All-optical control and visualization of ultrafast two-dimensional atomic motions in a single crystal of bismuth

Abstract

In a bulk solid, optical control of atomic motion provides a better understanding of its physical properties and functionalities. Such studies would benefit from active control and visualization of atomic motions in arbitrary directions, yet, so far, mostly only one-dimensional control has been shown. Here we demonstrate a novel method to optically control and visualize two-dimensional atomic motions in a bulk solid. We use a femtosecond laser pulse to coherently superpose two orthogonal atomic motions in crystalline bismuth. The relative amplitudes of those two motions are manipulated by modulating the intensity profile of the laser pulse, and these controlled motions are quantitatively visualized by density functional theory calculations. Our control-visualization scheme is based on the simple, robust and universal concept that in any physical system, two-dimensional particle motion is decomposed into two orthogonal one-dimensional motions, and thus it is applicable to a variety of condensed matter systems.

ARTICLE

Received 8 Jun 2013 | Accepted 22 Oct 2013 | Published 18 Nov 2013

H. Katsuki1,2,w, J.C. Delagnes1,w, K. Hosaka1,w, K. Ishioka3, H. Chiba1,4, E.S. Zijlstra5, M.E. Garcia5,H. Takahashi4,6, K. Watanabe7, M. Kitajima8,w, Y. Matsumoto7, K.G. Nakamura4,6 & K. Ohmori1,2,4

In a bulk solid, optical control of atomic motion provides a better understanding of its physical properties and functionalities. Such studies would benet from active control and visualization of atomic motions in arbitrary directions, yet, so far, mostly only one-dimensional control has been shown. Here we demonstrate a novel method to optically control and visualize two-dimensional atomic motions in a bulk solid. We use a femtosecond laser pulse to coherently superpose two orthogonal atomic motions in crystalline bismuth. The relative amplitudes of those two motions are manipulated by modulating the intensity prole of the laser pulse, and these controlled motions are quantitatively visualized by density functional theory calculations. Our control-visualization scheme is based on the simple, robust and universal concept that in any physical system, two-dimensional particle motion is decomposed into two orthogonal one-dimensional motions, and thus it is applicable to a variety of condensed matter systems.

1 Department of Photo-Molecular Science, Institute for Molecular Science, National Institutes of Natural Sciences, Myodaiji, Okazaki 444-8585, Japan.

2 The Graduate University for Advanced Studies (SOKENDAI), Myodaiji, Okazaki 444-8585, Japan. 3 Advanced Nano-Characterization Center, National Institute for Materials Science, Tsukuba 305-0047, Japan. 4 CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan. 5 Theoretical Physics and Center for Interdisciplinary Nanostructure Science and Technology (CINSaT), University of Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany. 6 Materials and Structures Laboratory, Tokyo Institute of Technology, Yokohama 226-8503, Japan. 7 Department of Chemistry, Kyoto University, Kyoto 606-8502, Japan. 8 National Defense Academy of Japan, Yokosuka 239-8686, Japan. w Present addresses: Nara Institute of Science and

Technology (NAIST), 8916-5 Takayama-cho, Ikoma 630-0192, Japan (H.K.); CELIA, CNRS-University of Bordeaux, 351 Cours de la Libration, F-33405 Talence cedex, France (J.C.D.); Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan (K.H.); Tokyo Institute of Technology, Yokohama 226-8503, Japan; LxRay Co., Ltd., Koshien-2Bancho, Nishinomiya, Hyogo, 663-8172, Japan; University of Tsukuba, Ten-nodai 1-1-1, Tsukuba, Ibaraki, 305-8574, Japan (M.K.). Correspondence and requests for materials should be addressed to K.O. (email: mailto:[email protected]

Web End [email protected] ).

