Owing to the coherence and the extreme peak power intensity, ultra short light pulses have been applied successfully to the coherent control of lattice oscillations in solids. Under strong photo-irradiation, the lattice distorts to the anharmonic region of the potential, which results many peculiar phenomena such as activation of zone boundary phonons [1] and an electronic phonon softening, the frequency shift against the excitation energy [2]. In these years, second-order Raman processes have attracted interests as a stage of coherent control of quantum oscillation. Hu and Nori proposed theoretically to squeeze phonons by exciting the second-order Raman bands[3, 4]. Squeezing is a coherence control of a physical parameter at the cost of the coherency of the conjugate parameter, which is well known in quantum optics. Experimental demonstrations of the phonon squeezing have been carried out soon[5, 6]. Recently, the spin squeezing has been reported also by exciting the two-magnon Raman band[7]. Furthermore, photo-irradiation of the second-order Raman bands excites two-phonon waves at k ₁ - k ₂ = K = q ₁ + q ₂, where k ₁ and k ₂ are the wavevectors of the incident light, K is that of the two-phonon wave, and q ₁ and q ₂ are those of the constituent one-phonon waves. Because k ₁ and k ₂ are sufficiently smaller than the inverse of the lattice parameters, the two-phonon wave is excited mostly at the Brillouin zone center. On the other hand, because q ₁ and q ₂ themselves can be arbitrary under k ₁ - k ₂ = q ₁ + q ₂, the phonons can be excited at whole Brillouin zone. These indicate that excitation of the second-order Raman bands may give a new pathway of coherent control of quantum oscillations which are not accessible by exciting the first-order Raman bands. We have shown that the impulsive excitation using two different color ultrashort light pulses are beneficial for selective and strong excitation of phonons[8, 9]. This technique is well known as Coherent anti-Stokes Raman Scattering (CARS). In this paper, results of exciting the second-order Raman bands by the CARS technique are reported. The CARS process under the excitation of the second-order Raman bands was discussed by Shen 25 years ago[10]. He discussed about not only the possibility of the two-phonon CARS, the stimulated Raman scattering resonant with the second-order Raman bands, but also the phonon parametric process, the amplification of one-phonon waves by exciting the second-order Raman bands. Experimentally, the parametric decay of a phonon under strong photo-irradiation was demonstrated by Colles and Giordmaine 35 years ago[11]. In their experiment, the fluorescence reduced at the frequency of the subharmonic under resonant pumping of a phonon using two strong laser pulses. After these works, it is strange that very few nonlinear spectroscopies of the second-order Raman processes have been carried out till the phonon squeezing. Indeed, even the CARS of the second-order Raman bands itself has been demonstrated only recently using one-color crossbeam excitation[12, 13]. Here, I show that Raman-inactive coherent phonons are generated by the phonon parametric amplification process from strongly excited second-order Raman bands using two-color cross-beam excitation. I would like to point out that the control of the quantum coherency is the basis of the quantum computation and that the parametric amplification process is one of the standard techniques to generate squeezed photon states[14]. The results presented here give a possibility of handling coherency of phonons similarly to photons.

The sample is a commercially available 1mm-thick SrTiO₃ plate with a (1,0,0) surface. It has a cubic perovskite structure with a space group Pm3m ($O_h^1$) above 110K. Its phonon modes at the Brillouin zone center at room temperature are made up of one T_u acoustic mode and three T _1u and one T _2u optical modes. Each polar T _1u optical mode splits into one LO and two doubly-degenerate TO modes. All the phonon modes are Raman-inactive but there are strong and broad second-order Raman bands, all of which consist of the two-phonon combination bands. The second-order Raman bands of SrTiO₃ locate at 300, 700, and 1000 cm^-1. The intensity decreases as the frequency increases [15].

