photon quantum mechanics and beam splitters

Photon quantum mechanics and beam splittersC. H. Holbrow,a) E. Galvez, and M. E. ParksDepartment of Physics and Astronomy, Colgate University, Hamilton, New York 13346

~Received 18 July 2001; accepted 8 November 2001!

We are developing materials for classroom teaching about the quantum behavior of photons in beamsplitters as part of a project to create five experiments that use correlated photons to exhibitnonclassical quantum effects vividly and directly. Pedagogical support of student understanding ofthese experiments requires modification of the usual quantum mechanics course in ways that areillustrated by the treatment of the beam splitter presented here. ©2002 American Association of Physics

Teachers.

@DOI: 10.1119/1.1432972#

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I. INTRODUCTION

By using recent advances in the production and detecof correlated photons, we are developing a sequence ofexperiments for undergraduates to demonstrate importanpects of quantum superposition. Doing these experimewill prepare students to understand issues of quantum ctography, quantum computing, and quantum teleportatMore directly, the experiments will show that the photexists, that a photon interferes with itself, that small modcations of the observing apparatus can erase and restorterference, and that there exist other kinds of photon inference than the one-photon sort. The apparatus will alsused to exhibit the nonlocal nature of quantum mechanicsshowing the violation of Bell’s inequalities. Chiao, Kwiaand Steinberg have written an excellent review of theserelated experiments.1 More information is also available oour website.2

Because we think that these experiments will makeunusual and unintuitive features of quantum mechanicsgible and vivid, we want to include them in our introductoquantum mechanics course~most recently taught from Grif-fiths’ text3!. However, for the experiments to be meaningfwe will need to expand the material on state vectors, stextensively the two-state system, go more deeply intolinear algebra needed to describe transformations of svectors by experimental apparatus, and show how supesition leads not just to conventional interference but alsoentanglement and interference of correlated pairs. Roomthis additional material will probably be at the expensewave functions and the hydrogen atom. Hence, the succful incorporation of our experiments into undergraduaquantum mechanics implies a substantial revision ofusual syllabus.

This paper analyzes a beam splitter to show the kindchange we have in mind for our syllabus. We omit discussof the quantum eraser1 and the violation of Bell’sinequalities4,5 even though the last experiment will be thclimax of our program. We concentrate on three experimefor which the beam splitter is a key part. We show howuse linear algebra to describe the operation of a beam splan approach that we believe will exercise students in theof mathematical tools important for the analysis of quantstate vectors. We show how these tools can be used tolyze a more complicated experimental apparatus, the MaZehnder interferometer, as a combination of beam splittWe then apply the same tools of analysis to predict the c

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pletely nonclassical output that can be produced with twphoton states which can be made with our apparatus.

II. BEAM SPLITTER EXPERIMENTS

A. Underlying technology needs

All of the experiments require correlated pairs of photoas their inputs. The possibility of doing any of our plannexperiments depends on the capability to generate, seand detect such photon pairs. Two devices, the down cverter and the avalanche photodiode, provide the technothat makes our experiments feasible.

Correlated pairs can be created with a down convertecrystal specially cut to exploit its birefringence and its nolinear properties so that a single entering photon, in the dblue or ultraviolet, results in the output of two lower enerphotons. The conversion efficiency of 10212– 10210 is verylow. Therefore, it is necessary to use a laser that can suenough input photons to compensate for this low efficienand also for inefficiencies in the selection of special pafrom among those generated. A laser with a power oforder of 100 mW is needed.~At 450 nm, 100 mW corre-sponds to 2.331017 photons/s.! The outgoing two photonsare correlated in time because they are produced at the sinstant; they are correlated in momentum and in frequeby the conservation laws; and with proper choice of cryscrystal orientation, and spatial filters, their polarization stacan be correlated. Any of these correlation properties canthe basis of coincidence experiments that dramaticallyhibit the quantum nature of light.

