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Transcript

OPTI 544 Final Exam, 29 April, 2015 Jessen I Consider a 4-level laser with "32 >> "10 >> "21 . The lasing mode is resonant with the lasing ! transition at " = 1.0 #10 $6 m , the lasing medium is unpolarized ( p21 oriented randomly with respect to the laser field), and the spontaneous decay rate for the lasing transition is A21 = 1.0 ! 10 6 s "1 . The atom density is N = 1.0 "1016 m #3 , the cavity length is L = 0.1m , the ! cavity cross sectional area is A = 0.5 "10 #6 m 2 , and the cavity mirrors have transmissions and ! ! reflectivities T1 = 0.00 , R1 = 1.00 (high reflector) and T2 = 0.01, R2 = 0.99 (output coupler). There are no cavity losses other than ! through the output coupler. You may ! use results from class or homework without!re-deriving them. ! averaged damping rate " , the ! threshold ! gain gt , and the threshold inversion (a) !Find the time "N t . Give expressions in terms of the laser parameters as well as numerical values. (10%) (b) Assume the laser is operated high above threshold, such that !10 >> P, ! " >> ! 21 . ! Keeping only leading order!terms, find an expression for the intracavity photon flux " . Calculate the output power for a pumping rate P = 1.0 "10 8 . For the latter give both an expression in terms of the laser parameters and a numerical value. (10%) (c) Calculate the laser line width in Hz using the Schawlow-Townes formula. (5%)! ! ! II Consider a homodyne detection setup as shown on the right, consisting of a 50/50 beam splitter and a coherent state input ! (2) in port 2. The detector measures the photon number difference between ports 3 and 4, and its output signal is thus proportional to the observable Mˆ = aˆ3+ aˆ3 ! aˆ4+ aˆ4 . (a) Assume the field in port 1 is a coherent state ! (1) . Write down the density operators !ˆ1 , !ˆ 2 and !ˆ in that describe the state of mode 1, mode 2, and the joint 2mode state, respectively. Then use the density operator formalism to derive an expression for the expectation value Mˆ (10%) (b) Next, generalize the result in (a) to the case where !ˆ1 is a statistical mixture of coherent states ! j (1) , each occurring with probability P(! j ) . (10%) (c) Finally, using the results from (a) and (b), find the expectation value Mˆ for an input state that is a superposition of two coherent states, !ˆ1 = ( " 1 (1) + " 2 (1) )( " 1 (1) + " 2 (1) ) # , ( and for an input state that is an incoherent mixture of those same states, !ˆ1 = 12 " 1 (1) " 1 (1) + 12 " 2 (1) " 2 (1) . Compare and discuss the two results! (10%) 1 ) ! = 2 1 + Re !" "1 (1) " 2 (1) #$ , III In this problem we explore a quantum version of the electron oscillator model, i. e. an atom where the internal motion of the electron relative to the nucleus is described by a onedimensional quantum mechanical harmonic oscillator, instead of the classical harmonic oscillator used in the Lorentz model. The quantum oscillator has frequency !0 , creation and annihilation ˆ . operators bˆ † , bˆ , and position operator xˆ = x0 (bˆ† + b) (a) Consider the dipole matrix elements pmn = m pˆ n between the harmonic oscillator states. What are the selection rules for electric-dipole transitions, i. e. which pmn 's are zero and which are non-zero? Write out the corresponding matrix for the dipole operator in the basis of oscillator states n , for n ! 5 . (10%) We put the quantum electron oscillator inside an optical cavity, where it interacts with a quantized mode of the electromagnetic field with frequency ! , creation and annihilation ˆ . For simplicity, we assume the operators aˆ † , aˆ , and electric field operator Eˆ = E0 (aˆ † + a) electric field is parallel to the electron oscillator motion. (b) Write down the Hamiltonian for the system consisting of the electron oscillator and the quantized field mode, including the electric dipole interaction between the oscillator and the field. Compare to the Jaynes-Cummings Hamiltonian for a two-level atom coupled to a quantized mode of the electromagnetic field. (10%) (c) Now assume ! = !0 . Based on what you found in (b), what do you expect will happen if the system at t = 0 has one quantum of excitation in the electron oscillator, and zero quanta of excitation in the field? Sketch the populations of the states nosc = 1 n field = 0 and nosc = 0 n field = 1 as function of time. (10%) IV A Cs atom with transition wavelength ! = 852nm and excited state decay rate A21 = 3.28 !10 7 s is located at an antinode of the field in a resonant optical cavity with length l = 100 µm and effective cross sectional area A = ! (2 µm)2 . We assume the field polarization is parallel to the electric dipole matrix element, and that the cavity mirrors are perfect, lossless reflectors. (a) Calculate the vacuum Rabi frequency and compare to A21 . Is the system in a regime that permits the observation of coherent vacuum Rabi oscillations? (10%) (b) A weak probe beam is passed through the cavity. Sketch the transmission as the probe frequency ! is tuned around the atom/cavity resonance frequency ! 0 . What phenomenon are you seeing? (5%) 2