 

Fig. 1. The setup of the measurement. OPA : optical parametric amplifier system, m : mirror (only one of the symbols is marked), VND : variable ND filter, T : translational stage with a retroreflector, L : lens, M : spectrometer and PC, F : optical fiber, ND : ND filter, R : rotational stage, S : sample (SrTiO₃), Ls : signal light (ω ₁), Li : idler light (ω ₂).

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The experimental setup is shown in Fig. 1. Photo-irradiation was carried out under two-color cross-beam configuration, which is the same as that of the CARS experiment, using signal (ω ₁) and idler (ω ₂) light pulses generated by a Ti:Sapphire-based femto-second optical parametric amplifier (OPA) system. Their pulse widths are 150 fs and the repetition rate is 1 kHz. They are focused nearly normally on to the sample through a lens with a crossing angle of approximately 5 deg. Their polarizations are vertical, which are in the same direction as one of the crystal axis. The size of the focused spots are 50 to 100 μm. The emitted light from the sample is picked up by an optical fiber with a 200-μm core radius. It is set on an arm which rotates around the sample by an electrically controlled rotational stage. The optical fiber is connected to a multichannel spectrometer. The distance between the sample and the end of fiber is about 40 mm. The angle of the emitted signals is measured from the direction of ω ₁ light. When photographs are taken, the optical fiber is replaced by a white paper and the bright spots on it are taken by a PC-based CCD camera. In order to measure the power dependence, part of higher-order signals are simultaneously picked-up by an optical fiber after collecting them using a lens. All the experiments are carried out at room temperature.

When the two incident light pulses do not overlap temporally, only two visible signals of third harmonic generation are observed by the naked eye. As the pulses overlap, a spatially chirped supercontinuum, the color of which changes from red to blue against the angle, appears at the outside of 3ω ₁. At the proper relative delay and for the frequency differences resonant with the second- and the third-strongest Raman peaks, a train of well-separated clear light spots ranging from infrared to blue region appears from the supercontinuum. These signals are sensitive to the relative delay of the incident light pulses. The spots are the clearest when the ω ₁ pulse arrives a few tens of fs earlier than the ω ₂ pulse. Trains of the light pulses are emitted symmetrically in both the forward and the backward direction of the sample. It is stressed that only supercontinuum and no clear spots are observed when the frequency difference is tuned to 390 cm^-1, the shoulder of the first strongest peak, which is the lowest for our OPA system, and 500 cm^-1, the valley between the second and the third Raman peaks. Figure 2 show the photographs taken by a PC-based CCD camera against irradiation powers. Because the focuses of the incident light are not optimized, the irradiation powers are higher than those in other measurements. The lower-order signals are invisible to the naked eye because they are infrared light. It is apparent that the power dependence is asymmetric against P₁ (signal light) and P₂ (idler light). The higher-order signals diminish faster than the lower-order ones as P₁ decreases. On the other hand, the number of visible spots does not change against P₂. Spectroscopic plots are shown in Fig. 3. Figures 3(a) and 3(c) are the spectra, the peak intensity of which are scaled. They show the change of the spectral shapes against P₁ with fixing P₂ to 1.3 mW and those against P₂ with fixing P₁ to 1.0 mW, respectively. Each spectrum is shifted upward by the value of the varying irradiation powers. The scaling factors are plotted against the powers in Figs. 3(b) and 3(d). One must remember that the absolute shape of the spectra have no meaning because only a part of the emitted light is collected by a lens. It is clearly seen that the change of the spectral shape is mostly independent of P₂ and the intensity dependence is linear to it except at very low power. On the other hand, the higher-order signals diminish faster than the lower-order ones as P₁ decreases and the intensity dependence has a clear threshold.

 

Fig. 2. The photographs of the signals, where ω ₁ = 7013 cm^-1 and ω ₂ = 5999 cm^-1 ( ∆ω = 1014 cm^-1). (a) P₁ = 27 mW, P₂ = 30, 20, 10, 5 mW and (b) P₂ = 10 mW, P₁ = 27, 25, 20, 15, 10, 5 mW from top to bottom, respectively. The two bright spots on the left-hand side are the ω ₁ and the ω ₂ light.

