| The development of ultrafast pulsed lasers formed the beginning of the field of ‘nano-acoustics’. Pressure waves with high amplitude can be generated using absorption of intense laser pulses in a metallic film. In this project it has been demonstrated that very short pressure packets are unstable upon propagation in a bulk crystal. Rather, they break up into a train of ultrashort soliton pulses with a thickness of only several nanometers. These acoustic solitons not only form an intriguing new application of elementary concepts of solid-state physics. They also open up a new avenue of ultrafast, high pressure dynamical experiments in condensed matter and nanostructures.
Solitons and nano-acoustics
The shortest pressure pulse that may be generated in matter has a thickness of only one interatomic distance. Although theoretically feasible, it has remained up to now an experimental challenge to study acoustic waves in this limit. Nano-acoustic pulses of several tens of nanometers thickness can be generated thermoelastically via absorption of femtosecond laser pulses in a thin metallic transducer. In this way the light pulse acts as a ‘hammer’, tapping the material and creating a pressure packet that propagates with the sound velocity through the medium. The reflections of the acoustic wavepacket at deeper lying structures can be monitored at the surface using acousto-optic pulse echo techniques, generally referred to as pump-probe spectroscopy. This nano-ultrasonic method has found broad application as an analysis tool in the production of semiconductor devices and nanostructures.
Figure 1 Simulation of the deformation of a nano-acoustic pulse in a sapphire crystal.
Most research in the field of nano-acoustics has been conducted on thin films, in which the packets travel over submicrometer distances. Propagation over macroscopic distances in a crystal is a novel and unexplored domain. In a pioneering experiment, Hao and Maris [1] succeeded in retrieving the acoustic wavepackets after long-distance propagation. They observed that low-amplitude pressure pulses are severely distorted by the dispersion of the crystalline lattice. This phonon dispersion results in the slowing down of higher-frequency components with respect to the lower frequencies in the wavepacket. Consequently, the short pulse is pulled apart into a string of oscillations with frequencies increasing toward the rear of the packet.
At sufficiently high pulse amplitudes, an additional force can balance the effect of phonon dispersion [2,3]. This effect is the nonlinearity of the crystal lattice, which manifests itself through an increase in sound velocity when a pressure is applied. The combination of dispersion and nonlinearity provides with a fascinating interplay, leading to the formation of extremely stable pulses: solitons. During my PhD-work, we have investigated the generation and detection of such solitons in sapphire and their interaction with a two-level electronic system in ruby. These studies combine experimental techniques using femtosecond lasers and nonlinear optics, and the analytical modelling of soliton wave generation and propagation.
From shock wave to soliton train
Figure 1 shows the simulated evolution of a typical pressure wave during its travel, under the combined influence of nonlinearity and dispersion of a sapphire crystal. The model that applies to the ultrashort acoustic pulses is a mathematical classic: the Korteweg - De Vries equation. In the wavepacket development we can separate three subsequent phases. The primary disturbance takes place when the peak of the pressure pulse overtakes the wavefront due to its nonlinear velocity difference. This leads to the formation of a steep shock wave and the concomitant formation of very high, terahertz acoustic frequency components in the wavepacket. The presence of these high frequencies strongly enhances the effect of phonon dispersion. The terahertz frequencies directly lag behind the wave front and give rise to an oscillating disturbance of the wave packet. This second phase in the development is denoted as the fragmentation of the acoustic packet.
The complicated interplay of shock formation and dispersion can only result in the formation of solitary waves. These are stable packets in which the intrinsical balance between the competitive effects provides with robustness and even a regenerative ability of the solitons under influence of weak external disturbances. The solitons are supersonic: their propagation velocity is higher than the velocity of linear sound.
An arbitrary compressive pulse develops into a train of soliton pulses, in which the number of solitons and their width, amplitude, and velocity are determined by a single parameter. This parameter is constituted of fundamental material constants, the pressure amplitude, and the width of the initial pulse. Figure 1 shows that our experimental conditions yield a soliton train of 13 individual pulses, in which the leading pulse is the shortest, with a time duration of 0.25 picoseconds, or a spatial width of only 2.5 nanometers!
Observation of a soliton train
In our research we demonstrate experimentally that nano-acoustic pulses can spontaneously break up into a train of solitons. We generate high-amplitude strain wavepackets by focusing amplified laser pulses onto a chromium transducer evaporated on a sapphire crystal. This way, one thousand times per second a compressive pulse of 80 nanometers is launched into the sapphire. The high power of the optical source allows us to achieve a maximum pressure of 10 kilobar over a spot of several millimeters. In the experiments, the sample is cooled down to several degrees Kelvin to reduce the damping of the wavepacket by the background of thermal vibrations.
