Ultrafast Quantum-path Interferometry Revealing the Generation Process of Coherent Phonons
Kazutaka G. Nakamura, Kensuke Yokota, Yuki Okuda, Rintaro Kase, Takashi Kitashima, Yu Mishima, Yutaka Shikano, Yosuke Kayanuma
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Ultrafast Quantum-path Interferometry Revealing the Generation Process ofCoherent Phonons
Kazutaka G. Nakamura, ∗ Kensuke Yokota, Yuki Okuda, Rintaro Kase, TakashiKitashima, Yu Mishima, Yutaka Shikano,
2, 3, 4, 5 and Yosuke Kayanuma † Laboratory for Materials and Structures, Institute of Innovative Research,Tokyo Institute of Technology, 4259 Nagatsuta, Yokohama 226-8503, Japan Quantum Computing Center, Keio University, 3-14-1 Gakuen-cho, Yokohama, 223-8522, Japan Research Center for Advanced Science and Technology (RCAST),The University of Tokyo, 4-6-1 Komaba, Meguro, Tokyo 153-8904, Japan Research Center of Integrative Molecular Systems (CIMoS),Institute for Molecular Science, National Institutes of Natural Sciences,38 Nishigo-Naka, Myodaiji, Okazaki, Aichi 444-8585, Japan Institute for Quantum Studies, Chapman University,1 University Dr., Orange, California 92866, USA Graduate School of Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka, 599-8531 Japan (Dated: May 22, 2019)Optical dual-pulse pumping actively creates quantum-mechanical superposition of the electronicand phononic states in a bulk solid. We here made transient reflectivity measurements in an n-GaAsusing a pair of relative-phase-locked femtosecond pulses and found characteristic interference fringes.This is a result of quantum-path interference peculiar to the dual-pulse excitation as indicated bytheoretical calculation. Our observation reveals that the pathway of coherent phonon generation inthe n-GaAs is impulsive stimulated Raman scattering at the displaced potential due to the surface-charge field, even though the photon energy lies in the opaque region.
Coherent control is a technique of manipulating quan-tum states in materials using optical pulses [1–3]. A wavepacket in quantum mechanical superposition is createdby the optical pulse via several quantum transition paths.In the case of a double-pulse excitation, wave packets cre-ated by transitions in each pulse and across the two pulsesinterfere and the generated superposition state is ma-nipulated by controlling a delay between the two pulses[4, 5]. A contribution of individual quantum paths can beextracted from the interference pattern, which is referredto as quantum-path interferometry [6].Coherent phonons are a temporally coherent oscilla-tion of the optical phonons induced by the impulsive ex-citation of an ultrashort optical pulse [7–12]. Using co-herent phonons and making a pump–probe-type opticalmeasurement, we can directly observe the dynamics ofthe electron–phonon coupled states in the time domainfor a wide variety of materials [13–26]. In this respect,the clarification of the generation mechanism of the co-herent phonon is a fundamental subject as an ultrafastdynamical process [27–29]. The generation mechanismsof coherent phonons are usually categorized as two types:a mechanism of impulsive stimulated Raman scattering(ISRS) [7] and a mechanism of displaced enhanced coher-ent phonons [13]. In addition, for polar semiconductorssuch as GaAs, the screening of the surface-space-chargefield [16, 27] is considered to be another generation mech-anism for opaque conditions. The generation mechanismof coherent phonons may become a controversial subjectin the case of opaque-region pumping because impulsiveabsorption (IA) and ISRS processes coexist as possible quantum mechanical transition paths [30]. A novel ex-perimental technique is needed to shed light on this sub-ject.In the present work, we apply quantum-path interfer-ometry to study the generation process of coherent op-tical phonons through the coherent control of electron-phonon coupled states in bulk solids. Coherent phononsare often coherently controlled using a pair of femtosec-ond pulses as pump pulses, and the phonon amplitudeis enhanced or suppressed via the constructive or de-structive interference of induced phonons [15, 31]. Un-like these earlier works, we used two relative-phase-lockedpump pulses (pulses 1 and 2) and a delayed probe pulse(pulse 3) [32], for quantum-path interferometry. If thedelay of the dual-pump pulses t was controlled withsubfemtosecond accuracy and if the electronic coherencewas maintained during the dual pulses, electronic excitedstates were created as a quantum mechanical superposi-tion; i.e., the electronic polarizations induced by pulses1 and 2 interfered with each other. Meanwhile, polariza-tion in the phonon system was coherently created withresulting interference within the phononic and electronicdegrees of freedom. The