Coherent control theory and experiment of optical phonons in diamond
Hiroya Sasaki, Riho Tanaka, Yasuaki Okano, Fujio Minami, Yosuke Kayanuma, Yutaka Shikano, Kazutaka G. Nakamura
aa r X i v : . [ c ond - m a t . o t h e r] J un Coherent control theory and experiment of optical phonons in diamond
Hiroya Sasaki, Riho Tanaka, Yasuaki Okano, ∗ Fujio Minami,
1, 3
YosukeKayanuma,
1, 4
Yutaka Shikano,
5, 6, 7, 8, † and Kazutaka G. Nakamura ‡ Laboratory for Materials and Structures, Institute of Innovative Research,Tokyo Institute of Technology, 4259 Nagatsuta, Yokohama 226-8503, Japan Center for Mesoscopic Sciences, Institute for Molecular Science,National Institutes of Natural Sciences, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585, Japan Department of Physics, Graduate School and Faculty of Engineering,Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501 Japan Graduate School of Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka, 599-8531 Japan Quantum Computing Center, Keio University, 3-14-1 Hiyoshi, Kohoku, 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 Institute for Quantum Studies, Chapman University,1 University Dr., Orange, California 92866, USA 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 (Dated: June 26, 2018)The coherent control of optical phonons has been experimentally demonstrated in various physicalsystems. While the transient dynamics for optical phonons can be explained by phenomenologicalmodels, the coherent control experiment cannot be explained due to the quantum interference.Here, we theoretically propose the generation and detection processes of the optical phonons andexperimentally confirm our theoretical model using the diamond optical phonon by the double-pump-probe type experiment.
PACS numbers: 78.47.J-, 74.25.Kc
INTRODUCTION
Coherent control was originally developed for controlling chemical reactions using coherent two-photon processes,in which an electronic excited state was used as an intermediary to assist the chemical reaction to the electronicground-state potential surface.
Coherent control has been performed for other physical properties, for example,electronic, vibrational and rotational states of atoms and molecules and excitons, spins, and phonons in the solidstate and the superconducting electrical circuits.
The coherent control of optical phonons was first demonstrated in the molecular crystals at cryogenic temperatureusing multiple femtosecond pulses. This was well explained by an impulsive stimulated Raman scattering (ISRS)mechanism to generate the coherent optical phonon. However, the similar coherent control experiments with doublefemtosecond pulses were performed on semimetal films to be explained by a displacive excitation mechanism. To understand the unified generation mechanism of coherent phonons, the microscopic theory based on quantumdynamics is required.
However, in the coherent control experiment, the amplitude and phase dependences havebeen not yet understood from these microscopic theories.The aim of this paper is to theoretically propose the unified process included the generation and detection of thecoherent optical phonons from quantum dynamics under a two-electronic-level and a displaced harmonic oscillatormodel under the off-resonant condition to extend Ref. 36. The effect of detuning and the control scheme are discussed.In addition, we demonstrate the coherent control of 40 THz optical phonons in diamond using a pair of sub-10-fsoptical pulses by the ISRS process since the band gap of diamond is well above the energy of a commonly usedfemtosecond laser pulse.As an application to quantum information technology, diamond is expected to be applied to quantum memory usingthe nitrogen or silicon-vacancy center in diamond and the optical phonon since the high-purity materialis available and it is working at room temperature. On the other hand, the phonon property of diamond has beendiscussed in the context of photophysics.
To understand the coherence of the optical phonon fundamentally andpractically, our coherent control scheme might be helpful.
RESULTSTwo-electric-level coherent-phonon generation and detection model
It was shown that the generation and detection processes of coherent phonons can be described by the two-banddensity matrix formalism with the optical response function. It was assumed that the band-gap energy is modulatedby the coherent oscillation of the optical phonon due to the deformation potential interaction. In the case of excitationto the transparent wavelength region in diamond treated in the present work, we may adopt a much simplified versionof this theory.Let us consider a two-level system for the electronic state coupled with a harmonic oscillator.
