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 7 Fall 2008 Jeffrey H. Shapiro c�2006, 2008 Date: Thursday, September 25, 2008 Reading: For quantum characteristic functions: • C.C. Gerry and P.L. Knight, Introductory Quantum Optics (Cambridge Uni-versity Press, Cambridge, 2005) Sect. 3.8. • W.H. Louisell, Quantum Statistical Properties of Radiation (McGraw-Hill, New York, 1973) Sect. 3.4. (On reserve at Barker library.) For positive operator-valued measurements: • M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Infor-mation (Cambridge University Press, Cambridge, 2000) Sect. 2.2.6. Introduction Today we will continue to explore the quadrature measurement statistics of the quan-tum harmonic oscillator, and use that exercise to introduce the notion of quantum characteristic functions. These characteristic functions, like their counterparts in classical probability theory, are useful calculational tools, as we will see when later when we study the quantum noise behavior of linear systems that have loss or gain, i.e., attenuators and amplifiers. Today we will also provide the positive operator-valued measurement (POVM) description for “measuring” the annihilation operator, aˆ. POVMs are extremely important in quantum information science, in that they are more general than observables. Lest you think that they are mere mathematical generalizations, it is worth noting now that, for a single-mode optical field, the ˆa POVM has a physical realization: optical heterodyne detection. Quadrature-Measurement Statistics Tables 1 and 2 summarize most of what we have learned so far about quantum har-monic oscillator’s quadrature-measurement statistics. Slides 4 and 5 give qualitative pictures of ˆa1(t) measurements when the oscillator is in a number state or coherent 1� State �aˆ(t)�0|n� αe−jωt |α� (µ∗β − νβ∗)e−jωt |β; µ, ν� Table 1: Mean value of ˆa(t) for number states, coherent states, and squeezed states. The real and imaginary parts of these table entries are the quadrature-measurement mean values, �aˆ1(t)� and �aˆ2(t)�, respectively. State �Δˆa12(t)� �Δˆa22(t)�(2n + 1)/4 (2n + 1)/4|n� 1/4 1/4|α� |µ − νe−2jωt|2/4 |µ + νe−2jωt|2/4|β; µ, ν� Table 2: Quadrature-measurement variances for number states, coherent states, and squeezed states. state (Slide 4), or an amplitude-squeezed or phase-squeezed state (Slide 5). In ad-dition to these results, we also know—from our wave function analysis of minimum uncertainty-product s tates—that the probability density functions for the quadrature measurements are Gaussian, when the oscillator is in a coherent state or a squeezed state. We have yet to determine what this probability density is when the oscillator is in a number state, nor have we given a very clear and explicit description for the phase space pictures shown on Slides 4 and 5. We will remedy both of these deficiencies by means of quantum characteristic functions. Quantum Characteristic Functions It is appropriate to begin our discussion of quantum characteristic functions by step-ping back to review what we know about classical characteristic functions. Suppose that x is a real-valued, classical random variable whose probability density function is px(X).1 The characteristic function of x, ∞ Mx(jv) ≡ �ejvx� = dX ejvX px(X), (1) −∞ 1Probability density functions (pdfs) are natural ways to specify the statistics of a continuous random variable. However, if we allow pdfs to contain impulses, then they can be used for discrete random variables as well. Hence, although we have chosen to use pdf notation here, our remarks apply to all real-valued, classical random variables. 2� � �� � �� � � � � � is equivalent to the pdf px(X) in that it provides a complete statistical characterization of the random variable. This can be seen from the inverse relation, ∞ dv (jv)e−jvX px(X) = Mx. (2) 2π−∞ Indeed, Eqs. (1) and (2) show that px(X) and Mx(jv) are a Fourier transform pair. On Problem Set 1 you exercised the key properties of the classical characteristic function, so no further review will be given here. What we shall do is use the char-acteristic equation approach to determine the quadrature-measurement statistics of the harmonic oscillator. Suppose that we measure aˆ1(t) = Re[ˆa(t)] = Re(ˆae−jωt) = ˆa1 cos(ωt) + ˆa2 sin(ωt), (3) where Re(ˆa) = aˆ1 and Im(ˆa) = aˆ2, for some fixed particular value of time, t. Let a1(t) denote the classical random variable that results from this measurement. Then the classical characteristic function for a1(t), when the oscillator’s state is |ψ�, can be found as follows, ∞ jvα1 ∞ jvα1Ma1(t)(jv) = dα1 e pa1(t)(α1) = �ψ| dα1 e |α1�t1 t1 �α1| |ψ�, (4) −∞ −∞ where {|α1�t1 } are the eigenkets of ˆa1(t), i.e., they are the (delta-function) orthonor-mal solutions to aˆ1(t)|α1�t1 = α1|α1�t1 , for −∞ < α1 < ∞. (5) Expanding ejvα1 in its Taylor series, we have that ∞∞(jv)n Ma1(t)(jv) = �ψ| dα1 n! αn 1 |α1�t1 t1 �α1| |ψ�. (6) n=0−∞ Next, we interchange the order of integration and summation, use the fact that the {|α1�t1 } diagonalize ˆan(t) for n = 0, 1, 2, . . . , and obtain 1 ∞(jv)n n jvaˆ1(t)Ma1(t)(jv) = �ψ| n! aˆ1 (t) |ψ� = �e �, (7) n=0 where the exponential of an operator (here aˆ1(t)) is defined by the Taylor series expansion. This final answer seems almost obvious. We said that a1(t) at time t is the classical random variable that results from measurement of ˆa1(t) at time t. It should follow that �ejva1(t)� = �ejvaˆ1(t)�. (8) 3� � � � Nevertheless, you should bear in mind that the averaging on the left-hand side of this equation is a classical probability average, see (4), whereas the averaging on the right-hand side of this equation is a quantum average, see (7). We’ll see more of this equivalence between the statistics of classical random variables and quantum