probe pulse was used to monitorthe interference fringe via heterodyne detection; i.e., viaa change in the reflectivity as a function of the pump–pump delay t and pump–probe delay t . We couldevaluate the electronic and phononic coherence times ofthe sample using this scheme. Furthermore, a theoreticalestimation predicts a decisive difference in the interfer-ence fringes between ISRS and IA, and it will be shownthat the dominant pathway of the generation of coherentphonons can be determined from the pump-pulse-delay-dependent interference pattern of generation efficiency. FIG. 1. Two-dimensional image map of the change in reflec-tion intensity with the pump–probe delay ( t ) and pump–pump delay ( t ). A femtosecond optical pulse (central wavelength of798 nm, pulse width of ∼
50 fs) was split with a par-tial beam splitter into two pulses (i.e., pump and probepulses). The pump pulse was introduced into a home-made Michelson-type interferometer to produce relative-phase-locked pump pulses (i.e., pulses 1 and 2), in whichstability was within 6 %. The probe pulse (i.e., pulse3) was irradiated with a controlled time delay. The op-tical bandpass filter (the center wavelength of 800 nmwith the band width of 10 nm) was used for detectingthe reflected probe pulse in order to reduce cancellationeffects of Stokes and anti-Stokes components [33]. Weset the two pump pulses in a collinear condition withparallel polarization in the present experiments. The k -vector direction of the two interfering pump pulses mayaffects to the interference fringes. The sample was a sin-gle crystal of n-GaAs with (100) orientation and kept at90 K in a cryostat. Details of the experimental settingare described in the Supplemental Material [34].Figure 1 is a two-dimensional map of the transientreflectivity change ∆ R/R plotted against the pump–probe delay t and pump–pump delay t . For a fixedvalue of t , ∆ R/R indicates an oscillation with peri-ods of 115 and 128 fs, which are equal to the periods ofthe longitudinal-optical (LO) phonon and LO phonon-plasmon coupled mode (LOPC) at the Γ point in GaAs[35–38]. For a fixed value of t , meanwhile, ∆ R/R hasa beat between a rapid oscillation with a period of 2.7 fs,which is nearly equal to that of the pump laser with awavelength of 798 nm, and a slow oscillation with vibra- tion periods of the LO phonon and LOPC.For fixed values of t , a Fourier transformation of thesignal ∆ R/R was carried out with respect to t . TheFourier transformation was performed over the intervalof 0 .
25 ps < t < .
25 ps after the irradiation of pump2 to avoid the spurious effect of excess charge in the veryearly stage and the effect of phonon decay at a late time.Figure 2 is a plot of the oscillation amplitude ∆ R of theFourier-transformed data against the frequency ω anddelay t . The figure shows that two modes of coherentoscillations are excited in the crystal. P u m p - P u m p d e l a y t (f s ) ω (THz) F T A m p lit ud e ( a r b . un it s ) FIG. 2. Two-dimensional image map of the Fourier spectraat various pump–probe delays ( t ). To see more clearly the t dependence of the oscilla-tion amplitude ∆ R of the transient reflectivity, we plot-ted ∆ R at peak values at frequency ω of 8.7 and 7.8THz for the LO and LOPC modes, respectively. Figure3 (a) presents results for the LO mode. The LOPC modeshown in Fig. 3(b) has qualitatively the same interfer-ence pattern as the LO mode. In Fig. 3 (a), the rapidoscillation with a period of ∼ . ∼
115 fs is the interference fringe due to the coherenceof phonons. The rapid interference fringes disappearedwhen we used the cross polarized pump pulses.Note that the electronic coherence survives well afterthe overlapping of the pump pulses ends, as can be seenfrom comparison with the linear optical interference ofthe dual pulses (Fig. 3 (c)). This means that the opticalphase of pulse 1 is imprinted on the electronic polariza-tion and interferes with that of pulse 2. The most im-portant feature of the interference pattern is the appar-ent collapse and revival of the electronic fringe at around t ∼