Even with 100 mW there are no photon pairs to waste, ait is necessary to detect coincidences with as much efficieas possible. Avalanche photodiodes are extremely sensdetectors that can register single photons with efficienciehigh as 80%. Their use makes it possible to measure coidences of correlated pairs and accumulate statisticallynificant numbers of counts within the duration of a typicundergraduate laboratory.6

B. The experiments: The photon exists

The layout of the experiment to show directly the extence of the photon is illustrated schematically in Fig. 1. Tphotons emerge simultaneously from the down converOne goes to detector D0 ; the other goes to the beam splittBS and then to detector D1 or D2 . If the photon were aclassical wave, it would split at the beam splitter, and so

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amplitude would appear in both D1 and D2. The experimentconsists of looking at coincidences D0– D1, D0– D2, andD1– D2. The observation is that D0– D1 or D0– D2 will oc-cur, but never D1– D2.7 The absence of D1– D2 coincidencesis what we mean when we say the photon exists. Clearlybeam splitter is the heart of the apparatus here.

C. The experiments: The photon interferes with itself

The next experiment uses a Mach–Zehnder interferomas shown in Fig. 2. Changing the length of one arm, saymoving mirror M1, will change the phase relationship btween the two paths from the input to the beam splitter B2 .As the phase is changed, the count rate in detector D1 willvary back and forth between some maximum and nezero, corresponding to variations from constructive tostructive interference. Such variations correspond to thepearance and disappearance of interference fringes asMichelson interferometer; therefore, in what follows we wuse the word ‘‘fringes’’ to refer to these variations in courate.

To show that a single photon interferes with itself, that‘‘takes both paths,’’ look at D0– D1 coincidences as a function of the change in phase. This arrangement guaranteesthe observed interference fringes have built up from couoccurring when there is only one photon in the interferoeter. Similar fringes occur in detector D2 but 90° out of phasewith those in D1 .

Fig. 1. Two photons emerge from XTL, the down converter—the incidlaser beam that excites the down converter is not shown. One photon godetector D0 ; the other goes by way of the beam splitter BS to D1 or D2 . Theabsence of coincidences between detectors D1 and D2 demonstrates the existence of the photon.

Fig. 2. The motion of the mirror M1 introduces a phase change in one aof the Mach–Zehnder interferometer. Interference fringes appear incounts in D1 and D2 . Detection in coincidence with D0 assures that there isonly one photon in the interferometer at the time of detection and, therethat a photon interferes with itself.

261 Am. J. Phys., Vol. 70, No. 3, March 2002

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Note that the Mach–Zehnder interferometer consiststwo mirrors and two beam splitters, so that once again besplitters are the heart of the apparatus.

D. The experiments: Two-photon interference

The quantum nature of light is made vividly apparewhen the apparatus is arranged so that both the correlphotons from the down converter simultaneously enterinputs of the beam splitter as shown schematically in FigQuantum mechanics predicts that in such a case the twotons will always arrive at the same output of the beam spter, either both at D1 or both at D2 , but never one photon aD1 and the other at D2 .

Moreover, when a phase difference is introduced betwthe two arms of the apparatus, interference fringes will ocin the D1– D2 coincidence counts, even though they are nobserved in the counts observed at D1 or at D2 alone.

III. WHAT DO STUDENTS NEED TO KNOW TOUNDERSTAND THE EXPERIMENTS?

A. Background

Our syllabus is intended for students who have hadthree-term introductory physics course plus a one-tecourse called ‘‘Waves and Modern Physics.’’ The wavcourse introduces students to the solution of the harmooscillator equation, the representation of waves by compexponentials, Fourier analysis, and to the one-dimensioSchrodinger equation applied to the particle in a box orpiecewise continuous potentials. The mathematics baground of the students is three semesters of calculus; ahave had a linear algebra course devoted largely to bproperties of matrices and their manipulation. Very fewany students have had a course in differential equations

B. Matrix mechanics

Students need a good grounding in the quantum mechics of two-state systems in order to understand these expments. As part of this grounding, we will spend more timthan is customary on state vectors and matrix operators.will also examine closely systems of identical particles,pecially entangled two-particle states that are central toproject.

Our plan is to introduce two-state systems using the poization states of photons as treated by French and Taylor8 andthe somewhat more complete treatment of Lipkin.9 We willthen treat the case of the spin 1/2 particle, adapting Feman’s Stern–Gerlach analysis10 and going on to a treatmensuch as Griffiths’.3 Feynman has discussed a rich variety

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Fig. 3. When two photons from a down converter are inputs to a besplitter, there can be interference fringes in the coincidences, even ththe counts in each counter show no evidence of interference.

