Phys.36,165(1995).[5]G.Molina-Terriza,J.P.Torres,and L.Torner,Phys.Rev.Lett.88,013601(2002).[6]M.Harris,C.A.Hill,P.R.Tapster,and J.M.Vaughan,Phys.Rev.A49,3119(1994).[7]M.J.Padgett,J.Arlt,N.B.Simpson,and L.Allen,Am.J.Phys.64,77(1996).[8]V.Y.Bazhenov,M.V.Vasnetsov,and M.S.Soskin,JETPLett.52,429(1990).[9]N.R.Heckenberg,R.McDuff, C.P.Smith,and A.G.White,Opt.Lett.17,221(1992).[10]G.Weihs and A.Zeilinger,Nature(London)412,313(2001).[11]V.V.Kotlyar,V.A.Soifer,and S.N.Khonina,J.Mod.Opt.44,1409(1997).[12]M.V.Vasnetsov,V.V.Slyusar,and M.S.Soskin,QuantumElectron.31,464(2001).[13]J.Courtial,D.A.Robertson,K.Dholakia,L.Allen,andM.J.Padgett,Phys.Rev.Lett.81,4828(1998).[14]M.J.Padgett and J.Courtial,Opt.Lett.24,430(1999).[15]M.Born and E.Wolf,Principles of Optics(PergamonPress,Oxford,1980),6th ed.[16]M.W.Beijersbergen,L.Allen,H.E.L.O.van der Veen,and J.P.Woerdman,mun.96,123(1993). [17]J.Arlt,K.Dholakia,L.Allen,and M.J.Padgett,J.Mod.Opt.45,1231(1998).[18]If the incident mode is an LG mode,which is of the formexp͑il f͒,the transmitted beam is no longer a pure LG mode,but is still of the form exp͑il0f͒(with l0෇l11).There is no fundamental reason limiting the holographic conversion of exp͑il f͒modes,but in practice they cur-rently convert at best about85%of the incident power. [19]T.J.Herzog,P.G.Kwiat,H.Weinfurter,and A.Zeilinger,Phys.Rev.Lett.75,3034(1995).[20]N.J.Cerf,C.Adami,and P.G.Kwiat,Phys.Rev.A57,R1477(1998).[21]C.H.Bennett and S.J.Wiesner,Phys.Rev.Lett.69,2881(1992).[22]C.H.Bennett,G.Brassard,R.Jozsa,A.Peres,and W.K.Wootters,Phys.Rev.Lett.70,1895(1993).[23]M.A.Nielsen and I.L.Chuang,Quantum Computationand Quantum Information(Cambridge University Press, Cambridge,United Kingdom,2000).。

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涡旋光束拓扑荷值的干涉测量方法

涡旋光束拓扑荷值的干涉测量方法

涡旋光束拓扑荷值的干涉测量方法周洋;李新忠;王静鸽;王辉;李贺贺【摘要】基于平面波或球面波与拉盖尔-高斯(Laguerre-Gaussian,LG)涡旋光束的干涉特性,提出了一种LG涡旋光束拓扑荷值的测量方法.利用改进的马赫-曾德尔干涉光路对LG涡旋光束的拓扑荷值进行测量,其中,干涉光路中的一支为平面波或球面波,另一支为LG光束.然后,通过干涉亮条纹中的分叉数(平面波干涉时)或螺旋线嵌套数(球面波干涉时)来确定LG涡旋光束的拓扑荷值.拓扑荷值的符号,通过分叉向上还是向下(平面波干涉时)或螺旋线为顺时针还是逆时针(球面波干涉时)来判断其正负.数值模拟和试验结果表明:该方法能有效测量LG涡旋光束的拓扑荷值和符号,并且具有快速、简洁的特点.【期刊名称】《河南科技大学学报（自然科学版）》【年(卷),期】2016(037)003【总页数】5页(P95-99)【关键词】涡旋光束;拓扑荷值;平面波;球面波;干涉【作者】周洋;李新忠;王静鸽;王辉;李贺贺【作者单位】河南科技大学物理工程学院,河南洛阳471023;河南科技大学物理工程学院,河南洛阳471023;河南科技大学物理工程学院,河南洛阳471023;河南科技大学物理工程学院,河南洛阳471023;河南科技大学物理工程学院,河南洛阳471023【正文语种】中文【中图分类】O436;O438.1涡旋光束是一种具有螺旋形相位波前且中心光强为0的暗中空光束[1]，其表达式中含有exp(imθ)的相位因子，其中，m是涡旋光束的拓扑荷值，每个光子携带mћ的轨道角动量。

