# hw09sol - Homework 9 Solution Prob. 9.1 For a two-level...

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Homework 9 Solution Prob. 9.1 For a two-level system with static couplingin Eq. (9.18), (a) Find A and B in terms of θ if c 1 (0) = 0 and c 2 (0) =1. (5 pts) A = sin ; B = cos . (b) Find c 1 (t) , c 2 (t) , p 1 (t) and p 2 (t) following the initial condition in (a). (5 pts) ( 29 ( 29 ( 29 ( 29 - - = - = - + - = - - - = - + - + + - + - t E E t p t E E t p t E i t E i t c t E i t E i t c B B B B B B 2 sin 2 sin 1 2 sin 2 sin exp cos exp sin exp exp cos sin 2 2 2 2 2 1 2 2 1 1 (c) Find A and B in terms of if c 1 (0) = c 2 (0) = 2 / 1 . (5 pts) ( 29 ( 29 sin cos 2 1 ; cos sin 2 1 - = + = B A (d) When is the first time p 1 (t) = 0 following the initial condition in (c). (10 pts) ( 29 ( 29 0 2 sin 4 sin 2 1 2 1 2 2 1 1 = - + = = - + t E E t c t p B In general, 4 sin 1 2 sin 2 - = - - + t E E B . Notice that this equality can ONLY be true for a specific that gives sin 4 = -1 where 8 3 1 1 cos 2 2 12 1 π = - + = - - H E V . If this condition is not met then p 1 (t) will never be zero (the oscillation is not full swing). If the condition is met, then the first time p1(t) is zero at - + - = E E t . Prob. 9.2 For a two-level system with dynamic coupling, assume that the evolution equations for the expansion coefficients c 1 (t) and c 2 (t) can be described by: 1

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## hw09sol - Homework 9 Solution Prob. 9.1 For a two-level...

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