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notes20 - 5.73 Lecture#20 Density Matrices II 20 1 Read...

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20 - 1 5.73 Lecture #20 revised October 21, 2002 Density Matrices II Last time: ψ,  ⟩, ρ =  ⟩⟨  coherent superposition vs. statistical mixture populations along diagonal, coherences off diagonal A = Trace( ρ A ) = Trace( A ρ ) Today: Quantum Beats prepared state ρ detection as projection operator D What part of D samples a specific off-diagonal element of ρ ? Optimize magnitude of beats [partial traces] system consisting of 2 parts — e.g. coupled oscillators motion in state-space vs. motion in coordinate space. Read CTDL, pages 643-652. The material on pages 20–2, –3, –5, and –7 is an exact duplication of pages 19–5, –6, –7 and –8.
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20 - 2 5.73 Lecture #20 revised October 21, 2002 Example: Quantum Beats Preparation, evolution, detection magically prepare some coherent superposition state Ψ (t) Ψ t ( ) = N a n ψ n e iE n t h n ρ t ( ) = Ψ (t) Ψ (t) Several eigenstates of H. Evolve freely without any time-dependent intervention N = a n 2 n −1 / 2 normalization Thus D = ψ 1 ψ 1 = 1 0 L 0 0 0 M 0 0 ρ = N 2 a 1 2 a 1 a 2 * e E 1 E 2 ( ) t h L a 2 2 a 3 2 O ρ 12 = 1 Ψ Ψ 2 ρ 12 = N 2 a 1 e E 1 t h a 2 * e + iE 2 t h D D t 1 2 1 2 * i t 2 1 = Trace Trace a a a e stuff 0 0 0 0 N a 12 ρ ( ) = = N 2 2 ω L M M M M D picks out only 1st row of ρ . Case (1): Detection: only one of the eigenstates, ψ 1 , in the superposition is capable of giving fluorescence that our detector can “see”. a projection operator (designed to project out only | ψ 1 part of state vector or ρ 11 part of ρ . no time dependence!
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20 - 3 5.73 Lecture #20 revised October 21, 2002 case (2): a particular linear combination of eigenstates is bright: the initial state 2 –1/2 ( ψ 1 + ψ 2 ) has D = 1. D = 1 2 ψ 1 + ψ 2 ( ) ψ 1 + ψ 2 ( ) = 1 2 ψ 1 ψ 1 + ψ 2 ψ 2 + ψ 1 ψ 2 + ψ 2 ψ 1 [ ] = 1 2 1 0 0 L 0 0 0 0 0 0 0 0 M 0 0 0 + 0 0 0 L 0 1 0 0 0 0 0 0 M 0 0 0 + 0 1 0 L 0 0 0 0 0 0 0 0 M 0 0 0 + 0 0 0 L 1 0 0 0 0 0 0 0 M 0 0 0 = 1 2 1 1 0 L 1 1 0 0 0 0 0 0 M 0 0 0 if the bright state had been 2 then = 1 1 -1 0 -1 1 0 0 0 -1/2 ψ ψ 1 2 0 0 0 0 0 ( ) , D
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