近年来，涡旋光束在量子信息编码[2-3]、光学扳手、粒子旋转与操纵[4-5]和图像处理[6-7]等方面获得了广泛的应用。

而在这些应用中，涡旋光束的拓扑荷值是一个关键参数。

因此，如何快速准确地对拓扑荷值进行测量是该领域研究中首要解决的问题。

从目前研究来看，涡旋光束拓扑荷值的测量方法主要分为干涉测量[8-10]和衍射测量[11-12]两种。

基于里德堡原子的微波全信息测量

基于里德堡原子的微波全信息测量

基于里德堡原子的微波全信息测量贾凤东1，郝建海1，崔越1，王宇翔1，刘宇晴1，王宇2，尤建琦2，白金海2，钟志萍1，3*（1.中国科学院大学 物理科学学院，北京 100049；2.航空工业北京长城计量测试技术研究所，北京 100095；3.中国科学院大学 中国科学院拓扑量子计算卓越中心，北京 100190）摘 要：介绍了里德堡原子微波电场传感器的工作原理，阐述了基于里德堡原子测量微波电场强度、相位、极化、频率等信息的技术特点，分析了基于里德堡原子的微波全信息测量的研究现状，探讨了当前绝对自校准测量和连续宽带高灵敏测量面临的困难，指出可以通过外场调控实现测量灵敏度提升和宽带连续频率测量；并可通过各种调制及解调手段简化相位、极化的测量和读取。

分析了在热原子系统中利用多光子激发消除多普勒展宽以及采用冷原子消除多普勒展宽对于提升微波测量灵敏度的潜在优势，提出未来可利用里德堡原子的高轨道角动量态、强关联等特性进一步提升里德堡原子微波电场传感器性能。

关键词：微波电场传感器；里德堡原子；电磁感应透明中图分类号：TB939；TP212；O56 文献标志码：A 文章编号：1674-5795（2024）01-0001-22Microwave full information measurement based on Rydberg atomsJIA Fengdong 1, HAO Jianhai 1, CUI Yue 1, WANG Yuxiang 1, LIU Yuqing 1,WANG Yu 2, YOU Jianqi 2, BAI Jinhai 2, ZHONG Zhiping 1,3*(1.School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China ；2.Changcheng Institute of Metrology & Measurement, Beijing 100095, China ；3.CAS Center for Excellence in TopologicalQuantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China)Abstract: This paper introduces the principles of the Rydberg atom microwave electric field sensor, elucidating the technical features of measuring the information of microwave electric field strength, phase, polarization, and frequency based on Rydberg atoms. Then it analyzes the current state of microwave full‐information measurement based on Rydberg atoms and discusses the challenges faced by absolute self‐calibration measurements and continuous broadband high‐sen‐sitivity measurements. It points out that measurement sensitivity enhancement and continuous broadband frequency mea‐surement can be achieved through external field modulation. Various modulation and demodulation techniques are sug‐gested to simplify phase and polarization measurements and data reading. The paper also explores the potential advan‐tages to eliminate Doppler broadening effect by adopting multiphoton excitation in atomic vapor cell systems and adoptingcold atomic systems to enhance microwave measurement sensitivity. It proposes the possibility for future exploration of such characteristics of Rydberg atoms as high orbital angular momentum states and strong correlations to further improvedoi ：10.11823/j.issn.1674-5795.2024.01.01收稿日期：2023-12-20；修回日期：2024-01-24基金项目：国家重点研发计划项目（2017YFA0304900、2017YFA0402300）；北京市自然科学基金项目（1212014）；中国科学院重点研究计划项目（XDPB08-3）；中央高校基本科研业务费专项资金项目引用格式：贾凤东， 郝建海， 崔越， 等. 基于里德堡原子的微波全信息测量［J ］. 计测技术， 2024， 44（1）： 1-22.Citation ：JIA F D ， HAO J H ， CUI Y ， et al. Microwave full information measurement based on Rydberg atoms ［J ］.Metrology & Measurement Technology ， 2024， 44（1）： 1-22.the performance of Rydberg atom microwave electric field sensors.Key words: microwave electric field sensor; Rydberg atom; electromagnetic induction transparency0　引言对量子状态的主动调控和操纵引发了第二次量子革命，发展出量子通信、量子计算和量子精密测量等前沿领域。

