By directing the input light into a particular mode it is possible to obtain
as output all of the input light for a beam splitter that is 50% absorbing.
This effect is also responsible for nonlinear quantum interference when two
photons are incident on the beam splitter.Comment: 10 pages, 2 figures, to appear in J. Mod. Op

J. Jeffers

Lai W. K.

Mandel L.

English

arXiv

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Interference and the lossless lossy beam splitter

By directing the input into a particular mode it is possible to obtain as output all of the input light for a beamsplitter which is 50% absorbing. This effect is also responsible for nonlinear quantum interference when two photons are incident on the beam splitter

Jeffers, J.

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryJeffers, J. (2000) Interference and the lossless lossy beam splitter. Journal of Modern Optics, 47(11). pp. 1819-1824. ISSN 0950-0340Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/  Jeffers, J.* (2000) Interference and the lossless lossy beam splitter. Journal of Modern Optics, 47 (11). pp. 1819-1824. ISSN 0950-0340      http://eprints.cdlr.strath.ac.uk/6168/   This is an author-produced version of a paper published in Journal of Modern Optics, 47 (11). pp. 1819-1824. ISSN 0950-0340. This version has been peer-reviewed, but does not include the final publisher proof corrections, published layout, or pagination.  Strathprints is designed to allow users to access the research output of the University of Strathclyde. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (http://eprints.cdlr.strath.ac.uk) and the content of this paper for research or study, educational, or not-for-profit purposes without prior permission or charge. You may freely distribute the url (http://eprints.cdlr.strath.ac.uk) of the Strathprints website.  Any correspondence concerning this service should be sent to The Strathprints Administrator: eprints@cis.strath.ac.uk  arXiv:quant-ph/0007025 v1   10 Jul 2000Interference and the lossless lossy beam splitterJOHN JEFFERSDepartment of Physics and Applied Physics,University of Strathclyde,107 Rottenrow,Glasgow G4 0NG, UK.email: john@phys.strath.ac.ukAbstract. By directing the input light into a particular mode it is possible to obtain as outputall of the input light for a beam splitter that is 50% absorbing. This effect is also responsible fornonlinear quantum interference when two photons are incident on the beam splitter.11 IntroductionThe beam splitter is the main component of many optical interferometers, both classical and quan-tum [1, 2]. Much of its usefulness in quantum optics is derived from the fact that an unentangledinput beam can emerge from the device in an entangled state. This state can then either have itsquantum correlations confirmed by measurement, or can itself be used in interference experiments.Two photon interference is a higher order quantum interference effect which has received muchattention [3–6]. In the classic experiments two identical photons from different input arms fallingsimultaneously upon a 50/50 beam splitter both exit the beam splitter from the same output arm;whereas if the photons arrive at the beam splitter in the same input they are distributed betweenthe output arms as if they obeyed the laws of classical probability, with each having a transmissionand reflection probability of 1/2.The relations between the input and output single-mode boson operators for the different armsof a lossless beam splitter (Fig. 1) are [7–10]aˆout = taˆin + rbˆin,bˆout = tbˆin + raˆin, (1)where t and r are the transmission and reflection coefficients. Throughout this paper discrete-modetheory is used. It is merely an approximation to the more correct continuous-mode formalism,but it is an adequate description here, provided that the reflection and transmission coefficientscan be treated as constants at the frequencies of interest. Conservation of the boson commutatorsand the requirement that independent inputs give independent outputs force the coefficients tosatisfy |t|2 + |r|2 = 1 and t = ±ir. It is useful to define superposition modes, the symmetric andantisymmetric combinations of aˆ and bˆ. The annihilation operators for these are (for either inputsor outputs)cˆ =aˆ+ bˆ√2, dˆ =bˆ− aˆ√2, (2)2ain^aout^bout^bin^Figure 1: The optical beam splitter. Two input fields are partially reflected, partially transmittedinto two output fields. In the lossy case there are further unobservable noise inputs and outputs,which are internal to the beam splitter.so that the input and output operators are related bycˆout = (t+ r)cˆin, dˆout = (t− r)dˆin. (3)It is clear from the above relations that these superposition operators are not mixed by the beamsplitter. The action of the device on any input state which is factorisable into a product state ofthe superposition modes, |ψin〉 = |ψc〉 ⊗ |ψd〉, is simply to produce a phase shift. For example, thephoton number state product|ψin〉 = |nc〉 ⊗ |md〉 ⇒ |ψout〉 = exp [±iθ(n −m)]|nc〉 ⊗ |md〉, (4)where cos θ = t and | sin θ| = r. For the 50/50 beam splitter two simple examples are the onephoton superpositions|1+〉 = |1c, 0d〉 = 1√2(|1a, 0b〉+ |0a, 1b〉),3|1−〉 = |0c, 1d〉 = 1√2(|1a, 0b〉 − |0a, 1b〉). (5)It is also possible to construct states for which the beam splitter is transparent with two photons.Consider an input in one of the superpositions|2±〉 = 1√2(|2a, 0b〉 ± |0a, 2b〉) . (6)Two photons falling on a beam splitter from the same input arm are usually distributed betweenthe outputs according to the laws of classical probability but this is not the case here. Photonsin the positive superposition are constrained to take different output arms. This is an illustrationof the reversibility of light beams in optics, as it is merely the standard two photon interferencephenomenon with the inputs and outputs reversed. The input state in this case is not factorisablein terms of states of the c and d modes. The negative superposition, on the other hand, is a twophoton example of a state which passes through the beam splitter unchanged. This state can alsobe written|2−〉 = 12(aˆ†2 − bˆ†2)|0a, 0b >= −cˆ†dˆ†|0c, 0d〉. (7)The reason that the state is passed is now clear. It is factorisable into a product state of thesuperposition modes.Much of the theory of the beam splitter has been done assuming that it is a lossless component[7–10], for which absorption is simply an experimental problem that hinders the observation ofinteresting effects. Absorption is unavoidable at some frequencies, however, as optical devices mustbe compatible with the Kramers-Kronig relations. In recent years loss has come to be seen as aquantum property in its own right, with its own effect on correlations [11]. There has been somework on loss in fibre couplers [12] but it is only recently that a quantum theory of the absorbingbeam splitter was proposed [13]. All of the features of the lossless device are retained but thereare additional quantum effects such as nonlinear absorption which are only possible when loss ispresent.In this paper a discrete-mode version of the lossy beam splitter theory is used to show severalinteresting effects. Firstly, the beam splitter may be rendered transparent or opaque, depending on4the phase relation of the transmission and reflection coefficients and the state of the input light; andsecondly, the nonlinear absorption effect discussed in reference [13] is shown to occur with othertwo photon input states.2 The lossy beam splitterIn this section the theory of the lossy beam splitter [13] will be outlined in discrete-mode form. Thedevice relates two inputs to two outputs as before (Fig. 1), but the presence of absorption forceschanges to the theory. The beam splitter input and output operators are related byaˆout = taˆin + rbˆin + fˆa,bˆout = tbˆin + raˆin + fˆb, (8)where in this case |t|2 + |r|2 < 1. The noise operators, fˆa and fˆb, are necessary to preserve thecommutators of the observable outputs. We will assume that the noise modes are unexcited, andcan thus make no contribution to output photocounts. Furthermore, the reflection and transmissioncoeffecients are not constrained by a phase relation similar to the one for the lossless beam splitter(that is, t 6= ir in general). The requirement that the beam splitter is lossy for coherent inputsdoes, however, give another relation which the reflection and transmission coefficients must satisfy,|t± r| < 1. (9)The superposition operators for the lossy beam splitter are simply the sum and difference operatorsas before. The output operators are appended by noise operators,cˆout = (t+ r)cˆin + fˆc,dˆout = (t− r)dˆin + fˆd, (10)where fˆc and fˆd are respectively the sum of and difference between the noise operators fˆa and fˆb.The theory outlined above is sufficient for this paper. A fuller account of the general theory in acontinuous mode formalism is given in reference [13].53 The lossless lossy beam splitterConsider a lossy beam splitter with transmission and reflection coefficients which are equal oropposite, t = ±r. Such a phase relation would not be allowed for the lossless beam splitter, but theextra freedom given by the noise operators associated with absorption means that it is an allowedchoice. There is, however, one restriction imposed by equation (9) which limits the moduli of tand r to be less than or equal to 1/2. The result of this is at best a 25/25 beam splitter with50% loss. The effects described also occur with reduced visibility for beam splitters with greaterabsorption than 50% but it is this ideal case which is assumed henceforth. It is clear that for thischoice of phase relation one of the superposition modes is passed completely and the other is fullyabsorbed. The |1±〉 states are the one-photon excitations of these superposition operators, andTable 1 summarises the effect of the beam splitter on these inputs.Table 1input beam splitter output|1+〉 t = r |1+〉t = −r |0〉|1−〉 t = r |0〉t = −r |1−〉Table 1. One photon superposition input states and output states for the 25/25 lossy beam splitter.Clearly the photons are passed or absorbed with either unit or zero probability. They arenot subject to the randomness normally associated with an absorbing medium that plagues otherinput states. Although the table illustrates the results with one photon states, the effect is notspecifically quantum in nature. The beam splitter will be rendered either transparent or opaque toany excitation of the superposition modes, even a classical excitation such as the coherent state.The device exhibits the remarkable property that even though half of the input light is normally lost,the full intensity can be obtained at the outputs if the input is in a particular mode. Furthermore,it is relatively easy to put input light into a beam splitter in one of the superposition modes. AMach-Zehnder interferometer with the second beam splitter as the lossy one could be set up to do6lossylosslessInputoutputFigure 2: Mach-Zehnder set up to obtain an input into the lossy beam splitter in one of thesuperposition modes. The first beam splitter is a standard lossless 50/50 device, the second is lossy,25/25.this for narrow band input light (Fig. 2).4 Two photon interference and nonlinear absorptionThe previous section described how a beam splitter can be made transparent (or opaque) for classicalor quantum excitations of the superposition modes. In this section the effects of the absorbingbeam splitter on purely quantum input states are described. The standard two photon interferenceexperiment, in which a pair of photons incident on a beam splitter from different input arms produceno coincidences at the outputs, survives with reduced visibility in the lossy case provided that thesame phase relation t = ±ir is upheld. The same is true of the extensions to two photon interferencedescribed in the Introduction. If, however, the beam splitter has equal or opposite t and r, nonlinearabsorption can take place. It has previously been pointed out that for an input state with one photonin each input arm it is not possible to obtain a one photon output [13]. Either both photons passthrough or both are absorbed. There are similar nonlinear phenomena which occur when the inputis in one of the two-photon superposition states (6). These are summarised in Table 2.7Table 2input output (t = ±r)|1a, 1b〉 mixture of |0a, 0b〉 and 1√2(|1a, 1b〉 ± |2+〉)|2+〉 mixture of |0a, 0b〉 and ±1√2(|1a, 1b〉 ± |2+〉)|2−〉 ∓|1±〉Table 2. Two photon input states and output states for the 25/25 lossy beam splitter.Clearly nonlinear absorption takes place in each case. For the input states specified in the firsttwo lines of Table 2 the outputs cannot consist of only one photon, whereas the input state in thethird line produces an output of one photon with certainty. In fact nonlinear interference occursfor two photon input superpositions with any relative phase angle (not merely 0 