55 fs. It will be shown below that this is due to aquantum-path interference peculiar to the ISRS process.In the time-region where t is large enough comparedwith t and the pulse-width, it is safely assumed thatthe generation process and the detection process of co-herent phonons are well separated. Hereafter, we con- I n t e n s it y ( a r b . un it s ) (fs)1.2x10 -3 A m p lit ud e ( a r b . un it s ) (fs)1.2x10 -3 A m p lit ud e ( a r b . un it s ) (fs) (a)(b)(c) FIG. 3. Interference fringe of LO phonon (a) and LOPC (b)and optical interference (c). centrate on the generation process of LO phonons. SeeSupplemental Material [34] for the theoretical treatmentof the probe processes. For microscopic interactions thatinduce the coherent oscillation of LO phonons throughthe irradiation of ultrashort optical pulses, several mod-els are conceivable, including the Fr¨ohlich interaction [39]and deformation-potential interaction [40]. In the case ofpolar materials, it is considered that electrostatic inter-action due to transient depletion field screening plays acentral role [27]. It is known that there are two types ofphotoinduced current, the usual injection current follow-ing the real excitation of carriers and the shift currentresulting from quantum mechanical polarization inducedby optical pulses [42, 43], in ionic semiconductors [41].The response of the shift current is usually faster thanthat of the injection current.We assume a model Hamiltonian that describes theelectron–phonon interaction as H = (cid:8) ǫ g + ~ ωb † b (cid:9) | g ih g | + X k (cid:8) ǫ k + ~ ωb † b + α ~ ω (cid:0) b + b † (cid:1)(cid:9) | k ih k | , (1)where | g i is the electronic ground state of the crystal withenergy ǫ g and | k i the excited state with energy ǫ k . Thecreation and annihilation operators of the LO phononat the Γ point with energy ~ ω are respectively denoted b † and b . It is assumed that the dimensionless electron-phonon coupling constant α is small and k -independent, assuming a rigid-band shift. The parameter α indicatesthe displacement of the potential, where all effects ondeformation of the potential, such as the surface-space-charge field, are included. See the Supplemental Material[34] for a detailed explanation.Within the rotating-wave approximation, the interac-tion Hamiltonian with a dual-pump pulse is given by H pump ( t ) = E pu ( t ) X k µ k | k ih g | + H.c., (2)where µ k is the transition dipole moment from | g i to | k i . E pu ( t ) is the temporal profile of the electric field of thepump pulse, E pu ( t ) = E (cid:16) f ( t ) e − i Ω t + f ( t − t ) e − i Ω ( t − t ) (cid:17) , (3)where Ω is the carrier frequency of the laser pulse. Here, f ( t ) is the pulse envelope, which is assumed to have aGaussian form, f ( t ) = (1 / √ πσ Ω ) e − t /σ , and E is theamplitude of the electric field. A fundamental quantityused to describe the optical properties of crystals is theelectric response function given by F ( t ) = X k | µ k | e − i ( ǫ k − ǫ g ) t/ ~ − η | t | / ~ , ( η = 0 + ) , (4)which is obtained via the Fourier transform of the effec-tive optical absorption spectrum I eff (Ω). t (3) t (1) t (4) t (2) t (3) t (2) t (1) t (4) (a) ISRS (b) IA FIG. 4. Double-sided Feynman diagrams for the density ma-trices corresponding to (a) the ISRS process and (b) the IAprocess. The thin and thick solid lines respectively representthe ground and excited states. The dashed curves representthe one-LO-phonon state. The red and blue Gaussian curvesrepresent the pulse envelope of the first and the second pulses,respectively, with the wavy lines their photon propagators.
We adopt the density matrix formalism to derive thegeneration amplitude of the coherent phonon. Thechange of the amplitude ∆ R of the reflectivity is pro-portional to the expectation value of the LO phonon co-ordinate Q = p ~ / ω (cid:0) b + b † (cid:1) except for constant factors.See Supplemental Material [34] for the formula of spec-trally resolved detection of reflectivity modulation. Fig-ure 4 presents double-sided Feynman diagrams for thegeneration by ISRS (Fig. 4 (a)) and IA (Fig. 4 (b)).In Fig. 4, the propagators shown by thin lines corre-spond to the ground state and those shown by bold linescorrespond to the excited state. The dashed lines rep-resent the one-phonon state. Note that the Hermitianconjugate terms arise from the processes in the diagramsin which the upper and the lower propagators are inter-changed, but these processes are ignored in Fig. 4 (a) forsimplicity.After a perturbation calculation, the amplitude of theoscillation of coherent phonons in the ISRS and IA pro-cesses, A ISRS and A IA are respectively given as A i ( t ) = L i (0) + e iωt L i (0) + e − i (Ω − ω ) t L i ( t )+ e i (Ω + ω ) t L i ( − t ) , (5)in which i = ISRS, IA and L ISRS ( x ) = 2 i Z ∞ du g ( u − x ) sin ωu e i Ω u F ( u ) , (6) L IA ( x ) = Z ∞−∞ du g ( u − x ) e i (Ω − ω ) u F ( u ) , (7)with g ( u ) = e − u / (2 σ ) , x = 0 , t , − t . The amplitude∆ R (0) is proportional to the absolute values of A i ( t ).The first, second, third and fourth terms in Eq. (5) cor-respond to the processes (1) to (4) in Fig. 4, respectively.Details of the calculation are shown in Supplemental Ma-terial [34].The actual calculation of the transient reflectiv-ity can be done for real materials if the electricresponse function F ( t ) is given. In the calcula-tion, we assumed a Lorentzian form, I eff (Ω) = I (Γ /π ) / (cid:8) (Ω − Ω ) + Γ (cid:9) , with ~ Ω = 1 .
55 eV andΓ = 0 .
