E B E tot
7.2 Results and Discussion
The 2D-TTR experiment (Fig. 7.1a) uses two intense carrier-envelope-phase (CEP) stable THz pump pulses followed by a weak 40 fs near-infrared (NIR) probe pulse.
By adjusting the delay t1 between THz pulses, we control the phase of the THz radiation at the sample, while maintaining a constant power. The heterodyne- detected transient birefringence is measured along the t2 axis with the NIR probe
pulse from the same laser system. By Fourier transforming over the t1and t2times, one can generate 2D plots in the frequency domain. In an isotropic medium, such as a liquid, the lowest order contribution to the measured nonlinear polarization is given by [37, 149]:
P(3)(t)=
∬
dt1dt2R(3)(t1,t2)EB(t−t2)EA(t−t1−t2)Enir(t), (7.1) whereR(3)(t1,t2)is the third-order response function of the liquid, andEA(t−t1−t2) and EB(t−t2)are the two THz pulses and Enir(t)is the NIR probe pulse. Weaker single-pulse third-order responses are removed by differential chopping, as shown in the Methods section.
We illustrate the principal components of a 2D-TTR response function with the birefringent signal from liquid bromoform at 295 K (Fig. 7.1 (lhs), Fig. 7.2a).
For the t1=0 fs trace (Fig. 7.1b (lhs), black line), the electronic response is visible as a sharp peak near t2=0 ps, which follows the (E-field)2(including phase) of the THz radiation [37]. An exponential decay extending to t2=2 ps results from the rotational diffusion of the molecules in the liquid as they reorient after aligning with the THz field [37]. Nonlinear vibrational coherences from the excitation of theν6 mode are visible as a damped oscillation out to t2=4 ps [148]. As t1 is shifted to 150 fs (red line), the transient birefringence is attenuated, while at 250 fs (blue line) the response becomes negative. A full 2D scan of this t1-dependent THz electric field-driven control is shown in Fig. 7.2a.
To understand the observed birefringence control, we return to the 2D-TTR pulse sequence (Fig. 7.1a). The polarizations of the THz pump pulses are orthogonal (shown in the black circles) and CEP stable. By changing the phase offset (∆φ) between the pulses, or t1, we change the polarization of the total THz field (Etot) when the pulses are overlapped in time (Fig. 7.1c). At t1= 0 fs, ∆φ=0, and Etot is oriented at +45 degrees with respect to the probe, while at t1= 250 fs, ∆φ=π and Etotis oriented at +135 degrees with respect to the probe.
The electronic and nuclear alignment of the molecules follows the polarization of the THz radiation, as shown in Chapter 6 [37, 148]. Thus, we posit that the change in the measured birefringence as a function of t1is due to angular control of the molecular alignment induced by the THz polarization (Fig. 7.1c). A simple orientational model of this behavior using the measured THz electric fields is shown in Fig. 7.1b (rhs). The parameters of the model are given in the Methods section.
t
1(ps )
1.0
0.0
-1.0 1.0 2.0
t2 (ps) 3.0
0.01
0.0
-0.01
(b) t
1(ps )
1.0
0.0
-1.0
0.0 1.0 2.0 t2 (ps)
0.1
0.0
-0.1
(a) TT R Sig nal ( A.U .) TT R Sig nal ( A.U .)
Figure 7.2: Representative time domain 2D-TTR signals. (a) The 2D time domain response of liquid bromoform (b) Cutting the response in (a) beyond t2=1 ps and detrending fits isolate the vibrational coherences.
The orientational model correctly predicts the sign changes and zero crossings of the birefringence, although it does not account for the nonlinear vibrational coherences.
Thus, by varying the phase of the THz polarization at the sample, one can control the orientation of molecules in a liquid.
We now consider the damped oscillation shown in Fig. 7.1b (lhs) and Fig. 7.2a.
This component is due to the excitation of intramolecular vibrational coherences, and provides information on the coupling and anharmonicity of vibrations in the liquid.
It is best visualized with a 2D Fourier transform along the t1and t2axes starting at t2=1 ps, after detrending the orientational response with a single exponential fit for each t2scan (Fig. 7.2b).