The Hamiltonianis given by H = H g | g ih g | + ( ǫ + H e ) | e ih e | ,H g = ~ ωb † b,H e = ~ ωb † b + α ~ ω (cid:0) b + b † (cid:1) , (1)where the state vector | g i refers to the electronic ground state of the crystal, and | e i refers to the electronic excitedstate with the excitation energy ǫ . In the case of diamond, | e i corresponds representatively to the electronic statesabove the direct band gap, and ǫ is approximately equal to the direct band gap energy 7 . The Hamiltonian H g and H e are the phonon Hamiltonians in the subspaces | g i and | e i . Here we have introducedthe annihilation and the creation operator b and b † for the interaction mode which is defined as a linear combinationof the normal modes lying close to the Γ point in the Brillouin zone. Because of the phase-matching condition, thewave vector of the phonon is equal to the wave vector of the incident photon modulated by the by refractive indexof the crystal. In the case of coherent phonons, the incident pulse is decomposed into a linear combination of planewaves around the central mode. Therefore, the wave vector of the coherent phonon is also distributed over a smallregion around Γ point in the Brillouin zone. The dispersion of the optical phonon energy near the Γ point is neglected,and we set the energy of the interaction mode ~ ω is equal to the optical phonon energy at Γ point. In the case ofdiamond, ω is evaluated as ∼ π × Note that the transverse and longitudinal optical phonon cannot bedistinguished at Γ point because of a non-polar material. The dimensionless coupling constant is denoted by α . Inthe bulk crystal, the Huang–Rhys factor α is considered to be small; α ≪
1. For simplicity, we consider a four-statemodel with two phonon states for each electronic state: | g, i and | g, i for phonon Fock states with n = 0 and 1 inthe electronic ground state and | e, i and | e, i for those in the electronic excited state. The interaction Hamiltonianwith the optical pulse is given by H I = µE f ( t )( e − i Ω t | e ih g | + e i Ω t | g ih e | ) , (2)in which µ is the transition dipole moment from | g i to | e i , and E , Ω and f ( t ) are the strength, central frequencyand temporal profile of the electric field of the pump pulse, respectively. The time evolution of the density operatorwas obtained by solving the quantum Liouville equation using a perturbative expansion in the lowest order.We restricted the well separated pulses for the two pump pulses (pump 1 and pump 2) and the probe pulse.Then the generation and detection processes were separately treated, which corresponded to a doorway-windowpicture in nonlinear spectroscopy. When pump 1 and 2 were well separated, the excitation of the optical phononsoccurred with each pulse. The pathway of electronic excitation by pump 1 and the deexcitation by pump 2 wasnot allowed for the off-resonant condition. We set the initial state in | g, i , then ρ ( −∞ ) = | g, ih g, | . This wasreasonable for the diamond case, because the population number in the n = 1 state was approximately 0 .
005 at 300K. There were four Liouville pathways for the exciting phonon polarization: | g, ih g, | → | e, ih g, | → | g, ih g, | , | g, ih g, | → | e, ih g, | → | g, ih g, | and Hermitian conjugates for each pump pulse.The density operator for the excitation by pump 1 ρ (2)1 ( t ) was obtained as ρ (2)1 ( t ) = α µ | E | ~ e − iωt Z ∞−∞ dt ′ Z t ′ −∞ dt ′′ f ( t ′ ) f ( t ′′ ) × (cid:16) e iωt ′ − e iωt ′′ (cid:17) e − i ∆( t ′ − t ′′ ) | g, ih g, | , (3)where | E | is the strength of the electric field of pump 1, ∆ ≡ ǫ/ ~ − Ω is the detuning, and we assumed that t is wellafter the passage of the pump pulse.In the case for far off-resonance excitation, the density matrix can be evaluated as follows. For simplicity, we assumea Gaussian pulse with pulse-width σ , f ( t ) = 1 √ πσ Ω exp (cid:18) − t σ (cid:19) , (4)where Ω is used to make the normalization factor, √ πσ Ω, dimensionless. Following the calculations to Ref. 36, weobtain ρ (2)1 ( t ) = iα µ | E | ~ e − iωt ω √ σ Ω ∆ e − σ ω / | g, ih g, | . (5)The density operator