Abstract
Fluctuations of the atomic positions are at the core of a large class of unusual material properties ranging from quantum paraelectricity to high temperature superconductivity. Their measurement in solids is the subject of an intense scientific debate focused on seeking a methodology capable of establishing a direct link between the variance of the atomic displacements and experimentally measurable observables. Here we address this issue by means of nonequilibrium optical experiments performed in shotnoiselimited regime. The variance of the timedependent atomic positions and momenta is directly mapped into the quantum fluctuations of the photon number of the scattered probing light. A fully quantum description of the nonlinear interaction between photonic and phononic fields is benchmarked by unveiling the squeezing of thermal phonons in αquartz.
Similar content being viewed by others
Introduction
In a classical description, the displacement of the atoms along the vibrational 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. While various evidences suggest that such quantum fluctuations may be of relevance in determining the onset of intriguing material properties, such as quantum paraelectricity, charge density waves or even high temperature superconductivity^{1,2,3,4,5,6,7}, the possibility of measuring quantum fluctuations in solids is the subject of an intense debate^{8,9,10,11,12,13,14,15,16,17,18,19,20,21}.
The time evolution of atomic positions in materials is usually addressed by means of nonequilibrium optical spectroscopy. An ultrashort light pulse (the pump) impulsively perturbs the lattice and a second one (the probe), properly delayed in time, measures a response that is proportional to the spatially averaged instantaneous atomic positions. In those experiments, the timedependent atomic displacements are often revealed by an oscillating response, commonly dubbed coherent phonon response^{22,23,24,25,26,27,28,29,30,31,32}, at frequencies characteristic of the vibrational modes of the material. In this framework, it has been shown that a nonlinear light–matter interaction can prepare nonclassical vibrational states^{8,13} such as squeezed states, where the fluctuations of the lattice position (or momentum) can fall below the thermal limit. A reduction below the vacuum limit is known as vacuum squeezing^{16}.
Here we propose a joint experimental and theoretical approach to access the fluctuations of the atomic positions in time domain studies. An experimental apparatus that allows for the measurement of the photon number quantum fluctuations of the scattered probe pulses in a pump and probe setup is adopted. 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, under 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, that combines nonlinear spectroscopic techniques with a quantum description of the electromagnetic fields, is benchmarked on the measurement of phonon squeezing in αquartz.
Results
Shotnoiselimited pump and probe experiments
In the nonlinear spectroscopy formalism, the excitation mechanism of phonon states in transparent materials is called impulsive stimulated Raman scattering (ISRS)^{29}. The susceptibility tensor χ^{(3)} connects the induced third order polarization P^{(3)} to three fields: , and ^{33}. In conventional two pulses pump and probe experiments, the fields and are two different frequency components of the pump laser pulse. In particular, all photon pairs such that ω_{3}−ω_{2}=Ω, where Ω is the frequency of the Raman active vibrational mode, contribute to ISRS^{34}. The interaction of the probe field with the photoexcited material induces an emitted field, , which depends on the pump–probe delay and carries information about the specific Raman mode excited in the crystal. Experimental details are reported in (Supplementary Note 1). We choose a polarization layout designed to excite Esymmetry Raman optical modes in αquartz at room temperature and get an emitted field with polarization orthogonal to the probe one^{35} (Supplementary Fig. 1).
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, which interacts with the photoexcited material at a delay time τ. The unique characteristics of our setup are: unlike standard experiments, where the response is integrated over many repeated measurements, our system can measure individual pulses; the apparatus operates in low noise conditions allowing for the measurement of intrinsic photon number quantum fluctuations. In detail, we adopt a differential acquisition scheme where each probe pulse is referenced with a second pulse which has not interacted with the sample. For each measurement, the differential voltage is digitized and integrated, giving the transmittance ΔT_{i} for the ith measurement. For every given pump and probe delay τ, we repeat this singlepulse measurements for N=4,000 consecutive pulses. Figure 1a gives a useful visual representation of the obtained data. For one pump and probe scan l, the normalized histogram of N=4,000 acquired pulses for each delay time is shown. Each histogram represents the distribution of the measured ΔT_{i} for a specific delay time τ. For a clearer visualization of the physically meaningful information in the time evolution of the statistical distribution, Fig. 1b reports the histogram centred at zero.