全球星卫星通信系统姿态和轨道控制系统设计说明书

全球星卫星通信系统姿态和轨道控制系统设计说明书

2.1 The AOeS Sensors
The AOCS sensors consist of three EDO-Barnes analog sunsensor blocks, an infrared earth sensor consisting of three analog EDO-Barnes static heads, an Institut Forster three-axis fluxgate magnetometer on a short boom, and a Space-SystemslLoral dual GPS Tensor receiver.
In addition to the sensors and actuators discussed below, the AOCS system is mechanized with a redundant digitial computer known as the OBPE (On-Board Processor Electronics). One half consists ofa 1750A CPU, 64K RAM, 12SK PROM, the sensor and actuator interfaces, and the telemetry and command logic. All equipment is cross strapped so that either half of the OBPE can talk to any sensor or any actuator.
The GPS Tensor in addition to being a space-qualifIed navigation receiver is also a full three-axis attitude measuring device. Four GPS antennas are mounted on the anti-earth face of the satellite, and attitude is determined by the phase difference of the incoming satellite signals. Since the antennas are on the anti-earth side of the satellite, GPS attitude is only possible during earth pointing. In partiCUlar it cannot be used for initial acquisition. Because there is very little flight experience with the GPS receiver as an attitude detector apart from an experimental flight on DASA's CRISTA-SPAS in November '94, the AOCS system has been designed to only use GPS attitude initially as a backup.
引力波观测原文PhysRevLett.116.061102