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Figure 4 show the angle-resolved spectra, where the frequency differences (∆ω = ω ₁ - ω ₂) are 660 and 1015 cm^-1, the values of which agree with the frequencies of the second- and the third-strongest second-order Raman bands. The relative delays are set as the spots become the clearest. For the purpose of making the peaks clear, each spectrum is scaled against the angle by the following method instead of calibrating by sensitivity. Firstly, an envelope of the strongest peak intensity as for the angle is calculated by taking nine points moving average. Then each spectrum is divided by the value of the envelope. There are series of different frequency spacing values, 475 and 530 cm^-1, in the spectrum of ∆ω = 660 cm^-1, and 475, 545, and 625 cm^-1, in the spectrum of ∆ω = 1015 cm^-1.

Figure 5 shows the wavevector component of the peaks of the emitted light to the direction of the sample surface against their frequencies, where the frequency difference is 714 cm^-1. The filled and the open markers are the data measured by a CCD and an InGaAs multi channel photodiode array, respectively. Several subsidiary peaks are also plotted. It is apparent that the dispersion relation is different from that of the ordinary multi-step CARS, k _n = k ₁ + n(k ₁ - k ₂) and ω_n = ω ₁ + n(ω ₁ - ω ₂) (broken line). The line starts from around the CARS and then curves slightly. At the same time, the step of k _∥ is half or one third of the k _∥ difference between the ω ₁ and the ω ₂ beams.

The observed frequencies are compared with the second-order Raman scattering data [15], the neutron scattering data [16], and the hyper-Raman scattering data [17]. It is summarized in Table 1. According to Nilsen and Skinner, the second-strongest Raman peaks (727 and 684) are assigned to TO₄ + TO₂ and the overtone of the TO₃ mode, and the third one (1038) to 2LO₂ and 2TO₄. The first-strongest one is assigned to TO₄ - TA, TO₄ - TO₁, and 2TO₂. This band is mainly made from the former two difference combination bands. The obtained frequencies 475 and 545 (or 530) cm^-1 agree well with those of LO₂ (480) and TO₄ (544). The frequency 625 cm^-1 agrees well with the second-order Raman peak of TO₄ + TO₁ or TO₄ + TA (629). I believe that the second-strongest Raman peak should include LO₂ + LO₁. Accepting these assignments, all the obtained frequencies in the spectra are assigned to those of the one-phonons and their combinations.

 

Fig. 3. Power dependence of the higher-order signals. Part of higher-order signals are simultaneously picked-up by an optical fiber after collecting them using a lens. (a) The change of the scaled spectra against P₁, where P₂ is fixed to 1.3 mW. (b) The change of the peak intensities of the spectra in Fig. 3(a) (scaling factors of Fig. 3(a)). (c) The change of the scaled spectra against P₁, where P₂ is fixed to 1.0 mW. (d) The change of the peak intensities of the spectra in Fig. 3(c). Each spectrum in Fig. 3(a) and 3(c) is shifted upward by the value of P₁ and P₂, respectively. The frequencies of excitation light are 6596 and 5993 cm^-1.

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Fig. 4. The angle resolved spectra. (a) ω ₁ = 6648 cm^-1 and ω ₂ = 5988 cm^-1 (∆ω= 660 cm^-1). P₁ = 2.0 mW, P₂ = 0.3 mW. (b) ω ₁ = 6819 cm^-1 and ω ₂ = 5804 cm^-1 (∆ω= 1015 cm^-1). P₁ = 2.1 mW, P₂ = 1.6 mW. Each spectrum is scaled so that the peak intensities are the same (see text). The angle is measured from the direction of ω ₁ light. The vertical lines indicate the peak positions. ∆’s are their spacing.

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Fig. 5. The wavevector component of the peaks of the emitted light to the direction of the sample surface against their frequencies, where the frequency difference is 714 cm^-1 (ω ₁ = 6667 cm^-1 and ω ₂ = 5953 cm^-1 , P₁ = 3.0 mW, and P₂ = 0.5 mW) . The large open circles indicate the incident light beams. The smooth line is a guide for the eye.