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systems that can be described using 232 matrices, and wewill use some of these to familiarize students with the termnology, properties, and use of matrices. We also wanteach students how to interpret quantum situations in teof probability amplitudes in the style of Feynman.11,12

We will emphasize the fundamental ideas that anynamical variable has a matrix representation, the eigenvaof that matrix are the only possible results of measuremeof the variable, and that any state of the system can beresented as a linear combination of eigenstates. We will nto teach how to find the eigenvalues and eigenvectorsmatrix. We must also teach the idea of representing a svector in terms of a basis, the idea of a complete set of stahow to transform from one basis to another, why the matriof observables must be Hermitian adjoint, and what unittransformations are and why they are important.

IV. PHOTONS ON A BEAM SPLITTER AS A TWO-STATE SYSTEM

Figure 4 represents an idealized model of a beam spliIt has two input ports and two output ports. We assume ibe lossless. For photons incident through port 1, we denthe reflection and transmission amplitudes asr andt, respec-tively. For photons incident through port 2, these amplitudare r 8 and t8. We simplify the analysis by assuming ththese fractions are independent of the photons’ angle ofcidence or their state of polarization.

A. Unitarity and the beam splitter matrix

The main point here is to recognize that the two possinputs of a photon into a beam splitter can be representea two-state system. One possible input stateuin& is a photonentering port 1; another is a photon entering port 2. Thinkthese as basis vectors (0

1) and (10), respectively. Similarly, we

choose an exit representation for the output states in w(0

1) represents a photon exiting port 3 and (10) represents one

exiting 4.Described in these terms, the beam splitter perform

linear transformationR that convertsuin& to uout&, that is,uout&5Ruin&. To find R in this representation, note that whethe reflection and transmission amplitudes arer and t, re-spectively, for an input state of (0

1), the output state will be(t

r) in the exit representation. A similar argument forr 8 andt8 and an input state of (1

0) yields

R5S r t 8

t r 8D .

Fig. 4. A schematic representation of an ideal beam splitter with input p1 and 2 and output ports 3 and 4. The amplitudes for reflection and trmission from port 1 are, respectively,r andt. The corresponding amplitudefor a photon entering 2 arer 8 and t8.


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Because of the conservation of probability, the matrixRmust be unitary. This condition means that the Hermitadjoint R† equals the inverseR21, a fact that yields usefurelationships among the elements of the matrix ofR. Theinverse ofR, like that of any matrix, is the transpose of icofactor matrix divided by its determinant. Therefore,

1

rr 82tt8 S r 8 2t8

2t r D 5S r * t*

t8* r 8* D . ~1!

The determinant of a unitary matrix has a modulus of onerr 82tt85eig. Because the factoreig multiplies every ele-ment of the matrix, it will not affect relative phases betweterms, and it may be assigned a convenient value. Ifchooseg50, the factorrr 82tt8 is just 1. By equating thecorresponding elements of the right- and left-hand sidesEq. ~1!, we obtain

r 5r 8* ~2a!

t52t8* . ~2b!

If we rewrite these factors as complex exponentials,ur ueidr,ur 8ueidr 8, utueid t, andut8ueid t8 and divide Eq.~2b! by Eq.~2a!,we obtain

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e2 i ~d t82dr 8!. ~3!

Equations ~2! show that utu5ut8u and ur u5ur 8u, so theseterms may be canceled in Eq.~3!. Then dividing Eq.~3! bythe right-hand side and using the fact that215eip, we ob-tain

d t2d r1d t82d r 85p, ~4!

as Zeilinger has shown.13

For the common case of a beam splitter that has the seffect on a beam incident through port 1 as on a beam ident through port 2, that is, a symmetric beam splitterr5r 8, t5t8, andd t2d r5d t82d r 85p/2, and the transmittedwave leads the reflected wave in phase byp/2 rad. Thisphase difference introduces an important factor ofi into thetransmission amplitudes, a factor that is usually introducwith no more explanation than ‘‘unitarity implies’’ it.