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DOI: 10.1038/ncomms3801 OPEN

All-optical control and visualization of ultrafast two-dimensional atomic motions in a single crystal of bismuth

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Atomic motions in the solid state can be triggered with optical pulses. If the pulse duration is shorter than the periods of lattice vibrations, the atoms start oscillating

collectively. This collective oscillation is referred to as coherent phonons1,2. Coherent phonons have been observed in a wide variety of materials316 and have been controlled one-dimensionally with a train of multiple laser pulses39. One of these control experiments was demonstrated with two different phonon modes. The directions of atomic motions are, however, parallel between those two different modes, so that the control was one-dimensional8. Two-dimensional (2D) control has been performed with a pair of identical laser pulses with different polarizations applied to energetically degenerate orthogonal atomic motions10. Another approach is necessary to control orthogonal non-degenerate atomic motions, which is more common in most materials. Such 2D control should be important in photo-induced processes17,18 and could lead to further development of optically engineered functionality of bulk solids19,20.

Here we demonstrate 2D control of atomic motions in crystalline bismuth. It is known that superposing two chirped femtosecond laser pulses causes intensity modulation whose characteristic period lies within the terahertz (THz) region21,22. This method is applied to atomic motions in the current study. Two orthogonal phonon modes of A1g and Eg symmetries23 are superposed with the modulated pulse and the transient relative reectivity change of the crystal surface is measured with another femtosecond probe pulse. The reectivity changes show beat structures whose temporal evolution changes drastically as we change the delay between the chirped femtosecond pulses on the attosecond timescale. We map these beat structures into a 2D space of atomic displacements with our density functional theory (DFT) calculations, demonstrating direct 2D control of atomic motion in each unit cell and its visualization.

ResultsCoherent phonons in crystalline bismuth. The crystal structure of bismuth is rhombohedral24 and its primitive unit cell consists of two atoms as shown in Fig. 1a. There exist three optical phonon modes with A1g and doubly degenerate Eg symmetries at the G-point in the reciprocal lattice space23. According to our denition of the axes as given in Fig. 1a, the A1g mode

corresponds to the longitudinal motion and the Eg mode corresponds to the lateral motion of the Bi atoms.

Experimental setup. Figure 1b schematically illustrates our pumpprobe experiments performed on the (0001) surface of a single-crystal Bi. The crystal was cooled in a liquid helium cryostat to 5.2 K to reduce damping and dephasing of the coherent phonons. The output of a Ti:sapphire laser (centre wavelength: B802 nm, bandwidth: B34 nm) was split with a partial beam splitter into two beams with 9:1 ratio, which were used as pump and probe pulses, respectively. The pump pulse was input to our homemade highly stabilized Michelson-type interferometer25,26 to produce a phased pair of laser pulses. The intensity of each pulse was B6 mW (uence: B2 mJ cm 2 on the sample). These pulses (pulse width: B30 fs in their Fourier transform (FT) limits) were chirped with transmitting optics inserted on their optical path and are hereafter referred to as chirped subpulses (CSPs)27. The group delay dispersion (GDD) parameter of each CSP is hereafter referred to as f00. Those CSPs were temporally superposed with their delay tmod to produce a modulated pump pulse whose temporal intensity envelope I(t) shows periodical undulations. The FT of I(t) shows a single peak in the THz region where its centre frequency and peak width are

functions of tmod and f00. A combination of these two degrees of freedom in the centre frequency and peak width allows for arbitrary tuning of the relative intensities of any two different frequency components in the THz region. Superposing CSPs thus serves as a practical technique for THz intensity modulation with near-infrared laser pulses. The train of femtosecond pulses has often been used in the previous experiments to control coherent phonons one-dimensionally39. The difference between such a pulse train and the current CSPs will be described later from the viewpoint of 2D control. The polarization of the modulated pump pulse was rotated by 90, so that the pump and probe pulses were focused on the Bi sample with their polarizations perpendicular to each other. The reectivity change DR of the probe pulse induced by the modulated pump pulse was monitored with a pair of balanced photodiodes (PD1 and PD2). The differential signal from the PDs was amplied with a current amplier (SRS SR570) and averaged in a digital oscilloscope. The temporal evolution of DR was measured by scanning the delay tprobe of the probe pulse

from the rst CSP repetitively at 20 Hz (APE Scan Delay) with a

z

x y

A1g (longitudinal)

[afii9838]/2

Eg (transverse)

Interferometer

[afii9848]mod

Pump pulse

SQ plates

Probe pulse

[afii9848]probe

Fast scan unit

Trigger

PD3

ND filter

PD2

CM

PD1

Lens (f=100)