for the excitation by pump 2, ρ (2)2 ( t ), was obtained in a similar calculation, and we obtain thedensity operator ρ (2) ( t ) = ρ (2)1 ( t ) + ρ (2)2 ( t ) ρ (2) ( t ) = iα µ ~ ω √ σ Ω ∆ e − σ ω / × (cid:16) | E | e − iωt + | E | e − iω ( t − τ ) (cid:17) | g, ih g, | = iA (cid:0) | E | + | E | e iωτ (cid:1) × e − iωt | g, ih g, | , (6)where A ≡ α µ ~ ω √ σ Ω ∆ e − σ ω / , (7) τ is the delay between the pump 1 and the pump 2, which is called the pump-pump delay, and E is the electric fieldstrength of pump 2.The coherent phonon dynamics can be investigated by calculating the mean value of the phonon coordinate h Q ( t ) i =Tr { Qρ (2) ( t ) } , where Q ≡ p ~ / ω (cid:0) b + b † (cid:1) and Tr indicate the trace. By considering the Hermitian conjugated paths,we obtain h Q ( t ) i = r ~ ω α µ ~ ω √ σ Ω ∆ e − σ ω / × (cid:8) | E | sin( ωt ) + | E | sin( ω ( t − τ ) (cid:9) = r ~ ω A (cid:8) | E | sin( ωt ) + | E | sin( ω ( t − τ ) (cid:9) . (8)Therefore, the amplitude of the phonon oscillation controlled by the two short pulses is expressed by a sum ofthe two sinusoidal functions. The phonon amplitude is enhanced by two times or canceled when the pump delaymatches an integer or half-integer multiple of the vibrational period through constructive or destructive interference,respectively, at the | E | = | E | condition. Note that the amplitude of oscillation is inversely proportional to thesquare of detuning from the excited state.When the heterodyne detection of the transmitted light is investigated, the detection intensity I h ( t ) should be I h ( t ) = Ω l × Im [ E ∗ ( t ) P s ( t )] , (9)where E ( t ) is the strength of the electronic field of the probe pulse, P s ( t ) is the polarization at time t, and l is thethickness of the sample. The probe pulse irradiates the sample at pump-probe delay t p . There are eight Liouvillepathways for the exciting phonon polarization: path 1: | g, ih g, | → | e, ih g, | → | g, ih g, | , path 2: | g, ih g, | → | e, ih g, | → | g, ih g, | , path 3: | g, ih g, | → | g, ih e, | → | g, ih g, | , path 4: | g, ih g, | → | g, ih e, | → | g, ih g, | ,and their Hermitian conjugates.We obtain ρ (3)1 ( t ′ ) for the path 1 as ρ (3)1 ( t ′ ) = iA ( | E | + | E | e iωτ ) iµ ~ E e − iωt p × Z t ′ −∞ dt ′′ f ( t ′′ ) e − iωt ′′ e − i Ω t ′′ e − i ( ǫ + ~ ω )( t ′ − t ′′ ) / ~ × | e, ih g, | , (10)where f ( t ′′ ) is the Gaussian pulse and t p is the pump-probe delay. For the polarization operator P op = µ | g ih e | + µ ∗ | e ih g | , the complex polarization at time t is given by P ( t ) = Tr { ρ (3) ( t ) P op } . Then the polarization ( P ( t ′ )) for thepath 1 is given by P ( t ′ ) = αA ( | E | + | E | e iωτ ) µ ~ E e − iωt p × Z t ′ −∞ dt ′′ f ( t ′′ ) e − iωt ′′ e − i Ω t ′′ e − i ( ǫ + ~ ω )( t ′ − t ′′ ) / ~ , (11)and the time-integrated intensity, I ( t p ), of the product between the probe light and polarization is I ( t p ) = Z ∞−∞ E f ∗ ( t ′ ) P ( t ′ ) dt ′ = αA ( | E | + | E | e iωτ ) µ ~ | E | e − iωt p × Z ∞−∞ dt ′ Z t ′ −∞ dt ′′ f ( t ′ ) e i Ω t ′ f ( t ′′ ) e − iωt ′′ × e − i Ω t ′′ e − i ( ǫ + ~ ω )( t ′ − t ′′ ) / ~ . (12)Using the Gaussian pulse (4), we obtain I ( t p ) = αA ( | E | + | E | e iωτ ) µ ~ πσ Ω | E | e − iωt p × Z ∞−∞ e − s /σ e − iωs ds Z ∞ due − u / (2 σ ) e i (∆ − ω/ u ≈ ( | E | + | E | e iωτ ) µ ~ πσ Ω | E | × e − iωt p √ π √ σ e − σ ω / iαA ∆ − ω/ iB ( | E | + | E | e iωτ ) e − iωt p − ω/ , (13)where s ≡ ( t ′ + t ′′ ) / , u ≡ t ′ − t ′′ , and B ≡ √ πωα µ σ Ω ∆ ~ | E | e − ω σ / . (14)A similar calculation shows that I ( t p ) = I ( t p ) and I ( t p ) = I ( t p )= − iB ( | E | + | E | e iωτ ) e − iωt p
1∆ + ω/ . (15)Then we find I ( t p ) = X I i ( t p )= ( | E | + | E | e iωτ ) e − iωt p iBω ∆ . (16)By considering the Hermitian conjugate paths, the time-integrated intensity of the heterodyne detection I h ( τ, t p ) isgiven by I h ( τ, t p ) = √ πω α µ σ Ω ∆ ~ | E | e − ω σ / × {| E | sin( ωt p ) + | E | sin( ω ( t p − τ )) } = C ( τ ) sin( ωt p − Θ( τ )) , (17)where C ( τ ) = √ πω α µ σ Ω ∆ ~ | E | e − ω σ / × | E | s | E | | E | cos( ωτ ) + (cid:18) | E | | E | (cid:19) , (18)Θ( τ ) = arctan sin( ωτ )cos( ωτ ) + | E | | E | . (19)The present model calculation clearly shows that the response of the transmitted light intensity measured withheterodyne detection exhibits the same dependence on the pump-pump delay as that of the mean value of the phononcoordinate. Single-pump transmission experiment