The 50–50 symmetric beam splitter is particularly simpFor this case not only do we haver 5r 8 and t5t8, but nowur u5utu5ur 8u5ut8u. It then follows from Eq.~2! that r mustbe real andt must be pure imaginary. Given thatt leadsr inphase byp/2 rad and that the determinantrr 82tt851, itfollows that r 51/& and t5 i /&. The matrixR is then14

R51

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i 1D . ~5!

B. Application to a Mach–Zehnder interferometer

As Fig. 2 shows, a Mach–Zehnder interferometer consessentially of two beam splitters. The mirrors can be ignobecause their effects in their respective arms balanceThe output is just two applications ofR: uout&5RRuin&. It ismade to function as an interferometer by inserting a phshifter into one arm. For example in Fig. 2, the phase shimight be just the motion of the mirror M1. In our outputrepresentation, a phase shifter can be represented as

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^outuout&5S sin2S f

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Interference fringes corresponding to these probabilitiescur asf is varied.

C. Two-photon interference

As noted above, our experimental setup is designedintroduce two photons into the apparatus at the same tWe consider the case for which the incident photons areidentical polarization states. There then result two-photonterference effects that can only be explained by quanmechanics.

To describe the states of a two-photon system, wedraw on the students’ earlier introduction to composite stems. They will have seen that the state of an assemblnoninteracting particles can be described as the producthe states of the individual particles. For photons leavinbeam splitter these states can be labeled in terms of theput port and the particle number. We denote particle 1 leing port 3 asu31&, particle 2 leaving port 4 asu42&, and soon. Obviously there can be just four distinct product staStudents will have seen that out of linear combinationsthese four product states, it is possible to construct four mtually orthogonal states that have a definite symmetry unthe exchange of two particles:

u31&u32& ~a! Exchange symmetric, bothphotons exit port 3.

1

&@u31&u42&1u32&u41&]

~b! Exchange symmetric, onephoton exits port 3, the otherport 4.

u41&u42& ~c! Exchange symmetric bothphotons exit port 4.

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&@u31&u42&2u32&u41&]

~d! Exchange antisymmetric, onephoton exits port 3, the otherport 4.

~8!

If we replace output port numbers 3 and 4 with input pnumbers 1 and 2, respectively, we obtain the analogousequations for the input states.


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If phase shifters are placed in the arms of the interferoeter and the output photons are detected in coincidenceinterference pattern will be observed in the coincideneven though none is observed in the counts of the individdetectors. Greenberger, Horne, and Zeilinger15 give a goodexplanation of the essential features of such two-phointerference, and Loudon16 gives a thorough treatmenusing the formalism of creation and annihilation operatoWe leave for another paper the discussion of our pedagogapproach to presenting this material to our students.

Two-photon coincidences can provide a dramatic examof the quantum effect of indistinguishability. The analysisalso a good application of Feynman’s rules for working wprobability amplitudes.1,10 For the setup shown in Fig. 3there are only two ways that coincidences between D1 andD2 can occur. Either each photon reflects at the beam splor each photon passes through it. The amplitude for treflections and two transmissions isrr andtt, respectively. Ifthe two path lengths through the apparatus are different, trr is physically distinguishable fromtt, and the probabilitythat there will be counts in both D1 and D2 is urr u21uttu2

51/2, the sum of the squares of the amplitudes of the tdistinguishable cases. But when the path lengths are mequal, the eventsrr and tt become indistinguishable, and thprobability is then the square of the sum of the individuamplitudes:urr 1ttu2. For a symmetric beam splitterr 51,t5 i , andrr 1tt512150. There is no amplitude for coincidences between D1 and D2. The effect is seenexperimentally17 as the appearance of a sharp minimumthe coincidence rate as the interferometer arms are adjuto be of equal length.

As noted, this result is for incident photons in identicpolarization states. When we take into account other poization states, a new and interesting possibility emerges.

D. Polarization

Because it can have either of two independent modepolarization, a single photon is itself a two-state system.example, with appropriate polarizers a photon can be puta state of vertical polarization,uV&, or horizontal polariza-tion, uH&. These states constitute a basis, and any genpolarization state can be expressed as a linear combinatiouH& and uV&. An arbitrary two-photon polarization state cathen be described in terms of four product states that areexact analogues of the spatial states described by Eq.~8!.These can be obtained from Eq.~8! by replacing port num-bers 3 and 4, respectively, with V and H.

