MIT OpenCourseWarehttp://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Gaussian-stochastic model for the absorption lineshape This worksheet plots the frequency correlation function Cωω, dephasing function F, and absorption lineshape σ. The parameters that determine the lineshape are the width of the frequency distribution Δ and the frequency correlation time τc. Defining κ = Δτc, we will investigate the fast (κ>1), intermediate (κ=1) and slow (κ <1) modulation cases. First define variables. Range variables for the time and frequency axes : ma := 212 i := 0.. ma − 1 ii := 0.. 213 − 1 i 0.1⋅time grid: t :=i We will perform all calculations for a fixed value of Δ, and calculate for three correlation times: z := 0 1.. 2, := 1 τc := z 0.2 1 10 ⎛⎜ ⎜⎝ ⎞⎟ ⎟⎠ Δ 0.2 1 10 κ := τc ⋅Δ κ zz = Define the lineshape function and dephasing function: −t exp − 1 Evaluate the correlation function and lineshape function ⎛⎜⎝ ⎡⎢⎣ ⎛⎜⎝ ⎞⎟⎠ ⎤⎥⎦ ⎞⎟⎠ 2 2 Δ t g(t , τc) )F(t , τc ( (−g t τexp , c ⎛⎜ ⎜⎝ exp − ) tτc1 ) := + :=τc ⋅ ⋅ τc τc − ( −tτc2 Ft⎛⎜ ⎜⎝ ( ⎛⎜exp exp⎜⎝ F 0 F 0 F 0:= := :=fast slow midFt Ftii ii ii ( ⎞⎟ ⎟⎠ ⎞⎟ ⎟⎠ ⎞⎟ ⎟⎠ ti i iCωfasti Cωslowi Cωmidi := := := τc0 ) ) ) Ffasti FslowFmid:= := :=,i τc0 ,i τc2 ,i τc1i ifast Δ = 1 τc =⎛⎜ ⎜⎝ 0.2 1 10 ⎞⎟ ⎟⎠ κ = ⎛⎜ ⎜⎝ 0.2 1 10 ⎞⎟ ⎟⎠ mid slow Frequency Correlation Function C(t) 0.8 Cωfast 0.6 Cωmid 0.4Cωslow 0.2 0 0 2 4 6 8 10 t Dephasing Function F(t) 0.8 Ffast 0.6 Fmid 0.4Fslow 0.2 0 0 2 4 6 8 10 tNow Fourier transform the dipole correlation function Cμμ and use the real part to obtain the lineshape. Cμfast:= Ft( i, τc) Cμslow:= Ft( i, τc) Cμmid:= Ft( i, τc)i 0 i 2 i 1 Sfast cfft Cμfast) Sslow := cfft Cμslow) Smid := cfft Cμmid):=( ( ( Some manipulations to wrap and normalize the Fourier transform: stack S( fast, Sfast stack Sslow, Sslow stack Smid , Smid)) ( ) (σfast := max Re S( fast) σslow := max Re Sslow)) σmid := ( (( ) ( ( max Re Smid)) Determine the frequency axis for the Fourier transform ⋅ freq := 1 tmax := tma−1 ωii := ii freqΩ := ω ma−1t1 − t0 π⋅tmax Absorption Lineshape − 2 − 1 0 1 2 ωΩ− 0 0.5 1 Re σfast( ) Re σmid( ) Re σslow(
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