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 5 Fall 2008 Issued: Tuesday, October 7, 2008 Due: Tuesday, October 14, 2008 Reading: For normally-ordered forms: • W.H. Louisell, Quantum Statistical Properties of Radiation (McGraw-Hill, New York, 1973) Sect. 3.2. Problem 5.1 Here we shall derive the signal-to-noise ratio (SNR) optimality of squeezed states for quadarature measurements. Let ˆa and ˆa† be the annihilation and creation operators, respectively, of a quantum harmonic oscillator, and let ˆa1 ≡ Re(ˆa) and ˆa2 ≡ Im(ˆa) be the associated quadrature operators. We want to find the state |ψ� that maximizes the SNR of the ˆa1 measurement, SNR ≡�aˆ1�2 , �Δˆa 21� when the mean of the quadrature measurement must be positive �aˆ1� > 0, and the state must satisfy the average photon-number constraint, �aˆ†aˆ� ≤ N. (a) Express �aˆ†aˆ� in terms of the squared-means and the variances of the ˆa1 and aˆ2 measurements. Use this result to argue that the optimum state should have �aˆ2� = 0 and �aˆ†aˆ� = N, and thus satisfy a SNR = N + 1/2 − �Δˆ22�− 1. �Δˆa 21� (b) Use the result of (a) to show that the optimum state must be a minimum uncertainty product state for the Heisenberg inequality �Δˆa12��Δˆa2 22� ≥ 1/16. Optimize your resulting SNR expression over 0 ≤ �Δˆa1�. (c) Show that your optimum SNR e xpression from (b) is achieved by the squeezed state |β : µ, ν�, where β = � N(N + 1), µ = (N + 1)/√2N + 1, and ν = N/√2N + 1. (d) Compare the SNR achieved by your optimum squeezed state from (c) with that of the coherent state, |√N�, of the same average photon number. 1� � � Problem 5.2 Here we shall introduce the notion of normally-ordered forms. Consider a quantum harmonic oscillator with annihilation operator ˆa and creation op e rator ˆa†. Operators built up from Taylor’s series of the form, �∞ ∞F (ˆa†, aˆ) ≡ fnmaˆ†naˆm , n=0 m=0 are said to be in normal order, because (in each nm-term) all the creation operators stand to the left of all the annihilation operators. On the other hand, operators built up from Taylor’s series of the form, �∞ ∞G(ˆa, aˆ†) ≡ gnmaˆnaˆ†m , n=0 m=0 are said to be in anti-normal order, because (in each nm-term) all the annihilation operators stand to the left of all the creation operators. By (repeated) use of the [ˆa, aˆ†] commutator it is possible to convert a normally-ordered operator into an equivalent anti-normally ordered operator. Normal order is very convenient, as we shall see, when calculations are performed using coherent states (a) Find the normally-ordered form, F (n)(ˆa†, aˆ), and the anti-normally ordered form, F (a)(ˆa, aˆ†), of the operator Fˆ≡ aˆaˆ†aˆ. Find �α|Fˆ|α�, where |α� is a coherent state, and verify that it satisfies, �α|Fˆ|α� = F (n)(α∗, α), i.e., it equals the normally-ordered form of Fˆwith the classical complex numbers α∗ and α replacing the creation and annihilation operators ˆa† and ˆa, respectively. (b) Use the fact that the coherent states resolve the identity, d2α Iˆ= π |α��α|, to show that any operator Gˆis completely determined by its coherent-state matrix elements, �α Gˆβ�, via, | |�� d2α d2β Gˆ= π π �α|Gˆ|β�|α��β|. (c) Suppose that we regard F (n)(α∗, α) = �α|F (n)(ˆa†, aˆ)|α� from (a) to be a de-terminstic function of two independent classical arguments, α∗ and α. Show that, �α|Fˆ|β� = F (n)(α∗, β)�α|β�, = F (n)(α∗, β)e−(|α|2+|β|2)/2+α∗β , 2� � � � � � � for any two coherent states |α� and |β�. In conjunction with (b), this implies that Fˆis completely determined by its diagonal elements in the coherent-state expansion. Moreover, these diagonal elements are immediately available from the normally-ordered form of Fˆ(and vice versa). (d) Of particular interest for quantum optics work is the normally-ordered represen-tation for the density operator, ˆρ, which specifies the state (pure or mixed) of the oscillator. Show that ρ(n)(α∗, α) ≡ �α|ρˆ|α� satisfies the following conditions: ρ(n)(α∗, α) ≥ 0, for all α, d2α ρ(n)(α∗, α) = 1. π Thus, p(α1, α2) ≡ ρ(n)(α∗, α)/π, where α ≡ α1 + jα2, is a possible classical joint-probability density for two real-valued random variables. Problem 5.3 In classical probability theory, probability densities and characteristic functions are Fourier transform pairs, and some calculations are easier to perform in one domain than the other. In quantum mechanics, the density operator takes the place of the probability density and, because of commutation rules, there are several different characteristic functions that can be defined. Here we will introduce the three most important of these characteristic functions. Consider a quantum harmonic oscillator with annihilation operator ˆa, creation operator ˆa†, and density operator ˆρ. The anti-normally ordered characteristic function is defined to be, χρ (ζ∗, ζ) ≡ tr ρe−ζ∗aˆe ζaˆ† ,ˆAthe normally-ordered chacteristic function is defined to be, χρ (ζ∗, ζ) ≡ tr ρeζaˆ† e−ζ∗aˆˆ ,N and the Wigner characteristic function is defined to be, χρ (ζ∗, ζ) ≡ tr ρe−ζ∗aˆ+ζaˆ† W ˆ , In these expressions, ζ is a complex number whose real and imaginary parts are ζ1 and ζ2, respectively. (a) Let Aˆand Bˆbe non-commuting ope rators that commute with their commutator, i.e., [ˆA, ˆB, [ˆB]] = 0. A, [ˆB]] = [ ˆA, ˆ3� � � It can be shown that e B = e A e B e−[ˆB]/2 = e B e A eA, ˆ. Aˆ+ˆ ˆ ˆA, ˆ ˆ ˆ[ˆB]/2 Use this result to relate the three characteristic functions to one another. Find all three characteristic functions for the pure-state density op e rator ˆρ = |α��α|, where |α� is a coherent state. (b) Let ρ(n)(α∗, α) ≡ �α|ρˆ|α� be the diagonal matrix elements