A complete description of the two-photon states of a besplitter can be given in terms of the sixteen products offour spatial states with the four polarization states. Thconstitute a complete set of orthogonal states for describtwo particles in a beam splitter. Because photons are bosthe only possible states are linear combinations of thestates that are symmetric under the exchange of the parlabels. Nine of these states are products of the exchasymmetric spatial states with the exchange symmetric poization states. None of these states yields a photon in eoutput channel. For six of the states, this null result iscause the spatial part of the state vector is eitheru31&u32& oru41&u42& that is, two photons leaving by the same outpport. The other three states have (1/&)@ u31&u42&


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1u32&u41&] as a factor, but this factor is zero because, asprevious section showed, the amplituderr 1tt is zero for a50–50 symmetric beam splitter.

The tenth exchange symmetric state is the product ofantisymmetric spatial state with the antisymmetric polarition state:

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&~ u31&u42&2u32&u41&)GF 1

&~ uV1&uH2&2uV2&uH1&)G . ~9!

Now because of the exchange antisymmetry, the two amtudes for producing coincidences subtract when the interometer arms are exactly equal in length, that is, the probaity for coincidences isPc5urr 2ttu251. Therefore, if theinitial two-photon state contains any antisymmetric spapart, there will be D1-D2 coincidences. This result is possibonly if the entering photons have different states of polarition.

Equation~9! is an example of what Schro¨dinger called an‘‘entangled state,’’ and in our course this will be a good plato begin a discussion of the properties and consequencesuch states. In such states photons do not have indiviidentity, although they are correlated. There is no waypredict the polarization state of a photon in a particular oput port, but once it has been measured and found to be,V, the photon in the other output port will always be mesured to beH, and conversely. This is the situation that Eistein, Podolsky, and Rosen called to our attention nearlyyears ago,18 and which Einstein deplored as giving rise‘‘spooky action at a distance.’’

We can demonstrate the correlation of the photon stexperimentally using polarizing beam splitters. The polarition state of photons exiting from such a device dependswhich output port they exit. For example, in Fig. 5 the aparatus can be arranged so that photons coming out upfrom PBS1 are in auH& state, while those going straight arethe uV& state. Then D1 will be detectinguV& photons and D2will be detectinguH& photons. Suppose further, for the saof example, that the apparatus is arranged so that D3 detectsuV& photons while D4 detectsuH& photons. Because of thpolarization correlation, an input entangled state of the foEq. ~9! would then give rise to coincidences only between1and D4 or between D2 and D3, and never between D1 and D3

or between D2 and D4. These properties of two-photon statof a beam splitter have been adroitly used to exhibit quanteleportation.19,20

Fig. 5. A Mach–Zehnder interferometer with polarizing beam splittePBS1 and PBS2 , to analyze photons exiting from the beam splitter BS1 .


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V. CONCLUSIONSDevelopment of the first experiment is under way, and

are preparing instructional materials to foster student undstanding of the experiments. The module on beam splitdescribed here is ready to be tried in our quantum mechacourse. We have also begun a module on Bell’s inequalithat has been used twice in the classroom. Further devement of that module is continuing along with the develoment of modules on the quantum eraser and on two-phointerference.

ACKNOWLEDGMENTS

We are grateful to the National Science Foundationsupport by Grant No. CCLI-DUE 9952626, and we thaPaul Kwiat, William K. Wootters, and David Mermin fotheir helpful suggestions.

a!Electronic mail: [email protected]. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, ‘‘Optical tests of quantumechanics,’’ inAdvances in Atomic, Molecular, and Optical Physics~Aca-demic, New York, 1994!, Vol. 34, pp. 35–83.

2Our Web site is at http://departments.colgate.edu/physics/research/Phphoton–quantum–mechanics.htm.

3D. Griffiths, Quantum Mechanics~Prentice–Hall, Englewood Cliffs, NJ1995!, pp. 154–176. Chapter 3 of this book supplies a nice summarythe concepts and terminology of linear algebra relevant to quantumchanics~pp. 75–120!. It may be too brief for our purposes.

