5.61 Fall 2017 Lecture#36 page 1 Lecture#36 Time Dependence...

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5.61 Fall 2017 Lecture #36 page 1 updated 12/12/17 12:57 PM Lecture #36: Time Dependence of Two-Level Systems: Density Matrix, Rotating Wave Approximation This lecture is based on Chapter 14 “The Density Matrix and Coherent Coupling of Molecules to Light” of the book Elements of Quantum Mechanics , Michael D. Fayer, Oxford University Press, 2001. Lecture #19 dealt with weak interactions of molecules with electromagnetic radiation. It is in the “linear response” region and illustrates the importance of the electric dipole approximation and, especially, “resonance”. What is “linear response”? The present lecture treats strong coherent interactions of electromagnetic radiation with two-level systems. The easily derived (by the chain rule for derivatives) equation of motion for the expecta- tion value of any Quantum Mechanical Operator is d dt A = i ! H , A [ ] + A t , derived by application of d dt to , A , and . One operator of particular importance is the density operator, ρ ( t ). It does more than simply repackage the information in Ψ ( x,t ). What is the density matrix? 1. ρ c = c c = c 1 c 2 ! c N c 1 * c 2 * " c N * ( ) 2. Repackaging an N × N matrix that contains all of the information in Ψ ( x,t ). 3. It minimizes the e iE j t ! factors of Ψ ( x,t ) = j c j ψ j ( x ) e iE j t ! . 4. It is directly observable. Its diagonal elements are populations and its off-diagonal elements are “coherences” that are observable as modulations at ω jk = E j E k ( ) ! with Fourier amplitudes c j c k * .
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5.61 Fall 2017 Lecture #36 page 2 updated 12/12/17 12:57 PM Ψ x , t ( ) t Ψ x , t ( ) = c n ( t ) n ψ n ( x ) t = c n ( t ) n n or t = c 1 c 2 ! c n n { } is a complete orthonormal set of basis states t is normalized to 1 c n ( t ) 2 n = 1 ρ ( t ) t t For a two-level system t = c 1 ( t ) 1 + c 2 ( t ) 2 ρ 11 = 1 t t 1 = 1 c 1 1 + c 2 2 ( ) c 1 * 1 + c 2 * 2 ( ) 1 = c 1 c 1 * ρ 12 = c 1 c 2 * ρ 21 = c 2 c 1 * ρ 22 = c 2 c 2 * ρ = c 1 c 1 * c 1 c 2 * c 2 c 1 * c 2 c 2 * . Since t is normalized to 1 1 = | c 1 | 2 + | c 2 | 2 = Trace ρ ( t ) = 1 and ρ ij = ρ ji * or ρ = ρ * ρ is Hermitian. Tr ρ ( t ) = 1 and ρ = ρ are general properties of N-dimensional ρ . For the 2-level system, we want to know the time dependence of ρ : d dt ρ = d dt t t + t d dt t i !
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