The pump and probe scan is repeated several times and each lth scan provides , and . Finally the averages of these two quantities are calculated over all M scans as and .
The time domain response, averaged over M=10 scans, is shown in Fig. 2a for a representative pump fluence of 14 mJ cm^{−2} (a pump fluencedependent study is reported later). The blue curve depicts the time evolution of the mean value of the transmittance ΔT_{mean}, whereas the red curve shows the time evolution of its variance ΔT_{var}. The Fourier transform of the mean (Fig. 2c, blue curve) has a single peak which is ascribed to the Esymmetry quartz vibrational mode with frequency Ω=128 cm^{−1}=3.84 THz^{36}. The same frequency component is observed in the Fourier transform of the variance (Fig. 2c, red curve). In addition, a second peak at twice the phonon frequency appears exclusively in the variance. A wavelet analysis of the variance oscillations allows for a time domain study of the two frequency components (Fig. 2b): one notices that while the fundamental frequency survives for roughly 7 ps, the 2Ω component vanishes within the first 2 ps. The different lifetimes between the Ω and 2Ω components of the variance are seen also by a close inspection of the raw data distribution plotted in Fig. 2b.
Note that the 2Ω in our data is visible only in experimental conditions where the noise is dominated by the quantum uncertainty, a situation which is known as shotnoise regime. In such conditions, ΔT_{var} measures the quantum variance of the scattered probe photon number. A full characterization of the detection system is reported in (Supplementary Note 2), including the shotnoise characterization (Supplementary Fig. 2) and the analysis of classical noise sources (Supplementary Figs 3 and 4). It should further be stressed that in experimental conditions where the noise is larger and dominated by classical sources, the 2Ω contribution to the noise becomes unmeasurable.
The presence of the 2Ω frequency component is suggestive of phonon squeezing, as it has been indicated by Raman tensor models^{8,9,12}. Nevertheless, the experimental evidences up to date lack a direct comparison with a reliable quantum noise reference^{10,11,13,20}. Hence, in these experiments the observation of the 2Ω frequency in the optical noise is considered as an indication of phonon squeezing, but not an unequivocal proof. In details, a 2Ω oscillating optical noise was reported in ref. 13, but later ascribed to an artifact^{15} due to the experimental amplification process. In particular, it has been demonstrated that amplification artifacts become more relevant when, using a lockin amplifierbased acquisition, the time constant of the lockin increases with respect to the time between steps in the pump–probe delay. This gives rise to maxima in the noise where the derivative of the mean signal is maximal^{15}. Here we use a pump power density, which is almost 3 orders of magnitude higher than in refs 13, 15. In addition, we observe a 2Ω frequency component in the optical variance which exhibits maxima in correspondence with the minima of the derivative of the mean signal, hence ruling out possible artifacts of the kind described in (ref. 15).
Fully quantum description of ISRS
To predict how the fluctuations of the atomic positions in a lattice can be mapped onto the photon number quantum fluctuations of the probe field, we develop a theoretical approach to time domain studies, which treats quantum mechanically both the material and the optical fields involved in the nonlinear processes. Several semiclassical models describe the possibility of generating ‘classical’ (coherent states) and nonclassical vibrational states by photoexcitation. In particular, for transparent materials like quartz, the most commonly used approach is to adopt Raman tensor models where the interaction between photons and phonons is not mediated by dipoleallowed electronic transitions. In this condition, interactions linear in the phonon operators allow for the generation of coherent vibrational states, while high order interactions are required for the generation of nonclassical squeezed states^{9,12,32}. In materials with allowed dipole transitions, as in presence of excitons, different models based on electron–phonon coupling Hamiltonians have been proposed. In those models it has been shown that squeezed phonon states can result only by successive excitations with a pair of pulses^{16,17}. All these models mainly adopt semiclassical approaches where the optical fields are described classically^{33}, and therefore are unable to reproduce the quantum proprieties of the probe optical field that can be measured with the shotnoiselimited pump and probe setup presented here. The key aspect of our approach, allowing us to bridge this gap, is to study both generation and detection of phonon states using a fully quantum formalism through an effective photon–phonon interaction, which is descriptive of experiments in transparent systems, such as αquartz. The basic tool is a quantum Hamiltonian able to describe both pump and probe processes. Being linear and bilinear in the photon and phonon operators, this Hamiltonian accounts for the possible generation of coherent and squeezed phonon states through the pump process. In particular, it models also the detection of the photoexcited phonon states, describing the probing process by a fully quantum approach, providing in this way a direct comparison with the experimentally measured photon number quantum fluctuations of the scattered probe pulses^{37}.