015 eV based on the absorption spectra [44, 45].The calculated fringe patterns ∆ R are shown for ISRS(Fig.5(a)) and IA (Fig.5(b)). We found that the featuresin the fringe shown in Fig. 3 (a) are well reproduced if itis assumed that only the ISRS process contributes to thegeneration of coherent phonons. Furthermore, the overallline shape is in good agreement with experimental data .Most important is the fact that the feature of thecollapse and revival of the electronic fringe at around t ∼
55 fs arises only from the ISRS process, while theIA signal does not yield any such feature. This is due tothe quantum-path interference peculiar to ISRS. In Fig.4 (a), the contribution arising from diagrams (1) and (2)gives rise only to the interference of the phonon, which isdescribed by the first and second terms on the right-hand
Pump-Pump Delay t (fs) A m p lit ud e ( a r b . un it s ) Pump-Pump Delay t (fs) A m p lit ud e ( a r b . un it s ) FIG. 5. (a) Theoretical curve for the interference fringe inthe oscillation amplitude of transient reflectivity due to theISRS process at the LO phonon frequency plotted againstthe pump–pump delay ( t ). (b) Same as (a) but for the IAprocess. side of Eq. (5). The electronic interference arises fromdiagram (3) and (4), which corresponds to the third andfourth terms respectively in Eq. (5). It should be notedthat the fourth term in A ISRS ( t ) is negligibly small for t >
0. Therefore, In ISRS, the electronic interferencefringe appears only from the cross term between (1) +(2) and (3). At t = π/ω , this term vanishes owingto the destructive interference of the phonon. The high-frequency oscillation of the electronic fringe therefore dis-appears at t = π/ω = 55 fs. This is a manifestationof the path interference of the electronic and phononicdegrees of freedom in the dual-pump process peculiar tothe ISRS. Note that in the IA process, both the thirdand fourth terms in A IA ( t ) make a finite contributionso that the electronic fringe does not vanish at t = π/ω .In Fig. 5 (a), the amplitude of the electronic fringe be-comes small for t >
130 fs. This is due to the dephasingcaused by the inhomogeneous broadening of the contin-uous spectrum in the excited states. In the experimentalcurve in Fig. 3(a), the electronic fringe disappears almostcompletely for t >
130 fs in contrast to the case in Fig.5. The finding of the ISRS dominance in coherent phonongeneration in the opaque region is surprising because, inthe opaque region, the phonon generation intensity inthe IA process is generally estimated to be higher thanthat for ISRS [29, 30, 46]. We conjecture that even if thecoherent phonon may be generated in the excited statesubspace, its coherence is quickly lost because of the ul-trafast deformation of the adiabatic potentials due to theelectronic relaxation in the excited state of bulk materi-als. This may be one of the differences in the atomic andmolecular dynamics of solids compared with those of thegas phase, in which the excited electronic states are longprotected from relaxation. In addition, it was revealedthat the generation of the coherent phonon in GaAs isa quick process as deduced from ISRS dominance evenin the opaque region. The underlying mechanism is thequantum mechanically induced shift current.In summary, we made transient reflectivity measure-ments for n-GaAs using relative-phase-locked femtosec-ond pulses and found characteristic interference fringes,which are assigned to quantum-path interference in thegeneration of coherent phonons. Our observations andtheory revealed that the pathway of coherent phonon gen-eration in n-GaAs is ISRS at the displaced potential dueto the surface-charge field, even though the photon en-ergy lies in the opaque region. We demonstrated thatoptical dual-pulse pumping actively creates quantum-mechanical superposition of the electronic and phononicstates in a bulk solid.The authors thank K. Norimatsu, K. Goto, H. Mat-sumoto, and F. Minami for their help with the ex-periments and calculation. K. G. N., Y. S., and Y.K. thank K. Ohmori, H. Chiba, H. Katsuki, and Y.Okano of the Institute of Molecular Science for theirvaluable advice on the experiments. This work was par-tially supported by Core Research for Evolutional Sci-ence and Technology of the Japan Science and Tech-nology Agency, JSPS KAKENHI under grant numbers25400330, 14J11318, 15K13377, 16K05396, 16K05410,17K19051, and 17H02797, the Collaborative ResearchProject of Laboratory for Materials and Structures, theJoint Studies Program of the Institute of Molecular Sci-ence, National Institutes of Natural Sciences, and ThePrecise Measurement Technology Promotion Foundation. ∗ Corresponding author: [email protected] † [email protected][1] P. Brumer and M. Shapiro, Chem. Phys. Lett. , 541(1986).