SECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C

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1 2752 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C2: LASER SCIENCE AND QUANTUM INFORMATION PROCESSING TRINITY TERM 2013 Friday, 14 June, 2.30 pm 5.45 pm 15 minutes reading time Answer four questions. Start the answer to each question in a fresh book. A list of physical constants and conversion factors accompanies this paper. The numbers in the margin indicate the weight that the Examiners anticipate assigning to each part of the question. Do NOT turn over until told that you may do so. 1

2 1. Discuss what is meant by the longitudinal and transverse modes of a laser cavity. How may laser oscillation be restricted to the lowest-order transverse mode of the cavity? [5] For a certain laser the gain exceeds the cavity and other losses for a range of frequencies ν osc. Show that single longitudinal mode operation is possible if the cavity length L c satisfies L c < L 0 c = c. ν osc You may assume that the longitudinal mode spacing of the cavity is the same as for a cavity comprising two plane, parallel mirrors separated by a distance L c. Sketch the frequency dependence of the laser gain and the frequencies of the longitudinal modes when L c = L 0 c. [4] Other considerations mean that the cavity length must be greater than L 0 c. Show that single longitudinal mode operation may still be achieved by inserting a Fabry-Perot etalon of finesse F = ν FSR / ν 1/2 if L c < (2F) L 0 c. State clearly the conditions on the etalon free-spectral range, ν FSR, and the full-width at half maximum, ν 1/2, of an etalon transmission peak which you have assumed to suppress unwanted longitudinal modes. Illustrate these conditions with the aid of a sketch. An inhomogeneously broadened laser can oscillate over a range of frequencies ν osc = 3 GHz and has a cavity length of 1 m. Fused silica etalon plates are available of thickness 5 mm, 10 mm and 20 mm. Which of these plates would be most suitable to restrict oscillation to a single longitudinal mode? For this plate, suggest a suitable value for the finesse. The refractive index of silica at the operating wavelength of the laser is [9] Describe qualitatively the fundamental processes which limit the linewidth of a laser operating on a single longitudinal mode. What other processes can increase the linewidth above this value in practice? Briefly describe one means by which the output of a single longitudinal mode laser can be locked to the frequency of an atomic or molecular absorption transition and sketch the apparatus required. [7]

3 2. List three crystal properties relevant for the efficient generation of a second harmonic wave. Explain with the aid of a sketch how the process of phase-matching can be achieved in a negative uniaxial crystal. [5] The output of a Nd:YAG laser at 1064 nm is to be converted to radiation of wavelength 532 nm using potassium dihydrogen phosphate (KDP). For this crystal the polarisation oscillating at the second harmonic frequency is given by P (2ω) = ε d d d 36 E 2 x E 2 y E 2 z 2E y E z 2E x E z 2E x E y. By means of a diagram show that for this negative uniaxial crystal type I phasematching, for which the fundamental propagates as the ordinary wave while the second harmonic propagates as an extraordinary wave, can be arranged if the fundamental has field components in the xy plane. Given that the variation of the extraordinary refractive index with the angle θ between the propagation direction and the optical axis can be expressed as 1 n 2 e (θ) = cos2 θ no 2 + sin2 θ n 2 e determine the value of θ and the angle ϕ to the x-axis which maximizes the generated second harmonic light. [7] Determine for this case the angular acceptance δθ 1/2 for which the conversion efficiency is half the peak value for a crystal of length 1 cm. [You may take (sin ς/ς) 2 = 0.5 when ς = 1.39.] [8] Why is the direction of Poynting s vector for the second harmonic wave not collinear with the k vector? Show that the walk-off angle ρ is given by tan(θ + ρ) = ( no n e ) 2 tan θ. Determine the angle ρ for this case. [5] [The relevant refractive indices for KDP are as follows: for λ = 1064 nm: n o = , n e = ; for λ = 532 nm: n o = , n e = ] [Turn over]

