Peter Moroshkin, Victor Lebedev and Antoine Weis, Department of Physics, University of Fribourg, Chemin du Musée 3, CH-1700 Fribourg
The physics of cavitation, nonlinear dynamics, and sound emission by bubbles in classical fluids is a vast and a very active domain of research that addresses a number of fundamental problems and a broad range of applications. Experimental studies (reviewed, e.g., in ) in this field usually focus on macroscopic bubbles with radii ranging from micrometers to millimetres. The dynamics of such large bubbles is well described by the classical Rayleigh-Plesset model for incompressible liquids or by more recent theoretical approaches  that take the finite compressibility of the liquid into account. To our knowledge, metal atoms in condensed helium represent the smallest known physical system that can be described in terms of a bubble. Indeed, metal atoms in liquid and solid 4He form nanometer-sized cavities that are called atomic bubbles. These bubble structures offer a unique possibility to study bubble dynamics and cavitation in the limit of the smallest possible bubble size. In particular, it is only in this system that one can study the resonant interaction between bubble vibrations and sound waves with a wavelength comparable to the (nanometer-sized) bubble diameter.
Atomic bubbles in liquid and solid He have been investigated experimentally and theoretically since the 1980ies (work reviewed in ). The interaction between a foreign atom and He atoms is in general dominated by the Pauli repulsion that arises due to the overlap of the electron density of the impurity atom with the closed electronic shell of the He atoms. This repulsive force pushes the He atoms out from the volume occupied by the valence electron of the dopant. The resulting cavity has a radius of several Angstroms and its interface consists of 20-30 He atoms. It is therefore not clear a priori whether such a microscopic system can be described by the same hydrodynamic equations that have been successfully applied to larger bubbles. At low temperatures, the He atoms are strongly delocalized and form a macroscopic quantum system, usually referred to as a quantum fluid or quantum solid. This delocalization makes it possible to describe liquid (solid) helium as a continuous medium down to the nanometer scale. In the past 20 years such hydrostatic atomic bubble models have been successfully applied to describe the properties of different metal atoms embedded in liquid and solid 4He cryomatrices. The predictions of this bubble model are in good agreement with the results of many experimental laser-spectroscopic studies.
Time-resolved experimental studies of the atomic bubble dynamics still remain a challenging task. So far, only the dynamics of the slightly larger bubble formed by the He2* excimer quasimolecule in superfluid He has been observed by means of femtosecond pump-probe laser spectroscopy . More recently, our group in Fribourg has developed a novel approach that allows us to infer information on bubble dynamics on the subpicosecond time scale from purely spectroscopic experimental data using nanosecond laser pulses or continuous laser radiation.
We have studied the laser-induced fluorescence spectra of the coinage-metal atoms Cu and Au in liquid and solid 4He cryomatrices. The experimental setup is described in detail in . A 200 cm3 large He matrix is produced in a high-pressure sample cell placed in a He-bath cryostat. The temperature of the cell can be varied in the range of 1.4 – 4 K by pumping on the He bath. The cell is filled with pressurized liquid or solid He. The matrix is then doped with Cu or Au atoms by means of laser ablation using frequency-tripled pulsed Nd:YAG laser radiation (355 nm, 50 mJ/pulse). The dopants are excited by the same laser which populates high-lying excited states. These states decay via a cascade of radiative and radiationless transitions towards the ground state. We collect the laser-induced fluorescence and analyse its spectrum by a grating spectrometer equipped with a CCD camera. In particular, we observe transitions of the valence electrons (Fig. 1a) as well as transitions of inner-shell electrons (Fig. 1b) which are forbidden in the free atom, but become observable in the He matrix .
Results and discussion
According to the Franck-Condon principle, electronic transitions in the dopant atoms occur in a fixed bubble configuration. Photon absorption takes place in a bubble whose size corresponds to the electronic ground state. It is followed by a bubble relaxation towards a larger equilibrium configuration that corresponds to the more extended electronic density distribution of the excited state. In a similar way, emission of the fluorescence photon occurs in the larger bubble and is followed by a relaxation of the bubble to the ground state configuration. The light-induced dynamics of such an absorption/reemission cycle thus represents a bubble expansion and contraction. Depending on the atomic transition under the investigation, the displacement of the bubble interface can be as large as 25% of the initial bubble size, which yields an efficient excitation of sound waves in the matrix.