[2] S.. A. Rice, D. J. Tannor, and R. Kosloff, J. Chem. Soc.,Faraday Trans. , 2423 (1986).[3] N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J.Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming,and S. A. Rice, J. Chem. Phys. , 1487 (1991).[4] H. Katsuki, N. Takei, C. Sommer and K. Ohmori, Acc.Chem. Res. , 1174 (2018).[5] H. Mashiko, Y. Chisuga, I. Katayama, K. Oguri, H. Ma-suda, J. Takeda, and H. Gotoh, Nat. Commun. , 1468(2018).[6] D. R. Austin and I. A. Walmsley, in CLEO/Europe andEQEC 2011 Conference Digest, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CG P1.[7] Y.-X. Yan, E. B. Gamble, and K. Nelson, J. Chem. Phys. , 5391 (1985).[8] R. Merlin, Solid State Commun. , 207 (1997).[9] T. Dekorsy, G. C. Cho, and H. Kurz, in Light Scatter-ing in Solids III , (eds.) M. Cardona and G. G¨untherodt,(Springer, Berlin, 2000) pp. 169–209.[10] O. V. Misochko, J. Exp. Theo. Phys. , 246 (2001).[11] F. Randi, M. Esposito, F. Giusti, O. Misochko, F. Parmi-giani, D. Fausti, and M. Eckstein, Phys. Rev. Lett. ,187403 (2017).[12] F. Glerean, S. Marcantoni, G. Sparapassi, A. Blason,M. Esposito, F. Benatti, and D. Fausti, J. Phys. Bdoi:10.1088/1361-6455/ab0bdc (in press).[13] H. J. Zeiger, J. Vidal, T.K. Cheng, E. P. Ippen, G.Dresselhaus and M.S. Dresselhaus, Phys. Rev. B , 768(1992).[14] M. F. DeCamp, D. A. Reis, P. H. Bucksbaum, and R.Merlin, Phys. Rev. B , 092301 (2001).[15] H. Katsuki, J. C. Delagnes, K. Hosaka, K. Ishioka, H.Chiba, E. S. Zijlstra, M. E. Garcia, H. Takahashi, K.Watanabe, M. Kitajima, Y. Matsumoto, K. G. Naka-mura, and K. Ohmori, Nat. Commun. , 2801 (2013).[16] G. C. Cho, W. K¨utt, and H. Kurz, Phys. Rev. Lett. ,764 (1990).[17] T. Dekorsy, T. Pfeifer, W. K¨utt, and H. Kurz, Phys. Rev.B , 3842 (1993).[18] G. A. Garrett, T. F. Albrecht, J. F. Whitaker and R.Merlin, Phys. Rev. Lett. , 3661 (1996).[19] O. V. Misochko, K. Kisoda, K. Sakai, and S. Nakashima,Phys. Rev. B , 4305 (2000).[20] M. Hase, M. Kitajima, A. M. Constantinescu, and H.Petek, Nature , 51 (2003).[21] A. Q. Wu, X. Xu, and R. Venkatasubramanian, Appl.Phys. Lett. , 011108 (2008).[22] N. Kamaraju, S. Kumar, and A. K. Sood, Europhys.Lett. , 47007 (2010).[23] K. Norimatsu, J. Hu, A. Goto, K. Igarashi, T. Sasagawa,and K. G. Nakamura, Solid State Commun. , 58(2013).[24] O. V. Misochko, J. Flock, and T. Dekorsy, Phys. Rev. B , 174303 (2015).[25] D. M. Riffe and A. J. Sabbah, Phys. Rev. B , 085207(2007).[26] F. Sun. Q. Wu, Y. L. Wu, H. Zhao, C. J. Yi, Y. C. Tian,H. W. Liu, Y. G. Shi, H. Ding, X. Dai, P. Richard, andJ. Zhao, Phys. Rev. B , 235108 (2017).[27] T. Pfeifer, T. Dekorsy, W. K¨utt, and H. Kurz, Appl.Phys. A , 482 (1992).[28] A. V. Kuznetsov and C. J. Stanton, Phys. Rev. B ,7555 (1995).[29] T. E. Stevens, J. Kuhl, and R. Merlin, Phys. Rev. B ,144304 (2002).[30] K. G. Nakamura, Y. Shikano, and Y. Kayanuma, Phys.Rev. B , 144304 (2015).[31] M. Hase, M, Mizoguchi, H. Harima, S. Nakashima, M.Tani, K. Sakai, and M. Hangyo, Appl. Phys. Lett. ,2474 (1996).[32] S. Hayashi, K. Kato, K. Norimatsu, M. Hada, Y.Kayanuma, and K. G. Nakamura, Sci. Rep. , 4456(2014).[33] K. G. Nakamura, K. Ohya, H. Takahashi, T. Tsuruta,H. Sasaki, S.-I. Uozumi, K. Norimatsu, M. Kitajima, Y.Shikano, and Y. Kayanuma, Phys. Rev. B , 024303 (2016).[34] See Supplementary Material for the details of experimen-tal setup and sample, deformed harmonic potential anddetails of calculation of coherent phonon generation andspectrally resolved detection.[35] A. Mooradian and A. L. McWhorter, Phys. Rev. Lett. , 849 (1967).[36] J. D. Lee and M. Hase, Phys. Rev. Lett. , 235501(2008).[37] K. Ishioka, A. K. Basak, and H. Petek, Phys. Rev. B ,235202 (2011).[38] J. Hu, O. V. Misochko, A. Goto, and K. G. Nakamura,Phys. Rev. B , 235145 (2012). [39] H. Fr¨ohlich, Adv. Phys. , 325 (1954).[40] Y. R. Shen and N. Bloembergen, Phys. Rev. , A1787(1965).[41] F. Nastos and J. E. Sipe, Phys. Rev. B , 035201 (2006).[42] J. E. Sipe and A. I. Shkrebtii, Phys. Rev. B , 5337(2000).[43] A. V. Kuznetsov and C. J. Stanton, Phys. Rev. B ,10828 (1993).[44] M. D. Sturge, Phys. Rev. , 768 (1962).[45] H. C. Casey Jr., D. D. Sell, and K. W. Wecht, J. Appl.Phys. , 250 (1974).[46] Y. Kayanuma and K. G. Nakamura, Phys. Rev. B ,104302 (2017). r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Supplementary Material: Ultrafast Quantum-path Interferometry Revealing theGeneration Process of Coherent Phonons
