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.p. 10-30 10.3. THIRD-ORDER NONLINEAR SPECTROSCOPIES Third-order nonlinear spectroscopies are the most widely used class of nonlinear methods, including the common pump-probe experiment. This section will discuss a number of these methods. The approach here is meant to be practical, with the emphasis on trying to connect the particular signals with their microscopic origin. This approach can be used for describing any experiment in terms of the wave-vector, frequency and time-ordering of the input fields, and the frequency and wavevector of the signal. Selecting signals by wavevector The question that arises is how to select particular contributions to the signal. Generally, it will not be possible to uniquely select particular diagrams. However you can use the properties of the incident and detected fields to help with selectivity. Here is a strategy for describing a particular experiment: 1) Start with the wavevector and frequency of the signal field of interest. 2) (a) Time-domain: Define a time-ordering along the incident wavevectors or (b) Frequency domain: Define the frequencies along the incident wavevectors. 3) Sum up diagrams for correlation functions that will scatter into the wave-vector matched direction, keeping only resonant terms (rotating wave approximation). In the frequency domain, use ladder diagrams to determine which correlation functions yield signals that pass through your filter/monochromator. Let’s start by discussing how one can distinguish a rephasing signal from a non-rephasing signal. Consider two degenerate third-order experiments (ω1 = ω2 = ω3 = ωsig) which are distinguished by the signal wave-vector for a particular time-ordering. We choose a box geometry, where the three incident fields (a,b,c) are crossed in the sample, incident from three corners of the box, as shown. (Note that the color in these figures is not meant sample ab c sig a b ck k k k=+ − + ak bk ck top-down view sig k ak+bk−ck+sig kto represent the frequency of the incident fields –which are all thep. 10-31 same – but rather is just there to distinguish them for the picture). Since the frequencies are the same, the length of the wavevector k = 2πn λ is equal for each field, only its direction varies. Vector addition of the contributing terms from the incident fields indicates that the signal ksig =+ka − kb + kc will be radiated in the direction of the last corner of the box when observed after the sample. (The colors in the figure do not represent frequency, but just serve to distinguish the beams). Now, comparing the wavevector matching condition for this signal with those predicted by the third-order Feynman diagrams, we see that we can select non-rephasing signals R1 and R4by setting the time ordering of pulses such that a = 1, b = 2, and c = 3. The rephasing signals R2and R3 are selected with the time-ordering a = 2, b = 1, and c = 3. Alternatively, we can recognize that both signals can be observed by simultaneously detecting signals in two different directions. If we set the time ordering to be a = 1, b = 2, and c = 3, then the rephasing and non-rephasing signals will be radiated as shown below: E2E1sample E3 kR =−k + k + k kNR =+k − k +sig 1 2 3 sig 1 2 3 21 1k+2k−3k+sig k NR 1k−2k+3k+sig k R R NR 3k In this case the wave-vector matching for the rephasing signal is imperfect. The vector sum of the incident fields ksig dictates the direction of propagation of the radiated signal (momentum conservation), whereas the magnitude of the signal wavevector ks′ ig is dictated by the radiated frequency (energy conservation). The efficiency of radiating the signal field falls of with the wave-vector mismatch k k − k′ ∝ P t()sinc (Δkl 2) where l is the path lengthEsig(t)Δ= sig sig , as (see eq. 1.10).p. 10-32 Photon Echo The photon echo experiment is most commonly used to distinguish static and dynamic line-broadening, and time-scales for energy gap fluctuations. The rephasing character of R2 and R3 allows you to separate homogeneous and inhomogeneous broadening. To demonstrate this let’s describe a photon echo experiment for an inhomogeneous lineshape, that is a convolution of a homogeneous line shape with width Γ with a static inhomogeneous distribution of width Δ. Remember that linear spectroscopy cannot distinguish the two: 2e−iωτ−g(τ)− cc.. (10.1)abR ()τ =μab For an inhomogeneous distribution, we could average the homogeneous response, gt()=Γbat , with an inhomogeneous distribution R =∫dωab G (ωab )R (ωab ) (10.2) which we take to be Gaussian ω G ( ) ωba =exp ⎜⎛−(ωba − 2 ba )2 ⎟⎞ . (10.3)⎜ 2Δ ⎟⎝ ⎠ Equivalently, since a convolution in the frequency domain is a product in the time domain, we can set 1 22gt()=Γ t + Δ t . (10.4)ba 2 So for the case that Δ>Γ, the absorption spectrum is a broad Gaussian lineshape centered at the mean frequency which just reflects the static distribution Δ rather than the dynamics in Γ.ωbaNow look at the experiment in which two pulses are crossed to generate a signal in the direction ksig =2k2 − k1 (10.5) This signal is a special case of the signal (k3 k2 1 )+−k where the second and third interactions are both derived from the same beam. Both non-rephasing diagrams contribute here, but since both second and third interactions are coincident, τ2 =0 and R2 = R3. The nonlinear signal can be obtained by integrating the homogeneous response,p. 10-33 Two-pulse photon echo 4 −iω (ττ3 ) −Γab 1 +3 )R 3 ωab = pae ab 1− e (τ τ (10.6)()( ) μab over the inhomogeneous distribution as in eq. (10.2). This leads to 4 −i ωab (τ τ )e−Γ (ττ+ )e−(ττ − )2 2 (10.7)3R()=1−3 ab 1 3 1 3 Δ /2pa eμab −−For Δ> R()3 is sharply peaked at τ, i.e. e (1 3 ) ≈δττ)>Γ ab , τ1 = 3 ττ 2 Δ2/2 (1 − 3 . The broad distribution of frequencies rapidly dephases during τ1, but is rephased (or refocused) during τ3, leading to a large constructive enhancement of the polarization at τ1=τ3. This rephasing enhancement is called an echo. In practice, the signal is observed with a integrating intensity-level detector placed into the signal scattering direction. For a given pulse separation τ (setting τ1=τ), we calculated the integrated