Cryostat

Ti:sapphire oscillator

Current amplifier

Bi

R

Oscilloscope

20

10

R/R(x106 )

0

10

20

30

2

0

2

4

6

8

10

12

Probe delay (ps)

Figure 1 | Scheme of the present experiment. (a) Crystal unit cell structure of Bi and the orientation of the A1g (longitudinal) and Eg (transverse) phonon motions. (b) Experimental conguration for the modulated pumpprobe reectivity measurement. CM, chirp mirror; PD, photodiode; ND lter, neutral density lter; SQ plates, window plates of synthesized quartz substrate. (c) Oscillatory coherent phonon signal observed with tmod B0 fs. The origin of the probe delay tprobe 0 ps is

dened as the top of the rst positive peak of the oscillatory phonon signal.

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xed tmod. A bandpass lter from 3 to 300 kHz was used in all of the measurements reported in this study, so that the non-oscillatory background in the temporal evolution, which corresponds to the featureless electronic response, was ltered out. The intensity of the modulated pump pulse was monitored with a photodiode (PD3) at each tmod. Other details of signal processing are given in the Methods section.

The temporal full width at half maximum (FWHM) of the probe pulse at the sample was measured to be B50 fs, giving its

GDD parameter f00B430 fs2. The FWHM and GDD parameter of the CSPs were estimated from the cross-correlation between one of the CSPs and the probe pulse to be B260 fs and f00B2,600 fs2, respectively.

Control of 2D atomic motions. Figure 1c shows an example of the temporal evolution DR(tprobe)/R with tprobe scanned and tmod

xed to B0 fs. The beat structure with a period of B330 fs arises from coherent phonon motion and is described with a linear combination of two damped harmonic oscillators corresponding

to A1g (longitudinal) and Eg (transverse) phonons23:

DRt=R A0

DRa

R exp Gat

sin 2pnat da

DRe

R exp Get

sin 2pnet de

;

1

where A0 is the baseline, and DRi/R, Gi, ni and di with {i} {a, e}

are the phonon amplitudes, damping factors, oscillation frequencies and initial phases of A1g and Eg phonons, respectively.

Assuming each CSP has a Gaussian spectral prole with the GDD parameter f00, the spectrum of the electric eld is described as28,

CSPo E0e 2 ln 2

o oL

Do

f00

2e i

2 o oL

2 ; 2

where oL is the central angular frequency of the pulse and Do is its spectral FWHM. The modulated pump pulse is the superposition of two CSPs with their interpulse delay tmod, so

that its intensity envelope I(t) is

It j ECSPt E CSPt tmod j 2; 3

70

Intensity (a.u.)

100

80

60

40

20

0

4

3

2

1

0

60

3.0

2.0

1.0

0.0

R/R(x106 )

50

40

30

20

10

0

10

600

400

200

200

400

600

0

Time (fs)

-2

0

2

4

6

8

10

12

Intensity (a.u.)

(x5)

[afii9840](Eg)

Intensity (a.u.)

[afii9840](A1g)

2

0

2

4

6

8

10

12

4.0

3.0

2.0Frequency (THz)

0

4

6

8

Probe delay (ps)

(x5)

2 THz

1.0

1.0

1.0

Longitudinal displacement (fm)

(A 1gmode)

0.5

0.5

0.5

0.0

0.0

0.0

0.5

0.5

0.5

1.0

1.0

1.0

0.2 0.1 0.0 0.1 0.2

Longitudinal displacement (fm)

(A 1gmode)

0.2 0.1 0.0 0.1 0.2

Longitudinal displacement (fm)

(A 1gmode)

0.2 0.1 0.0 0.1 0.2

Lateral displacement (fm) (Eg mode)

Lateral displacement (fm) (Eg mode)

Lateral displacement (fm) (Eg mode)