In this framework, the first step is to adopt a quantized description for the modelocked pulsed laser fields^{38}. Each mode of frequency ω_{j}=ω_{0}+jδ, where ω_{0} is the pulse central frequency, δ is a constant depending on the laser repetition rate and j is an integer, is quantized and described by singlemode creation and annihilation operators and . In this framework, ISRS can be modelled by means of an effective impulsive interaction Hamiltonian, which is descriptive of both the pumping and the probing processes. In both processes, two optical fields with orthogonal polarizations (denoted with subscript x or y) are involved: two pump fields in the pumping process and the probe and the emitted field in the probing process. The interaction Hamiltonian has the form
where 2J+1 is the total number of modes within a modelocked optical pulse, and are the phonon annihilation and creation operators; μ_{d} and μ_{s} are coupling constants and the function takes into account the relations between the frequencies of the involved fields,
with Ω the phonon frequency. A complete interaction Hamiltonian should contain also second order processes involving phonons with opposite momenta. However, since the probe detects only the k≃0 optical transition, we can make use of an effective Hamiltonian that accounts only for this kind of process.
The whole theoretical description of the experiment can be rationalized in a fourstep process as sketched in Fig. 3: (i) generation of phonon states in the pumping process, (ii) evolution of the produced vibrational state, (iii) probing process and (iv) read out of the emitted photon observables.
(i) Initially, the sample is in thermal equilibrium and it is described by a thermal phonon state , at inverse temperature β. The laser pump pulse is described by a multimode coherent state of high intensity , where are singlemode coherent states associated with all the frequency components within the pulse. Each is an eigenstate of the annihilation operator of photons in the mode of frequency ω_{j}, . We indicate with the vector whose components are the amplitudes ν_{j}. The equilibrium (prepump) photon–phonon state is instantaneously transformed into by means of the unitary operator . Since the pumping operator acts on a high intensity photon coherent state , we can use the mean field approximation for the photon degrees of freedom and replace with ν and with ν* for both pump modes involved in equation (1), thus replacing by
The evolution operator generates coherent and squeezed phonon states, respectively, through the linear and quadratic terms in the phonon operators and . The initial state contains information about both photons and phonons. Tracing over the photon degrees of freedom, the resulting state describes the excited phonons brought out of equilibrium by the impulsive pump process.
(ii) The time evolution of the excited phonons is described by using an open quantum systems approach, namely by means of a suitable master equation of Lindblad form^{39,40} that takes into account, besides the quantum unitary evolution, also the dissipative and noisy effects due to the interaction with a thermal environment.
(iii) The incoming probe pulses are in the multimode coherent state . The probing process at time τ is described by the same impulsive unitary operator used for the pump. However, in this case we can apply the mean field approximation only to the probe photon operators with x polarization, which correspond to a much more intense field than those with y polarization. Moreover, since the probe field is much weaker than the pump field, the quadratic terms in the interaction Hamiltonian in equation (1) can now be neglected. The resulting unitary operator is
where is the vector with components and is a collective photon annihilation operator such that .
The latter unitary operator acts on a state of the form . The information about the phonons are extracted by measuring the emitted field photons. In particular, the emitted photon state is obtained by tracing away the phonon degrees of freedom.
