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 5 Fall 2008 Jeffrey H. Shapiro c�2006, 2008 Date: Thursday, September 18, 2008 Reading: For coherent states and minimum uncertainty states: • C.C. Gerry and P.L. Knight, Introductory Quantum Optics (Cambridge Uni-versity Press, Cambridge, 2005) Sects. 3.1, 3.5, 3.6. • R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 1973) chapter 7. • L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995) Sects. 11.1–11.6. Introduction Today we continue our development of the quantum harmonic oscillator, with a pri-mary focus on measurement statistics and the transition to the classical limit of noiseless oscillation. In particular, we’ll work with the time-dependent annihilation operator, aˆ(t) = ˆae−jωt, for t ≥ 0, (1) its quadrature components1 aˆ1(t) ≡ Re[ˆa(t)] = Re(ˆae−jωt) and aˆ2(t) ≡ Im[ˆa(t)] = Im(ˆae−jωt), (2) and the number operator Nˆ= ˆa†(t)ˆ a†ˆ (3) a(t) = ˆ a. 1There are three equivalent representations for a real-valued classical sinusoid, x(t), of frequency ω: (1) the phasor (complex-amplitude) representation, x(t) = Re(xe−jωt), where x is a complex number; (2) the quadrature-component representation, x(t) = xc cos(ωt) + xs sin(ωt), where xc and xs are real numbers; and (3) the amplitude and phase representation, x(t) = A cos(ωt − φ), where A is a non-negative real number and φ is a real number. Taking x = xc + jxs = Aejφ establishes the connections between these representations. We are using the first two in our quantum treatment of the harmonic oscillator. There are subtleties—which we may go into later—in trying to use the amplitude and phase representation for the quantum harmonic oscillator. 1� � � In terms of the number operator’s orthonormal eigenkets, {|n�}, and associated eigen-values, {n}, we have �∞ ∞Nˆ= n|n��n| and Iˆ= |n��n|, (4) n=0 n=0 as well as aˆ = �∞√n |n − 1��n| and aˆ† = ∞√n + 1 |n + 1��n|, (5) n=1 n=0 which will also be of use in what follows. Although we will not make much use of the Hamiltonian in today’s lecture, we note that its eigenket-eigenvalue expansion is ∞Hˆ= �ω(Nˆ+ 1/2) = �ω(n + 1/2)|n��n|, (6) n=0 where the minimum energy, �ω/2, which is associated with the zero-quantum (zero-photon) state |0�, is called the zero-point energy. What we will develop today is very much in keeping with a basic principle of quantum mechanics: the state of a quantum system and the measurement that is made on that system determine the statistics of the resulting measurement outcomes. We will see that the zero-point energy plays a key role in the quadrature-measurement statistics. Quadrature-Measurement Statistics for Number States Slide 5 reprises the classical versus quantum picture that we presented last time for the quadrature behavior of classical and quantum harmonic oscillators. We were a little vague, last time, about the meaning of the phasor and time-evolution plots for the quantum case, so let’s try to make them prec ise for the case of a quantum harmonic oscillator that is in its number state What we’d like to see is that |n�. classical physics—noiseless sinusoidal oscillation—emerges as quantum behavior in the limit of large quantum numbers. So, we’ll derive the quadrature-measurement statistics when the state is |n� and see what happens as n → ∞. Before doing so, let’s note that desired classical limit behavior is already exhibited by the number state |n� insofar as energy-measurement statistics are concerned. Because |n� is an eigenket of the Hamiltonian with eigenvalue �ω(n + 1/2) we know that Pr(Hˆmeasurement = �ω(n + 1/2) | state = |n�) = 1, (7) so the energy is always quantize d. However, as n → ∞ the �ω granularity becomes imperceptibly small, compared to the energy in the state. At this point in our development, we don’t have enough theoretical machinery to fully characterize the quadrature measurement statistics. So, we will limit our 2� � � � � attention to the mean values and variances of the the quadrature measurements. For the mean values we have that �n|aˆ(t)|n� = �n |aˆ|n�e−jωt = �n| ∞√m |m − 1��m| |n�e−jωt = 0, (8) m=0 from which it follows that �aˆ1(t)� = �aˆ2(t)� = 0 when the oscillator is in a number state. Evidently, the number state cannot give us noiseless classical oscillation in the limit n → ∞, because its mean value for both quadratures is always zero. Despite this failure, it is still worth looking into the variance of the quadrature measurements when the oscillator is in a number state. Now we find that [ˆa(t) + ˆa†(t)]2 �n|Δˆa12(t)|n� = �n|aˆ12(t)|n� = �n| 4 |n� (9) = �n|aˆ2(t)|n� + �n|aˆ(t)ˆa†(t)|n� + �n|aˆ†(t)ˆa(t)|n� + �n|aˆ†2(t)|n� (10) 4 =2�n|aˆ†aˆ|n� + 1 = 2n + 1 . (11) 4 4 A similar calculation—left as an exercise for the reader—leads to 2n + 1 �n|Δˆa 22(t)|n� =4 . (12) Thus we see that the number state has equal uncertainties is each quadrature with an uncertainty product, � �22n + 1 1 �Δˆa12(t)��Δˆa22(t)� =4 ≥ 16, (13) with equality if and only if n = 0. So, the zero-photon (vacuum) state |0� is a min-imum uncertainty-product state for the quadrature components of the annihilation operator, but all the other number states have higher than minimum uncertainty products. Slide 7 is a pictorial summary of what we have just learned. Classically, the oscillator can undergo noiseless sinusoidal oscillation, as illustrated by the phase space and time-evolution plots shown on the left-hand side of this slide. For a quantum oscillator that’s in a number state |n�, the mean value of the annihilation operator is zero, and the variances of the quadratures are equal and their product is larger (for n ≥ 1) than that for a