Kazutaka G. Nakamura, ∗ Kensuke Yokota, Yuki Okuda, Rintaro Kase, TakashiKitashima, Yu Mishima, Yutaka Shikano,
2, 3, 4, 5 and Yosuke Kayanuma † Laboratory for Materials and Structures, Institute of Innovative Research,Tokyo Institute of Technology, 4259 Nagatsuta, Yokohama 226-8503, Japan Quantum Computing Center, Keio University, 3-14-1 Gakuen-cho, Yokohama, 223-8522, Japan Research Center for Advanced Science and Technology (RCAST),The University of Tokyo, 4-6-1 Komaba, Meguro, Tokyo 153-8904, Japan Research Center of Integrative Molecular Systems (CIMoS),Institute for Molecular Science, National Institutes of Natural Sciences,38 Nishigo-Naka, Myodaiji, Okazaki, Aichi 444-8585, Japan Institute for Quantum Studies, Chapman University,1 University Dr., Orange, California 92866, USA Graduate School of Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka, 599-8531 Japan (Dated: May 22, 2019)
I. EXPERIMENTAL SETUP
Ti:sapphirelaserCMShakerInterferometer Lens SamplePDBBO Band pass filter
PBS
PD1 PD2 (cid:16109)(cid:16153) (cid:16156)
Oscilloscope
Pump1 Pump2Probe t t Optical monitor (cid:16089)
RefrigeratorPD
Wave plate PolarizerWave platePolarizer
Lens
FIG. 1. Schematic of the experimental setup. CM, PD andBBO denotes chirped mirror, photodiode and BBO nonlinearoptical crystal.
A schematic of the pump-probe setup is shown in Fig.1. The output of the Ti:sapphire oscillator (center wave-length = 798 nm (1.54 eV), pulse width ∼
50 fs) wassplit with a partial beam splitter into two pulses; thesepulses were used as pump and probe pulses. The pumppulse was introduces into a scan delay unit (APE Co.,Scan Delay 150) to control the delay between the pumpand probe pulses. It was then introduced into a home-made Michelson-type interferometer to produce two rel-ative phase-locked pump pulses (pulse 1 and 2). Oneoptical arm of the interferometer was equipped with anautomatic positioning stage (Sigma Tech Co. Ltd., FS-1050SPX). The stage used is an active-feedback con-trolled stage (with a feedback stage controller FC-601A) ∗ Corresponding author: [email protected] † [email protected] with a minimum resolution of 5 nm and repetition-position accuracy of ± π for the 800-nmlight. The expanded figure of the optical interference be-tween 2 and 10 fs presented with circles and lines in Fig.2. I n t e n s it y ( a r b . un it s ) (fs) FIG. 2. Expanded figure of the optical interference between2 and 10 fs presented with circles and lines.
The power of pulses 1 and 2 were 18 and 21 mW, re-spectively. The power of the probe pulse was 8 mW.The relative phase-locked pump pulses were focused onthe sample using a lens, which also focused the probepulse (pulse 3). The sample was a single crystal of n-type GaAs with (100) orientation and set in a cryostatcooled by a refrigerator. The n-type GaAs was obtainedfrom DOWA Semiconductor Co. (Si doped with a car-rier concentration of 9 . × cm − , 350 − µ m thick-ness and mirror-polished surface). The refrigerator usedis Pascal-OP-S101 AD, which is the closed type GM cry-ocooler with a low-nose option in which the sample holderis apart from the cold head motor and attached to thelaser table. Temperature in the range between 10 and300 K was controlled by using Model 9700 (Scientific In-struments inc.). The pulse width was estimated fromthe linear optical interference and the frequency resolvedauto correlation (FRAC) for the pump pulses. The pumpand probe pulses were linearly polarized, and set alongthe (010) and (100) axis of GaAs, respectively. The re-flected pulse from the probe pulse was fed into a polariza-tion beam splitter. The parallel and perpendicular polar-ized light was monitored with two balanced photodiodes(PD1 and PD2). The differential signal from the PDswas amplified with a current amplifier (SRS Co., SR570)and averaged in a digital oscilloscope (Iwatsu Co., DS-4354M). The temporal evolution of the reflectivity change(∆ R/R ) was measured by scanning the delay ( t ) be-tween the pump 1 and probe pulses, repetitively, at 20Hz with a fixed t . To filter non-oscillatory backgroundin the temporal evolution, an electric bandpass filter (3-300 kHz) was used to amplify the oscillatory signals. Theoptical bandpass filter (the center wavelength of 800 nmwith the band width of 10 nm) was used for detectingthe reflected probe pulse in order to reduce cancellationeffects of Stokes and anti-Stokes components [1]. II. DEFORMED HARMONIC POTENTIAL