In the followings, the experimental detection of the optical phonons was performed using a pump-probe typetransient transmittance measurement with femtosecond pump pulses, see the details in Methods. The transienttransmittance change induced by only pump 1 or pump 2 were measured in Fig. 1 (a) and (b), respectively, againstthe pump-probe delay t p between −
200 and 1000 fs. It is noted that time zero was set at the time when pump2 irradiates the sample; the minimum portion of the sharp response. After the sharp peak, which arose from thenonlinear response for overlapped pump and probe pulses, there was a modulation caused by the coherent opticalphonons in diamond. The oscillation period was 25 . ± .
03 fs (frequency of 39 . ± .
05 THz). The coherent oscillationin the transmitted pulse intensity arising from the optical phonons was the same as that obtained by the reflectionexperiments. !!"!! $ % & ’ & ’ ( ) * + ,- . / ( ( $.*+(+&607(8 89(:(*; FIG. 1. Transient transmittance change of diamond. The oscillation (a) is excited by the first pump pulse only, and theoscillation (b) is excited by the second pump pulse only. It is noted that the baseline in our previous experiment, seems tomore flat compared to the present one. This is because the experimental data shown in Ref. 49 has been already subtractedby the smoothing curve of the obtained experimental data to easily analyze this. ! "!" $ % & ’ ( ) * ++ & ’ , - . , / & ’ - . " . " . ; .2.A( FIG. 2. The pump power dependence of the amplitude of the 40 THz oscillation. It is noted that the statistical average was4 ,
800 signals.
To verify the theoretical treatment on the detection process in the previous section, Figure 2 shows a pump laserpower dependence of the oscillation amplitude of the 40 THz oscillation in the transmittance spectrum. Accordingto Eq. (17) with | E | = 0, the oscillation amplitude is a linear dependence on the pump laser intensity | E | . Thisis well agreement with our experiment data. It is noted that the deviation between our prediction and experimentaldata has not yet been identified such as the laser power and measurement-setup stability. Coherent control experiment
Figure 3 shows typical examples of the transient transmittance changes induced by the pair of pump pulses (pump1 and 2). Pump 1 induces a coherent oscillation in the transmission intensity with a frequency of 39 . ± .
05 THz.This oscillation amplitude was controlled by pump 2. It was reduced at τ = 237 . τ = 251 . τ = 263 . τ in Fig. 4(a). We also estimatedthe initial phase of the oscillation after pump 2 by extrapolation of the fitted sinusoidal function at the timing of thepump 2 irradiation. The estimated initial phase is plotted in Fig. 4 (b). Measurement error to define the timingof the pump 2 irradiation was approximately ± . ± . π for the phase of the 39 . !" % & ’ ( ) *+ , - . / ) ( *+ . - /
3$ 3" $$ $" 4$ 4" 5$ 5" = > + + ) * / ? ) ; - / @ / / A B)CBDC
FIG. 3. Transient transmittance change along the pump-probe delay (between pump 2 and the probe) for several pump-pumpdelays ( τ between pump 1 and pump 2): at 237 . . . . . In Fig. 4, the phonon amplitude was normalized by the phonon amplitude excited by pump 1, which was observedat the pump 2-probe delay between −
270 and 0 fs. The phonon amplitude was enhanced almost twice at τ = 251 . τ = 237 . . . Therefore, the coherent optical phonons should be excited by the ISRS process at an off-resonant condition. Theinitial phonon state at room temperature was well expressed with the n = 0 state, because the phonon energy (39 . ≃
135 meV) was higher than the thermal energy ( ∼
25 meV) and the population ratio between the groundand excited state of the optical phonon was 0 . | E | / | E | = 1 .