Figure 2 | Temporal evolution of coherent phonons and 2D atomic motions controlled with the CSP technique. (a) Simulated temporal pulse envelopes for three different CSP delays. Green: tmod 49.4 fs. Blue: tmod 92.2 fs. Red: tmod 93.6 fs. The dotted curve is the temporal intensity of CSP pair with

tmod

0 fs. The pulse parameters are 2pc/oL 802 nm (c is the speed of light in air), Do/2p 15.7 THz (corresponding to 34 nm FWHM), and

f00 2,600 fs2, respectively. (b) Fourier-transformed THz intensity spectra calculated from the temporal pulse envelopes of a. (c) Observed coherent

phonon signals for three different tmod similar to those for the simulations shown in a and b. Inset: 5 expanded view of the blue trace. The origin of the

probe delay tprobe 0 ps is dened as the top of the rst positive peak of the green trace. (d) Fourier-transformed THz spectra of the coherent

phonon signals given in c. The blue trace is multiplied by 5. The peak at B2.1 THz corresponds to the Eg motion and the peak at B3.0 THz corresponds to the A1g motion. (eg) Traces of the atomic motions within a unit cell converted, respectively, from the green, blue and red phonon signals shown in c. Each trace represents the trajectory within the time window from tprobe 0.82 to 10.48 ps. Note the scaling difference between the lateral and longitudinal axes.

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where E CSPt is given by

E CSPt

1

2p

0

Z 1

1

15

Optical conductivity (1015 s1 )

CSPoe iotdo: 4

The envelope I(t) is Fourier-transformed to give its THz spectrum ~I(o) as

~Io Z

[afii9840] (THz)

10

1

0

0.02 0.04 0.06

1

1

Iteiotdt: 5

Figure 2a shows the intensity envelopes I(t) of the modulated pump pulse simulated with three different CSP delays tmod

49.4, 92.2 and 93.6 fs combined with the CSP parameters oL, Do and f00, taken from the current experimental conditions. The dotted curve represents the intensity envelope of a CSP pair with the delay tmod 0 fs. Figure 2b shows the THz spectra

0.03

0 0 0.1 0.2 0.3 [afii9840] (THz)

5

R non-osc/R

0 0 1 2 3 4

Photon energy (eV)

zeq ()

~I(o)

Fourier-transformed from those simulated envelopes I(t), demonstrating that the spectrum ~I(o) is modulated by changing the CSP delay tmod, so that we can control the

relative intensities of two different THz components resonant with A1g and Eg phonons, respectively.

Figure 2c shows the observed temporal evolutions DR(tprobe)/R

with three different CSP delays tmod similar to those for the simulations shown in Fig. 2a,b. It is clearly seen that the beat structure changes drastically as we change the CSP delay. The FTs of the traces in Fig. 2c are plotted in Fig. 2d in which two peaks at B3.0 and B2.1 THz correspond to the A1g and Eg phonons, respectively, demonstrating that their relative intensities are clearly controlled.

Visualization of 2D atomic motions. The real-space atomic motions have been reconstructed in Fig. 2eg from the measured beats shown in Fig. 2c, based on the correspondence between the nuclear displacement2931 and the reectivity change DR. This correspondence has been obtained by linear response DFT calculations32 with the full-potential linearized augmented plane wave programme WIEN2k33,34, which is among the most accurate DFT codes presently available. The two quantities that we have computed are the potential energy surface as a function of the absorbed laser energy and optical properties in dependence of the Bi atomic coordinates. Details of our total-energy computations have been published in ref. 35. For the optical calculations, we have used the module described in refs 36 and 37, which is based on the Lindhard equation. With this package, we computed the imaginary part of the dielectric tensor for photon energies up to 27 eV. We obtained the real part using the KramersKronig relation, where we added an imaginary part of0.1 eV to the energy to account for the nite lifetime of excited states. It is important to realize that both our total-energy computations and the optical calculations rely on the interpretation of the KohnSham energies as single-electron excitation energies34,36,37, which, although not strictly valid within DFT, is widely used and appears to work well for many materials including bismuth34,37. As the optical calculations are very sensitive to the quality of the k mesh, in this part of our theoretical work we have included 40 40 40 k points.