S – P transitions of the valence electron in alkali (Rb, Cs) and coinage (Cu, Au) atoms are accompanied by such a particularly large displacement of the bubble interface and therefore induce large amplitude sound waves that can be well described by classical hydrodynamics. A sudden change of the electronic state of the embedded atom triggers a sudden displacement of the bubble interface that is followed by a damped oscillation at the characteristic bubble frequency ωb. The vibrating bubble emits a spherical sound wave oscillating at the same frequency (Fig. 2).
The resonance frequency ωb strongly depends on the bubble size, for nanometer-sized atomic bubbles it lies in the range of 0.1–1 THz. The corresponding (classical) sound wave can be described as the sum of a large number of elementary excitations (phonons). Their wavelengths, as determined by the dispersion relation of liquid/solid helium, are on the order of several nanometers, i.e., comparable to the bubble size. The resonant coupling between the bubble vibrations and the phonons leads to a very efficient damping of the bubble oscillations, as shown in Fig. 3. The oscillations are overdamped on the time scale of 1 ps, which is shorter than a single period and the bubble interface stabilizes at the new equilibrium position.
Our analysis shows that the fluorescence spectra of this type of transitions are strongly broadened and have no substructure that can be associated with the bubble vibration frequency. The fluorescence lines experience a strong shift towards shorter wavelengths that is proportional to the He density. The atomic bubble model calculations successfully reproduce the observed lineshapes as we have shown earlier for the 6S1/2 – 6P1/2 transition of Cs in liquid and solid He.
We have found that the transitions of the inner-shell electrons, e.g., the forbidden (n-1)d10ns - (n-1)d9ns2 transitions of Cu and Au leave the bubble radius almost unchanged . The (n-1)d electrons are screened from the surrounding He atoms by the ns valence electron and do practically not contribute to the dopant-He interaction that defines the bubble size. In this case the resulting phonon wavepacket has a very small amplitude and contains only a small number of quanta. There is a significant finite probability for an electronic transition that is not accompanied by phonon creation. In the fluorescence spectra such a transition manifests itself as a sharp peak close to the unperturbed atomic line, a feature called a zero-phonon line (ZPL). ZPLs are usually accompanied by transitions that excite a very small number of phonons. These processes produce a broader spectral feature that is redshifted with respect to the unperturbed line and is called the phonon wing (PW).
In the experimental spectrum shown in Fig. 1b, the width of the ZPL is limited by the resolution of our spectrometer. On the other hand, the lineshape of the PW is well resolved and provides information on the spectrum of the phonon wavepacket that is generated by the bubble oscillations. Our experiments  show that the phonon wings produced by different inner-shell transitions of Cu and Au have identical lineshapes. On the other hand, we observed a marked difference between the shapes of the PWs in liquid and solid helium matrices which reflects the different properties of the elementary excitations (phonons) in the two phases.
The spectrum of the phonon wavepacket is encoded into the PW spectrum. Assuming that the dispersion relation of doped condensed He is the same as that of pure helium and that all the phonons constituting the wavepacket are excited simultaneously at the instant of the electronic transition, we can reconstruct the dynamics of the wavepacket and compare it with the predictions of the hydrodynamic model (Fig. 3). The reasonable agreement (on an absolute time scale) between the theoretically predicted bubble interface dynamics and the reconstruction of that dynamics from our (quasi-CW) spectroscopic data validates the applicability of the hydrodynamic treatment to the subpicosecond dynamics of nanometer-sized bubbles in quantum fluids/solids.
 W. Lauterborn and T. Kurz, Rep. Prog. Phys., 73, 106501 (2010).
 J. B. Keller and M. Miksis, J. Acoust. Soc. Am., 68, 628 (1980).
 P. Moroshkin, A. Hofer, and A. Weis, Phys. Rep., 469, 1 (2008)
 A. V. Benderskii, J. Eloranta, R. Zadoyan, and V. A. Apkarian, J. Chem. Phys., 117, 1201 (2002).
 P. Moroshkin, V. Lebedev, and A. Weis, J. Low Temp. Phys., 162, 710 (2011)
 P. Moroshkin, V. Lebedev, and A. Weis, to be published in Europhys. Lett.
[Released: September 2011]