引力波观测原文PhysRevLett.116.061102

Observation of Gravitational Waves from a Binary Black Hole MergerB.P.Abbott et al.*(LIGO Scientific Collaboration and Virgo Collaboration)(Received21January2016;published11February2016)On September14,2015at09:50:45UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal.The signal sweeps upwards in frequency from35to250Hz with a peak gravitational-wave strain of1.0×10−21.It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole.The signal was observed with a matched-filter signal-to-noise ratio of24and a false alarm rate estimated to be less than1event per203000years,equivalent to a significance greaterthan5.1σ.The source lies at a luminosity distance of410þ160−180Mpc corresponding to a redshift z¼0.09þ0.03−0.04.In the source frame,the initial black hole masses are36þ5−4M⊙and29þ4−4M⊙,and the final black hole mass is62þ4−4M⊙,with3.0þ0.5−0.5M⊙c2radiated in gravitational waves.All uncertainties define90%credible intervals.These observations demonstrate the existence of binary stellar-mass black hole systems.This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.DOI:10.1103/PhysRevLett.116.061102I.INTRODUCTIONIn1916,the year after the final formulation of the field equations of general relativity,Albert Einstein predicted the existence of gravitational waves.He found that the linearized weak-field equations had wave solutions: transverse waves of spatial strain that travel at the speed of light,generated by time variations of the mass quadrupole moment of the source[1,2].Einstein understood that gravitational-wave amplitudes would be remarkably small;moreover,until the Chapel Hill conference in 1957there was significant debate about the physical reality of gravitational waves[3].Also in1916,Schwarzschild published a solution for the field equations[4]that was later understood to describe a black hole[5,6],and in1963Kerr generalized the solution to rotating black holes[7].Starting in the1970s theoretical work led to the understanding of black hole quasinormal modes[8–10],and in the1990s higher-order post-Newtonian calculations[11]preceded extensive analytical studies of relativistic two-body dynamics[12,13].These advances,together with numerical relativity breakthroughs in the past decade[14–16],have enabled modeling of binary black hole mergers and accurate predictions of their gravitational waveforms.While numerous black hole candidates have now been identified through electromag-netic observations[17–19],black hole mergers have not previously been observed.The discovery of the binary pulsar system PSR B1913þ16 by Hulse and Taylor[20]and subsequent observations of its energy loss by Taylor and Weisberg[21]demonstrated the existence of gravitational waves.This discovery, along with emerging astrophysical understanding[22], led to the recognition that direct observations of the amplitude and phase of gravitational waves would enable studies of additional relativistic systems and provide new tests of general relativity,especially in the dynamic strong-field regime.Experiments to detect gravitational waves began with Weber and his resonant mass detectors in the1960s[23], followed by an international network of cryogenic reso-nant detectors[24].Interferometric detectors were first suggested in the early1960s[25]and the1970s[26].A study of the noise and performance of such detectors[27], and further concepts to improve them[28],led to proposals for long-baseline broadband laser interferome-ters with the potential for significantly increased sensi-tivity[29–32].By the early2000s,a set of initial detectors was completed,including TAMA300in Japan,GEO600 in Germany,the Laser Interferometer Gravitational-Wave Observatory(LIGO)in the United States,and Virgo in binations of these detectors made joint obser-vations from2002through2011,setting upper limits on a variety of gravitational-wave sources while evolving into a global network.In2015,Advanced LIGO became the first of a significantly more sensitive network of advanced detectors to begin observations[33–36].A century after the fundamental predictions of Einstein and Schwarzschild,we report the first direct detection of gravitational waves and the first direct observation of a binary black hole system merging to form a single black hole.Our observations provide unique access to the*Full author list given at the end of the article.Published by the American Physical Society under the terms of the Creative Commons Attribution3.0License.Further distri-bution of this work must maintain attribution to the author(s)and the published article’s title,journal citation,and DOI.properties of space-time in the strong-field,high-velocity regime and confirm predictions of general relativity for the nonlinear dynamics of highly disturbed black holes.II.OBSERVATIONOn September14,2015at09:50:45UTC,the LIGO Hanford,W A,and Livingston,LA,observatories detected the coincident signal GW150914shown in Fig.1.The initial detection was made by low-latency searches for generic gravitational-wave transients[41]and was reported within three minutes of data acquisition[43].Subsequently, matched-filter analyses that use relativistic models of com-pact binary waveforms[44]recovered GW150914as the most significant event from each detector for the observa-tions reported here.Occurring within the10-msintersite FIG.1.The gravitational-wave event GW150914observed by the LIGO Hanford(H1,left column panels)and Livingston(L1,rightcolumn panels)detectors.Times are shown relative to September14,2015at09:50:45UTC.For visualization,all time series are filtered with a35–350Hz bandpass filter to suppress large fluctuations outside the detectors’most sensitive frequency band,and band-reject filters to remove the strong instrumental spectral lines seen in the Fig.3spectra.Top row,left:H1strain.Top row,right:L1strain.GW150914arrived first at L1and6.9þ0.5−0.4ms later at H1;for a visual comparison,the H1data are also shown,shifted in time by