QUANTUM THEORY OF LIGHT, Summaries of Quantum Mechanics

Download QUANTUM THEORY OF LIGHT and more Summaries Quantum Mechanics in PDF only on Docsity! QUANTUM THEORY OF LIGHT EECS 638/PHYS 542/AP609 FINAL EXAMINATION Instructor: Professor S.C. Rand Date: April 25, 2001 Duration: 2.5 hours PLEASE read over the entire examination before you start. DO ALL QUESTIONS and show all your work in submitted material to be eligible for partial credit. Useful formulae: cos(2x)=cos2(x)-sin2(x) sin(2x)=2sin(x)cos(x) The Pauli spin matrices: σ z = −       1 0 0 1 ; σ x =       0 1 1 0 ; σ y i i= −      0 0 ; σ σ σ± = ± 1 2 ( )x yi ________________________________________________________________________ Honor Code: I have neither given nor received aid on this exam. __ _____ Name (Print) _____ Signature 1. (15 marks total) A system with two closely-spaced excited levels 2 and 3 lying above the ground state 1 has allowed transitions 1 ↔ 2 and 1 ↔ 3 , with corresponding dipole moments µ12≠0 and µ13≠0. Assume µ23=0. The energy separation of 2 and 3 is ∆ω ω ω= −3 2 . An ultrashort pulse of light with a frequency bandwidth in excess of ∆ω excites the system at a carrier frequency ωL at time t=0 (ω ω ω ω ωL ≅ − ≅ −2 1 3 1 ), generating several first-order coherences. (a) With reference to the nine density matrix elements of this 3-level system, identify the non-zero, first-order coherences. (3 marks) (b) Give a formal expression for the microscopic polarization field p as a trace over appropriate operators, and write out the non-zero terms explicitly. (3 marks) (c) Given the microscopic polarization of the medium, from what equations must the radiative signal field Es be calculated in general to account for macroscopic properties of the medium or the field, and what is the value of Es here (assuming for simplicity that there are no inhomogeneities such as position-dependent saturation to take into account)? (2 marks) ∆ω 1 2 3 wL Problem 2 (Cont'd): (c) Show that electric-dipole transitions involving the +,n state are not allowed at the Rabi frequency Ωn g n= +2 1 . (2 marks) (d) Suppose that the laser is modulated* to produce two frequency components at ωL and (ωL - Ωn). Draw a diagram in the dressed state picture of a 2-photon transition −,n → +,n utilizing one photon at each frequency. (1 mark) _____________ * Assume the modulation frequency falls within the single mode bandwidth (Ω<<c∆k), so the problem still involves only the original mode. Problem 2 (Cont'd): (e) Determine whether the 2-photon transition of part (d) is allowed. (4 marks) (f) For the sake of argument, suppose the frequency component of the modulated laser field at (ωL - Ωn) is intense, like the component at ωL, and that the −,n → +,n transition can be strongly driven on resonance by the 2-photon transition. Could the quasi-stationary states −,n and +,n of the dressed atom themselves each split under these circumstances? Why or why not? (2 marks) 3. General solutions for the probability amplitudes ca,n(t) and cb,n+1(t) of a 2-level atom interacting with a single mode light field (where the atom may be in the excited state with n photons present or in the ground state with n+1 photons respectively) are: c t c t i t ig n c t ea n a n n n n n b n n i t , , , /( ) ( ) cos sin ( ) sin=     −           − +           +0 2 2 2 1 0 21 2Ω ∆ Ω Ω Ω Ω ∆ , and c t c t i t ig n c t eb n b n n n n n a n n i t , , , /( ) ( ) cos sin ( ) sin+ + −=     +           − +           1 1 20 2 2 2 1 0 2 Ω ∆ Ω Ω Ω Ω ∆ , Ω ∆n g n2 2 24 1= + +( ) is the square of the Rabi frequency at detuning ∆ for a field containing n photons in the one mode, and g is the transition coupling constant. Assume the atom is initially in the excited state, so that c ca n n, ( ) ( )0 0= and cb n, ( )+ =1 0 0 , where cn(0) is the probability amplitude that the field contains n photons. (a) Write out simplified probability amplitudes for ca,n(t) and cb,n+1(t) using the initial conditions and calculate c ta n, ( ) 2 and c tb n, ( ) 2 . (3 marks) (b) Using the results of part (a), calculate the population inversion W(t), given by the excited state population density minus the ground state population density, summed over photon number n. (1 mark)