To check the validity of our approach, in Fig. 3 we have plotted a number of calculated dependencies and we have compared our predictions to published experimental data. In particular, the quasiequilibrium coordinate zeq, which is associated with the A1g phonons, changes with increasing pumplaser uence. Simultaneously, the A1g phonon frequency softens. In Fig. 3a, we have plotted the functional dependence of both effects, which can also be experimentally assessed through time-resolved X-ray

spectroscopy29,30. Clearly, the agreement with these experiments is very good. In Fig. 3b, we show the computed optical conductivity for the electronic ground state. It qualitatively reproduces the essential features of the measured data of ref. 38, in particular it has a maximum around 0.95 eV, a shoulder at 1.5 eV, a dip at 2.1 eV, followed by a maximum and a decay. At the photon energy of 1.55 eV (800 nm), to which we shall restrict ourselves in the following, the computed optical conductivity is in good quantitative agreement with the data of ref. 38. As a nal check, in Fig. 3c, we show how the average reectivity (sometimes called background reectivity, because it ignores the superimposed oscillatory effect due to excited coherent phonons) and the A1g phonon frequency change simultaneously. Our prediction of Fig. 3c is compared with optical measurements of refs 4,39,40. The good agreement between our calculations and the experimental data conrms that our approach reliably reproduces the experimental results.

We now turn to a description of our computed results for the optical reectivity R of Bi at a wavelength of 800 nm. For the ground state (zeq 0.2344c), where c is the lattice parameter, we

have obtained R (Rxx Ryy Rzz)/3 74.4%, in perfect agree

ment with the experimental value of 74.3% (ref. 39). In Fig. 4a, we show the relative change in the reectivity Rxx for small

displacements of the atoms in the z direction. For comparison, a value of Dz 18.4 pm would undo the Peierls distortion in Bi.

The calculated reectivity change shows a clear dependence on the displacement in the z direction with a tted slope of q(DR/R)/ qz 0.0164 per pm. Moving the Bi nuclei in the x direction

changes the reectivity tensor anisotropically with a slope of 0.0046 per pm, as shown in Fig. 4b.

Utilizing these parameters, we have converted our measured beats DR(tprobe)/R shown in Fig. 2c to the real-time atomic

motions in the z and x directions. Details of the conversion process are described in the Methods section. Figure 2eg shows

Figure 3 | Comparison between theoretical results and published experimental data. The red lines show our calculated results. The symbols reproduce published experimental data. (a) Relation between the softening of the A1g phonon mode and the change in the quasiequilibrium coordinate zeq of the Bi atoms. (b) Optical conductivity as a function of photon energy. The blue vertical line indicates the photon energy of 1.55 eV (800 nm) used in the present experiments. (c) Relative change in the non-oscillatory (non-osc; or background) component of the reectivity at 800 nm as a function of the phonon softening. The experimental data points and their error bars were taken from the following publications: lled circles, ref. 30; lled diamonds, ref. 29; lled squares, ref. 38; open square, ref. 39; open diamonds, ref. 4; and open circle, ref. 40.

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those atomic motions in the Bi unit cell converted from the observed beats shown in Fig. 2c. In this conversion, the observed beats between tprobe 0.82 and 10.48 ps are tted with the

oscillatory function in equation (1), giving the amplitude parameters, DRa/R and DRe/R, transformed to the absolute displacement of the Bi atom. Details of this tting procedure and the relevant parameters are given in the Methods section. It is clearly seen from Fig. 2eg that the ultrafast 2D atomic motions in crystalline bismuth have been actively controlled by shaping the pump pulse.

Comparison between CSP and pulse-train techniques. The train of femtosecond pulses has often been used in the previous experiments to control coherent phonons36,8. It should be interesting to understand the difference between those pulse-train techniques and the current CSP technique from the viewpoint of THz intensity modulation. Figure 5a shows the intensity envelopes I(t) of the two superposed CSPs and the train of FT-limited pulses with their similar peak-to-peak intervals.

Figure 5b shows the THz spectra ~I(o) Fourier-transformed from the envelopes I(t) shown in Fig. 5a. The clear difference is seen in the high-frequency region above 10 THz, where the second and third harmonics of the lowest frequency component around 6 THz are ltered out in the CSP case. The CSP technique thus allows for more exible tuning of the relative intensities of different frequency components in the THz region than the conventional pulse-train approach, making CSPs ideal for controlling ultrafast 2D atomic motions in bulk solids. Spatial light modulators or acousto-optic modulators could be other candidates to produce temporal proles similar to the one of the CSPs.