4 3. The Helmholtz equation describing the spatial propagation in two dimensions of an optical signal E(x, z) with angular frequency ω is [ 2 x + 2 z + k 2 ]E = 0, where k = ω/c is the wavevector and c is the speed of light. For a collimated beam travelling along the z-axis, we have E(x, z) = U(x, z)e ikz, where U is a slowly varying amplitude such that z U ku. Derive the paraxial wave equation satisfied by U, and solve it by Fourier transformation to arrive at the Kirchhoff one-dimensional diffraction integral U(x, z) = ( ik/z) 1/2 U(x, 0)e ik(x x ) 2 /2z dx. [10] What condition on the initial transverse size δx of the beam must apply such that the transverse intensity profile I U(x, z) 2 of the diffracted field is given by the squared modulus of the Fourier transform of the initial field U(x, 0). What is the name of this regime? [3] Figure 1: (a) A collimated beam diffracts as it propagates along the z-axis. (b) An optical pulse is directed through a phase modulator. We now neglect diffraction but consider a plane wave optical pulse with timevarying amplitude E(τ), where τ = t z/c is the local time (the time in a frame travelling with the pulse). The pulse enters an electro-optic phase modulator (see part (b) of Figure 1). The phase modulator imparts a sinusoidally oscillating temporal phase ϕ(τ) = A cos(ωτ). The pulse is centred at τ = 0 and its duration is much smaller than the modulation period 1/Ω. Show that the field amplitude emerging from the modulator has the form E (τ) = E(τ) exp( i ϕτ 2 /2), where ϕ = AΩ 2 and we have dropped an unimportant constant phase. [3] The pulse E now enters a spectrometer. Show by Fourier transforming E (τ) Ẽ (ω) that measuring the spectral intensity I Ẽ (ω) 2 of the pulse reveals its temporal shape, provided that ϕ δω 2 /8π, where δω is the spectral bandwidth of the pulse. How might this observation be useful? [9] [You may find the following formula helpful: e iαx2 e ixy dx = ( iπ/α) 1/2 e iy2 /4α.]

5 4. Write down the formula for a single mode optical coherent state and prove that it is a minimum uncertainty state. [5] We are given a beamsplitter with the real reflection and transmission amplitudes r and t respectively. Prove that if a coherent state α enters one input port (the other one being the vacuum) it splits at the output into two coherent states rα and tα. [7] A thermal state, ρ T, of the single frequency mode ω of light is a Boltzmann-Gibbs mixture of number states of the form ρ T = n p n n n, where p n = e n ω/kbt /Z and Z = (1 e ω/kbt ) 1, k B being Boltzmann s constant and T the temperature. Show that in terms of coherent states the thermal state can be written as ρ T = P (α) α α d 2 α. Find the function P (α). You may use the integral e α 2 π 2 (1 + n) d 2 β e β 2 1+1/ n e βα +β α = 1 α 2 nπ e n where n = 1/(e ω/kt 1) is the average number of photons in ρ T. [7] Hence write down the output from the above beamsplitter if a thermal state enters one port (and the vacuum the other). Using the above results, or otherwise, explain the behaviour of a coherent state entering a Mach-Zehnder interferometer (where both beamsplitters are fifty-fifty) and compare this with the behaviour of an input thermal state. [6] [Turn over]

6 5. State the five DiVincenzo criteria for implementing a quantum computer and describe briefly how they can be achieved (or why they cannot be achieved) in liquid state NMR. [10] Consider the propagator for a rotation of a qubit by an angle θ around an axis in the xy plane at an angle of ϕ from the x axis U(θ, ϕ) = exp [ iθ (σ x cos ϕ + σ y sin ϕ) /2]. Show that a 180 rotation with phase angle ϕ 1 followed by a 180 rotation with phase angle ϕ 2 is equivalent to a z rotation, and find the rotation angle. Hence give explicit networks that implement the quantum logic gates H and T using only 90 rotations about axes in the xy plane. (You may ignore global phase terms.) [6] Consider a system of two spin-1/2 nuclei with gyromagnetic ratios γ 1 and γ 2 in a magnetic field of strength B along the z axis. Write down the Hamiltonian (you may neglect the spin spin coupling) and hence find the thermal equilibrium density matrix in the high temperature approximation. [4] During a crush gradient pulse the main magnetic field is made inhomogeneous over the NMR sample, so that the Hamiltonian varies over the spatial ensemble of spins. Show that in a heteronuclear two-spin system this will remove any off-diagonal elements from the density matrix written in the Zeeman basis (the computational basis). Explain why this is not true for a homonuclear spin system, and comment on the significance of this. [5]

7 6. Calculate the state of the qubit after each gate in the network below, where the single-qubit phase gate ϕ z performs the operations 0 0 and 1 e iϕ 1. 0 H ϕ z H How can this network be used to estimate the value of ϕ? [7] The sensitivity of phase measurements can be enhanced using entangled states. In some physical systems it is simple to implement parallel gates between a single central qubit and n ancilla qubits, while the individual ancillas cannot be individually addressed. Such systems can be described using a compact network notation, illustrated below for the case n = 2, where the central qubit is drawn on the upper line and the ancilla qubits on the lower line(s). ψ X ψ X 0 H = 0 0 H H Find the state of the system after each gate in the network below for the case n = α z β z and hence deduce the final state of the system for any value of n. [7] It is sometimes desirable to suppress evolution under the phase gate applied to the first qubit. Find the state of the system after each gate for the network below for the cases n = 1 and n = α z β z and hence deduce the final state of the system for any value of n. [11] [Turn over]