this amount and inverted(to account for the detectors’relative orientations).Second row:Gravitational-wave strain projected onto each detector in the35–350Hz band.Solid lines show a numerical relativity waveform for a system with parameters consistent with those recovered from GW150914[37,38]confirmed to99.9%by an independent calculation based on[15].Shaded areas show90%credible regions for two independent waveform reconstructions.One(dark gray)models the signal using binary black hole template waveforms [39].The other(light gray)does not use an astrophysical model,but instead calculates the strain signal as a linear combination of sine-Gaussian wavelets[40,41].These reconstructions have a94%overlap,as shown in[39].Third row:Residuals after subtracting the filtered numerical relativity waveform from the filtered detector time series.Bottom row:A time-frequency representation[42]of the strain data,showing the signal frequency increasing over time.propagation time,the events have a combined signal-to-noise ratio(SNR)of24[45].Only the LIGO detectors were observing at the time of GW150914.The Virgo detector was being upgraded, and GEO600,though not sufficiently sensitive to detect this event,was operating but not in observational mode.With only two detectors the source position is primarily determined by the relative arrival time and localized to an area of approximately600deg2(90% credible region)[39,46].The basic features of GW150914point to it being produced by the coalescence of two black holes—i.e., their orbital inspiral and merger,and subsequent final black hole ringdown.Over0.2s,the signal increases in frequency and amplitude in about8cycles from35to150Hz,where the amplitude reaches a maximum.The most plausible explanation for this evolution is the inspiral of two orbiting masses,m1and m2,due to gravitational-wave emission.At the lower frequencies,such evolution is characterized by the chirp mass[11]M¼ðm1m2Þ3=5121=5¼c3G596π−8=3f−11=3_f3=5;where f and_f are the observed frequency and its time derivative and G and c are the gravitational constant and speed of light.Estimating f and_f from the data in Fig.1, we obtain a chirp mass of M≃30M⊙,implying that the total mass M¼m1þm2is≳70M⊙in the detector frame. This bounds the sum of the Schwarzschild radii of thebinary components to2GM=c2≳210km.To reach an orbital frequency of75Hz(half the gravitational-wave frequency)the objects must have been very close and very compact;equal Newtonian point masses orbiting at this frequency would be only≃350km apart.A pair of neutron stars,while compact,would not have the required mass,while a black hole neutron star binary with the deduced chirp mass would have a very large total mass, and would thus merge at much lower frequency.This leaves black holes as the only known objects compact enough to reach an orbital frequency of75Hz without contact.Furthermore,the decay of the waveform after it peaks is consistent with the damped oscillations of a black hole relaxing to a final stationary Kerr configuration. Below,we present a general-relativistic analysis of GW150914;Fig.2shows the calculated waveform using the resulting source parameters.III.DETECTORSGravitational-wave astronomy exploits multiple,widely separated detectors to distinguish gravitational waves from local instrumental and environmental noise,to provide source sky localization,and to measure wave polarizations. The LIGO sites each operate a single Advanced LIGO detector[33],a modified Michelson interferometer(see Fig.3)that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms.Each arm is formed by two mirrors,acting as test masses,separated by L x¼L y¼L¼4km.A passing gravitational wave effec-tively alters the arm lengths such that the measured difference isΔLðtÞ¼δL x−δL y¼hðtÞL,where h is the gravitational-wave strain amplitude projected onto the detector.This differential length variation alters the phase difference between the two light fields returning to the beam splitter,transmitting an optical signal proportional to the gravitational-wave strain to the output photodetector. To achieve sufficient sensitivity to measure gravitational waves,the detectors include several enhancements to the basic Michelson interferometer.First,each arm contains a resonant optical cavity,formed by its two test mass mirrors, that multiplies the effect of a gravitational wave on the light phase by a factor of300[48].Second,a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometer as a whole[49,50]:20W of laser input is increased to700W incident on the beam splitter,which is further increased to 100kW circulating in each arm cavity.Third,a partially transmissive signal-recycling mirror at the outputoptimizes FIG. 2.Top:Estimated gravitational-wave strain amplitude from GW150914projected onto H1.This shows the full bandwidth of the waveforms,without the filtering used for Fig.1. The inset images show numerical relativity models of the black hole horizons as the black holes coalesce.Bottom:The Keplerian effective black hole separation in units of Schwarzschild radii (R S¼2GM=c2)and the effective relative velocity given by the post-Newtonian parameter v=c¼ðGMπf=c3Þ1=3,where f is the gravitational-wave frequency calculated with numerical relativity and M is the total mass(value from Table I).the gravitational-wave signal extraction by broadening the bandwidth of the arm cavities [51,52].The interferometer is illuminated with a 1064-nm wavelength Nd:Y AG laser,stabilized in amplitude,frequency,and beam geometry [53,54].The gravitational-wave signal is extracted at the output port using a homodyne readout [55].These interferometry techniques are designed to maxi-mize the conversion of strain to optical signal,thereby minimizing the impact of photon shot noise (the principal noise at high frequencies).High strain sensitivity also requires that the test masses have low displacement noise,which is achieved by isolating them from seismic noise (low