Introduction Todays lecture has two purposes. First, we will - PDF document

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.453 Quantum Optical Communication Lecture Number 10 Fall 2016 Jeffrey H. Shapiro � c 2006, 2008, 2012, 2014 Date: Thursday, October 13, 2016 Single-mode signatures of non-classical light, and the squeezed-state waveguide tap Introduction Today’s lecture has two purposes. First, we will summarize our results on single-mode semiclassical and quantum photodetection theory by focusing on the non-classical sig- natures that can be seen in direct, homodyne, and heterodyne detection. This will include showing how a set of measurements on an ensemble of identically prepared quantum harmonic oscillators can be used to infer their state. Our second task, in today’s lecture, will be to revisit the squeezed-state waveguide tap that was described in Lecture 1. Armed with our understanding of single-mode photodetection, we will be able to derive the signal-to-noise ratio results that were presented in that open- ing lecture. Furthermore, we shall take this opportunity to extend our single-mode photodetection theory to include photodetectors whose quantum efficiencies are less than unity. Semiclassical versus Quantum Photodetection Slide 3 summarizes what we have learned about semiclassical photodetection of a single-mode field. Here, to simplify the notation, we have suppressed the spatial de- | ( t ) | 2 is now the energy T � pendence that we included in last lecture, so that � ω d t E 0 illuminating the photodetector’s light-sensitive region during the measurement inter- val 0 ≤ t ≤ T . Direct detection yields a final count N that, conditioned on knowledge of the phasor a , is a Poisson-distributed random variable with mean | a | 2 . Homo- dyne detection gives a quadrature value α θ that, conditioned on knowledge of a , is a variance-1/4 Gaussian-distributed random variable with mean value a θ = Re( ae − jθ ). Here, θ is √ the phase shift of the strong local oscillator relative to the signal, i.e., √ N LO e − j ( ωt − θ ) / E LO ( t ) = T for 0 ≤ t ≤ T , with N LO → ∞ . Heterodyne detection yields a pair of quadrature values, α 1 and α 2 , that, conditioned on knowledge of a , are statistically independent, variance-1/2, Gaussian-distributed random variables with mean values a 1 = Re( a ) and a 2 = Im( a ), respectively. Slide 4 summarizes what we have learned about quantum photodetection of a single-mode field. Here too we have simplified our notation by suppressing the spatial 1

dependence � that we included last time. Thus, a ˆ is still a photon annihilation operator, T ˆ ˆ d t E † so that � ω ( t ) E ( t ) is still the observable representing the total energy (above 0 the zero-point energy) illuminating the photodetector’s light-sensitive region during the measurement interval 0 ≤ t ≤ T . Direct detection yields a final count N that is ˆ ˆ † a equivalent to the quantum measurement of N ≡ a ˆ, the number operator associated with the excited mode. 1 Homodyne detection gives a quadrature value α θ that is ˆ − jθ ), where θ is again the equivalent to the quantum measurement of a ˆ θ = Re( ae phase shift of the strong local oscillator relative to the signal. Heterodyne detection yields a pair of quadrature values, α 1 and α 2 , such that α = α 1 + jα 2 is equivalent to the positive operator-valued measurement of a ˆ. Alternatively, we can say that α ˆ † is equivalent to the quantum measurement of a ˆ + a I . Here, a ˆ I is the annihilation √ ˆ I e − j ( ω − 2 ω IF ) t / operator of the unexcited (vacuum-state) image mode a T for 0 ≤ t ≤ ˆ † + a ˆ † † T . Because [ a ˆ+ a I , a ˆ I ] = 0, the real and imaginary parts of a ˆ+ a ˆ I are commuting observables. Thus they can be measured simultaneously, and their measurements are equivalent to the heterodyne outputs α 1 and α 2 , respectively, when the image mode is in its vacuum state. Non-classical Signatures in Photodetection Variances Three of the most important non-classical signatures—ways in which quantum pho- todetection theory makes predictions that are impossible to reproduce from semi- classical photodetection theory—appear in the variances of direct, homodyne, and heterodyne detection. Because it is very difficult, experimentally, to produce an opti- cal field that is completely deterministic (in classical electromagnetism) or in a pure state (quantum mechanically) it is important for us to make these variance compar- isons when the classical field on Slide 3 is allowed to have a be a complex-valued random variable whose joint probability density, for its real and imaginary parts, is p a ( α ). Likewise, we will take the quantum field on Slide 4 to be in a mixed state characterized by the density operator ρ ˆ a for its single excited mode. We now need to perform a little exercise in iterated expectation before we can compare and contrast the semiclassical and quantum photodetection variances of direct, homodyne, and heterodyne detection. Consider semiclassical direct detection when a is a complex-valued random vari- 1 Recall, from last lecture, that, in general, the final count N is equivalent to measurement of the ˆ T ˆ ˆ � d t E † ( t ) E ( t ). When the single mode shown on Slide 4 total photon number observable, N T = 0 ˆ is the only non-vacuum mode in the field, this reduces the outcome of the N T measurement to the ˆ † a outcome of the a ˆ measurement. 2

able with pdf p a ( α ). We then have that � d 2 α p a ( α ) Pr( N = n | a = α ) Pr( N = n ) = (1) d 2 α p a ( α ) | α | 2 n e −| α | 2 � = , for n = 0 , 1 , 2 , . . . (2) n ! From this it follows that the mean, mean-square, and variance of N are: ∞ ∞ α 2 n e −| α | 2 � n | | � � d 2 α p a ( α ) � N � = n Pr( N = n ) = (3) n ! n =0 n =0 � d 2 α | α | 2 p a ( α ) = �| a | 2 � , = (4) ∞ ∞ | α | 2 n e −| α | 2 � n 2 Pr( N = n ) = � � � N 2 � d 2 α p a ( α ) n 2 = (5) n ! n =0 n =0 � d 2 α ( | α | 2 + | α | 4 ) p a ( α ) = �| a | � + �| a | 4 � = � N � + �| a | 4 � , 2 = (6) and � ∆ N 2 � = � N 2 � − � N � 2 = � N � + ( �| a | 4 � − �| a | 2 � 2 ) = � N � + var( | a | 2 ) . (7) Now consider semiclassical homodyne detection when a is a complex-valued ran- dom variable with pdf p a ( α ). In this case, because α θ is Gaussian with mean a θ and variance 1/4 when a θ is known, we find the following results for the mean, mean- square, and variance of α θ : � d 2 � α � = β p a ( β ) β θ = � a θ � (8) θ � d 2 β p ( β )( β 2 + 1 / 4) = � a 2 � α 2 θ � = θ � + 1 / 4 (9) a θ � ∆ α 2 � α 2 2 2 θ � = θ � − � α θ � = � ∆ a θ � + 1 / 4 . (10) Finally, consider semiclassical heterodyne detection when a is a complex-valued random variable with pdf p a ( α ). Here, using the conditional statistics given earlier, the mean, mean-square, and variance of the quadratures, α k for k = 1 , 2, are found 3