(iv) The possible quantum features of the phonon state, for example, squeezing, can be read off as they are imprinted into . In particular for each time delay τ, we can compute the quantities and , which correspond to the observables measured in the experiment that are the mean value and the variance of the number of photons of the emitted field. The details of the theoretical model are reported in (Supplementary Note 3). The theoretical results for μ_{s}=0 and μ_{s}≠0 are shown in Fig. 4 together with the corresponding wavelet analysis for the variance of the number of emitted photons. The calculations reproduce the experimental results in Fig. 2, revealing a 2Ω frequency component in the variance, only when the pump creates squeezed phonon states (μ_{s}≠0). In particular, for μ_{s}≠0, the model reproduces the different lifetimes between the Ω and 2Ω components in the variance observed in the experiments. The explicit expressions for the theoretically predicted amplitudes of both the frequency components in the variance are reported in (Supplementary Note 3), showing that the same damping constant, characterizing the dissipative phonon time evolution, contributes differently to the two frequency components giving rise to different decay times.
Discussion
The proposed effective interaction model is further validated by a pump fluence dependence study. Figure 5 shows the amplitude of the 2Ω peak in the Fourier transform of the variance, ΔT_{var}, as a function of the pump fluence. A fluence dependence study of the Ω peak is reported in (Supplementary Fig. 5). The functional behaviour obtained from the model predictions (continuous line in Fig. 5) agrees with the experimental data only in presence of a pumpinduced squeezing of the phonon mode (μ_{s}≠0). The increase of the 2Ω peak amplitude with the pump fluence allows us to give a direct estimation of the uncertainties of the phononconjugated quadratures, which are reported in the inset of Fig. 5 for the different excitation fluences (calculation details are given in (Supplementary Note 3)). For high pump fluences, the uncertainty on one of the phonon quadratures falls below the thermal limit at the equilibrium, indicating the squeezed nature of the photoexcited thermal vibrational states. Our 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. The comparison of the predicted noise with the experimental photon number quantum uncertainty, measured in shotnoise conditions, allows us to unveil nonclassical vibrational states produced by photoexcitation. A future extension of the model taking into account the role of the electronic degrees of freedom would allow to extend such a study from transparent materials to complex absorbing systems.
In conclusion, a Raman active phonon mode has been impulsively excited via ISRS in a αquartz by means of a pump and probe transmittance experiment with singlepulse differential acquisition in noise conditions limited by intrinsic probe photon number fluctuations. A fully quantum mechanical effective model where both phonons generation and detection are studied through the same effective coupling Hamiltonian establishes a direct connection between the measured photon number quantum fluctuations of the emitted probe field and the fluctuations of the atomic positions in a real material. Our approach is used here to reveal distinctive quantum properties of vibrational states in matter, in particular the squeezed nature of photoexcited phonon states in αquartz. Finally, we stress that our approach can be applied in future studies addressing the role of unconventional vibrational states in complex systems^{3,6}, and the thermodynamics of vibrational states^{41,42} possibly in the quantum regime.
Additional information
How to cite this article: Esposito, M. et al. Photon number statistics uncover the uctuations in nonequilibrium lattice dynamics. Nat. Commun. 6:10249 doi: 10.1038/ncomms10249 (2015).
Change history
01 February 2016
A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has not been fixed in the paper.
References
Sachdev, S. Quantum criticality: competing ground states in low dimensions. Science 288, 475–480 (2000).
Nozawa, S., Iwazumi, T. & Osawa, H. Direct observation of the quantum fluctuation controlled by ultraviolet irradiation in SrTiO3 . Phys. Rev. B 72, 121101 (2005).
Newns, D. M. & Tsuei, C. C. Fluctuating CuOCu bond model of hightemperature superconductivity. Nat. Phys. 184, 1745–2473 (2007).
Hashimoto, K. et al. A sharp peak of the zerotemperature penetration depth at optimal composition in BaFe2(As1−xPx)2 . Science 336, 1554–1557 (2012).
Castellani, C., Di Castro, C. & Grilli, M. Nonfermiliquid behavior and dwave superconductivity near the chargedensitywave quantum critical point. Z. Phys. B: Condens. Matter 103, 137–144 (1996).