The harmonic potential representing the opticalphonon is deformed under a long range external field(e.g. the surface-charge field). Here we treat a simplecase for a uniform electric field ( F ( q ) = − dq ) actingalong the phonon coordinate ( q ). Although in realis-tic depletion layers the surface-charge field in nonuni-form and decreases on the length scale of a few microm-eters, the coherent carriers and phonon dynamics occurtypically much shorter length scale. Then the effectivecharge field can be treated as uniform in such a shortscale. When the external field potential is applied to aharmonic potential U ( q ) = kq /
2, the potential changesto U ′ ( q ) = U ( q ) + F ( q ); U ′ ( q ) = k q − dq = k (cid:18) q − dk (cid:19) − d k . (1)Then the potential minimum position and energy shift d/k and − d / (2 k ), respectively. The slope d of the ex-ternal fields in the electronic excite state is lower thanthat in the ground state, because the surface screening issuppressed by a electron-hole pair or electronic polariza-tion. Then the effective harmonic potential in the excitedstate is displaced from that in the ground state. E ne r g y ( a r b . un i t ) Position (arb. unit)
FIG. 3. The harmonic potential shifted by the external elec-tric field potential. The solid black curve, dotted red line andsolid red curve are the original harmonic potential ( U ( q )),external electronic field ( F ( q )) and the effective harmonic po-tential ( U ′ ( q )), respectively. III. CALCULATION OF COHERENT PHONONGENERATION
Here we describe details of the perturbation calculationof the density matrices for the coherent phonon genera-tion by dual pulses, and see the origin of the differencein the interference fringes due to ISRS and IA processes.The equation of motion for the wave function | ψ ( t ) i inthe Schr¨odinge picture, i ~ ddt | ψ ( t ) i = { H + H pump ( t ) } | ψ ( t ) i (2)is transformed into that in the interaction picture by thesubstitution | ˜ ψ ( t ) i ≡ exp[ iHt/ ~ ] | ψ ( t ) i as i ~ ddt | ˜ ψ ( t ) i = ˜ H pump ( t ) | ˜ ψ ( t ) i , (3)in which˜ H pump ( t ) = e iHt/ ~ H pump ( t ) e − iHt/ ~ = E pu ( t ) X k µ k exp[ i (cid:8) ǫ k + ~ ωb † b + α ~ ω ( b + b † ) (cid:9) t/ ~ ] × exp[ − i (cid:8) ǫ g + ~ ωb † b (cid:9) t/ ~ ] | k ih g | + H.c.. (4)Using a property of Boson operators,exp [ iωt (cid:8) b † b + α ( b + b † ) (cid:9) ] exp[ − iωtb † b ]= exp[ − iα ( ωt − sin ωt )] exp[ g ∗ ( t ) b − g ( t ) b † ] , (5)with g ( t ) = α (1 − e iωt ), we find˜ H pump ( t ) = E n f ( t ) e − i Ω t + f ( t − t ) e − i Ω( t − t ) o × X k µ k exp[ i ( ǫ k − ǫ g ) t/ ~ − iα ( ωt − sin ωt )] × exp[ g ∗ ( t ) b − g ( t ) b † ] | k ih g | + H.c.. (6)We calculate the density matrix to the second orderwith respect to E and to the first order with respect to α under the condition that it is given by | g, ih g, | at t = −∞ , where | ξ, n i denotes the ket vector for the n -phononstate ( n = 0 ,
1) in the ground state ( ξ = g ) or the excitedstate ( ξ = e ). For demonstration, the term correspondingto the paths (3) of ISRS shown in Fig. 4 (a) of the maintext will be derived. The other terms are calculated byapplication of the same technique. In the ISRS paths, weneed only to calculate the upper propagator (ket vector)since the system stays always in the ground states inthe lower one. In the path (3) of ISRS, the system isexcited to | e, i and | e, i by the first pulse at time t ′ anddeexcited to | g, i by the second pulse at t ′′ . The formalsolution to Eq. (3) written as | ˜ ψ ( t ) i = exp + [ − i ~ Z t −∞ ˜ H pump ( τ ) dτ ] | ˜ ψ ( −∞ ) i (7)is expanded to the second order terms. The term corre-sponding to the path (3) of ISRS is given by | ˜ ψ ( t ) i = α (cid:16) E ~ (cid:17) Z t −∞ dt ′′ Z t ′′ −∞ dt ′ f ( t ′′ − t ) f ( t ′ ) × F ( t ′′ − t ′ ) e i Ω ( t ′′ − t ′ − t ) ( e iωt ′′ − e iωt ′ ) | g, i . (8)The time-ordered integral can be reduced to a single in-tegral by introducing a pair of new variables ( s, u ) de-fined as s = ( t ′′ + t ′ ) / , u = ( t ′′ − t ′ ) /
