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 2 Fall 2008 Issued: Thursday, September 11, 2008 Due: Thursday, September 18, 2008 Supplementary Reading: For basic Dirac notation quantum mechanics: • Section 2.2 of M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information • Sections 1.1–1.16 of W.H. Louisell, Quantum Statistical Properties of Radiation. Problem 2.1 Here we shall explore the use of wave plates to perform polarization transformations on a single photon. The polarization state of a +z-propagating, frequency-ω photon at z = 0 is characterized by a complex-valued unit vector, αxi ≡ αy , (1) such that Re[ie−jωt] describes the time evolution of the photon at z = 0 w here i†i = |αx|2 + |αy|2 = 1, with i† ≡ αx ∗ αy ∗ , is the unit-length condition for i. (a) For our monochromatic photon, propagation through L m of material in which light of arbitrary polarization propagates at velocity c/n, where n is the ma-terial’s refractive index at frequency ω, leads to a phase delay φ = ωnL/c. Thus the time evolution of the photon at z = L is given by Re[ie−jω(t−nL/c)] = Re[i�e−jωt], where i� ≡ iejφ. Show that the polarization state i� is identical to the polarization state i, i.e., the contour traced out by Re[ie−jωt] in the x-y plane is identical to that traced out by Re[i�e−jωt]. (b) Wave plates are made of birefringent materials, i.e., materials which have differ-ent velocities of propagation for light polarized along their principal axes. When these axes are aligned with x and y, respectively, propagation of a monochro-matic photon—whose polarization at z = 0 is given by Eq. (1)—results in a new polarization at z = L, � � αxejφx i� = αyejφy , (2) 1� � � � � � � � � � where φx ≡ ωnxL/c and φy ≡ ωnyL/c give the respective phase shifts in terms of the propagation velocities c/nx and c/ny along the x and the y axes. A quarter-wave plate (QWP) is one for which φx − φy = π/2. Suppose that a photon of +45◦ linear polarization, 1/√2 i = 1/√2 is the input to a QWP whose principal axes are aligned w ith x and y, respec-tive ly. Show that the output of this QWP is circularly polarized. Suppose that this circularly polarized output is the input to another QWP whose principal axes are aligned with x and y, respectively. What is the result-ing polarization of the output from this QWP? (c) A half-wave plate (HWP) is one for which the phase difference between propa-gation along its principal axes is π rad. Suppose that a photon of polarization 1 i = 0 is the input to an HWP whose “fast” (low refractive index) axis is parallel to the unit vector �ifast = �ix cos(θ) +�iy sin(θ), and whose “slow” (high refractive index) axis is parallel to the unit vector �islow = −�ix sin(θ) +�iy cos(θ). What is the polarization state at the output of the HWP? (d) Suppose we wish to transform an x-polarized input photon, 1 iin = 0 into an output photon of polarization state, αxiout = αy Show that this can be done by first using a half-wave plate to transform iin to iHWP = |ααxy| ,| | and then using another wave plate, whose principal axes are aligned with x and y respectively, and whose propagation phase difference φx − φy is chosen appropriately, to transform iHWP into iout. 2� � � � (e) The polarization transformation scheme you verified in (d) is not a convenient experimental approach, b e cause it requires a phase plate with a controllable propagation phase difference φx − φy. He re we consider an alternative approach that only needs a QWP and an HWP. Suppose that we wish to transform an arbitrary given input polarization αxiin = ,αy into horizontal polarization � � 1 iout = . 0 Because iin is, in general, an elliptical polarization, there must be a Cartesian coordinate system, (x�, y�), in which this input polarization takes the form α�xiin = ,αy� with αy� = jkα�, for k a positive constant. Use this fact to argue that a QWP, xwith its fast axis aligned in the y� direction, will convert iin into linear polariza-tion, after which an HWP can be used to obtain an iout that is linearly polarized in the x direction. Using these results, explain how propagation through an HWP and a QWP can be used to transform an initially x-polarized photon into any desired polarization state. Problem 2.2 Here we shall introduce the Poincar´e sphere, viz., a 3-D real representation for the 2-D polarization state � � αxi = ,αy of a +z-propagating, frequency-ω photon. Define a real-valued 3-vector, r as follows, ⎡ ⎤ ⎡ ⎤ r1 2Re[αx∗ αy] r ≡ ⎣ r2 ⎦ = ⎣ 2Im[αx∗ αy] ⎦ . r3 |αx|2 − |αy|2 (a) Show that knowledge of r is equivalent to knowledge of i, i.e., r completely describes photon’s polarization. (b) Show that i†i = 1 implies that rT r ≡ r2 + r2 + r2 = 1, i.e., the photon’s 1 2 3 polarization-state lies on the unit-sphere (called the Poincar´e sphere) in r space. (c) Where do x and y polarizations appear on the Poincar´e sphere? Where do left and right circular polarizations appear on this sphere? 3� � � � �� � � Problem 2.3 Let Aˆbe a linear operator that maps kets in the Hilbert space H into other kets in this space, i.e., for every |x� ∈ H, there is a |y� ∈ H that satisfies |y� = Aˆ|x�. Let {|φn� : n = 1, 2, . . . , } be an arbitrary complete orthonormal (CON) set of kets in H, i.e., 1, for n = m, �φn|φm� = δnm ≡ 0, for n =� m. ∞Iˆ= ,|φn��φn|n=1 where Iˆis the identity operator on H. (a) Show that the operator Aˆis completely characterized by its {φn} matrix ele-ments, viz., {�φm|Aˆ|φn� : 1 ≤ n, m ≤ ∞}, by proving that ∞ ∞Aˆ= �φm|Aˆ|φn�|φm��φn|m=1 n=1 (b) Let |x� = ∞ n=1 xn|φn� be an arbitrary ket in H and let |y� = Aˆ|x�. Show that ∞ ∞|y� = m=1 ym|φm� with ym = n=1 �φm|Aˆ|φn�xn, for 1 ≤ n, m < ∞. (c)