MIT 6 453 - Quantum Optical Communication (18 pages)

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Quantum Optical Communication



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Quantum Optical Communication

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School:
Massachusetts Institute of Technology
Course:
6 453 - Quantum Optical Communication
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MIT OpenCourseWare http ocw mit edu 6 453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 453 Quantum Optical Communication Lecture Number 22 Fall 2008 Je rey H Shapiro c 2008 Date Thursday December 4 2008 Introduction Last time we established the quantum version of coupled mode theory for sponta neous parametric downconversion SPDC We exhibited the exact solutions for the output signal and idler beams their jointly Gaussian state characterization when the input beams are in their vacuum states and the low gain regime approximations for the correlation functions that characterize that state We also introduced the lumped element coupled mode equations for the optical parametric ampli er OPA presented their solutions described their jointly Gaussian state when the signal and idler inputs are unexcited and showed that the OPA produced quadrature noise squeezing Today we shall nish our survey of the nonclassical signatures produced by 2 interactions by considering Hong Ou Mandel interferometry the generation of polarization entangled photon pairs from SPDC and the photon twins behavior of the signal and idler beams from an OPA Along the way we will learn about quantum interference and photon indistinguishability Quantum Interference Let us get started with a simple single mode description in order to introduce quantum interference Consider the 50 50 beam splitter arrangement shown on slide 3 Here the only excited modes at the input ports are the co polarized pure tone plane wave j 0 t j 0 t pulses a Sin e T and a Iin e T for 0 t T The resulting excited modes at the beam splitter s output then have annihilation operators given by1 a Sout ja Sin a Iin 2 and a Iout 1 a Sin ja Iin 2 1 The reader should check that this is indeed a unitary transformation and that it conserves energy and commutator brackets It di ers from the 50 50 beam splitter relation a Sout a Sin a Iin 2 and a Iin a Sin a Iin 2 that we have previously employed That di erence however is one of phase angle choices that amount to simple changes in the input and output reference planes on which the elds are de ned The new choices make the transformation symmetrical which lends itself to greater insight into the quantum interference process 1 We shall assume that the input modes are each in their single photon state so that their joint state is the product state in 1 Sin 1 Iin What then is the joint state of the output modes We know that it must be a pure state because we are starting from a pure state and the beam splitter transformation is a unitary evolution We know that it must contain exactly two photons because the beam splitter transformation conserves energy and there are exactly two photons present at its input Thus we can safely postulate out c20 2 Sout 0 Iout c11 1 Sout 1 Iout c20 0 Sout 2 Iout 2 for the output state s number state representation where c20 2 c11 2 c02 2 1 Furthermore treating each input mode s input state as an independent billiard ball photon that is equally likely to be transmitted or re ected by the beam splitter we could easily be led to conclude that c20 2 c02 2 1 4 and c11 2 1 2 3 so that Pr NSout nS NIout 1 4 for nS 2 nI 0 1 2 for n 1 n 1 S I nI 1 4 for nS 0 nI 2 0 otherwise 4 for ideal unity quantum e ciency photon counting measurements on the output modes These results seem quite reasonable There is only one way for both photons to emerge in the a Sout mode the a Sin photon gets transmitted and the a Iin photon gets re ected Similarly there is only one way for them to both emerge in the a Iout mode the a Sin photon gets re ected and the a Iin photon gets transmitted On the other hand there are two ways for one photon to emerge in each mode i e both input photons are transmitted or both are re ected by the beam splitter Because this billiard ball picture says photon transmission and re ection is equally likely to occur at the 50 50 beam splitter we get the photon counting distribution given above Photons however are not billiard balls as we know from our work on polarization entanglement In the present context their wave like properties cause them to interfere at the 50 50 beam splitter leading as we will soon show to the following output state out 2 Sout 0 Iout 0 Sout 2 Iout 2 5 Two things are worth noting before proceeding to the derivation the input state was a product state but the output state is entangled and both photons always leave through the same output port Why is it impossible to get one photon to appear in each output port Quantum interference is the answer In particular we 2 must add the complex amplitudes for the two possible ways in which one photon can appear in each output port before taking the squared magnitude to calculate the photon counting probability for the event in which one photon is present at each output port It is the nature of 50 50 beam splitting that the complex amplitudes for these two possibilities b oth input photons transmitted or both re ected ha ve equal magnitudes but are radians out of phase Hence their complex amplitudes sum to zero and we never get one photon emerging from each of the beam splitter s output ports To verify that the output state is as given in Eq 5 let us assume that this equation is correct The normally ordered characteristic function for the output state then obeys S a S I a I S a Sout I a Iout out out e out N S I S I e 6 e S a Sout e I a Iout e S a Sout e I a Iout 7 2 2 2 S Sout 1 S2 Sout 0 Iout 0 Iout 2 2 I Iout 1 I2 Iout 0 Sout 0 Sout 2 2 2 2 2 Sout 2 S 1 Sout S2 0 Sout 0 Iout 2 Iout 2 I 1 Iout I 2 0 Iout 0 Sout 2 1 S 2 I 2 S2 I2 2 4 8 where the second equality follows because a Sout and a Iout commute and the third equality follows from series expansion of the exponentials plus the assumed output state Now let us show that we can get this same result by starting from the input state and the beam splitter transformation From Eq 1 we can easily show that Sin j S I j S I Iin j I S j I S out N S I S I N N 9 2 2 2 2 By series expansion of the exponentials in the characteristic …


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