Müller, K. A., Berlinger, W. & Tosatti, E. Indication for a novel phase in the quantum paraelectric regime of SrTiO3 . Z. Phys. B: Condens. Matter 84, 277–283 (1991).
Cohen, J. D. et al. Phonon counting and intensity interferometry of a nanomechanical resonator. Nature 520, 522–525 (2015).
Hu, X. & Nori, F. Quantum phonon optics: coherent and squeezed atomic displacements. Phys. Rev. B 53, 2419–2424 (1996).
Hu, X. & Nori, F. Squeezed phonon states: modulating quantum fluctuations of atomic displacements. Phys. Rev. Lett. 76, 2294–2297 (1996).
Garrett, G., Rojo, A., Sood, A., Whitaker, J. & Merlin, R. Vacuum squeezing of solids: macroscopic quantum states driven by light pulses. Science 275, 1638–1640 (1997).
Garrett, G., Whitaker, J., Sood, A. & Merlin, R. Ultrafast optical excitation of a combined coherentsqueezed phonon field in SrTiO3 . Opt. Express 1, 385–389 (1997).
Hu, X. & Nori, F. Phonon squeezed states: quantum noise reduction in solids. Phys. B 263264, 16–29 (1999).
Misochko, O. V., Sakai, K. & Nakashima, S. Phasedependent noise in femtosecond pump–probe experiments on Bi and GaAs. Phys. Rev. B 61, 11225–11228 (2000).
Bartels, A., Dekorsy, T. & Kurz, H. Impulsive excitation of phononpair combination states by secondorder Raman Scattering. Phys. Rev. Lett. 84, 2981–2984 (2000).
Hussain, A. & Andrews, S. R. Absence of phasedependent noise in timedomain reflectivity studies of impulsively excited phonons. Phys. Rev. B 81, 224304 (2010).
Sauer, S. et al. Lattice fluctuations at a double phonon frequency with and without squeezing: an exactly solvable model of an optically excited quantum dot. Phys. Rev. Lett. 105, 157401 (2010).
Reiter, D. E., Wigger, D., Axt, V. M. & Kuhn, T. Generation and dynamics of phononic cat states after optical excitation of a quantum dot. Phys. Rev. B 88, 195327 (2011).
Misochko, O. V., Hu, J. & Nakamura, K. G. Controlling phonon squeezing and correlation via one and twophonon interference. Phys. Lett. A 375, 4141 (2011).
Hu, J., Misochko, O. V. & Nakamura, K. G. Direct observation of twophonon bound states in ZnTe. Phys. Rev. B 84, 224304 (2011).
Misochko, O. V. Nonclassical states of lattice excitations: squeezed states and entangled phonons. Phys. Usp. 183, 917–933 (2013).
Riek, C. et al. Direct sampling of electricfield vacuum fluctuations. Science 350, 420–423 (2015).
Dhar, L., Rogers, J. A. & Nelson, K. A. Timeresolved vibrational spectroscopy in the impulsive limit. Chem. Rev. 94, 157–193 (1994).
Johnson, S. L. et al. Directly observing squeezed phonon states with femtosecond Xray diffraction. Phys. Rev. Lett. 102, 175503 (2009).
Henighan, T. et al. How to distinguish squeezed and coherent phonons in femtosecond xray diffuse scattering. Preprint at http://arxiv.org/abs/1510.02403 (2015).
Trigo, M. et al. Fouriertransform inelastic Xray scattering from time and momentumdependent phononphonon correlations. Nat. Phys. 9, 790–794 (2013).
Merlin, R. Generating coherent THz phonons with light pulses. Solid State Commun. 102, 207–220 (1997).
Papalazarou, E. et al. Coherent phonon coupling to individual Bloch states in photoexcited bismuth. Phys. Rev. Lett. 108, 256808 (2012).
Li, J. J., Chen, J., Reis, D. A., Fahy, S. & Merlin, R. Optical probing of ultrafast electronic decay in Bi and Sb with slow phonons. Phys. Rev. Lett. 110, 047401 (2013).