8 7. Alice and Bob share classical messages x i and y j with joint probabilities p(x i, y j ). Give definitions of the random variables X and Y describing Alice s and Bob s messages, respectively. Define the conditional entropy H(X Y ) and the mutual information H(X : Y ). State the mutual information for the case where Alice and Bob each possess the message 0 and 1 with equal probability and Bob s message is (i) perfectly correlated and (ii) completely uncorrelated to Alice s messages. [6] When Alice and Bob share quantum messages described by the density operator ρ AB the mutual information S(ρ A : ρ B ) can be written as S(ρ A : ρ B ) = S(ρ A ) + S(ρ B ) S(ρ AB ). Give definitions of ρ A, ρ B and S(ρ) in this expression. Calculate S(ρ A : ρ B ) for the state ρ AB = ( )/2, where ± = ( 0 ± 1 )/ 2. Is ρ AB a pure state or a mixed state? Is it entangled? [7] Now consider the case where Alice and Bob measure their qubits in the computational basis with the measurement outcomes forming classical messages x i and y j. What is the mutual information between these classical messages? How does the mutual information change if Bob decides to apply (i) a Hadamard gate or (ii) a 45 y rotation to his qubit which transforms + c c 1, and 1 c 0 + c + 1, where c ± = 1 2 ± 2, 2 before the measurements in the computational basis? [8] It can be shown that the application of the 45 y rotation before measurement in the computational basis yields the maximum possible mutual information H(X : Y ) between classical messages obtainable from ρ AB. Hence calculate the minimum possible difference δ min = S(ρ A : ρ B ) H(X : Y ) and discuss the physical implications of δ min > 0. The quantity δ min is the quantum discord of ρ AB. [4]

9 8. An observable T takes on two distinct values ±1 and can be measured at three times t 1, t 2 and t 3. Consider the correlation function K formed from measurement outcomes T (t i ) and given by K = T (t 2 )T (t 1 ) + T (t 3 )T (t 2 ) T (t 3 )T (t 1 ), where denotes the expectation value obtained when averaging measurements over a large number of identically prepared samples. Assume that T has a definite value, independently of whether it is measured or not (realism), and is measured non-invasively. Show that these assumptions lead to the Leggett-Garg inequality K 1, starting from a table with all eight possible combinations of measured values T (t i ). [6] A two level atom with ground state 0 g, excited state 1 e and Hamiltonian H atom = ω eg σ z /2 evolves in a resonant laser field with interaction Hamiltonian H int = Ω cos(ω L t)σ x. Write down the resonance condition relating ω eg and ω L and list two physical quantities which determine the Rabi frequency Ω. Briefly describe the transformation to the rotating frame and any approximation that is necessary to arrive at the Hamiltonian H = Ω 2 σ x, which describes the dynamics of the atom in the rotating frame. Express the evolution operator U(t, t 0 ) = e ih(t t 0)/ induced by this Hamiltonian in terms of Pauli matrices. [6] The atom starts at time t 0 in state g. Calculate the correlation function σ z (t 0 + τ)σ z (t 0 ) g g σ z (t 0 + τ)σ z (t 0 ) g, where σ z (t) is the Heisenberg-picture operator σ z (t) = U (t, t 0 )σ z U(t, t 0 ). Similarly work out σ z (t 0 + τ)σ z (t 0 ) e for the situation where the atom starts in state e at time t 0. Hence or otherwise show that the correlation function σ z (t 0 + τ)σ z (t 0 ) = cos(ωτ) for an atom which is measured in the computational basis at time t 0 and a later time t 0 + τ. [6] Identify T = σ z and calculate K for an atom in a resonant laser field. Consider the value of K in the case t 1 = 0, t 2 = τ and t 3 = 2τ. Work out the values of τ for which the maximum K is achieved, and show that this maximum value of K violates the Leggett-Garg inequality. [7] [LAST PAGE]

Edward S. Rogers Sr. Department of Electrical and Computer Engineering. ECE318S Fundamentals of Optics. Final Exam. April 16, 2007.

Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE318S Fundamentals of Optics Final Exam April 16, 2007 Exam Type: D (Close-book + two double-sided aid sheets + a non-programmable