DiscussionThe intensity of the Eg mode in Fig. 2d should be correlated with the THz intensity at its resonance frequency, because the Eg mode is excited directly by the pump pulse1,2,15,16. The coupling between the A1g and Eg phonons, which has been discussed in ref. 34, is of third order in the atomic displacements and can be safely neglected at the low uence in the present experiments.

The A1g phonons originate from a laser-induced shift of the potential energy surface minimum in the z direction in Fig. 1a (displacive excitation of coherent phonons (DECP) model)11, whereas the Eg phonons originate from impulsive stimulated

Raman scattering (ISRS model)41. When irradiated with a FT-limited laser pulse, the DECP mechanism launches coherent phonons that are cosine-like with an initial phase of zero degree, whereas the ISRS launches sine-like ones that are phase-shifted by 90 (ref. 11). The present initial phases of the A1g and Eg phonons are determined by a combination of their generation mechanisms (DECP and ISRS, respectively) and the intensity envelope I(t) of our modulated pump pulse. The effect of an intensity envelope on the initial phase of coherent phonons has been recently discussed by Shimada et al.9. The trajectory starting with these initial phases will eventually reach any point in the 2D conguration space accessible with a given ratio of A1g and Eg amplitudes as long as the phonon lifetime is innite. However, the lifetime is nite, so that there could be points that are not reached by this trajectory. It is important to note that this trajectory can be modied to reach an arbitrary point in that accessible conguration space if the initial phases of those two phonon modes are actively controlled independently. This control could be implemented by tuning the chirp rates of the two CSPs independently with a dispersive optic inserted in each of the two optical paths of the interferometer. This additional control should be useful when the phonon lifetime is short, and the trajectory needs to be driven to a particular atomic conguration that is effective in inducing a specic dynamical process such as a phase transition.

R xx/R (103 )

R ii /R (104 )

0

2

1

1

2

0

100

0 100

2

1

1

2

40 20

z (fm)

0 20 40

x (fm)

Figure 4 | Calculated reectivity as a function of nuclear displacement. (a) Change in the reectivity Rxx at 800 nm as a function of the nuclear displacement in the z coordinate. The change in Ryy is equal. (b) Change in the reectivity components Rxx (crosses) and Ryy (pluses) as a function of the nuclear displacement in the x coordinate. Solid lines represent linear ts to the computed data points.

2.0

1.5

1.0

0.5

0.0800 400 0 400 800

1.0

0.8

0.6

0.4

0.2

0.0

Intensity (a.u.)

CSPs

FT-pulse train

Intensity (a.u.)

FT-pulse train

CSPs

0

5

10

15

20

Time (fs)

THz

Figure 5 | Comparison of the CSP and the pulse-train techniques. (a) Simulated temporal pulse envelopes for the CSPs and a pulse train of three FT pulses. Temporal separation of the pulse train is 160 fs and tmod for the CSPs is 93.6 fs. The pulse parameters of 2pc/oL 802 nm (c is the speed of light in

air), Do/2p 15.7 THz (corresponding to 34 nm FWHM) are common for both cases, and f00 2,600 fs2 for CSPs and 0 fs2 for the pulse train,

respectively. (b) The Fourier-transformed THz intensity spectra calculated from a.

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It should be also noted, however, that the relative phase of A1g and Eg modes recurs to its initial value every 1.1 ps, which is shorter than the lifetimes of those phonons by a factor of B10 in the present case. Each of the controlled trajectories shown in Fig. 2eg thus covers almost all possible trajectories within the possible 2D space given by each relative amplitude of those two phonon modes. The current amplitude control, therefore, allows for almost full control of 2D atomic motions.

The present combination of the ultrafast optical measurements and the DFT calculation of optical response as a function of atomic displacements offers a new method to visualize ultrafast structural changes in solids, all-optical alternative to time-resolved X-ray and electron diffraction techniques30,42.