Weiner, A. M., Wiederrecht, G. P., Nelson, K. A. & Leaird, D. E. Femtosecond multiplepulse impulsive stimulated Raman scattering spectroscopy. J. Opt. Soc. Am. B: Opt. Phys. 8, 1264–1275 (1991).
Zeiger, H. J. et al. Theory for displacive excitation of coherent phonons. Phys. Rev. B 45, 768–778 (1992).
Lobad, A. I. & Taylor, A. J. Coherent phonon generation mechanism in solids. Phys. Rev. B 64, 180301 (2001).
Hu, X. & Nori, F. Phonon squeezed states generated by secondorder Raman scattering. Phys. Rev. Lett. 79, 4605–4608 (1997).
Mukamel, S. Principles of Nonlinear Optical Spectroscopy Oxford Univ. Press (1995).
Stevens, T. E., Kuhl, J. & Merlin, R. Coherent phonon generation and the two stimulated Raman tensors. Phys. Rev. B 65, 144304 (2002).
Rundquist, A., Broman, J., Underwood, D. & Blank, D. Polarizationdependent detection of impulsive stimulated Raman scattering in αquartz. J. Mod. Opt. 52, 2501–2510 (2006).
Scott, J. F. & Porto, S. P. S. Longitudinal and transverse optical lattice vibrations in quartz. Phys. Rev. 161, 903–910 (1967).
Titimbo, K. Creation and detection of squeezed phonons in pump and probe experiments: a fully quantum treatment (PhD thesis, Univ. Trieste (2015).
Esposito, M. et al. Pulsed homodyne gaussian quantum tomography with low detection efficiency. New J. Phys. 16, 043004 (2014).
Alicki, R. & Lendi, K. Quantum Dynamical Semigroups and Applications 717, (SpringerVerlag (2007).
Breuer, H. P. & Petruccione, F. The Theory of Open Quantum Systems Oxford Univ. Press (2002).
Campisi, M., Hänggi, P. & Talkner, P. Colloquium: quantum fluctuation relations: foundations and applications. Rev. Mod. Phys. 83, 771–791 (2011).
Goold, J., Huber, M., Riera, A., Del Rio, L. & Skrzypczyk, P. The role of quantum information in thermodynamics—a topical review. Preprint at http://arxiv.org/abs/1505.07835 (2015).
Acknowledgements
We are grateful to John Goold, Roberto Merlin, Keith Nelson, Mauro Paternostro and Charles Shank for the insightful discussions and critical reading of the manuscript. We thank the CAEN company for the project and the realization of the differential detector used in the experiments. We acknowledge Giovanni Franchi for the design of the detector and Riccardo Tommasini for support during its development. The experimental activities have been carried out at the TRex labs within the Fermi project at Trieste’s synchrotron facility. This work has been supported by a grant from the University of Trieste (FRA 2013) and a grant from the Italian Ministry of Education Universities and Research MIUR (SIR 2015, Controlling quantum Coherent Phases of matter by THz light pulses).
Author information
Authors and Affiliations
Contributions
M.E., F.G., F.R., D.B. and D.F. performed the experiments. F.R. and M.E. developed the acquisition system. K.T., K.Z., R.F. and F.B. developed the theoretical description. M.E., F.G. and D.F. analysed the experimental data. M.E, D.F., F.B. and F.P. coordinated the project and wrote the manuscript with contributions from all the coauthors. D.B. proposed the αquartz case of study and provided the sample. The experiment has been conceived by D.F and F.P.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 15, Supplementary Notes 13 and Supplementary References (PDF 396 kb)
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Esposito, M., Titimbo, K., Zimmermann, K. et al. Photon number statistics uncover the fluctuations in nonequilibrium lattice dynamics. Nat Commun 6, 10249 (2015). https://doi.org/10.1038/ncomms10249
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms10249
This article is cited by

Subcycle quantum electrodynamics
Nature (2017)

Correction: Corrigendum: Photon number statistics uncover the fluctuations in nonequilibrium lattice dynamics
Nature Communications (2016)

Quantum dynamics of optical phonons generated by optical excitation of a quantum dot
Journal of Computational Electronics (2016)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.