The origins of the 2D-TTR signatures can be derived using third-order perturbation theory on a three-level system, yielding 24 rephasing and non-rephasing Liouville
|𝒂𝒂⟩
|𝒄𝒄⟩
|𝒃𝒃⟩
|𝒃𝒃⟩⟨𝒃𝒃|
|𝒄𝒄⟩⟨𝒃𝒃|
|𝒄𝒄⟩⟨𝒂𝒂|
|𝒂𝒂⟩⟨𝒂𝒂|
t1 t2
|𝒂𝒂⟩
|𝒄𝒄⟩
|𝒃𝒃⟩
|𝒃𝒃⟩⟨𝒃𝒃|
|𝒃𝒃⟩⟨𝒂𝒂|
|𝒄𝒄⟩⟨𝒂𝒂|
|𝒂𝒂⟩⟨𝒂𝒂|
t1 t2
CHBr
3CCl
4f1=ωbc
f2=ωac f1=ωab f2=ωac
|𝒂𝒂⟩ |𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎⟩
|𝟐𝟐𝟎𝟎⟩
|𝒃𝒃⟩
|𝒄𝒄⟩
2.8 THz 1.9 THz 4.8 THz
|𝟎𝟎𝟎𝟎⟩
|𝒂𝒂⟩ |𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎⟩
|𝟐𝟐𝟎𝟎⟩
|𝒃𝒃⟩
|𝒄𝒄⟩
3.7 THz 2.8 THz 6.5 THz
|𝟎𝟎𝟎𝟎⟩
(a) (b)
(c)
|𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎⟩
1.1 THz
Bath States 1.3 THz
Energy/h (THz) Energy/h (THz)
0 5 10
0 5 10
|𝟎𝟎𝟎𝟎⟩
𝒗𝒗𝒗𝒗
|𝟎𝟎𝟎𝟎⟩
𝒗𝒗𝒗𝒗
𝒗𝒗𝟐𝟐 𝒗𝒗𝒗𝒗
𝒌𝒌𝐁𝐁𝑻𝑻 𝒉𝒉
𝒌𝒌𝐁𝐁𝑻𝑻 𝒉𝒉
Bath States
Figure 7.3: A quantum mechanical description of the measured signals. (a) 2 of 24 Liouville pathways in the 2D-TTR experiment for generalized eigenstates |ai, |bi, and |ci. (b) The energy levels and observed couplings in CHBr3, and for CCl4 in (c). THz excitations are shown in black, Raman in red.
pathways, as shown in Figs. 7.14 and 7.15 in the Methods section. With phase- sensitive heterodyne detection, rephasing and non-rephasing Liouville pathways are differentiated in a 2D Fourier transform. Signals in the first quadrant(f1=±, f2= ±) are non-rephasing, while signals in the second quadrant(f1=±, f2= ∓)arise from rephasing pathways [149, 163]. The bandwidth (0.5-4 THz) of the THz pulses restricts the experiment to 8 of the 24 pathways, all non-rephasing, 2 of which are shown in Fig.7.3. The sum of the 8 pathways yields a response function R(3)(t1,t2) given by:
R(3)(t1,t2) ∝ µabµbcΠac(pabe−iωabt1e−iωact2−pbce−iωbct1e−iωact2). (7.2) Here dipole and polarizability couplings are labeled as µxy andΠxy, and the equi- librium population differences between states y and x as pxy for the generalized three-level system. Thus, each peak in a 2D-THz-THz-Raman spectrum results from a closed-loop between three molecular eigenstates, hereafter denoted|ai, |bi,
|ci. Coherences are initiated by THz pump A and seen on the t1 time- or f1
Experiment (a) CHBr3 RDM Simulation
(b) CCl4
(c) CBr2Cl2
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4 6 f1 (THz)
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4 6 f1 (THz)
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4
-2 6
-4
-6 f2 (THz)
0.35 0.7 1.0
0.35 0.7 1.0
0.35 0.7 1.0
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4 6
f1 (THz) Magnitude (A.U.) Magnitude (A.U.) Magnitude (A.U.)
Figure 7.4: Experimental and RDM simulated 2D-TTR spectra of CHBr3(a), CCl4 (b) and CBr2Cl2 (c). The f1 axis corresponds to the THz pump, while the f2 axis corresponds to the optical Raman probe.
frequency-axis. THz pump B moves these coherences to a new eigenstate, analo- gous to a coherence transfer pulse in 2D-NMR. The loop is closed with a |ci →
|ai or |ai → |ci Raman transition on the t2 time, or f2 frequency, axis. Similar to 2D-Raman and 2D-Raman-THz spectroscopy, every closed-loop contains one or more transitionsforbidden in the harmonic approximation, allowing us to directly measure the molecular anharmonicity [12, 30].
In Fig. 7.4, we demonstrate the capabilities of 2D-TTR in measuring vibrational anharmonicities with the spectra of bromoform (CHBr3, 295 K), carbon tetrachlo- ride (CCl4, 295 K), and dibromodichloromethane (CBr2Cl2, 313 K). For all three liquids, we observe peaks on the f2Raman axis that match the known lowest energy intramolecular vibrations: ν6 in CHBr3 (Fig. 7.5); ν2 in CCl4[169, 170]; ν4 and ν5 for CBr2Cl2 [171]. Returning to the closed-loop picture, the|ci ↔ |aiRaman
ν6
ν3
Figure 7.5: The linear spectrum of bromoform, taken with a plasma filamentation THz source, and a 200 micron thick GaP crystal for EO detection.
transition is clearly assigned to changes of one quantum in each of these low energy modes. The remaining mystery is the identity of the intermediate state |bi in the closed-loop, which can be determined by examining the f1positions of the peaks.
