Msschusetts Institute of Technology Deprtment of Electricl Engineering nd Computer Science 6.453 Quntum Opticl Communiction Problem Set 6 Fll 2004 Issued: Wednesdy, October 13, 2004 Due: Wednesdy, October 20, 2004 Reding: For eigenkets of yˆ ˆA I ˆ B + Î A ˆB: J.H. Shpiro nd S.S. Wgner, Phse nd Amplitude Uncertinties in Heterodyne Detection, IEEE J. Quntum Electron., QE-20, 803 813 (1984). For quntum photodetection: L. Mndel nd E. Wolf, Opticl Coherence nd Quntum Optics (Cmbridge University Press, Cmbridge, 1995), Sects. 12.1 12.4, 12.9, 12.10. H.P. Yuen, H.P. nd J.H. Shpiro, Opticl Communiction with Two-Photon Coherent Sttes Prt III: Quntum Mesurements Relizble with Photoemissive Detectors, IEEE Trns. Inform. Theory, IT-26, 78 92 (1980). For semiclssicl photodetection: L. Mndel nd E. Wolf, Opticl Coherence nd Quntum Optics (Cmbridge University Press, Cmbridge, 1995), Sects. 9.1 9.8. R.M. Gglirdi nd S. Krp, Opticl Communictions (New York, Wiley, 1976) Chp. 2. Problem 6.1 Here we begin the nlysis of quntum liner trnsformtions by treting the singlefrequency quntum theory of the bem splitter. Consider the rrngement shown in Fig. 1. Here, â IN nd ˆb IN re the nnihiltion opertors of the frequency-ω components of the quntum fields entering the two input ports of the bem splitter, nd â nd ˆb re the corresponding frequency-ω nnihiltion opertors t the two output ports. The input-output reltion for this bem splitter is the following: ˆ = ǫ IN ˆ + 1 ǫˆbin ˆb = 1 ǫ ˆIN + ǫˆbin, where 0 < ǫ < 1 is the power-trnsmission of the bem splitter, i.e., the frction of the incident photon flux tht psses stright through the device (from ˆ IN to ˆ or from ˆb IN to ˆb ). 1
^b ^ IN ^ ^ b IN Figure 1: Single-frequency bem splitter configurtion () Show tht the bem splitter s input-output reltion is lossless, i.e., prove tht ˆ ˆ ˆb = ˆ b ˆ + ˆb ˆIN IN + ˆIN b IN, so tht regrdless of the joint stte of the â IN nd ˆb IN modes, the totl photon number in the output modes is the sme s the totl photon number in the input modes. (b) The inputs to the bem splitter hve the usul commuttors for nnihiltion opertors of independent modes: [ˆ IN, ˆbIN ] = [ˆ IN, ˆb IN ] = 0 [ˆ IN, ˆIN ] = [ˆb IN, ˆb IN ] = 1. Show tht the bem splitter s input-output reltion is commuttor preserving, i.e., prove tht [ˆ, ˆb ] = [ˆ, ˆb ] = 0 [ˆ, ˆ ] = [ˆb, ˆb ] = 1. (c) The joint stte of the input modes, ˆ IN nd ˆb IN, is their density opertor, ρˆ IN. This density opertor is fully chrcterized by its normlly-ordered form, ρ (n), β IN (α ; α, β) IN β IN α ρˆin α IN β IN, where α IN nd β IN re the coherent sttes of the ˆ IN nd ˆb IN modes. The 4 D Fourier trnsform of ρ (n) IN (α, β ; α, β) is then the nti-normlly ordered joint chrcteristic function, ( ) ˆ ˆ IN ζ b b IN eζ +ζ b b χ ρ IN (ζ ζ, ζ IN IN b ; ζ, ζ b ) tr ρˆin e ˆ ˆ, A 2
where ζ nd ζ b re complex numbers. Relte the nti-normlly ordered chrcteristic function of the output modes, ( ) ˆ ζ ˆ b b ˆ eζ +ζ b b χ ρ (ζ ζ, ζ b ; ζ, ζ b ) tr ρˆ e ˆ, A to tht for the input modes by: (1) using the bem splitter s input-output reltion to write the exponentil terms in the χ ρ A (ζ, ζ b ; ζ, ζ b ) definition in terms of the input-mode nnihiltion nd cretion opertors, nd (2) tking the expecttion of the product of the resulting exponentil terms by mutliplying by the joint density opertor of the input modes nd tking the trce. (d) Suppose tht the joint stte of ˆ IN nd ˆbIN is the two-mode coherent stte αin IN β IN IN. Use the result of (c) to show tht the joint stte of â nd ˆb is the two-mode coherent stte α β where α = ǫ αin + 1 ǫ βin, β = 1 ǫ α IN + ǫ βin. Problem 6.2 Here we shll develop moment-generting function pproch to the quntum sttistics of single-mode direct detection. Suppose tht n idel photodetector is used to mke the number-opertor mesurement, N ˆ ˆ ˆ, on single-mode field whose stte is given by the density opertor ˆρ nd let N denote the clssicl rndom vrible outcome of this quntum mesurement. The moment-generting function of N is sn M N (s) e Pr(N = n) = e n=0 n=0 sn n ρˆ n, for s rel, (1) where the second equlity follows from Problem 3.2(b). (The moment-generting function of rndom vrible, from clssicl probbility theory, is the Lplce trnsform of the probbility density function of tht rndom vrible cf. the chrcteristic function, which is the Fourier trnsform of the probbility density nd hence provides complete chrcteriztion of the rndom vrible. In other words, the probbility density function cn be recovered from the moment-generting function by n inverse Lplce trnsform opertion.) () Define function Q(λ) s follows, Q N (λ) = (1 λ) n n ρˆ n, for λ rel. (2) n=0 Show how M N (s) cn be found from Q N (λ). 3