2. If one notes( t ′′ − t ) + t ′ = 2( s − t ) + ( u − t ) , the integra-tion over s is carried out exactly and we find, except forconstant factors, | ˜ ψ ( t ) i = 2 ie − i (Ω − ω ) t Z ∞ due − ( u − t ) / (2 σ ) × sin ωu e i Ω u F ( u ) | g, i . (9)The amplitude of LO phonon oscillation is proportionalto the expectation value h g, | Q | ˜ ψ ( t ) i . This reproducesthe result in agreement with Eq. (6) in the main text.The process of path (4) in ISRS is anomalous , sincethe system is excited by the second pulse and deexcitedby the first pulse. Therefore, its contribution to thephonon generation is negligible except for the time regionwhere the delay t is very small. This is the reason whythe apparent collapse and revival of the electronic fringeemerges in ISRS. Since, in the ISRS, the rapid frequencyterm e − i Ω t appears alone without counter-rotatingterm, the rapid oscillation is realized only through thecross term with the slowly oscillating terms due to paths (1) and (2). Thus it disappears for t ≃ π/ω at whichthe phonon oscillation becomes suppressed through thedestructive interference of phonon. This is in sharp con-trast to the IA process. In the IA, both of the path (3)and (4) have the same order of amplitude so that theelectronic fringe does not disappear. The difference be-tween the ISRS and IA originates from the difference inthe integral domain over u = ( t ′′ − t ′ ) / IV. SPECTRALLY RESOLVED DETECTION OFREFLECTIVITY MODULATION
The interaction Hamiltonian for the probe pulse hasthe same form as Eq. (2) in the main text with theonly change that E pu ( t ) is replaced by E pr ( t ) = E f ( t − t ) e − i Ω t , where t is the delay time of the probe pulseand E is its amplitude, which is usually much smallerthan E . The coherent phonons generated by the pumppulses give rise to the periodic modulation ∆ P ( t ) of thepolarization due to the probe pulse. This results in theexcess loss or gain of the reflected probe pulse. As shownin Ref. [1], the spectrally resolved reflection modulation∆ R (Ω) for the detection frequency Ω and the delay t is given by∆ R (Ω , t ) = Im (cid:8) E ∗ pr (Ω)∆ P (Ω) (cid:9) , (10)where E pr (Ω) and ∆ P (Ω) are respectively the Fouriertransforms of E pr ( t ) and ∆ P ( t ). The spectrally re-solved reflectivity ∆ R (Ω , t ) is a quantity of order of | E | | E | α , and can be divided in two, ∆ R (Ω , t ) =∆ R ( a ) (Ω , t )+∆ R ( s ) (Ω , t ), where ∆ R ( a ) (Ω , t ) makesa main contribution to the anti-Stokes side, Ω ≃ Ω + ω ,while ∆ R ( s ) (Ω , t ) makes a main contribution to theStokes side, Ω ≃ Ω − ω . This is detected through thereflectivity change for the probe pulse that oscillates withthe frequency of the LO phonon as a function of t .After a perturbation calculation, we find, aside fromcommon constant factors,∆ R ( a ) i (Ω , t ) = − exp[ − σ { (Ω − Ω ) + (Ω − Ω − ω ) } ] × Re (cid:8) A i ( t ) e − iωt { χ (Ω − ω ) − χ ( ω ) } (cid:9) , (11)∆ R ( s ) i (Ω , t ) = − exp[ − σ { (Ω − Ω ) + (Ω − Ω + ω ) } ] × Re (cid:8) A ∗ i ( t ) e iωt { χ (Ω + ω ) − χ (Ω) } (cid:9) , (12)where χ (Ω) ≡ R ∞ e i Ω u F ( u ) du is the optical suscepti-bility and i = ISRS, IA correspond to the generationprocess ISRS and IA. In the above equations, A i ( t ) isa quantity proportional to the expectation value of theamplitude of coherent phonon after the irradiation of thedual pulses, which are given in Eq. (5) in the main text.As shown by the exponential factors, the above resultsindicate that the transient reflectivity oscillates with fre-quency ω both in the Stokes side and the anti Stokesside of the central frequency Ω of the probe pulse. Theamplitude of the oscillation is proportional to the abso-lute values of A ISRS ( t ) and A IA ( t ). But it’s relativephase is generally different depending on the complexamplitudes of A i ( t ) and χ (Ω). If χ (Ω) is a smoothlyvarying function, we may set χ (Ω ± ω ) − χ (Ω) = ± ∂χ (Ω) ∂ Ω ω . Furthermore in the transparent region, χ (Ω) is a realquantity. Then, the contribution from ∆ R ( a ) i (Ω , t ) and∆ R ( s ) i (Ω , t ) almost cancels out in the spectrally inte-grated measurement [1]. This is approximately true evenin the opaque region because of the cancellation due tothe mismatching of the phase. This is the reason whythe spectrally resolved measurement of transient reflec-tivity gives better quality of signals than the ordinaryspectrally integrated measurement. [1] K. G. Nakamura, K. Ohya, H. Takahashi, T. Tsuruta,H. Sasaki, S. Uozumi, K. Norimatsu, M. Kitajima, Y. Shikano, and Y. Kayanuma, Phys. Rev. B94