At ∼300 K (kBT/h=6.2 THz) there is significant population in the low energy vibrational and bath states of these liquids. Using linear Raman and THz spectra and Eq. 7.2, we can predict the precise peak positions for coupling to different|bi states. For CHBr3, we observe a doublet f1peak that is consistent with anharmonic coupling betweenν6 andν3, but not with anharmonic coupling ofν6 to bath modes (Fig. 7.3b, Fig. 7.4a). CCl4exhibits a similar doublet on the f1axis that matches a ν2-ν4 coupling (Fig. 7.3c, 7.4b). Simulated spectra from a reduced density matrix (RDM) method (see Methods section) show good agreement with the experimental results (Fig. 7.4a,b, rhs).
The thermally populated vibrational manifolds of CBr2Cl2 (Fig. 7.6) are more complicated than those of CCl4and CHBr3, which leads to further possibilities for
4 |𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
𝒍𝒍𝒂𝒂𝒃𝒃𝒍𝒍𝒍𝒍𝒍𝒍: |𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗⟩
|𝟎𝟎𝟐𝟐𝟎𝟎𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
Energy/h (THz)
|𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
|𝟐𝟐𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
|𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⟩
0 2 6 8 10
𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗
𝒗𝒗𝒗𝒗
𝒌𝒌𝐁𝐁𝑻𝑻 𝒉𝒉
𝒗𝒗𝒗𝒗
𝒗𝒗𝒗𝒗
Figure 7.6: The vibrational energy levels of CBr2Cl2. Observed THz excitations are shown as black arrows, those for Raman excitations in red.
vibrational coupling. However, a simple RDM simulation with eigenstates from the linear Raman spectrum and weak anharmonic ν4↔ν7 and ν5↔ν7 coupling reproduces all of the observed peaks (Fig. 7.4), and gives us insight into the relative coupling strengths in this liquid. Specifically, we see that the two lowest energy vibrational modesν4 andν5 are more strongly coupled toν7 than ν3 andν9.
To test the robustness of the CBr2Cl2 simulations, we have plotted the RDM pre- dicted spectra for several different coupling schemes in Fig. 7.7. Withν3, ν9, or ν7/ν9/ν3 coupling, the RDM spectra are qualitatively different than the experimental spectrum, while there is good agreement with onlyν7 coupling.
We can further differentiate theν3 coupling schemes by considering the symmetry
(c) RDM 𝒗𝒗𝟑𝟑 coupling – with symmetry
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4 6 f1 (THz)
(d) RDM 𝒗𝒗𝒗𝒗 coupling
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4 6 f1 (THz)
(a) Experiment (b) RDM 𝒗𝒗𝒗𝒗 coupling
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4
-2 6
-4
-6 f2 (THz)
0 2 4 6
f1 (THz)
0.35 0.7 1.0 Magnitude (A.U.)
(e) RDM 𝒗𝒗𝟑𝟑 coupling – no symmetry
0 2 4
-2 6
-4
-6 f2 (THz)
0.35 0.7 1.0 Magnitude (A.U.)
0 2 4
-2 6
-4
-6 f2 (THz)
0.35 0.7 1.0 Magnitude (A.U.) (f) RDM 𝒗𝒗𝒗𝒗,𝒗𝒗𝒗𝒗,𝒗𝒗𝟑𝟑 coupling
Figure 7.7: The experimental and RDM simulated spectra for various couplings in CBr2Cl2.
of the modes. For any transition dipole matrix element to be non-zero, the following must be true
hΨa|
bµ|Ψbi,0⇔Γa⊗Γµ⊗Γb⊃ A1,
where Ψj are vibrational eigenstates, bµis the dipole operator, Γi denotes the irre- ducible representation and A1 refers to the totally symmetric representation. The point group of CBr2Cl2 is C2v, with irreducible representations of the dipole op- erator A1, B1, and B2. Thus, the transition between |01000i (A2 symmetry) and
|00010i (A1 symmetry) in Fig. 7.6 is forbidden, while all others are allowed. In Fig. 7.7c we set this forbidden coupling to zero, but still do not see good agreement with the experimental spectrum. Finally, we also performed this analysis on CCl4 and CHBr3, but found no symmetry forbidden transitions.