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Unformatted text preview: HOMEWORK ASSIGNMENT 10 PHYS852 Quantum Mechanics II, Spring 2008 1. Dark State Adiabatic Passage: [Should be fairly easy] An atomic -system consists of two ground- state hyperfine sub-levels couple via an electronically excited state. Let | 1 ) and | 2 ) refer to the two lower levels and | e ) be the upper level. If | 1 ) corresponds to an m f = 1 state, | e ) to an m f = 0 state, and | 2 ) to an m f = 1 state, then due to angular momentum conservation, a right-circularly polarized laser will couple | 1 ) to | e ) and a left-circularly polarized laser will couple | 2 ) to | e ) . In this way, the following Hamiltonian can be realized (in a suitable rotating frame and in the rotating wave approximation): H = i planckover2pi1 | e )( e | + planckover2pi1 R ( | 1 )( e | + | e )( 1 | ) + planckover2pi1 L ( | 2 )( e | + | e )( 2 | ) , (1) where the i planckover2pi1 term gives the excited state a finite lifetime due to spontaneous emission. Ignoring the fact that H is not Hermitian, find the eigenvalues and eigenvectors of H . In general, the eigenvalues will be complex, of the from n i n . Then we can think of n as the inverse lifetime of the eigenstate before it decays due to spontaneous emission. We want to consider the so-called counter-intuitive pulse sequence, where the system starts out in state | 1 ) , with R = 0 and L = . Then R is slowly increased and L is decreased, maintaining 2 R + 2 L = 2 . Plot the real parts of the eigenvalues versus R during this pulse-sequence. If the two Rabi-frequencies are varied very slowly, what will the final state of the system be? What is thetwo Rabi-frequencies are varied very slowly, what will the final state of the system be?...
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