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.Andrei Tokmakoff, MIT Department of Chemistry, 5/10/2007 p. 11-44 11.5. CHARACTERIZING FLUCTUATIONS Eigenstate vs. system/bath perspectives From our earlier work on electronic spectroscopy, we found that there are two equivalent ways of describing spectroscopic problems, which can be classified as the eigenstate and system/bath perspectives. Let’s summarize these before turning back to nonlinear spectroscopy, using electronic spectroscopy as the example: 1) Eigenstate: The interaction of light and matter is treated with the interaction picture Hamiltonian HH= 0 +Vt (). H0 is the full material Hamiltonian, expressed as a function of nuclear and electronic coordinates, and is characterized by eigenstates which are the solution to 0Hn = E n . In the electronic case n = en n 1 2,, K represent n labels for a particular vibronic state. The dipole operator in Vt()couples these states. Given that we have such detailed knowledge of the matter, we can obtain an absorption spectrum in two ways. In the time domain, we know 2 e−iωmnt (1)Ct()=∑pn nμ()t ( ) μ 0 n =∑pn μμμ nm n ,nm The absorption lineshape is then related to the Fourier transform of Ct(), 21 σω()=∑pn μ (2)nm ωω −Γi−nm, nm nm where the phenomenological damping constant Γnm was first added into eq. (1). This approach works well if you have an intimate knowledge of the Hamiltonian if your spectrum is highly structured and if irreversible relaxation processes are of minor importance. 2) System/Bath: In condensed phases, irreversible dynamics and featureless lineshapes suggest a different approach. In the system/bath or energy gap representation, we separate our Hamiltonian into two parts: the system Hs contains a few degrees of freedom Q which we treat in detail, and the remaining degrees of freedom (q) are in the bath HB. Ideally, the interaction between the two sets HSB(qQ)is weak. H0 =HS + HB + HSB . (3)Andrei Tokmakoff, MIT Department of Chemistry, 5/10/2007 p. 11-45 Spectroscopically we usually think of the dipole operator as acting on the system state, i.e. the dipole operator is a function of Q. If we then know the eigenstates of HS , Hn = E n where n = g or e for the electronic case, the dipole correlationS n function is 2 −i ω tegCt()=μ e exp ⎡⎢⎣−i∫0 t HSB ()t′ dt′⎤⎥⎦ (4)μμ eg The influence of the dark states in HB is to modulate or change the spectroscopic energy gap ωeg in a form dictated by the time-dependent system-bath interaction. The system-bath approach is a natural way of treating condensed phase problems where you can’t treat all of the nuclear motions (liquid/lattice) explicitly. Also, you can imagine hybrid approaches if there are several system states that you wish to investigate spectroscopically. Energy Gap Fluctuations How do transition energy gap fluctuations enter into the nonlinear response? As we did in the case of linear experiments, we will make use of the second cumulants approximation to relate dipole correlation functions to the energy gap correlation function Ceg(τ). Remembering that for the case of a system-bath interaction that that linearly couples the system and bath nuclear coordinates, the cumulant expansion allows the linear spectroscopy to be expressed in terms of the lineshape function gt()Cμμ()t =μ 2 e −iωegt e −g()t (5)eg (6)g t =t dt′′′′dt′ 1 δH t′δH 0()∫∫t 2 () eg ( )eg0 0 h14444244443Cteg()′ Ceg (τ)=δωeg (τ δω ) eg (0) (7) gt() is a complex function for which the imaginary components describe nuclear motion modulating or shifting the energy gap, whereas the real part describes the fluctuations and damping that lead to line broadening. When Ceg(τ) takes on an undamped oscillatory formAndrei Tokmakoff, MIT Department of Chemistry, 5/10/2007 p. 11-46 Ceg()τ = Deiω0τ, as we might expect for coupling of the electronic transition to a nuclear mode with frequency ω0, we recover the expressions that we originally derived for the electronic absorption lineshape in which D is the coupling strength and related to the Frank-Condon factor. Here we are interested in discerning line-broadening mechanisms, and the time scale of random fluctuations that influence the transition energy gap. Summarizing our earlier results, we can express the lineshape functions for energy gap fluctuations in the homogeneous and imhomogeneous limit as 1) Homogeneous. The bath fluctuations are infinitely fast, and only characterized by a magnitude: Ceg(τ) (). =Γδτ (8) In this limit, we obtain the phenomenological damping result gt()=Γt (9) Which leads to homogeneous Lorentzian lineshapes with width Γ. 2) Inhomogeneous. The bath fluctuations are infinitely slow, and again characterized by a magnitude, but there is no decay of the correlations Ceg(τ)=Δ2 . (10) This limit recovers the Gaussian static limit, and the Gaussian inhomogeneous lineshape where Δ is the distribution of frequencies. 1 22gt()= 2 Δ t . (11) 3) The intermediate regime is when the energy gap fluctuates on the same time scale as the experiment. The simplest description is the stochastic model which describes the loss of correlation with a time scale τc Ceg(τ)=Δ2 exp (−t /τc) (12) which leads to gt()=Δ2 τc 2 ⎣⎡exp(−t /τc)+ t /τc−1⎦⎤ (13) For an arbitrary form of the dynamics of the bath, we can construct gt() as a sum over independent modes gt()=ig t . Or for a continuous distribution for modes, we can ∑ i()Andrei Tokmakoff, MIT Department of Chemistry, 5/10/2007 p. 11-47 describe the bath in terms of the spectral density ρ(ω)that describes the coupled nuclear motions () 1 Im ⎡C%eg()ω⎤ (14)ρω= 2πω2 ⎣⎦ +∞ 1 %gt()=∫−∞ dω 2 Ceg ()ω⎡⎣exp(−iωt )+ iωt −1⎤⎦2πω (15) =∫+∞dωρω ⎛ ⎛βω⎟⎞ ω ω ⎟⎞()⎜coth ⎜ h(1−cosωt ) ( + i sin t − t )−∞ ⎝⎝ 2 ⎠⎠ To construct an arbitrary form of the bath, the phenomenological Brownian oscillator model allows us to construct a bath of i damped oscillators, Ceg ′′()ω=∑ξiCi ′′()ω i hωΓ (16)C ′′ωi ()= (i 2 2 )2 i 2 i 2mi ω ω − +4ωΓ Here ξi is the coupling coefficient for oscillator i. Nonlinear Response with the Energy Gap Hamiltonian In a manner that parallels our