or pi as in theTable 2). There is a simple explanantion for the nonlinear interference in terms of the superpositionoperators. The input state from the first line of the Table 2 can be written|1a, 1b〉 = aˆ†inbˆ†in|0a, 0b〉 =12(cˆ†2in − dˆ†2in)|0c, 0d〉, (11)when expressed in terms of the superposition operators for the inputs. One of cˆ or dˆ is completelyabsorbed by the beam splitter, so that the part of the output which passes is either cˆ†2out/2 or −dˆ†2out/2acting on the vacuum. The output state, found by re-expressing these operators in terms of thestandard output operators, is as given in Table 2. Similarly the input state in the second line ofTable 2, |2+〉, can be written 12(cˆ†2in + dˆ†2in)|0c, 0d〉, so the explanation, and the output state are thesame (except for a simple phase change in one case). The odd state out is the input in the thirdline of the Table 2, which from equation (7) is −|1c, 1d〉. There is therefore one photon in each ofthe input superposition modes, one of which is absorbed and the other passes. Hence the outputmust contain one photon in the passed superposition mode.5 ConclusionsIn this paper it has been shown that the amount of absorption produced by a lossy beam splittercan be reduced if the transmission and reflection coefficients have an appropriate phase relation andthe input light is directed into particular modes. For a 50% lossy beam splitter, which ordinarily8would absorb half of the light, it is possible to obtain all of the input light in the output, thusrendering the device transparent.The modes which are affected are the superposition modes, the sum and difference of the twoinputs. One photon excitations of these modes are quantum states which are useful in quantuminformation [14]. The lossy beam splitter described here could therefore be used as a filter, absorbingmerely that part of the input which is not in the required superposition.A further consequence of the different amount of loss in each superposition is the possibility ofnonlinear absorption from a linear device. For some two photon inputs the chance of a one photonoutput can be eliminated. On the other hand for the two photon input which is factorisable interms of superposition input states, a single photon output is obtained with certainty. It is clearfrom the above that, in both the classical and quantum cases, interference is not simply washedout by absorption. Loss can enhance interference, and it even has effects of its own which are notpresent in lossless environments.AcknowledgmentsThis research was supported by the Engineering and Physical Sciences Research Council. I wouldlike to thank Steve Barnett for useful comments.9References[1] see, for example, Mandel, L. and Wolf, E., 1995, Optical Coherence and Quantum Optics,(Cambridge: Cambridge University Press).[2] Hanbury-Brown, R. and Twiss, R.Q., 1956, Nature 177, 27 .[3] Hong, C. K., Ou, Z. Y., and Mandel, L., 1987, Phys. Rev. Lett. 59, 2044.[4] Rarity, J. and Tapster, P. R., 1990, in Coherence and Quantum Optics VI, Proceedings of theSixth Rochester Conference on Coherence and Quantum Optics, edited by J.H. Eberly, L. Man-del and E. Wolf (New York: Plenum); 1988, in Photon Localisation, Detection, Amplificationand Antibunching, edited by E.R. Pike and H. Walther (Bristol: Hilger).[5] Steinberg, A.M., Kwiat, P.G. and Chiao, R.Y., 1992, Phys. Rev. Lett. 68, 2421.[6] Loudon, R., 1990, in Coherence and Quantum Optics VI, Proceedings of the Sixth RochesterConference on Coherence and Quantum Optics, edited by J.H. Eberly, L. Mandel and E. Wolf(New York: Plenum); 1998, Phys. Rev. A, 58, 4904.[7] Prasad, S., Scully, M.O. and Martienssen, W., 1987, Opt. Commun. 62, 139.[8] Ou, Z.Y., Hong, C.K. and Mandel, L., 1987, Opt. Commun. 63, 118.[9] Fearn, H. and Loudon, R., 1987, Opt. Commun. 64, 485.[10] Campos, R.A., Saleh, B.E.A. and Teich, M.C., 1989, Phys. Rev. A 40, 1371.[11] Jeffers, J. and Barnett, S. M., 1993, Phys. Rev. A, 47, 3291; 1994, J. Mod. Opt., 41, 1121.[12] Lai, W.K., Buzˇek, V. and Knight, P.L., 1991, Phys. Rev. A 43, 6323.[13] Barnett, S. M., Jeffers, J., Gatti, A. and Loudon, R., 1998, Phys. Rev. A, 57, 2134.[14] Bruss, D., Ekert, A., Huelga, S.F., Pan, J.-W. and Zeilinger, A., 1997, Phil. Trans. Roy. Soc.A, 355, 2259.10

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