frequencies)and designing them to have low thermal noise (intermediate frequencies).Each test mass is suspended as the final stage of a quadruple-pendulum system [56],supported by an active seismic isolation platform [57].These systems collectively provide more than 10orders of magnitude of isolation from ground motion for frequen-cies above 10Hz.Thermal noise is minimized by using low-mechanical-loss materials in the test masses and their suspensions:the test masses are 40-kg fused silica substrates with low-loss dielectric optical coatings [58,59],and are suspended with fused silica fibers from the stage above [60].To minimize additional noise sources,all components other than the laser source are mounted on vibration isolation stages in ultrahigh vacuum.To reduce optical phase fluctuations caused by Rayleigh scattering,the pressure in the 1.2-m diameter tubes containing the arm-cavity beams is maintained below 1μPa.Servo controls are used to hold the arm cavities on resonance [61]and maintain proper alignment of the optical components [62].The detector output is calibrated in strain by measuring its response to test mass motion induced by photon pressure from a modulated calibration laser beam [63].The calibration is established to an uncertainty (1σ)of less than 10%in amplitude and 10degrees in phase,and is continuously monitored with calibration laser excitations at selected frequencies.Two alternative methods are used to validate the absolute calibration,one referenced to the main laser wavelength and the other to a radio-frequencyoscillator(a)FIG.3.Simplified diagram of an Advanced LIGO detector (not to scale).A gravitational wave propagating orthogonally to the detector plane and linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shortening the other during one half-cycle of the wave;these length changes are reversed during the other half-cycle.The output photodetector records these differential cavity length variations.While a detector ’s directional response is maximal for this case,it is still significant for most other angles of incidence or polarizations (gravitational waves propagate freely through the Earth).Inset (a):Location and orientation of the LIGO detectors at Hanford,WA (H1)and Livingston,LA (L1).Inset (b):The instrument noise for each detector near the time of the signal detection;this is an amplitude spectral density,expressed in terms of equivalent gravitational-wave strain amplitude.The sensitivity is limited by photon shot noise at frequencies above 150Hz,and by a superposition of other noise sources at lower frequencies [47].Narrow-band features include calibration lines (33–38,330,and 1080Hz),vibrational modes of suspension fibers (500Hz and harmonics),and 60Hz electric power grid harmonics.[64].Additionally,the detector response to gravitational waves is tested by injecting simulated waveforms with the calibration laser.To monitor environmental disturbances and their influ-ence on the detectors,each observatory site is equipped with an array of sensors:seismometers,accelerometers, microphones,magnetometers,radio receivers,weather sensors,ac-power line monitors,and a cosmic-ray detector [65].Another∼105channels record the interferometer’s operating point and the state of the control systems.Data collection is synchronized to Global Positioning System (GPS)time to better than10μs[66].Timing accuracy is verified with an atomic clock and a secondary GPS receiver at each observatory site.In their most sensitive band,100–300Hz,the current LIGO detectors are3to5times more sensitive to strain than initial LIGO[67];at lower frequencies,the improvement is even greater,with more than ten times better sensitivity below60Hz.Because the detectors respond proportionally to gravitational-wave amplitude,at low redshift the volume of space to which they are sensitive increases as the cube of strain sensitivity.For binary black holes with masses similar to GW150914,the space-time volume surveyed by the observations reported here surpasses previous obser-vations by an order of magnitude[68].IV.DETECTOR VALIDATIONBoth detectors were in steady state operation for several hours around GW150914.All performance measures,in particular their average sensitivity and transient noise behavior,were typical of the full analysis period[69,70]. Exhaustive investigations of instrumental and environ-mental disturbances were performed,giving no evidence to suggest that GW150914could be an instrumental artifact [69].The detectors’susceptibility to environmental disturb-ances was quantified by measuring their response to spe-cially generated magnetic,radio-frequency,acoustic,and vibration excitations.These tests indicated that any external disturbance large enough to have caused the observed signal would have been clearly recorded by the array of environ-mental sensors.None of the environmental sensors recorded any disturbances that evolved in time and frequency like GW150914,and all environmental fluctuations during the second that contained GW150914were too small to account for more than6%of its strain amplitude.Special care was taken to search for long-range correlated disturbances that might produce nearly simultaneous signals at the two sites. No significant disturbances were found.The detector strain data exhibit non-Gaussian noise transients that arise from a variety of instrumental mecha-nisms.Many have distinct signatures,visible in auxiliary data channels that are not sensitive to gravitational waves; such instrumental transients are removed from our analyses [69].Any instrumental transients that remain in the data are accounted for in the estimated detector backgrounds described below.There is no evidence for instrumental transients that are temporally correlated between the two detectors.V.SEARCHESWe present the analysis of16days of coincident observations between the two LIGO detectors from September12to October20,2015.This is a subset of the data from