Our control-visualization scheme is based on the simple, robust and universal concept that in any physical system, 2D motion of a particle is determined by the projections of its pathway onto two orthogonal axes. The scheme is thus applicable to a variety of condensed matter systems.

Methods

Data processing. The reectivity change obtained as the differential signal of the two balanced PDs was rst amplied by a current amplier and then transferred to a digital oscilloscope at each pumpprobe delay tprobe scanned with the fast scan mechanical stage. The displacement of this stage was proportional to the voltage applied, and the voltage was swept sinusoidally in each scan. This temporal evolution of the voltage was associated with electric noises and was tted with a sine function to calibrate tprobe. The data points thus sampled along the tprobe axis

were not equidistantly distributed. We interpolated these measured data points with cubic spline functions to give equidistant points to be fast-Fourier-transformed with the Hanning window function and zero padding. The temporal evolutions of the reectivity change shown in this study are composed of these interpolated data.

The calibration of the reectivity change signal was performed as follows. The phonon signal accumulated by the digital oscilloscope was transferred to the personal computer with a 16-bit integer format. To convert these integer-formatted data to the voltage unit, we independently performed a reference signal acquisition of a sine curve generated by a function generator as an input source. The range of the oscilloscope was set to the same scale as we have used in the phonon measurement, and data were saved in both the raw voltage format and 16-bit integer format. Each data was tted with a sine function, and by comparing the tted amplitude parameter of each data we obtain the proportionality parameter to transform the integer data to voltage unit. Using this parameter, the phonon signal in 16-bit integer format was rescaled to the voltage unit and then further transformed to the current unit according to the current amplier setting. This value was divided by the recorded current value of each PD to give the reectivity change data as plotted in the main text.

Conversion process from phonon signals to 2D atomic motions. The coherent phonon signals obtained with three different tmod timings as shown in Fig. 2c were tted with the model function equation (1) to give the amplitude of each of the A1g

(longitudinal) and Eg (transverse) phonon modes. Part of the phonon signal within the time window from tprobe 0.82 to 10.48 ps was used for the tting procedure.

The frequency and decay rate of each phonon mode was determined as follows. First, the green trace of Fig. 2c was tted by equation (1), neglecting the contribution of Eg mode. Next, using the parameters Ga and na xed to the value determined by this process, we tted the blue and red traces of Fig. 2c. The parameters Ge and ne obtained from the blue and red traces were averaged to give the nal value of Ge and ne. Finally, we performed the tting procedure again with these four parameters xed to determine other parameters. SupplementaryTable S1 summarizes the resulting tting parameters for each phonon signal. The experimental phonon signals and the best-tting results are shown in Supplementary Fig. S1. The amplitude parameters, DRa/R and DRe/R, were then converted to the displacements of the Bi atom in the z and x directions by using the coefcient q(DR/R)/qz 0.0164 per pm and q(DR/R)/qx 0.0046 per pm. These

2D displacements are plotted in Fig. 2eg.

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2. Dekorsy, T., Cho, G. C. & Kurz, H. Light Scattering in Solids VIII 76, 169209 (Springer, Berlin, 2000).

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Acknowledgements

E.Z. and M.G. thank the BMBF and the DFG for nancial support. H.K. and K.O. acknowledge partial supports by Grant-in-Aid for Scientic Research by JSPS and Photon-Frontier-Consortium Project by MEXT of Japan.

Author contributions

H.K., M.K., K.G.N. and K.O. designed the experiment. H.K., H.C., K.I., H.T. andK.O. constructed the apparatus. H.K., J.C.D., K.H., K.I., H.C. and H.T. made the measurements. H.K., J.C.D., K.H. and K.I. analysed the data. All authors discussed the results. E.S.Z. and M.E.G. performed the DFT calculations to convert the reectivity changes to the atomic displacements. H.K., E.S.Z., M.E.G. and K.O. wrote the manuscript, which was polished by M.K. and K.G.N. The project was conceived and supervised by K.O.

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How to cite this article: Katsuki, H. et al. All-optical control and visualization of ultrafast two-dimensional atomic motions in a single crystal of bismuth. Nat. Commun. 4:2801 doi: 10.1038/ncomms3801 (2013).

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