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arXiv:2005.05736

Quadrature-variances in high-harmonics generation

Ákos Gombkötő

Published May 12, 2020• 0 citations
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Abstract

We analyse the quadrature variance of high harmonic photons, using a simplified quantum optical model associated with high-harmonic generation (HHG) in the case of an elementary quantum source, modelled with a two-level atom. Assuming classical excitation, we find weak, oscillating (anti-)squeezing in the scattered radiation, with the variance-spectrum following similar evolution to the photon-number spectrum. Investigating the phase-relations, we conclude that the squeezed states of each mode are {most of the time{ close to being amplitude-squeezed, and squeezed states appear periodically within each attosecond pulse. We have also defined an approximate single noise-ellipse based on the collective field, and its analysis shows that it is anti-squeezed.

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We analyse the quadrature variance of high harmonic photons, using a simplified quantum optical model associated with high-harmonic generation (HHG) in the case of an elementary quantum source, modelled with a two-level atom. Assuming classical excitation, we find weak, oscillating (anti-)squeezing in the scattered radiation, with the variance-spectrum following similar evolution to the photon-number spectrum. Investigating the phase-relations, we conclude that the squeezed states of each mode are {most of the time{ close to being amplitude-squeezed, and squeezed states appear periodically within each attosecond pulse. We have also defined an approximate single noise-ellipse based on the collective field, and its analysis shows that it is anti-squeezed.
We analyse the quadrature mean values and variances of high harmonic photons, emitted by a classically driven elementary quantum source of radiation. This fundamental system is modelled with a two-level atom. The quadrature mean values and variances are calculated for harmonic modes in the near-field. Numerical results imply the presence of weak squeezing in each harmonic mode, more pronounced in even harmonics. Approximate time-evolution of the collective electromagnetic field is calculated, and the electric field-variance of the scattered pulse-train is given.
We present a study of photon statistics associated with high-order harmonic generation (HHG) involving one-mode and intermodal correlations of the high harmonic photons. The aim of the paper is to give insight into the nonclassical properties of high-order harmonic modes. To this end, we use a simplified model describing an elementary quantum emitter: the model of a two-level atom. While the material system is extremely simplified in this description, the conclusions and the methods may be generalized for more complex cases. Our primary interest is an effective model of HHG in which the exciting pulse is classical and the harmonics are quantized, although we touch upon the more generalized, fully quantized model as well. Evolution of the Mandel parameter, photon antibunching, squeezing, and cross correlations are calculated. Results imply that with respect to a single quantized emitter, nonclassicality of the harmonics is present: sub-Poissonian photon statistic and squeezing can characterize certain optical modes, while strong anticorrelation can also be present.
Strong-field high-harmonic generation (HHG) is conventionally described semiclassically, with the driving laser treated as a classical field. This approximation becomes insufficient in nanoscale interaction geometries, where extreme spatial confinement raises the vacuum-field amplitude to the ~10^-2 level relative to the driving-field amplitude. When the quantum fluctuations of the driving field are redistributed between conjugate quadratures by squeezing, they can be directly imprinted onto macroscopic HHG observables. To model this interaction, we employ a Wigner phase-space approach that maps the quantum-optical driver onto a stochastic ensemble of time-dependent Schrodinger equation realizations. Although each realization remains classically simulable, the sub-vacuum quadrature covariance structure of squeezed states cannot be reproduced by any field admitting a non-negative Glauber-Sudarshan P-representation. Within this single-mode Gaussian framework, we show that amplitude squeezing suppresses the shot-to-shot variance of the HHG cutoff below the standard quantum limit (SQL). To leading order in the vacuum-to-driver ratio, the normalized cutoff variance decays exponentially with the squeezing parameter, independent of the absolute vacuum-field amplitude and therefore robust against uncertainties in the effective nanoscale mode volume. A subleading phase-noise contribution from the anti-squeezed quadrature produces a variance minimum near r_opt ~ 1.6, providing a concrete experimental target. These results establish the HHG cutoff variance ratio as a nonlinear, self-calibrating Heisenberg witness in which sub-SQL scaling directly reflects the redistribution of quantum uncertainty in the driving field.
High harmonic generation (HHG) is an extremely nonlinear effect generating coherent broadband radiation and pulse durations reaching attosecond timescales. Conventional models of HHG that treat the driving and emitted fields classically are usually very successful but inherently cannot capture the quantum-optical nature of the process. Although prior work considered quantum HHG, it remains unknown in what conditions the spectral and statistical properties of the radiation depart considerably from the known phenomenology of HHG. The discovery of such conditions could lead to novel sources of attosecond light having squeezing and entanglement. Here, we present a fully-quantum theory of extreme nonlinear optics, predicting quantum effects that alter both the spectrum and photon statistics of HHG, thus departing from all previous approaches. We predict the emission of shifted frequency combs and identify spectral features arising from the breakdown of the dipole approximation for the emission. Our results show that each frequency component of HHG can be bunched and squeezed and that each emitted photon is a superposition of all frequencies in the spectrum, i.e., each photon is a comb. Our general approach is applicable to a wide range of nonlinear optical processes, paving the way towards novel quantum phenomena in extreme nonlinear optics.
Recently, the intersection of quantum optics and attosecond physics has attracted growing interest. Here, we study the quantum optical signatures of high-harmonic generation (HHG) from solid-state material driven by squeezed coherent states. We find that squeezed coherent states can either enhance or suppress the HHG yields compared to that driven by coherent light, depending on the photon bunching or antibunching characteristics of driving light. The analysis on semiclassical trajectories indicates that the amplitude fluctuation alters the recombination energy, while the phase fluctuation affects the birth and recombination time of electron-hole pairs. By calculating the Husimi Q distributions of HHG, we demonstrate that the squeezing of the two conjugate quadratures of driving light asymmetrically influence the squeezing phase of harmonics. Surprisingly, we find that even under coherent driving, the harmonics still exhibit squeezing due to the highly nonlinear response of solid-state HHG, which was overlooked previously. Numerical calculations of the Mandel parameter are performed to characterize the photon statistics of emitted HHG arising from different microscopic mechanisms. The results indicate that intraband radiation can exhibit a super-Poissonian distribution, while interband emission exhibits sub-Poissonian distribution under coherent driving.
In recent years, strong-field processes such as high-order harmonic generation (HHG) and above-threshold ionization driven by nonclassical states of light have become an increasingly popular field of study. The theoretical modeling of these processes often applies an approximate phase-space expansion of the nonclassical driving field in terms of coherent states, which has been shown to accurately predict the harmonic spectrum. However, its accuracy for the computation of quantum optical observables like the degree of squeezing and photon statistics has not been thoroughly considered. In this work, we introduce this approximative phase-space description and discuss its accuracy, and we find that it mischaracterizes the quantum optical properties of the driving laser by making it an incoherent mixture of classical states. We further show that this error in the driving field description maps onto the light emitted from HHG, as neither sub-Poissonian photon statistics nor quadrature squeezing below vacuum fluctuations can be captured by the approximative phase-space description. Lastly, to benchmark the approximative phase-space description, we consider the quantum HHG from a one-band model, which yields an exact analytical solution. Using the approximative phase-space representation with this specific model, we find a small quantitative error in the quadrature variance of the emitted field that scales with pulse duration and emitter density. Our results show that using this approximative phase-space description can mischaracterize quantum optical observables. Attributing physical meaning to such results should therefore be accompanied by a quantitative analysis of the error.