Advanced LIGO’s first observational period that ended on January12,2016.GW150914is confidently detected by two different types of searches.One aims to recover signals from the coalescence of compact objects,using optimal matched filtering with waveforms predicted by general relativity. The other search targets a broad range of generic transient signals,with minimal assumptions about waveforms.These searches use independent methods,and their response to detector noise consists of different,uncorrelated,events. However,strong signals from binary black hole mergers are expected to be detected by both searches.Each search identifies candidate events that are detected at both observatories consistent with the intersite propa-gation time.Events are assigned a detection-statistic value that ranks their likelihood of being a gravitational-wave signal.The significance of a candidate event is determined by the search background—the rate at which detector noise produces events with a detection-statistic value equal to or higher than the candidate event.Estimating this back-ground is challenging for two reasons:the detector noise is nonstationary and non-Gaussian,so its properties must be empirically determined;and it is not possible to shield the detector from gravitational waves to directly measure a signal-free background.The specific procedure used to estimate the background is slightly different for the two searches,but both use a time-shift technique:the time stamps of one detector’s data are artificially shifted by an offset that is large compared to the intersite propagation time,and a new set of events is produced based on this time-shifted data set.For instrumental noise that is uncor-related between detectors this is an effective way to estimate the background.In this process a gravitational-wave signal in one detector may coincide with time-shifted noise transients in the other detector,thereby contributing to the background estimate.This leads to an overestimate of the noise background and therefore to a more conservative assessment of the significance of candidate events.The characteristics of non-Gaussian noise vary between different time-frequency regions.This means that the search backgrounds are not uniform across the space of signals being searched.To maximize sensitivity and provide a better estimate of event significance,the searches sort both their background estimates and their event candidates into differ-ent classes according to their time-frequency morphology. The significance of a candidate event is measured against the background of its class.To account for having searchedmultiple classes,this significance is decreased by a trials factor equal to the number of classes [71].A.Generic transient searchDesigned to operate without a specific waveform model,this search identifies coincident excess power in time-frequency representations of the detector strain data [43,72],for signal frequencies up to 1kHz and durations up to a few seconds.The search reconstructs signal waveforms consistent with a common gravitational-wave signal in both detectors using a multidetector maximum likelihood method.Each event is ranked according to the detection statistic ηc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E c =ð1þE n =E c Þp ,where E c is the dimensionless coherent signal energy obtained by cross-correlating the two reconstructed waveforms,and E n is the dimensionless residual noise energy after the reconstructed signal is subtracted from the data.The statistic ηc thus quantifies the SNR of the event and the consistency of the data between the two detectors.Based on their time-frequency morphology,the events are divided into three mutually exclusive search classes,as described in [41]:events with time-frequency morphology of known populations of noise transients (class C1),events with frequency that increases with time (class C3),and all remaining events (class C2).Detected with ηc ¼20.0,GW150914is the strongest event of the entire search.Consistent with its coalescence signal signature,it is found in the search class C3of events with increasing time-frequency evolution.Measured on a background equivalent to over 67400years of data and including a trials factor of 3to account for the search classes,its false alarm rate is lower than 1in 22500years.This corresponds to a probability <2×10−6of observing one or more noise events as strong as GW150914during the analysis time,equivalent to 4.6σ.The left panel of Fig.4shows the C3class results and background.The selection criteria that define the search class C3reduce the background by introducing a constraint on the signal morphology.In order to illustrate the significance of GW150914against a background of events with arbitrary shapes,we also show the results of a search that uses the same set of events as the one described above but without this constraint.Specifically,we use only two search classes:the C1class and the union of C2and C3classes (C 2þC 3).In this two-class search the GW150914event is found in the C 2þC 3class.The left panel of Fig.4shows the C 2þC 3class results and background.In the background of this class there are four events with ηc ≥32.1,yielding a false alarm rate for GW150914of 1in 8400years.This corresponds to a false alarm probability of 5×10−6equivalent to 4.4σ.FIG.4.Search results from the generic transient search (left)and the binary coalescence search (right).These histograms show the number of candidate events (orange markers)and the mean number of background events (black lines)in the search class where GW150914was found as a function of the search detection statistic and with a bin width of 0.2.The scales on the top give the significance of an event in Gaussian standard deviations based on the corresponding noise background.The significance of GW150914is greater than 5.1σand 4.6σfor the binary coalescence and the generic transient searches,respectively.Left:Along with the primary search (C3)we also show the results (blue markers)and background (green curve)for an alternative search that treats events independently of their frequency evolution (C 2þC 3).The classes C2and C3are defined in the text.Right:The tail in the black-line background of the binary coalescence search is due to random coincidences of GW150914in one detector with noise in the other detector.