4J. S. Bell,Speakable and Unspeakable in Quantum Mechanics~CambridgeU.P., Cambridge, 1987!. See particularly ‘‘On the Einstein-PodolskyRosen paradox,’’ pp. 14–21, and ‘‘Bertlmann’s socks and the naturereality,’’ pp. 139–158.

5D. Mermin, ‘‘Is the moon there when nobody looks? Reality and quanttheory,’’ Phys. Today38~4!, 38–47~1985!.

6An important long-range goal of our project is to achieve a design texploits advances in technology to bring the cost of the apparatus wthe budgets of colleges.

7A significant part of this laboratory experiment will be devoted to intrducing students to the statistics of counting.

8A. P. French and Edwin F. Taylor,Introduction to Quantum Physics~Chap-man and Hall, London, 1974!, pp. 231–278.

9H. J. Lipkin, ‘‘Polarized photons for pedestrians,’’ inQuantum Mechanics:New Approaches to Selected Topics~North-Holland, New York, 1973!, pp.2–29.

10R. P. Feynman, R. B. Leighton, and M. L. Sands,The Feynman Lectureson Physics~Addison–Wesley, Reading, MA, 1963!, Vol. 3, Chaps. 5 and6. Feynman develops quantum mechanics largely in terms of two-ssystems. We will make extensive use of his material.

11R. P. Feynman,QED, the Strange Theory of Light and Matter~PrincetonU.P., Princeton, 1985!.

12D. F. Styer,The Strange World of Quantum Mechanics~Cambridge U.P.,New York, 2000!.

13A. Zeilinger, ‘‘General properties of lossless beam splitters in interferoetry,’’ Am. J. Phys.49, 882–883~1981!. This article expresses the gener232 unitary matrix in terms of Pauli spin matrices. This approach wobe particularly useful for further familiarizing students in a quantum mchanics course with these important mathematical tools.

14The exit probabilities are independent of whether the transmitted bleads or lags the reflected beam byp/2 rad. In other words, a matrixR inwhich t52 i is equivalent to one witht5 i .

15D. M. Greenberger, M. A. Horne, and A. Zeilinger, ‘‘Multiparticle interferometry and the superposition principle,’’ Phys. Today46~8!, 22–29~1993!. This article is an excellent source of explanations of the kindsquantum effects that our project is intended to display.

16R. Loudon, ‘‘Spatially-localized two-particle interference at a beam sp

,


nd

de

.and

A.

ter,’’ in Coherence and Quantum Optics VI, edited by J. H. Eberlyet al.~Plenum, New York, 1990!, pp. 703–708.

17C. K. Hong, Z. Y. Ou, and L. Mandel, ‘‘Measurement of subpicosecotime intervals between two photons by interference,’’ Phys. Rev. Lett.59,2044–2046~1987!.

18A. Einstein, N. Rosen, and B. Podolsky, ‘‘Can quantum-mechanicalscription of physical reality be considered complete?,’’ Phys. Rev.47,777–780~1935!.


-

19C. H. Bennett, G. Brassard, C. Cre´peau, R. Jozsa, A. Peres, and W. KWootters, ‘‘Teleporting an unknown quantum state via dual classicEinstein–Podolsky–Rosen channels,’’ Phys. Rev. Lett.70, 1895–1899

~1993!.20D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and

Zeilinger, ‘‘Experimental quantum teleportation,’’ Nature~London! 390,575–579~1997!.

EXISTENCE OF ATOMS

‘‘I Don’t believe that atoms exist!’’This blunt declaration of disbelief came, in fact, in January 1897 at a meeting of the Imperial

Academy of Sciences in Vienna. The skeptic was Ernst Mach, not quite 60 years old, who hadbeen for many years a professor of physics at the University of Prague and who was now aprofessor of history and philosophy of science in Vienna. He pronounced his uncompromisingopinion in the discussion following a lecture delivered by Ludwig Boltzmann, a theoretical physi-cist. Boltzmann, a few years younger than Mach, had likewise recently returned to Vienna aftermany years at other universities in Austria and Germany. He was an unabashed believer in theatomic hypothesis—indeed, his life’s work had centered on that single theme.

Davie Lindley,Boltzmann’s Atom~The Free Press, New York, NY, 2001!, page 1 of the introduction.

Submitted by Louise Moll Claver.


photon quantum mechanics and beam splitters

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