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Table 1. Assignments of the observed phonon frequencies [15, 16, 17]. The values in the parenthesis are the frequencies at the zone center. The spectral widths of the incident light pulses are about 200 cm^-1.

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There are three problems to understand the phenomenon, why the one-phonon modes are excited from the two-phonon bands, why the dispersion relation of the emitted light does not satisfy that of the multi-step CARS, and why the Raman-inactive phonons can be observed.

When the non-degenerate light pulses, |E ₁|exp(iω ₁ t) and |E ₂|exp(iω ₂ t), are injected, they interfere with each other and generate a beating electric field, |E ₁ ∙ ${\mathbf{E}}_2^{*}$|exp(i(ω ₁ - ω ₂)t). When ω ₁ - ω ₂ is tuned to the second-order Raman bands, the beating electric field excites the two-phonon wave resonantly. When the amplitude of the two-phonon wave becomes sufficiently large, nonlinear wave-wave interaction among two-phonon wave and one-phonon waves comes to the central stage and the power of the two-phonon wave is transferred to the one-phonon waves. In order to model the experimental result, the phonon Hamiltonian must have a product of at least four creation or annihilation operators of phonon. In the case of the product of three terms, because two of them are excited simultaneously by the resonant two-phonon CARS process, the observed one-phonon wave must be generated directly from the two-phonon wave. This is impossible due to energy and momentum conservation law. Similar to the discussion of phonon-phonon scattering by third-order phonon-phonon interaction[18], the fourth-order phonon-phonon interaction is expressed as $H_{\mathit{\text{phonon}}}^{\mathit{\text{NL}}}$ = ħKΣ_{q,i,q′,j;q″,k;q‴,l} δ _{q+q′+q″+q‴}($b_{\mathbf{q},i}^{†}$ - b _-q,i)($b_{\mathbf{q}\prime ,j}^{†}$ - b _-q′,j)(b†_q″,k - b _-q″,k)($b_{\mathrm{q}‴,l}^{†}$ - b _-q‴,l) , where b _q,i ($b_{\mathbf{q},i}^{†}$) are the phonon annihilation (creation) operators with wavevector q and eigen frequency Ω_i and K is the strength of the lattice nonlinearity. In the case of our experimental situation, a two-phonon wave is excited externally by the beating electric field with the frequency of Ω₁ + Ω₂ and the wavevector of k ₁ - k ₂. The two-phonon wave is composed of two one-phonon waves with Ω₁, q and Ω₂, k ₁ - k ₂ - q, respectively, where q distributes over whole Brillouin zone. Retaining the terms which satisfy energy and momentum conservation, the dominant nonlinear terms become

For the externally driven large amplitude two-phonon wave, its operator commutes with b ^† and b ^†, which is rewritten as a single operator 〈b _q,1 b _k1-k2-q,2〉 = 〈b ₁ b ₂〉₀ exp(-i(Ω₁ + Ω₂)t), where b _q,1 and b _k1-k2-q,2 are abbreviated to b ₁ and b ₂. Then the fourth-order nonlinear term is rewritten as (1\2)(〈$b_1^{†}$ $b_2^{†}$〉b ₁ b ₂+$b_1^{†}$ b ^† ₂〈b ₁ b ₂〉). This formulation stems that oscillation of one-phonon waves are modulated under the existence of an elastic oscillation which is driven externally by two-phonon CARS process. Under some condition, the one-phonon waves are amplified from quantum fluctuation by this modulation, which is well known as the parametric amplification process. Combining the harmonic terms, the equations of motion of the one-phonon waves have a form of a nonlinear three-wave interaction,

where Λ = (1/2)KΣ_q〈b ₁ b ₂〉 = Λ₀ exp(-i(Ω₁ + Ω₂)t). Here, it is assumed for the simplicity that the two-phonon wave is a plane wave and its amplitude does not change during the interaction. The equations have solutions of plane waves. The eigen frequencies are