09 and approximately agreed with the amplitude ratio of the coherent phonon,∆ T / ∆ T = 0 . ± .
03 according to the single-pump experiment in Fig. 1. The transmission intensity changedepending on the pump-pump delay τ was calculated using Eq. (17) and the frequency ( ω = 2 π × . !""! ’ ( ) * + ,- . / - / - FIG. 4. The amplitude (a) and phase (b) of the controlled oscillation after pump 2 against the pump-pump delay τ . Theamplitude is normalized using that obtained after excitation after only pump 1; oscillation between the pump 1 and pump 2irradiation timing. Solid circles are the experimental data and the solid curves are obtained by calculation using Eqs. (18) and(19) with | E | / | E | = 1 . ω = 2 π × . − . π . was also obtained from the calculation and shown in Fig. 4(b). Our proposed model for the coherent control of theoptical phonons reasonably represents the experimental data. It is noted that the initial phase shift − . π , whichcorresponds to 7 . DISCUSSIONS
In summary, we investigated the coherent control of the optical phonons using a pair of optical pulses with twoelectronic levels and two harmonic phonon levels. The calculations showed that the controlled phonon amplitudeand transmission intensity can be expressed by the sum of two sinusoidal functions. Furthermore, we demonstrateda coherent control of the optical phonons in a single crystal diamond. We used a pump and probe protocol andthe change in the transmitted light intensity was determined with heterodyne detection. The phonon amplitude wascoherently controlled by changing the the pump-pump delay from 230 fs to 270 fs. The control scheme was wellexplained by our theoretical generation and detection model with the interference between the two phonon statesexcited by each pump pulse.The wave packet dynamics of the coherent optical phonons is only measured in the transmission intensity change.Therefore, the amplitude and the phase of the wave packet cannot be individually controlled. To reproduce thewave packet of the coherent phonon, the transmittance and reflectivity changes should be simultaneously measured.According to the Kramers-Kr¨onig relation, the transient complex dielectric constant can be measured. The opticalphonon amplitude is also measured by combining to the Raman spectroscopy. Furthermore, there still are openquestions on the nonlinear response of the optical phonon. The coherent control of the optical phonon around τ ∼ METHODS
The experimental setup has been described in the previous paper in addition to that the optical pulses were gen-erated by using a home-made Michelson-type interferometer. While the transient reflectivity change was measuredin Ref. 49, in this experiment, the transient transmittance change is measured. According to Fig. 5, the ultrafastlaser conditions measured immediately behind the output port were a maximum-intensity wavelength of 792 nm witha estimated pulse width of 8 . ,
200 signals were averaged and taken asthe measured value. By converting the temporal motion of the scan delay unit to the pump-probe delay, the temporalevolution of the change in the transmitted light intensity, ∆ T ( t p ) /T , was obtained. Here we used the heterodynedetection technique. The powers of the pump 1 and 2 and the probe were 19 . . . × and 0 . − ! "!" $ % & ’ ( ) * ++ & ’ , - . , / & ’ - . " . " . ; .2.A( FIG. 5. The measured spectrum property of ultrafast laser.
ACKNOWLEDGMENTS
The authors thank Yuki Okuda and Mayuko Kato for technical assistance. This work was supported in part by CoreResearch for Evolutional Science and Technology (CREST) of the Japan Science and Technology Agency (JST), JSTERATO (Grant No. JPMJER1601), JSPS KAKENHI Grant Numbers 15H02103, 15K13377, 16K05410, 17K19051,and 17H02797, Collaborative Research Project of Laboratory for Materials and Structures, Institute of InnovativeResearch, Tokyo Institute of Technology, and Joint Studies Program of Institute for Molecular Science.
Author Contribution
Y.S. and K.G.N. conducted the project. H.S. and R.T. measured the experimental data. H.S., R.T., and Y.O.analyzed the experimental data under the guidance of F.M. and Y.S.. K.G.N. calculated the theoretical model withthe help of Y.K. and Y.S.. Y.S. and K.G.N. mainly wrote the manuscript. All authors discussed the results andcommented on the manuscript.0
Competing Interests
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