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, 6/15/2009 p. 11-54 11.6. TWO-DIMENSIONAL CORRELATION SPECTROSCOPY Our examination of pump-probe experiments indicates that the third-order response reports on the correlation between different spectral features. Let’s look at this in more detail using a system with two excited states as an example, for which the absorption spectrum shows two spectral features at ωba and ωca. Imagine a double resonance (pump-probe) experiment in which we choose a tunable excitation frequency ωpump , and for each pump frequency we measure changes in the absorption spectrum as a function of ωprobe . Generally speaking, we expect resonant excitation to induce a change of absorbance. The question is: what do we observe if we pump at ωba and probe at ωca? If nothing happens, then we can conclude that microscopically, there is no interaction between the degrees of freedom that give rise to the ba and ca transitions. However, a change of absorbance at ωcaindicates that in some manner the excitation of ωba is correlated with ωca. Microscopically, there is a coupling or chemical conversion that allows deposited energy to flow between the coordinates. Alternatively, we can say that the observed transitions occur between eigenstates whose character and energy encode molecular interactions between the coupled degrees of freedom (here β and χ):Andrei Tokmakoff, MIT Department of Chemistry, 6/15/2009 p. 11-55 Now imagine that you perform this double resonance experiment measuring the change in absorption for all possible values of ωpump and ωprobe , and plot these as a two-dimensional contour plot:1 This is a two-dimensional spectrum that reports on the correlation of spectral features observed in the absorption spectrum. Diagonal peaks reflect the case where the same resonance is pumped and probed. Cross peaks indicate a cross-correlation that arises from pumping one feature and observing a change in the other. The principles of correlation spectroscopy in this form were initially developed in the area of magnetic resonance, but are finding increasing use in the areas of optical and infrared spectroscopy. Double resonance analogies such as these illustrate the power of a two-dimensional spectrum to visualize the molecular interactions in a complex system with many degrees of freedom. Similarly, we can see how a 2D spectrum can separate components of a mixture through the presence or absence of cross peaks.Andrei Tokmakoff, MIT Department of Chemistry, 6/15/2009 p. 11-56 Also, it becomes clear how an inhomogeneous lineshape can be decomposed into the distribution of configurations, and the underlying dynamics within the ensemble. Take an inhomogeneous lineshape with width Δ and mean frequency ωab , which is composed of a distribution of homogeneous transitions of width Γ. We will now subject the system to the same narrow band excitation followed by probing the differential absorption ΔA at all probe frequencies. Here we observe that the contours of a two-dimensional lineshape report on the inhomogeneous broadening. We observe that the lineshape is elongated along the diagonal axis (ω1=ω3). The diagonal linewidth is related to the inhomogeneous width Δ whereas the antidiagonal width ωω ωab⎡+ = /2⎤⎦ is determined by the homogeneous linewidth Γ.⎣1 3 2D Spectroscopy from Third Order Response These examples indicate that narrow band pump-probe experiments can be used to construct 2D spectra, so in fact the third-order nonlinear response should describe 2D spectra. To describe these spectra, we can think of the excitation as a third-order process arising from a sequence of interactions with the system eigenstates. For instance, taking our initial example with three levels, one of the contributing factors is of the form R2:Andrei Tokmakoff, MIT Department of Chemistry, 6/15/2009 p. 11-57 Setting τ2 = 0 and neglecting damping, the response function is 2 ωτ ωτ R ττ, = p μ 2 e−i ba 1 −i ca 3 (5.1)2 (1 3 ) a μab ac The time domain behavior describes the evolution from one coherent state to another—driven by the light fields: A more intuitive description is in the frequency domain, which we obtain by Fourier transforming eq. (5.1): %∞ ∞ 11 33 R , ∫∫ R (ττ d2 (ωω1 3 )= −∞ −∞ eiωτ+iωτ 2 1,3 )dτ τ 1 3 2 2 = pa μ δω ωδωω ) (5.2)(− )( −μab ac 3 ca 1 ba 2 2≡ pa μ Ρ ,;ωτ ω μab ac (3 2 1 ) The function P looks just like the covariance xy that describes the correlation of two variables x and y . In fact P is a joint probability function that describes the probability of exciting the system at ωba and observing the system at ωca (after waiting a time τ2 ). In particular, this diagram describes the cross peak in the upper left of the initial example we discussed. Fourier transform spectroscopy The last example underscores the close relationship between time and frequency domain representations of the data. Similar information to the frequency-domain double resonanceAndrei Tokmakoff, MIT Department of Chemistry, 6/15/2009 p. 11-58 experiment is obtained by Fourier transformation of the coherent evolution periods in a time domain experiment with short broadband pulses. In practice, the use of Fourier transforms requires a phase-sensitive measure of the radiated signal field, rather than the intensity measured by photodetectors. This can be obtained by beating the signal against a reference pulse (or local oscillator) on a photodetector. If we measure the cross term between a weak signal and strong local oscillator: 2 2−I ( ) =δLO τLO Esig + ELO ELO . (5.3)+∞dE () ( τ E ττ − )≈2Re ∫−∞ τ3 sig 3 LO 3 LO For a short pulse ELO , δI ( ) τLO ∝ Esig (τLO ). By acquiring the signal as a function of τ1 and τLO we can obtain the time domain signal and numerically Fourier transform to obtain a 2D spectrum. Alternatively, we can perform these operations in reverse order, using a grating or other dispersive optic to spatially disperse the frequency components of the signal. This is in essence an analog Fourier Transform. The interference between the spatially dispersed Fourier components of the signal and LO are subsequently detected. 2 2δωI ()= −() () ()3