Because the eigen frequencies have an imaginary term, the one-phonon waves grow exponentially under existence of the two-phonon wave. From Eq. (1) to Eq. (4), the sum combination bands are taken into account. When the difference combination bands are excited, the two-phonon wave operator is replaced by 〈b^† ₁ b ₂〉. Then Eq.(1) and (2) are replaced by

where Λ′ = (1/2)KΣ_q〈$b_1^{†}$ b ₂〉 = Λ′₀ exp(i(Ω₁ - Ω₂)t) and the eigen frequencies become

Because these are pure real values, they have no solution of parametric amplification, which agrees with the case of 390 cm^-1 excitation, where no clear spots are found.

Next, the wavevector is considered. The dispersion relation of the emitted signals is far from linear as shown in Fig. 5. This means that their wavevectors are not dominated by only those of the incident light. Now, what is the origin of the additional wavevectors? Because the phonon waves which are excited from the second-order Raman bands satisfy the momentum conservation, K = q ₁ + q ₂, each constituent phonon wave can have arbitrary wavevector under this constraint. Then each phonon can couple with light with any wavevectors. It should be noted that the multi-colored signals accompany with a supercontinuum which satisfies the same dispersion relation of the higher-order signals. The origin of the supercontinuum is thought to be the cross-phase modulation of the two incident light pulses. The ω ₁ light modulates the second-order refractive index by the optical Kerr effect. The ω ₂ light is injected to this modulated region from a different direction of the ω ₁ light and is scattered from the region. The phase modulation of the ω ₂ light is ∆ϕ(t) = (ω ₂/c)∫n ₂|E ₁(t)|² dl and the frequency modulation is ∆ω = -∂(∆ϕ)/∂t, where n ₂ is the second-order refractive index [19]. The integration is carried out along the path of the ω ₂ beam. The frequency modulation generates a supercontinuum covering visible regions when femto-second light pulses are used. This mechanism of supercontinuum generation is well known in the self-phase modulation. It is expected from the above equation that the spectrum of the supercontinuum depends only on |E ₁(t)|²and its shortest wavelength becomes longer as |E ₁(t)|² max decreases. This is consistent with the experimental results as shown in Fig. 2 and Fig. 3. The shortest wavelength of the multi-colored signals becomes longer as P₁ decreases and is independent of P₂. If the signals come from the multi-step CARS, i.e. the multiple diffraction from the dynamic grating generated by the incident two light, the dispersion relation must be linear [8, 12], which is not the case. I believe that the higher-order signals are the stimulated amplification of light from sufficiently pumped phonon bath rather than the multi-step CARS. The supercontinuum generated by cross-phase modulation supplies seed light of the higher-order signals. The wavevectors of phonons may be selected adaptively at each step of the wave-mixing process on the dispersion relation of the supercontinuum so as to satisfy the energy conservation. Then the light is amplified resonantly to be seen as discrete coherent light. Even though the three-wave interaction amplifies two one-phonon waves simultaneously, when one of them establishes the coupling with the seed light, the other can not satisfy the phase matching condition necessarily.

Finally, one more question remains why the Raman-inactive phonons can couple with the light. A symmetry breaking is necessary to probe these Raman-inactive one-phonon waves. I have no clear idea more than speculation. Here, I point out one possibility. Under strong photo-irradiation, the lattice distorts due to the modulation of higher-order polarizability or strong lattice anharmonicity, and the light feels the symmetry of the distorted conformation. The phonons would be able to couple with light under this dynamically induced new symmetry. Once the Raman-inactive phonon is amplified, the broken symmetry would give a positive feedback to the coupling between light and lattice oscillation.

The author acknowledges to Prof. HANAMURA E., Prof. KAWABE Y., Prof. INOUE K., and Prof. YAMANAKA A. of Chitose Institute of Science and Technology for valuable discussion.