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MIT OpenCourseWarehttp://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 Problem Set 6 Fall 2008 Issued: Tueday, October 14, 2008 Due: Tuesday, October 21, 2008 Problem 6.1 Here we b egin the analysis of quantum linear transformations by treating the single-frequency quantum theory of the beam splitter. Consider the arrangement shown in Fig. 1. Here, ˆaIN and ˆbIN are the annihilation operato r s of the frequency-ω components of the quantum fields entering the two input ports of the beam splitter, and ˆaOUT and ˆbOUT are the corresponding frequency- ω annihilation operators at the two output ports. The input-output relation for this beam splitter is the following: ^bOUT a a IN OUT ^ ^ ^ bIN Figure 1: Single-frequency beam splitter configuration aˆOUT = √ǫ aˆIN + √1 − ǫˆbIN ˆbOUT = −√1 − ǫ aˆIN + √ǫˆbIN, where 0 < ǫ < 1 is the power-transmission of the beam splitter, i.e., the fraction of the incident photon flux that passes straight through the device (from ˆaIN to ˆaOUT or from ˆbIN to ˆbOUT). (a) Show that the beam splitter’s input-output relation is lossless, i.e., prove that aˆ† OUTaˆOUT +ˆbOUT † ˆbOUT = aˆ† INaˆIN +ˆbIN † ˆbIN, so that regardless of the joint state of the ˆaIN and ˆbIN modes, the total photon number in the output modes is the same as the total photon number in the input modes. 1� � � � (b) The inputs to the beam splitter have the usual commutators for a nnihilation operators of independent modes: [ˆaIN, ˆbIN] = [ˆaIN, ˆb† IN] = 0 [ˆaIN, aˆIN† ] = [ˆbIN, ˆbIN† ] = 1 . Show that the beam splitter’s input-output relation is commutator preserving, i.e., prove that [ˆˆ ˆb† ] = 0aOUT, bOUT] = [ˆaOUT, OUT[ˆaOUT, aˆ† OUT] = [ˆbOUT, ˆb† OUT] = 1 . (c) The jo int state of the input modes, ˆaIN and ˆbIN, is their density operator, ˆρIN. This density operator is fully characterized by its normally-ordered form, ρ(n)(α∗, β∗ ; α, β) ≡ ˆIN INhβ|INhα|ρIN|αiIN|βiIN, where αiIN and βiIN are the coherent states of the ˆaIN and ˆbIN modes. The 4-| |(n)D Fourier transform of ρIN (α∗, β∗ ; α, β) is then the ant i-normally ordered joint characteristic function, χρIN (ζ∗ −ζa∗ aˆIN−ζb ∗ˆbINζaaˆIN† +ζb ˆb† IN Aa, ζb∗ ; ζa, ζb) ≡ tr ρˆINe e, where ζa and ζb are complex numbers. Relate the anti-normally ordered char-acteristic function of the o utput modes, ab OUTOUT χρOUT (ζ∗, ζ∗ ; ζa, ζb) ≡ tr ˆ−ζ∗ aˆOUT−ζ∗ˆbOUTe ζaaˆ† +ζb ˆb† ,A a bρOUTe to that for the input modes by: (1) using the beam splitter’s input-output relation to write the exponential terms in the χρOUT (ζ∗, ζ∗ ; ζa, ζb) definition in A abterms of the input-mode annihilation and creation operators, and (2) taking the expectation of the product of the resulting exponential terms by multiplying by the joint density operator of t he input modes and taking the trace. (d) Suppose that the j oint state of ˆaIN and ˆbIN is the two- mode coherent state |αINiIN|βINiIN. Use the result of (c) t o show that the joint state o f ˆaOUT and ˆbOUT is the two-mode coherent state |αOUTiOUT|βOUTiOUT where αOUT = √ǫ αIN + √1 − ǫ βIN, βOUT = −√1 − ǫ αIN + √ǫ βIN. 2� � � � � Problem 6.2 Here we shall develop a moment-generating function approach to the quantum statis-tics of single-mode direct detection. Suppose that an ideal photodetector is used to make the number-operato r measurement, Nˆ≡ aˆ†aˆ, on a single-mode field whose state is given by the density operator ˆρ and let N denote the classical random variable outcome of this quantum measurement. The moment-generating function of N is ∞ ∞ MN(s) ≡ e sn Pr(N = n) = e sn hn|ρˆ|ni, for s real, (1) n=0 n=0 where the second equality follows from Problem 3.2(b). (The moment-g enerating function of a random variable, from classical probability theory, is the Laplace trans-form of the probability density function of that random variable—cf. the character-istic function, which is t he Fourier transform of the probability density—and hence provides a complete characterization of the random variable. In other words, the probability density function can be recovered fr om the moment-generating function by an inverse Laplace transform operation.) (a) Define a function QN(λ) as follows, ∞ QN(λ) = (1 − λ)n hn|ρˆ|ni, for λ real. (2) n=0 Show how MN(s) can be found from QN(λ). (b) Show that dkQN(λ)�� ∞ dλk �� λ=0 = n=k (−1)k n(n −1)(n − 2) ··· (n − k + 1)hn|ρˆ|ni = (−1)k haˆ†kaˆk i, for k = 1, 2, 3 . . . (The last equalilty explains why haˆ†kaˆki is called the kt h fa ctoria l moment of the photon count.) (c) Suppose that ˆρ = , i.e., that the field mode is in the mth number state. |mihm|†kˆkFind the factorial moments {haˆ a i : k = 1, 2, 3, . . . }. Use the Taylo r series, ∞ � �� QN(λ) = � 1 dkQN(λ) �� λk k! dλk k=0 λ=0 to find QN(λ) and then use the result of part (a) to find MN(s). Verify that this moment-generating function agrees with what you would find directly fro m Eq. (1). 3� � � � (d) Suppose that ˆρ = , i.e., that the field mode is in a coherent state with |αihα|†kˆkeigenvalue α. Find the factorial moments {haˆ a i : k = 1, 2, 3, . . . }. Use the Taylor series, ∞ � �� QN(λ) = � 1 dkQN(λ) �� λk k! dλk k=0 λ=0 to find QN(λ) and then use the result of part (a) to find MN(s). Verify that this moment-generating function agrees with what you would find directly fro m Eq. (1). Problem 6.3 Here we shall examine a quantum photodetection model for single-mode direct de-tection with sub-unity


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MIT 6 453 - Problem Set 6

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