The vibrations in a solid can be analyzed in terms of collective modes of motion of the atoms which are dubbed phonons. In a classical description the displacement of the atoms along the phonon eigenmodes of a crystal can be measured with unlimited precision. Conversely, in the quantum formalism positions and momenta of the atoms can be determined simultaneously only within the boundary given by the Heisenberg uncertainty principle. For this reason, in real materials, in addition to the thermal disorder, the atomic displacements are subject to fluctuations which are intrinsic to their quantum nature. The motivation of studying the quantum proprieties of phonons in crystals comes from various evidences, suggesting that quantum fluctuations of the atoms in solids may be of relevance in determining the onset of intriguing and still not completely understood material properties, such as quantum paraelectricity, charge density waves, or high temperature superconductivity.The time evolution the atomic position in crystals is usually addressed in the framework of ultrafast optical spectroscopy by means of pumpprobe experiments. In these experiments the phonon dynamics is driven by an intense ultrashort laser pulse (the pump), and then the collective excitation is investigated in time domain through the interaction with a weaker pulse (the probe). Unfortunately this method typically provides information only about the average position of the atoms at a certain time after the excitation. On the other hand different static techniques give access (indirectly) to a time integrated statistical distribution of the atomic position.
The possibility of measuring the time evolution of fluctuations of the atomic positions (quantum or thermal), beyond a time integrates statistical dystribution, is the subject of an intense scientific debate.In our recent research a new approach to investigate quantum fluctuations of collective atomic vibrations in crystals is proposed. An original experimental apparatus that allows for the measurement of the photon number quantum fluctuations of the probe pulses in a pump and probe setup has been developed. The connection between the measured photon number uncertainty and the fluctuations of the atomic positions is given by a fully quantum mechanical theoretical description of the time domain process. Overall we prove that, in appropriate experimental conditions, the fluctuations of the lattice displacements can be directly linked to the photon number quantum fluctuations of the scattered probe pulses. Our methodology, which combines nonlinear spectroscopic techniques with a quantum description of the electromagnetic fields, is benchmarked on the measurement of phonon squeezing in αquartz. The experimental layout is similar to standard pump and probe experiments. The sample is excited by an ultrashort pump pulse and the time evolution of the response is measured by means of a second much weaker probe pulse, that interacts with the photoexcited material at a given delay time. Both pump and probe come from the same laser source, a 250 kHz modelocked amplified Ti:Sapphire system.
The unique characteristics of our setup are:
I. unlike standard experiments, where the response is integrated over many repeated measurements, our system can measure individual pulses;
II. the apparatus operates in low noise conditions (shot noise limited) allowing for the measurement of intrinsic photon number quantum fluctuations.
The time domain response is shown in Figure 1 for a representative pump fluence of 14 mJ cm^{2}.
Figure 1. Histogram plot of repeated shot noise limited pump and probe experiments. For every time step, the histogram represents the distribution of the outcome for 4000 experiments. (a) Measured distribution and (b) distribution centered at their mean value. A time dependent noise is revealed by the modulation of the width of the distribution at positive times..
Our novel experimental approach allows for the direct measurement of the photon number quantum fluctuations of the probing light in the shotnoise regime and our fully quantum model for time domain experiments maps the phonon quantum fluctuations into such photon number quantum fluctuations, thereby providing an absolute reference for the vibrational quantum noise. A quantitative analysis of noise (Figure 2) and mean values allowed for a comparison of the experimental results with the predictions of the model unveiling non classical vibrational states (squeezed states) produced by photoexcitation. In particular, we demonstrated that the observation of an oscillating component in the variance of the optical transmittance at twice the phonon frequency is indicative of a squeezed phonon state.
This research put at test a new spectroscopic approach based on the photon number statistics by investigating quantum fluctuations of simple excitations, Raman active atomic vibrational modes, in a prototype transparent system. The approach can be in principle generalized to the study of quantum fluctuations of any collective excitations in crystals, included  for example  excitations of electronic origin.
Figure 2. Time domain transmittance mean and variance. (a) Mean (blue curve) and variance (red curve) of the transmittance calculated over 4000 acquired pulses. The zero time is the instant in which pump and probe arrive simultaneously on the sample. In the inset a zoom of the variance for the first 3 ps is shown. (b) Wavelet analysis (Morlet power spectrum) of the variance oscillating part. (c) Fourier transforms of the oscillating parts of mean (blue curve) and variance (red curve). The dashed lines indicate the phonon frequency and twice the phonon frequency.
Daniele Fausti, email: daniele.fausti@elettra.eu
Martina Esposito, email: martina.esposito@elettra.eu
M. Esposito, K. Titimbo, K. Zimmermann, F. Giusti, F. Randi, D. Boschetto, F. Parmigiani, R. Floreanini, F. Benatti, and Fausti D.“Photon number statistics uncover the fluctuations in nonequilibrium lattice dynamics” Nature Communication 6, 10249 (2015). DOI: 10.1038/ncomms10249. 

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