We developed two independent methods to detect the solitons in the bulk of the crystal. From these experiments and the theoretical model of the Korteweg - De Vries equation we can retrieve the behavior of the strain profile.
Traveling mirror
In the first method, light of a well-defined wavelength is reflected from the pressure waves in the crystal. The strain packet acts as a reflecting mirror traveling through the material with the velocity of sound. Due to this velocity, the reflected light is frequency-shifted analogue to the classical Doppler effect. The frequency shift is used to spectrally separate this light from the background. Information on the development of the wavepacket is obtained by measuring the scattered intensity as a function of the traveled distance in the crystal.
Figure 2 Scattered light intensity as a function of traveled distance of the acoustic wave, with (bottom) the resonant conditions for constructive and destructive interferences of the light reflected from the moving solitons
Figure 2 shows a typical development of the scattered light intensity against the propagated distance of the pressure wave, for two different temperatures. Both curves have an initial decrease of the scattered intensity in the crystal. The high-temperature measurement subsequently shows a slow and weak oscillation, while the low-temperature data has a typical fast oscillation pattern over several millimeters.
This behavior can be reproduced with excellent precision for both temperatures by the theoretical model, using the known material parameters of sapphire. The only free parameter is the peak amplitude of the wavepacket. The line through the low-temperature data corresponds to the Korteweg - De Vries model calculations, in which the packet breaks up into the soliton train of Fig. 1. At elevated temperatures, an additional damping term is added into the model (the Burgers term) with a known value for the viscosity of sapphire. This damping destroys the terahertz frequency components before these can disturb the shock front via dispersion. The packet does not break up and travels in the form of a damped shock-wave.
Soliton train and diffraction grating
The oscillating pattern at low temperatures demonstrates that in our experiment we are sensitive to the velocities in the wave packet. We can simply explain the oscillation from the interference of light scattered from a train of mirrors traveling with different velocities. For this we make use of Bragg’s law, stating that a maximum in the scattered intensity occurs when a resonant condition occurs for the distances between the solitons (c.f. Fig. 2). As the solitons propagate with their own relative velocity differences, the separation between the mirrors varies linearly with the propagated distance. This corresponds quantitatively with the oscillation in the experiment, for a velocity difference of 10 meters per second.
On a similar vein we can describe the train of mirrors as a diffraction grating. As the grating period increases with the traveled distance, the trace of Fig. 2 can be identified as the grating function of the soliton train! As the pulses are not exactly equally spaced, the grating function is not periodical, but has the complex structure as shown in the experiment.
Resonant two-level systems
The second method to detect strain solitons makes use of resonant electronic transitions. Electronic two-level systems that strongly interact with the acoustic field can be found in optically pumped chromium ions embedded in a ruby crystal. The relevant levels,
Figure 3 (left) Artist impression of the soliton detector in ruby, (right) time-resolved luminescence from the two-level system, (inset) transverse profile of the soliton beam for two distances z.
 and  in the level scheme of Fig. 3, have an energy splitting of 0.87 terahertz. In the interaction with the acoustic field an energy quantum is transferred from the strain packet to the electrons in the ion. The ion in fact vibrates on the beat of the sound waves and emits this radiation in the form of new pressure waves.
We have succeeded in mapping out the interaction between the two-level systems and the solitons. For this purpose we use the fact that, next to sound emission, the chromium ion can also emit a light quantum after interaction with the strain field. Using time-resolved detection, this luminescence can be used as a measure of the excitation of the two-level medium. A typical time-resolved luminescence curve is depicted in Fig. 4. We distinguish two contributions, one non-directional emerging from the heat generated in the metal film, and one directional component due to the soliton pulses. The inset of the figure shows the transverse profile and demonstrates that this is a well collimated beam. With this experiment we have directly proven the generation of terahertz acoustical frequencies in the wavepacket.
Conclusion
In our experiments we show that nano-acoustic packets break up in extremely short pressure pulses with soliton properties. This opens up new avenues for the research to ultrafast pressure dynamics, imaging of nanostructures or even local manipulation of materials (nano-explosions).
The interplay between solitons and two-level systems can be used to amplify sound waves by means of energy transfer from the electronic system to the strain packet. This requires the inversion of the two-level medium, which can be achieved using strong optical pumping of the  -state. A soliton-based terahertz acoustic laser appears well within the range of possibilities.
[1] H.-Y. Hao and H. J. Maris, Phys. Rev. Lett. 84, 5556 (2000)
[2] H.-Y. Hao and H. J. Maris, Phys. Rev. B 64, 063402 (2001)
[3] O. L. Muskens and J. I. Dijkhuis, Phys. Rev. Lett. 89, 285504 (2002)
[4] O. L. Muskens, A. V. Akimov, and J. I. Dijkhuis, Phys. Rev. Lett. 92, 035503 (2004)
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