(This type of event is practically absent in the generic transient search background because they do not pass the time-frequency consistency requirements used in that search.)The purple curve is the background excluding those coincidences,which is used to assess the significance of the second strongest event.For robustness and validation,we also use other generic transient search algorithms[41].A different search[73]and a parameter estimation follow-up[74]detected GW150914 with consistent significance and signal parameters.B.Binary coalescence searchThis search targets gravitational-wave emission from binary systems with individual masses from1to99M⊙, total mass less than100M⊙,and dimensionless spins up to 0.99[44].To model systems with total mass larger than 4M⊙,we use the effective-one-body formalism[75],whichcombines results from the post-Newtonian approach [11,76]with results from black hole perturbation theory and numerical relativity.The waveform model[77,78] assumes that the spins of the merging objects are alignedwith the orbital angular momentum,but the resultingtemplates can,nonetheless,effectively recover systemswith misaligned spins in the parameter region ofGW150914[44].Approximately250000template wave-forms are used to cover this parameter space.The search calculates the matched-filter signal-to-noiseratioρðtÞfor each template in each detector and identifiesmaxima ofρðtÞwith respect to the time of arrival of the signal[79–81].For each maximum we calculate a chi-squared statisticχ2r to test whether the data in several differentfrequency bands are consistent with the matching template [82].Values ofχ2r near unity indicate that the signal is consistent with a coalescence.Ifχ2r is greater than unity,ρðtÞis reweighted asˆρ¼ρ=f½1þðχ2rÞ3 =2g1=6[83,84].The final step enforces coincidence between detectors by selectingevent pairs that occur within a15-ms window and come fromthe same template.The15-ms window is determined by the10-ms intersite propagation time plus5ms for uncertainty inarrival time of weak signals.We rank coincident events basedon the quadrature sumˆρc of theˆρfrom both detectors[45]. To produce background data for this search the SNR maxima of one detector are time shifted and a new set of coincident events is computed.Repeating this procedure ∼107times produces a noise background analysis time equivalent to608000years.To account for the search background noise varying acrossthe target signal space,candidate and background events aredivided into three search classes based on template length.The right panel of Fig.4shows the background for thesearch class of GW150914.The GW150914detection-statistic value ofˆρc¼23.6is larger than any background event,so only an upper bound can be placed on its false alarm rate.Across the three search classes this bound is1in 203000years.This translates to a false alarm probability <2×10−7,corresponding to5.1σ.A second,independent matched-filter analysis that uses adifferent method for estimating the significance of itsevents[85,86],also detected GW150914with identicalsignal parameters and consistent significance.When an event is confidently identified as a real gravitational-wave signal,as for GW150914,the back-ground used to determine the significance of other events is reestimated without the contribution of this event.This is the background distribution shown as a purple line in the right panel of Fig.4.Based on this,the second most significant event has a false alarm rate of1per2.3years and corresponding Poissonian false alarm probability of0.02. Waveform analysis of this event indicates that if it is astrophysical in origin it is also a binary black hole merger[44].VI.SOURCE DISCUSSIONThe matched-filter search is optimized for detecting signals,but it provides only approximate estimates of the source parameters.To refine them we use general relativity-based models[77,78,87,88],some of which include spin precession,and for each model perform a coherent Bayesian analysis to derive posterior distributions of the source parameters[89].The initial and final masses, final spin,distance,and redshift of the source are shown in Table I.The spin of the primary black hole is constrained to be<0.7(90%credible interval)indicating it is not maximally spinning,while the spin of the secondary is only weakly constrained.These source parameters are discussed in detail in[39].The parameter uncertainties include statistical errors and systematic errors from averaging the results of different waveform models.Using the fits to numerical simulations of binary black hole mergers in[92,93],we provide estimates of the mass and spin of the final black hole,the total energy radiated in gravitational waves,and the peak gravitational-wave luminosity[39].The estimated total energy radiated in gravitational waves is3.0þ0.5−0.5M⊙c2.The system reached apeak gravitational-wave luminosity of3.6þ0.5−0.4×1056erg=s,equivalent to200þ30−20M⊙c2=s.Several analyses have been performed to determine whether or not GW150914is consistent with a binary TABLE I.Source parameters for GW150914.We report median values with90%credible intervals that include statistical errors,and systematic errors from averaging the results of different waveform models.Masses are given in the source frame;to convert to the detector frame multiply by(1þz) [90].The source redshift assumes standard cosmology[91]. Primary black hole mass36þ5−4M⊙Secondary black hole mass29þ4−4M⊙Final black hole mass62þ4−4M⊙Final black hole spin0.67þ0.05−0.07 Luminosity distance410þ160−180MpcSource redshift z0.09þ0.03−0.04。