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 Lecture Number 11 Fall 2008 Jeffrey H. Shapiro c�2006, 2008 Date: Thursday, October 16, 2008 Reading: For the quantum theory of beam splitters: • C.C. Gerry and P.L. Knight, Introductory Quantum Optics (Cambridge Uni-versity Press, Cambridge, 2005) Sect. 6.2. For the quantum theory of linear amplifiers: • C.M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982). • L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), Chap. 20. • H.A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer Verlag, Berlin, 2000), Chap. 11. Introduction In this lecture we will begin by reprising the work done last time for the squeezed-state waveguide tap, focusing on the case in which the photodetectors used in the homodyne measurements at the output ports have quantum efficiencies that are less than unity. We will use this as a springboard from which to address the classical versus quantum theories for single-mode linear attenuation and single-mode linear amplification. Optical Waveguide Tap with Ideal Photodetectors Slide 3 reprises the quantum photodetection theory of the optical waveguide tap that was introduced in Lecture 1 and analyzed in Lecture 10. Assuming that the input signal is a coherent state |asin � whose eigenvalue is real, and that the tap input mode 1is in a squeezed-vacuum state |0; µ, ν� with µ, ν > 0, we found that the s ignal input, signal output, and tap output signal-to-noise ratios were, SNRin = 4|asin |2 (1) SNRout =4T |asin |2 (2) T + (1 − T )(µ − ν)2 SNRtap = 4(1 − T )|asin |2 , (3) (1 − T ) + T (µ − ν)2 so that for ν sufficiently large—i.e., when there is sufficient quadrature noise squeez-ing to make the tap input’s noise contribution an insignificant component of the quadrature noise seen at the two output ports—we get SNRout ≈ SNRtap ≈ SNRin = 4|asin |2 . (4) This result is beyond the realm of semiclassical photodetection, in that semiclassical photodetection predicts SNRout + SNRtap = SNRin, (5) which is the performance that is obtained from the quantum theory when the tap input is in the vacuum state. The contrast between semiclassical and quantum be-havior of the optical waveguide tap is illustrated on Slide 4, which compares the SNR tradeoffs—for the semiclassical (vacuum-state tap input) and squeezed-state (10 dB squeezed tap input)—as the tap transmissivity is varied from T = 0 to T = 1. Un-fortunately, as we quickly showed in Lecture 10, sub-unity photodetector quantum efficiency can easily wash out the desirable non-classical behavior of the squeezed-state waveguide tap. Before quantifying that SNR-behavior loss, let us provide a more complete discussion of photodetection at sub-unity quantum efficiency. Single-Mode Photodetection with η < 1 Photodetectors Last time we introduced the following single-mode quantum photo detection mo de l for a detector whose quantum efficiency, η, was less than one: • Direct detection measures the number operator Nˆ� ≡ aˆ�†aˆ� associated with the photon annihilation operator aˆ� ≡√η aˆ + � 1 − η aˆη, (6) where 0 ≤ η < 1 is the photodetector’s quantum efficiency, ˆa is the annihila-tion operator of the single-mode field that is illuminating the photodetector’s light-sensitive region over the measurement interval, and ˆaη is the annihilation operator of a fictitious mode representing the loss associated with η < 1 pre-vailing. This fictitious mode is in its vacuum state, and its annihilation and creation operators commute with those associated with the illuminating field. 2• Balanced homodyne detection measures the quadrature operator ˆa�θ ≡ Re(ˆa�e−jθ), where θ is the phase of the local oscillator relative to the signal mode. • Balanced heterodyne detection realizes the probability operator-valued mea-surement associated with the annihilation operator ˆa�. Equivalently, balanced heterodyne detection can be said to provide a simultaneous measurement of the commuting observables that are the real and imaginary parts of √η (ˆa + ˆa†I ) + √1 − η (ˆaη + ˆa†), where ˆaI is the annihilation operator of the image-band field Iη that is illuminating the photodetector’s light-sensitive region over the measure-ment interval,1 and ˆaIη is its associated η < 1 loss operator All four of the modal annihilation operators—ˆa, ˆaI , ˆaη, and ˆaIη —commute with each other and with each other’s adjoint (creation) operator. Last time we were not particularly explicit about the semiclassical theory for single-mode photodetection with quantum efficiency η < 1, so let us list its specifica-tions now: • Semiclassical direct detection—for a s ingle-mode classical field with phasor a illuminating the photodetector’s light-sensitive region over the measurement interval—yields a final count, N�, that, conditioned on knowledge of a, is a Poisson-distributed random variable with mean η|a|2 , i.e., Pr( N� = n | a = α ) = (η|α|2)nn! e−η|α|2 , for n = 0, 1, 2, . . . (7) • Semiclassical balanced homodyne detection—for a single-mode classical field with phasor a illuminating the photodetector’s light-sensitive region over the measurement interval—produces a quadrature measurement outcome αθ�that, conditioned on knowledge of a, is a variance-1/4 Gaussian random variable with mean value √η aθ = √η Re(ae−jθ) • Semiclassical balanced heterodyne detection—for a single-mode classical field with phasor a illuminating the photodetector’s light-sensitive region over the measurement interval—yields quadrature measurement outcomes α1�and α�2 that, conditioned on knowledge of a, are a pair of statistically independent variance-1/2 Gaussian random variables with mean values √η a1 = √η Re(a) and √η a2 = √η Im(a), respectively. Because experiments