(b) Show tht d k Q N (λ) = ( 1) k n(n 1)(n 2) (n k + 1) n ρˆ n dλ k λ=0 n=k kˆk = ( 1) k ˆ, for k = 1, 2, 3... (The lst equlilty explins why ˆ kˆk is clled the kth fctoril moment of the photon count.) (c) Suppose tht ρ ˆ = m m, i.e., tht the field mode is in the mth number stte. Find the fctoril moments { ˆ kˆk : k = 1, 2, 3,... }. Use the Tylor series, ( 1 d k Q N (λ) Q N (λ) = k! dλ k k=0 λ=0 ) λ k to find Q N (λ) nd then use the result of prt () to find M N (s). Verify tht this moment-generting function grees with wht you would find directly from Eq. (1). (d) Suppose tht ρ ˆ = α α, i.e., tht the field mode is in coherent stte with eigenvlue α. Find the fctoril moments { ˆ kˆk : k = 1, 2, 3,... }. Use the Tylor series, ( ) 1 d k Q N (λ) Q N (λ) = λ k k! dλ k k=0 λ=0 to find Q N (λ) nd then use the result of prt () to find M N (s). Verify tht this moment-generting function grees with wht you would find directly from Eq. (1). Problem 6.3 Here we shll exmine quntum photodetection model for single-mode direct detection with sub-unity quntum efficiency. Suppose tht the sensitive region, A, of quntum-efficiency-η photodetector is illuminted by photon-units, positivefrequency quntum field opertor Ê(x, y, t) whose only excited, i.e., non-vcuumstte, mode is âe jωt / AT for 0 t T where A is the re of A, s shown in Fig. 2. The output of this detector is clssicl rndom vrible N whose sttistics coincide with those of the number opertor ˆ N ˆ ˆ where ˆ ηˆ + 1 ηˆ η. (3) In Eq. (3), ˆ η is photon nnihiltion opertor tht commutes with ˆ nd ˆ ; ˆ η is in its vcuum stte 0 η. 4
^ E(x,y,t) η i(t) 1 _ q T dt N 0 Figure 2: Sub-unity-quntum efficiency photon counter () Find the fctoril moments { ˆ { ˆ kˆk : k = 1, 2, 3,... }. kˆ k : k = 1, 2, 3,... } in terms of η nd (b) Use the result of prt () to relte Q N (λ) to Q N (λ) from Eq. (2). (c) Use the result of prt (b) to relte M N (s) to M N (s) from Eq. (1). (d) Verify tht your nswer to prt (c) stisfies, [ ] ( ) sn k MN (s) = e η n (1 η) k n k ρˆ k, n n=0 k=n where ( ) k k!, n n!(k n)! is the binomil coefficient. [Hint: Interchnge the orders of summtion over n nd k nd use the binomil theorem on the resulting inner sum.] (e) Use the result of prt (d) to find Pr(N = n) for the quntum-efficiency-η photodetector in terms of Pr(N = n), the photon counting probbility distribution of unity-quntum-efficiency detector, when the stte of the single-mode illumintion field is ˆρ. Problem 6.4 Here we shll continue our investigtion of quntum liner trnsformtions by treting the single-frequency quntum theory of the degenerte prmetric mplifier (DPA), i.e., the Bogoliubov trnsformtion tht produces squeezed sttes. Let ˆ IN be the nnihiltion opertor of the frequency-ω quntum field t the input to the DPA. This opertor hs the usul commuttor brcket with its djoint, viz., [ˆ IN, ˆIN ] = 1. The nnihiltion opertor of the frequency-ω output from the DPA is, ˆ µˆ IN + νˆin, where µ nd ν re complex numbers tht stisfy µ 2 ν 2 = 1. () Show tht the DPA trnsformtion is commuttor preserving, i.e., prove tht [ˆ, ˆ ] = 1. 5
(b) Suppose tht the input mode s density opertor is ˆρ IN = α IN ININ α IN, where α IN IN is coherent stte. Find the Wigner chrcteristic function, ( ) ζ ˆ χ ρ IN W (ζ, ζ) tr ρˆ IN e IN +ζˆin, of ˆρ IN. (c) Find χ ρ W (ζ, ζ), the Wigner chrcteristic function of the output mode ˆ by: (1) using the DPA s input-output reltion to write the exponentil term in the output-mode s chrcteristic function in terms of the input-mode s nnihiltion nd cretion opertors, nd (2) tking the expecttion of the resulting exponentil term by multiplying by the input-mode density opertor nd tking the trce. (d) Suppose tht µ nd ν re rel-vlued nd positive. Use the result of (c) to find the mrginl probbility densities for the outcome of the output-mode qudrture mesurements, ˆ + ˆ ˆ ˆ1 nd ˆ ˆ 2. 2 2j 6